structural optimisation in building design practice: case-studies in
TRANSCRIPT
STRUCTURAL OPTIMISATION
IN BUILDING DESIGN PRACTICE:
CASE-STUDIES IN TOPOLOGY OPTIMISATION
OF BRACING SYSTEMS
Robert Baldock
Corpus Christi College
June 2007
A dissertation submitted for the Degree of Doctor of Philosophy
Cambridge University Engineering Department
Declaration
Except where otherwise stated, this thesis is the result of my own research and does not
include the outcome of work done in collaboration.
This thesis has not been submitted in whole or in part for consideration for any other
degree of qualification at the University or any other institute of learning.
The thesis contains 49 figure, 14 tables and less than 42,000 words.
Robert Baldock
Corpus Christi College
Cambridge
June 2007
Abstract
Keywords: structural topology optimisation, structural design practice, bracing
design, Evolutionary Structural Optimisation, Pattern Search, Optimality Criteria,
Genetic Programming, computer-aided design, large-scale structural size optimisation
This thesis aims to contribute to the reduction of the significant gap between the state-
of-the-art of structural design optimisation in research and its practical application in
the building industry. The research has focused on structural topology optimisation,
investigating three distinct methods through the common example of bracing design
for lateral stability of steel building frameworks. The research objective has been
aided by collaboration with structural designers at Arup.
It is shown how Evolutionary Structural Optimisation can be adapted to improve
applicability to practical bracing design problems by considering symmetry
constraints, rules for element removal and addition, as well as the definition of element
groups to enable inclusion of aesthetic requirements. Size optimisation is added in the
optimisation method to improve global optimality of solutions.
A modified Pattern Search algorithm is developed, suitable for the parameterised,
grid-based, topological design problem of a live, freeform tower design project. The
alternative objectives of minimising bracing member piece count or bracing volume
are considered alongside an efficient simultaneous size and topology optimisation
approach, through integration of an Optimality Criteria method. A range of alternative
optimised designs, suitable for assessment according to unmodelled criteria, are
generated by stochastic search, parametric studies and changes in the initial design.
This study is significant in highlighting practical issues in the application of structural
optimisation in the building industry.
A Genetic Programming formulation is presented, using design modification operators
as modular "programmes", and shown to be capable of synthesising a range of novel,
optimally-directed designs. The method developed consistently finds the global
optimum for a small 2D planar test problem, generates high-performance designs for
larger scale tasks and shows the potential to generate designs meeting user-defined
aesthetic requirements.
The research and results presented contribute to establishing a structural optimisation
toolbox for design practice, demonstrating necessary method extensions and
considerations and practical results that are directly applicable to building projects.
Acknowledgements
I wish to thank my academic supervisor, Kristina Shea, for her dedicated support,
guidance and encouragement throughout the course of this project. I am also greatly
indebted to Geoff Parks for his efficiency and advice in the role of advisor and
subsequently as administrative supervisor. Thanks are due to Marina Gourtovaia and
Andrew Flintham for their valuable assistance in computing matters and to all my
friends and colleagues in the Engineering Design Centre, Cambridge for many
stimulating discussions.
The collaboration with Arup has been fundamental to this research. I therefore wish to
express my sincere gratitude to Ed Clark and Alvise Simondetti, as industrial
supervisors, as well as Damian Eley, Chris Neighbour, Steve McKechnie, Martin Holt,
Colin Jackson, Jan-Peter Koppitz, Chris Carroll, Pat Dallard and Peter Young, all of
whom generously gave time to aid me in various aspects of this project. Additionally,
the support of Chris Kaethner and Stephen Hendry, in relation to Oasys GSA, has been
very beneficial.
I could not have completed this thesis without the fantastic friends who have inspired,
distracted and kept me sane.
I have been blessed with loving and loyal parents who have supported me from my
first steps to the conclusion of this thesis. I owe them the greatest thanks of all.
This research has been made possible through funding by the Engineering and Physical
Sciences Research Council and an Industrial CASE studentship from Arup. Additional
financial support from Cambridge University Engineering Department, Corpus Christi
College, Cambridge and the Royal Commission for the Exhibition of 1851 is also
gratefully acknowledged.
Contents
1. INTRODUCTION ………………………………………………………….. 1
1.1. The nature of design optimisation ……………………………………. 1
1.2. Optimisation of structures ……………………………………………. 3
1.3. The design process for building structures ………………………… ... 5
1.4. Drivers and barriers for structural optimisation in the building industry 7
1.5. Summary of research contributions .………………………….……… 10
1.6. Thesis structure …………………………………………….………… 11
2. STATE-OF-THE-ART: RESEARCH AND PRACTICE OF DESIGN
OPTIMISATION IN STRUCTURAL ENGINEERING …………………... 12
2.1. Structural design optimisation research ……………………………… 12
2.1.1. Section-size optimisation ………………………………………... 14
Optimality Criteria ………………………………………………… 14
Mathematical Programming ………………………………………. 14
Fully Stressed Design ……………………………………………… 15
Additional considerations ………………………………………….. 15
2.1.2. Discrete topology optimisation methods ………………………… 16
Ground structure approach ………………………………………… 16
Ruled-based approaches …………………………………………… 18
2.1.3. Evolutionary Algorithms in topology optimisation ……………… 19
Genetic Algorithms …………………………………………...….... 19
Genetic Programming ……………………………………………… 20
Evolutionary Strategies …………………………………………….. 21
Evolutionary Programming ……………………………………….... 21
2.1.4. Continuum-based optimisation methods …………………………. 21
Homogenisation …………………………………………………….. 22
Bubble method ……………………………………………………… 22
Evolutionary Structural Optimisation ……………………………… 22
2.1.5. Computer-based conceptual design methods …………………….. 24
2.2. Optimisation in building engineering design practice ………………… 25
2.2.1. Comparison of structural design in the automotive and aeronautical
industries versus the building industry …………………………… 25
2.2.2. Commercial optimisation software ……………………………….. 27
2.2.3. Published literature on industrial applications ……………………. 29
Section-size optimisation …………………………………………… 29
Evolutionary Structural Optimisation (ESO) ……………………….. 30
Parametric optimisation …………………………………………….. 31
Non-parametric optimisation ……………………………………….. 31
2.2.4. Facilitating structural optimisation ……………………………….. 32
Software …………………………………………………………….. 32
Parametric optimisation case studies ……………………………….. 33
Non-parametric discrete optimisation and design generation case studies
………………………………………………………………………. 33
2.3. Conclusions …………………………………………………………….. 34
2.4. Justification of case study ……………………………………………… 35
2.5. Context of research contributions ……………………………………... 37
Evolutionary Structural Optimisation ………………………………. 37
Pattern Search and Optimality Criteria ……………………………... 37
Genetic Programming using design modification operators ………... 38
3. CONTINUUM TOPOLOGY OPTIMISATION OF BRACED STEEL FRAMES
……………………………………………………………………….. 40
3.1. Introduction …………………………………………………………….. 40
3.2. Background …………………………………………………………….. 40
4.7.1 Method overview ………………………………………………….. 40
3.2.1. Addition considerations and extensions …………………………... 41
3.2.2. Evolutionary Structural Optimisation (ESO) for stiffness and
displacement constraints ………………………………………….. 43
3.2.3. Bi-directional Evolutionary Structural Optimisation (BESO) …… 45
3.3. Benchmark problem: structural model specifications ………………… 45
3.4. Optimisation for minimal mean compliance ………………………….. 46
3.5. Optimisation for displacement constraint …… ……………………….. 47
3.6. Including optimisation of domain thickness …………………………... 53
3.7. Including architectural requirements and pattern definition …………... 58
3.8. Discrete interpretation of continuum topologies ………………………. 60
3.9. Conclusions ……………………………………………………………. 64
3.10. Guidelines for practical use …………………………………………… 64
4. BRACING TOPOLOGY AND SECTION-SIZE OPTIMISATION BY A HYBRID
ALGORITHM: AN INDUSTRIAL CASE-STUDY ………………………... 67
4.1. Introduction …………………………………………………………… 67
4.2. Background ………………………………………… ………………… 68
4.4.1. Overview of studies ……………………………………………… 69
4.3. Design task definition …………………………………………………. 69
4.3.1. Structural models ………………………………………………… 69
4.3.2. Topology optimisation models …………………………………… 72
Optimisation model A ……………………………………………… 72
Optimisation model B ……………………………………………… 72
4.4. Pattern Search method ………………………………………………… 73
4.5. Live project optimisation ……………………………………………… 75
4.5.1. Topology optimisation by Modified Pattern Search ……………... 75
4.5.2. Parametric studies ………………………………………………… 76
4.5.3. Outline proposals …………………………………………………. 77
4.6. Characterisation of design space ………………………………………. 78
4.7. Topology optimisation method development …………………………. 80
4.7.1. Objective function formulation …………………………………... 83
Formulation 1 ………………… ……………………………………. 83
Formulation 2 ………………… ……………………………………. 83
4.7.2. Comparative investigation ………………………………………... 84
Evolving designs from fully-braced initial configuration ………….. 85
Alternative objective function formulations ……………………….. 86
Scheduling of exploratory moves ………………………………….. 86
Performance of designs evolved from randomly generated initial
configurations …………………………………………………… 86
Use of pattern moves ………………………………………………. 87
4.8. Topology optimisation: structural model B …………………………… 87
4.8.1. Results ……………………………………………………………. 89
4.8.2. Observations ……………………………………………………… 91
4.8.3. Diversity ………………………………………………………….. 91
4.9. Size optimisation ………………… …………………………………… 92
4.9.1 Overview ………………… ………………………………………. 92
4.9.2. Derivation of iterative approach from Optimality Criteria ……….. 92
4.9.3. Pitfalls ……………………………………………………………. 97
Complex values of Ai.………………………………………………. 97
Convergence failure ………………… ……………………………... 97
Negative values of Cj* and Cj.………………… ……………………. 97
4.9.4. Assignment of discrete sections .…………………………………. 98
4.9.5. Size optimisation of fully braced configuration .…………………. 99
4.9.6. Size optimisation by Optimality Criteria with bending moments ... 102
4.10. Integration of topology and size optimisation.………………………... 102
4.10.1. Results ……………………………………………………………. 106
4.10.2. Observations ……………………………………………………… 107
4.11. Summary of results from optimisation model B ……………………… 108
4.12. Conclusions.…………………………………………………………… 112
5. STRUCTURAL TOPOLOGY OPTIMISATION OF BRACED STEEL
FRAMEWORKS USING GENETIC PROGRAMMING ………………….. 114
5.1. Introduction …………………………………………………………… 114
5.2. Background …………………………………………………………… 115
5.3. Genetic Programming method ………………………………………... 115
5.3.1. Introduction ……………………………………………………… 115
5.3.2. GP for bracing design ……………………………………………. 116
Creating initial designs …………………………………………….. 118
Analysis and fitness ………………………………………………... 120
Generating subsequent populations ………………………………... 120
Handling geometrically infeasible designs ………………………… 121
5.4. Bracing design for a 2x6 framework ………………………………….. 125
5.5. Bracing design for a 6x30 framework ………………………………… 130
5.6. Defining aesthetic style ……………………………………………….. 138
5.7. Further work …………… …………………………………………….. 139
5.8. Conclusions …………………………………………………………… 139
6. CONCLUDING REMARKS ……………………………………………….. 141
6.1. Review of contributions ………………………………………………. 141
6.2. Recommendations for future work ……………………………………. 145
Evolutionary Structural Optimisation …………………………….... 145
Pattern Search - Optimality Criteria ……………………………….. 145
Genetic Programming ……………………………………………… 146
6.3. Application of structural optimisation in practice …………………….. 147
6.4. Projected trends in structural design automation and optimisation in practice
……………………………………………………………………… 148
6.5. Closing notes ………………………………………………………….. 149
APPENDIX 1. STRUCTURAL ANALYSIS …………………………………. 151
APPENDIX 2. SOFTWARE DEVELOPMENT AND PROTOTYPING ……. 152
REFERENCES …………………………………………………………………. 153
List of Figures
Figure 1.1: Structural optimisation tasks illustrated through the example of the design
of a simply-supported, centrally point-loaded structure .………………. 5
Figure 2.1: Michell truss subjected to load F at point A and fixed at a circular support
at point B, after Michell (1904) .……………………………………….. 13
Figure 2.2: Optimal self-adjoint cantilever trusses with six and eleven joints, subjected
to load F at point A and fixed at support points B (Prager 1977) ……… 13
Figure 2.3: Fully-connected ground structure for a relatively simple (3x6) grid . 17
Figure 2.4: Concept sketches for bracing design of 122 Leadenhall St. Building
(reproduced by kind permission of Chris Neighbour, Arup) …………… 34
Figure 3.1: Weighting factors used for averaging sensitivity numbers across elements
to avoid checkerboarding ……………………………………………….. 42
Figure 3.2: Real loads (left), including member groupings and geometric
specifications, and virtual load (right). ASCE standard section specifications
(below) ………………………………………………………………….. 46
Figure 3.3: Design topology of Liang et al. (2000): δ=0.024, element retention = 22%
(left) Comparative result to Liang et al. (2000) topology: δ=0.024, element
retention = 23% (right) …………………………………………………. 47
Figure 3.4: Elements removed in the top left bay unit in the first iteration, based on
maximum cross strain energy (left) and sum of strain energies (right) in pairs of
elements grouped by the horizontal symmetry condition ………………. 48
Figure 3.5: ESO results with element removal determined by cross-strain energy.
25.4mm designable domain, 8 elements removed per iteration (left: maximum
of sensitivity number in pairs of elements, right: sum of sensitivity number in
pairs of elements) ……………………………………………………….. 49
Figure 3.6: Minimum volume designs satisfying the displacement constraint derived by
BESO for varying domain thickness and starting configuration ……….. 51
Figure 3.7: Process flowchart for ESO with domain thickness optimisation …... 54
Figure 3.8: Flowchart for domain thickness optimisation loop ………………… 55
Figure 3.9: Process history for simultaneous topology and domain thickness
optimisation with a single thickness group …………………………….. 57
Figure 3.10: Best designs derived by simultaneous thickness and topology
optimisation, with one, three and six thickness groups ………………… 58
Figure 3.11: Evolving topologies with prescribed symmetry, using simultaneous
thickness optimisation of appropriate groups …………………………... 60
Figure 3.12: Discrete bracing topologies (with circular solid sections) optimised for
minimum mass satisfaction of displacement constraint ………………… 63
Figure 4.1: Fully-braced analysis model (left to right): plan view; side elevation;
isometric view (shown with two spirals highlighted); isometric split sections
…………………………………………………………………………… 70
Figure 4.2: Split elevation view of the upper section of structural model 1, with spiral
numbering and bracing members at the tip of each element highlighted . 75
Figure 4.3: Parametric Studies ………………………………………………….. 77
Figure 4.4: Designs generated for consideration for outline proposal ………….. 78
Figure 4.5: 2D simplified representation of design domain, model A ………….. 80
Figure 4.6. A sample exploratory move ………………………………………… 81
Figure 4.7: Pattern Search topology optimisation flowchart …………………… 82
Figure 4.8: Design solutions from topology optimisation of structural model 2 .. 89
Figure 4.9: Size optimisation flowchart ………………………………………… 99
Figure 4.10: Convergence of size optimisation algorithm from maximum section sizes
in fully braced design …………………………………………………… 101
Figure 4.11: Convergence of size optimisation algorithm from minimum section sizes
in fully braced design …………………………………………………… 101
Figure 4.12: Flowchart for combined size and topology optimisation algorithm . 105
Figure 4.13: Arup design proposal, without requirement for bracing members to be
grouped in continuous spirals …………………………………………… 109
Figure 4.14: Volume reduction by simultaneous versus sequential topology and size
optimisation routines ……………………………………………………. 111
Figure 5.1: Tree representation of mathematical equation: y = 4/(X*X) + 5*(7-X) 116
Figure 5.2: Function set for GP trees representing bracing designs ……………. 117
Figure 5.3: Seeded framework ………………………………………………….. 118
Figure 5.4: Development of an initial design by application of design modification
operators ………………………………………………………………… 119
Figure 5.5: Linear rank-based weighting system for parent selection ………….. 120
Figure 5.6: Genetic programming evolutionary process flowchart …………….. 121
Figure 5.7: Example of geometric infeasibility, with units overlapping and extending
beyond the orthogonal framework ……………………………………… 122
Figure 5.8: Repair algorithm in the context of mutation or crossover ………….. 124
Figure 5.9: Example of initial population, penalised designs shown in grey (Population
size = 30, Crossover ratio = 0.9, run number 22) ………………………. 129
Figure 5.10: Example of final population, penalised designs shown in grey (Population
size = 30, Crossover ratio = 0.9, run number 22), best-of-run design top-left
…………………………………………………………………………... 129
Figure 5.11: Example of evolution history (Population size = 30, Crossover ratio = 0.9,
Run number 22) ………………………………………………………… 130
Figure 5.12: Example of most efficient tree representation of the optimal double
echelon design (left), with the actual representation found in Run number 22,
Population size = 30, Crossover ratio = 0.9 (right) …………………….. 130
Figure 5.13: Geometry and cross-section groupings for 6x30 framework ……... 132
Figure 5.14: Initial population of randomly generated designs ……………….... 134
Figure 5.15: Final generation of designs (run 1), including best-of-run design (top-left)
…………………………………………………………………………... 135
Figure 5.16: Evolution history of run 1 ………………………………………… 136
Figure 5.17: Best-of-run designs and performance (* indicates displacement constraint
violation) ………………………………………………………………... 136
Figure A1.1: Structural analysis flowchart ……………………………………... 151
List of Tables
Table 2.1 Design and production of a typical automotive component versus a steel
building structure ……………………………………………………….. 27
Table 4.1 Comparison of structural models ……………………………………. 71
Table 4.2: Statistical analysis of 10000 randomly generated designs ………….. 81
Table 4.3: Statistical summary of 20 runs per case …………………………….. 85
Table 4.4: Performance of designs derived from fully braced initial configuration 90
Table 4.5: Performance of randomly generated initial designs and solutions derived
from them through bi-directional topology optimisation ………………. 90
Table 4.6: Catalogue of circular hollow sections and corresponding areas available for
bracing members ……………………………………………………….. 93
Table 4.7: Size optimisation of fully-braced design from different initial distributions
…………………………………………………………………………… 100
Table 4.8: Performance of optimised designs derived from fully braced initial
configuration ……………………………………………………………. 106
Table 4.9: Performance of initial and optimised designs derived from random initial
configurations …………………………………………………………... 107
Table 4.10: Summary of best designs from structural model 2 ……………….... 109
Table 5.1: Batch characteristics in parametric study …………………………… 127
Table 5.2: Cross-sections made available and selected in fully-stressed design .. 133
Table 6.1: Summary of methods used in this thesis ……………………………. 143
Structural Optimisation in Building Design Practice: Case-studies in topology optimisation of bracing systems
ROBERT BALDOCK
Summary
Keywords: structural topology optimisation, structural design practice, bracing design,
Evolutionary Structural Optimisation, Pattern Search, Optimality Criteria, Genetic
Programming, computer-aided design, large-scale structural size optimisation
This thesis aims to contribute to the reduction of the significant gap between the state-of-the-
art of structural design optimisation in research and its practical application in the building
industry. The hypothesis that optimisation can be successfully and appropriately applied in
practice through consideration of industry specific issues is explored. The research has
focused on structural topology optimisation, investigating three distinct methods through the
common example of bracing design for lateral stability of steel building frameworks. The
research has been aided by collaboration with structural designers at Arup.
It is shown how Evolutionary Structural Optimisation can be adapted to improve applicability
to practical bracing design problems by considering symmetry constraints, rules for element
removal and addition, as well as the definition of element groups to enable inclusion of
aesthetic requirements. Size optimisation is added in the optimisation method to improve
global optimality of solutions.
A modified Pattern Search algorithm is developed, suitable for the parameterised, grid-based,
topological design problem of a live, freeform tower design project. The alternative objectives
of minimising bracing member piece count or bracing volume are considered alongside an
efficient simultaneous size and topology optimisation approach, through integration of an
Optimality Criteria method. A range of alternative optimised designs, suitable for assessment
according to unmodelled criteria, are generated by stochastic search, parametric studies and
changes in the initial design. This study is significant in highlighting practical issues in the
application of structural optimisation in the building industry.
A Genetic Programming formulation is presented, using design modification operators as
modular "programmes", and shown to be capable of synthesising a range of novel, optimally-
directed designs. The method developed consistently finds the global optimum for a small 2D
planar test problem, generates high-performance designs for larger scale tasks and shows the
potential to generate designs meeting user-defined aesthetic requirements.
The research and results presented help to establish a structural optimisation toolbox for
design practice, demonstrating necessary method extensions and considerations and practical
results that are directly applicable to building projects. The research hypothesis is hence strongly supported.
1
1. Introduction
The research presented in this thesis is motivated by the disparity between the vast
volume of academic literature in the field of structural optimisation and the very
modest uptake of these methods in building design practice. The core research
objective is therefore to contribute towards reducing the gap between research and
industry. The accompanying central hypothesis is that optimisation can be
successfully and appropriately applied in practice through consideration of industry
specific issues. Collaboration with the structural engineering consultants, Arup, has
allowed observation of and involvement in live projects, providing useful insights into
the pertinent issues to be addressed in furthering the application of optimisation in the
building industry. The research objective is achieved through the investigation of
three optimisation methods: Evolutionary Structural Optimisation, Pattern Search
with Optimality Criteria for simultaneous section-size optimisation and Genetic
Programming using design modification operators, all applied to test problems in the
field of topological bracing design for lateral stability of steel building frameworks.
At the start of each chapter, research questions are posed, with corresponding
proposals stated and subsequently developed in detail. Significant research
contributions are made in each of these studies, as stated in section 2.5, following a
discussion of the state-of-the-art of structural optimisation in research and practice.
Major themes in this work are generating a range or selection of high performance
designs for assessment according to unmodelled criteria, such as aesthetics and the
integration of size and topology optimisation.
This introductory chapter begins to explore the issues associated with the research
objective, considering the nature of optimisation in general, specific characteristics of
optimisation of structures and the design process in the building industry. Benefits
and barriers to optimisation are discussed, aided by the opinions of practising
structural designers. The structure of the thesis is then presented with an overview of
each subsequent chapter.
2
1.1. THE NATURE OF DESIGN OPTIMISATION
Design optimisation is loosely defined by Papalambros and Wilde (2000) as the
selection of the "best" design within the available means. When stated so simply,
optimisation seems an obvious objective of any design task. Yet when the problem is
ill-structured (defined by Simon (1973) as lacking definition in some respect),
including a possible absence of appropriate tools and knowledge, or if the expenditure
in finding an optimal solution places a high premium on the design cost, a good
design that meets a defined tolerance on all requirements is generally accepted. The
American political scientist and pioneer of Artificial Intelligence, Herbert Simon,
coined the term "satisficing" to describe the process of finding such designs (Simon
1955). Indeed, this is the standard approach adopted in manual design.
Automated and directed search are often considered under the name of optimisation.
In this case, referred to as design exploration, the process is more focused on
examining a broad range of the design space, in search of diverse and novel high-
performance designs, without emphasis on strict global optimality. Here the search
may be seeking satisficing designs where none was previously known.
Papalambros and Wilde (2000) observe that design optimisation involves:
"1. The selection of a set of variables to describe the design alternatives.
2. The selection of an objective (criterion), expressed in terms of the design variables,
which we seek to minimise or maximise.
3. The determination of a set of constraints, expressed in terms of the design
variables, which must be satisfied by any acceptable design.
4. The determination of a set of values for the design variables, which minimise (or
maximise) the objective, while satisfying all the constraints."
A corresponding mathematical definition of a classical optimisation model with
equality and inequality constraints and mixed discrete-continuous design variables is
as follows:
3
Deviations from this form of definition are frequently observed, for example the
number of variables in the problem may not be fixed or multiple conflicting
objectives may exist. This latter class of problem is known as multi-objective
optimisation and has been extensively researched (Collette and Siarry 2003). It
should also be noted that objectives and constraints may not always be readily
mathematically defined nor their values quantifiable. An obvious example with
respect to the subject matter of this thesis is aesthetic appeal.
Complications arise in practical design problems on account of multi-modal design
spaces with numerous local optima, from which a small deviation of any combination
of variables will increase the objective value, despite the existence of a better solution
elsewhere in the design space. The design space may also be fragmented, with
several disconnected feasible regions, surrounded by infeasible space.
1.2. OPTIMISATION OF STRUCTURES
Prior to considering existing optimisation methods, it is useful to define the
framework associated with structural design optimisation. This section presents a
classification of the design tasks themselves and is followed by a discussion of the
typical phases of the structural design process. In increasing order of complexity,
structural design optimisation tasks are generally considered to be:
Minimise: f(x), x = (x1, x2,…, xn)T objective function
Eq 1.1
subject to: gj(x) ≤ 0, j = 1,…,p inequality constraints
Eq 1.2
hj(x) = 0, j = 1,…,m equality constraints
Eq 1.3
xi∈Di, Di = (di1,di2,…,diqi); i = 1,…nd discrete variable set
Eq 1.4
where f is the objective function of x, a set of n design variables, nd of which are
discrete, the remainder being continuous. qi is the number of available discrete
values within Di for each xi, gj is the set of p inequality constraints (including bounds
on continuous variables), hj is the set of m equality constraints.
4
- Optimisation of size (and shape) of cross-section for discrete structural members,
such as beams and columns, or thickness of continuous material, such as panels or
floor slabs. This is often referred to as size optimisation.
- Shape Optimisation, varying positioning of nodes or connections and definition of
lines, curves and surfaces that describe structural form.
- Topology Optimisation, varying the configuration and connectivity of members or
material.
These tasks are illustrated in figure 1.1, noting the trend in the stage of the design
process at which the tasks are addressed.
Whilst it is possible to assign a fixed set of variables in defining an optimisation
model for size and shape optimisation, this is generally not the case for topological
optimisation, hence an infinite number of solutions may exist. The requirement for
modelling member connectivity in topology design is a significant barrier to
application of many classical optimisation methods, as noted by Deb (2001). Shape
optimisation is often considered to include cross-sectional size optimisation; in turn
topology optimisation may include both shape and cross-sectional size optimisation.
It is possible to define shape and topology optimisation tasks parametrically, for
example by defining control points on a curve or varying the number of columns on
the perimeter of a building, although this obviously places restrictions on the search
space. Additionally, it is possible to consider optimisation of plan layout, for example
for maximising potential letting revenue, type of structural system or material
selection.
The field of structural design optimisation includes a number of unique characteristics
and corresponding methods. Many structural design tasks are ill-structured,
especially those in the earlier stages of the design process, where decisions carry the
greatest influence on final efficiency. A crucial part of a potential optimisation
process in the building industry is the evaluation of structural designs, generally by
finite element analysis, which often carries a significant time cost.
5
Initial Design
Size
Optimisation
Shape
(+ Size)
Optimisation
Topology
(+ Shape
+ Size)
Optimisation
Progression of design process
Increased ease of task formulation
Increased generality of design task
Figure 1.1: Structural optimisation tasks illustrated through the example of the
design of a simply-supported, centrally point-loaded structure.
1.3. THE DESIGN PROCESS FOR BUILDING STRUCTURES
It is vital to the successful implementation of optimisation in structural design that the
optimisation tasks detailed above are linked to the appropriate phase of the design
process. The structural design process essentially follows the same progression as
any other design task. However, the interdisciplinary nature of building design, with
input from clients, architects and structural and building services engineers, serves to
complicate the process and may lead to a large number of iterations and revisions,
even revisiting earlier design phases.
With reference to design of topology and form and section allocation, it is useful to
consider the corresponding stage in the design process for each of these tasks.
Structural systems and topologies are developed earlier in the design process, with the
optimisation problem less well-defined, the design space larger and hence a greater
range of possible solutions. Section sizes are not finalised until the latter stages of the
6
design process. Although section-size optimisation is a much more straightforward
task, a strong driver for optimisation prior to this stage is provided by estimates
suggesting that up to four-fifths of the total resources in an engineering project are
committed in the early design stages (Deiman 1993).
The classical design process follows the following stages:
In the conceptual design stage, a set of initial concepts is generated in an attempt to
satisfy the broad design requirements prescribed, in the case of design of buildings, by
the architect or client.
The preliminary design stage further develops one (or more) conceptual design(s).
At this point, the general building system functionalities that were determined
previously will be subject to further refinement in order to furnish a more accurate
cost estimate for the project.
The detailed design stage finalises all information required for construction. In these
latter stages, member-sizing, joint-detailing and similar well-defined tasks are
undertaken in structural design.
Whilst these design stage definitions are widely used throughout the design
community, the Plan of Work Stages 1999 as described by the Royal Institute of
British Architects (RIBA 1999), (Phillips 2000) is recognised and implemented
throughout the construction industry. Stages A to L include tasks undertaken both
before and after the design stages described above, e.g. tendering, construction and
completion. However, the following stages roughly correspond to those detailed
above:
"B: Strategic Briefing Preparation of Strategic Brief by, or on behalf of, the client
confirming key requirements and constraints. Identification of procedures,
organisational structure and range of consultants and others to be engaged for the
project. [Identifies the strategic brief (as CIB Guide) which becomes the clear
responsibility of the client.]
C: Outline proposals. Commence development of strategic brief into full project
brief. Preparation of outline proposals and estimate of cost. Review of procurement
route.
D: Detailed proposals. Complete development of the project brief. Preparation of
detailed proposals. Application for full development control approval.
E: Final proposals. Preparation of final proposals for the Project sufficient for co-
ordination of all components and elements of the Project."
7
1.4. DRIVERS AND BARRIERS FOR STRUCTURAL OPTIMISATION IN THE
BUILDING INDUSTRY
It is necessary to validate the overriding objective of aiding the adoption of structural
design optimisation in the building industry, demonstrating it to be a worthwhile
endeavour. This can be achieved by highlighting the drivers for computational design
optimisation and search in the building industry. In the validation process it is also
necessary to establish that it is possible to overcome the common barriers to the use of
optimisation in structural design practice. The subsequent discussion is augmented by
the opinions of Arup engineers relating to application of structural engineering in the
building industry, as canvassed by Shea (2003) and the author.
Drivers
- Rapid generation and evaluation of a large number of design alternatives. This is
desirable in the early stages of a project, permitting a wider and more thorough
exploration of the design space than could be achieved manually.
