A Comparative Study of the Isocontour- and the

Level Set Based Evolutionary Structural

Optimization Method

Alexander Verbart Guerrero

December 7, 2009

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Abstract

This thesis presents a comparative study of two Evolutionary Structural Op-timization (ESO) methods: the Isocontour ESO Method (ISOESO) and theLevel Set Based ESO Method (LS-ESO). These methods are both variantsof the classical Evolutionary Structural Optimization (ESO) method whichis based on removing gradually low-stressed material. Both variants are sim-ilar in their heuristic approach taking the von Mises stress as a local removalcriterion. However they differ in their topological description: ISOESO is anone-directional method with an explicit boundary representation; the struc-tural boundary coincides with a von Mises threshold stress contour. Mate-rial is removed by increasing this threshold value. In LS ESO the structuralboundary is represented in an implicit manner by the level set model whichis bi-directional and thus able to admit previously removed material. Thesetwo methods are compared quantitatively on different benchmark problems.

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Preface

The thesis is the final work of my MSc study Engineering Dynamics atthe Precision and Microsystems Engineering (PME) department, Facultyof 3ME, TU Delft. The main part of this work was carried out during anexchange year at the laboratory of Applied Mechanics at EAFIT Universityin Medellin, Colombia. Here I worked under the supervision of prof. ManuelGarcia.

Originally I went to EAFIT University for only one semester to do a lit-erature study on Topology Optimization, and in particular Level Set BasedOptimization. This semester was such a great experience, I ended up stayingfor more than one year, doing this final research.

Special thanks go to my supervisor prof. Manuel Garcia for his hospi-tality, the weekly discussions and the facilities I was given. Thanks to mysupervisors at my home University, Matthijs Langelaar and prof. Fred vanKeulen. Matthijs for his advise during the skype meetings concerning mywork and prof. Fred van Keulen for giving the most possible freedom ofallowing me to study abroad for such a long time. Thanks to prof. van Wo-erkom offering lots of help concerning my MSc track. Thanks to Adelaidafor revising my work. Thanks to all the fellow students and researchers atthe lab of ’Mecanica Aplicada’ who have become good friends during thisexchange year.

Thanks to my mom, brother and sister.

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Contents

Abstract iii

Preface iv

I Introduction 1

II Preliminaries 4

1 Fixed Grid Finite Element Analysis (FG FEA) 41.1 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Fixed Grid Approximation . . . . . . . . . . . . . . . . . . . . 51.3 Bilinear Quadrilateral Elements . . . . . . . . . . . . . . . . . 61.4 Element Classifications (I,O,NIO) . . . . . . . . . . . . . . . . 81.5 Stress Recovery . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Evolutionary Structural Optimization 122.1 Objective Function and Removal Criterion . . . . . . . . . . . 122.2 Evolutionary Parameters . . . . . . . . . . . . . . . . . . . . . 122.3 ESO Optimization Algorithm . . . . . . . . . . . . . . . . . . 13

3 Level Set Method 153.1 Implicit Representation of an Interface . . . . . . . . . . . . . 16

3.1.1 Implicit Function . . . . . . . . . . . . . . . . . . . . . 163.1.2 Example in 1-D. . . . . . . . . . . . . . . . . . . . . . 163.1.3 Example in 2-D. . . . . . . . . . . . . . . . . . . . . . 173.1.4 Implicit vs. Explicit Representation. . . . . . . . . . . 18

3.2 Level Set Equation . . . . . . . . . . . . . . . . . . . . . . . . 193.2.1 Hyperbolic Level Set Equation . . . . . . . . . . . . . 193.2.2 Parabolic Level Set Equation . . . . . . . . . . . . . . 20

3.3 Discretization of the Level Set Equation . . . . . . . . . . . . 213.3.1 Time Discretization . . . . . . . . . . . . . . . . . . . 213.3.2 Spatial Discretization Hyperbolic Level Set Equation . 223.3.3 Spatial Discretization Parabolic Level Set Equation . 26

3.4 Definition of the Implicit Function: Signed Distance Function 273.4.1 Reinitialization to Signed Distance . . . . . . . . . . . 30

III Methodology 32

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4 Isocontour Evolutionary Structural Optimization (ISOESO) 334.1 ISOESO Optimization Algorithm . . . . . . . . . . . . . . . . 34

5 Level Set ESO Optimization (LS-ESO) 385.1 Definition of the Velocity Field . . . . . . . . . . . . . . . . . 385.2 LS-ESO Optimization Algorithm . . . . . . . . . . . . . . . . 39

IV Results 45

6 Centrally Loaded Cantilever 476.1 Optimized Cantilever . . . . . . . . . . . . . . . . . . . . . . . 476.2 Objective function and Von Mises stress distribution . . . . . 506.3 Computational Effort . . . . . . . . . . . . . . . . . . . . . . 52

7 Michell’s Beam 537.1 Optimized Michell Beam . . . . . . . . . . . . . . . . . . . . . 537.2 Objective function and Von Mises stress distribution . . . . . 547.3 Computational Effort . . . . . . . . . . . . . . . . . . . . . . 56

8 MBB Beam 578.1 Optimized MBB Beam . . . . . . . . . . . . . . . . . . . . . . 578.2 Objective function and Von Mises stress distribution . . . . . 588.3 Computational Effort . . . . . . . . . . . . . . . . . . . . . . 60

V Discussion 61

9 Optima 619.1 Final design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619.2 Optimization history . . . . . . . . . . . . . . . . . . . . . . . 629.3 Computational Effort . . . . . . . . . . . . . . . . . . . . . . 63

10 Robustness 6510.1 Mesh Dependency . . . . . . . . . . . . . . . . . . . . . . . . 6510.2 Influence Evolutionary Parameters . . . . . . . . . . . . . . . 66

11 Smooth Boundary Representation 6811.1 ISOESO: Invalid NIO Elements . . . . . . . . . . . . . . . . . 6811.2 LS-ESO: Conserving a Smooth Boundary . . . . . . . . . . . 68

12 A critical view on the ESO algorithm 7212.1 Proposal: An Improved ISOESO algorithm . . . . . . . . . . 7312.2 Proposal: An Adaptive Level Set ESO Algorithm . . . . . . . 75

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VI Conclusion and Recommendations 76

13 Conclusions 77

14 Recommendations for Further Research 78

A Principle of Virtual Work for Solid Body Mechanics 85A.1 Internal Equilibrium . . . . . . . . . . . . . . . . . . . . . . . 85A.2 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . 86A.3 Boundary Value Problem for an Elastic Body . . . . . . . . . 87A.4 Weak formulation . . . . . . . . . . . . . . . . . . . . . . . . . 88

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Part I

Introduction

In structural design often there exists a conflict of interest in the designphase. A typical problem is stiffness versus weight. For example is a bridgedesign for which material reduction is of interest for cost reduction. Anotherexample is weight reduction in aviation to reduce fuel consumption. In bothexamples the structural performance has to be maintained, i.e. conflictinginterests.

These problems can be tackled by Structural Optimization methods.Note that a structural domain contains information about three geometricalproperties: size, shape and topology. The topology of a structure representsthe connection among parts within an object, i.e. the number of holes of anobject determines its topology. In Structural Optimization the main goal isto find the optimal structural design by changing any of these three proper-ties. Therefore, the field of Structural Optimization can be subdivided intothe following group of methods which are based on each of theses geometricproperties:

• Sizing optimization: Find an optimum design by taking its size asdesign variables (shape and topology remain fixed).

• Shape optimization: Find an optimum design by taking the shape ofthe domain itself as a design variable (topology remain fixed).

• Topology optimization: Find the optimum distribution of materialinside a general domain independent of a predetermined shape (size,shape and final topology of the structure are unknown).

In this comparative study the focus is on Topology Optimization meth-ods. In Figure 1 an practical example is given of a buggy frame optimizedby Topology Optimization.

Figure 1: Example of Topology Optimization in automotive applications:(a) buggy, (b) optimized frame

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In Figure 2 an overview is given of the field of topology optimization. Adistinction is made between the methods of design parameterization and thesolution methods. The first group describes the topology of the structure.The second group are methods used to solve the optimization problem af-ter parameterizing the design domain. Furthermore, the solution methodsgroup can be subdivided in two different groups of solution methods whichare based either on heuristics or gradient information.

Figure 2: Field of Topology Optimization

The homogenization method, introduced by Bendsøe and Kikuchi, [4] ini-tiated the field of topology optimization. In this method anisotropic materialwith an infinite amount of micro-scale voids is used to represent void/full ma-terial. Under the applied loads and restricitions the optimal design is foundby optimizing the composition of the composite material to obtain maxi-mum stiffness. Later, Bendsøe and Sigmund introduced the Solid IsotropicMaterial with Penalization (SIMP) Method [6] in which isotropic materialis used and intermediate density values are penalized by the SIMP model toobtain the desired 0-1 design. This SIMP model describes a relation betweenthe density and the stiffness tensor which makes intermediate densities lessattractive since they have a relatively low stiffness. The homogenizationmethod and SIMP are both density-based approaches in which the topol-ogy of the design is represented by the densities in the elements. Bothmethods provides promising results and can deal with multiple objectives.However, problems occur with grey elements (dependent on the power law)and checkerboards. Furthermore the boundary displayed are jagged edgesand thus non-smooth.

A different approach in Topology Optimization, introduced by Xie andSteven [35], is known as Evolutionary Structural Optimization (ESO) whichis based on removing gradually low stress material (taking the Von Misesstress or strain energy as local removal criterion). ESO is based on heuristic

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since no relationship is proven between the objective function (Compliance-Volume product) and the removal criterion. The topology of the structure isdescribed by a finite element model from which elements are removed undera certain threshold value (hard kill). The advantages of ESO are its simpleapproach and its easy implementation in FEA packages. Drawbacks are itslack of theoretical basis, limited to single load cases and like in SIMP: jaggededges and checkerboard patterns are present.

A recent development in the field of topology optimization is the use ofa level set model to describe the topology of a structure and its evolution.In this approach the structural boundary is represented by the zero levelset of an implicit function describing the whole structural domain. The ad-vantages of this approach is that the boundary representation is smooth.This avoid checkerboard problems and jagged edges. So far this method ismainly used in combination with gradient based solution methods in whichthe shape gradient and/or topological gradient are used.

The aim of this research is to compare two evolutionary type methods:

• Isocontour Evolutionary Structural Optimization (ISOESO) [9]

• Level Set Evolutionary Structural Optimization method (LS-ESO)

These methods are both variants of the classical ESO method. Bothmethods are similar in their heuristic approach removing gradually lowstressed material taking the Von Mises stress as the local removal criterion.However they differ in their boundary representation.

ISOESO, an one-directional method in which the low stress material issimply cut off along the Von Mises iso-stress contours. The topology of thestructure is described by the Fixed Grid Finite Element Analysis [7] (FGFEA) using boundary elements for a smooth boundary representation. LS-ESO is bi-directional method based on Sethian and Wiegmann’s method [28]in which the structural boundary is represented in an implicit way by thelevel set model. These two methods are compared quantitatively on differ-ent benchmark problems by their final designs, the objective function valuefor these final designs (Compliance-Volume product), the iteration history,computational effort and robustness.

This thesis is built up as follows: Part II are preliminaries in which thefundamentals for the optimization algorithms are discussed. Such conceptsare: fixed grid FEA, classic Evolutionary Structural Optimization and thelevel set model. In Part III the two optimization algorithms (ISOESO andLS-ESO) are discussed in detail. In Part IV the results of optimizing threedifferent standard problems are listed and in Part V these results are dis-cussed. Finally in Part VI conclusions and recommendations are made.

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Part II

Preliminaries

In this Part concepts are discussed which are used in both optimizationmethods. In Section 1 the fixed grid finite element method is used with itsboundary elements. In Section 2, the method used is the classic EvolutionaryStructural Optimization method. Finally, in Section 3 the general level setmodel is discussed for a smooth boundary description.

1 Fixed Grid Finite Element Analysis (FG FEA)

FG FEA has been used in problems in which the geometry and/or phys-ical properties of an object change in time. In elasticity problems it hasbeen used to develop micro-structural models for studying the behavior ofcomposite materials.

FG FEA for elasticity problems was introduced by Garcia and Stevenin [7]. In the fixed grid approach a rectangular grid with equally sized ele-ments is superimposed over the structural domain instead of a finite elementmesh that fits the structural geometry. The structural geometry is definedby identifying the individual elements in three groups: Outside (O), Inside(I) and Neither Inside Nor Outside (NIO) as depicted in Figure 3.

Figure 3: Different type of elements in fixed grid finite element analysis:Inside elements (I), Outside Element (O) and Neither Inside or Outsideelements (NIO) (Figure taken from [8])

For a change in geometry (e.g. optimization step) the stiffness matrix isupdated by the elements that changed from group. This saves computationaltime in comparison with classic FEA in which a new stiffness matrix isconstructed (see Figure 4).

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A disadvantage of the FG FEA approach is a loss of accuracy for theFEA element approximation. Its main advantages are simplicity and speed.Due to its ability to handle changing geometries it is attractive for the earlydesign stages, interactive design and Structural Optimization.

Figure 4: Comparative graph of FG FEA with classical FEA (Figure takenfrom [8])

In this thesis the FG FEA is used for the finite element step in thedifferent optimization algorithms.

1.1 Discretization

The finite element equation derived in Appendix A is discretized whichresults in the linear system of equations:

Ku = f (1.1)

Where K denotes the global stiffness matrix, u denotes the displacementvector and f denotes the load vector.

1.2 Fixed Grid Approximation

In the FG FEA a structured mesh is superimposed over the structural do-main as depicted in Figure 5. This is different from classical FEA in which

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an unstructured mesh fits the structural domain. The fixed grid domainencloses the structural domain. The material properties depend on how anelement is classified as Inside (I), Outside or Neither Inside nor Outside(NIO).