- Discovering previously unknown feasible solutions. In the case of some highly
complex structures, there may be uncertainty as to the existence of a feasible
design within the defined constraints. Starting from an initial infeasible design,
optimisation or heuristic search methods have the potential to find feasible
solutions where none was previously known, as seen in section 2.2.3.
- Cost benefits such as reduced material or construction cost and increased
potential revenue. Financial savings are an obvious potential driver for use of
optimisation methods. Cost is often simplistically equated to structural weight,
especially in research investigations, whereas piece-count and connection
detailing are also important in construction costing. Further, maximisation of
floor space and quality will control the potential letting revenue in office and
residential buildings, influencing the financial feasibility of a project as a whole.
- Time savings through computer-assisted search. A well-executed optimisation
process has the potential to save design time, avoiding the need to iterate a design
by hand to find structurally or financially feasible solutions, as well as reducing
the tedium of routine tasks.
- Marketability of optimisation capabilities. Substantial interest has been reported
in using computational optimisation for gaining a market-edge by offering clients,
8
for example, maximisation of net lettable floor space or minimisation of steel
tonnage.
- Decision support in the design process. Decisions in the earlier stages of design
are frequently made based on previous experience or intuition of the engineer, but
without rigorous justification. Increased rigour provided by optimisation or
search could provide leverage for the structural designer in multidisciplinary
design decisions.
Barriers
- Problems may be ill-structured. Topology design problems, for example, do not
present a fixed set of variables, which creates difficulties in implementing
numerical optimisation techniques. However, various methods are capable of
handling such complications, an example being domain-specific methods, such as
those discussed in Chapter 2.
- Architectural constraints often heavily influence the structural design of
buildings. These are difficult to incorporate into an optimisation model and often
not conducive to using gradient-based optimisation methods. However, stochastic
methods, with a random component to the search method, can be used to present a
range of high-performance options, for assessment according to unmodelled
criteria.
- Time required for modelling, method customisation and running optimisation
processes on a specific project may be prohibitive. Since design time is often at a
premium, optimisation cannot be allowed to become a critical path with large
amounts of time devoted to method development or parameter tuning. Each
structural analysis can also be computationally expensive, although analysis time
can be reduced by appropriate approximations and simplifications to improve
efficiency. Repeated use of appropriate optimisation and analysis techniques for
routine tasks within a company will develop a skills base, with increasing
efficiency of implementation. It is highly desirable that in-house tools, techniques
or expertise developed on one project should be reusable. Thus initial investments
may need to be made to develop capabilities, before cost-benefits are achieved.
- Issues of scale in extending small research case-study problems to real world
design scenarios. A number of optimisation methods detailed in research
literature could not feasibly handle large-scale industrial problems due to the large
9
numbers of design variables, especially discrete variables, possible lack of a
feasible initial solution and the number of function evaluations required.
Adopting simplified models for the design task may ease these issues, otherwise
alternative methods may be required.
- Specifications for a project may change rapidly at certain points in the design
process. This possibility presents the danger that the results of an optimisation
process may be obsolete before they are even generated, due to the rapid
information exchange between architects and engineers. Optimisation tools must
therefore be versatile and adaptable to facilitate rapid incorporation of
specification changes.
- Designers lack tools, experience and knowledge required to implement
optimisation methods. Commercial optimisation programmes require technical
and theoretical expertise and hence can be inaccessible to structural designers who
do not use them on a regular basis. As will be discussed in chapter 2, many
commercial tools are targeted at the aerospace and automotive industries.
Practical concerns also include the cost of software licenses, which are a
substantial expenditure, especially for a single project.
- User scepticism. 20 years after Berke commented on the doubts held by designers
(Sobieski et al. 1986) regarding effective benefits, reliability and appropriate
methodologies of structural optimisation, these reservations still persist. Cohn
(1994) notes that "structural engineers willing to optimise their designs have no
alternative but to learn many optimisation procedures and then decide which of
these fit their real problems". However, an increased volume of successful case
studies on "live" building projects is likely to reduce scepticism and accelerate
uptake.
- Optimisation results can be hard to verify. Although it may be hard to
demonstrate that a design is globally optimal with respect to the defined
optimisation problem, an improvement on a manual design may provide sufficient
endorsement for optimisation results to be accepted in practice.
In summary, the obstacles highlighted present important issues to be addressed,
without negating the potential for structural optimisation in the building industry.
10
1.5. SUMMARY OF RESEARCH CONTRIBUTIONS
A thorough discussion of the research contributions of this thesis, in the context of
previous work, is presented in section 2.5 following a review of the state-of-the-art in
academic research and industry. This section summarises them as follows:
- The methods discussed in research chapters 3 to 5 contribute to the development
of a structural optimisation toolbox appropriate for use in the building industry.
- The practical consideration of constraints, repetition and symmetry given
precedence in consideration of Evolutionary Structural Optimisation.
- A unique example of "live" structural topology optimisation is presented on a full-
scale building project, with associated method development. A modified Pattern
Search algorithm is used for this purpose. Constraint handling methods are
developed to suit the problem formulation.
- Means of efficiently integrating size optimisation into topology optimisation
processes are presented for the methods relating to the previous two points, with
modest increases in structural analysis requirements.
- Genetic Programming is used to rapidly generate novel and diverse design
concepts. Through the use of design modification operators in the development of
design blueprints, a new tool is developed for optimally directed and controlled
pattern generation.
1.6. THESIS STRUCTURE
Chapter 2 presents a review and comparison of the state-of-the-art in academic
research and building engineering practice, to explore the reasons for the low transfer
of techniques and tools. This section also compares structural optimisation in the
context of the building industry with that in the automotive and aerospace industries,
where the state-of-the-art is significantly more developed. The chapter closes with a
clear statement of the research contributions to be subsequently presented and proven.
In research chapters 3 to 5, three distinct optimisation and search methods are
investigated, developed and results presented. Their present or future potential impact
on structural design of buildings will be assessed. These methods are all applied to
the common topological design problem of bracing configuration in steel building
frameworks to meet lateral stability requirements, the choice of which is justified in
Chapter 2. This area of application is not intended to be exclusive, since generality is
11
desirable in any method, but rather to provide a unifying theme to the distinct
elements of this thesis. Further threads are provided through investigating the
integration of size and topology optimisation in two of these cases and considering
control over aesthetic form of solutions.
Chapter 3 considers Evolutionary Structural Optimisation (Xie and Steven 1997), a
form of continuum-based optimisation, which has attracted interest in research and
practice on account of its comprehensibility and versatility. Development in this
thesis is based on more practice-orientated criteria than in previous research. The
effect of variation of domain thickness is considered, both manually in a parametric
study and within the context of size optimisation. Capacity for control of form is
demonstrated through defining repetition patterns and symmetry lines. Finally, the
importance of discrete design interpretation is discussed.
Chapter 4 presents work undertaken both live and retrospectively on an Arup project,
in close collaboration with the design team. The topological bracing design problem
is parameterised and tackled using a discrete Pattern Search algorithm (Hooke and
Jeeves 1961). Simultaneous size optimisation is introduced using the Optimality
Criteria method (Rozvany 1989) at each iteration, assuming unchanged force-moment
distribution in the structure. Valuable insight into complications faced in applying
optimisation in practice was gained through close collaboration with the design team.
Chapter 5 introduces the application of function-based Genetic Programming for the
development of optimally-directed bracing designs. Branch points within the tree
representation take the form of design modification operators, in a manner more
closely analogous than any previous structural application to the "programme
routines" on which the original Genetic Programming method (Koza 1992) was
based. The potential for restricting the search space according to aesthetic style is
explored.
Chapter 6 concludes the thesis by summarising the results of the preceding research
and discussing future work required in developing these methods.
The two appendices provide notes on the integration of structural analysis into the
prototype tools developed during the course of the current research, as well as
discussing software development and prototyping for structural optimisation more
generally.
12
2. State-of-the-art: research and practice of design
optimisation in structural engineering
This chapter first presents a review of academic literature, structured to consider the
areas to which this thesis contributes: size optimisation, discrete topology
optimisation and continuous topology optimisation. This section also includes a
discussion of research in computer-based conceptual design methods for structural
design, both with and without optimisation.
The subsequent section presents the current state of the building industry with regard
to the use of structural optimisation. The building industry is first juxtaposed against
the automotive and aeronautical industries where structural design optimisation is
relatively well established. A review of commercial software is presented, followed
by details of published examples of structural optimisation in building design.
Potential for further use of structural optimisation is considered through a selection of
case studies.
Conclusions are drawn from the preceding discussion, regarding the disparity between
research and practice. This leads to the establishment and justification of the design
task tackled in the test-cases used in this thesis: the topological design of bracing
systems for lateral stability in steel framed buildings. Finally the research
contributions of the subsequent studies are stated.
2.1. STRUCTURAL DESIGN OPTIMISATION RESEARCH
Optimisation of structural shape and topology is rooted in the work of Michell (1904).
His pioneering studies in the field derived conditions for limits of material economy
in truss structures, developing structural concepts originally demonstrated by Maxwell
(1864). In the case of a point-loaded cantilever truss with a circular support, and
disregarding the weight of joints, the minimum weight is achieved by a truss-like
continuum with an infinite number of joints and bars, the form of which is shown in
figure 2.1.
13
A B
F
Figure 2.1: Michell truss subjected to load F at point A and fixed at a circular
support at point B, after Michell (1904)
Prager (1977) refined this approach by modelling the weight of joints in his
minimisation problem and employing optimality criteria and the concept of adjoint
trusses to derive simple and practical cantilever structures. Nevertheless, these
structures, shown in figure 2.2, are limited in their treatment of constraints and
loadcases.
F F
A
B
B
B
B
A
Figure 2.2: Optimal self-adjoint cantilever trusses with six and eleven joints,
subjected to load F at point A and fixed at support points B (Prager 1977)
Although many popular general optimisation methods have been applied or adapted to
structural design optimisation tasks, various problem specific methods also exist, in
particular to tackle the unique challenges of topological design.
Methods may be divided according to the representation type (discrete or continuous
material), the search type (deterministic or stochastic1) or the specific design task to
which they are applied, as presented in section 1.2.
Reviews of structural topology optimisation approaches are provided by Bendsøe and
Sigmund (2004) and Papalambros and Shea (2002).
1 A deterministic method, under fixed parameters, will always yield the same solution from a given
starting point. A stochastic search includes a probabilistic component, permitting multiple design
solutions to be obtained from a fixed starting point and fixed search parameters.
14
2.1.1. Size optimisation
Vast quantities of academic literature exist testing a diverse range of optimisation
methods on benchmark sizing problems. This section attempts to highlight the most
pertinent of these to the issue of integrating size and topology optimisation, since this
is the primary role of size optimisation in this thesis.
Size optimisation problems can easily be expressed mathematically and are
traditionally solved by deterministic methods. However, for statically indeterminate
structures subject to multiple displacement constraints, the design space is almost
invariably multi-modal.
Fully Stressed Design
Fully Stressed Design is most commonly used for structures in which strength
considerations govern over stiffness, such as small and medium size frames. Maxwell
(1864) recognised that in a statically indeterminate structure, in which members can
be resized without influencing the load path of the structure, the minimum weight
design is the one in which every member is subjected to the maximum permissible
stress in at least one loadcase, i.e. fully stressed. For indeterminate structures, the
number of distinct fully stressed designs can be very large. Conventional procedures
increase the size of over-stressed members and reduce the size of under-stressed
members, reanalysing and iterating until convergence is achieved. However, Mueller
and Burns (2001) demonstrate that this approach excludes a set of repelling fully
stressed designs, in which some members will respond to an increase in size by
attracting greater stress. This causes an initial sizing solution in the vicinity of a
repelling fully stressed design to rapidly move away from this area of the design
space. Mueller and Burns (2001) employ a series of non-linear equations to define
the fully stressed state and solve with a hybrid Newton-Monomial method to find sets
of fully stressed designs, commencing from randomly generated initial designs.
Mueller et al. (2002) consider the trade-off between material volume and maximum
lateral drift in such a set and classify the load-bearing systems observed.
Mathematical Programming
Mathematical Programming methods (Borkowski and Jendo 1990) take an iterative
approach in seeking an optimal solution. A search direction within the design space is
15
first calculated, then the distance to travel in this direction, often referred to as step-
size, is determined. A wide range of techniques exist for determining the search
direction and step-size, according to the assumed characteristics of the objective
function, constraints and variables. This selection is generally, although not
necessarily, based on gradient information, hence defining the set of gradient-based
methods. Linear Programming may be used if functions exhibit linearity, otherwise
quadratic and general non-linear methods exist. Integer Programming will account
for the requirement for variables to take integer values, as required in discrete section
size optimisation.
Optimality Criteria
This group of methods incorporates problem-specific knowledge, such as the
principle of virtual work, into the Kuhn-Tucker conditions for optimality. A set of
necessary and sufficient conditions are derived to describe optimal solutions to
convex problems. Applied to structural section sizing, optimal solutions are found
iteratively with reanalysis to account for changes in load distribution. A
comprehensive introduction to optimality criteria methods in structural optimisation is
presented by Rozvany (1989). Grierson and Chan (1993) present an approach tailored
to the design of tall buildings.
The efficiency of Optimality Criteria (OC) methods is strongly dependent on the
number of global constraints, such as permissible displacements, and only weakly
dependent on the number of design variables. OC methods hold a further advantage
over mathematical programming techniques in that they are not restricted to locally
optimal solutions in the vicinity of the initial design. However, in structures with a
high degree of statical indeterminacy, changes in load distribution may mean the
approach still fails to locate the global optimum. This led to the hybrid OC-GA
method (Chan and Liu 2000), developed to combine the robustness of Genetic
Algorithms (GA), discussed in section 2.1.3, with the computational efficiency of OC.
OC methods sacrifice an element of generality on account of the requirement for
problem-specific physical laws.
Additional considerations
The problem of handling discrete variables is recurrent in structural optimisation,
since member sections must frequently be selected from catalogues, or fabricated
16
from standard gauge sheets. This issue is considered by Arora (2002) through
discussion and review of recent approaches.
Shea et al. (1997) consider the practical issue of dynamically assigning members to
groups, based on cross-sectional area and address this in the shape annealing method.
Recent trends have seen hybridisation of optimisation methods: in addition to the OC-
GA method previously discussed, various approaches use one method for topology
optimisation and another more appropriate technique for simultaneous size
optimisation. Examples include Simulated Annealing with Fully-Stressed Design in
cases with only stress constraints (Shea 1997), Genetic Algorithms with Optimality
Criteria (Sakamoto and Oda 1993), (Kicinger 2004) and Genetic Programming with
Optimality Criteria (Liu 2000).
2.1.2. Discrete topology optimisation methods
Ground Structure approach
The Ground Structure approach, first proposed by Dorn et al. (1964), effectively
reduces the complexity of a topology optimisation problem by considering a fixed
grid of nodes, initially with a high degree of connectivity (in extreme cases each node
may be connected to every other node) as exemplified in figure 2.3. These links
between nodes are potential structural members. They may take binary "on"/"off"
states, or be iteratively assigned a section-size and removed altogether if found to be
under-utilised, for example if assigned a section-size less than a prescribed minimum.
Various techniques have been used in obtaining a reduced, "optimal" structural
configuration from the initial system: Bendsøe and Sigmund (2003) discuss
deterministic methods including Optimality Criteria and Linear Programming,
Bennage and Dhingra (1995) use a meta-heuristic Tabu Search method, Xie and
Steven (1997) extend the principles of Evolutionary Structural Optimisation to the
design of pin- and rigid-jointed frames. Stochastic approaches include Simulated
Annealing (Topping et al. 1996) and numerous examples of the application of Genetic
Alogrithms, e.g. (Hajela and Lee 1995). Despite widespread interest in the ground
structure approach, there exist a number of significant limitations (Ohsaki and Swan
2002):
- The number of elements required for complete connectivity increases factorially
with the number of nodes in the ground structure. However, performance of
17
"optimal" structures is heavily dependent on the initial design and is compromised
by using simplistic ground structures. It is possible that much better solutions
may exist beyond the restrictions imposed by the grid framework. Most
implementations do not allow movement of nodes, or addition of nodes and
elements. The method is best suited to problems of modest scale, as opposed to
the design of complex structures.
- Unrealistic "optimal" solutions may be generated, with the possibility of
instability arising from removing too many members in a region of the structure.
- Complications arise in interpreting the connectivity of overlapping members for
analytical purposes.
- High sensitivity to multiple loadcases can be observed, as seen in a simplified case
based on the Eiffel Tower by Bendsøe and Sigmund (2004).
Figure 2.3: Fully-connected ground structure for a relatively simple (3x6) grid
In attempting to address some of the above issues, Smith (1998) has investigated
methods of generation of appropriate ground structures in two and three dimensions.
Bendsøe et al. (1994) construct a 3-D cantilever ground structure for which optimal
topology and member sizes are found. Performing a subsequent shape optimisation,
allowing nodal locations to vary, reduces grid dependence. It is worth noting that the
shape-optimised structure does not necessarily constitute the globally optimal design,
since it was derived indirectly and a different optimal topology may have been found
if the nodal locations in the ground structure had been different. Current research by
Gilbert and Tyas (2003) has used low connectivity initial designs (still based on a
fixed nodal grid), introducing new members to stiffen the structure at each iteration,
by considering the relative nodal displacements. This approach permits the use of
18
much larger grids and many more potential members (in excess of 100 million), but is
nevertheless prone to many of the standard complications of this class of method.
Ruled-based approaches
This section discusses examples of heuristics and grammars used to govern the
development of structures.
Grammars are a production system capable of describing a set of designs through the
transformations that map one design to another (Stiny 1980). An innovative approach
to optimally directed topology design is offered by Shape Annealing (Cagan and
Mitchell 1993), combining a defined shape grammar with Simulated Annealing
optimisation techniques. The shape grammar is a set of permissible design
transformations that may include the addition, removal or reorientation of structural
members, adjustment of nodal co-ordinates and resizing of components. Shea and
Cagan (1999) consider the design specifications as a syntax encoded in the structural
shape grammar with objectives and constraints as semantics, to produce a language
that encompasses a set of valid and meaningful structural designs. Applications
include planar trusses, 3D space-trusses (domes), truss beams and practical
transmission towers (Shea and Smith 2006)
McKeown (1998) considers growing least-volume trusses from the simplest structure
required to transfer prescribed loads to the supports. Auxiliary joints and members
are sequentially added, optimising joint locations and member sizes at each stage,
until the additional weight of a new joint outweighs the saving from reduced total
member volume, or an alternative termination criterion is met.
Rule (1994) presents a deterministic rule-based generative approach to truss design.
The final structure "evolves" in a number of stages, with the level of complexity
increasing at each advancement. A small base structure is generated, determined by
the number and location of loads and supports and the number of design stages to be
used. At each stage, the positions of free nodes (i.e. those unloaded and not acting as
supports) are optimised by a Mathematical Programming method, with new members
added by bisecting the longest members, thus ensuring structural members are of a
similar length and that the truss remains fully triangulated. The design of a 2D
representation of a high-voltage cable-support tower is detailed.
19
2.1.3. Evolutionary Algorithms in topology optimisation
Although not pure optimisation algorithms (De Jong 1993), Evolutionary Algorithms
(EAs) are versatile, stochastic, problem-solving methods, alternatively classified
under the name of Evolutionary Computing (EC). This class of methods (not to be
confused with Evolutionary Structural Optimisation) is so-called due to its mimicry of
natural biological evolution as postulated originally by Charles Darwin (1859). In
general, the performance of a population of individual solutions in solving the
prescribed problem is assessed according to one or more quantifiable criteria. In turn,
performance, commonly referred to as fitness, influences the chances of an
individual's involvement in populating the subsequent generation of solutions, by
some combination of the genetic operators of reproduction, crossover (or
recombination) and mutation. Around the 1960s, three sub-classes of EA were
developed independently: Genetic Algorithms (GA) (Holland 1975), Evolutionary
Programming (EP) (Fogel et al. 1966), and Evolutionary Strategies (ES) (Rechenburg
1965). However, these were not brought together under the name of EAs until the
1990s. A fourth class, Genetic Programming (GP) (Koza 1992), also emerged in the
1990s.
On account of operating on populations of solutions, as well as their stochastic nature,
EAs tend to be computationally expensive and hence cannot compete with numerical
methods in such tasks as regular continuous parametric optimisation (Eiben and
Schoenauer 2002). However, there are a number of problem types for which EAs, and
stochastic search methods in general, are particularly appropriate:
- multi-objective design space exploration (Deb 2001)
- problems with mixed (discrete and continuous) variables
- problems with a discontinuous space of feasible or legal solutions
- unconventional problems for which the representational flexibility of EAs can be
exploited.
A brief introduction to the field is presented by Eiben and Schoenauer (2002) and a
more comprehensive review is given by Bäck (1996).
Genetic Algorithms
In recent years the GA has been one of the most widely used computational search
techniques within the engineering research community. The genetic representation, or
genotype, of a physical design, or phenotype, is central to the concept of the GA. The
20
original, simple genetic algorithm (Holland 1975) uses a binary bit-string encryption,
although a wide range of alternative representations has since been used. Notable to
the field of structural topology optimisation are the real-valued representation used for
topology optimisation of three-dimensional trusses (Azid et al. 2002), voxel
representation used for determining the optimal cross section of beams (Griffiths and
Miles 2003) and graph representation (Borkowski et al. 2002) capturing structural
connectivity. Kicinger (2004) uses a GA to evolve Cellular Automata for generating
bracing schema in tall buildings. Rajeev and Krishnamoorthy (1997) define a
variable string length Genetic Algorithm (VGA) capable of considering
simultaneously a number of alternative pre-defined topologies through the use of
control variables, as well as encoding nodal positioning and section sizes in a standard
manner. A more conventional representation is used by Shrestha and Ghaboussi
(1998), with individuals encoded as a lengthy set of sub-strings representing a node,
its spatial co-ordinates and its connecting members. Significant in addressing the
issue of computational intensity is the Micro-Genetic Algorithm (µGA)
(Krishnakumar 1989), designed to operate on very small populations, normally of five
individuals.
As mentioned in section 2.1.1, GAs can and have been hybridised very effectively
with other optimisation methods to combine a broad exploration of the design space
in general with efficient exploitation of high performance regions.
Pezeshk and Camp (2002) present a chronological survey of research work conducted
in the 1990s related to the use of GAs in structural steel design.
Genetic Programming
The newest member of the class of EAs, Genetic Programming (Koza 1992) evolves
"programmes" composed of a sequence of low-level functions, traditionally
represented as a tree structure, to perform a prescribed task. Within the tree structure,
input variables or constants appear as "leaves" at the extremities of the tree. These are
operated on by functions at branch points, with the output passed to further functions
closer to the root of the tree. Hence, unlike the GA, GP methods do not require an
encryption scheme to convert the representation on which the genetic operators act
into the true representation. Graph structures can also be used within the context of
GP.
21
Genetic Programming is predominantly used in computer science for creating
programmatic solutions for tasks and has also found substantial application in design
of electrical circuits (Koza et al. 2003), reproducing or improving on patented
solutions. A review of use of GP in civil engineering is presented by Shaw et al.
(2003). Within structural engineering and topological design in particular, the work
of Soh and Yang (2000) and Liu (2000) serve as the sole examples known to the
author using the tree-based representation scheme. The former make appropriate use
of this representation to dispense with the need for a prescribed set of variables, but
retain a GA-style encryption scheme to map between genotype and phenotype.
Indeed, branch points take the form of section properties rather than any form of
operator, making the classification of the method as Genetic Programming debatable.
Liu (2000) uses a graph structure to represent the hierarchical decomposition of
structural system, again without the use of true functions at branch points.
Genetic Programming will be discussed in greater detail in Chapter 5.
Application of ES and EP methods to structural topology design problems has been
minimal, but these classes are mentioned below for completeness.
Evolutionary Strategies
Evolutionary Strategies (ES) (Rechenburg 1965) were developed in the 1960s as a
tool for continuous parameter optimisation. With greater emphasis on mutation,
offspring are generated by adding a mutation vector, with normally distributed,
randomly selected components, to a current design (Beyer and Schwefel 2002),
aiming to improve on previous solutions.
Evolutionary Programming
Evolutionary Programming (EP) (Fogel at al. 1966), develops populations of Finite
State Automata that transform an input sequence to different output sequences.
Fitness is assessed according to accuracy of response and offspring generated by
mutation of parents.
2.1.4. Continuum-based optimisation methods
This class of method considers a continuous designable domain, discretised into a
mesh of elements that are defined individually in a structural analysis model. The
properties of the continuum elements, such as porosity or thickness, can be varied
22
individually for size optimisation, or they can be removed or considered of vanishing
thickness for shape and topology optimisation. Regions of the analysis model may be
designated as non-designable.
Eschenauer and Olhoff (2001) and Bendsøe and Sigmund (2004) present a
comprehensive review of the research advances and state-of-the-art in this field. This
section presents three significant forms of continuum-based optimisation method,
with particular focus on Evolutionary Structural Optimisation on account of the
interest it has received in the building industry, due to its simplicity and intuitive
nature, and its development in Chapter 3 of this thesis. It should be noted that other
generic optimisation methods, such as GA have also been used with continuum
representations.
Homogenisation
This method, pioneered by Bendsøe and Kikuchi (1988) and expounded in detail by
Bendsøe and Sigmund (2004), defines individual materials for each element in the
mesh, each containing infinite microscopic voids. The porosity of the medium is
optimised according to some objective function. Each element-material may have its
own hole-size and orientation. Commonly, intermediate densities are penalised, to
encourage elements to become either fully solid or fully void. A target volume
fraction is generally set and variation of such modelling parameters admits a range of
types of solution from truss-like structures and plate type solutions to composites and
stiffened composites. This method provides the foundations for a number of
commercial packages, discussed in the next section 2.2.
Bubble method
The bubble method of Eschenauer et al. (1993) iteratively places voids or bubbles
within a continuum domain through a definite function before subsequently
performing a shape optimisation on each topology. Whilst this method may be
suitable for small components, its applicability to structures with high topological
complexity appears low, on account of the procedure required for the addition of each
void.
Evolutionary Structural Optimisation
This method was originally developed by Xie and Steven (1993), (1997). In its
simplest form elements that are under-utilised, as defined by some metric such as
23
strain energy density, are removed from a continuous finite element mesh, to reduce
the designable domain to an efficient optimal topology. The name is misleading,
since it is not evolutionary in the same sense as Evolutionary Algorithms, based on
Darwinian principles, nor is it strictly optimisation, rather it is a design technique
seeking uniformity of parameters within a structure, such as stress or strain energy
density. However, although the term coined by Rozvany (Rozvany 2001), Sequential
Element Rejection and Admission (SERA) techniques, is more accurate, it will be
referred to as Evolutionary Structural Optimisation (ESO) throughout this thesis, in
line with the majority of literature on the subject. ESO can be considered a hard kill
method, in that a step-function is used in defining the elastic modulus of elements as
opposed to the soft kill homogenisation methods where a continuous range of values
is permitted.
Since its conception, ESO has proven to be a versatile method, readily understood and
simple to implement.
There have been a number of developments on the basic ESO method: Additive
Evolutionary Structural Optimisation (AESO) (Querin 2000) adds new elements
adjacent to the most suitable existing perimeter elements, to solve boundary problems
through shape optimisation. This method has low relevance to building design on
account of its limitation to shape optimisation. Bi-directional Evolutionary Structural
Optimisation (BESO) (Querin 2000a) permits removal and addition of elements, the
latter either by extrapolation of "sensitivity number" (a performance parameter such
as strain energy density), or by considering the sensitivity number of perimeter
elements. BESO methods have the significant advantage that material that was
removed early in the evolutionary process can be replaced later if found to be
structurally advantageous, hence offering improved design space exploration and
increasing the probability of finding globally optimal solutions. BESO also allows
development of solutions from simple initial designs (Querin 2000a) that are therefore
less computationally expensive. However, on account of bi-directionality, a greater
number of iterations are likely to be required than in the basic algorithm. Extended
Evolutionary Structural Optimisation (XESO) (Cui et al. 2003) works by constructing
contour lines of stress, or some other property, within the designable domain at each
iteration. Material with stress below the critical threshold is removed, with material
added in areas where extrapolated contours lines predict high stress values. The finite
element model is revised by remeshing for each step of the process. It is claimed that
24
XESO allows evolution of configurations that cannot be obtained by the original ESO
method, such as a "suspension-arch" design for a uniformly vertically loaded beam
with simple supports at each end.
2.1.5. Computer-based conceptual design methods
It is important to appreciate the iterative and multidisciplinary nature of the
conceptual design of buildings. Architects, structural engineers and building services
engineers all interact in developing ideas of how a structure should look and behave.
The development of a design tool to support the interests of all of these groups is an
ambitious goal, but one that has nevertheless been attempted in a number of research
initiatives. Sisk et al. (2003) state that the focus of development for computing tools
in conceptual design should be towards Decision Support Systems (DSS) rather than
optimisers, suggesting the use of the GA as a search tool, but also highlighting the
importance of human-computer interactivity. This point is crucial to gaining
acceptance in engineering practice, since designers will inevitably be dismissive of a
black-box process that produces a single "optimised" solution to a given problem,
without insight into its development. Rapid interpretation and understanding of
results is essential. Hence the role of computer-assisted search methods should be to
enable designers to consider a wider range of design alternatives, with more
indication of their projected performance.
Grierson and Khajehpour (2002) present a major review of computer-based
conceptual design research, both with and without optimisation, between 1989-1999.
Some of the more relevant of these are mentioned below.
McCarthy (2002) provides a review of the application of Knowledge-Based Expert
Systems (KBESs) to structural steelwork design, used for eliciting and applying
"facts" and "rules" from pre-programmed information. HI-RISE and further KBESs
developed at Carnegie Mellon University are summarised by Maher (1987). The HI-
RISE system (Maher 1984), amongst the most cited of early KBESs, performs the
preliminary structural design of high-rise buildings, without use of optimisation.
Given a 3D grid defining the space planning of the building, the system will present
feasible structural systems. The aim is to support the designer by increasing the
number of designs available for consideration for further development. Smith (1996)
and Rafiq et al. (2003) comment on the limitations of KBESs, notably the difficulty in
developing individual or novel design solutions that are unlikely to be conceived by a
25
human design team. Limitations are attributed to problems in combining computer
system heuristics with human knowledge and a lack of flexibility.