In this Section the FG FEA for this research is discussed: the finiteelements used, the element classification (I,O,NIO) and stress recovery.

Figure 5: Fixed grid mesh: the different element types are indicated in thestructure. The NIO elements are the elements crossed by the structuralboundary (Figure taken from [8])

1.3 Bilinear Quadrilateral Elements

For the FG FEA bilinear quadrilateral elements are used as depicted inFigure 6. Shape functions are only used for the displacement interpolation.

Where ξ and eta are natural coordinates.

The displacement field over the element is approximated by:

u(x) =4∑i=1

Niui (1.2)

Where ui are the nodal displacements and Ni are the bilinear shape func-tions. The bilinear interpolation is defined as:

u = a+ bξ + cη + dξη (1.3)

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Figure 6: Quadrilateral element

The nodal values are given by:

u = u1 at ξ, η = −1,−1u = u2 at ξ, η = 1,−1u = u3 at ξ, η = 1, 1u = u4 at ξ, η = −1, 1 (1.4)

Substituting this into Equation (1.3) and solving for a, b, c and d gives:

a =u1 + u2 + u3 + u4

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b = −u1 − u2 − u3 + u4

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c = −u1 + u2 − u3 − u4

4

d =u1 − u2 + u3 − u4

4(1.5)

Substituting into Equation (1.2) and write this in terms of the nodal valuesgive the bilinear shape functions as:

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N1 =14

(1− ξ)(1− η)

N2 =14

(1 + ξ)(1− η)

N3 =14

(1 + ξ)(1 + η)

N4 =14

(1− ξ)(1 + η)

(1.6)

for −1 ≤ ξ ≤ 1 and −1 ≤ η ≤ 1These shape function are bilinear. The ξη term is linear in η for ξ =constantand vice versa.

1.4 Element Classifications (I,O,NIO)

In this Section the different element classifications are discussed and thederivation of the corresponding material properties.

Inside (I) and Outside (O) Elements. For I and O elements the entireelement lies respectively inside and outside the structural domain. Insideelements are the most simple elements, which have the material propertiesof the material of the structure. For the outside materials the materialproperties are chosen as of non-interactive material. In fact the materialproperties are zero but this would lead to singularity of the stiffness matrix.Therefore the Young’s modulus is chosen relatively small to the young’smodulus of the structural material/Inside elements: EO

EI<< 1.

In this thesis this ratio is chosen as: EOEI

= 10−6.

Neither Inside nor Outside (NIO) Elements. NIO elements are infact boundary elements and in contrary to the the other element types thematerial properties are not constant over the element. There are differentmethods to approximate the material properties over the boundary elements:discrete approach versus weighted average.

Discrete approach. Initially the NIO elements in FG FEA were approx-imated by a discrete approach. The elements were classified either as I oras O depending which volume fraction was greater. If more than 50% of theelement lies inside the structural domain than the element is classified asI-element and vice versa. This method was used for in the classical Evolu-tionary Structural Optimization method discussed in Section 2.

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Weighted average. Another way of approximating NIO elements is todescribe it as a continuous homogenous (grey) element with material prop-erties which depend on the ratio α which is the volume fraction of materialin the NIO element. The material properties are then taken as a weightedaverage of the material properties of inside and outside material. In the com-parative study in thesis, the contribution of these two materials is chosen tobe directly proportional to their volume fraction.

KNIO = αKI + (1− α)KO

Where α =AI

ATOT(1.7)

For example, consider the NIO element in Figure 7 which is a piecewiselinear approximation of the real NIO element. Once the polygon 1-2-3-4-5,which encloses the inside material, is determined the volume fraction can bedetermined.

Figure 7: NIO element

The volume fraction of inside material is determined by summing thevector cross product for all sides of the polygon.

AI =12

n∑i=1

vi × vi+1 and vn = v1 (1.8)

Where vi is the vector location of point i and n is the number of pointsenclosing the polygon.

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1.5 Stress Recovery

The stress field is obtained from the displacement field which is the primerquantity. The stress is thus a derived quantity which is less accurate. Differ-ent methods exist for deriving this stress field and are called Stress recoverymethods:

• Averaging.Note that the stresses can be computed over the individual elementsusing the displacement field. However, the stresses from element toelement sharing a nodal point will not be the same since the displace-ment field is not C1 continuous between elements. A solution is tocompute the stress on the element nodes and then compute the stresson the nodal points by averaging.

The stresses on the element nodes can be derived in different ways:

– Directly on the element nodes.

– In the Gauss points and extrapolating to the element nodes usingbilinear shape functions. However, in this case for rectangular el-ement this gives the same results as computing the stress directlyon the element nodes.

Once the stress is known on the element nodes the stress on the nodalpoints can be determined by averaging the stress of adjacent elementssharing a node. This can by weighted and unweighted averaging:

– Unweighted average: the same weight is assigned to all elementssharing a nodal point.

– Weighted average: different weights are assigned to the elementssharing a nodal point, this weight can be a function of the stressvalue, geometry of the element or element type.

Different stress recovery methods can be made by combining thesemethods.

• Superconvergent Patch Recovery Method.A different and more accurate approach came from by Zienkiwicz [36]and is known as the Superconvergent Patch Recovery Method. Inthis method the stress values in a nodal point is calculated by a leastsquares approximation using the Gauss points of the element patch.The element patch is an union of of elements surrounding the nodalpoint under consideration. In Figure 8 such a patch is depicted. Thecentral node is determined by a least squares fit using the Gauss pointsof the surrounding patch. The number of elements in a patch dependon the desired order of the least squares approximation.

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Figure 8: Patch for quadrilateral bilinear element. (Figure taken from [36])

In the phd thesis of M.J. Garcia [8] a test was performed on an infiniteplate and the stress errors for the weighted average and this superconvergentpatch method were compared. It was shown that the average and maximumstress error both decreased significantly by using the superconvergent patchmethod. For a mesh of 2000 elements the average stress dropped down from2.5% to 1% and the maximum error dropped down from 16% to less than10%. Furthermore, the stress error shows no oscillatory behavior.

However, for reasons of simplicity in this thesis the stress is computeddirectly into the element nodes and the nodal point stress is computed by aweighted average as in Equation (1.9):

σwa =∑n

i=1 αkσk∑ni=1 αk

(1.9)

Where α =AreaIAreaelem

is the volume fraction of inside material.

A weighted approach is used here since the elements in FG FEA are con-sidere homogeneous for which the material properties are taken proportionalto the area of inside material.

Note: The Superconvergence Patch Recovery Method is not used here butits implementation in the methods used in this thesis is most likely to giveimproved results since the stress recovery is not only used for post-processingpurposes but the stress is actually the local criterion on which material isremoved. Therefore an improved accuracy is expected to give improvedresults.

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2 Evolutionary Structural Optimization

Evolutionary Structural Optimization (ESO) is a Topology Optimizationmethod introduced by Xie and Stephen in [35]. ESO is an intuitive opti-mization method based on gradually removing low stressed material until apredetermined target volume is reached. The material is removed using theVon Mises stress criterion. In the classic approach entire elements with aVon Mises stress below a certain threshold value σth are removed. Further-more, the boundary elements in the FG FEA were determined by a discreteapproach as discussed in Section 1.4. The NIO elements are classified asInside or Outside depending on the volume fraction of inside material. Thenew Von Mises stress for the changed structure is then determined by per-forming a FG FEA step and the process is repeated until the target volumeis achieved.

2.1 Objective Function and Removal Criterion

The objective function used in ESO methods is the Compliance-Volumeproduct. ESO is assumed to minimize this product. In general the localremoval criteria used in ESO methods is the Von Mises stress. The VonMises stress is a scalar stress value which can be derived from the symmetricstress tensor which is related to the strain tensor by Hooke’s law for linearelastic material (considered in this research). The Von Mises stress givesvaluable information about the state of a structure since the Von MisesYield Criterion indicates the critical value for the Von Mises stress whenyielding begins. The Von Mises stress is given by the following relation:

σvm =1√2

√(σ1 − σ2)2 + (σ2 − σ3)2 + (σ3 − σ1)2 (2.1)

Where σ1, σ2 and σ3 are the principal stresses.

Note: There is no proven relationship between the Von Mises stress asremoval criteria and the CV-product as objective function. This is a weak-ness of this method.

2.2 Evolutionary Parameters

The so-called evolutionary parameters are three parameters which determinethe performance of the optimization algorithm since they determine theamount of material to be removed per iteration step. These parametersRR,4RR and 4V have the following purposes:

• RR⇒ A removal rate which determines the threshold value as a frac-tion of the maximum Von Mises stress: σth = RR× σth

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• 4RR⇒ A redistribution factor by which the removal rate is increasedwhen there is not sufficient removal of material by the current σth. Thisminimal amount of material is the third evolutionary parameter:

• 4V ⇒ The minimum amount of material to be removed per iteration.Its minimum value is that of the size of one element.

2.3 ESO Optimization Algorithm

The actual optimization algorithm is depicted in the flow chart in Figure 9.It can be seen that the optimization process is divided in two parts: initial-ization (blue dotted line) and the optimization loop (red dotted line).

Figure 9: Flow chart for Evolutionary Structural Optimization (ESO) algo-rithm

During initialization the fixed grid is constructed, the load and boundaryconditions are applied and the first FG FEA is preformed to determine theinitial stress distribution. The evolutionary parameters are given as inputby the user.

The red dotted line encloses the optimization loop. In this loop, materialis removed by rejecting entire elements with a stress below σth. Then, a newFG FEA is performed and a new threshold value σth is determined. Thisloop is repeated until the predefined target volume is reached.

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Example: Cantilever Beam Optimized by ESO. In Figure 10 a can-tilever beam optimized by ESO is depicted. This is an early result fromthe mid-nineties by the founders of ESO, Xie and Steven in [35]. In thisFigure the appearance of checkerboards in encircled in red. This is one ofthe problems that arises using this method.

Figure 10: Optimized cantilever beam by ’Classical’ ESO. Checkerboards areencircled in red

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3 Level Set Method

The Level Set Method is a mathematical method for describing the evolutionof an interface in an implicit manner, developed by Sethian and Osher in [18].The evolution of an interface in n dimensions is described by the isocontourof an implicit function in (n+ 1) dimensions. An example of such a level setmodel is depicted in Figure 11.

Figure 11: Level set model (Figure taken from [34])

In this Figure the two circles which represent the interface are describedby the zero level set of the implicit function φ. Furthermore, the inside areais then described by the domain in which the implicit function φ takes apositive value. The outside area is described by the domain for which φtakes a negative value, i.e. the structural domain can be described as:

φ(x) < 0 ∀ x ∈ D\Ωφ(x) > 0 ∀ x ∈ Ωφ(x) = 0 ∀ x ∈ ∂Ω

The main advantages of this interface representation is that it gives a smoothrepresentation of the boundary which easily handles merging/breaking andlarge deformations. In Figure 12 is depicted how the evolution of the implicitfunction results in a fluent change in topology.

Figure 12: Evolution of interface by the level set model. (Figure takenfrom [27])

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This property makes it also a useful tool for describing the interface inTopology Optimization. In this Part the general Level Set Method is dis-cussed developed by Osher and Sethian [18]. In Section 3.1 the implicitrepresentation of an interface is discussed. In Section 3.1 the implicit repre-sentation of interfaces is considered. In Section 3.2 dynamics is add to thisimplicit representation by the Level Set equation. Furthermore, differentrepresentations of the Level Set equation are distinguished depending onthe definition of the velocity field. Finally, in Section 3.4 the signed distancefunction, which could be used to describe an interface and its evolution,is discussed. The signed distance function is an important tool in solvingthe Level Set equations numerically, since it was shown by Mulder et al.that this gives less numerical errors than initialize φ as a Heaviside func-tion [15], furthermore it simplifies the parabolic level set equation into theheat equation.

3.1 Implicit Representation of an Interface

The key-idea of the level set method is to describe an interface using animplicit function. In this Section some examples of an implicit representationin several dimensions are given. Furthermore, it will be compared with theexplicit representation of an interface.

3.1.1 Implicit Function

A traditional technique for representing the evolution of an interface is todiscretize first the interface into a finite set of points. The evolution of thisinterface is then solved numerically by discretizing this parameterization andsolving the equations of motion in a Lagrangian way.

Instead of this explicit representation it is also possible to use an implicitrepresentation. Note that an interface in n dimensions could be describedimplicitly by the isocontour of a function in (n + 1) dimensions. This isthe basic concept from which the level set method departs. Following, someexamples in 1-D and 2-D of such an implicit representation.

3.1.2 Example in 1-D.

Consider the 1-Dimensional region Ω in Figure 13, defined by the domainx = [−1, 1]. The two points x = −1 and x = 1 represent then the zero-dimensional interface ∂Ω of this domain. Note that if the inside/outsideregions are n-dimensional then the interface is (n− 1)-dimensional. An ex-plicit way to describe this interface is ∂Ω = −1, 1.An implicit representation of ∂Ω = −1, 1 could be for example: the zeroisocontour φ(x) = 0 of the function φ(x) = x2 − 1. One can see that theisocontour is an implicit representation of ∂Ω = −1, 1 since this is thesolution for x2 − 1 = 0 (see Figure 14).

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Figure 13: Interior region Ω = [−1, 1] with interface ∂Ω = −1, 1

Figure 14: Implicit representation of ∂Ω

Note that an infinite amount of implicit functions can be used. Further-more, the implicit function is defined such that φ = 0 is the isocontour thatdescribes ∂Ω, this is done for reasons of simplicity. In that case the isocon-tours that lie outside are of opposite sign of isocontours which lie inside. Tochoose φ(x) for which φ(x) = 0 describes ∂Ω is done in the following way:consider the arbitrary function φ for which φ = a represents the isocontourthat describes ∂Ω. Then one can define φ(x) = ˆφ(x)− a such that φ(x) = 0represents ∂Ω.

3.1.3 Example in 2-D.

Next the interface ∂Ω is considered of a circle with radius r = 1 (see Fig-ure 15). Now let’s define the function φ(x) = x2 + y2 − 1 in R2. One cansee that this interface can be described in an implicit way by the isocontourφ(x) = 0.In R2 one can see clearly that ∂Ω is closed and that this interface separatesan interior region from an exterior region. Furthermore, note that by the

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Figure 15: Implicit representation of ∂Ω

definition φ(x), the interior and exterior region are described respectivelyby φ(x) < 0 and φ(x) > 0.

3.1.4 Implicit vs. Explicit Representation.

For these examples ∂Ω could be described easily in an explicit way like:∂Ω = x s.t. |x| = 1. But for general shapes of the interface, problemscould arise by parameterizing the interface ∂Ω, since the interface is requiredto be a closed curve and one has to find a parameterization x(s) where s isin the domain [s0, sf ] for which x(s0) = x(sf ). Such a parameterization isin general difficult to find. An advantage of the implicit representation isthat the interface is not parameterized. This is also a big advantage of theLevel Set Method where dynamics is add to an implicit representation of theinterface and merging/splitting and large displacements are easily handledwithout the need of reparameterization

Another advantage of the implicit representation is that one avoids prob-lems of connectivity. In R3 connectivity is difficult to determine if the ana-lytical representation of the boundary is not known. For dynamic interfacesit is still more difficult due to merging and breaking. An important prop-erty of the implicit representation is that one does not need to determineconnectivity (an uniform grid can be used).

A drawback of the implicit representation is that more nodal points arenecessary for discretization since discretized region is of one order higher.In the explicit case x(s) is discretized in R into a finite set of points. Inthe implicit case ∂Ω is discretized in R2, which is unbounded. However,in

18

most cases the only interest is the motion of the interface. However thediscretized region can be reduced by using a narrow band method describedin the textbook of Sethian [27].

3.2 Level Set Equation

The next step is to add dynamics to this implicit function to describe theevolution of this interface.

Suppose φ(x, t) = 0 is the isocontour describing the interface ∂Ω. Theevolution of this interface in time is then described by taking the time deriva-tive of φ(x, t) = 0 and using the chain rule, this gives:

∂φ(x, t)∂t

+ V · ∇φ(x, t) = 0 (where V =dxdt

) (3.1)

Equation (3.1) is known as the Level Set equation, which is an Eulerian wayto describe the evolution of the interface. As discussed before, there areseveral advantages of this Level Set representation:

• The interface represented by the zero isocontour could handle topo-logical changes easily without the need of reparameterization.

• Intrinsic geometric properties of the interface could be determinedfrom the Level Set Equation. The normal unit vector to the interfaceis given by N = ∇φ

|∇φ| and the curvature of the level sets are then given

by κ = ∇ · ∇φ|∇φ| .

• This representation remains unchanged for higher dimensions.

The evolution of the interface depends on the definition of V. In thissection two different forms of the level set equation are discussed in R2: thehyperbolic- and the parabolic level set equation.

3.2.1 Hyperbolic Level Set Equation

In case the interface motion is driven by a random external velocity fieldV = 〈u, v〉 which can come from different sources. For example, when φ = 0is the interface between two fluid the interface velocity is derived by thetwo-phase Navier Stokes equations. Another example is the velocity profileused in this thesis where it is related to the Von Mises stress criterion. Inboth cases the velocity has an external source i.e. it does not depend on theimplicit function itself, the level set equation is then given by:

φt + uφx + vφy + wφz = 0 (3.2)

(where φt = ∂φ∂t , φx = ∂φ

∂u , φy = ∂φ∂v )

19

This equation is hyperbolic and therefore velocity field determines the di-rection in which the quantity φ propagates. This form (3.2) of the Level Setequation is a Hamilton-Jacobi equation, since HJ-equations are of the form:

φt +H(φx, φy) = 0 (3.3)

So that one use schemes for solving HJ-Equations to solve this hyperbolicLevel Set Equation (3.2). For the time discretization part one could useForward Euler and for the spatial discretization one uses upwinding meth-ods since it is hyperbolic. In these upwinding methods numerical stabilityis achieved by satisfying the Courant-Friedrichs-Lewy condition for a hyper-bolic PDE:

4tmax

u

4x+

v

4y

= α (3.4)

where 0 < α < 1.

This time step restriction comes from the fact that the numerical wavesshould propagate at least as fast as the physical waves, e.g. to enforcestability in x-direction, u should be smaller then 4x4t .

3.2.2 Parabolic Level Set Equation

In this Section the parabolic level set equation is considered. This is thecase when a self-generated velocity field is used which depends on the meancurvature κ. First, the level set equation is rewritten for reasons of simplicity.Note that the velocity field can be defined as V = VnN + VtT, where thefirst term on the right hand side is the velocity in normal direction and thesecond term is the velocity in tangential direction. Substituting this intothe Level Set Equation (3.1) gives:

φt + (VnN + VtT) · ∇φ = 0 (3.5)

This equation can be simplified by noting that the velocity term in tangentialdirection is cancelled out by the inner product and that N ·∇φ = ∇φ

|∇φ| ·∇φ =|∇φ|, this gives:

φt + Vn|∇φ| = 0 (3.6)

Now suppose the velocity is defined as a function of the mean curvature:

Vn = −bκ where κ = ∇ · ∇φ|∇φ|

(3.7)

where b is a constant which is defined as positive to have the interfacemoving in the direction of concavity so that circles shrink to a single point.In Figure 16 the evolution of this interface is depicted.

20

Substituting the internal velocity field (3.7) into the Level Set Equa-tion (3.6) gives:

φt = bκ|∇φ| (3.8)

This is a parabolic equation which cannot be solved by upwinding methodslike the hyperbolic level set equation since information is not propagatingin one direction but coming from all spatial directions. This equation canbe solved using central differencing. In the next Section is discussed how tosolve these different level set equations numerically.

Figure 16: Evolution interface for internal velocity field proportional to meancurvature

3.3 Discretization of the Level Set Equation

In this Section the numerical solution of these different forms of the level setequations is discussed. The hyperbolic level set equation can be solved byupwinding schemes for the spatial discretization. For the parabolic level setequation it central differencing is used instead. First the time discretizationwill be discussed briefly.

3.3.1 Time Discretization

For the time discretization one could use different methods from low orderto high order. However, in [19] was shown that Forward Euler in generalgives a good approximation and that upgrading the order of the time dis-cretization scheme give not significantly better results. For this reason inthis research only forward Euler is used. The discretized version of Equa-tion (3.2) becomes:

φk+1 − φk

4t+ ukφkx = 0 (3.9)

where,

21

φk = φ(tk)φk+1 = φ(tk+1)with tk+1 = tk +4tk = current time step

The Forward Euler method is a first order method so that the error is oforder O(4t). In case one needs a more accurate time discretization scheme,one could choose a higher order method like: Total Variation DiminishingRunge-Kutta (TVD RK).

3.3.2 Spatial Discretization Hyperbolic Level Set Equation

In this Section the spatial discretization of the Hyperbolic Level Set Equa-tion (3.2) is discussed. For the spatial discretization HJ ENO upwindingmethods are used. For upwinding methods the type of spatial discretizationdepends on the direction the information is flowing to. This depends onthe sign of the velocity value at each grid point. Shown is how to discretizethe second term of (3.9) unφnx at an arbitrary grid point xi by upwindingmethods. For simplicity reasons the level set equation is considered in onespatial dimension.

First-order Upwinding. To approximate the spatial derivative φx at anarbitrary point xi in the domain one first determines the sign of ui. If ui > 0the values of φ are moving from left to right. So that, to determine (φx)i,one has to use information which lies in the region to the left of xi. Sincethis region contains the information that determines the value of φ later intime. So that for ui > 0, one has to use thus backward differencing whichis notated as (φ−x )i. So that φ−x is defined as:

(φ−x )i =φi − φi−1

4x(3.10)

where 4x is the space between two grid points.

Conversely, if ui < 0 the information is flowing from right to left and φxshould be approximated by (φ+

x )i, which is forward differencing by notationand defined as:

(φ+x )i =

φi+1 − φi4x

(3.11)

Equations (3.10) and (3.11) are first-order approximations of φx. Next, highorder methods are discussed.

22

HJ ENO. A higher order approximation for φ−x and φ+x can be achieved

by a Hamilton Jacobi-equation Essentially Nonoscillatory Polynomial Inter-polation method (HJ ENO). The ENO method was originally introduced forconservation laws by Harten et al. in [12] and was extended to HJ-Equations.In this method the smoothest possible Newton polynomial is reconstructedby divided differences. The implicit function φ is then approximated bythis polynomial which can be differentiated to determine the numerical fluxfunction φx. The polynomial used is a Newton Polynomial of the followingform:

φ(x) = Q0(x) +Q1(x) +Q2(x) +Q3(x) (3.12)

Where the terms on the right hand side Q1 to Q3 are determined by respec-tively the zeroth order to third order divided difference.

Differentiating this polynomial (3.12) to x will give an approximation forthe spatial derivative φx. Evaluation at xi gives the approximation φ−x orφ+x (depending on the sign of ui). Equation (3.12) is called a HJ ENO3

approximation since it gives a third order approximation. Next, for reasonsof clarity an example is given in which such a polynomial is constructed byHJ ENO3.

Example: Constructing Polynomial by Divided Differences. Inthis example a polynomial is constructed for the case that ui < 0. Thefollowing notation is used: Dn

i where n indicates that this is the n − thdivided difference of φ, i indicates that this divided difference is evaluatedat xi. The divided differences are then given as:

• 0th-order divided difference:

The 0th-order divided difference is simply the value of φ in xi:

D0i = φi (3.13)

This gives:

Q0(x) = D0i

Which is the 0th-order term of the Newton polynomial.

• 1st-order divided difference:

The 1st-order divided difference is:

D1i+ 1

2

=D0i+1 −D0

i

4x=φi+1 − φi4x

(3.14)

23

Note that this is the same as the 1st-order forward differencing. Thisgives:

Q1(x) = D1i+ 1

2

(x− xi)

Which is the 1st-order term of the Newton polynomial.

• 2nd-order divided difference:

For the 2nd-order divided differences exist two possible options:

D2i =

D1i+ 1

2

−D1i− 1

2

4xor D2

i =D1i+ 3

2

−D1i+ 1

2

4x(3.15)

A polynomial as smooth as possible is desired since steep gradientscan lead to large numerical errors. Therefore, |D2

i | is compared with|D2

i+1| and smallest value is chosen. One departs from this point todetermine the 3rd-order divided difference.

By assumption |D2i+1| is taken as the smallest. This gives:

Q2(x) = D2i+1(x− xi)(x− xi+1)

Which is the 2nd-order Newton polynomial.

• 3rd-order divided difference:

For the 3rd-order divided difference, again there exists two options:

D3i+ 1

2

=D2i+1 −D2

i

4xor D3

i+ 32

=D2i+2 −D2

i+1

4x(3.16)

From |D3i+ 1

2

| and |D3i+ 3

2

| the divided difference with the smallest norm

should be picked to get the smoothest polynomial. Assume now |D3i+ 3

2

|is the smallest. This gives:

Q3(x) = D3i+ 3

2

(x− xi)(x− xi+1)(x− xi+2)

Which is the 3rd-order Newton polynomial.

HJ ENO3 polynomialWith these divided differences (3.13)-(3.16) the Newton polynomial is then:

φ(x) = Q0(x) +Q1(x) +Q2(x) +Q3(x)

24

Where

Q0(x) = D0i

Q1(x) = D1i+ 1

2

(x− xi)

Q2(x) = D2i+1(x− xi)(x− xi+1)

Q3(x) = D3i+ 3

2

(x− xi)(x− xi+1)(x− xi+2)

(3.17)

Differentiating this polynomial to x and evaluating in xi gives the 3rd-orderaccurate approximation for φ+

x .

Figure 17: Divided difference scheme for upwinding when ui < 0

Note that for the 2nd and 3rd-order divided difference exist two possiblepaths. These paths can be visualized in a scheme (see Figure 17) whereinterconnection points indicate the possible polynomials for φ+

x (for ui < 0).Since upwinding is considered, one is restricted by the sign of ui, so that inthis case (ui < 0) the three possible paths lay in the region to the right of xi.

Discretized Form Hyperbolic Level Set equation. Combining For-ward Euler for the time discretization Part and upwinding for the spatial dis-cretization gives then the discretized version of the Level Set Equation (3.9)as:

φn+1i − φni4t

+ max(u, 0)∇φ+i + min(u, 0)∇φ−i = 0 (3.18)

One can see that the direction of discretizing the right hand side of thisequation depends on the sign of u.

25

∇φ+i =

√max(φ−x , 0)2 + min(φ+

x , 0)2 + max(φ−y , 0)2 + min(φ+y , 0)2

∇φ−i =√

min(φ−x , 0)2 + max(φ+x , 0)2 + min(φ−y , 0)2 + max(φ+

y , 0)2

This way of describing the spatial discretization is done since the velocityis a local description and for that reason backward or forward differencingcannot be used over the whole domain.

3.3.3 Spatial Discretization Parabolic Level Set Equation

In this case two spatial dimensions x and y are assumed. The parabolic levelset equation Equation (3.8) can be written in terms of the first and secondorder derivatives as:

φt = bκ|∇φ| (3.19)= b(φ2

xφyy + φ2yφxx − 2φxφyφxy)/|∇φ|2

To discretize the right hand side of the parabolic Level Set equationEquation (3.8) one uses central differencing since the information comesfrom all spatial directions. The approximations for φx and φxx (= ∂2φ

∂x2 ) arenow defined as:

(φx)i ≈φi+1 − φi−1

24x; (φxx)i ≈

φi+1 − 2φ+ φi−1

4x2(3.20)

The other Partial derivatives (φy, φz, φyy) are derived in the same manner.The CFL stability criterion in this case (using forward Euler) becomes:

4t 2b4x2

+2b4y2 = α (3.21)

where 0 < α < 1.Thus 4t should be of order O((4x)2). This is a more restringent con-

dition than the stability condition in the case of an external velocity fieldwhere the time step is of order 4t = O(4x) (see Equation (3.4)).