Park and Grierson (1999) use a Multicriteria Genetic Algorithm (MGA) to generate
pareto-optimal conceptual designs of office buildings under the competing objectives
of minimising building project cost and maximising flexibility of floor space. Design
variables of plan dimensions and storey height were used, with the possibility of four
non-rectangular floor plans, and a set of feasible floor systems. This approach was
developed by Khajehpour (2001) by considering rectangular plan high-rise office
buildings with multicriteria optimisation minimising capital and operating costs and
maximising income revenue for a given project. Colour filtering was used to mark
variable values of the pareto-optimal (Pareto 1896) design set in the three-
dimensional criteria space. Substantial effort was expended in researching costs and
accounting for architectural, structural, mechanical and electrical systems.
Recommendations for future work include accounting for alternative floor plan shapes
and changing size with height as well as development of structural design criteria.
Currently material costs are based on relatively approximate member sizing
techniques. A further study by Khajehpour and Grierson (2003), motivated by the
progressive collapse failure of the World Trade Center Towers, extended the previous
work to consider the trade-off between profitability and safety of high-rise office
buildings. Load-path safety against progressive collapse is determined by the degree
of force redundancy in the structural system.
The SEED system (Software Environment to support Early phases in building
Design) of Rivard and Fenves (2000) is divided into three main modules, supporting
the generation of an architectural program, generation of scheme layout and design of
3D building configuration. Crucial to the third of these modules, SEED-Config, is the
definition of a building design representation. The following requirements are noted
in such a representation: extensibility, ability to integrate multiple views and support
design evolution and favouring design exploration. The authors acknowledge the
tendency for designers to break down complex problems into small subproblems and
this is reflected in the representation developed. Of particular note are the four
structural subsystems: the foundation (transfers all loads to the ground), the vertical
gravity subsystem (transfers vertical loads to foundations), vertical lateral subsystem
(transfers horizontal load, e.g. wind to foundations) and horizontal subsystem
(transfers live, dead and snow loads to vertical gravity subsystem). This subdivision
26
of the structural volume permits independent consideration of the subsystems. Each
subsystem consists of a set of structural assemblies (e.g. floors, roofs, frames, walls,
column stacks or 3D systems such as cores or tubes). These in turn are made up of
basic elements.
2.2. OPTIMISATION IN BUILDING ENGINEERING DESIGN PRACTICE
2.2.1. Comparison of structural design in the automotive and
aeronautical industries versus the building industry
When considering the issues associated with successful application of optimisation
methods to structural design in the building industry, it is worthwhile contrasting this
sector against the automotive and aeronautical industries where, in recent years,
optimisation has become increasingly widespread, aided by the implementation of
sophisticated optimisation methods in commercial software.
Table 2.1 compares aspects of design and production of a typical automotive
component and a steel building structure. Examples of topological design
optimisation of automotive components are provided by Rousseau (2004), considering
a steering wheel, and Wieloch and Taslim (2004), considering a pump bracket. The
independent consideration of detailing and minimising weight of small components in
the automotive industry stands in contrast to the standard sections used in most
building projects where it is harder to define substructures and design components.
Keer and Sturt (2007) discuss optimisation of the car body, or "body-in-white", as a
whole. They consider gauge (panel thickness) optimisation and shape optimisation,
noting that the former is better developed and more widely used. A shape
optimisation example is provided by Paas and Hilman (2006).
In summary, the high premium on weight in automotive and aerospace industries,
coupled with the economies-of-scale offered by producing vast numbers of identical
parts, mean that investment in optimisation techniques capable of saving small
proportions of the total mass are generally rewarded. Additionally, aesthetics are
commonly of lesser concern than in building structures and the continuous material
layout task is well suited to material distribution techniques such as Homogenisation
and Evolutionary Structural Optimisation.
27
An assessment and discussion of the potential for application of optimisation in the
building industry will be made at the end of this chapter after considering existing and
potential examples. Leubkemann and Shea (2005) have previously highlighted the
benefits and potential of computational design and optimisation in building practice.
Table 2.1 Design and production of a typical automotive component versus a steel
building structure
Characteristic Automotive/aerospace component
Steel building structure
Topological complexity
Generally low, higher for whole body
High, but often ordered, hence parameterisation may be possible
Production volume Large (up to 100,000s) Low (generally one-off)
Principal cost considerations
Design, material, knock-on costs of weight
Design, construction, life-cycle costs
Knock on effects of excess mass beyond material cost
Reduces efficiency and vehicle speed
Extra dead-weight increases loads elsewhere in structure
Typical constraints Stiffness, strength, natural frequency, fatigue
Stiffness (global), strength (local, often buckling-related)
Manufacture Purpose designed process (e.g. casting) allows flexibility and irregularity of component form
Discrete catalogued structural members from steel supplier
Aesthetic considerations
Minimal Often critical for topology design
2.2.2. Commercial optimisation software
A number of increasingly sophisticated software suites are commercially available for
tackling structural optimisation problems.
Material distribution approaches have attracted significant interest and are
implemented in TOSCA2 (FE-Design), GENESIS
3 (Vanderplaats Research and
Development) and OptiStruct4 (Altair), alongside other optimisation methods. A
2 http://www.fe-design.de/en/tosca/tosca.html
3 http://www.vrand.com
4 http://www.uk.altair.com/software/optistruct.htm
28
beta-capability for topology optimisation by a material distribution method is also
included in Nastran5 (MSC). Leiva (2001) presents examples of the application of
GENESIS optimisation capabilities to design of automotive sub-structures.
Other programs integrate with CAD/CAE systems and finite element analysis
software to automate simulation tasks, with the aim of converging to optimal designs.
Optimus6 (Noesis) and LS-OPT
7 (Livermore Software Technology Corp.) offer these
capabilities through Design of Experiments (DOE) and Response Surface Modelling
(RSM) (Myers and Montgomery 1995) and a selection of other modules such as
global optimisation through stochastic methods, discrete variable and multiobjective
optimisation, robust design and parallelisation for computational efficiency.
Nastran is a well-established and powerful general-purpose finite element solver,
including the BIGDOT optimiser (Vanderplaats 2004), also used by GENESIS, as
well as a range of gradient-based methods in the Automated Design Synthesis
program.
SODA8 (Structural Optimization Design Analysis, Acronym Software Inc.) is the
commercial realisation of extensive research at the University of Waterloo, e.g.
(Grierson and Chan 1993), implementing the Optimality Criteria method for least-
weight section-size optimisation and including code-checking capabilities. SODA has
benefited from the feedback of practising engineers across North America, as well as
being used in academic research, e.g. (Kicinger et al. 2007).
The application of continuous design domain or distributed material methods will be
discussed in more detail in Chapter 3. However, it should be noted at this point that,
in general, the discrete nature and large scale of building structures means they are not
as suited as an automotive component to design by distributed material methods.
The use of deterministic optimisation methods and automated search for assigning
structural section sizes appears to be becoming increasingly frequent in design of
complex structures. In-house and small-scale commercial software, as well as
5 http://www.mscsoftware.com/support/prod_support/nastran/documentation/rg_2005.pdf
6 http://noesissolutions.com/index.php?col=/products&doc=optimus
7 http://www.lstc.com/pages/lsopt.pdf
8 http://www.acronym.ca/
29
spreadsheets are often used for this task. Section 2.2.3 discusses specific examples in
more detail.
2.2.3. Published literature on industrial applications
Section-size optimisation
As discussed previously, size optimisation is the easiest structural design task to
define mathematically and has been the focus of vast amounts of academic research
and literature. Whilst the use of optimisation and automated search for section-sizing
in industry practice is not commonplace, it is increasing, primarily using deterministic
or hybrid methods.
The OPTIMA system developed at HKUST (Chan 2004) has been used for the
section-size optimisation of designs for a number of tall buildings in Hong Kong. The
software uses techniques based on combinations of OC methods, GA and exhaustive
search, interfacing with most structural analysis software. Optimal member sizes are
found for tall buildings of mixed construction, subject to all design performance
criteria and various codes and requirements. In the design of the Kowloon Mega
Tower, due to be the world's second tallest building at 474m, optimisation was used
to minimise construction material cost whilst maximising value of floor space,
considering the area of vertical walls and columns. Structural layout changes were
recommended for the Park Central Development by the hybrid OC-GA method,
affording greater cost savings than by section-size optimisation alone.
An earlier collaboration between Ove Arup and Partners Hong Kong Ltd and HKUST
in the size optimisation of the North East Tower, Hong Kong Station (Chan et al.
1998) considered optimisation by OC methods for minimum overall weight,
minimum material cost and minimum overall cost allowing for the benefits of
increased useable floor area. An overall cost saving of 9% was reported considering
the third of these objectives.
Various intuitive optimisation and automated section sizing procedures have been
implemented by engineers at Arup in response to the demands of highly complex
design specifications, a selection of which are reported in an internal Arup seminar
report (Shea and Baldock 2004) or discussed below.
Each of the 25,000 steel beams in the irregular long-span cellular moment frame of
the Beijing National Swimming Centre (Stansfield 2004), (Bull 2004) was assigned
30
to one of three groups and a section size chosen from the allowable set for the group,
subject to strength checks and constraints on slenderness ratio. A heuristic tool was
developed, combining an Excel spreadsheet with a Chinese code checker purpose-
written using Visual Basic macros and the Strand7 (finite element analysis) Advanced
Programming Interface (API). From an initial design, analysis results guided the
increase or decrease in section sizes, with the process repeated until convergence.
Since there is no explicit objective function, this is a constraint satisfaction task using
a deterministic, heuristic search. Minimum tonnage was attained using the smallest
available sections in the initial design and using an optimised range of cross sections
for each of the three groups.
An iterative graphical approach was taken in determining the most efficient material
distribution for meeting stiffness requirements in the design of the 299m
Commerzbank HQ, Frankfurt (Wise et al. 1996).
Evolutionary Structural Optimisation (ESO)
Of the distributed material approaches, Evolutionary Structural Optimisation has
received the greatest attention in the building industry. This can be attributed to the
intuitive and readily comprehensible method through which it provides the designer
with an insight into load-paths within the structure. The cases below use the method
for the design of free-form or "concept" structures.
ESO has recently been applied to the design of the Akutagwa River Side Building at
Takatsuki JR station, in Japan (Ohmori et al. 2005). This is a rectangular four-
storey office building, the construction of which was completed in April 2004. The
designable domain defines the free-form external concrete wall/column system. A
glass façade is integrated with this structural system to form the building's enclosure.
Of particular note in this project is the fact that a discrete interpretation of the ESO
design was not required, although a smoothing algorithm was applied.
In 2002, a Japanese collaboration between architect Arata Isozaki and engineer
Mutsoro Sasaki (Sasaki 2005) resulted in the development of a competition design for
the Florence New Station, Italy. The design of the supporting structure for a 150m
long, 26m wide and 15m high roof was developed using the Extended ESO (XESO)
method (Cui 2003). Again, the ESO-derived topology was not strictly translated into
a standard section discrete design, but in this case the shape was subdivided to form a
structural grid of steel hollow sections. The structural engineer (Sasaki 2005) noted
31
the importance of the "man-machine interface" in absorbing the judgements of the
designers in the development of the design solution.
The Convention Hall at the Qatar International Exhibition Centre was also
designed using ESO, by Mutsuro Sasaki and coworkers (Cui et al. 2005).
In his recent thesis, Holzer (2006) uses ESO as a focal method in a discourse on the
architect-engineer interaction in the early stages of design. An architectural project
brief for a 12 storey tower building is used as an exploratory study.
The application of ESO to the design of discrete external bracing systems for tall
buildings will be explored in greater detail in chapter 3.
Parametric optimisation
A retrospective study was carried out on the design of the pedestrian footbridge at
the Selfridges store in Birmingham, UK (Maher and Burry 2003) as a collaborative
effort between Arup, RMIT and Future Systems architects. This served to illustrate
the potential of integrating the CATIA V5 geometric model with finite element
analysis software and the in-built optimiser to generate a range of parametrically
diverse solutions. Parametric variables, such as location of cable restraint, horizontal
and vertical location of the main tube as well as its diameter, were used to define
families of configurations of the bridge. Optimisation used a combination of a local
gradient-based method and simulated annealing, provided by the CATIA module (see
below). Parametric studies in three or four free parameters took around seven hours,
illustrating the premium on computational efficiency in such tasks.
Repeated ad hoc computer analysis was used for similar parametric analysis of the
roof structure of the Wellness Center for Shaw University, North Carolina USA
(Place 2001). The roof is formed from two intersecting sections cut from inclined
cylinders. In over a hundred distinct analyses, issues explored included varying
element spacing, section shape and size, using different enclosure materials and
considering possibilities for element removal to reduce complexity and blocking of
light. Whilst this is not true algorithmic optimisation, it seeks to find the best design
possible within the design domain defined by the constraints.
32
Non-parametric topology optimisation
Shea and Zhao (2004) detail the design optimisation of a recently installed noon-mark
cantilever, carried out in collaboration with Eric Parry architects, in London. A
Structural Topology and Shape Annealing (STSA) method, using discrete
representation, was used to create a selection of novel designs subject to stringent
structural and architectural constraints. Solutions could then be considered according
to unmodelled criteria. This project demonstrated a number of challenges faced by
applying optimisation methods in practice, including changing design specifications,
incorporating architectural constraints and applying design codes.
2.2.4. Facilitating structural optimisation
The purpose of this section is to briefly present examples of software suites that are
likely to facilitate the application of structural optimisation in building engineering
practice in the future. It also considers project cases studies to which these tools and
others presented in this thesis could be applied.
Software
CATIA V59 (Computer Aided Three dimensional Interactive Application, Dassault
Systemes) is a multi-platform Product Lifecycle Management (PLM) / Computer-
Aided Design (CAD) / Computer-Aided Manufacture (CAM) / Computer-Aided
Engineering (CAE) suite with 3D parametric design functionality. CATIA includes
structural analysis capabilities and an optimisation module capable of gradient-based
search and simulated annealing.
GenerativeComponents10
(Bentley Systems) is described as a parametric and
associative design system, capable of capturing and displaying design components
and their interrelationships. The intention to provide a tool for both architects and
engineers is important in improving documentation exchange and common
understanding, which could, in turn, aid the uptake of optimisation.
9 http://www-306.ibm.com/software/applications/plm/catiav5/
10 http://www.bentley.com/en-US/Markets/Building/GenerativeComponents.htm
33
Parametric optimisation case studies
During the design of the Swiss Re Building at 30 St Mary Axe, London, the external
form was defined parametrically in terms of shape variables (Foster and Partners
2005). This approach allowed Foster and Partners to rapidly regenerate complex
geometric models, which would otherwise require days to construct. Bentley
Systems' GenerativeComponents software was used in this project. The method has
many advantages, including aiding information exchange between architects and
engineers, facilitating panel definition for ease of construction and offering the
potential of integrating an iterative optimisation routine to automatically explore the
defined design space.
Another example of potential for parametric optimisation can be seen in the design
development of the SAGE Music Centre, Gateshead (Cook et al. 2006). Parametric
modelling of geometric associations between member segments facilitated exploration
of primary and secondary arch profile configurations for the structure's free-flowing
roof form.
Although computational optimisation was not used in either of these projects, it would
have been a relatively straightforward and potentially beneficial extension.
Non-parametric discrete optimisation and design generation case studies
The superstructure of the 234m tower of the CCTV Headquarters, Beijing, China is
essentially a continuous braced tube, with patterned diagonal bracing on a regular grid
of columns and beams (Carroll et al. 2005). The bracing was required to visually
express the force distribution within the structure, whilst providing sufficient stiffness
during construction and service as well as robustness and redundancy in the event of
the removal of key elements. The bracing mesh was manually and iteratively
modified in response to the observed force distribution in the analysis model and
close consultation with the architect. Mesh density was increased in areas of high
force and reduced where forces were low. The process was complicated by the highly
indeterminate behaviour of the structure, with substantial changes to force
distribution, stiffness and dynamic behaviour resulting from changes to the bracing
patterning. With over 10,000 elements in the full structural model, analysis time was
also significant. An optimisation routine or automated method of implementing the
iterative, heuristic bracing pattern modifications would have been valuable on this
project, although incorporating all necessary considerations would have been a major
34
challenge. A stochastic component within an automated search could have been
useful in avoiding convergence to locally optimal solutions.
The proposed 122 Leadenhall Street building is a 48-storey, 225m wedge-shaped
tower with a braced perimeter tube stability system. Although the configuration
submitted in planning applications adopted a regular pattern with a diagrid on the
inclined face, in the early stages of the project a large number of bracing
configurations were sketched by structural designers, as seen in figure 2.4. These
were not subjected to any form of structural evaluation, but simply presented for
consideration on aesthetic grounds. However, a tool for rapid generation and
evaluation of diverse bracing schema could have been very informative on this project
and given weight to the case for one design over another.
Figure 2.4: Concept sketches for bracing design of 122 Leadenhall St. Building
(reproduced by kind permission of Chris Neighbour, Arup)
The two examples in this section were motivating factors for the research in this
thesis, as they show how topology optimisation could offer significant benefit if
appropriate, practical tools were available.
2.3. CONCLUSIONS
High potential cost benefits, relatively small task size and the availability of suitable
commercial software have driven a significant number of cases of the application of
35
structural design optimisation in the automotive and aerospace industries.
Commercial software for continuous topology optimisation is closely linked to current
research and generally adopts cutting-edge techniques. However, successes in
automotive and aerospace industries have not been matched in the building sector.
It is apparent that assignment of member section-sizes is the most readily solvable of
structural optimisation problems, due to its mathematically well-defined nature. This
is reflected in the vast amount of academic research in the area and its prevalence in
existing practical examples of structural optimisation. Nevertheless, this class of task
is not always straightforward, with issues of local minima, discrete variables on
account of section catalogues, multiple loadcases and constraints, multiple variables
for a given cross-section, potentially thousands of variables and possible non-linearity
(e.g. in the case of concrete structures). Despite examples in high-profile projects and
increasing use in recent years, size optimisation is not common-place in structural
design practice. Further, the efficient integration of size optimisation into topology
optimisation routines presents a relevant challenge that is arguably under-researched.
Increasing use of electronic document interchange between architects and engineers,
and CAD software with appropriate capabilities, has raised the potential for
parametric design investigation and ultimately computational parametric optimisation.
Optimisation modules exist within tools such as CATIA, or alternatively could be
purpose written.
Since decisions in conceptual and topological design have a great impact on final
design efficiency (Deiman et al. 1993) and the exploration of novel designs at this
stage is a difficult task, research in the field of non-parameterised topological design
is highly justified. With the exception of the ESO examples detailed previously, the
author is unaware of cases in which this task has been successfully performed on an
entire building in practice. It is likely that successful examples will fuel further
interest and accelerate the uptake of design optimisation in this sector.
2.4. JUSTIFICATION OF CASE STUDY
A large proportion of the examples outlined in the above discussion have featured
high profile tall buildings, generally irregular in some shape or form. Such structures
are often iconic for a city and become flagship projects for the architects and
engineers. The challenges presented by pushing the boundaries of structural
36
engineering demand innovative solutions. Topology optimisation in tall buildings is
therefore an appropriate choice of case-study problem for the methods considered in
this thesis.
In very tall buildings, providing lateral stability through a central concrete core can
require unacceptable loss of lettable floor space. For this reason, various forms of
external stability systems are popular in this class of structure. A steel tubular system
consists of columns and beams defining an orthogonal grid around the perimeter of
the building. In isolation, this vierendeel framework relies primarily on its bending
stiffness and is therefore susceptible to unacceptably high displacements. Diagonal
bracing is generally used to triangulate the structure, so that loads are carried
primarily axially, thus greatly increasing the stiffness. The bracing configuration
often plays a prominent role in the aesthetic impression of the building, examples
including the Swiss Re Building and the Bank of China Tower, Hong Kong
(Robertson 2004). However, an efficient design can offer great savings in material
and construction cost, as well as increasing the potential revenue from letting of floor
space.
This thesis therefore adopts as its central test problem the design of bracing for lateral
stability in tubular steel building frameworks. As discussed, this choice is appropriate
on account of its relevance to practical building design, but also as a common proof-
of-concept problem in the academic literature. This section will highlight a selection
of previous academic work tackling the bracing design problem.
Continuum topology optimisation has been previously explored by Mijar et al. (1998)
and Liang et al. (2000) and whilst these studies exhibit limitations, as will be
discussed presently, they provide useful benchmark cases. Mention should also be
made of a practice-driven study by Holzer (2006) who uses ESO for the design of a
tower structure from a real architectural project brief, subject to lateral and other
loads.
Kicinger (2004) used the design problem to illustrate methods developed in
distributed evolutionary design, considering various types of bracing and beam and
support fixity. Arciszewski (1994), (Arciszewski et al. 1997) explored the learning of
design rules through examples in this field, also postulating the possibility of
parameterising bracing designs according to the size of a bracing unit (number of bays
and storeys) and the type of bracing therein (e.g. K or X bracing) in conjunction with,
for example, a Genetic Algorithm.
37
Bracing design also features as part of a number of wider-ranging tasks in
optimisation research, such as those addressed by Liu (2000) and Khajehpour and
Grierson (2003).
Neidle-Cornejo (2004) conducted a comparative study into forms of bracing design in
tall buildings.
2.5. CONTEXT OF RESEARCH CONTRIBUTIONS
The contributions of this thesis to the field of structural optimisation research are
stated at this point, in advance of a full presentation of the work undertaken.
Evolutionary Structural Optimisation
The primary objective of this investigation is to demonstrate means of making ESO
more useful to the building industry.
The grouping of elements is a significant development on previous ESO research in
topology optimisation and is done in two ways:
(i) to have the same thickness. This form of group definition is required for the
integration of the topological ESO process with optimisation of domain thickness. To
the author's knowledge, previous integration of thickness and topology optimisation in
ESO has involved only linear scaling of the thickness of a single designable domain
region to meet a displacement constraint (Liang et al. 2000a). In the current research,
at each iteration of the ESO process, the thickness in each group is adjusted to obtain
uniform average strain energy density in each group, whilst still meeting the
displacement constraint. The Bi-directional Evolutionary Structural Optimisation
algorithm is modified to adapt to the constraint tracking scenario whilst moving
towards efficient designs retaining a low proportion of the full set of possible
elements.
(ii) to be removed together. This has been done previously with pairs of elements
grouped to define symmetry in a vertical centre line (Liang et al. 2000). However,
defining larger groups allows designs to be developed according to preconceived
aesthetic requirements such as pattern repetition or more complex symmetry.
Increased practicality is also achieved by focusing on displacement constraints as
opposed to overall compliance of the structure. Whilst ESO with displacement
constraints is well established (Liang et al. 2000a), this was not applied to previous
work in bracing design (Liang et al. 2000), where compliance was considered instead.
38
Pattern Search and Optimality Criteria
This section demonstrates how pattern search can be applied to a parameterised, grid-
based topological design problem in the context of a live industrial project. To the
author's knowledge the example is unique in the scale of the topology problem
addressed "live". It illustrates the selection of a method appropriate to the task being
considered, following the "problem-seeks-solution" model discussed by Cohn (1994)
as opposed to the "solution-seeks-problem" approach more commonly seen in
academic research. The documentation of topology optimisation on a "live" project is
a valuable contribution in itself, especially with the associated discussion of the
design issues encountered during the project development to which the optimisation
model and method must adapt. The study includes using optimisation in a parametric
constraint sensitivity investigation, as well as presenting a selection of high-
performance, locally optimal designs at different stages of the design process. From a
single starting point, the potential for generating a range of designs by randomly
selecting the order of exploratory moves is seen. However, diversity of designs is
shown to increase by using different starting points.
An innovative method of constraint handling is developed, scheduling the
convergence towards constraint boundaries through the use of penalty functions,
preventing premature process termination due to acceptance of disadvantageous
design modifications.
Computational efficiency is highlighted as being important and the classic Hooke and
Jeeves Pattern Search algorithm (1961) is modified in order to improve this. An
efficient means of integrating topology and size optimisation is established, finding an
approximation for the optimal set of section sizes at each topology iteration, thus
minimising the amount of additional structural analysis required. This approach
effectively considers strength and stiffness requirements in the member sizing
operation.
Genetic Programming using design modification operators
Chapter 5 presents an innovative application of Genetic Programming to structural
topology design. This approach is a unique and powerful development since the tree
representation includes design modification operators as functions at internal nodes,
rather than an encryption scheme for mapping genotype to phenotype (Yang and Soh
2000).
39
With appropriate optimisation parameters, consistent convergence to a global
optimum is observed on a small benchmark problem, with acceptable computational
efficiency. On a scale that is more realistic from an industrial perspective, novel high
performance designs are obtained, including one that outperformed known designs for
the defined optimisation model.
Simple but powerful extensions are detailed to allow tighter control over design
aesthetics, with the aim of empowering the user and improving convergence on
account of the reduced design space. This form of design tool holds great potential
for assisting designers in the conceptual design stage, when the ability to generate and
evaluate a diverse range of design solutions is highly beneficial. The general
approach shown is likely to be applicable beyond bracing design tasks.
3. Continuum topology optimisation of braced steel
frames
3.1. INTRODUCTION
As discussed in the preceding chapter considering the state-of-the-art in research and
practice, Evolutionary Structural Optimisation (ESO), in its various forms, has gained
significant interest due, in part, to its simplicity, elegance and ease of implementation.
This chapter considers benchmark problems proposed by Mijar et al. (1998) and
Liang et al. (2000), to address the following:
Research Question:
– How can the practicality of ESO be improved to make it more useful to the
building industry?
Proposals:
- use practical design criteria and objectives, such as displacement constraints
for tall buildings, with design heuristics to meet aesthetic criteria of pattern
repetition and symmetry.
- attempt to improve global optimality of solutions, independent of domain
thickness, mesh density and initial design.
- introduce simultaneous topology and thickness optimisation of defined
designable regions.
3.2. BACKGROUND
3.2.1. Method overview
The following description of the canonical form of ESO and the argument presented
in section 3.2.3 is based on the seminal book by Xie and Steven (1997). The method
offers parameter-free optimisation of shape and topology by gradually removing
inefficient material from a continuous representation of a structural domain.
40
Finite element analysis is performed on an initial design model, in which a material
continuum is divided into a fine mesh of finite elements, generally quadrilaterals for a
two-dimensional problem. The model includes loads and boundary conditions. An
efficiency metric, referred to as the sensitivity number, α, is defined, examples being
average von Mises stress or strain energy. The sensitivity number can then be
calculated for each element and a selection of the elements with the lowest sensitivity
numbers, according to a defined criterion, are then removed from the structure.
Examples of removal criteria are elements with sensitivity number less than a
prescribed proportion of the maximum value observed within the structure, referred to
by Xie and Steven (1997) as the rejection ratio, RR, or a prescribed number of
elements with the lowest sensitivity number. The reduced structure is reanalysed and
the process repeated. When no further elements can be removed according to the
rejection ratio removal criterion, RR is increased by adding to it a defined
evolutionary rate, ER, thus allowing the removal process to continue. Termination is
arbitrarily defined by, for example, a set proportion of the original material having
been removed or a displacement or compliance constraint being violated. Xie and
Steven (1997) note that the evolutionary approach gives the designer the opportunity
to select any intermediate stage in the process as a basis for further design
development.
3.2.2. Addition considerations and extensions
Li et al. (1999) demonstrate equivalence in ESO between stiffness, using some form
of strain energy as sensitivity number, and stress based element removal criteria. This
is significant through the implication that stiffness optimisation will yield strength
efficient designs. From a practical perspective, von Mises stress provides an
alternative to considering strain energy, or strain energy density, for stiffness
optimisation if the element stiffness matrix formulation is not known when using a
commercial package. The paper further notes that some differences in evolved
topology may occur, on account of numerical error, such as may arise from different
numbers of Gauss points used to estimate the von Mises stress in an element. Despite
the equivalence between strain energy and stress based element removal, the research
41
described in this chapter will be based on the former value, in line with previous,
closely related work.
Checkerboarding is a commonly observed phenomenon in continuum structural
topology optimisation (Sigmund and Petersson 1998). Patterns of elements and voids
may emerge in a checkerboard formation, presumed to be due to numerical errors in
the finite element approximation causing the design criterion to be alternately
overestimated and underestimated. Such regions are unacceptable in practice, since
they inhibit interpretation of the design as a discrete or manufacturable structure.
Various solutions have been proposed, notably:
– use of higher order elements (Manickarajah et al. 1998), which greatly increases
computational time.
– cavity control (Kim et al. 2000), whereby the number of cavities within the final
topology is defined.
– perimeter control (Yang et al. 2003), in which, for Bi-directional Evolutionary
Structural Optimisation (BESO, detailed in 3.2.4), element addition and removal
is restricted to prevent the total topological perimeter length from exceeding a
prescribed value. Perimeter control is also capable of eliminating mesh
dependency and provides influence over topological complexity, hence
manufacturability and, by extension, cost of design.
The current research uses a first-order weighted averaging algorithm as detailed by Li
et al. (2001) to avoid checkerboarding. For square elements, as used for the problem
considered in this chapter, this reduces to averaging the sensitivity numbers of the
element in question and its immediate neighbours. A weighting coefficient of 4 is
assigned to the element itself, 2 to active elements with a common edge and 1 to
active elements with a common corner, as shown in figure 3.1.
Figure 3.1: Weighting factors used for averaging sensitivity numbers across
elements to avoid checkerboarding.
42
4
2
2
2
2
1
1
1
1
3.2.3. Evolutionary Structural Optimisation (ESO) for stiffness and
displacement constraints
Linear static finite element analysis solves the equilibrium equation:
[ ]{ } { }PuK = Eq. 3.1
where:
[K] = global stiffness matrix;
{u} = nodal displacement vector;
{P} = nodal load vector;
The strain energy, or compliance, of the structure as a whole can be expressed as:
{ } { }uPCT
2
1= Eq. 3.2
or as the sum of the strain energy of each constituent element of the finite element
model.
{ } [ ]{ } ∑∑∑===
===N
i
i
N
i
i
N
i
iiTi cuKuC1112
1α Eq. 3.3
where the sub- or superscript i indicates that the term refers to the ith element in the
structure. Thus, the sensitivity number for this problem, αi, is the element strain
energy. In cases where the designable domain is divided into unequal elements, either
in terms of area or thickness, strain energy density should be used as the sensitivity
number on which element removals are based, obtained by dividing the element's
strain energy by its volume or weight.