Note: One could avoid this stringent time step restriction using BackwardEuler for the time discretization part since this implicit method is stable forany time step. The time step is then chosen for accuracy reasons only. How-ever in this thesis in all cases Forward Euler is used since for the moderategrids used increased computational time by the CFL conditions was not anissue.

26

3.4 Definition of the Implicit Function: Signed Distance Func-tion

It was seen that an interface could be described by the implicit functionφ and that the evolution of this interface is then described by the LevelSet equation, in which the velocity field(s) applicable to the problem areplugged-in. Note that when in the the level set method the shape of theimplicit function off the boundary has no physical representation. However,for accuracy reasons when taking the spatial gradient, smoothness of φ isdesired.

A question arises: how to choose a proper function representing theinterface? Unfortunately there is no closed answer to this question. Thereexist an infinite amount of functions that could represent a given interface.But due to the desired smoothness of the implicit function it is commonto use a signed distance function since it gives less numerical error in thespatial discretization as was shown by Mulder et al. [15]. Another benefit ofusing signed distance is the simplification of the parabolic level set equation.Assuming signed distance simplifies this equation to the heat equation.

In this Section the signed distance function and its application in theLevel Set method is discussed. There are different ways to construct asigned distance function and to preserve signed distance by reinitialization.The signed distance function d(x) is defined as:

Figure 18: Signed distance function

d(x) = min(|x− xj |) ∀xj ∈ Γ (3.22)

Suppose that for an arbitrary point x, the closest point to x on Γ is xc, thenEquation (3.22) becomes:

d(x) = |x− xc| (3.23)

27

This function has the property that |∇d(x)| = 1.

Next the level set function φ is defined a signed distance function as (inFigure 18:

φ((x)) = d(x) ∀x ∈ Ω+

φ((x)) = −d(x) ∀x ∈ Ω−

φ((x)) = 0 ∀x ∈ Γ

So that |∇φ(x)| = 1. Following two examples of signed distance functionsto describe an interface (the same interface as in Section 3.1).

Example in R1. For the one-dimensional example of an implicit represen-tation of an interface in Section 3.1.2 the zero-dimensional interface −1, 1was defined as the zero isocontour of the level set function φ = x2−1. Now asigned distance function will be used instead, to represent the same interfaceimplicitely. This signed distance function is defined as: φSD(x) = |x|−1. InFigure 19 one can see that the signed distance function represents the sameinterface Γ = −1, 1 as before.

Figure 19: Signed distance function in R1

Example in R2. In Section 3.1.3 an example was shown in which theinterface x2+y2 = 1 was defined by the zero isocontour of φ(x) = x2+y2−1.Now this interface is defined by the signed distance function: φSD(x) =|x| − 1. In Figure 20 this signed distance function is depicted.

One can see that the signed distance functions have a kink at x = 0 incontrary to the implicit functions used before which where smooth. There-fore, one would expect singularities solving the Level Set equations (since

28

Figure 20: Signed distance function in R2

the derivative of φ in this kink is undefined). But in numerical approxima-tions this kink will give no problems due to singularities because the kinkwill be smeared out on the grid points. The only disadvantage is that closeto this kink the gradient will be no longer equal to one (|∇φ| 6= 1). Thiswill give errors using the simplified equations based on the assumption ofsigned distance. But, in general these kinks are far away from the φ = 0isocontour and there will be no problem since in Structural Optimizationthe only interest is the evolution of this interface. For this same reason, onecould save computational time by solving the Level Set equation only in anarrow band around the interface.

Figure 21: Deviation from signed distance. On the left the initial signeddistance function. (Figure taken from [1])

Next, the need of reinitialization of the signed distance function is dis-

29

cussed.Reinitialization is performed for several reasons. The first reason is that

the velocity field is defined in such a way that φ deviates from signed distanceas the interface evolves in time (like in Figure 21). The second reason forreinitialization is that the interface reaches one of the edges of the narrowband (in case of the narrow band method).

In the first case φ deviates from signed distance due to the velocity fieldwhich varies in the normal direction to the interface. Suppose that φ isinitially signed distance then the Level Sets are parallel to each other. Theonly case in which these Level Sets stay parallel to each other is when thevelocity in normal direction is constant. In general this is not the case.

Suppose for example that the velocity is proportional to the mean cur-vature. This would imply that the velocity varies in the normal direction.Then after one time step φ is no longer a signed distance function sincein general the velocity field on the isocontours is different. Deviation fromsigned distance leads to an increase in numerical errors in solving the levelset equation and in the geometric properties of the interface.

To prevent this, the function φ has to be reinitialized to a signed distancefunction. One resets the narrow band surface around the interface andconstructs a signed distance function again.

3.4.1 Reinitialization to Signed Distance

For the reinitialization step, one can choose between different methods: di-rect upwinding method versus an iterative scheme . In this research a simplebut computationally more expensive iterative scheme was used. It is basedon constructing signed distance by finding the steady-state solution to thereinitialization equation. The reinitialization equation is defined as:

φt + sign(φ)(|∇φ| − 1) = 0 (3.24)

One can see that if this equation converges to the steady state solution, thissolution is a signed distance function since |∇φ| = 1. This method can beused without explicitly finding the zero level set. Since φ < 0 inside andφ > 0 outside, the sign function controls the way of solving the reinitializa-tion Equation (3.24). This balances out around the zero isocontour sincein the inside region Equation (3.24) flows in outward direction and in theoutside region Equation (3.24) flows inward in the direction of the zero levelset. The steady state solution is then a signed distance function. This workswell if φ initially is close to signed distance and smooth since steep gradi-ents can cause the interface to move incorrectly over a gridpoint. A way toimprove this is to use instead of sign(φ) a continuous function s(φ) which issmooth around the interface (numerically smeared out), so that the veloc-ity decreases close to the interface and balancing out circular dependencies

30

around the interface. One could define s(φ) for instance as:

s(φ) =φ√

φ2 + |∇φ|2(4x)2(3.25)

Where s(φ) also depends on the the gradient of φ.Altough this iterative procedure can work very well and give good ap-

proximation for signed distance in the reinitialization step, in [27] is sug-gested to avoid reinitialization as much as possible. This could be accom-plished by building an extended velocity field. With this velocity field signeddistance is conserved and their is no need to reinitialize the signed distancefunction for the reason of deviation from signed distance (however, for thethe narrow band method reinitialization is still needed when the interfaceapproaches an edge).

31

Part III

Methodology

The aim of this research is to compare two evolutionary type methods; vari-ants of the classical Evolutionary Structural Optimization method (ESO)which was discussed in Section 2. ESO is based on gradually removing lowstress material towards a certain target volume (more optimal design). Inboth methods the objective function is the compliance-volume product andthe removal criterion is the Von Mises stress. Furthermore, the finite elementanalysis is performed by a FG FEA model discussed in Section 1.

The first method is the Isocontour Evolutionary Structural Optimizationmethod (ISOESO). This is an one-directional method which is based on re-moving low stress material along the Von Mises iso-stress contours. Thesecond method is the level set Evolutionary Structural Optimization method(LS-ESO) in which the level set model is used for an implicit boundary rep-resentation. LS-ESO is based on Sethian and Wiegmann’s approach in [28]where the level set model was combined with immersed interface methods.In the level set function is evolved using a velocity field which is directlyrelated to the Von Mises stress. The velocity field is defined in such a waythat below a certain threshold value material is removed and above thisthreshold value material is added to the structure. In contrary to ISOESOthis method is bi-directional.

In this Part these two methods are discussed in detail. In the nextPart IV both methods are applied on three different benchmark problemsfor different mesh sizes. In Part V these results are discussed and comparedby the final designs, the objective function value for these final designs (CV-product), the iteration history, computational effort and robustness.

32

4 Isocontour Evolutionary Structural Optimization(ISOESO)

The ISOESO method is a soft-kill variant of the classical Evolutionary Struc-tural Optimization method (ESO) developed by Xie and Steven [35] anddiscussed in Section 2. In ESO low stress material is removed by a hard-killapproach. Entire elements below a certain stress threshold value are subse-quentially removed from the finite element grid until a predetermined targetvolume is reached. In Figure 10 an example is depicted of a cantilever beamoptimized by ESO. One can see the jagged edges and also checkerboardpatterns appear which are encircled in red.

Figure 22: ISOESO iteration step

In ISOESO a so-called soft-kill approach is used instead by using FGFEA and using boundary elements (NIO); low stress material is removedgradually along the Von Mises iso-stress contours, instead of removing entireelements. In Figure 22 an example of one iteration step is shown: the redline indicates the Von Mises stress threshold along which material is cut.ISOESO has several advantages over the classical ESO method:

• Higher accuracy due the more accurate approximation of boundaryelement.

• Smooth boundary.

• No checkerboard patterns.

However, ISOESO method still has some drawbacks inherited from theclassical ESO method, including the fact that it does not admit previouslyremoved material (a bidirectional method BESO [22] exists but is limited toremoving/admitting entire elements).

33

4.1 ISOESO Optimization Algorithm

In Figure 23 a flow chart is depicted for the ISOESO algorithm. The op-timization algorithm can be divided in two Parts. The blue dotted lineindicates the initialization Part and the red dotted line indicates the opti-mization loop. In this Section the optimization algorithm is discussed indetail.

Figure 23: Flow Chart ISOESO Optimization Algorithm

Initialization.

• Fixed grid mesh construction over the structural domain.

• Apply load and boundary conditions

• Initial FG FEA is performed to determine the initial stress distribu-tion. Bilinear four-node quadrilateral elements are used and the VonMises stress is determined by weighted average approach. FG FEA isdiscussed in Section 1.

• Evolutionary parameters 4V,RR,4RR. These parameters are pro-vided by the user and influence the accuracy and speed of the opti-mization algorithm.

In Figure 24 is depicted a centrally loaded cantilever and its initial stresscontribution determined by FG FEA for a coarse mesh of 40× 20.

34

Figure 24: Von Mises stress distribution for cantilever beam on 40×20 mesh

Optimization loop.

• Determine the threshold value for the Von Mises stress below whichmaterial is removed. This threshold value is a certain percentage ofthe maximum Von Mises stress: σth = RR× σmax.

• Remove material along the Von Mises stress isocontour σth. Materialis eliminated by considering each element separately and check theelement classification. The elements are classified as I,O and NIO bycomparing the Von Mises stress in the element nodes with σth. Thisis shown in algorithm 1.

Algorithm 1 Element Classification (I,O and NIO) ISOESOfor i = 1 to nelem do

if σimin > σth thenElement typei = IKielem = KI

V i = Areaelemelse if σimax < σth then

Element typei = OKielem = KO

V i = 0else if σimin ≤ σth ≤ σmax then

Element typei = NIO

Kielem = αKI + (1− α)KO where α =

AreaIAreaelem

V i = AreaIend ifSubstitute Ki

elem into Kglobal if the element changed from group.V = V + V i

end for

Algorithm 1 gives the new shape after removing material below σth.

35

Furthermore, it updates the global stiffness matrix for the altered el-ements.

The element stiffness matrix for elements classified as I or O is thesame for all elements in the group of I and O. The NIO elements area separate group since the stiffness of the NIO element depends onits volume fraction. As discussed in Section 1.4 in this research thestiffness of the NIO element is approximated to be proportional tothe volume fraction of inside material. The procedure for elementsclassified as NIO is:

– Determine the points on the element boundaries where σvm = σth.These points are determined by linear interpolation between thenodal values over the element boundary.

– Reconstruct a polygon connecting element nodes which lie insidethe structural domain and intersection points. An example isgiven of such a polygon is depicted in Figure 25 where it is formedby the points 1-2-3-4-5.

– Finally the area is determined by the cross product like in Equa-tion (1.8) and then the volume fraction α is determined.

Figure 25: NIO element

• Check volume change. Check if the removed volume is more or lessthan the predetermined value of 4V . 4V is the minimal volume tobe removed per iteration step and its minimum value is the size ofone element. The value of 4V has a large effect on the number of

36

iterations of the optimization process. Furthermore, for fine meshesthis value is chosen larger than for coarse meshes since the value isof order 4V ≈ O(4x 4 y)−1. The volume check can result in twodifferent cases:

– If the volume change is less than 4RR then: the removal rateRR is increased by 4RR and a new σth is determined to removemore material.

– If the volume change is more than 4RR then: the next check isperformed:

• Check if target volume is reached.

– If Vcurrent > Vtarget then a FG FEA is performed for the changedshape and the new stress distribution is determined.

– If Vcurrent ≤ Vtarget then STOP

Example: Optimization of cantilever by ISOESO, mesh: 40×20.The centrally loaded cantilever beam of Figure 24 with its initial stressdistribution is now optimized towards a target volume of 50% by ISOESO.This results in the final state depicted in Figure 26. A relatively coarse meshof 40× 20 was used.

Figure 26: Optimized Cantilever by ISOESO for a coarse 40× 20 mesh anda target volume of 50%. The color map is the Von Mises stress distribution.

In this Figure the stress distribution is shown and it is clear that thisfinal result is closer to the fully stressed design than the initial shape. Fur-thermore is was measured that the final design has a compliance-volumeproduct lower than the initial shape. Details on the behavior of the ob-jective function and the optimality of such an optimum are discussed inSection V.

37

5 Level Set ESO Optimization (LS-ESO)

A solution to admit lost material is to use of the level set model for theboundary representation. This level set model is discussed in detail in Sec-tion 3 and its bi-directionality enables to admit erroneously removed mate-rial. The motion is described by the level set equation (5).