The change in the global structural stiffness matrix by removing a single element
from the structure is exactly the negative of the corresponding element stiffness
matrix:
[ ] [ ] [ ] [ ]iKKKK −=−=∆ * Eq. 3.4
where K* is the global stiffness matrix of the resulting structure. Making the
approximation that element removal does not change the load vector {P} and by
neglecting a higher order term:
[ ] [ ] }{}{1
uKKu ∆−=∆ −Eq. 3.5
By extension, the change in compliance of the structure is thus predicted to be
identical to the strain energy in the element to be removed:
43
{ } { } { } [ ]{ }iiTiTuKuuPC
2
1
2
1=∆=∆ Eq. 3.6
It is therefore proposed that by removing elements with the lowest strain energy at
each iteration of the evolutionary process, optimal structures will be developed,
capable of meeting a compliance constraint with minimal material volume.
It is more common to consider displacement rather than compliance constraints in the
design of building structures. A set of j displacement constraints applied to a
structure may be expressed as:
*jj uu ≤ Eq. 3.7
where uj* is the prescribed limit for |uj|. The principle of virtual work can be used to
demonstrate that the change in uj due to the removal of element i can be expressed as:
{ } [ ]{ } ij
iiTij
j uKuu α==∆ Eq. 3.8
where {uij} is the element displacement vector resulting from a unit load in the
relevant direction, applied to the node at which displacement is constrained and {ui}
is the element displacement vector corresponding to the real applied loads. It is of
note that αij may be either positive or negative. Previous work (Xie and Steven 1997)
suggests that the modulus of this value should be used as the sensitivity number in
determining element removals, although it is actually more appropriate to select
elements with the most negative value of αij for removal to effect the most negative
possible change in displacement of the structure as a whole. Hereafter, this sensitivity
number will be referred to as the cross strain energy with the corresponding cross
strain energy density found by dividing by volume or weight. The sum of the cross
strain energy in every element in the structure, including the orthogonal framework,
should be equal to the displacement at the point of interest, in the direction of the
virtual load.
A central assumption of the method described is that the change in distribution of
displacements within the structure is negligible when a small number of elements are
removed, otherwise it is not possible to accurately predict which element removals are
most appropriate. The number of elements to remove at each iteration therefore
becomes an important process parameter, influencing the trade-off between numerical
accuracy and computational efficiency.
44
3.2.4. Bi-directional Evolutionary Structural Optimisation (BESO)
Different algorithms have been proposed for determining element rejection and
addition in BESO (Young et al. 1998), (Querin 2000a). The algorithm developed in
this research adopts a fixed modification ratio, MR, (0.01), which dictates the
approximate total number of removals and additions. The relative proportion of
removals and additions is governed by the proximity of the maximum lateral
displacement in the current design to the constraint value. The addition ratio, ARit, in
iteration it is:
=
*2,1min
j
j
itu
uAR Eq. 3.9
The number of elements to be added, NAit, is then the product of the addition ratio,
modification ratio and maximum number of elements, Nmax:
maxitit NMRARNA ××= Eq. 3.10
and by extension, the number of elements to be removed, NAi,, is:
( ) maxitit NMRARNR ××−= 1 Eq. 3.11
Elements to be added are selected by working through the list of elements in
decreasing order of sensitivity number, placing new elements immediately adjacent to
currently active elements where possible, until the quota has been attained.
According to this algorithm, if a design meets the displacement constraint exactly, an
equal number of elements are added and removed.
3.3. BENCHMARK PROBLEM: STRUCTURAL MODEL SPECIFICATIONS
Figure 3.2 defines the two dimensional skeleton steel framework of the two-bay, six-
storey structure to be considered in this benchmark problem, previously used in
continuum topology optimisation research by Mijar et al. (1998) and Liang et al.
(2000). Mijar et al. (1998) sized the base framework to act alone in carrying vertical
live and dead loads. For the ESO process, a continuum mesh of square four-noded
quadrilateral elements is added to the framework, with the two components acting
together to resist lateral loads. Each bay-storey unit is 15 elements wide and 9
elements high. This problem is unusual in the context of ESO literature, since it
includes a non-designable framework. Connection to the base framework is at corner
45
nodes of the removable elements. Whilst connection of a quadrilateral to the
framework is always along the side of the removable element, a string of
quadrilaterals may be connected solely at a corner node, causing stress concentrations
and moment-free interfaces.
Figure 3.2: Benchmark problem specifications (Mijar et al. 1998).
Real loads (left), including member groupings and geometric specifications, and
virtual load (right). ASCE standard section specifications (below).
1 W8x21 2 W8x28 3 W10x26 4 W12x26 5 W14x26 6 W13x22 7 W10x17
8 W8x10 9 W12x19 10 W12x14 11 W14x22 12 W16x26 13 W16x31 14 W24x62
3.4. OPTIMISATION FOR MINIMAL MEAN COMPLIANCE
The problem addressed by Liang et al. (2000) was to maximise the performance
index, PI, expressed as:
itit
oo
WC
WCPI = Eq. 3.12
where:
Wo, Wit = initial and current (at the itth iteration) weight of the continuum design
domain;
46
1N64.1kN
128.2kN
122.7kN
117.3kN
103.6kN
90.4kN
1
2
3
4
5
6
1
2
3
4
5
6
7
8
10
11
12
12
8
9
11
13
13
14
7
8
10
11
12
12
6.096m 6.096m
6 @ 3.658m
Co, Cit = initial and current mean compliance of the braced framework, calculated
according to equation 3.2.
The continuum design domain took a thickness, t = 0.0254m, with material properties
of Young's modulus E = 200GPa, Poisson's ratio = 0.3. The optimisation process is
terminated when the performance index falls below unity.
An attempt to reproduce the design topology published by Liang et al. (2000) is
shown in figure 3.3. Although the exact shape is not identical, the topology is nearly
the same and performance is close to identical. The former inconsistency may be
attributed to subtle variations such as numerical inconsistencies in different analysis
software. It should be noticed that in both cases the top storey has had all 2D
elements removed, except for at beam-column intersections, where they serve to add
significant stiffness to the region and substantially reduce rotation and lateral
displacement of the top corner node. This effect would not be transferred to a discrete
interpretation of the design, as demonstrated in section 3.8.
Figure 3.3: Design topology of Liang et al. (2000): δ=0.024, element retention =
22% (left) Comparative result to Liang et al. (2000) topology: δ=0.024, element
retention = 23% (right)
3.5. OPTIMISATION FOR DISPLACEMENT CONSTRAINT
Although the work of Liang et al. (2000) uses a performance index based on mean
structural compliance, mention is made of the more practical performance metric of
maximum lateral displacement, quoting a value of 0.024m for the best design. This
value is substantially below any standard maximum lateral displacement limit (e.g.
47
h/500 = 0.044m), but for purposes of comparison 0.024m will be adopted in the
subsequent discussion as a target or constraint value for maximum lateral
displacement. Quality of solutions can then be assessed by comparing weight of
bracing material required to meet the constraint. A simple optimisation model could
therefore be stated as:
Minimise: N Eq. 3.13
Subject to: *δδ ≤ Eq. 3.14
where:
N = total number of elements retained in the current design;
δ, δ* = current and largest permissible maximum lateral displacement respectively.
Careful consideration of the strain energy in pairs of elements grouped together by the
horizontal symmetry condition is required. Liang et al. (2000) suggest that two load-
cases should be considered, identical in magnitude but acting on opposite faces.
Elements with the lowest maximum strain energy in the two loadcases in either of the
pair are removed. This approach may be beneficial for fully stressed design, for
which every element should be at full capacity in one or more loadcases. However, as
previously described, we wish to remove symmetric pairs of elements to minimise the
increase in displacement (or compliance). For the current task this entails summing
the strain energies of each element of the pair. It is thus only necessary to consider a
single loadcase, since the strain energy in an element in loadcase 2 will be identical to
that of its symmetry-partner in loadcase 1. Reconsidering equation 3.8, the lowest
predicted compliance increase is achieved by removing pairs of elements with the
lowest combined strain energy under a single loadcase. Figure 3.4 compares the
elements removed according to the requirements of Liang et al. and the requirement
proposed here. Elements are removed from the same area of the structure, but with
some difference in the exact elements removed. An increase in maximum lateral
displacement of 6.8x10-6m is predicted from removing elements selected on the basis
of the sum of strain energies, compared to 7.4x10-6m from removing elements
selected on the basis of maximum strain energy of either of the elements in the pair.
Figure 3.4: Elements removed in the top left bay unit in the first iteration, based
48
As a consequence, differences in topology occur, as observed in the designs presented
in figure 3.5. Although the above argument suggests that topologies evolved by
removing elements with the smallest sum of strain energies should be more efficient,
this will not necessarily be true in all cases, due to differences between predicted and
actual displacement and the search path that is taken.
Figure 3.5: ESO results with element removal determined by cross-strain energy.
25.4mm designable domain, 8 elements removed per iteration (left: maximum of
sensitivity number in pairs of elements, right: sum of sensitivity number in pairs
of elements)
The Bi-directional Evolutionary Structural Optimisation algorithm presented
previously in this chapter was used to tackle the same problem. Bi-directionality has
the obvious advantage that elements that were removed early in the process may be
reintroduced if found to be advantageous at a later point, rather than pursuing a sub-
optimal path. It also allows the process to commence from virtually any design,
subject to a minimal number of elements being present. Hence a more thorough
design space exploration is afforded by BESO.
Figure 3.6 shows the best designs evolved starting from a full initial continuum
designable domain, a minimal configuration, with 2D elements only present around
the 1D structure and two randomly generated initial configurations, for bracing
element thicknesses of 2.5mm, 5mm, 10mm and 25.4mm (the last of these being used
by Liang et al. (2000)). It is logical to expect that a change in thickness and hence the
relative stiffness of frame structure and the 2D elements of the continuum domain will
cause corresponding changes in optimal topology. Analysis of randomly generated
initial configurations is made possible by the fact that, for the purposes of the
49
structural model, elements to be removed are actually assigned a thickness value of
10-10m, thus avoiding floating elements causing matrix singularities, whilst not
significantly affecting the results.
A chevron form, or inverse 'V', in at least the lower section of the evolved bracing
structure is common to all designs. The exact topology exhibits some variation with
both starting point and domain thickness, but the most common and practically
realisable solutions are the single chevron (25mm, Full Start; 10mm, Random Cloud
2) and double chevron designs (5mm, Full Start, Perimeter Start and Random Cloud
2; 10mm, Full Start, Perimeter Start and Random Cloud 1), the latter either being two
units of equal height, or a four storey chevron above a two storey chevron.
Comparing the number of elements retained in the optimal solutions derived using an
element thickness of 25.4mm in figure 3.6 against the solution of Liang et al. (2000)
in figure 3.3, all of the current solutions are more efficient in meeting the lateral
displacement constraint on the top corner node of 0.024m. The best solution, derived
from the Full Start configuration, retains 14% of original elements, compared to 22%
in the benchmark solution, representing a material saving of 37%. This is primarily
due to retaining bracing in the top storey.
Figure 3.6 suggests that with the BESO algorithm used, although volume varies
considerably according to the domain thickness, the optimal evolved topology is
relatively insensitive both to change in thickness and starting point. One point of note
is that although the lateral displacement of the top storey node is within the prescribed
limit, it is possible that the global maximum lateral displacement may be lower in the
structure. This could be addressed by placing displacement constraints at every storey
level, or considering inter-storey drift, in a multiple constraint formulation as
presented by Xie and Steven (1997). However, this will require additional loadcase
definitions, lengthening analysis times. Additionally, care must be taken in task
formulation, since elements in upper storeys can have negligible sensitivity values
with respect to constraints on displacement in lower storeys and be inappropriately
removed.
50
Figure 3.6: Minimum volume designs satisfying the displacement constraint
derived by BESO for varying domain thickness and starting configuration
3.6. INCLUDING OPTIMISATION OF DOMAIN THICKNESS
The topological diversity observed in designs resulting from variation of domain
thickness indicates that solutions are not globally optimal. In this section we explore
the concept of varying domain thicknesses in such a way as to maintain the maximum
51
lateral displacement of the structure at its constraint value throughout the ESO
process. The continuous designable domain elements may be divided into a number
of groups, with a single thickness variable used for all elements in any one group.
Variation of domain thickness was previously considered by Liang et al. (2000a), but
in the simple context of exploiting the linear relationship between displacement and
thickness in a structure consisting purely of a single designable domain. This
approach is invalid for any design which includes a non-modifiable framework or
multiple groups of elements which may be assigned different thicknesses.
The section sizing of the orthogonal framework was previously conducted based on
vertical loads. For the purposes of this study, these sections remain fixed, essentially
making the conservative assumption that diagonal bracing elements do not contribute
to the vertical stability of the structure. However, it would be straightforward to
integrate optimisation of framework section sizes with thickness and topology
optimisation of the designable domain, by minor extension to the method outlined
below.
Domain thickness optimisation is incorporated into the process flowchart of figure
3.7. Five iterations of thickness modification are performed to find near-optimal
thicknesses for the starting topology. A further single loop of thickness modifications
is performed at each iteration to continuously update the domain thicknesses as
elements are added and removed. This is only intended to give an approximation to
the optimal thicknesses for the current topology. However, this is justified by the
relatively small thickness changes occurring and the fact that this approach requires
no more analysis time than for fixed thickness ESO. Performing multiple iterations
with reanalysis of the thickness optimisation loop would greatly increase the run time
for the ESO process.
Section 4.8.2 of this thesis demonstrates the simple result that, when considering a
single displacement constraint, optimal solutions require equal (cross) strain energy
density in all structural members, or in the current case, equal average (cross) strain
energy density in all groups of elements. This is derived by considering Optimality
Criteria (Borkowski and Jendo 1990). Working on the simple approximation that the
(cross) strain energy in the orthogonal framework is unaffected by modifications to
thickness of designable domain groups, a discrepancy between the maximum lateral
52
displacement and the constraint value must be absorbed by changing the strain energy
in the designable domain. The process adopted is outlined in the flowchart of figure
3.8 and corresponding calculations defined by equations 3.14 to 3.16. The BESO
algorithm presented in section 3.2.2 determines the proportions of elements to be
removed and added on the basis of the ratio of the maximum lateral displacement in
the current design to the corresponding maximum permissible value. However, in the
current method, including thickness optimisation, the algorithm aims to maintain
displacement exactly at the constraint value, hence the previous approach is
unsuitable. Since bi-directionality is considered beneficial, the iterative process of
element modification is split into two phases. In each of 5 iterations, a set number of
elements with the lowest cross strain energy density are removed. Thereafter, for 2
iterations, the same number of elements are added. Hence a staggered bi-
directionality is introduced, with a general trend for reduction in the number of
elements. Choice of termination criterion is arbitrary, but may be defined as the point
at which a group thickness exceeds a prescribed value, a prescribed proportion of the
original elements have been removed, or volume exceeds a set value or proportion of
the original optimised volume.
53
Figure 3.7: Process flowchart for ESO with domain thickness optimisation
54
Analyse structural model
Retrieve nodal displacement information
Calculate strain energy in each element, using
element stiffness matrix
Calculate strain energy density in symmetric
pairs and groups of elements
Adjust group thicknesses for uniform average
strain energy density in all groups and
meeting displacement constraint
Remove 10 element pairs with
lowest strain energy density
(average over both elements)
Add 10 elements adjacent to
element pairs with highest
strain energy density
START
Write structural model
GO TO THICKNESS OPTIMISATION LOOP
Initialisation complete?
Topology Phase?
Termination criterion met?
END
YES
NO
AdditionRemoval
YES
NO
Figure 3.8: Flowchart for domain thickness optimisation loop
( )*__ jjcurrentbracingreqdbracing uuXSEXSE −−= Eq. 3.14
tot
reqdbracing
projV
XSEXSED
_= Eq. 3.15
=
proj
g
ggXSED
XSEDtt ' Eq. 3.16
where:
XSEbracing_current, XSEbracing_reqd = current cross strain energy in the designable bracing
domain and corresponding cross strain energy required to meet the displacement
constraint, assuming the cross strain energy in the orthogonal framework to be
unchanged by domain thickness modifications;
XSEDproj = average cross strain energy density predicted to correspond to the required
cross strain energy, calculated using the current total volume of bracing elements, Vtot;
XSEDg = current average cross strain energy density in group g;
tg, tg' = current and revised thicknesses for group g, respectively.
55
Calculate required cross strain energy in bracing from current
value and violation of constraint (equation 3.14)
Calculate required cross strain energy density in
bracing from required cross-strain energy and total
volume (equation 3.15)
Double thickness of all element groups
Modify group thicknesses to meet target cross strain
energy density requirement (equation 3.16)
Update total volume and hence required cross strain
energy density
Thickness change < 1% for all groups?
Required cross strain energy < 0?
YES NO
YES
NO
ENTER THICKNESS
OPTIMISATION LOOP
EXIT THICKNESS
OPTIMISATION LOOP
ITERATIVE THICKNESS
ADJUSTMENT LOOP
The cross strain energy density corresponding to the required cross strain energy is
not fixed, since it will be affected by group thickness modifications changing the
domain volume. An iterative loop is therefore adopted to find the thickness required
for each of the designable domain element groups, such that:
– all groups have the same projected cross strain energy density
– the sum of the total projected cross strain energy in the designable domain and
that observed in the orthogonal framework is equal to the displacement constraint
value.
The progress of the process detailed using a single thickness variable for all elements
is charted in figure 3.9. No elements are added for the first 5 iterations while the
thickness required to meet the displacement constraint with all elements active is
found. As the overall trend for element removal proceeds, the volume decreases
slightly, before increasing above its original value. Until iteration 275, bracing
volume remains within 10% of the initial volume. The structure adapts to removal of
critical elements around iteration 280, after which, until iteration 310 the volume
remains around 25% above its initial value. Topology plots beneath the graph
demonstrate that it is unrealistic to use minimum volume as the sole measure of
efficiency since the minimum volume designs, between iteration 50 and 100, do not
offer a discrete interpretation. In practice, a chart such as this should be inspected
alongside the design topologies in order to select appropriate topologies for further
consideration and discrete member interpretation. In this case, the topology of
iteration 300 appears a good compromise between structural efficiency and discrete
interpretability.
56
Figure 3.9: Process history for simultaneous topology and domain thickness
optimisation with a single thickness group.
Figure 3.10 shows the most appropriate topologies evolved by allowing domain
thickness to vary in 1, 3 or 6 groups. It should be restated that these are not minimum
volume topologies, but rather those that are most suitable for discrete interpretation.
The designs are all similar in performance and appearance, adopting a compound 'X-
chevron' form, similar to that evolved from the random cloud designs with a fixed
thickness of 2.5mm (figure 3.6). Material volume for the three designs in the table is
similar to those evolved using a fixed domain thickness of 2.5mm, but are arguably
more readily interpreted as discrete structures. It is interesting to note that there is not
a continuous reduction in thickness with ascending height in the evolved topologies.
For example, with 3 groups, more elements are present in the middle group than either
of the others, hence thinner elements in this group achieve the same average cross
strain energy density as other groups in the structure.
57
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
0 50 100 150 200 250 300 350Iteration
Normalised Value
Thickness (/t(0))
Displacement (/d*)
Volume (/V(0))
Elements (/E(0))
Full domainoptimisedthicknesses
0.00061m
No. of groups 1
No. of iterations 310
Volume 0.196
Displacement 0.0249
No. of elements 358
Evolvedtopologyoptimisedthickness
0.00331m
0.00033mNo. of groups 3
No. of iterations 3860.00320m
0.00059mVolume 0.195
Displacement 0.02450.00289m
0.00076mNo. of elements 310
0.00562m
0.00023m No. of groups 6 0.00311m
0.00041m No. of iterations 336 0.00415m
0.00053m Volume 0.208 0.00515m
0.00063m Displacement 0.0248 0.00427m
0.00073m No. of elements 228 0.00858m
0.00078m 0.00901m
Figure 3.10: Best designs derived by simultaneous thickness and topology
optimisation, with one, three and six thickness groups.
3.7. INCLUDING ARCHITECTURAL REQUIREMENTS AND PATTERN
DEFINITION
The form of solutions presented by ESO can frequently be irregular or inelegant, as
seen in some of the designs presented to this point. Often an architect or client will
have a preconception of the form, symmetry or degree of repetition that is sought in
bracing forms.
This chapter has previously considered elements in pairs defined by horizontal
symmetry. It is a simple extension then to consider larger "families" of elements,
defined by vertical symmetry or repetition, which will be simultaneously removed
from or added to the structure. Defining groups of elements corresponding to lines of
mirror or translational symmetry or patterns of repetition for simultaneous
58
optimisation of domain thickness presents the opportunity for high utilisation amongst
elements retained in final solutions.
The topologies evolved from defining three different symmetry systems are shown in
figure 3.11:
(A) reflection symmetry with mirror-line at half building height,
(B) translational symmetry with two-bay three-storey unit repeated once vertically,
(C) translational symmetry with two-bay two-storey unit repeated twice vertically.
All designs use the simultaneous topology and thickness optimisation strategy
discussed in the previous section, with groups corresponding to the sections defined
by the symmetry patterns. Alternative topologies are shown for case (A), occurring at
different stages of the evolutionary process. When defining translational symmetry
up the building, as in cases (B) and (C), the chevron patterns evolved in earlier
investigations in this chapter are again observed. However, in the symmetry study
with thickness optimisation, the lower chevrons are thicker with the same number of
elements, as opposed to having more elements than those higher in the structure.
59
A No. of groups 2
Iteration 239
Volume 0.218
Evolved topologyoptimisedthicknesses
0.0027m
Displacement 0.0246
No. of elements 352 0.0048m
No. of groups 2
Iteration 287
Volume 0.427
0.00785m
Displacement 0.0242
No. of elements 204 0.01752m
B No. of groups 2
Iteration 277
Volume 0.265
0.00460m
Displacement 0.0206
No. of elements 240 0.00874m
C No. of groups 3
Iteration 3020.00416m
Volume 0.204
Displacement 0.02500.00769m
No. of elements 1860.00936m
Figure 3.11: Evolving topologies with prescribed symmetry, using simultaneous
thickness optimisation of appropriate groups
3.8. DISCRETE INTERPRETATION OF CONTINUUM TOPOLOGIES
The topological designs presented in this chapter to this point require discrete
interpretation before they can be considered feasible structures. It is possible that the
performance observed in the discrete interpretation may be quite different from that of
the ESO design. The majority of topologies discussed, including that of Liang et al.
(2000) and some notable others appear in discrete form in figure 3.12. Bracing
members will all take the form of circular solid sections, with members assigned to
60
groups in the same manner as for the thickness optimisation previously presented.
As is standard practice, end connections of the bracing members are modelled as
having no moment capacity. Once again, uniform average strain energy density is
required across the different groups and an iterative approach (similar to that shown in
figure 3.8) is adopted in achieving this.
Of significance is the fact that, despite achieving a maximum lateral displacement of
0.024m in the ESO process, this performance could not be repeated in the discrete
interpretation of Liang et al.'s topology (M). Due to the complete absence of bracing
in the top storey, rather a small strip of corner stiffening 2D elements, even infinitely
large bracing members are unable to provide sufficient stiffness to meet 0.024m or
0.012m displacement constraints, without increasing the size of members in the
orthogonal framework.
The double and triple chevron topologies (B and C) are the most regular topologies in
a high performance group that includes variations on the compound 'X-chevron' form
(H, I, K). The volume of steel required in bracing members to meet the δ* = 0.024m
constraint for these designs varies by only around 10%. When other factors such as
piece-count, aesthetics, impingement on view and the effect on the rest of the
structure is considered, this variation is relatively insignificant. It can also be seen
that the least volume topology for one displacement constraint is not necessarily the
best for another: solution C best meets the δ* = 0.044m constraint, whilst the other
constraints are best met by solution I. This is indicated by large font size in figure
3.12.
The material volumes required in the optimised discrete versions of solutions are
generally within 25% of the corresponding design found through the thickness
optimising BESO process, with the exception of the single 'X' form resulting from the
definition of a horizontal mirror-line.
61
δ* = 0.044 δ* = 0.024 δ* = 0.012
A Upper Diameter (m) 0.0716 0.0983 0.1402
Lower Diameter (m) 0.0828 0.1155 0.1655
Bracing Volume (m3) 0.2142 0.4116 0.8417
Bracing XSE (m) 0.0393 0.0224 0.0113
B Upper Diameter (m) 0.0437 0.0634 0.1045
Lower Diameter (m) 0.0565 0.0831 0.1384
Bracing Volume (m3) 0.1007 0.2155 0.5933
Bracing XSE (m) 0.0359 0.0184 0.0070
C Upper Diameter (m) 0.0345 0.0533 0.1190
Middle Diameter (m) 0.0452 0.0700 0.1571
Lower Diameter (m) 0.0493 0.0778 0.1768
Bracing Volume (m3) 0.0847 0.2063 1.0486
Bracing XSE (m) 0.0324 0.0146 0.0030
D Upper Diameter (m) 0.0457 0.0696 0.1401
Lower Diameter (m) 0.0594 0.0918 0.1869
Bracing Volume (m3) 0.1108 0.2618 1.0757
Bracing XSE (m) 0.0331 0.0153 0.0040
E Upper Diameter (m) 0.0537 0.0813 0.1518
Lower Diameter (m) 0.0544 0.0818 0.1524
Bracing Volume (m3) 0.1168 0.2657 0.9211
Bracing XSE (m) 0.0342 0.0164 0.0050
F Upper Diameter (m) 0.0375 0.0631 0.2938
Lower Diameter (m) 0.0476 0.0812 0.5749
Bracing Volume (m3) 0.0947 0.2722 10.7415
Bracing XSE (m) 0.0293 0.0112 0.0011
G Upper Diameter (m) 0.0344 0.0600 0.2526
Middle Diameter (m) 0.0447 0.0788 0.3320
Lower Diameter (m) 0.0484 0.0860 0.8468
Bracing Volume (m3) 0.1234 0.3809 19.9
Bracing XSE (m) 0.0281 0.0101 0.0020
62
H Upper Diameter (m) 0.0354 0.0515 0.0868
Lower Diameter (m) 0.0552 0.0818 0.1390
Bracing Volume (m3) 0.0924 0.2005 0.5752
Bracing XSE (m) 0.0353 0.0179 0.0066
I Upper Diameter (m) 0.0246 0.0357 0.0593
Middle Diameter (m) 0.0406 0.0585 0.0953
Lower Diameter (m) 0.0546 0.0804 0.1326
Bracing Volume (m3) 0.0903 0.1928 0.5221
Bracing XSE (m) 0.0358 0.0185 0.0072
J Upper Diameter (m) 0.0408 0.0769 0.1500
Middle Diameter (m) 0.0436 0.0819 0.2924
Lower Diameter (m) 0.0599 0.0172 0.2727
Bracing Volume (m3) 0.1241 0.4632 3.0746
Bracing XSE (m) 0.0254 0.0080 0.0033
K Upper Diameter (m) 0.0354 0.0515 0.0868
Middle Diameter (m) 0.0526 0.0788 0.1355
Lower Diameter (m) 0.0177 0.0226 0.0292
Bracing Volume (m3) 0.0915 0.1983 0.5690
Bracing XSE (m) 0.0354 0.0179 0.0066
L Upper Diameter (m) 0.0343 0.0722 0.1740
Lower Diameter (m) 0.0448 0.0542 0.1323
Bracing Volume (m3) 0.0954 0.2439 1.4297
Bracing XSE (m) 0.0310 0.0137 0.0027
M Diameter 5 (m) - - 0.1172
Diameter 4 (m) - - 0.1294
Diameter 3 (m) - - 0.0911
Diameter 2 (m) - - 0.1215
Diameter 1 (m) - - 0.1334
Bracing Volume (m3) - - 0.8753
Bracing XSE (m) - - 0.0064
Figure 3.12: Discrete bracing topologies (with circular solid sections) optimised
for minimum mass satisfaction of displacement constraint
63
3.9. CONCLUSIONS
– The bi-directional form of ESO has a beneficial effect in achieving more practical
solutions, that can be readily interpreted as discrete structures. However, this is
accompanied by an increase in computation time on account of more analysis
iterations.
– Some variation occurs on account of domain thickness (if fixed) and starting
point. This suggests that the result of a single process cannot be considered as
globally optimal. Further variation is likely on account of numerical
inconsistencies arising from process parameters such as mesh density and
modification ratio.
– Simultaneous optimisation of domain thickness and topology has a significant
effect on the form of solutions generated. This technique works particularly well
when defining symmetry patterns in the structure.
– ESO has substantial limitations: the overall process is complex, requiring discrete
interpretation and optimisation to assess practical performance of solutions; the
structural model is expensive to analyse compared to a discrete model of the same
structure using one-dimensional elements, on account of the large number of two-
dimensional elements in each unit.
– It is difficult to consider strength and buckling constraints alongside stiffness
within the ESO process. Although Xie et al. (2002) consider ESO for
optimisation against buckling, this is in the context of thickness optimisation only.
The most practical way to consider strength and buckling is likely to be as a check
in optimisation of the discrete interpretation.
3.10. GUIDELINES FOR PRACTICAL USE
Referring back to the original research question relating to improving the usefulness
of ESO for the building industry and the proposals subsequently put forward, we can
state the following:
– Consideration of appropriate constraints is essential to successful use of any
optimisation or pseudo-optimisation tool, including ESO and its variants. In this
case, maximum lateral displacement at the highest point of the structure is likely
to govern, but the user should be aware that it is possible that displacement may
64
actually be greater elsewhere. Further, other forms of constraints, such as strength
and buckling may be relevant, both in the bracing domain and the orthogonal
framework. These are difficult to consider in the ESO process itself, but should
be included in optimisation of the corresponding discrete structure.
– BESO offers the ability to start from alternative configurations to that with all
elements active. Running the process from different configurations, in the optimal
thickness region (for this problem approximately between 3 and 10mm) most
designs are similar (generally based on a double chevron), but consistent
convergence to a single optimum is not observed. This may be beneficial in
creating different design options and since performance of discrete and continuous
design interpretations is often different. BESO yields higher performance and
more regular designs than unidirectional ESO.
– Defining all elements to be equal size and shape permits the use of a single
element stiffness matrix. Different thicknesses are readily accommodated by
linear factoring.
– Simultaneous topology and thickness optimisation gives a reasonable indication of
what material volume is likely to be required, providing there is an obvious
discrete interpretation. This technique ensures appropriate thickness is used and
provides a means of assigning different thicknesses to different regions of the
structure, thus promoting structural efficiency.
– Defining symmetry conditions with corresponding thickness grouping allows
tailoring of designs to preconceived aesthetic requirements, whilst retaining high
performance.
Further noteworthy observations:
– Using a “film” of very thin elements in place of inactive elements will stabilise the
ESO process, eliminating the possibility of singularities in the global stiffness
matrix causing the computational process to crash. However, this does require
additional analysis time due to the extra elements.
– In considering the results of an ESO process with thickness optimisation, it is
valuable to inspect topologies generated throughout the history, alongside a chart
of the form shown in figure 3.9. This offers the option to trade-off structural
65
efficiency, as indicated by the bracing volume required, against interpretability of
the design as a discrete structure.
– A number of ESO solutions should be given discrete interpretation since
performance of continuous and discrete solutions may vary. This also allows
strength and buckling constraints to be considered.
66
4. Bracing topology and section-size optimisation by
a hybrid algorithm: an industrial case-study
4.1. INTRODUCTION
This chapter presents an industrial case-study, considering issues associated with
application of optimisation to the design of the lateral stability system of a specific tall
building structure in the scheme design phase. The phases of research work
presented include “live” project involvement, retrospective method development and
convergence studies. These phases incorporate changes to the structural and
optimisation models, reflecting the progression of the project. Topology optimisation
is conducted by parameterising the design task and applying variants of the Pattern
Search method (Hooke and Jeeves 1961). The Optimality Criteria method is used for
section sizing. It is seen that minimum piece-count topologies are obtained by fixing
member section sizes at their maximum value and performing topology optimisation,
whilst minimum volume solutions are obtained by the simultaneous optimisation of
topology and section size.
Research Questions:
– How can existing optimisation techniques be adapted to practical topology
problems in scheme design, accommodating considerations that are difficult to
model, such as aesthetics and design intent?
– How can simultaneous optimisation of size and topology be efficiently
integrated?
– Does simultaneous optimisation of size and topology offer improved solutions
compared to performing tasks separately?
Proposals:
- Use appropriate methods for the problem in question, combining these where
suitable.
67
- Consider practical issues such as computation time, adaptability to changes in
problem specification and model complexity.
- Use stochastic methods to generate a number of designs, avoiding a single local
minimum and enabling choice according to unmodelled criteria.
4.2. BACKGROUND
As discussed in chapter 2, use of optimisation techniques in building engineering
practice remains low. Collaboration with Arup presented the opportunity to consider
a real-world design problem, observing real-time developments and attempting to
support decision-making with suitable optimised design solutions and parametric
studies. It was therefore possible to assess the feasibility of applying topology
optimisation in practice.
A structural design team from Arup worked in conjunction with architects Kohn
Pedersen Fox Associates on the design of the Pinnacle Tower, London (known
previously as the DIFA Bishopsgate Tower (Baldock et al. 2005)), which is planned
to stand at around 300m. As with all towers of this height, and slender structures in
general, lateral stability is a key concern. This is provided by a tubular bracing
system: a steel framework that wraps around the perimeter of the building and is
irregular in both plan and elevation. For much of the design time, it was envisaged
that individual bracing members would be grouped in spirals of fixed angle of
inclination. The spirals emanate from the base of external columns, wrapping
diagonally around the perimeter of the building and terminating at different, visually
varied, heights up the building. Spirals rise three floors in height for every bay
spanned, with individual bracing members defined by the intersections of spirals and
columns. Later in the project and beyond the scope of the work presented in this
chapter, the design team concluded that the requirement for continuous spirals was, in
fact, an unnecessary constraint.
A high premium is placed on minimising the total number of bracing members
required. A high piece count will increase material cost and construction time and
cost, hence losing potential letting revenue. Bracing members may also impinge on
floor space and restrict view, thus reducing letting value of the corresponding floor.
Adopting such a bracing system removes the need for an intrusive central core, which
68
tends to be impractical in buildings in excess of 200m. Pin-jointed internal columns
serve to transfer some proportion of the vertical loads downwards. Triangulation
introduced by adding diagonal bracing to the perimeter framework aims to reduce the
bending moments occurring in what would otherwise be a Vierendeel framework,
with a stiffening effect to reduce lateral displacement to an acceptable level. This
work develops from a particular selected structural system: for a concise discussion of
structural systems used in tall buildings, including tubular and horizontal load
resisting systems, the reader is referred to Khajehpour (2001).
4.2.1. Overview of studies
Baldock et al. (2005) report on live work undertaken alongside the structural design
team who, in collaboration with the architectural team, were seeking to develop an
appropriate and efficient bracing pattern for the building. Following the “problem-
seeks-design” approach (Cohn 1994), a variant on the Hooke and Jeeves “Pattern
Search” method (Hooke and Jeeves 1961) was rapidly implemented to meet industrial
requirements. Alongside further work to refine this method and test for convergence,
studies were made to sample the landscape of the design space and the nature of the
feasible region within it. The structural model used in this work was subject to minor
alterations during this phase, but for the purposes of this chapter this set of models
will be referred to as Model A.
Almost two years later, following a submission to planning authorities and a lengthy
review procedure, further optimisation studies were conducted. The structural model
had been subject to a number of revisions by this point, but the model on which this
second phase of optimisation research is based will be referred to as Model B for the
purposes of this chapter. At this stage, section-size optimisation is considered and
integrated into the pattern search algorithm.
4.3. DESIGN TASK DEFINITION
4.3.1. Structural models
Within each bracing spiral, individual pin-jointed bracing members are defined
spanning adjacent columns, rising three floors in height. It is required that bracing
69
spirals should terminate at various heights up the building. Some potential spirals are
omitted entirely due to peculiarities around the base of the building, with transfer
areas defined at access points. However, spirals should be continuous from the base
of the building to their termination point. Due to the highly iterative nature of the
direct search optimisation method to be presented, a simplified finite element model is
required for the evaluation of alternative bracing configurations to reduce
computation time. Most internal columns are excluded from each structural model, so
vertical loads applied to the model are only a proportion of the total loads. The very
stiff concrete floor plates are approximated in different ways in the two structural
models (see table 4.1).
Constant parameters in optimisation models include:
– design of orthogonal framework (topology and member sections)
– definition of potential bracing spirals
– member section sizes (variables in section sizing algorithm)
– angle of inclination and location of potential bracing members
– applied loads
– nodal positions defining column locations and floor heights.
The primary structural considerations in the design of the bracing system are modeled
as constraints in the optimisation model. Secondary structural considerations were
raised by structural designers to be checked after the optimisation process.
A fully braced configuration of structural model B is shown in figure 4.1.
Figure 4.1: Fully-braced analysis model (left to right): plan view; side elevation;
isometric view (shown with two spirals highlighted); isometric split sections
70
Structural model B
“Spider” systems of dummy beams at each storey used to
approximate the stiffness contribution of concrete floor
plates
Uniform
ly distributed loads applied to structural members
on perimeter framework
Circular hollow sections in all members, except horizontal
beams in the perimeter framework, which are I-sections
Self-weight, dead load and reduceable and fixed live-load
analysis cases combined with representative wind loads
define an ultimate limit state envelope for local strength
assessment. Five indicative wind loading directions for
stiffness assessment
Stiffness: maximum lateral displacement of the structure at
263m AOD
1, inter-storey drift (difference in lateral
displacement of central nodes at 3 storey separation)
Strength: buckling capacity and cross-section capacity
(tension and compression) in bracing members
Transfer forces in floor plates, forces in pin-jointed
horizontal beams in the perimeter framework and angle of
rotation of the structure as a whole under various
loadcases due to asymmetry
Table 4.1:C
omparison of structural models
Structural model A
“Rigid linking” constrains all nodes on a given floor of the
building to move together in-plane.
Horizontal loads applied at the centre of area of each floor
to a fictitious node within the relevant rigid-linking system
Standard I-beam or rectangular hollow sections in all
columns, beams and bracing m
embers
Six analysis cases are considered in two stages: self-
weight and superimposed dead load in a construction
stage; self-weight, live load and wind-loads in two
orthogonal directions on the final structure. 23 load-case
combinations defined, enveloped for "worst-case"
combination in each member
Strength: maximum in-plane force passing through a floor
plate limited by constraint on axial force in bracing
elements; moments in connections limited by constraint on
bending moment in horizontal beams.
Out-of-plane horizontal forces on floor plates, arising from
sharp changes of direction of bracing elements; maximum
lateral displacements
Concrete floor plate
modelling
Load application
Member section shape
Load cases
Primary structural
considerations
Secondary structural
considerations
1 AOD = Above Ordnance Datum: height relative to mean sea level at New
lyn, Cornwall
71
4.3.2. Topology optimisation models
Two optimisation models were developed corresponding to the distinct structural
models A and B, with the same fundamental objective of minimising the total number
of bracing elements, but subject to different constraints arising from different primary
structural considerations.
Optimisation model A:
Minimise: ∑=
=SP
sp
spnN1
Eq. 4.1
Subject to: 1max
≤F
Fi Eq. 4.2
1max
≤M
M b Eq. 4.3
where:
N = number of bracing members in current design;
nsp = number of bracing members in spiral sp;
SP = total number of spirals, 45;
Fi = maximum observed absolute axial force occurring in bracing member i in load
case combination envelope;
Fmax = maximum permissible axial force in any bracing member;
Mb = maximum observed bending moment occurring in horizontal beam b in load
case combination envelope;
Mmax = maximum permissible bending moment in any horizontal beam.
Optimisation model B:
Minimise: ∑=
=SP
sp
spnN1
Eq. 4.1
Subject to: 1≤c
i
yi
i
M
M
pA
F + Eq. 4.4
15.01 ≤
++
c
ci
c
i
c
ci
P
F
M
mM
P
FEq. 4.5
72
300
11
sssj
sj
hhdd
−≤−
++ Eq. 4.6
500
maxmax h
d j ≤ Eq. 4.7
where:
N = number of bracing members in current design;
nsp = number of bracing members in spiral sp;
SP = total number of spirals, 48;
Ai = cross-sectional area of member i;
Fi = maximum absolute axial force occurring in bracing member i in ultimate limit
state loadcases;
Fci = maximum compressive axial force occurring in bracing member i in ultimate
limit state loadcases;
Mi = maximum bending moment occurring in bracing member i in ultimate limit state
loadcases;
py = section design strength;
Mc = section moment capacity;
Pc = section compression resistance;
m = equivalent uniform moment factor;
djs+1 - dj
s = inter-storey drift between storeys s and s+1 under loadcase j;
hs+1 - hs = height of storey s;
djmax = maximum lateral displacement under loadcase j;
hmax = total height of building.
Equation 4.4 models the cross-sectional capacity requirement, equation 4.5 models
the buckling resistance requirement, both of which are applicable to circular hollow
sections (BSi 2000, Section 4.8.3).
4.4. PATTERN SEARCH METHOD
The Pattern Search method proposed by Hooke and Jeeves (1961) is a simple
gradient-free search technique, which in its conventional form is applicable to
optimisation tasks with a defined set of continuous variables. The search follows the
steps outlined below.
73
1. Set base-point at initial location in the design space. Select initial distance to
move in each variable direction (step-size).
2. Attempt positive and negative changes (exploratory moves) equal to the step-size
to each of the variables in turn, accepting the move if the objective function is
reduced.
3. After exploratory moves have been made for all variables, set a new base point at
the current location in the design space.
4. Apply a pattern move, corresponding to the vector between the current and
previous base-points. This move is accepted if the objective function is reduced.
5. Reduce step-size if no exploratory move was made in the last iteration.
6. Terminate if step-size has become less than a prescribed convergence value, else
repeat steps 2 to 6.
The use of pattern moves attempts to accelerate the search by exploiting known good
directions in the search space, thus reducing computational expenditure.
This basic method was adapted to the topology optimisation models defined in section
4.3, with different variants at different stages of the project. The number of bracing
members in each spiral becomes the set of variables for the pattern search, with a
reduction in the value of a variable corresponding to the removal of the relevant
number of bracing members from the top of the spiral. Figure 4.2 shows a close-up
elevation of the fully-braced upper section of structural model 2, with tip members
highlighted. With step-size set to a single bracing member, the removal of these
members would be attempted in the first set of exploratory moves.
In all optimisation processes described in this chapter, structural analysis is performed
in Oasys GSA as explained in Appendix 1 and the method implementation
programmed in C++.
74
Figure 4.2: Split elevation view of the upper section of structural model 1, with
spiral numbering and bracing members at the tip of each element highlighted
4.5. LIVE PROJECT OPTIMISATION
4.5.1. Topology optimisation by Modified Pattern Search
The initial approach adopted for finding design solutions, subject to optimisation
model A in the context of live project work, involved the piecewise removal of
bracing members from a fully braced configuration. In this respect, the method bears
a resemblance to the ESO methods discussed in Chapter 4. The technique can be
formalised as a variant of the Pattern Search algorithm described in the previous
section. In each iteration loop, the removal of a single bracing member, from the tip
of a different spiral in turn, is attempted in the set of exploratory moves. The order is
determined either randomly, or in order of increasing axial force in the tip elements.
On completion of this phase, the pattern move attempts the removal of a further
member from spirals which were reduced in length during the exploratory moves.
However, an additional base point is set after successful pattern moves, so that only
one bracing member is removed from any one spiral at a time. The sequence of
exploratory and pattern moves is repeated until no further members can be removed
without constraint violation. At this stage in the project, the step-size was fixed at one
75
19
1718 36
3738
1516 32
3334
35
14 2930
31
27
28
4
2 3 5
1 62122
720
8
9
10 13
26
11 12
23
2524
bracing member and the Pattern Search was unidirectional, since members cannot be
replaced.
The significant challenge in this approach was in handling the constraints on axial
force in bracing members and bending moment in horizontal beams, since the
removal of any bracing member will improve the objective function. In the method
developed, subject to the time constraints of the project, acceptance of a member
removal required a change in constraint value to be less than a defined proportion of
the distance to the constraint boundary. This can be stated as:
Accept individual move if: ( )( )
tCC
CC
i
ii ≤−
−+
lim
1Eq. 4.8
where:
Ci = previous constraint value;
Ci+1
= new constraint value;
Clim = constraint limit;
t = current tolerance (<1);
The tolerance, or limit on change as a proportion of the distance from the constraint,
is the same for both constraints. Initially set to a small value, this is increased when
no further moves can be made at the current tolerance. When the tolerance exceeds
unity, only the absolute value of the constraint function need be considered. The
procedure terminates when no further moves can be made at this maximum tolerance.
Moves that increase the distance to both constraint boundaries are always accepted.
4.5.2. Parametric studies
Following initial method development, the design team requested the investigation of
the relative sensitivity of the number of members in minimum bracing designs to
changes in parameter values governing the structural constraint limits. Such changes
could be implemented in the design by reconsidering other aspects of the structure, if
the new values proved to have a major positive effect on the achievable minimum
number of bracing members.
Figure 4.3 shows a set of designs generated using deterministic Pattern Search. The
initial tolerance value, t, controlling move acceptance is set to 0.005. This is doubled
76
when no moves are possible at the current value. Fbracing
and M
beams for each final
design are given in parentheses. Adjusting either parameter limit produces a
significant change in the number of elements that may be removed, with the greater
effect seen in adjusting the axial force limit.
Figure 4.3: Parametric studies
4.5.3. Outline proposals
Aided by the parametric study, the design team selected limits of 750kNm for bending
moment in horizontal beams and 8500kN for axial force in bracing members. Three
alternative designs, shown in Figure 4.4, were generated by applying these constraints
with deterministic element removal: one from a fully braced initial design and the two
best feasible designs from a randomly generated set. The appearance of these designs
is notably different, with lighter designs obtained from random starting points. The
design team welcomed the opportunity to have alternative high performance solutions
available for aesthetic consideration.
77
384bracing
elements
(6976kN
436kNm)
296bracing
elements
(8486kN
480kNm)
303bracing
elements
(8335kN
379kNm)
262bracing
elements
(8490kN
730kNm)
248bracing
elements
(9961kN
481kNm)
216bracing
elements
(9923kN
725kNm)
500 kNmBending limit
400 kNmBending
limit
750 kNmBending limit
7,000 kNBracing force limit
8,500 kNBracing force limit
10,000 kNBracing force limit
A feasible initial solution is a valuable asset, as it provides a starting point for design
improvement, but such a design may not be known in other structural design
problems.
Design A (from fully braced)
271 elements684kNm, 8398kN
Design B (from design 433)251 elements
744kNm, 8489kN
Design C (from design 1248)
251 elements726kNm, 8475kN
Figure 4.4: Designs generated for consideration for outline proposal.
4.6. CHARACTERISATION OF DESIGN SPACE
When developing an optimisation procedure, it is useful to gain an understanding of
the design space as a whole, the relative size of the feasible region and the behaviour
of constraint boundaries. The topology optimisation models define vast design
spaces: for example in optimisation model A, with 45 potential spirals in the structural
model, each with an integer number of members between 0 and at most 21, a total of
3x1048 possible designs exist in the design space. It is immediately obvious that
exhaustive search is impossible.
Using structural model A, a 10,000-point domain sample was taken. For each design,
a random length is assigned to each of the constituent spirals, with equal probability
of all integer values between zero and the particular maximum spiral length. The
performance of each design is determined by finite element analysis and a statistical
overview of the designs generated is displayed in Table 4.2. Adopting the constraint
values of 750kNm on maximum bending moment in horizontal beams and 8500kN on
78
maximum force in bracing members, it is observed that 0.42% of designs are feasible.
The most lightly braced of this small set has 337 elements (just over half of all
possible elements).
Table 4.2: Statistical analysis of 10000 randomly generated designs
Maximum bending moment (kNm)
<500 <625 <750 <875 <1000 unlimited
Maximum axial force (kN) <7000 0 0 0 0 0 0
<7750 0 3 5 10 12 23
<8500 0 9 42 75 111 288
<9250 0 25 105 224 364 1087
<10000 0 36 169 398 692 2419
unlimited 0 53 274 711 1367 10000
A: Number of designs with constraint values less than the maximum
Maximum bending moment (kNm)
<500 <625 <750 <875 <1000 unlimited
Maximum axial force (kN) <7000 - - - - - -
<7750 - 405 402 412 407 393
<8500 - 384 382 379 375 366
<9250 - 364 365 361 358 349
<10000 - 360 359 353 349 338
unlimited - 355 349 341 335 306
B: Mean number of elements in designs with constraint values less than the maximum
Maximum bending moment (kNm)
<500 <625 <750 <875 <1000 unlimited
Maximum axial force (kN ) <7000 - - - - - -
<7750 - 379 379 379 369 344
<8500 - 345 337 317 317 304
<9250 - 323 307 299 299 289
<10000 - 321 307 288 282 259
unlimited - 305 282 256 242 174
C: Minimum number of elements in designs with constraint values less than the limit
79
An insight into constraint behaviour is gained from the simplified two-dimensional
representation of the design space, shown in figure 4.5. Using a known feasible
design as a starting point, two spirals were selected. The lengths of the spirals were
varied such that every combination of lengths was analysed. For each combination,
values of maximum axial force in bracing elements and maximum bending moment in
horizontal beams were retrieved and used to put values to a set of grid points. These
values were used to plot constraint boundaries in the 2D design space as shown
above. The resulting space is observed to be non-convex.
Figure 4.5: 2D simplified representation of design domain, model A
4.7. TOPOLOGY OPTIMISATION METHOD DEVELOPMENT
Following the live project involvement detailed in section 4.5, a more rigorous
investigation was undertaken, continuing to work with structural model A. This
explored alternative Pattern Search strategies, assessing their effect on the number of
bracing members required in the final design, computational efficiency in arriving at
these designs, diversity of designs generated and reliability of finding feasible
solutions.
Bi-directionality was introduced to the Pattern Search, with variable step-size. In this
approach, exploratory moves are made, considering each of the bracing spirals in turn,
by first attempting to remove a number of elements equal to the current step-size and,
if this is unsuccessful, then attempting to add the same number of elements. A move
cannot take a spiral length beyond its maximum or minimum value: in this case, a
move smaller than the current step-size is attempted and if successful, the attempted
80
Maximum Bending
Moment Constraint
Maximum Axial
Force Constraint
Maximum Spiral
Length Constraint
Length of spiral 1
Length of spiral 2
Feasible
Design Space
pattern move would not include this direction. Bi-directional exploratory moves are
demonstrated in figure 4.6.
1
2
3 5
6
7
37
9
10
12
11
14
15
16
17
22
19
24
39
22
38
23
26
27
27
28
29
30
32
31
3440
35
8 13
40
43
41
42
44
20
21
33
1
2
3 5
6
7
37
9
10
12
11
14
15
16
17
22
19
2420
39
22
21
38
23
26
27
27
28
29
30
32
31
33
3440
35
8 13
40
43
41
42
44
1
2
3 5
6
7
37
9
10
12
11
14
15
16
17
22
19
24
39
22
38
23
26
27
27
28
29
30
32
31
3440
35
8 13
40
43
41
42
44
20
21
33
(i) The schedule forattempting exploratorymoves reaches spiral 38(shown as double line). Current step-size is 5
elements.
(ii) The structure isreanalysed with 5 elementsremoved from spiral 38.
(iii) If the element removalmove is rejected, the
structure is reanalysed with 4elements added (reachingmaximum spiral length)
Figure 4.6. A sample exploratory move
An alternative acceptance criterion must be formulated to include the possibility of
beneficial additive moves. This is discussed in the next section.
Pattern moves attempt to speed the progress of the search in a known good direction.
A base point is fixed after each set of exploratory moves, then a pattern move is
attempted with a vector equal to that defined between the current and previous base
points, subject to constraints on spiral length. In this way, pattern moves may grow
with consecutive acceptance. The current implementation modifies the classic
method with regards to control of step-size. Traditionally, with continuous variables,
step-size is reduced by a prescribed factor when no further moves are possible at the
current size. In the current implementation, using integer variables, the step-size is
adjusted through integer-valued modifications. Moreover, in seeking to minimise the
total number of analyses required, step-size is reduced when the number of successful
exploratory moves in an iteration falls below a critical value (nominally set to five).
Similarly, step-size is increased if more than a requisite number of moves (nominally
set to 10) are accepted. Figure 4.7 presents a diagrammatic representation of the
evolutionary design process. In this case, structural performance criteria are
81
maximum bending moment in horizontal beams and maximum axial force in bracing
members (constraints) and maximum axial force in members at the tip of each bracing
spiral.
Figure 4.7: Pattern Search topology optimisation flowchart
82
Analyse design
START: INITIAL DESIGN
Remove relevant bracing member(s) and analyse
Retrieve structural performance characteristics
Calculate objective function
END: OPTIMISED DESIGN
Current Spiral Length > 0?
YES
NO
Set step-size and weighting factors
Order spirals:
(a) in order of increasing axial force (deterministic) or
(b) in random order (stochastic)
Retrieve structural performance characteristics
Calculate objective function
Move accepted?
Add relevant bracing member(s) and analyse
Current spiral length <
Maximum spiral length
Move accepted?
Select next spiral in list
Retrieve structural performance characteristics
Calculate objective function
Termination criteria met?
Attempt pattern move and analyse
Retrieve structural performance characteristics
Calculate objective function
Move accepted?
Update objective function
and current best design
YES
NO
NO
YES
NO
YES
End of exploratory moves
NO
YES
Start of exploratory moves
YES
4.7.1. Objective function formulation
Objective functions in optimisation are commonly formulated to include soft
constraints, whereby a penalty function is included to move the search away from
infeasible designs. However, in the current application, as discussed previously, due
to the general correlation between member reduction and increase in constraint
values, it is beneficial to move slowly towards constraint boundaries.
Two formulations are proposed, both including constraint function terms in the
objective function:
Formulation 1:
{ }( )∑∑
=
= +
++=
2
1maxmax
11 ,0max
i
ibi
it
orig
SP
s
sp
gpM
M
F
FW
N
n
X Eq. 4.9
where:
max
max2
max
max1 ;
M
MMg
F
FFg bi −
=−
= Eq. 4.10
Norig = number of members in initial configuration.
The first term of equation 4.9 represents the true objective of minimising number of
bracing members, normalised by dividing by the original number of members. The
second term represents the optimisation constraints converted to objectives in a multi-
objective function with weighting, Wit, scheduled to linearly reduce from 1 to 0 over
10 iterations. The final term applies a penalty to infeasible designs, with p = 10.
Formulation 2:
{ }( )∑∑
=
= +=2
1
1
2 ,0maxi
i
orig
SP
s
sp
gpN
n
XEq. 4.11
where:
max
2
max
1 ;M
MMg
F
FFg schedulebschedulei −
=−
= Eq. 4.12
The first term of equation 4.11 is identical to that of the objective function of
formulation 1. The second term applies a penalty to designs with constraint values
above prescribed targets, Fschedule and Mschedule, for the current iteration. The targets are
83
scheduled to converge linearly over 10 iterations from their values in the initial design
to the values of the constraint limits. With p=10, the formulations become identical
after 10 iterations.
In both cases a move is accepted if the value of the relevant objective function of the
resulting design is less than that of the current design. It should be noted that these
formulations define search spaces which will change over the course of the
optimisation process as the weighting factors are reduced.
Further process parameters involved in using the above objective function
formulations in a Pattern Search procedure include:
– Schedule of weights in formulation 1
– Schedule of targets in formulation 2
– Value of penalty factor in both formulations
– Initial step size
– Conditions for decrement or increment of step size (simply decrementing step size
when no moves are possible at the current size is wasteful of analysis time).
4.7.2. Comparative investigation
Table 4.3 summarises the results of five distinct algorithmic variations, with 20
optimisation runs in each case, as well as the salient details of a randomly created
initial design set used as starting points in cases 3-5.
84
Table 4.3: Statistical summary of 20 runs per case
Evolving designs from fully-braced initial configuration
Two sets of twenty optimisation runs were performed using a fully braced initial
design, with random ordering of exploratory moves. The first set (case 1) permitted
only element removal, whilst the second set (case 2) also allowed elements to be
added. Objective function formulation 1 was used, with an initial step-size of 7
elements. Unsurprisingly, the population of final designs evolved using bi-directional
search exhibited better characteristics: an average of 9 fewer bracing elements; a best
design with 10 fewer elements and a smaller standard deviation on element number
when compared with the population evolved by uni-directional search. The only
disadvantage was computational efficiency: using bi-directional search requires
almost twice as many analyses on average. A gross mean number of 18 elements
were added in each bi-directional optimisation run. These may be largely accounted
for by multiple-element search steps in the early stages of evolution, although design
domain non-linearity may also be a contributing factor.
85
Random
design setCase 1 Case 2 Case 3 Case 4 Case 5
-Fully
braced
Fully
braced
Random
set
Random
set
Random
set
- 1 1 1 2 1
- Uni- Bi- Bi- Bi- Bi-
- Stoch. Stoch. Det. Det. Stoch.
- 7 7 7 7 7
Mean 314.9 260.5 251.7 244.8 253.7 245.7
Standard deviation 39.8 14.4 11.9 10.5 13.8 7.1
1322.6 730 730.1 724.7 735 727.3
11471 8478 8485 8490 8479 8488
N/A 555.4 1031.5 933.6 1102.5 865.2
N/A 0/20 0/20 1/20 2/20 2/20
5.38 4.63 4.68 5.58 5.41 5.61
390 243 233 213 230 234
555.5 743.2 748.1 743.1 742.4 742.6
8246 8456 8489 8503 8447 8461
Optimisation parameters Initial configuration(s)
Optimisation formulation
Exploratory move
directionality
Exploratory move
scheduling method
Initial step size
Population characteristics
Number of
elements
Mean maximum
bending moment
Mean maximum axial force
Mean number of analyses
Failed optimisation runs
Diversity metric value
Best design Number of elements
Maximum bending moment
Maximum axial force
Alternative objective function formulations
Two sets of twenty optimisation runs (cases 3 and 4) were performed using an
identical set of randomly generated initial designs, all but one of which violate at least
one of the constraints, each set adopting one of the formulations presented above. Bi-
directional search is necessitated by infeasibility of initial designs. Initial step-size
was set to 7 elements, with exploratory moves scheduled by increasing axial force in
the tip element of the corresponding spirals. Formulation 1 performed significantly
better: the corresponding population of final designs has an average of 9 fewer
elements, whilst comparing best designs 17 fewer elements are observed. The
standard deviation of number of bracing members is also smaller in formulation 1 and
an average of 15% fewer analyses were required.
It should be noted that not all optimisation runs were successful in locating a feasible
design. Occasionally in intermediary infeasible designs, no exploratory move was
capable of reducing the relevant objective function. One such optimisation run was
encountered using formulation 1 and two using formulation 2.
Scheduling of exploratory moves
As previously discussed, a random, or stochastic, scheduling of exploratory moves is
necessary to create diversity in designs evolved from a single fully-braced initial
design. However, the use of randomly generated initial designs offers an alternative
mode of diversification. The question therefore arises as to whether the choice of
method for scheduling of exploratory moves affects the quality of final designs. The
concept of attempting element removal in increasing order of axial force in tip
elements is derived from the Evolutionary Structural Optimisation methods of Liang
et al. (2000). However, in the current procedure, with all possible exploratory moves
attempted and including constraints on bending moment in beams, there is no clear
advantage in this method of scheduling, as seen by comparing the results of cases 3
and 5. One extremely light design was evolved using deterministic scheduling, but
this is not statistically significant.
Performance of designs evolved from randomly generated initial configurations
Comparing the set of designs evolved through stochastic bi-directional search from a
single fully-braced initial design (case 2) against the set evolved through deterministic
86
bi-directional search from randomly generated initial designs (case 3), both using
formulation 1, significant advantages are observed in the latter case. On average, 7
fewer elements are present, with a best design of 213 members, compared to 233
members. Moreover, greater diversity is noted in the population evolved from
different starting points. The value of this is demonstrated by the strong bias towards
clockwise spirals by the best design of case 3, which may be considered unappealing.
Alternative designs with slightly more members offer a more uniform distribution of
elements. Diversity can be measured in a simplistic but quantitative way: two
designs are compared by averaging the absolute differences in length of each spiral
and the resulting sum averaged over all possible pairs of designs within the
population. This diversity metric is 20% greater in the population evolved from
randomly generated initial configurations (see Table 4.3).
Use of pattern moves
A pattern move attempts to move further in a direction that has been found to be
profitable by previous exploratory moves. If successive pattern moves are accepted,
the magnitude grows to accelerate in a direction that reduces the objective function.