φt + Vn|∇φ| = 0 (5.1)

The approach used here is based on Sethian and Wiegmann’s approachin [28]. However, Sethian and Wiegmann used the Jump Immersed Interfacemethod to determine the stress distribution. Here the FG FEA will be usedinstead. The definition of velocity field Vn in (5) is an important featuresince it determines the motion of the implicit function and therefore also ofthe structural boundary. In this Section first the velocity field used in LS-ESO will be discussed and then the optimization algorithm will be discussedin detail.

5.1 Definition of the Velocity Field

In LS-ESO the velocity field Vn is directly related to the Von Mises stress.The definition of the velocity field is based on the principal idea of removingand adding material around a certain threshold Von Mises stress σth =RR × σmax where RR is a removal rate. The velocity field used here is thesame as Sethian and Wiegmann used in their initial paper introducing thelevel set model in Structural Optimization [28]. This velocity profile Vn isdepicted in Figure 27:

Figure 27: Scaled Velocity field Vn as a function of Von Mises stress σVM

38

This profile follows the relation±(1+cos)/2 over the intervals [s, s] and [S, S]which were rescaled to [0, π]. The three intervals can be characterized:

• [s, s] in this region the stress is below σth and therefore the velocity isnegative which results in the boundary moving inward.

• [s, S] in this interval around σth the velocity is set to zero to avoidshearing. Shearing in this case means that the boundary would evolvein opposite direction around the point on the boundary where thestress is equal to σth. Resulting in a discontinuity on the boundary.

• [S, S] in this region the stress is above σth and therefore the velocityis positive which results in the boundary moving outward.

The values of s, s, S and S are determined experimentally. Intuitively onewould choose the domain [S, S] larger than [s, s] because the idea is to removemore material than to add. In this research the following relationship is used:

• s = 0

• s = 0.9RR× σmax

• S = 1.1RR× σmax

• S = 5RR× σmax

It can be seen that the velocity is negative for low stress regions and positivefor stress regions above σth. This results in the desired behavior that for lowstress regions the boundary moves in inward direction to remove materialand for stresses above the threshold value the boundary moves in outwarddirection and thus adds material. The definition of the velocity profile andthe relation above are completely heuristic. The author tested two otherprofiles: constant velocity and a direct proportional velocity. It was clearthat the velocity profile of Figure 27 gave best results. This led to the deci-sion to use this velocity profile in this comparative study.

Note: the author expects that different velocity profiles can be found whichgive improved results. Therefore, the definition of the velocity profile seemsto be an interesting topic for further research. However, this is not in thescope of this thesis.

5.2 LS-ESO Optimization Algorithm

In Figure 28 a flow chart is depicted for the LS-ESO algorithm. The op-timization algorithm can be divided in two Parts. The blue dotted lineindicates the initialization Part and the red dotted line indicates the opti-mization loop. In this Section the optimization algorithm is discussed indetail.

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Figure 28: Flow Chart LS-ESO Optimization Algorithm

Initialization. The initialization Part in LS-ESO is similar to that of theof ISOESO but extended by an additional initialization routine in which theimplicit function φ is constructed.

Again a fixed grid mesh is constructed over the reference domain andthe load and boundary conditions are applied. The implicit function φ isnow constructed over the same fixed grid mesh before performing the initialFG FEA. The shape of the initial implicit function depends on the desiredinitial shape of the structure.

In this case a structural domain that contains a certain number of holesis used. These holes are present in this case because of the definition of thevelocity field. This definition does not allow the creation of holes duringoptimization. i.e. Topological changes are only possible by merging. Hence,the implicit function will have some kind of a bubble shape where the holesare created initially by the bubbles intersecting the zero level set.

Furthermore signed distance is required. So that finally the implicitfunction φ is constructed by a sequence of matrix operation using booleanoperators. For example a circle of radius r with its center located at x =a, b is given by the zero level set of the following signed distance function:

φsd1 =√

(x− a)2 + (y − b)2 − r (5.2)

Combining this with other signed distance functions describing the otherholes and the outer boundary will result in the final implicit function φ overthe hole reference domain. This combined function is not a signed distance

40

function, however, this will give no problems since signed distance is approx-imated along the boundary which is of interest. In Figure 29 an initial levelset function and its corresponding structure are depicted for the problem ofa centrally loaded cantilever inclined on the left hand side as in Figure 24.

Figure 29: Initial state implicit function, mesh: 80× 40

In Figure 29 the Von Mises stress distribution is given under the ap-plied load and boundary conditions. again the FG FEA was used for thestructural analysis. Before performing FG FEA the elements are classifiedinto three groups: I, O and NIO. In this case this depends by the value ofφ in the nodes. A different approach than ISOESO where a threshold VonMises stress determined the classification for each element. The elementsare classified by algorithm 2:

Algorithm 2 Element Classification (I,O and NIO) LS-ESOfor i = 1 to nelem do

if σimin > 0 thenElement typei = IKielem = KI

else if σimax < 0 thenElement typei = OKielem = KO

else if σimin ≤ 0 ≤ σmax thenElement typei = NIOKielem = αKI + (1− α)KO where α = AI

Aelem

end ifend for

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Optimization Loop The optimization loop now includes an update schemefor φ and a reinitialization part to conserve signed distance. The differentsteps are listed below:

• Determine the threshold value for the Von Mises stress σth = RR ×σmax. This is the σth from the velocity profile in Figure 27. Materialis removed and added around this value.

• The velocity profile Vn(σ) is derived around this σth as discussed inthe previous Section. The velocity field is constructed by assigning thevelocities to the nodes.

• The implicit function is reinitialized to signed distance. The signeddistance function is constructed by solving the reinitialization Equa-tion (5.3) as discussed in detail in Section 3.4.1:

φt +φ√

φ2 + |∇φ|2(4x)2(|∇φ| − 1) = 0 (5.3)

Where the first term on the right hand side is a smeared sign function.One can see that the steady state solution of this reinitialization equa-tion is a signed distance function. This equation is also a HamiltonJacobi type equation and therefore the HJ ENO schemes discussed inSection 3.3.2 are used to solve this equation numerically. This reini-tialization works fine when the implicit function is close to signed dis-tance. Since signed distance is required for numerical accuracy in thespatial discretization every update of the implicit function is followedby reinitialization.

Note: The signed distance function can also be constructed by theso-called Fast Marching method developed by Sethian [27]. In thismethod the construction of signed distance is treated as a boundaryvalue problem: the Eikonal equation with the explicit location of theinterface as the boundary condition. This boundary value problem isthen solved by marching out from the interface in upwind direction. Aheapsort technique is used to determine the rank of the nodes passedby the interface. This method is faster than the iterative procedure.Furthermore there is no restriction on the implicit function to be closeto signed distance before reinitialization. A detailed description of theFast Marching method can be found in the text book of Sethian [27]and in An Introduction to Level Set Based Optimization by the au-thor [31]

However, in this research the reinitialization equation is used for itssimplicity because it works directly on the nodal values (no need of to

42

explicitly find the interface). Furthermore the time saved using fastmarching is minimal for the relatively simple benchmark problemswith a moderate number of finite elements considered here.

• Update of the level set function φ. The new level set function is deter-mined by solving the level set Equation (5) numerically. In this casethe level set equation is a hyperbolic equation and thus discretized as:

φn+1i − φni4t

+ max(u, 0)∇φ+i + min(u, 0)∇φ−i = 0 (5.4)

Where ∇φ+i and ∇φ−i are expressed as:

∇φ+i =

√max(φ−x , 0)2 + min(φ+

x , 0)2 + max(φ−y , 0)2 + min(φ+y , 0)2

∇φ−i =√

min(φ−x , 0)2 + max(φ+x , 0)2 + min(φ−y , 0)2 + max(φ+

y , 0)2

This method is an upwinding method since the direction of discretiza-tion depends on the sign of the velocity u. For the time discretizationforward Euler method is used and for the spatial derivatives HJ ENOschemes are used. The desired accuracy determines which order HJENO scheme is the most appropriate. Referred is to Section 3.3 for adetailed description of the discretization of the level set equation.

Once the new level set function is determined the next stages in thealgorithm are similar for both methods.

• Check volume change. Now that the updated φ is known the currentvolume is again determined by algorithm 2. Check if the removedvolume is more or less than the predetermined value of 4V . Twocases follow:

– If volume change is less than 4RR then: the removal rate RRis increased by 4RR and returned is to the first step in theoptimization loop. A new σth is determined by the increasedremoval rate. This threshold value is thus a larger fraction ofthe maximum Von Mises stress σmax. The velocity field is nowconstructed around this new value which will lead to the removalof more material since more material will lie below this thresholdvalue and will thus move inwards.

– If volume change is more than 4RR then: the next check isperformed:

• Check if target volume is reached.

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– If Vcurrent > Vtarget then a FG FEA is performed for the changedshape and the new stress distribution is determined.

– If Vcurrent ≤ Vtarget then STOP

Example: optimization of cantilever beam, mesh 80× 40. In theprevious Sections the centrally loaded cantilever beam was optimized usingISOESO. In this example the cantilever beam is optimized using LS-ESO.However, a finer grid is used of 80×40. This is done to get smoother resultswhile in the ISOESO case the focus was on the fixed grid. The purposehere is to give a simple example of the application of LS-ESO and not aquantitative comparison with ISOESO. This will be the principal subject ofSection IV.

In Figure 30 the final result is depicted for the centrally loaded cantileveroptimized towards a target volume of 40%. The initial state was depictedin Figure 29.

Figure 30: End state implicit function

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Part IV

Results

In this Part both methods are applied on optimizing three benchmark prob-lems: a centrally loaded cantilever beam, a MBB beam and a Michell beam.The results are presented here and will be discussed in Part V.

For each benchmark problem the methods are applied on different meshsizes to be able to study the effect on he final design. The objective func-tion is the compliance-volume product. Furthermore, the average CPU periteration time for both methods is discussed.

The FG FEA, the evolutionary parameters and the discretization of thelevel set model used are listed next:

FG FEA.

• Material properties: Young’s modulus E of 1 N/m2 for inside materialand 10−6 N/m2 for outside material and a Poisson’s ratio of 0.3.

• Stress: plane stress model and the von Mises stress is assigned tothe nodes of the bilinear quadrilateral elements by a weighted directmethod discussed in Section 1.5.

• Boundary elements (NIO): Assumed to have a Young’s modulus directproportional to the volume fraction of inside material.

Evolutionary Parameters. The choice of the evolutionary parametershave a large influence on the final results. Furthermore, they differ betweenbenchmark problems; e.g. parameters that result in a feasible design forthe cantilever could result in failure for the Michell beam and vi. There-fore, these parameters are determined empirically for each individual case.Chosen in such a way that material is gradually removed towards the targetvolume and that failure caused by disconnected regions is avoided.

Discretization of the level set model. The level set function is con-structed over the finite element grid. In this Section the discretizationschemes for LS-ESO have a predetermined fixed order. The following dis-cretization schemes are used:

• Forward Euler for the time discretization

• HJ ENO2 for the spatial discretization

• HJ ENO1 for the reinitialization

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The HJ ENO2 scheme gives second order accuracy which is an intermediateaccuracy but sufficient for evolving a level set equation for which the velocityfield is based on the stress field over the bi-linear quadrilateral elements. Forthe reinitialization a low scheme is used to obtain a smoothing effect on theboundary. Smoothening is needed and discussed in detail in Section 11.Furthermore, note that the stability of these upwinding schemes is ensuredby limiting the time step to the Courant-Friedrichs-Lewy condition which isgiven by Equation 5.5.

4tmax u4x

+v

4y = α (5.5)

where 0 < α < 1.

In this case the restriction is chosen as α = 0.5 to ensure numerical sta-bility.

Boundary smoothing. The level set model is used for its smooth bound-ary representation so conserving this property is important. For all casessome additional boundary smoothing was desired since chaotic behavior ofthe boundary was displayed. Therefore the boundary was smoothened byadding an additional velocity term which depends on the mean curvature.This type of smoothening is discussed in Section 11.

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6 Centrally Loaded Cantilever

The first benchmark problem is a cantilever subjected to the load and bound-ary conditions depicted Figure 31. This cantilever is optimized towards atarget volume of 40% using ISOESO and LS-ESO. This value is chosen sinceit is approximately the lower limit for which the optimization works fine inthe range of mesh sizes used. For smaller target volumes the structures havethe tendency to break down. Here the prime objective is to compare bothworking methods.

Figure 31: Cantilever centrally loaded on right end

6.1 Optimized Cantilever

In Figures 32 and 33 the optimization of the cantilever beam towards avolume fraction of 40%is depicted for a 80× 40 mesh. For reasons of clarityfour design stages are displayed: the initial design, two intermediate designsand the final design. The colors in these structures correspond to the VonMises stress distribution.

In ISOESO the structural boundary in each design stage is the VonMises stress isocontour determined by the threshold value at that designstage. While in LS-ESO the structural boundary is the zero level set of thelevel set function.

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(a) (b)

(c) (d)

Figure 32: ISOESO Optimization of a cantilever, mesh 80 × 40. (a) initialdesign, (b) iteration 40, (c) iteration 60, (d) iteration 95.

(a) (b)

(c) (d)

Figure 33: LS-ESO Optimization of a cantilever, mesh 80 × 40. (a) initialdesign, (b) iteration 11, (c) iteration 35, (d) iteration 70

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In Figure 34 the corresponding implicit function for the LS-ESO resultis depicted. One can see the evolution of the implicit function in the initialdesign (a) towards the final design (b).

(a)

(b)

Figure 34: the implicit function: (a) initial state, (b) end state

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6.2 Objective function and Von Mises stress distribution

For the objective function the compliance-volume product is chosen. InFigure 35 the behavior of the normalized objective function, volume andcompliance energy, during optimization is depicted for the results consideredin Figures 32 and 33.

(a) (b)

Figure 35: Iteration history of normalized compliance, volume andcompliance-volume product : (a) ISOESO (b) Level Set ESO

The optimization procedure was also applied on a coarse 40 × 20 meshand a fine 120 × 60 mesh. It appeared that the objective function duringiteration and the Von Mises stress distribution graphs for these mesh sizesare similar. Therefore, these graphs are not included here. The initial dataand final results for all mesh sizes are listed in Table 1.

Case No. CV [Nm] CV [Nm] σmaxIter. Initial Final [N/m2]

ISOESO-40× 20 41 73.73 56.36 16.67LS-ESO-40× 20 49 86.05 59.40 15.61ISOESO-80× 40 102 75.83 57.60 22.82LS-ESO-80× 40 88 93.55 61.94 19.93

ISOESO-120× 60 49 76.73 57.68 23.92LS-ESO-120× 60 46 96.13 66.50 20.66

Table 1: Centrally loaded cantilever for different mesh sizes

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The final designs for the different mesh sizes are depicted in figure 36.

(a) (b)

(c) (d)

(e) (f)

Figure 36: ISOESO: (a) 40× 20, (c) 80× 40, (e) 120× 60,LS-ESO: (b) 40× 20, (d) 80× 40, (f) 120× 60

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6.3 Computational Effort

In Table 2 the computational time for both methods are listed for the dif-ferent mesh sizes used.

Case No. LS steps per Avg. CPU-time [s] per it. step TotalIter. FEA step FEA LS Reinit [s]

ISOESO-40× 20 41 - 0.54 - - 22.14LS-ESO-40× 20 49 1 0.64 0.19 0.14 47.53ISOESO-80× 40 102 - 4.90 - - 499.80LS-ESO-80× 40 88 1 6.10 0.24 0.22 577.28

ISOESO-120× 60 49 - 28.10 - - 1367.90LS-ESO-120× 60 46 2 29.40 0.31 0.39 1398.85

Table 2: Computational Effort for both methods and different mesh sizes

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7 Michell’s Beam

In this Section Michell’s beam in Figure 37 is considered as benchmarkproblem. The beam is optimized towards a volume fraction of 30%.

Figure 37: Michell Beam

7.1 Optimized Michell Beam

In Figure 38 the final design is depicted for both methods using a 80 × 40mesh. The colors inside the structural domain correspond to the Von Misesstress distribution.

(a) (b)

Figure 38: Final design Michell Beam, volume fraction: 30%, mesh 80× 40(a) ISOESO, (b) LS-ESO

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The initial design in the case of LS-ESO (Figure 38 (b)) is the same asfor the cantilever: 4 × 3 holes. The end state of the implicit function isdepicted in Figure 39.

Figure 39: End state implicit function for the Michell beam

7.2 Objective function and Von Mises stress distribution

The normalized objective function, volume and compliance during optimiza-tion for the results depicted in Figure 38 mesh are depicted in Figure 40.

(a) (b)

Figure 40: Iteration history of normalized compliance, volume andcompliance-volume product : (a) ISOESO (b) Level Set ESO

54

The optimization procedure was also applied on a coarse 40 × 20 meshand a fine 120 × 60 mesh. The final results for the different mesh sizes arelisted in Figure 3.

Case No. CV [Nm] CV [Nm] σmaxIter. Initial Final [N/m2]

ISOESO-40× 20 76 15.01 11.32 20.43LS-ESO-40× 20 152 15.67 11.20 20.48ISOESO-80× 40 98 17.37 11.23 40.85LS-ESO-80× 40 104 18.37 12.38 40.91

ISOESO-120× 60 80 18.74 11.63 61.25LS-ESO-120× 60 155 19.72 16.06 60.81

Table 3: Michell’s beam for different mesh sizes

The final designs for these different mesh sizes are depicted in Figure 41.

(a) (b)

(c) (d)

(e) (f)

Figure 41: ISOESO: (a) 40× 20, (c) 80× 40, (e) 120× 60,LS-ESO: (b) 40× 20, (d) 80× 40, (f) 120× 60

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7.3 Computational Effort

In Table 4 the computational time for both methods are listed for the caseof the Michell beam.

Case No. LS steps per Avg. CPU-time [s] per it. step TotalIter. FEA step FEA LS Reinit [s]

ISOESO-40× 20 76 - 0.61 - - 46.36LS-ESO-40× 20 152 1 0.69 0.19 0.08 145.92ISOESO-80× 40 98 - 5.60 - - 548.83LS-ESO-80× 40 104 1 6.07 0.24 0.17 673.88

ISOESO-120× 60 80 - 30.13 - - 21410.37LS-ESO-120× 60 155 2 30.42 0.31 0.24 4848.31

Table 4: Computational Effort for both methods and different mesh sizes

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8 MBB Beam

In this Section the MBB beam in Figure 42 is considered as benchmark.The MBB beam is optimized towards a volume fraction of 40%. In this casesymmetry was used and only the right half was considered with symmetricboundary conditions on the cut off boundary.

Figure 42: MBB Beam

8.1 Optimized MBB Beam

The final designs for a mesh of 90× 30 are depicted in Figure 43.

(a)

(b)

Figure 43: Final design MBB Beam, volume fraction: 40%, mesh 90 × 30(a) ISOESO, (b) LS-ESO

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The end state of the implicit function for the LS-ESO optimization isdepicted in figure 44.

Figure 44: End state implicit function for the MBB beam

8.2 Objective function and Von Mises stress distribution

The objective function during optimization is depicted in figure 45.

(a) (b)

Figure 45: Iteration history of normalized compliance, volume andcompliance-volume product : (a) ISOESO (b) Level Set ESO

The final results are listed in Table 5.

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Case No. CV [Nm] CV [Nm] σmaxIter. Initial Final [N/m2]

ISOESO-60× 20 148 23.93 16.64 10.29LS-ESO-60× 20 111 25.87 15.85 10.34ISOESO-90× 30 118 24.10 16.57 15.37LS-ESO-90× 30 145 27.89 17.26 15.41

ISOESO-120× 40 115 24.23 16.64 20.51LS-ESO-120× 40 101 28.44 19.48 20.49

Table 5: MBB beam for different mesh sizes

The corresponding final designs are depicted in Figure 46.

(a) (b)

(c) (d)

(e) (f)

Figure 46: ISOESO: (a) 60× 20, (c) 90× 30, (e) 120× 40,LS-ESO: (b) 60× 20, (d) 90× 30, (f) 120× 40

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8.3 Computational Effort

In Table 6 the computational time for both methods are listed for the caseof the MBB beam.

Case No. LS steps per Avg. CPU-time [s] per it. step TotalIter. FEA step FEA LS Reinit [s]

ISOESO-60× 20 148 - 1.05 - - 155.38LS-ESO-60× 20 111 1 1.11 0.20 0.14 160.95ISOESO-90× 30 118 - 3.77 - - 444.90LS-ESO-90× 30 145 1 4.60 0.21 0.20 726.44

ISOESO-120× 40 115 - 12.31 - - 1415.61LS-ESO-120× 40 101 2 13.40 0.23 0.27 1427.41

Table 6: Computational Effort for both methods and different mesh sizes

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Part V

Discussion

In this Part the results of both methods are discussed and compared. Thefound optima for both methods are compared by the final objective func-tion value and behavior of the objective function during optimization. Fur-thermore, both methods are discussed by different aspects like: computa-tional effort, mesh-dependency, robustness and the evolution of the struc-tural boundary. Followed by a critical view on both methods and a criticalview on the validity of ESO methods in general.

9 Optima

In this Section the final designs and the optimization process are compared.

9.1 Final design

When the objective function values for the final designs in Tables 1- 5 arecompared it shows that ISOESO gives a more optimal design than LS-ESO.In general the found optima by LS-ESO have an objective function value5% − 10% above the optima found by ISOESO. This does not correspondwith the expected result that LS-ESO would give a more optimal design dueto its bi-directionality.

The observation that ISOESO leads to a more optimal design than LS-ESO is confirmed by looking at the Von Mises stress in the structure inFigures 36, 41 and 46. The Von Mises stress distribution is represented bya color map and it can be seen that the final designs obtained by ISOESOhave a more uniform stress distribution and are thus closer to the optimalfully stressed design. In Figure 47 the final designs for the cantilever for an80×40 mesh are depicted. In the final design for LS-ESO low stress regionsremain present for this volume fraction (see red encircled region).

(a) (b)

Figure 47: Low stressed region in the final cantilever LS-ESO design

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The reason that LS-ESO lacks to give a more optimal design than ISOESOis that the level set model is limited to move freely inside the computationaldomain in search of an optimum. This is a results of the formulation of Evo-lutionary Structural Optimization in which material is gradually removedtowards a target volume. In this approach the algorithm is driven by lo-cal removal criterion in a certain direction and stopped suddenly when thetarget volume is reached. The author believes that this is a flaw in the for-mulation of ESO methods, since this direction is based on local criteria; itis thus not ensured to be a direction in which objective function decreasesmost. Furthermore, these local criteria are not taken into considerationaround the found solution. LS-ESO suffers more from this limitation due toits fundamentally different boundary representation.

In LS-ESO the new boundary after on optimization step is the evolvedstate of the previous boundary. While in ISOESO the new boundary ischosen from an infinite set of possible Von Mises iso-stress contours inside astructural domain enclosed by the previous boundary.

This results in the fact that the final design in LS-ESO depends more onthe initial state (the location and the number of holes) than ISOESO whichcan cut directly in low stress regions. Hence, in the current formulation ofthe optimization problem, LS-ESO does not take full advantage of its bi-directional character. In Section 12 an adaptive algorithm is presented bythe author, in which the structure is optimized further around this targetvolume.

Note: When considering the maximum values of the von Mises stress itappears that the stress increases when the mesh is chosen finer. This hap-pened because the author applied the load conditions on two or three ele-ments. For a finer mesh this correspond with a smaller area and thus higherlocal stress values.

9.2 Optimization history

For each benchmark problem the normalized objective function (Compliance-volume product), volume and compliance are plotted versus the iterationsteps (see Figures 35, 40 and 45). The graphs for both methods show dif-ferent behavior during optimization.

When considering the volume graphs it appears that for the ISOESOcase this displays a somewhat chaotic and unpredictable behavior comparedto the graphs of LS-ESO. In the volume graph for LS-ESO more jumpsare present which indicate that material is not removed gradually. Insteadsudden design changes appear during optimization. These irregularities aredirectly related to the evolutionary parameters. The problem of these sud-den design change is the effect on the state of the structure. Removing toomuch material in one iteration step can lead to breakage of bars that con-

62

tributed in the overall stiffness of the structure. In LS-ESO there are nojumps due to sudden increase in material removal. Hence,the choice of theevolutionary parameters have less influence on the optimization process.

When the objective function is considered jumps are inherited from thevolume changes for ISOESO. Furthermore it appears that neither ISOESOnor LS-ESO has monotonically decreasing functions for all cases. Littlehumps in the objective function graphs appear near the target volume. Thisis exactly one of main drawbacks using this local heuristic removal criterion:it is not guaranteed that the next solution is better than the previous one.These little humps indicate breakage of certain bars that contributed inoverall stiffness and there the structure becomes less optimal.

In conclusion the behavior of the objective function and the volumeduring optimization is far from a desired monotonically decreasing func-tion. However the author believes that the the smooth behavior of theoptimization process for LS-ESO is an advantage above the irregular be-havior displayed in ISOESO. Slow removal of material is desired since usingthese local criteria the effect of this removed material on the stress distri-bution is not known a priori. Furthermore, irregular behavior implies a lessrobust optimization process with respect to the evolutionary parameters.Choosing these parameters large will lead to breakage and thus failure ofthe optimization process. While choosing these parameters too small willlead to less irregular behavior but will increase the number of iterationssignificantly. In practice a good choice of the evolutionary parameters iscritical. In Section 10 the effect of the evolutionary parameters is discussedin detail and in Section 12.1 a new ISOESO algorithm is proposed which ismore stable.

9.3 Computational Effort

The computational effort for all cases are listed in Tables 2, 4 and 6. Allcomputations where performed on the same machine. Furthermore, the FEAstep is performed in both methods in the same way by Gaussian eliminationby the backslash operator in Matlab.

LS-ESO is computational more expensive than ISOESO as expected.This was expected since LS-ESO is the FEA step for ISOESO plus twoadditional steps; x level set updates and a reinitialization step. The data inthe Tables show that the average difference per iteration between ISOESOand LS-ESO step is not that large since solving the level set equation andthe reinitialization equation costs a little compared to the FEA step.

However, this a bit misleading since the FEA step is no performed inan efficient way. The assembly of the global stiffness matrix is done by aloop and subsequentially substituting the element stiffness matrix for eachelement. There exist a more efficient way in which changes in the elementsstiffness matrix point directly to the global stiffness matrix. In conclusion,

63

the author believes that the additional computation time by applying the LS-model is more significant than it appears to be from the data in the tables,when a more efficient way of these algorithms is performed. Furthermore thenumber of level set steps per FEA steps becomes larger for fine meshes. Thisis related to the fact that the time step is restricted by the CFL condition.

Furthermore it should be noticed that the reinitialization part takes thesame time as updating the level set equation. Therefore it is interesting toinvestigate Fast Marching methods [26] as a faster alternative to (re-)builta signed distance function. Furthermore, an extended velocity field haspotential since it reduces the need of reinitialization due to deviation fromsigned distance.

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10 Robustness

In this Section the robustness of the algorithms against the evolutionaryparameters and the mesh size is discussed.

10.1 Mesh Dependency

ISOESO was claimed to be mesh independent by Victoria in [32]. In thisthesis neither ISOESO nor LS-ESO appeared to be mesh independent. Meshdependency was checked by simply counting the number of holes for the finaldesigns in Figures 36,41 and 46 and comparing. It appeared that the numberof holes in most cased increase when the mesh size is chosen finer.