Pattern moves will be particularly successful in design domains with smooth objective
functions and relatively small exploratory moves. The current procedure operates in a
highly non-linear design domain with exploratory moves that are initially of the order
of half of the permissible range of the design variables. Nevertheless, pattern moves
removing up to 112 members have been accepted in case 2 (see table 4.2). On
average, approximately 1 pattern move (of around 15 attempted) is accepted in each
optimisation run, with a mean of around 20 elements added or removed.
4.8. TOPOLOGY OPTIMISATION: STRUCTURAL MODEL B
A later design review on the project considered a revised structural model, with
different primary structural considerations, detailed in Table 4.1. The previous
method development section demonstrated an efficient constraint handling method
(formulation 1). The versatility of this formulation is seen through its straightforward
adaptation to handle the constraints on utilisation factor, lateral displacement and
inter-storey drift, defined in optimisation model 2, with no domain knowledge
assumed. The revised objective function is expressed as:
87
( )
( ) ( )
−
−
−+
−
+−
+
−
−+
++=
+
+
+
+=∑
1300/
,0max1500/
,0max1,0max
300/500/
max
1
1
max
max
max
max
max
1
1
max
max
max
max1
1
ss
s
j
s
jj
ss
s
j
s
jj
it
orig
SP
s
sp
hh
dd
h
dUp
hh
dd
h
dUW
N
n
X
Eq. 4.13
where:
Norig = total number of bracing elements in the initial configuration;
Wit = weighting factor at iteration it;
Umax = maximum utilisation factor, the larger of the values of the left hand sides of
equations 4.4 and 4.5, occurring in any bracing member;
p = penalty factor applied to constraint violations.
A range of standard circular hollow sections from STD CHS 508 15.9 (area =
0.0246m2) to STD CHS 609 50.8 (area = 0.0891m2) were made available for
selection. A full list is presented in the following section. In the current topology
optimisation problem, piece-count is of primary concern and therefore becomes the
primary term in the objective function. A simple premise would be to assume that
optimal solutions are obtained by assigning the largest permissible section size to each
member. The STD circular hollow section 609 50.8 is therefore used for all members
in the topology only optimisation process. It should be noted that if a bracing member
makes a negative contribution to maximum lateral displacement or inter-storey drift,
then the negative contribution is increased (and thus total displacement reduced) by
reducing the section size. A negative contribution is made, as determined by the
principle of virtual work, if forces in real and virtual load-cases are of opposite sign.
This concept is addressed further in the next section. However, using maximum
section sizes should provide near-optimal results, which can be fine-tuned if required,
as well as simplifying fabrication and connection detailing.
The Pattern Search topology optimisation flowchart of figure 4.7 again applies to this
formulation, with structural performance characteristics of maximum lateral
displacement, inter-storey drifts and forces and moments in bracing members.
88
4.8.1. Results
Two sets of optimisation runs were performed using the algorithm previously
described, again comparing results obtained from a fully-braced configuration against
those derived from alternative randomly generated bracing configuration. In both
cases, exploratory moves were scheduling randomly. Full details are shown in tables
4.4 and 4.5, with three of the best designs displayed in figure 4.8.
From fully braced initialdesign: case 1
From randomlygenerated initial design:
case 3
From randomlygenerated initial design:
case 10
Figure 4.8: Design solutions from topology optimisation of structural model 2
89
Table 4.4: Performance of designs derived from fully braced initial configuration.
RunNumber
ofmembers
Inter-storey drift(m)
(limit = 0.038)
Max lat disp (m)(limit = 0.5192)
Max util(limit = 1)
Additions RemovalsSuccessful
pattern moves(number (size))
Fullybraced
648 0.0228 0.3432 1.00572 - - -
1 194 0.0379 0.5186 0.98131 50 500 1 (4)
2 197 0.0351 0.5182 1.00001 12 449 1 (14)
3 197 0.0374 0.5188 0.98866 33 477 1 (7)
4 207 0.0344 0.5191 0.99293 22 440 2 (6, 17)
5 205 0.0378 0.5191 0.96831 26 469 0
6 203 0.0366 0.5192 0.9829 5 450 0
7 212 0.0377 0.5182 0.98562 46 482 0
8 214 0.0353 0.5188 0.99483 15 449 0
9 206 0.0378 0.5192 0.96623 16 452 1 (6)
10 195 0.0373 0.5186 0.99035 41 477 1 (17)
Mean 203 0.0367 0.5188 0.98512 26.6 464.5 0.7 (10)
Standarddeviation
7.055 0.001313 0.000379 0.010934 15.35 19.28 0.675 (5.64)
Diversity 3.73
Table 4.5: Performance of randomly generated initial designs and solutions derived
from them through bi-directional topology optimisation
RunNumber
ofmembers
Inter-storey drift(m)
(limit = 0.038)
Max lat disp (m)(limit = 0.5192)
Max util(limit = 1)
Additions RemovalsSuccessful
pattern moves(number (size))
1Start 387 0.0291 0.4271 0.99926
Final 213 0.0357 0.5192 0.96306 109 283 3 (62, 2, 9)
2Start 351 0.0575 0.4894 1.02336
Final 213 0.0351 0.5192 0.96854 88 226 1 (4)
3Start 385 0.0494 0.4917 1.12656
Final 189 0.0310 0.519 0.98418 125 321 0
4Start 330 0.0378 0.4605 0.96264
Final 209 0.377 0.5190 0.98260 118 239 1(7)
5Start 326 0.0449 0.4871 1.00931
Final 196 0.0321 0.5192 0.99854 132 262 2 (16, 2)
6Start 326 0.146 0.8498 0.97501
Final 203 0.0349 0.5191 0.98034 164 287 0
7Start 338 0.1165 0.6916 1.0840
Final 196 0.0329 0.5187 0.99889 85 227 0
8Start 336 0.0531 0.5521 1.06283
Final 197 0.035 0.5190 0.99791 204 333 2 (83, 1)
9Start 306 0.1751 0.7583 0.98346
Final 217 0.037 0.5191 0.98729 203 292 1 (78)
10Start 347 0.0731 0.5969 1.18026
Final 193 0.0351 0.5189 0.98416 133 287 2 (3, 4)
Mean 202.6 0.03465 0.51904 0.984551 136 276 1.2 (23)
Standarddeviation
9.778 0.108383 0.000158 0.012144 42.16 36.94 1.033 (31.8)
Diversity 4.69
90
4.8.2. Observations
The pattern search algorithm using structural model 2 can be seen to perform equally
well starting from a fully-braced initial configuration as from a randomly generated
infeasible configuration, according to the results in table 4.5. This trend was not seen
in the previous structural model (section 4.7.2). A 13% variation in number of
elements in the final design is observed. Moreover, the best design (run 2 from the
randomly generated initial design set), has approximately 25% fewer elements than
the proposed Arup design at the corresponding stage in the design process. Each run
takes around 300mins to complete on a PC Pentium 4 CPU 2.66GHz, 512MB RAM
desktop computer. In general, Pattern Search moves do not make a significant
contribution to the topology optimisation process, accounting for an average of just
2% of the total member removals and additions. This is likely to be due to a
combination of high-dimensionality and unevenness of the search space. However,
the low acceptance of pattern moves does not negate the effectiveness of the process
of exploratory moves with gradual refinement.
4.8.3. Diversity
Results show that there is no discernible difference in piece-count or average
structural performance between individual solutions derived from randomly generated
initial starting points and those derived from a fully braced configuration. A
secondary objective of the optimisation procedure is to be able to generate a diverse
set of solutions. The relative diversity of the two sets of solutions can be assessed
using a diversity metric, introduced in section 4.7.2. It can be seen from tables 4.4
and 4.5 that the value of the diversity metric is substantially increased, from 3.73 to
4.74, by using different randomly generated starting points, as opposed to the fully
braced design. It is also of note that the average diversity value when comparing each
random starting point to the corresponding final design is 4.69. This implies that
optimised designs are almost as similar in appearance to each other as to their
corresponding origin. The use of a set of randomly generated starting points is
therefore a good way of ensuring aesthetically diverse final designs.
91
4.9. SIZE OPTIMISATION
4.9.1. Overview
This section considers the section size optimisation of bracing members for a given
topological configuration through the Optimality Criteria method (Borkowski 1990).
It would be possible to extend the method to include size optimisation of the
remainder of the structure, but the illustrative purposes of this section are adequately
served by keeping sections in the skeleton structure fixed. The subsequent argument
considers pin-jointed bracing members, carrying only axial force under applied wind-
loading, although they will also carry bending moment and shear force in ultimate
limit state loadcases due to directly applied distributed loading. A note on application
of this method to members carrying bending moment is presented in section 4.9.6.
4.9.2. Derivation of iterative approach from Optimality Criteria
For a fixed topology, the size-optimisation problem can be stated as:
Minimise: ∑=
=N
i
ivV1
Eq. 4.13
)( iii LAv =
Subject to: A∈iA Eq. 4.14
1≤c
i
yi
i
M
M
pA
F + Eq. 4.15
15.01 ≤
++
c
ci
c
i
c
ci
P
F
M
mM
P
FEq. 4.16
300
11
sssj
sj
hhdd
−≤−
++ Eq. 4.17
500
maxmax h
d j ≤ Eq. 4.18
where:
V = total volume of bracing members in the structure, equivalent to steel mass;
vi = volume of individual member i;
N = number of members in current design;
Ai = cross sectional area of member i;
92
Li = length of member i;
A = discrete set of cross sectional areas available from catalogue (table 4.6);
S = number of bracing spirals in the structure (up to 48);
Fi = maximum absolute axial force occurring in bracing member i in ultimate limit
state envelope;
Fci = maximum compressive axial force occurring in bracing member i in ultimate
limit state envelope;
Mi = maximum bending moment occurring in bracing member i in ultimate limit state
envelope;
py = section design strength;
Mc = section moment capacity;
Pc = section compression resistance;
m = equivalent uniform moment factor;
djs+1 - dj
s = inter-storey drift between storeys s and s+1 under loadcase j;
hs+1 - hs = height of storey s;
djmax = maximum lateral displacement under loadcase j;
hmax = total height of building.
Table 4.6: Catalogue of circular hollow sections and corresponding areas available
for bracing members
Catalogue listing Area (m2) Assign if continuous area (m2):
STD CHS 508 15.9 0.024581 A < 0.026
STD CHS 508 19.1 0.029336 0.026 < A < 0.031
STD CHS 508 22.2 0.033881 0.031 < A < 0.035
STD CHS 508 25.4 0.03851 0.035 < A < 0.040
STD CHS 508 31.8 0.047574 0.040 < A < 0.049
STD CHS 508 34.9 0.051871 0.049 < A < 0.053
STD CHS 508 38.1 0.056245 0.053 < A < 0.058
STD CHS 508 40.5 0.059482 0.058 < A < 0.061
STD CHS 508 44.5 0.064798 0.061 < A < 0.066
STD CHS 609 38.1 0.068334 0.066 < A < 0.070
STD CHS 609 40.5 0.072333 0.070 < A < 0.074
STD CHS 609 44.5 0.078918 0.074 < A < 0.080
STD CHS 609 47.6 0.083952 0.080 < A < 0.085
STD CHS 609 50.8 0.089085 0.085 < A
93
Equations 4.15 and 4.16 are complex constraints, since py, Mc and Fc must be
calculated when considering any given prospective section size. However,
considering observed peak forces and moments in member, i, these inequalities can be
used to set a minimum section size, Aimin,, above which one would expect the
equations to always be satisfied. For the purposes of generating Optimality Criteria,
a continuous range of cross-sectional areas is made available for each bracing
member, between Aimin and Amax, the latter corresponding to the largest section size in
the catalogue. Equations 4.17 and 4.18 are constraints on inter-storey drift and
maximum lateral displacement respectively, both under wind loading. In practice
only the three most critical drift or displacement constraints (of a total of 115) are
considered at any one time. This reduces the number of virtual load cases that need to
be analysed, the processing time in calculating member contributions to displacement
and the complexity of the size optimisation algorithm. The critical cases are
recalculated at each iteration.
Inequality constraints, as required in establishing a Lagrangian function, are therefore
stated as:
011*
≤−≡ ∑i
ij
j
j eC
g (j=1,...,L) Eq. 4.19
01
min
≤−≡+i
iiL
A
Ag (i=1,...,E) Eq. 4.20
01max
≤−≡++A
Ag i
iEL (i=1,...,E) Eq. 4.21
where:
i
iijij
ijEA
lUFe = Eq. 4.22
(contribution to maximum lateral displacement)
or ( )
i
iijijij
ijEA
lUUFe
1−−= Eq. 4.23
(contribution to inter-storey drift)
−−= ∑
i
ijjj edh
C maxmax
*
500 Eq. 4.24
94
or ( )
−−−
−= ∑+
+
i
ij
s
j
s
j
ss
j eddhh
C 11
*
300Eq. 4.25
eij = contribution of ith member (of a total of E) to inter-storey drift or displacement in
case j, as calculated by the principle of virtual work. eij will take a negative value if
the forces it carries in the real load case, Fij and virtual load case, Uij or Uij-Uij-1, are of
opposite sign. In this case, the negative contribution to total displacement or drift is
increased by reducing the cross-sectional area of the member, which also has the
benefit of reducing the volume of the structure.
E = Young's modulus of steel.
Cj* = maximum permissible contribution of bracing members to inter-storey drift or
displacement in the jth case (of a total of L).
The Lagrangian function is then expressed as:
( )
∑∑ ∑∑
∑∑∑
−+
−+
−+=
+++=
+++
++++++
i i
iiEL
i
iiL
j i
ij
j
j
i
i
i
iELiELiLiLj
j
j
i
i
A
A
A
Ae
Cv
gggvV
1111
max
min
*
*
µµµ
µµµ
Eq. 4.26
where µ values are undetermined multipliers taking the value of zero for inactive
constraints or a positive value for active constraints, where the term inside the bracket
should be equal to zero. Hence for a converged, optimal design, only the first term on
the right hand side of equation 4.26 contributes to the objective function.
For optimality, the partial derivative of the Lagrangian function with respect to the
section area of each member should be zero, i.e.:
max2
min
*
*
0i
iEL
i
iiL
j i
ij
j
j
i
i
i AA
A
A
e
CA
v
A
V +++ +−∂
∂+
∂
∂==
∂∂
∑µµµ
Eq. 4.27
Replacing vi and eij with the corresponding parameters that are independent of area:
length, li, and ēij, gives:
0max2
min
2*=+−− +++∑
i
iEL
i
iiL
j i
ij
j
j
iAA
A
A
e
Cl
µµµEq. 4.28
=
i
iij
A
ee
; ( )iii Alv =
95
Hence:
+
+
=++
+∑
max
min
*
i
iELi
iiL
j j
ijj
i
Al
AC
e
Aµ
µµ
Eq. 4.29
It is noteworthy that in the case where there exists a single active displacement
constraint and for members for which size constraints are not active, equation 4.29
can be expressed as:
ii
i
ii
i
lA
e
lA
eC==
2
*
µEq. 4.30
Since the term on the right hand side of the above equation is exactly the “cross”
strain energy density of the member (the displacement contribution of the member, as
calculated by virtual work, per unit volume), it is apparent that cross strain energy
density should be constant for all members with inactive size constraints. There is no
such elegant result when multiple displacement constraints are active.
The current method uses an iterative approach to solve the above problem. For each
loadcase, since Ai is a function of the square root of µj and Cj is a function of the
reciprocal of Ai , this information can be combined in the recursive formula:
vj
j
jvj
C
Cµµ
2
*
1
=+ Eq. 4.31
where v and v+1 denote successive iterations. Each revised value of µj is used to
calculate a new set of Ai values and subsequently predict Cj, on the fundamental
assumption that ēij values are unchanged.
Using a similar rationale, multipliers corresponding to constraints on section size are
iterated according to equations 4.32 and 4.33.
viL
i
iviL
A
A+
++
= µµ
2min
1 Eq. 4.32
viEL
i
iviEL
A
A++
+++
= µµ
2
max
1Eq. 4.33
96
4.9.3. Pitfalls
Complex values of A i.
Since eij values can be negative, it is conceivable that during the course of
convergence towards an optimal solution, the value to be square-rooted on the right-
hand side of equation 4.29 may be negative, giving a nonsensical complex value for
Ai. Clearly this should not be the case for a converged, feasible design, since the
value of the multiplier corresponding to the minimum section size constraint would
become very large in order to prevent this. With reference to equation 4.29, this
scenario can be satisfactorily avoided by setting the initial values of µL+i and µL+E+i to
be substantially bigger than that for µj. If the value to be square-rooted does become
negative, the value of Ai is set to Amin, and the corresponding Kuhn-Tucker multiplier
is doubled.
Convergence failure
For many bracing topologies, it is not possible to find a feasible set of section sizes to
meet displacement constraints. In these cases, intuition suggests that for the most
critical loadcase, section areas should be set to maximum for members with positive
displacement contribution, or minimum for members with negative contribution. In
practice, using the iterative algorithm detailed previously, section areas will increase
beyond their maximum permissible value, due to continuing escalation of the µj
multiplier. However, after the prescribed maximum number of iterations, such
members will simply be assigned the maximum section area.
Negative values of C j* and C j
According to equations 4.22 and 4.23, if the contribution of the non-designable
structure to the overall displacement or lateral drift in a given loadcase is greater than
the corresponding limiting value, Cj* will be negative. It is also possible that Cj
values may be negative, if there is a negative contribution to total displacement of a
large number of bracing members. If one or both of Cj* or Cj
are negative, the
behaviour of µj may be erratic. This situation may be avoided by adding a constant
value, k, to both Cj* and Cj
(calculated as the summation of eij), to ensure both are
positive at all times, without changing the fundamental problem to be solved.
97
However, selecting a value of k that is excessively large will greatly slow
convergence of the algorithm, since the ratio of Cj to Cj* in equation 4.31 will be
close to unity. k is therefore assigned on a case-by-case basis according to the
following equation:
( )[ ]jGlobalj CCk −= *_*1.0,0max Eq. 4.34
4.9.4. Assignment of discrete sections
On convergence of the above iterative procedure, discrete sections must be assigned
to the Ai values. Extensive research has been conducted into discrete variable
structural optimisation (Arora 2002) focusing on achieving strict global optima for
this problem. However, this will generally involve much reanalysis, which, in
practice, becomes too computationally expensive. Instead, an approximate method,
which will be neither overly conservative nor cause constraint violation is required. If
section size constraints are active, the choice of discrete section is obvious, otherwise
the allocation method shown in table 4.6 is generally found to adequately meet the
requirements.
98
Figure 4.9: Size optimisation flowchart
4.9.5. Size optimisation of fully braced configuration
The bracing design with all members present is optimised using the above method,
with various starting points:
(i) all members take maximum section sizes;
(ii) all members take minimum section sizes;
(iii)five cases in which members take randomly assigned section sizes.
The results are detailed in table 4.7.
99
Analyse design
INITIAL DESIGN
Determine critical loadcases
Retrieve:
- Maximum forces in ULS envelope
- Forces in real wind and virtual loadcases
Reanalyse with required virtual loadcases
Calculate for each critical displacement/drift case:
- Total contribution of bracing members to
displacement/drift
- Required contribution
- k value to ensure required contribution and projected
contribution remain positive at all times
Calculate for each bracing member:
- Minimum permissible area
- Contribution to displacement/drift in each critical case
Overall
Convergence?
END
Assign:
- Initial multiplier values
Calculate for each bracing member:
- Revised section area
- Revised multipliers corresponding to minimum and
maximum permissible areas
Calculate for each displacement/drift case:
- New projected contribution from bracing members on
the basis of revised section areas
- Revised multipliers
Convergence or
maximum iterations reached?
APPLICATION OF OPTIMALITY CRITERIA
Assign:
- Catalogue sections to all bracing members
YES NO
YES
NO
Table 4.7: Size optimisation of fully-braced design from different initial
distributions
CASE InitialVolume(m3)
Normalised Initial Max.
Displacement/Drift
InitialUtilisationFactor
FinalVolume(m3)
Normalised Final Max.
Displacement/Drift
FinalUtilisationFactor
(i) 880.1 0.661 1.007 298.2 0.852 0.989
(ii) 242.9 0.939 1.978 286.1 0.862 0.999
(iii).1 575.4 0.749 1.864 294.3 0.856 0.998
(iii).2 564.3 0.752 1.459 295.1 0.854 0.992
(iii).3 567.7 0.752 1.838 293.5 0.856 0.998
(iii).4 570.4 0.743 1.588 295.1 0.855 1.000
(iii).5 557.4 0.763 1.940 293.5 0.856 0.999
In all cases, despite comfortably satisfying displacement criteria, the initial design is
infeasible due to the utilisation factor for buckling being greater than unity. Figures
4.10 and 4.11 show how the algorithm converges on a locally optimal solution in
around 8 iterations, for cases (i) and (ii). A 4.2% variation in bracing volume in
locally optimal solutions is observed, with case (i) producing the highest volume
design and case (ii) producing the lowest volume design. All final designs meet the
constraint on buckling capacity. This is particularly noteworthy in case (i), from
which one might presume that no feasible set of section sizes exists for this topology.
This is not a direct consequence of the sizing algorithm, but of unpredictable
redistribution of forces in the structure resulting from section size modifications
affecting local stiffnesses. There is clearly no guarantee of global optimality in sizing
an indeterminate structure by this method.
100
Figure 4.10: Convergence of size optimisation algorithm from maximum section
sizes in fully braced design
Figure 4.11: Convergence of size optimisation algorithm from minimum section
sizes in fully braced design
101
Section Size Optimisation: Fully Braced Topology from Minimum Sections
0
1
2
0 1 2 3 4 5 6 7 8 9 10
Iteration
Norm
alised Value
Normalised Displacement/Drift Buckling Capacity Utilisation Normalised Volume
Section Size Optimisation: Fully Braced Topology from Maximum Sections
0
1
2
0 1 2 3 4 5 6 7 8 9 10
Iteration
Norm
alised Value
Normalised Displacement/Drift Buckling Capacity Utilisation Normalised Volume
4.9.6. Size optimisation by Optimality Criteria with bending
moments
Without bending moment in bracing members in critical load-cases, a major
complication in the generic section-sizing task is removed, since the cross-sectional
area is the only relevant section property. In a more general formulation, changes in
bending stiffnesses, shear factors and torsional stiffnesses and their effect on the
contribution to displacement of the member in question, must be considered, as well
as the cross-sectional area. This clearly introduces a large number of semi-dependent
variables that for commercial standard steel sections are linked, but not proportional
to cross-sectional area. Grierson and Chan (1993) propose the use of linear regression
analysis to define coefficients allowing the additional cross-sectional properties to be
approximately expressed in terms of the cross-sectional area of a given member. A
similar iterative method to that described above can then be adopted to perform the
optimisation task.
4.10. INTEGRATION OF TOPOLOGY AND SIZE OPTIMISATION
In many structural design problems, minimising material volume or mass may be of
primary concern. Although this was not the case in the original Pinnacle Tower
design, minimising bracing volume is considered in this section for two reasons:
– Offering designers an understanding of the trade-off between minimum mass and
minimum piece count.
– Research value and to provide a practical example of the potential of the proposed
method.
The following pertinent questions should be addressed:
– Can size optimisation be incorporated into a topology optimisation algorithm
without a prohibitive increase in computation time?
– By optimising size and topology simultaneously, can we obtain a lower volume
design than by performing size optimisation on an optimal topology?
– How does simultaneous size and topology optimisation affect the resulting
topology solutions?
Section 4.8.5 indicates that convergence of the size optimisation algorithm requires
around 8 analysis iterations using a starting point that is far from optimal. However,
102
if the starting point is closer to satisfying the optimality conditions, with very little
redistribution of forces and moments in the structure required, convergence can be
expected to be significantly faster and a single iteration should achieve a near optimal
design. If the projected behaviour of the structure subject to revised section sizes,
assuming no redistribution of forces and moments, is consistently accurate, only a
single structural analysis is required per topological exploratory move. This would
add very little computational expenditure to the integrated optimisation process,
compared with optimisation of topology only. Unfortunately, the assumption of
unchanged force and moment distribution is not always sufficiently accurate, and
determining acceptance on the basis of projected behaviour can often lead to an
increase in objective function value due to constraint violations. It is therefore
necessary to introduce a further structural analysis with revised section sizes, to
validate the projected behaviour. This also allows required changes to the set of
critical loadcases to be detected.
The objective function used for optimisation of topology alone is adapted for the
integrated optimisation problem. However, the primary objective is now to minimise
the sum of the volumes of the spirals, rather than the number of members therein.
The pure topology optimisation algorithm included terms with an iteration-dependent
weighting coefficient designed to keep solutions away from constraint boundaries in
early iterations. In the current hybrid algorithm, we attempt to minimise the volume
for each solution by meeting the most critical displacement-drift constraint exactly,
hence the term favouring designs that are distant from displacement constraint
boundaries is discarded. The term favouring designs that are distant from utilisation
constraint boundaries is modified to consider the maximum utilisation factor value
that would occur if all members took the largest section-size. This projected
utilisation factor using maximum sections is calculated for each member using the
current force-moment distribution in the structure. Violation of utilisation factor
constraints is penalised as before, although drift and displacement constraints are
considered together in a single term, since these constraints are equivalent in the
section size optimisation algorithm.
Hence the objective function is defined as:
103
( )
( ) ( )
−
−
−−
+−+
+=
+
+
=∑
1300
,1500
,0max1,0max
,...
max
1
1
max
max
max
max
max1
11
ss
s
j
s
jj
capit
orig
SP
s
s
SP
hh
dd
h
dUp
UWV
v
vvX
Eq. 4.35
with the following symbols introduced:
vs = volume of spiral s;
Vorig = total volume of bracing members in initial design;
maxU = maximum utilisation factor, as defined by the left hand sides of equations 4.15
and 4.16 .
= maximum utilisation factor capacity: the largest utilisation factor that would
occur in any bracing member if all took maximum section size (with current force-
moment distribution).
Figure 4.12 illustrates the integration of size optimisation into the topology
optimisation routine.
104
maxcapU
Figure 4.12: Flowchart for combined size and topology optimisation algorithm
105
Analyse design
INITIAL DESIGN (size optimised)
Retrieve structural performance characteristics:
- Maximum lateral displacement in wind loadcases
- Maximum interstorey drift in wind loadcases
- Maximum forces and moments in ULS envelope
Determine critical displacement/drift cases
Remove relevant bracing member(s) and analyse
Calculate objective function
END
Current Spiral Length > 0?
YES
NO
Set step-size and weighting factor
Generate randomly ordered list of spirals
Retrieve structural performance characteristics
Move accepted?
Add relevant bracing member(s) and analyse
Current spiral length <
Maximum spiral length
Move accepted?
Select next spiral in random list
Retrieve structural performance characteristics
Calculate objective function
Termination criteria met?
Attempt pattern move and analyse
Retrieve structural performance characteristics
Calculate objective function
Move accepted?
Update objective function,
critical displacement/drift
cases
and current best design
YES
NO
NO
Retrieve structural performance characteristics.
Calculate objective function
Retrieve structural performance characteristics
Perform single iteration of optimality criteria
SIZING ALGORITHM and reanalyse
Perform single iteration of optimality criteria
SIZING ALGORITHM and reanalyse
End of exploratory moves
Start of exploratory moves
YESNO
YES
NO
YES
NO
Retrieve structural performance characteristics
Perform single iteration of Optimality Criteria
SIZING ALGORITHM and reanalyse
4.10.1. Results
Two sets of optimisation runs were performed using the hybrid algorithm described.
In the first set, ten runs were performed starting from an optimised fully braced
configuration, each using a different random number seed to determine the order in
which bracing spirals are considered for member removal or addition. In the second
set, ten runs were performed each starting from a different randomly generated
bracing configuration. These are the same as those considered for topology
optimisation only, but in the current case full size optimisation is performed before
modification of topology is introduced.
Table 4.8: Performance of optimised designs derived from fully braced initial
configuration
Design Description Members Optimisedvolume (m3)
Max.utilisation
Normalised max.disp/drift
Comments
Fullybraced
Optimisedfeasible
648 294.6 0.970 0.726
1 Final 348 194.0 0.990 0.990
2 Final 320 192.3 0.998 0.988 Fewest members
3 Final 342 193.8 0.986 0.991
4 Final 343 192.7 0.985 0.989
5 Final 330 189.2 0.989 0.990 Lowest volume
6 Final 348 196.5 0.978 0.990
7 Final
8 Final
9 Final
10 Final
322 191.1 1.000 0.991
377 200.6 0.968 0.996
320 191.6 0.979 0.990
368 197.8 0.976 0.997
Mean 341.8 194.0 0.985 0.991
Standard
deviation
11.98 2.318 0.00761 0.00107
106
Table 4.9: Performance of initial and optimised designs derived from random
initial configurations.
Design Description Members OptimisedVolume (m3)
Max.Utilisation
Normalised Max.disp/drift
Comments
1 Start: Infeasible 387 398.1 1.001 1.245
Final 294 189.8 0.989 0.989
2 Start: Infeasible 351 407.3 1.030 1.453
Final 313 198.5 0.999 0.988
3 Start: Infeasible 385 414.0 1.097 1.247
Final 357 250.5 1.055 0.989 Infeasible Solution
4 Start: Feasible 330 233.5 0.983 0.995
Final 299 191.1 0.968 0.988
5 Start: Infeasible 326 350.7 1.020 1.142
Final 303 189.9 0.993 0.989
6 Start: Infeasible 326 377.2 0.972 3.763
Final 338 372.9 1.006 1.105 Infeasible Solution
7 Start: Infeasible 338 367.4 1.101 2.997
Final 255 189.3 0.996 0.884 Fewest members
8 Start: Infeasible 336 361.2 1.003 1.350
Final 272 186.1 0.996 0.905 Lowest volume
9 Start: Infeasible 306 351.7 1.015 4.534
Final 349 227.7 0.970 0.993
10 Start: Infeasible 347 391.1 1.739 1.861
Final 329 379.9 0.997 1.076 Infeasible Solution
Mean 310.9 237.6 0.997 0.991
Standard
deviation
32.75 68.45 0.0267 0.0639
4.10.2. Observations
Two solutions derived from randomly generated initial designs (7 and 8) form a
Pareto-dominant set (Pareto 1896) from the solutions evolved from both fully braced
and randomly generated initial designs. That is to say, compared to any other design,
these two solutions have either lower volume or piece count. In fact, with the
exception of solution 5 in the set starting from the fully braced solution, the two
solutions discussed are superior in both respects to all other designs. Generally,
starting from randomly generated infeasible designs appears to offer slightly lower
volume solutions with appreciably lower piece-counts, when compared with solutions
derived from the fully-braced configuration. However, in a minority of cases
problems have occurred in finding feasible or high quality solutions from infeasible
initial designs. Possible causes include:
107
– Inter-storey drift governs at a high storey level. No additions at current step-size
will aid this local structural performance issue. Removal of members lower in the
structure will not change the critical wind-case and its value.