LS-ESO depends more on the mesh size since more holes appeared in thefinal design than in ISOESO. LS-ESO has the tendency for fine meshes tooptimize towards a design in which the initial number of holes are conserved.This is a drawback but the countereffect is that it makes it easy to controlthe number of holes in the final design.

In ISOESO there are no initial number of holes but when the mesh sizebecomes finer the final design tends to structures with small bars whichtogether give more stiffness and thus a more optimal design.

The reason that these designs are mesh-dependent are:

• More accurate approximation for the stress field when the mesh size isincreased. This has influence on the final design since both methodsuse the stress as removal criterion.

• When the mesh size is increased the finite element model is able torepresent thin bars. This results in final design that appear to be somekind of organic structure with a large number of thin bars. These thinbars together give more stiffness for the same amount of material inone bar and therefore in theory this is correct. However, these type ofstructures containing a large amount of thin bars are not suitable formanufacturing and therefore unwanted.

To control this behavior different techniques exist like perimeter constraintsand filtering methods as described in the textbook of Bendsoe [6]. In thisthesis these techniques were not used since the objective was not to opti-mize the algorithms as considered, but compare them in their basic con-cepts. However, when the evolutionary parameters are scaled when increas-ing the mesh size has some reducing effect on the mesh-dependency. Scalethe parmeters by their relation to the mesh size:

• 4V ∼= O( 14x2 ) since the minimal volume to be removed is put equal

to a number of elements and not a certain area.

• RR ∼= O( 14x2 ) since the load conditions where applied on some ele-

ments and not on an area.

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10.2 Influence Evolutionary Parameters

The behavior of the objective function in Figures 35, 40 and 45 is directlyrelated to the choice of the evolutionary parameters. If the volume removedis less than 4V the removal rate RR is increased by 4RR.

In the objective function graphs for ISOESO this resulted in jumps aswas discussed in Section 9.2. These jumps happen whenRR is increased with4RR. Hence, the frequency of these jumps depend on 4V and the heightof the jumps correspond to the value of 4RR. In practice the performanceof the ISOESO algorithm is a direct effect of the choice of these parameters.The problem is choosing these parameters correctly since it is impossible topredict how these behave during optimization.

In LS-ESO the graphs of the objective function are more smooth and theevolutionary parameters used are not directly recognized in the graphs like inISOESO. This can be explained by the fact that the evolutionary parametershave a indirect effect on the optimization process trough the definition ofthe velocity field. The increment of RR by 4RR results in a shift in thevelocity field which has less effect than in ISOESO where it directly relatesto an area to be removed. However, the initial RR is important since itdetermines the initial velocity field. But this can be checked relatively easyby performing some iterations and see if material is removed or added andthus choosing an appropriate RR.

Example: Different 4RR in Cantilever problem An example is de-picted in Figure 48 where is shown that choosing different values for 4RRin a small range, has a large effect on the final design.

The precise effect is not easy to determine since it differs between the prob-lems considered. The author is aware of the limitations of such a particularexample for just three values of 4RR. However, it should be emphasizedthat the author experienced that the ISOESO algorithm gives much moreproblems during optimization. These problems include breakage by remov-ing too much material which result in a increment in stress and thereforethe objective function increases instead of decreasing. Another problem thatappears in ISOESO are disconnected regions by removing too much mate-rial. An exaggerated example is depicted in Figure 49. All the final designsachieved by ISOESO where the results of a long process of tweaking theevolutionary parameters.

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(a) (b)

(c) (d)

(e) (f)

Figure 48: (a) ISOESO4RR = 0.01, (b) LS-ESO4RR = 0.01, (c) ISOESO4RR = 0.015, (d) LS-ESO 4RR = 0.015, (e) ISOESO 4RR = 0.02, (f)LS-ESO 4RR = 0.02

Figure 49: Disconnected region problems with ISOESO

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11 Smooth Boundary Representation

The two methods have a smooth boundary representation. Different fromthe density based methods or the classical ESO where the boundaries werejagged edges and checkerboard problems appeared. However, still there existsome particular problems in the boundary representation for both methods.In ISOESO particular problems arise for the definition of the boundaryelements (NIO). In LS-ESO problems arise conserving a smooth boundaryrepresentation. In this Section these problems are discussed.

11.1 ISOESO: Invalid NIO Elements

In ISOESO problems arise with the definition of the NIO elements. Themain problem is to cut correct along the threshold stress and form a properNIO element, i.e. a NIO which is formed by a polygon which points are notinside the element but along the element boundary like in Figure 7. Someof the new NIO’s had points which layed inside the original quadrilateralelement (and thus not on the element sides). This resulted in boundary ele-ments with lots of sides and sharp corners. This was resolved by surpressingthe inside points.

Another problem which is still unsolved is depicted in Figure 50. Thebar appeared to be broken but the bar remains and is not eliminated. Thiscan be explained by the fact that if the densities were depicted here thisbar would be still valid. The NIO’s are do not use information how theyare connected between elements, .i.e. the polygon are reconstructed at anelement level.

(a) (b)

Figure 50: (a) MBB beam optimized by ISOESO, mesh: 120×40, (b) InvalidNIO’s

11.2 LS-ESO: Conserving a Smooth Boundary

In LS-ESO, problems arise conserving a smooth boundary. The boundaryloses much of its smoothness during optimization. The author believes thatthis effect comes from the definition of the Von Mises stress field used in thisthesis. As was discussed in Section 1.5 the Von Mises stress is determinedin the grid nodes by a weighted average approach. In this method the

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Von Mises stress is determined on the nodes of the individual quadrilateralelements. Afterwards a weighted average σwa is calculated on the grid nodesdepending on the number of elements sharing that particular node and thevolume fraction of these elements.

σwa =∑n

i=1 αkσk∑ni=1 αk

(11.1)

Where α is the volume fraction of inside material (which is one in thecase of internal elements) and n is the number of elements sharing the node.

The problem of this definition is that the derivative of the stress field isdiscontinuous between elements. The stress field will show large local varia-tions due to this discontinuity. Since the velocity field is directly related tothe Von Mises stress this will result in local variations of the velocity fieldwhich finally results in these non-smooth boundaries. A beneficial effectcame from taking a low first order reinitialization but did not resolve theproblem and the lack of smoothness gave problems for the representation ofsmall bars.

Smoothing Effect by Additional Velocity Term. In this researchthese smoothness problems were solved by adding an extra velocity termVmc to the velocity field Vn. This velocity field is defined in Equation (11.2):

Vmc = −bκ (11.2)

where κ is the mean curvature and b is a constant which can be chosenfreely and depends on the amount of smoothing needed. Here this value ischosen in the following range: b = 0.001...0.01. The effect of this of thisvelocity field was discussed in Section 3.2.2 where it was shown that theinterface for such a velocity field moves in the direction of concavity as wasdepicted in Figure 16. The effect of this artificial smoothing is depicted inFigure 51 where it is applied on the MBB Beam.

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(a) (b)

Figure 51: (a) no additional smoothing, (b) smoothing by additional velocityterm

Two problems can arise using this method. The first is that this artificialsmoothness can lead to a different final design like in Figure 51. This canbe the results of too much artificial smoothing. The second problem isthat Courant-Friedreichs-Lewy conditions becomes stricter since the levelset equation becomes parabolic:

4t 2b(4x)2

+2b

(4y)2+

u

4x+

v

4y = α (11.3)

where 0 < α < 1.

The time step 4t is now of order (4x)2. More level set steps are neededfor fine meshes and this will increase computational time. However, in theexample of the MBB Beam and Michell beam only a little artificial smooth-ing is added (b = 0.002) and this has little influence for a moderate meshsince in that case 2b

(4x)2<< u

4x .

Other techniques that will have a smoothing effect on the boundary andwhich are recommended for further research are listed next:

• Filtering techniques. Filtering can be used to limit the variationsin the velocity field. Similar to the filtering technique used for thedensities in SIMP [6].

• More accurate and smooth stress field. Use an alternative amore accurate approximation of the Von Mises stress field e.g. by theSuperconvergent patch recovery method discussed in Section 1.5.

• Different level set grid. Use a level set grid which is not aligned overthe FEA grid but instead the grid nodes are placed on the center pointsof the finite elements. In that case the current weighted approach canbe used for deriving the nodal stress values but the level set function isevolved by the more accurate stress values in the Gauss points whichcoincide with the level set grid nodes.

Especially the additional smoothing effect that the author expects ofusing a more accurate stress field is interesting. Since it would be a con-

70

sequence of using a more accurate approximation and no artificial way ofrepairing problems.

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12 A critical view on the ESO algorithm

The validity of ESO as an optimization method is often questioned (e.g. Roz-vany in [23], [24]). In this Section these fundamental questions are avoidedand the algorithm is critically reviewed under the assumption that graduallyremoving low stress material indeed leads to an optimum design.

The algorithm 3 shows the general ESO algorithm. For a detailed de-scription is referred to Section 2 and the corresponding flow chart 9.

Algorithm 3 Classical ESO Algorithmwhile V > Vtarget do

Perform FG FEA −→ σkV onMises

σkth = RR× σkmaxRemove material σk ≤ σkthif V k − V k+1 ≤ 4V thenRR = RR+4RRGo back to line 2 and determine new σkth

end ifk = k + 1

end while

Where σk is the Von Mises stress field for the iteration step k. V denotesthe volume. σth is the threshold value which is a certain fraction of themaximum Von Mises stress σmax. Material below this threshold value isremoved. RR,4RR and 4V are the evolutionary parameters as discussedbefore.

Unstable formulation. An observation is that the increment of the thresh-old value σth is at least direct proportional to the increment of σmax, sincefor (V k − V k+1 ≥ 4V ):

σkth = RR× σkmaxσk+1th = RR× σk+1

max

=⇒ σk+1th =

σk+1max

σkmax× σkth

It should be noticed that this formulation can lead easily to failure of theoptimization process. Stability problems arise particularly when there is asudden increase in stress concentration. Due to this increment in stress,more material is removed than before. It is clear that this leads to an evenmore increased stress concentration. Finally this cyclic process will leadto failure in the optimization process due to breakage/disconnected regions(see Figure 52).

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(a) (b)

Figure 52: Failure due to sudden stress increment

Another drawback of this formulation is that if the material removed be-comes less than 4V the removal rate RR is increased by 4RR to continueto optimize. This sudden increment in RR can have the same disastrous ef-fect by removing too much material. In the compliance-volume graphs (e.g.Figure 35 (b)) this resulted in jumps in the compliance and volume. There-fore the optimization parameters have to be chosen very carefully since theyhave a large effect on the optimization process. When the parameters arevery small ISOESO gives best results w.r.t failure and the objective functionvalue in the final design. However, this leads to a large number of iterationsand to structures with large numbers of holes. In the next Subsection 12.1a new improved ISOESO algorithm is proposed by the author. This algo-rithm deals with these issues and avoids the use of many parameters whichall have their influence on the final design.

Questionable Optimum. A drawback of both methods is that the op-timization process stops suddenly when the target volume is reached. Theobjective function value at this target volume is taken as its optimum state.Apart from the question if the final design found by ESO can be consideredas an optimum; the author strongly believes that this ’optimum’ is not eventhe optimal design within the field of evolutionary methods. Regardless ofthe validity of the sensitivities used here, these should be zero to consider adesign as ’optimal’. This is clearly not the case for the these ESO methodsin which optimization is suddenly stopped when reaching a target volume.Therefore in Section 12.2 a new and more ’correct’ formulation is presentedin the form of an adaptive level set evolutionary method.

12.1 Proposal: An Improved ISOESO algorithm

In this Section a different formulation for ISOESO is presented. The prin-cipal idea is to control/limit to amount of removed material. This will limitthe changes in geometry between iterations and therefore avoid failure ofthe structure.

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The algorithms removes material in equal fractions. In this case the re-moval rate RR represents the volume fraction which will removed betweeniterations with respect to the volume at that iteration. Optimizing the struc-ture towards a target volume Vtarget in n iteration steps is then described interms of the initial volume Vinit as:

Vtarget = RRn × Vinit where 0 < RR < 1 (12.1)

The optimization parameter to be chosen in this case is the removal rateRR or the number of iterations n. Both are related by Equation (12.1). Itdepends on the preferences of the user which parameter is provided as aninput value.

The author believes that removing material by a fixed volume fraction isa more plausible way of removing material. The absolute amount of removedmaterial becomes less during optimization since RRi+1Vinit ≤ RRiVinit ∀i ∈N.

In the classical ISOESO the volume fraction was not regulated nor lim-ited during optimization since it depended on the maximum von Mises stress.The removed volume did not decrease when reaching a target volume. Thisproblem is avoided by this new optimization algorithm 4 which is an alter-native for ISOESO. The corresponding flow chart is depicted in Figure 53.

Algorithm 4 Alternative ISOESO Algorithmfor i = 1 to n do

Perform FG FEA −→ σVMSort σVMDetermine which σth correspond to a material removal of RRi×VinitialRemove material σVM ≤ σth

end for

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Figure 53: Proposed ISOESO algorithm

12.2 Proposal: An Adaptive Level Set ESO Algorithm

In this Section an improved extended algorithm is proposed for LS-ESO.This will be an adaptive evolutionary approach in which the sensitivities aretaking into consideration around the design state. In the current formulationfor both methods, the sensitivities (Von Mises stress along the boundary)are only used as a removal criterion. This new formulation could be animprovement on both methods.

The principal idea of the adaptive method proposed here is to optimizetowards a target volume and to ensure the sensitivities are zero at thistarget volume. The optimization algorithm is similar as LS-ESO until thetarget volume is reached. However, when this target volume is reachedin this adaptive algorithm the structure is optimized further until a state isreached where the sensitivities converged to zero. This state of the structurecorresponds to a design change (DC) that converged to zero. The designchange can be described for example as:

DC =|ρk − ρk−1||ρk−1|

Where ρ denotes the density vector in which the entries correspond tothe density of the individual elements. k denotes the iteration step. Thenorm of the difference between two design stages gives an indication of thedesign change.