Possible Solution: starting with a very large step-size will ensure bracing
members can be added towards the top of the structure, reducing inter-storey drift
at any height.
– Addition of bracing members to an infeasible design may allow the new topology
to meet drift constraints. In the section-sizing algorithm, a major reallocation of
section sizes may occur, causing a substantial redistribution of load-paths within
the structure. On the first iteration, this may make the new design infeasible on
account of utilisation factor, despite a superior topological configuration.
Possible Solutions: determine move acceptance purely on topology change or
perform multiple size-optimisation iterations whenever a substantial change in
constraint values occurs.
– A size-optimisation step gives false superiority to a design. Targets for bracing
displacement contribution are intentionally conservative to avoid a design being
flagged as infeasible due to load-path redistribution, when a feasible solution is
possible. So if bracing contribution is higher than intended, but the design is
feasible (but very close to the constraint), it can be hard for further designs to
improve on this.
A comparison of the best designs generated by simultaneous topology and size
optimisation and those generated by topology optimisation with subsequent size
optimisation is presented in the following section.
4.11. SUMMARY OF RESULTS FROM OPTIMISATION MODEL B
Table 4.10 presents a summary of the characteristics of the best designs developed
from structural model B and considering optimisation model B. This includes a
manually evolved design proposal, shown in figure 4.13, developed by Arup
designers from structural model B, but having removed the requirement for bracing
members to be grouped in continuous spirals starting from the base of the vertical
columns. It should be noted that 22 of the bracing members in the basement storey,
that were fixed in the work previously described, have been removed. This will have
108
a detrimental effect on stiffness, hence comparison is not entirely fair. Bracing
members are concentrated around the narrow edge of the building, to increase the
resistance to minor-axis bending.
Figure 4.13: Arup design proposal, without requirement for bracing members to
be grouped in continuous spirals.
Table 4.10: Summary of best designs from structural model 2
Design descriptionBracingmembers
Optimisedbracing
volume (m3)
Max.utilisation
Normalisedmax disp/drift
Solutionnumber
(figure 4.14)
Arup design 241 328.5 1.050 0.966 -
Size-optimised fully braceddesign
648 286.1 0.999 0.862 6
Lowest piece-counttopology-optimised solutionfrom fully-braced design(subsequently size-
optimised)
194 267.3
(252.5)
0.981 0.999 1
(2)
Lowest piece-counttopology-optimised solutionfrom random initial design
(subsequently size-optimised)
189 261.7
(247.0)
0.984 1.000 4
(5)
Lowest volume topologyand size optimised solutionfrom fully-braced design
330 189.2 0.989 0.990 7
Lowest volume topologyand size optimised solutionfrom random initial design
272 186.1 0.996 0.905 10
109
Even accounting for the removal of bracing members in the basement of the Arup
design, the topology optimised designs show a significant reduction in number of
bracing members, despite the design constraint requiring continuous bracing spirals.
A reduction in bracing volume of over 25% is achieved in the best designs generated
by simultaneous topology and size optimisation, compared to those from sequential
topology and size optimisation. However, since there is no incentive towards low
piece-count designs, final solutions from the hybrid method have around 50% more
members. The increased reduction in bracing volume achieved by integrated size-
topology optimisation over topology optimisation followed by size optimisation is
illustrated in figure 4.14. Lines at 45 degrees to the axes are contours of constant
volume. Movement along the y-axis (downwards), indicates reduction in volume
resulting from topology reduction, whereas movement along the x-axis results from
size-optimisation. Starting from an infeasible initial design, volume may be increased
in an effort to locate a feasible design before the general trend of volume reduction is
seen.
A designer could tune trade-off between piece-count and total volume by combining
these terms in the objective function of equation 4.36:
( )
( ) ( )
−
−
−−
+−
+++=
+
+
==∑∑
1300
,1500
,0max1,0max
,...;,...
max
1
1
max
max
max
max
max11111
ss
sj
sjj
capit
orig
SP
s
s
V
orig
SP
s
s
NSPSP
hh
dd
h
dUp
UWV
v
WN
n
WnnvvX
Eq. 4.36
Position on a Pareto front could be changed by adjusting the weighting of the two
terms: WN and WV (=1-WN) are weighting coefficients on the total number of bracing
members and the total volume of bracing members respectively (other terms in this
expression have been previously used in sections 4.9 and 4.10). Alternatively a
Pareto archive could be introduced, with one primary objective, such as piece-count,
driving the search and a secondary objective, such as bracing volume, controlling the
archive. Starling (2004) proposes the introduction of this concept to Pattern Search
for grammar-based synthesis of artefacts.
110
0: Datum Point: Fully Braced, Maximum Sections. 648 bracing members, 880.1m3
1: Lowest piece-count solution from fully-braced starting point (topology only, with maximumsection sizes). 194 bracing members, 267.3m3
2: Size-optimised design from solution 1. 194 bracing members 252.5m3
3: Randomly generated infeasible starting point (maximum sections sizes) 385 bracingmembers, 522.8m3
4: Lowest piece-count solution (topology only, with maximum section sizes) developed fromsolution 3. 189 bracing members, 261.7m3
5: Size-optimised design from solution 4. 189 bracing members 247.0m3
6: Size-optimised design from datum point (0). 648 bracing members, 286.1m3
7: Lowest volume design developed from size optimised fully-braced starting point (solution 6)by simultaneous topology and size optimisation. 330 bracing elements, 189.2m3
8: Randomly generated infeasible starting point (maximum section sizes) 336 bracingmembers, 456.3m3
9: Infeasible size-optimised design from solution 8. (no feasible section-size solution exists)336 bracing members, 361.2m3 10: Lowest volume design developed from a size optimised random starting point (in this casesolution 9) by simultaneous topology and size optimisation. 272 bracing members, 186.1m3
Figure 4.14: Volume reduction by simultaneous versus sequential topology and
size optimisation routines.
111
Volume change by size-optimisation
Volume change by topology optimisation
Datum Point
Solution 0
850m3
800m3
750m3
700m3
650m3
600m3
550m3
500m3
450m3
400m3
350m3
300m3
250m3200m3150m3100m350m30m3
850m3800m3750m3700m3650m3600m3550m3500m3450m3400m3350m3300m3250m3
150m3
100m3
50m3
0m3
Simultaneous size-
topology optimisation
Solution 10
Solution 1Solution 2
Solution 5 Solution 4
Solution 3
Solution 6
Solution 7
Solution 9 Solution 8
Size optimisation
Simultaneous
size-topology
optimisation
Size optimisation
Topology optimisation
Topology optimisation
200m3
4.12. CONCLUSIONS
In response to the research question posed at the start of this chapter, it has been
demonstrated how the established Hooke and Jeeves (1961) search method can be
applied to a practical topology problem, by simplifying the task to consider a fixed set
of variables. Through stochastic search and varying starting points a range of
optimally-directed designs can be found, avoiding single local optima and allowing
unmodelled criteria, such as aesthetics, to influence final design selection.
Simultaneous optimisation of size and topology was efficiently carried out by
performing a single iteration of the Optimality Criteria method at each topological
step. This integrated approach offered substantial volume reductions when compared
to sequential topology and size optimisation.
Focusing this research on a practical problem has revealed a number of crucial
considerations and obstacles relevant to the application of structural optimisation
techniques in the building industry. These problems are either neglected or not
apparent when considering small scale benchmarks problems and include:
– Adaptability of an optimisation model to changes in geometric specifications,
constraints, objectives and aesthetic requirements. This is vital since external
modifications are inevitable and it would rarely be acceptable for an optimisation
procedure to hinder progress of a project. During the period of involvement in
and observation of the design process for the Pinnacle Tower, a number of
revisions and modifications were made, influencing the structural models and
constraints used in the optimisation routine. This necessitated careful
reconsideration of the adaptation and subsequent behaviour of the optimisation
algorithms, since approximations and simplifications must be carefully justified.
– Irregularity of design spaces. The formulation of objective functions have been
shown to define design spaces which are very irregular and have many local
minima. Finding the global optimum, and proving it so, within a set design space
is therefore virtually impossible. However, the use of stochastic methods mean
that a range of locally optimal solutions, diverse in appearance but with similar
high performance, can be offered for further consideration.
– Constraint handling. In the early stages of topology optimisation, a means of
repelling the search path from constraint boundaries was required, in order to
112
avoid acceptance of disadvantageous moves and subsequently becoming trapped
in poor local optima. Addition of appropriate terms with diminishing weighting in
successive iterations was successful in meeting this need.
– Approximations and simplified structural models. Improving algorithm efficiency
by these and other means allows more solutions to be generated in a given time.
– Section sizing considerations. In considering section-size optimisation as an
independent problem, best results were observed by adopting the lightest possible
initial solution and allowing sizes to increase. Maximum section sizes may not
always provide the best solution in terms of local strength, since larger sections
attract a higher share of the load due to higher stiffness.
Optimisation in industry should ideally be driven by cost models, as considered in
detail by Khajehpour (2001). The holistic objective in a generic project is to
minimise the total cost incurred in design, material and labour purchase, construction,
provision of services and maintenance and where applicable, maximising the revenue
from letting of floor space or other sources. This problem must be simplified and
varying degrees of complexity may be considered in cost modelling. A crude
approximation of cost being proportional to weight can be developed to consideration
of component cost, construction time and cost, revenue (lettable area and
corresponding value), maintenance cost and even design time costs. However, in
practice, collating and establishing suitable costs for an optimisation model can prove
difficult.
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5. Structural topology optimisation of braced steel
frameworks using Genetic Programming
5.1. INTRODUCTION
In the early stages of a design project it is desirable to generate and assess a number
of alternative concepts. In the structural design of buildings, this might involve
considering alternative structural systems, such as a concrete core or steel tubular
framework for providing lateral stability. Otherwise, one might consider the design of
a given structural system, for example the configuration of bracing in a steel tubular
framework. Ideally assessment of solutions is from both a structural and architectural
perspective. This chapter presents the application of Genetic Programming (GP) to
tackle this problem, attempting to include an element of design rationale and the
ability to influence design aesthetics.
Research Questions:
– How can multiple, novel, high performance designs be rapidly generated for a
structural topology problem?
– How can aesthetic constraints be defined and their effect on performance be
investigated?
– How can Genetic Programming be applied in a pure sense to structural
topology optimisation and what benefits can this offer?
Proposals:
- The tree-based representation of design development in genetic programming
avoids the requirement of a fixed number of variables common to many
optimisation methods.
- The functional basis of Genetic Programming trees allows pattern development
for large scale structures from simple representations.
114
5.2. BACKGROUND
Genetic Programming (GP) is a class of evolutionary algorithm developed in the early
1990s (Koza 1992), which in its seminal form manipulates tree structures containing
instructions for solving a task, such as a design problem. Despite various attempts at
using GP in civil engineering (Shaw 2003), in the field of structural topology
optimisation, to the author's best knowledge, the full potential of GP has not been
fully exploited. This is because the trees have taken the form of encrypted
representations of a design (Yang and Soh 2002) or an assembly of lower level
components without functionality defined at branch points (Liu 2000). The current
research uses tree structures containing design modification operators as internal
nodes, thus detailing the development of a design from fundamental components.
These tree structures can be manipulated by genetic operations to evolve optimally
directed solutions. Evolved programme trees can be considered as "blue-prints" for a
design to play back the development of a given solution by sequential execution of
the branch functions.
In common with other evolutionary algorithms, GP is population-based and
stochastic. This facilitates the generation of a set of optimally-directed designs for
further consideration according to criteria that are difficult to model computationally,
such as aesthetic value. The potential for concurrent evaluation of solutions through
parallel computing can be valuable in offsetting the often prohibitive number of
solution evaluations required by such methods. EAs require neither any domain
knowledge, nor gradient information, but are effective at global search. Despite
generally poor performance in precise local search, the ability to hybridise with
domain dependent heuristics or deterministic local search methods makes EAs very
versatile. Successive populations are developed through the genetic operations of
reproduction, crossover and mutation.
5.3. GENETIC PROGRAMMING METHOD
5.3.1. Introduction
Genetic Programming uses tree structures to define solutions, with fundamental
components as terminals or "leaves", operated on by internal function nodes. This has
115
been applied literally in developing high performance computer programmes (Koza
1992) as well as in electronic circuit design and other fields (Koza et al. 2003).
Roston (1994) presents a Genetic Design (GD) methodology applied to artifact
design, adopting the tree representation and form of genetic operations associated
with GP, whilst appreciating that the encoding of design information meant that this
was “essentially a generalisation of GA”. In Genetic Programing, terminals take the
form of constants or variables, while functions operate on given inputs from lower in
the tree and pass a result up. The number of inputs on which a function operates is its
arity and for a given function may be fixed or variable. Figure 5.1 demonstrates a tree
representation of the mathematical equation y = 4/(X*X) + 5*(7-X) evolved to fit a set
of experimental data. Terminals may be constant integer values or the variable, X.
Functions are chosen from the arithmetic operators: add '+', subtract '-', multiply '*',
divide '/'.
Figure 5.1: Tree representation of mathematical equation: y = 4/(X*X) + 5*(7-X)
5.3.2 GP FOR BRACING DESIGN
This chapter proposes the application of Genetic Programming to the problem of
bracing system design with terminals taking the form of bracing units occupying a
single bay-storey cell within an orthogonal framework. A range of functions, or
design modification operators, are defined to operate on terminals and outputs of
functions lower in the tree. Thus the basic bracing units are developed through
scaling and patterning to a state where they may represent a high-performance
stability system. Each operator has an associated vector with parameters defining the
direction, frequency or magnitude of the operation relative to the instance on which it
acts. The set of functions is shown in figure 5.2, along with details of their associated
vectors. Functions are either reversible or unidirectional, as indicated by the
arrowheads. The vector(s) associated with each operator is shown above or below the
arrow. As an example, the unite function combines bracing units located in a three-
116
+
*/
5 -4 *
7 xx x
bay, three-storey “neighbourhood”, the bottom left hand corner of which is indexed as
(1,1). Every function is unary (arity = 1), with the exception of the unite function,
which takes two or more inputs.
The set of design modification operators can be considered as an informal form of
grammar (Stiny 1980), which, along with a vocabulary of bracing types, defines a set
of designs or language. Previous use of grammars within structural topology design
is demonstrated by Shea (1997) through integration with simulated annealing to the
optimally-directed design of planar and space trusses.
Figure 5.2: Function set for GP trees representing bracing designs
117
SCALEADJUST
SPLITS
ORTHOGONAL
REPEATROTATE TRANSLATE
UNITEIRREGULAR
REPEATREFLECT
[X enlargement,
Y enlargement]
[X divisions,
Y divisions]
[X spacing,
Y spacing,
frequency]
[angle of rotation] [X translation,
Y translation]
[X centre-line,
Y centre-line]
[X spacing,
Y spacing,
frequency]
[X index, Y index;
X neighbourhood
size,
Y neighbourhood
size]
[-1,-2]
[1,2] [1,1]
[1,3]
[0,1,2] [90o] [0,1]
[0,-1]
[1,1;3,3] [1,3,2] [2.5,-]
[2.5,-]
1 2 3 4 5 6
1
2
3
4
5
6
7
8
9
10
1 2 3 4 5 6
1
2
3
4
5
6
7
8
9
10
1 2 3 4 5 6
1
2
3
4
5
6
7
8
9
10
1 2 3 4 5 6
1
2
3
4
5
6
7
8
9
10
Creating initial designs
An orthogonal framework, with insufficient lateral rigidity, is taken as the starting
point. An initial population of bracing system designs is created by applying the
following steps to each individual:
1. Seed the framework with a number of bracing units (or instances) each occupying
a single bay-storey cell, with column (x) and row (y) indices, as seen in figure 5.3:
Figure 5.3: Seeded framework
2. Sequentially apply a number of design modification operations, selected
randomly, to single or united groups of instances, to assemble a tree representation
of a design (figure 5.4).
118
ROOT
X: 1,10 X: 1,2 X: 2,3 X: 3,12
112
35
78
29
46
10
11
1 2 3
Figure 5.4: Development of an initial design by application of design
modification operators
119
Unite
(1,2; 2,2)
Irregular Rep
(0,4,1)
Orthogonal
Rep (0,-1,5)
Adjust Splits
(1,2)
Rotate
Translate
(0,-1)
Scale
(0,1)
ROOT
Irregular Rep
(1,-1,1)
X: 1,10 X: 1,2 X: 2,3
X: 3,12
112
35
78
29
46
10
11
ROOT
X: 1,10
X: 1,2 X: 2,3
X: 3,12Unite
(1,2; 2,2)
NO CHANGE
FROM ORIGINAL
PHENOTYPE
ROOT
X: 1,10
X: 1,2 X: 2,3 X: 3,12
Unite
(1,2; 2,2)
Orthogonal
Rep (0,-1,5)
12
35
78
94
610
11
12
ROOT
X: 1,10 X: 1,2 X: 2,3 X: 3,12
Unite
(1,2; 2,2)
Orthogonal
Rep (0,-1,5)
12
35
78
94
610
11
12
Scale
(0,1)
ROOT
X: 1,10
X: 1,2 X: 2,3
X: 3,12Unite
(1,2; 2,2)
Orthogonal
Rep (0,-1,5)
12
35
78
94
610
11
12
Scale
(0,1)
Irregular
Rep (1,-1,1)
Application of design
modification
operators omitted
Analysis and fitness
In the test problems presented subsequently, all individuals in a population are
analysed subject to prescribed wind-loading. Each design is then assigned a “fitness”
based on bracing length and structural performance.
Generating subsequent populations
The selection process employed in the test problems in the following section is elitist,
coupled with linear weighting of selection probability based on rank. Individuals are
ranked relative to others in the population according to their fitness. This ranking is
used to determine the individual's likelihood of selection as a parent in generating the
next population. As seen from figure 5.5, the fittest individual has approximately
twice the probability of selection in a given operation than the individual ranked at the
fifty-percentile. Elitism ensures that the best known solution(s) is passed from one
generation to the next.
Figure 5.5: Linear rank-based weighting system for parent selection
Subsequent generations are populated by the application of genetic operators:
– the best R% of designs from the previous generation are directly reproduced
through the elitism strategy.
– Crossover is applied with probability Pc to pairs of parent designs selected from
the previous generation. The same design cannot be chosen as both parents.
“Cuttings” from each parent are selected at random and exchanged to form new
120
1 2 3 ... N-2... N-1 N
Fitness-based rank (of N individuals in current generation)
Probability of selection
2/N
1/N
trees. The cutting may be a terminal alone or a number of functions applied on a
terminal(s).
– Mutation is applied with probability Pm (=1-P
c) to a single parent, passing one
offspring design into the next generation. The mutation operator may take four
forms:
1. mutation of a single terminal
2. mutation of a branch-section (terminal(s) and functions)
3. removal of a branch (subject to retaining at least one branch in the tree
representation
4. addition of a branch.
Mutation forms 2 and 4 involve generating a new branch. In the implementations
described in this chapter, the new branch has a single terminal, with between 0
and 4 functions.
A summary of the Genetic Programming search process is presented in the flowchart
of figure 5.6.
Figure 5.6: Genetic Programming evolutionary process flowchart
Handling geometrically infeasible designs
A substantial proportion of offspring from both crossover and mutation operations are
found to be geometrically infeasible, either due to bracing units overlapping or
extending beyond the prescribed framework. An example is shown in figure 5.7,
121
START
Generate initial
population
Termination
criteria satisfied?
Reproduction
Select genetic
operation
Select parent(s)
Population
complete?
Select crossover/
mutation point(s)Create offspring
ENDYES
YES
NO
Evaluate
population and
sort by fitness
NO
where, in attempting to accommodate a new branch into the parent tree (see figure
5.8(D)), the function “Irregular repeat (1,-1,1)” causes overlap of bracing units and
the bracing system to extend outside the underlying orthogonal framework.
Figure 5.7: Example of geometric infeasibility, with units overlapping and
extending beyond the orthogonal framework.
The Structured Genetic Algorithm (Dasgupta and McGregor 1991) includes a genetic
hierarchy to account for the necessity or infeasibility of certain design combinations
(Rafiq et al. 2003). Unfortunately this is not applicable in the current GP method on
account of the form of the design representation. Various alternative techniques have
been previously adopted for handling infeasible individuals (Michaelewicz 1996):
– Rejection of infeasible solutions may help focus the search on the feasible region,
but can also lead to the loss of potentially valuable “genetic” information. In the
context of the current application, rejection leads to populations of sparsely braced
individuals and low diversity.
– Prevention of illegal solutions through careful design of evolutionary operators
may be possible in some cases, though this is not applicable to the current
application.
– Penalisation can be applied to solutions violating constraints by reducing their
fitness. This is the approach taken in the current case with designs with excessive
lateral displacement under the given loads, as detailed in the following section.
However, geometric infeasibility may mean that solutions cannot be assigned a
fitness since they are structurally nonsensical, hence penalisation is not possible in
this case.
– Repair operations can be performed on infeasible solutions. Although in some
applications this may be very difficult, it is readily achieved in the current case by
122
X: 2,2
Scale
(1,1)
ROOT
X: 1,2
Unite
(1,2; 3,2)
Irregular Rep
(1,-1,1)
112
35
78
29
46
10
11
removing or making appropriate adjustments to relevant design modification
operators, as illustrated in figure 5.8.
(continued overleaf)
123
Unite
(1,2; 2,2)
Irregular Rep
(0,4,1)
Orthogonal Rep
(0,-1,5)
Adjust Splits
(1,2)
Rotate
Translate
(0,-1)
Scale
(0,1)
ROOT
Irregular Rep
(1,-1,1)
X: 1,9 X: 1,2 X: 2,3
X: 3,12
112
35
78
29
46
10
11
Unite
(1,2; 2,2)
Irregular Rep
(0,4,1)
Orthogonal Rep
(0,-1,5)
Adjust Splits
(1,2)
Rotate
Translate
(0,-1)
Scale
(0,1)
ROOT
Irregular Rep
(1,-1,1)
X: 1,9 X: 1,2
X: 3,12
ROOT
X: 2,2
Scale
(1,1)
112
35
78
29
46
10
11
A: Parent Design (phenotype and genotype)
B: A cut is made in the parent tree
C: A new branch is created
(mutation) or taken from a
second parent (crossover)
1 2 3
Figure 5.8: Repair algorithm in the context of mutation or crossover
124
ROOT
X: 1,2
X: 2,2
Scale
(1,1)
ROOT
X: 1,2
Unite
(1,2; 3,2)
X: 2,2
Scale
(1,1)
X: 2,2
Scale
(1,1)
ROOT
X: 1,2
Unite
(1,2; 3,2)
Irregular Rep
(0,4,1)
X: 2,2
Scale
(1,1)
ROOT
X: 1,2
Unite
(1,2; 3,2)
Irregular Rep
(0,4,1)X: 1,9 X: 3,12
Orthogonal Rep
(0,-1,4)
X: 2,2
Scale
(1,1)
ROOT
X: 1,2
Unite
(1,2; 3,2)
Irregular Rep
(0,4,1)
X: 1,9
X: 3,12Rotate
Scale
(0,1)
112
35
78
29
46
10
11
112
35
78
29
46
10
11
112
35
78
29
46
10
11
D: The new branch is developed by adding functions above the cut
made in the original tree, where possible. N.B. “Irregular repeat (1,-1,1)”
was rejected as this would have caused geometric infeasibility
Translate (0,-1)
E: Other terminals appearing in the original parent tree are replaced
where possible.
F: These terminals are then traced back to the root of the parent tree,
replacing functions where possible. N.B. In right hand branch,
“Orthogonal repeat” is reduced to (0,-1,4) to avoid overlap. The
subsequent “Split Adjust (1,2)” function is then infeasible and is therefore
removed.
112
35
78
29
46
10
11
112
35
78
29
46
10
11
1 2 3 1 2 3 1 2 3
1 2 3
1 2 3
It is important to note that the mapping between physical structural representation, or
phenotype, and the GP tree representation, or genotype, is not one-to-one. That is to
say, a given bracing framework in the solution space can be represented by a
potentially infinite number of distinct trees in the representation space.
Termination of the optimisation process is determined by improvement of best-of-
generation individual fitness and lowest average fitness of a generation. The method
is implemented in Matlab.
5.4. BRACING DESIGN FOR A 2X6 FRAMEWORK
Validation of the GP method will be conducted using the same two-bay, six-storey
framework test problem as was used for the exploration of ESO techniques in chapter
3. By using the same loading conditions and displacement constraint, it is possible to
select a fixed diameter for circular solid sections for all bracing members such that the
globally optimal design solution is known with reasonable certainty.
Referring to table 3.12, taking a fixed section diameter of 0.08m for bracing members
of circular hollow section, it is reasonable to expect that the double echelon
configuration of solution B would meet the displacement constraint of 0.024m with
minimal total bracing length. It is readily demonstrated that this design is indeed
acceptable, with a top corner displacement of 0.023m. As previously, bracing
members are modelled with simple supports.
The optimisation model for the problem established can be expressed as an
unconstrained minimisation problem (using a penalty function):
Minimise: ( )( )*,0max max
1
ddpLLn
e
e −+=∑=
Eq. 5.1
where:
L = design fitness (equivalent to total length of bracing members for feasible designs)
Le = length of bracing member, e
n = total number of bracing members
d* = limit on maximum lateral displacement
dmax
= maximum lateral displacement observed in structure
125
p = penalty factor imposed on designs violating constraint on top storey lateral
displacement, nominally chosen to be 5000 for all studies detailed in this chapter.
The total number, length and location of bracing members are variable in the
evolutionary process. Fixed parameters in the structural model include framework
geometry, applied loads, vocabulary of basic bracing types and section size of bracing
members (Ae), beams and columns. Issues of strength and buckling are recognised as
important but not included at this stage for means of comparison.
A symmetry constraint is applied in this case. This simplifies the problem to one of a
single bay, six storey framework, which is reflected in the horizontal centre-line and
has the added advantage that only a single structural analysis loadcase need be
considered for each design. An asymmetric design must be analysed in two
loadcases, with identical loads applied to each face. Defining three basic bracing
units: “X”, “/”, “\” will allow the majority of the designs shown in figure 3.12 to be
represented (with the exception of designs F, H, J, K and M).
Stochastic optimisation and search methods can often be inefficient on account of
generating a large number of poor quality designs, the structural analysis of which is a
waste of computational resources. A means of filtering out, or improving, such
designs can greatly improve the efficiency of the search. In the case of braced steel
frameworks, it is known that bracing is required in every available storey in order to
maximise structural efficiency (Ji 2003). A “fill” algorithm is therefore proposed,
which will add functions or fresh terminals to the tree representation until bracing is
present in every storey. It is calculated that this reduces the size of the search space in
this test problem by more than a third, from over 10,000 distinct designs to 3072.
These values are calculated by considering the combinatorial problem of filling the
six cells on one side of the structure with bracing units of different size and form. The
“fill” algorithm also compensates for the fact that complexity is generally lost in
implementing the “repair” algorithm.
A study was made to investigate the effect of varying the optimisation parameters,
using the specifications described above. Population size took values of 10, 30 and
50, combined with crossover probabilities equivalent to 0, 0.75 or 0.9 of new
offspring generated by crossover operation. Twenty runs were conducted for each
pairing of parameters. Since this suggested that a population size of 30 offers a good
126
compromise between accuracy and computational efficiency, further runs were then
performed for a population size of 30, making up to a total of 50 runs for each
crossover probability listed previously, as well as 0.5. The convergence criterion
requires that no improvement is seen in either best-of-generation individual fitness or
mean population fitness for 10 generations, with zero tolerance. At this point the
process is terminated. The objective of this investigation is to understand sensitivity
to these parameters and how maximum computational efficiency can be achieved
without compromising accuracy in finding the global optimum solution. Ideally, a
good level of diversity in high performance designs archived over the course of each
process is still achieved.
Table 5.1 shows the performance of runs with each combination of parameters. The
best solution found in any of these runs was indeed the double echelon configuration
of solution B in table 3.12, with a bracing length, equal to the fitness value, of 50.2m.
This design is therefore referred to as the global optimum in the right hand column of
table 5.1. The 119 function evaluations required, on average, to locate the optimal
solution using a population size of 10 and a crossover ratio of 0.9 is less than 4% of
the 3072 function evaluations required to exhaustively assess all possible design
solutions defined by the current formulation with the fill algorithm.
Table 5.1: Batch characteristics in parametric study
Popn.size
Crossoverprobability
Number ofgenerations to
find localoptimum
Mean S.D.
Functionevaluations to findlocal optimum:
(Population size xmean generations
to optimum)
% runsfindingglobal
optimum
Final populationfitness statistics (m)
Meanof S.D.
Meanof
means
S.D. ofmeans
50
(20 runs
each)
0 5.4 15.9 270 95% 29.6 114.8 3.96
0.75 3.6 6.2 178 100% 31.2 98.8 4.20
0.9 3.8 11.3 190 100% 34.4 94.0 6.10
30
(50 runs
each)
0 6.2 26.5 187 94% 33.0 115.6 5.64
0.5 7.3 45.6 220 98% 32.2 108.2 5.65
0.75 7.1 25.5 212 100% 35.0 99.0 8.36
0.9 5.7 25.7 172 100% 32.8 93.2 6.16
10
(20 runs
each)
0 11.1 81.8 110 65% 45.0 113.8 14.34
0.75 12.4 97.2 124 90% 33.2 98.2 7.56
0.9 11.9 65.4 119 100% 32.0 88.8 8.90
127
The primary conclusion from these results is that the method exhibits relatively low
sensitivity to the optimisation parameters of population size and crossover ratio.
Traditionally in EAs, excessively small population size creates a danger of
insufficient genetic material in the gene pool and hence convergence to a false
optimum, whereas overly large population size leads to computational inefficiency.
Mutation should prevent potentially important genetic information being lost, whilst
crossover exploits beneficial characteristics already in the population. With reference
to table 5.1, small population size or mutation-only runs may yield false optima, at
least within the convergence criteria specified, as seen in 35% of runs with a
population size of 10 and without use of crossover operation. It should be noted that
with the use of the “fill” algorithm, new genetic information is introduced into
virtually every new offspring. This may hinder convergence and cause deviation
from conventional trends observed in EAs. The final three columns present details of
design fitnesses in the final populations: standard deviation of fitness within a
population, averaged across a batch of runs; average fitness of designs in the final
populations of a batch and standard deviation of mean fitness within a run. Of these,
the most significant trend is the decrease in mean fitness with increased crossover
ratio, for a constant population size.