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The implementation of this idea is as follows: the level set function is evolvedunder a fixed removal rate until it converges to a steady state solution whereDC < ε where ε is a very small number. At this point the current volume isevaluated with respect to the target volume and the removal rate is increaseor decreased depending on the sign of Vk − Vtarget. If Vk − Vtarget is positiveRR is increased since more material has to be removed to reach the targetvolume. If negative, RR is decreased to add material. In algorithm 5 andthe Adaptive LS-ESO method is depicted in 5.

Algorithm 5 Adaptive LS-ESO algorithmwhile |V − Vtarget| > ε1 or DC > ε2 do

if DC > ε2 thenRR = RR

else if DC < ε2 and V < Vtarget thenRR = RR−4RR

else if DC < ε2 and V > Vtarget thenRR = RR+4RR

end ifend while

Figure 54: Adaptive Level Set ESO algorithm

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Part VI

Conclusion and Recommendations

In this Part, conclusions of the results and discussion presented in Part IVand Part V are made. Furthermore, recommendations are presented forfurther research.

13 Conclusions

The results indicated that ISOESO gave more optimal designs than LS-ESO(5−10% more optimal w.r.t objective function value). This contradicted theexpected improvement by LS-ESO as a bi-directional method. It appearedthat the reason that LS-ESO lacked to give a more optimal design is exactlya weakness in the formulation of all ESO methods:

• Removing material by local criteria and stop when a certain targetvolume is reached.

• Using local criteria which do not ensure movement in a direction forwhich the objective function decreases.

These drawbacks had an even worse effect on LS-ESO since the Levelset model could not take full advantage of its bi-directionality due to thissudden stop criterion. Furthermore the heuristic chosen velocity profile isbased on these heuristic removal criteria from which it is known that it doesnot guarantee to move in the right direction.

A possible solution to improve LS-ESO is to use an alternative adaptiveLS-ESO. In this method the the sensitivities are taking into considerationaround the design state. The algorithm is similar to LS-ESO except thatan optimum is found for which the velocities on the boundary converged tozero. In this formulation the motion of the interface described by the levelset model is less restricted.

However, the author emphasizes that this new adaptive LS-ESO mayresolve this subproblem in the field of ESO but other limitations of ESO(limited to single load case, no theoretical basis, etc.) will remain a prob-lem. The author believes that drawbacks, limitations and of ESO can beavoided by considering the topology optimization problem by gradient basedsolution, e.g. a level set based optimization method using global sensitivi-ties as the shape or topological gradient. The main advantage is that thesensitivity information ensures to move the interface in a direction for whichthe objective function decreases. Furthermore it can be applied to multipleload cases and is not limited to fully stressed design problems.

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14 Recommendations for Further Research

Recommendations for further research include direct improvements on thelevel set algorithm by the implementation of different techniques and othermore general recommendations on the algorithms used.

Figure 55: Narrow band

Narrow Band Method. An important drawback of the use of the levelset method is that the computational domain for the boundary represen-tation one order higher than in the explicit representation. A method todecrease the number of computations was proposed by Sethian [27] and isthe so-called narrow band method. This method takes benefit of the non-physical meaning of the implicit function off the interface. However, toevolve the interface on needs to be able to take a spatial derivative near andalong the interface. Therefore one could construct a narrow band aroundthis interface so that only the grid points inside this narrow band form Partof the computational domain. Its advantage is that it reduces the number ofoperations significantly. E.g. solving the HJ-equation on a two dimensionalN2 grid, the narrow band method will reduce the computational domainto an order of O(kN), where k is the number of grid cells in the width ofthe narrow band (see Figure 55). One problem that arises using the narrowband approach is the need of reconstruction of the narrow band when theinterface approaches the inner or outer edge of the narrow band. Choosing ktoo width will increase the computational domain while a too small k leadsto more reinitialization. In the textbook of Sethian [27] a bandwidth of 12grid points is proposed.

The author proposes a Narrow Band Method in which the bandwidthdepends on the velocity field on the interface. Since Vn determines how fastthe interface moves into the direction of the edge of the narrow band.

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Fast Marching method. In this thesis the signed distance function wasconstructed by solving the reinitialization equation which was discussed inSection 3.4.1. The reinitialization equation was solved iteratively by a HJ-ENO scheme.

A technique recommended to reduce computational time for the con-struction of the signed distance is the Fast Marching method [26]. Thismethod is a fast technique of solving the Eikonal equation and is usedfor the description of an interface that only move in one direction (in-wards/outwards):

|∇φ|F = 1 φ = 0 ∈ Γ (14.1)

Where F is the velocity and φ the implicit function. Signed distance canbe constructed around the interface by solving this equation for a normalvelocity of F = 1 or F = −1. One can see that the construction of signeddistance is now described as a boundary value problem with the locationof the interface as boundary condition. Instead of an iterative procedureto solve this equation the fast marching method is used which uses upwindinformation to solve this equation rapidly. Since information propagatesfrom smaller values of φ to larger values of φ one could built the solutionmarching out from the interface, from the smallest value. Due to the upwindcharacter larger values have no influence on smaller values. If the order of φfrom small too large was known a priori, on a grid of M nodes this methodwould only need M steps. In reality the smallest value has to be found inthe set of outside nodes that are adjacent to inside node (nodes next to theboundary). However, a fast heap sort technique is used which makes thismethod fast (O(M logM) in comparison with iterative schemes.

For a detailed description of this method is referred to the text book ofSethian [27] and in an Introduction to Level Set Based Optimization by theauthor [31]

Extended Velocity Field. Another method to reduce computationaltime is to conserve signed distance to avoid reinitialization. This can beachieved by constructing an extended velocity field. Remember that theprincipal reason for the deviation from signed distance of the level set func-tion is the variation of the velocity field in normal direction. The solutionpresented here is based on the fact that the only interest is the motion of thezero isocontour which represents the interface. The velocity field given onthis interface determines the motion of this interface. The other isocontoursφ = k ∀ k ∈ R are meaningless and are defined in such a way that it givesless numerical errors for the spatial discretization. Therefore, one is free todefine φ as a signed distance function as discussed before. One is also freeto choose the velocity field for the nodes off the interface. Therefore, theextended velocity field Vext is built which satisfies:

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∇Vext ·N = 0 where Vext = Vn ∀ x on Γ (14.2)

Which restricts Vext to be constant in normal direction.

Test Different Velocity Profiles. In this thesis the evolutionary ap-proach uses local criteria. The velocity field is related to this local VonMises stress criteria by the velocity profile described in Section 5.1. Thechoice of this velocity field is completely heuristic and the same as Sethianused in [28]. A recommendation for further research is test different profilesand their effect on the final design.

Use more accurate stress recovery method. The stress recovery methodwas a weighted average method discussed in Section 1.5. In this formulationthe derivative of the stress field is discontinuous between the elements. Theauthor believes that this is the main reason the boundary did not conserveits smoothness during optimization (see Section 11). Using a stress recoveryscheme as the Superconvergent patch recovery [36] method gives a continu-ous and more accurate stress field. This results will results in conservation ofthe smoothness of the boundary so that no artificial smoothing is required.

Extension to 3D using Marching Cubes. The extension to a three-dimensional model can be achieved by using Marching Cubes. The MarchingCubes algorithm is a fast way of representing a three-dimensional (iso)surface.The algorithm reconstructs a surface by discretizing the computational do-main in cubes. Then the cubes are distinguished by the way they are crossedby the isosurface. There are 28 = 256 possible ways of this isosurface to passtrough the cube. These different configurations can be distinguished by cou-pling bits to the eight nodes of the cube. The bit takes 0 or 1 depending ifthe nodal value is lower or higher than the isosurface value. Using symmetryand rotations the 256 cases can be reduced to 15 individual cases. Thesepredetermined cases are stored in an array which makes the algorithm fastsince the cubes are identified and not reconstructed separately. For level setbased optimization this algorithm can be used over the three dimensionallevel set grid. For a detailed description is referred to [13]

Test the algorithms proposed in Section 12. In Section 12 two alter-native methods were presented:

• An ISOESO algorithm which is based on controlling the amount ofremoved material.

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• An adaptive LS-ESO algorithm for which the design state at the targetvolume is optimized further to a state where the sensitivities convergeto zero.

These methods are expected to give better results. In particular theadaptive LS-ESO method seems promising since it includes sensitivity in-formation to determine the optimal state.

Combination of LS-ESO and ISOESO. A last recommendation withrespect to these ESO methods is to test a combination of both methods. Anew method could be one in which holes are cut along the von Mises iso-stress contours and evolved by the level set model until there is no designchange. When there is no design change new holes are cut and the sameprocedure is repeated until a target volume is reached.

The implementation of this idea should be combined with a fast marchingtechnique for the construction of the level set function explicitly from fromthe holes. Therefore, it is not easy to implement in the current code wherethe level set function is initially constructed containing a number of holesand evolved in an Eulerian way on the nodes.

Switch to gradient-based solution methods The ESO method hassome significant problems and limitations. A general and fundamentallymore correct way of solving optimization problems can be achieved by gra-dient based methods. The author proposes to focus on these methods infuture research.

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Appendices

A Principle of Virtual Work for Solid Body Me-chanics

Linear elastic behavior is considered in this research. In the linear elasticregion a body returns to its original state after deformation. Furthermore,the stress is proportional to its deformation following Hooke’s law.

The state of a body can be described by a set of differential equations.These differential equations can be formulated into a variational form. Thisvariational/weak formulation is the starting point for the finite element for-mulation. The structural domain is discretized in finite elements using shapefunctions. In this Section, the finite element formulation is derived.

A.1 Internal Equilibrium

In Figure 56 internal equilibrium in x-direction is depicted for a differentialelement. σix for i = x, y, z are the components in the stress tensor workingin x-direction on the planes with a normal i = x, y, z. fx is a volume forceworking in the x-direction on the differential element.

Figure 56: Equilibrium in x-direction for a differential element

σxx(x+ dx)− σxx(x)dydz + σyx(y + dy)− σyx(y)dxdz + ...

...+ σzx(z + dz)− σzx(z)dxdy + fxdxdydz = 0(A.1)

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Further simplification can be achieved by a Taylor expansion. In x-directionthis is:

σxx(x+ dx) ≈ σxx(x) +∂σxx∂x

dx+O(dx2) + ...

⇒ ∂σxx∂x

dx ≈ σxx(x+ dx)− σxx(x)

(A.2)

Using this Taylor expansion (and the two other spatial dimensions) in Equa-tion (A.1) and dividing by dV = dxdydz gives:

∂σxx∂x

+∂σyx∂y

+∂σzx∂z

+ fx = 0 (A.3)

The same force equilibrium can be obtained in the other two spatial direc-tions. The general internal equilibrium in index notation becomes then:

σji,j + fi = 0 for i, j = x, y, z (A.4)

Where the notation is used ∂∂xj

=,j .

Finally, moment equilibrium results in:

σji = σij (A.5)

Which is the symmetry condition for the stress tensor.

A.2 Constitutive Equations

The constitutive equations give the stress-strain relationship for the materialconsidered. In this case the material is linear elastic and the stress-strainrelationship is described by Hooke’s law (A.6).

σij = Eijklεkl (A.6)

In which εkl is Cauchy’s strain tensor:

εkl =12

(∂uk∂xl

+∂ul∂xk

) (A.7)

Eijkl is the Elasticity tensor. For an isotropic material it is given by:

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Figure 57: Design Domain

σij = λεkkδij + 2µεij (A.8)

Where λ and µ are:

λ =Eν

(1− 2ν)(1 + ν), µ =

E

(2(1 + ν)(A.9)

These are the Lame constants which depend on the Young’s modulusand Poisson ratio.

A.3 Boundary Value Problem for an Elastic Body

The boundary value problem for a linear elastic body as in Figure 57 is thengiven by the equilibrium Equation (A.4), constitutive Equation (A.6) andboundary conditions as:

−σji,j = fi for i, j = x, y, z

σij = Eijklεkl

With boundary conditions:

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ui = gi on Γuσjinj = hi on Γt

(A.10)

Where Γ = Γu⋃

Γt, Γu⋂

Γt

Which are the Dirichlet and von Neumann boundary conditions, respec-tively. The Dirichlet boundary conditions represents a restriction on thedisplacement. The von Neumann condition represents the boundary trac-tion.

A.4 Weak formulation

For the finite element method one needs the boundary value problem inintegral form. This is done by transforming equations (A.10) into the weakformulation by multiplying it by the test function (virtual displacement) vwhich satisfies the boundary conditions and integrating over the structuraldomain Ω. ∫

Ω−∂σji∂xj

vidΩ =∫

ΩfividΩ (A.11)

Using the product rule (σjivi),j = σji,jvi + σjivi,j and Gauss divergencetheorem gives: ∫

Ωσjivi,jdΩ =

∫ΩfividΩ +

∫ΓσjinjvidΓ (A.12)

The left hand side of Equation (A.12) can be simplified by using the sym-metry of the stress tensor:

σji(u)vi,j = σij12

(vi,j + vj,i)

= σijεij(v) (A.13)

So that Equation (A.12) becomes:∫Ωσij(u)εij(v)dΩ =

∫ΩfividΩ +

∫Γσji(u)njvidΓ (A.14)

Substituting the boundary conditions (A.10) and constitutive relation (A.6)into (A.14) give:

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∫ΩEijklεkl(u)εij(v)dΩ =

∫ΩfividΩ +

∫Γt

hividΓ

ui = gi on Γuvi = 0 on Γu (A.15)

This is the weak form of the boundary value problem and the starting pointof the finite element method. This formulation is known as the Principleof virtual work for solid body dynamics. This principle states that equilib-rium of a body requires that for any compatible small virtual displacementimposed on the body in its state of equilibrium the total internal virtualwork is equal to the total external virtual work. In Equation (A.15) the lefthand side represents the internal virtual work and the right hand side theexternal virtual work.

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