Figures 5.9 to 5.11 characterise a representative run from the set with population size
of 30 individuals and crossover probability of 0.9. In the initial population a mix of
the three bracing types, “X”, “/” and “\”, is observed. More than half (16 out of 30) of
these randomly generated designs do not satisfy the lateral displacement constraint.
Average and best individual fitness in a generation improves rapidly in the early
stages, until the optimum design is found in the seventh generation, as seen in figure
5.11 (where the initial population is marked as 0). Thereafter the average fitness
within a generation fluctuates, whilst the best solution is unchanged on account of the
elitist strategy. The final population, seen in figure 5.10, still contains many
infeasible designs (14 out of 30), and although the optimum solution is repeated
several times, other good solutions are found. In this population, the '/' bracing unit
dominates, although due to the mutation operator and fill algorithms, other types still
appear.
128
The n:1 mapping between genotype and phenotype means that many different trees
can represent the optimal design. Figure 5.12 shows that the method described finds
relatively simple tree-representations, without the phenomenon of uncontrolled
program growth, or “bloat” (Langdon and Poli 1997), observed in some previous
applications of GP.
A typical run with a population size of 30 with 18 generations, hence a total of around
500 function evaluations, took around 40 minutes to run on a PC with Pentium® 4
CPU 2.66 GHz and 512 MB RAM.
Figure 5.9: Example of initial population, penalised designs shown in grey
(Population size = 30, Crossover ratio = 0.9, run number 22)
Figure 5.10: Example of final population, penalised designs shown in grey
(Population size = 30, Crossover ratio = 0.9, run number 22), best-of-run design
top-left
129
Figure 5.11: Example of evolution history (Population size = 30, Crossover ratio
= 0.9, Run number 22)
Figure 5.12: Example of most efficient tree representation of the optimal double
echelon design (left), with the actual representation found in Run number 22,
Population size = 30, Crossover ratio = 0.9 (right)
5.5. BRACING DESIGN FOR A 6X30 FRAMEWORK
The potential of the proposed method for larger structures is now demonstrated
through the example of bracing design for a six-bay, thirty-storey framework. Each
unit cell is 6.096m wide and 4.267m high. The orthogonal framework of beams and
columns was sized from a selection of standard circular hollow sections (as per the
130
ROOT
/: 1,4
Adjust Splits
(1,1)
Irregular Repeat
(0,1,1)
Scale (0,1)
Irregular Repeat
(0,-3,1)
ROOT
/: 1,4
Scale (0,2)
Irregular Repeat
(0,-3,1)
0 5 10 15 20 25 300
25
50
75
100
125
150
Generation
Fitness value
best of generation
average of generation
Pinnacle Tower, considered in chapter 4) by fully-stressed design. The members are
grouped in three storey blocks, as shown in figure 5.13. Within each block, all
horizontal beams are required to take the same section, as are the four outermost
columns and the three inner columns. Table 5.2 lists the sections made available in
the fully stressed design process, along with the groups assigned to each section in the
converged solution. The sizing process considered the combination of two loadcases:
uniform vertical loading of 40kN/m acting downwards on all horizontal beams and
horizontal loading of 50kN applied as a point load at all column-beam intersections
on the left hand face of the structure. Since the structure is designed to have
horizontal symmetry it is not necessary to define a loadcase with horizontal loads on
the opposite face.
The bracing configuration is to be designed such that the maximum lateral
displacement in the structure does not exceed 0.256m (1/500 of the total height of the
structure) under extreme wind-loading: three times the horizontal loadcase
considered in the framework sizing. Hence point loads of 150kN are applied at each
storey height on the left hand face. All bracing members are to take the same circular
hollow section, selected such that a known good design of repeated X-bracing, as
shown in figure 5.17, satisfies the displacement constraint. This is achieved using the
standard CHS 273 16.0 section, whereby maximum lateral displacement of 0.248m is
observed at the top left hand corner node.
Based on the assumption that larger search spaces are better explored with larger
population sizes, a population size of 50 was selected for a set of three runs. The
crossover ratio of 0.75 performed well in the previous study (see table 5.1) and is
hence adopted for tackling the current problem. The convergence criterion was
modified such that termination occurs when no improvement, again with zero
tolerance, is seen in the best-of-generation fitness for 30 generations. Initial and final
populations are shown for the first of these runs in figures 5.14 and 5.15 respectively.
The corresponding evolution history is shown in figure 5.16. The best solution found
by each run is seen in figure 5.17, alongside the datum design of repeated X-bracing,
with performance characteristics included.
131
Figure 5.13: Geometry and cross-section groupings for 6x30 framework
132
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
2 2 2 2 2 2
2 2 2 2 2 2
2 2 2 2 2 2
3 3 3 3 3 3
3 3 3 3 3 3
3 3 3 3 3 3
4 4 4 4 4 4
4 4 4 4 4 4
4 4 4 4 4 4
5 5 5 5 5 5
5 5 5 5 5 5
5 5 5 5 5 5
6
7
8
9
10 10 10 10 10 10
10 10 10 10 10 10
10 10 10 10 10 10
9 9 9 9 9
9 9 9 9 9 9
9 9 9 9 9 9
8 8 8 8 8
8 8 8 8 8 8
8 8 8 8 8 8
7 7 7 7 7
7 7 7 7 7 7
7 7 7 7 7 7
6 6 6 6 6
6 6 6 6 6 6
6 6 6 6 6 6
11 11
11 11
11 11
11 11
11 11
11 11
12
12
12
12
12
12
12
12
12
12
12
12
13
13
13
13
13
13
13
13
13
13
13
13
14
14
14
14
14
14
14
14
14
14
14
14
15
15
15
15
15
15
15
15
15
15
15
15
16
16
16
16
16
16
16
16
16
16
16
16
17 17
17 17
17 17
17 17
17 17
17 17
18
18
18
18
18
18
18
18
18
18
18
18
19 19
19 19
19 19 19 19
19 19
19 19
20
20
20
20
20
20
20
20
20
20
20
2030 30 30
30 30 30
30 30 30
29 29 29
29 29 29
29 29 29
28 28 28
28 28 28
28 28 28
27 27 27
27 27 27
27 27 27
26 26 26
26 26 26
26 26 26
25 25 25
25 25 25
25 25 25
24 24 24
24 24 24
24 24 24
23 23 23
23 23 23
23 23 23
22 22 22
22 22 22
22 22 22
21 21 21
21 21 21
21 21 21
6 x 6.096m
30 x 4.267m
Table 5.2: Cross-sections made available and selected in fully-stressed design.
Section
ID
Section catalogue
listing
Groups assigned
to section
1 STD CHS 273 16.0 20, 30, bracing
2 STD CHS 273 20.0 -
3 STD CHS 273 25.0 -
4 STD CHS 323 16.0 10
5 STD CHS 323 16.0 19, 29
6 STD CHS 323 25.0 -
7 STD CHS 355 20.0 9
8 STD CHS 355 25.0 8, 28
9 STD CHS 406 20.0 -
10 STD CHS 406 25.0 6, 7 ,18, 27
11 STD CHS 406 32.0 3, 4, 5, 16, 17, 26
12 STD CHS 457 25.0 2
13 STD CHS 457 32.0 1, 15, 25
14 STD CHS 457 40.0 14, 24
15 STD CHS 508 32.0 -
16 STD CHS 508 40.0 12, 13, 22, 23
17 STD CHS 508 40.0 11, 21
133
Figure 5.14: Initial population of randomly generated designs
134
Figure 5.15: Final generation of designs (run 1), including best-of-run design
(top-left).
135
Figure 5.16: Evolution history of run 1.
Knowndesign
Run 1 Run 2 Run 3Projecteddesign
Total bracing length (m) 446.4 434.4 506.2 353.2 314.6
Maximum lateraldisplacement, dmax (m)
0.248 0.257* 0.252 0.258* 0.273*
Constraint violation (%) 0 0.3 0 0.8 6.8
Fitness 446.4 439.8 506.2 355.2 366.8
Generations to optimum - 68 58 113 - -
Figure 5.17: Best-of-run designs and performance (* indicates displacement
constraint violation)
136
0 10 20 30 40 50 60 70 80 90 1000
200
400
600
800
1000
1200
Fitness Value
Generation
best of generation
mean of generation
From figure 5.14 it is seen that a high degree of diversity is attained in the generation
of random designs. Figure 5.15 shows that diversity remains high in the final
population, although occurrences of the 'X' bracing unit are greatly reduced, since
intuitively this is detrimental to the objective of meeting the displacement constraint
with minimal bracing material. The evolution history of figure 5.16 follows a similar
trend to that of the simpler 2x6 framework example, but on a longer timescale. The
mean-of-generation values reach a noisy plateau after around 15 generations, with the
continuous addition of new genetic material from the “fill” algorithm preventing any
further reduction. The best-of generation solution improves until generation 68.
Inspection of the form and performance of best-of-run designs (figure 5.17) reveals
that two of the three solutions present a reduction in steel tonnage when compared
with the datum known design, despite their unconventional form. In the case of the
design from run 3, a material saving of 20% is offered over a popular solution in
building design practice. The penalty function has permitted these solutions, despite a
violation of the displacement constraint of less than 1% in both cases. This is
beneficial, since a nominal addition of material would add sufficient stiffness to the
structure to meet the constraint. The best-of-run designs exhibit a high proportion of
diagonal bracing units which rise two storeys for each bay spanned. Arrangement of
these units into chevrons, as per the optimal 2x6 framework bracing solution, also
appears highly beneficial.
The right-hand design of figure 5.17 presents a design of more regular appearance
than might be considered by designers as a result of the above discussion. However,
despite offering a reduction in bracing length of 11% over the best design of run 3, the
larger constraint violation of 6.8%, as opposed to 0.8%, means that it has a poorer
fitness according to the defined optimisation model. Hence in three runs, the genetic
programming method has found a design superior to the regular designs proposed, as
evaluated by the fitness function used.
Each run performs of the order of 5000 structural analyses in calculating the objective
function value for each solution. Whilst this number of analyses would have
permitted an exhaustive search of the design space defined for the 2x6 framework
problem, it will only cover a minute proportion of the solutions included in the vast
design space for the 6x30 framework problem. It is therefore unsurprising that a
137
stochastic method with restricted computational resources is unable to consistently
locate a single optimal solution.
5.6. CONTROL OF AESTHETIC STYLE
The methodology previously described offers great versatility on account of the
capacity for prescribing aesthetic style. This can be achieved through exerting control
over:
unit types – examples above allowed 'X', '/' and '\' units. This list could be extended or
restricted.
maximum and minimum bracing size – intricate designs may be generated by
specifying a maximum size to which bracing units may be scaled. Alternatively,
designs in the search space may be simplified by requiring bracing units to be greater
than a minimum size. Search may also be simplified by, for example, considering a
6x30 framework to be reduced to a 1x10 grid, with each basic unit being 3-bay x 3-
storey, reflected in the centre-line of the structure
aspect ratio – scaling operations could be constrained to keep units in a certain
proportion, such that, for example, units must have equal height and width.
pattern definition – repetition could be constrained to follow prescribed vectors, e.g.
orthogonal or rising one storey for every bay.
As an example, considering the aesthetic requirements on the Pinnacle Tower on
which the work in Chapter 4 is based, one might define the following constraints:
– unit types: '/' or '\', must be anchored to the base of the building.
– minimum bracing size: 3 storey by 1 bay units
– aspect ratio: 3 storey rise for each bay spanned. This influences permissible scale
and adjust splits function vectors.
– Repeat vectors must be of the form (1, 3, r) or (x, 0, r)
However, the pattern generating behaviour of the functions defined in the Genetic
Programming methodology would not be very compatible with the desire for a
randomised visual effect on this project.
This form of control can also be used to restrict the design space to be searched in an
attempt to improve convergence and performance of solutions. For example, figure
5.17 suggests that 'X' bracing could be removed from the list of available unit types
138
and 2-storey, 1-bay bracing used as a basic unit, due to its frequency in best-of-run
solutions. This would constitute a learning strategy and could be incorporated into the
algorithm as machine learning. It should also be noted that it would be very straight-
forward to define further symmetry-lines and repeating units as used in Chapter 3.
5.7. FURTHER WORK
It is a notable omission that the proof-of-concept studies do not include section-size
optimisation nor any form of strength consideration. Section-sizing may be
overlooked in topology design if the minimisation of piece-count or bracing length
rather than steel tonnage is the primary objective, as in the project-based work of
Chapter 4. Alternatively, approximate section-sizing operations could be performed
on every solution, or on selected individuals using a Lamarkian operator, as employed
in the pseudo-Genetic Programming method of Liu (2000). The former approach
would add significant computation time to each run, whilst the latter is likely to slow
convergence.
5.8. CONCLUSIONS
This chapter has described a methodology for optimally directed search using genetic
programming. This approach exploits the capacity of GP for evolving instructions to
create a solution, as opposed to evolving solution representations. Through the use of
two proof-of-concept examples, the potential of this method for generating high
performance, pattern-based topological designs has been demonstrated. Further, the
versatility of the method offers potential for defining subsets of solutions based on
aesthetic criteria.
The initial research question is answered by employing an Evolutionary Algorithm:
the use of populations and a stochastic method allow multiple diverse solutions to be
obtained, whilst the GP algorithm removes the need for a fixed set of variables.
Section 5.6 presents a framework for defining a desired aesthetic form, allowing
different styles to be explored in response to the second research question. Finally,
the use of design modification operators offers a more pure interpretation of GP for
structural topology design than is seen in previous studies and allows a range of
complexity in the structural designs, but with simple tree-representations.
139
The form of design modification operators used offer potential well beyond bracing
design, to any design problem in which pattern emergence is relevant and
performance may be quantified.
140
141
6. Concluding Remarks
This chapter commences with a reassessment of each of the three methods presented
in the preceding chapters. Recommendations are then put forward for future research.
There follows a summary of issues raised relating to the application of structural
optimisation in building design practice. Finally a discussion of potential future
trends in structural design automation and optimisation practice is given.
6.1. REVIEW OF CONTRIBUTIONS
Table 6.1 presents a summary of the three distinct structural optimisation and search
methods investigated in the preceding research chapters. The method type (e.g.
stochastic or deterministic, discrete or continuous material representation) is stated,
alongside the type of solution obtained (single or multiple, form of optimality). The
most appropriate design stage for application of each method is also stated, along with
requirements and practical implications for appropriate use. The research
contribution of each was stated in Chapter 2 and is briefly summarised in the table.
The wider context of these contributions, realisation of the research objectives and
validity of the central hypothesis are now considered.
There is no single method that is applicable to, let alone optimal for, all structural
design problems, since the nature of design task specifications vary greatly from one
project to another. For example, ESO is well suited to the free-form structures to
which it has been applied, detailed in section 2.2.3, but when the layout of discrete
structural members is more tightly controlled, as in the bracing design of the Pinnacle
Tower (Chapter 4), ESO is inefficient and impractical. The Genetic Programming
methodology is arguably most powerful when more concept generation and pattern
emergence is desirable and in larger scale structures. Fogel (1999) states, in a
discussion of the No Free Lunch theorem that:
"…for an algorithm to perform better than even random search (which is simply
another algorithm) it must reflect something about the structure of the problem it
faces. By consequence, it mismatches the structure of some other problem…"
Therefore a structural optimisation toolbox is required for structural topology, shape
and size optimisation, of which the methods presented in this thesis could form a part.
142
Chapter 3 demonstrated how the practicality of Evolutionary Structural Optimisation
can be improved by using appropriate design criteria, such as displacement, to govern
element removal and addition. The consideration of aesthetic issues such as
symmetry and repetition through element grouping also offers advances on previous
work. The integration of element thickness optimisation by defined groups removes
the dependence of final solutions on prescribed thickness. This method could readily
be used in structural topology optimisation problems in general, especially where
aesthetic considerations are concerned.
Chapter 4 applied Hooke and Jeeves' Pattern Search method to tackle a parameterised
topology optimisation problem, in an example of a "problem-seeks-design" scenario.
It demonstrated how existing optimisation techniques may be adapted to practical
topology problems in preliminary or detailed design. This study also showed how
size and topology optimisation can be efficiently integrated by performing a single
iteration of the optimality criteria sizing algorithm for each topology change. This
only increases the number of analyses required by a factor of two, compared to the
optimisation of topology alone. It was observed that simultaneously optimising size
and topology offered a volume reduction of more than 20% when compared to the
staged process of performing topology optimisation with maximum sections, followed
by a separate size optimisation. The use of stochastic search allowed a range of
distinct high-performance designs to be generated for appraisal by designers
considering unmodelled criteria. The Pattern Search method, with integration of the
Optimality Criteria sizing approach, has potential for any structural topology
optimisation problem that can be parameterised in some way.
Chapter 5 demonstrated the potential for generating multiple, novel, high performance
conceptual designs for a topological design problem, through the use of a Genetic
Programming method using design modification operators as internal nodes. This
approach avoids the need for a fixed set of variables and allows complex bracing
patterns to be developed from very simple blue-prints. Control over the form of the
design modification operators allows the user to influence the form of the solutions
obtained. Use of this GP method in other structural topology design problems would
be eminently possible, but dependent on the definition of an appropriate set of design
modification operators.
The central hypothesis, as stated in chapter 1, is that optimisation can be successfully
and appropriately applied in practice through consideration of industry specific issues.
143
This has been validated most directly in chapter 4, with optimised designs being used
directly in outline proposals. Relevant industry specific issues included aesthetic
requirements, desire for alternative proposals, adaptability to specification changes
and multiple loadcases. The ESO and GP methods developed in chapters 3 and 5
respectively have shown potential for successful application in practice, through the
ability to include aesthetic considerations and in the case of GP, multiple proposed
solutions, with possible decision support and interactivity.
Table 6.1: Summary of methods used in this thesis
Chapter 3: Evolutionary
Structural Optimisation
Chapter 4: Pattern Search
- Optimality Criteria
Chapter 5: Grammar-based
Genetic Programming
Method type
Deterministic, continuous material
representation
Stochastic (or deterministic),
discrete material representation,
mathematical programming (PS);
Deterministic, continuous /
discrete variables (OC)
Stochastic Evolutionary Algorithm
Tree-based representation
without encryption.
Design modification operation
functions.
Appropriate design
stage
Conceptual / Preliminary design
Preliminary / Detailed design
Conceptual / Preliminary design
Requirements
Definition of criteria for element
addition and removal
Ability to parameterise
optimisation problem
Grid-based framework
Type of solution
Single or multiple solutions from
different start points and domain
thicknesses.
Uncertainty regarding degree of
optimality
Local optima, improved by
constraint handling method (PS).
Local optima on account of
force-moment redistribution (OC).
Multiple optimally-directed
solutions.
Global optimum for small-scale
tasks.
Primary research
contributions
Practical consideration of
constraints, repetition and
symmetry
Thickness optimisation of
element groups
Efficient integration of topology
and size optimisation
Constraint handling
Successful industrial application
on live project
Use of GP to generate
"programmes" that in turn
generate structural designs.
New tool for optimally directed
and controllable bracing pattern
generation
Practical
implications
Modest practical applicability to
bracing design due to difficulty in
discrete interpretation. Large
numbers of elements required
leads to large computational time.
General method applicability to
any parameterised problem.
Constraint handling requires
problem specific modification
Example of potential for optimally-
directed, rapid design generation
and evaluation methods
Currently bracing frame specific
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6.2. RECOMMENDATIONS FOR FUTURE WORK
This section focuses on future work that would be appropriate in further developing
the methods and themes presented in this thesis.
Evolutionary Structural Optimisation
- Increase of scale: The scale and complexity of the structures considered here is
substantially less than in some studies, for example the three dimensional, multi-
storey Docklands Tower model of Holzer (2006). However, the results presented
in Chapter 3 are arguably more regular and more appropriate for discrete
interpretation and assignment of standard sections than those found elsewhere in
the literature. It would therefore be a worthwhile extension to increase the scale
of the structural models considered to gain both benefits.
- Frame design: The described integration of size optimisation into the ESO process
considered only the thickness of the two-dimensional elements in the designable
domain. A logical extension would be to also allow the section sizes of the
orthogonal framework to adapt to changes in load path in meeting the
displacement constraint. This would require the inclusion of the vertical load-
case, originally defined by Mijar et al. (1998), to size the framework.
- Inclusion of inter-storey drift constraints: Considering inter-storey drift
constraints in addition to top-storey lateral displacement will complicate the ESO
process, but avoid the possibility of the profile of the framework bowing out lower
down the structure.
- Design process integration: The integration of ESO into the overall design
process also warrants further consideration. In evolving a structure to efficiently
meet a stiffness requirement, one must consider how the solution is interpreted as
a discrete design and at what stage to consider strength requirements, including
buckling.
Pattern Search - Optimality Criteria
- Investigation of the performance compromise in hybridisation: The hybrid method
described executes an Optimality Criteria sizing algorithm for each topological
change, assuming the force-moment distribution within the structure to be
unchanged by these modifications. Performing a full sizing operation for each
topological stage is likely to yield improved designs accompanied by an order of
magnitude increase in the number of structural analyses required. A study into the
compromise made by the approximate method would be very informative.
Various possible alternative strategies exist for efficient simultaneous size and
topology optimisation, such as reapplying the OC algorithm until the changes
made are less than a defined threshold.
- Cost modelling: The possibility of developing a cost model to quantify the trade-
off between piece-count and volume minimisation was discussed with Arup
designers. This would include estimates of construction time per additional
member and associated costs, reduction in letting revenue on account of view
impingement, material costs, etc. A cost modelling approach for conceptual
design was previously adopted by Khajehpour (2001). However, it is crucial to be
aware of the liability of the cost model to change throughout the project.
- Wider method application: The "problem-seeks-solution" approach adopted for
the design task on this industrial project means that it requires validation through
use on further problems and more conventional building forms.
Genetic Programming
- More thorough testing on structures of a practical scale: The 2x6 benchmark case
used in Chapter 5 is a relatively simple problem. Further validation and possible
method development is required on design tasks of larger scale, such as the 6x30
framework also considered in Chapter 5, or the concept sketches of figure 2.4 that
were inspirational to the development of this tool.
- Strength (including buckling) considerations: In the work presented, maximum
lateral displacement is the only constraint considered. This is a simplification and
should be extended to include consideration of inter-storey drift and strength
constraints, potentially in conjunction with the introduction of section-size
optimisation.
- Section-size optimisation: Although detailed consideration of section-sizing is
often neglected in conceptual design, the integration of section sizing into the GP
algorithm is of significant research interest. This could be achieved in a similar
manner to the Pattern Search - Optimality Criteria hybrid method of chapter 4,
although topology changes in the GP process are likely to be more significant,
making the approximations less acceptable.
- Control over aesthetic style: Section 5.5 presents a framework for offering the
user control over the aesthetic style of solutions generated, by defining available
unit-types, restrictions on aspect-ratio and maximum/minimum size of bracing
units, pattern repetition characteristics, symmetry and regions of structure in
which bracing is prohibited. This approach offers the potential to incorporate
project specific requirements and alternatives.
- Graphical User Interface: For use beyond prototyping, this tool requires the
development of a graphical user interface to become a "user-seductive" program
(Cohn 1994) appropriate to mainstream structural designers.
6.3. APPLICATION OF STRUCTURAL OPTIMISATION IN
PRACTICE
This thesis has highlighted a number of issues relevant to the practical application of
structural optimisation in industry, most significantly through the involvement in the
Pinnacle Tower project of Chapter 4, as well as the discussion of drivers and barriers
in section 1.4. The following points have become apparent:
- Current commercially available optimisation software is more suited to design of
components in automotive and aerospace engineering and is regarded as a
specialist tool, rather than being accessible to the majority of structural designers.
- Aesthetic issues influence a substantial proportion of structural design decisions
and should generally be either incorporated into the optimisation model or used to
assess a range of high-performance solutions produced by the optimisation
process. These alternative solutions may be generated using a stochastic process
or through parametric variation.
- Decision support tools, potentially including optimisation methods such as the
Genetic Programming tool presented in Chapter 5, show potential to be valuable
in the early stages of structural design. Currently decisions are often based on
experience and intuition, with a limited range of alternatives considered.
- Substantial interest in the use of structural design optimisation exists within the
building industry. However, this is tempered by the barriers discussed in section
1.4. Optimisation should be an interactive process with a high level of control
offered to the user, with easy customisation and the ability to rapidly incorporate
model changes.
- Design "freezes" (Eger et al. 2005), marking the end of a development stage and
fixing some aspect of the design, occur in the building engineering industry in
much the same way as any other design process, although the architect-engineer
interaction can lead to frequent "thaws". Use of optimisation must account for
these freezes, as well as offering the versatility to adapt to frequent changes in
specification of geometry, constraints, objectives and variables.
Questions that should be addressed when considering the use of structural design
optimisation in practice, and more specifically when selecting an optimisation
method, include:
- Is the problem suitably well-defined to yield meaningful optimal solutions? What
are the design objectives, variables, constraints and parameters in the optimisation
task?
- Is it possible to gauge the size and complexity of the design space?
- How will constraints be incorporated?
- How long will each objective function evaluation or structural analysis take?
What simplifications and approximations can be made?
- For a chosen method, is it possible to predict how many function evaluations are
likely to be required for acceptable convergence? Hence is the total projected run
time realistic?
- Is the goal to find a single best solution, a range of good designs or to understand
trade-offs between multiple objectives?
- What starting point will be used? Is this point feasible according to the
constraints? Is it appropriate to use a range of starting points?
6.4. PROJECTED TRENDS IN STRUCTURAL DESIGN
AUTOMATION AND OPTIMISATION IN PRACTICE
Over 20 years ago, Templeman (1983) stated that,
"… unless researchers are prepared to step back from the research frontiers of
structural optimisation and become involved in providing practical design software
and unless practicing design organisations are prepared to step forward and sponsor
the writing of such software, both sides of the engineering profession are likely to
miss out on an exciting development within the profession…"
In the years since this observation, several commercial software packages, examples
of which can be found in chapter 2, have developed directly from academic research
and practicing design organisations have sponsored and collaborated in research
investigations, such as that in this thesis. However the full potential of structural
optimisation in design practice remains to be realised.
In the context of structural design of buildings, the potential of methods using a
continuum design domain is limited, on account of the difficulties discussed earlier in
the thesis. However, it should continue to find use in small scale, novel structures and
isolated sub-systems of larger structures as well as in providing insight into load path
distribution.
With the exception of certain ESO examples, significant use of optimisation in non-
parameterised topological design remains a long-term goal, despite the strong
incentives. However, if a small number of high-profile examples can be established,
this could serve as proof or potential and increase interest and further use.
It is likely that the use of section-size automation and optimisation will continue to
increase, in response to the demands of complex design tasks, aided by dissemination
of rigorous methods, e.g. Optimality Criteria and software, and inspired by existing
examples. Naturally, this advancement could be hastened by the inclusion of relevant
material in undergraduate structural engineering teaching to spread awareness of
methods.
Increasing use of appropriate supporting software, such as parametric CAD
modelling, and digitalisation of document interchange between architects and
engineers mean that the path has been laid for parametric shape and topology
optimisation. This could become widely practised in the near future on account of the
modest risk and high potential return.
6.5. CLOSING NOTES
In summary, it has been shown that this research has satisfied the objective of
contributing towards reducing the gap between research and industry, through the
steps made towards establishing a structural optimisation toolbox for the building
industry, as discussed in section 6.1. The thesis has demonstrated, primarily in
Chapter 4, the potential benefits of applying structural topology optimisation to "live",
full-scale projects, hence validating the central hypothesis. These include rapid
generation and optimisation of a range of solutions, customisation of methods such
that results are directly applicable and gaining understanding of feasible design
spaces. As demonstrated in chapters 3 to 5, using appropriate methods it is possible to
successfully accommodate aesthetic design considerations, amongst other practical
issues, either explicitly in an optimisation model, or by generating multiple optimally
directed designs.
151
Appendix 1: Structural analysis
Common to each of the methods presented in this thesis is the requirement for
structural analysis to evaluate the performance of a proposed design. In all cases,
finite element analysis is carried out using Oasys1 GSA (General Structural Analysis)
(Oasys 2003). In the prototype implementation of each optimisation algorithm,
structural models are automatically written as GSA text input files, to include requests
for the required results files to be written when analysis is conducted. From the
prototype programme, GSA is called, the relevant file is opened and analysed and
results files are written. These can then be read by the programme and processed as a
required. This process is illustrated in figure A1.1.
Read and process results files
Call batch file <filename.bat> file to open GSA and
execute instructions listed in GSA command file
<filename.gwc>:- open structural model <filename.gwa>
- analyse model
- write text results files <filename(x).txt>
- close <filename.gwa>
- exit GSA
Write structural model as GSA text input file,
<filename.gwa>, to include results file request details.
Figure A1.1: Structural analysis flowchart
1 Oasys Limited is the software house of Arup.
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Appendix 2: Software development and prototyping
Each of the three search methods investigated in this thesis was implemented in
prototype form in either C++ (Pattern Search with Optimality Criteria) or MATLAB
(Evolutionary Structural Optimisation and Genetic Programming). As previously
discussed, minimising the time spent on software development is crucial to the
successful application of optimisation methods in industry. Although the precise
nature of the design problem may require tailoring of algorithms and corresponding
code, it is clearly desirable to recycle and adapt existing code where possible in an
effort to drive down development time. This applies to adapting to changes in
specification within a project, as seen in Chapter 4, as well as transfer from one
project to another.
In the author's experience, MATLAB is an excellent numerical computing
environment and programming language for development of prototype tools. It is also
optimised for matrix manipulation and is easy to learn and intuitive to use. Plotting of
results and development of Graphical User Interfaces (GUIs) is very straightforward.
MATLAB is widely used within the research community. Licensing for industry is an
additional overhead, although free open-source alternatives such as SciLab2 and
Octave3 are available.
C++, Java etc. are free, although licensing may be required for Microsoft
development environments. They are potentially considerably more powerful than
MATLAB for general programming purposes, but harder to learn and more time
consuming for tool development. These programming languages are arguably better
suited to development of full-scale systems.
2 http://www.scilab.org/
3 http://www.gnu.org/software/octave/
153
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