a nodal based evolutionary structural … nodal based evolutionary structural optimisation algorithm...

A nodal based evolutionary structural optimisation algorithm

Y.-M. Chen1, A. J. Keane2 & C. Hsiao1

1National Space Program Office (NSPO), Taiwan 2Computational Engineering & Design Centre, UK

Abstract

This paper proposes a simple and interesting algorithm dealing with structural shape, topology and thickness optimisation simultaneously. The proposed approach is based on an interesting idea of migrating boundary nodes in an iterative manner. An intuitive nodal-based evolutionary structural algorithm drives the optimisation process. Finite element analysis is required at each stage to reveal the relative stress distribution of the evolving structure, from which the lowly stressed edge nodes are identified as design variables and shift towards the higher stressed areas within the design domain during optimisation. Migrating the geometry boundary nodes directly changes the structural shape and thus perform shape optimisation. By introducing a fixed shape circular cavity into the evolving structure during optimisation initiates topology optimisation. Further shifting the cavity boundary nodes during optimisation gradually reveals the structural topology. In addition, nodal thickness is gradually decreased to enhance the reduction of structural weight. One feature of this nodal based approach is that by employing boundary nodal coordinates as design variables the interior finite element mesh becomes irrelevant to the shape definition of structures, thus unstructured mesh may be used. This enhances the compatibility on integrating the proposed algorithm with auto-meshing capability which is incorporated in most commercial finite element software today. A couple of benchmark problems, including the classical MBB beam, are used for illustration purposes. Keywords: nodal based approach, structural optimisation, evolutionary optimisation, finite element, topology optimisation.

Computer Aided Optimum Design in Engineering IX 55

© 2005 WIT Press WIT Transactions on The Built Environment, Vol 80, www.witpress.com, ISSN 1743-3509 (on-line)

1 Introduction

Predicting ideal topologies is an active research subject within the field of structural optimization. Currently, the discretised optimality criteria method (DCOC) of Zhou and Rozvany [1,2] is considered to be highly efficient in handling a structure with a very large number of elements and active stress constraints. The DCOC method works well in multipurpose generalized shape optimization problems by making use of the so-called SIMP method (Solid Isotropic Microstructure with Penalty for intermediate densities) (see [3] and [4]). The SIMP approach models the material properties as the relative material density raised to some power times the material properties of the solid material [4] producing solid-empty (SE) solutions showing very good approximations to known exact solutions. However, the approach has been criticized because the solutions obtained can lack easy physical interpretation. Bendsoe and Sigmund [5] suggests a physical interpretation to the power-law approach by giving a few simple conditions that need to be satisfied. In contrast, the so-called evolutionary structural optimization (ESO) method [6] is based on repeatedly removing low stress elements from the design model during an evolutionary process. The evolutionary approach has been much criticized: Zhao et al. [7] note that the classical stress based rejection ratio and evolutionary rate were the main weaknesses of the ESO approach. As a result, the use of nodal displacement has been suggested to replace the classical ESO stress based criteria. In addition, Zhou and Rozvany [8] show that the ESO rejection criteria may result in a highly non-optimal design through a simple test example. Researchers within the field of topology optimization have investigated various ways of finding the optimum distribution of the selected elements or materials within a design domain. There is a commonality shared by most of the existing topology optimization algorithms, i.e., the use of repetitive (periodic) elements (microstructure) to discretise the design domain. The optimization is then based on stresses, displacements, material properties (densities) or mesh-densities. As a result, the solution converged to is represented by an optimum distribution of elements within the domain. In other words, the final geometry is made up of building blocks of the elements chosen. However, the objectives of structural optimization are usually to reduce structural weight and at the same time maintain boundary smoothness. Most topology optimization approaches are capable of obtaining impressive solutions that satisfy functional requirements, but the boundary smoothness criterion is usually satisfied only by image post processing or interpolations (see [5] and [9]). In a previous paper [10], the author proposed a method which combined the idea of moving boundary nodes and evolutionary optimization together, producing the Nodal-Based Evolutionary Structural Optimization (NESO) approach. In addition, a parallel optimization algorithm was introduced which extended the capabilities of the NESO method to topology optimization. Here, a refined and extended algorithm is presented. NESO algorithm that integrates shape, topology and thickness optimization together is presented in this paper. Nodal migration strategy has been published (see [10]); details will not be

56 Computer Aided Optimum Design in Engineering IX


discussed here again. The subject of this paper is to present the NESO algorithm which is outlined in the subsequent sections.

2 Nodal based evolutionary structural optimisation algorithm

The NESO algorithm is an infinite loop that drives the optimization. The idea behind the NESO algorithm is to move low stress nodes using the NESO method iteratively. These low stress nodes are identified by comparing the nodal stress values with the maximum nodal stress value maxVM

Nσ in the design domain.

Figure 1: NESO algorithm.

Creation of an initial oversize design domain

FEA

yes Upper limit of optimum ratio (OR) reached?

no

Any external edge or cavity nodes with stress less than ORVM

Nmaxσ ?

no

OR=OR+δOR yes

External edge movement and Cavity edge movement

FEA

Any internal ineffective points?

yes

Cavity Formation Algorithm

no

Thickness Optimisation Algorithm FEA

Has the minimum nodal stress of structure edges reached ORVM

Nmaxσ ?

no

yes

Stop



All the edge nodes with nodal stress values less than the product of ORVMN

maxσ are classified as ineffective nodes. Ineffective nodes are allowed to participate in the node shifting process. As the low stress nodes gradually shift towards the high stress locations within the design domain, the stress level increases. When there are no more ineffective nodes, i.e., no more nodes with stress values less than the product of ORVM

Nmaxσ the OR value is increased by a small amount δOR

until ineffective nodes are found, and then the optimization continues. The NESO algorithm terminates when OR=1. This criterion is never attained because the minimum stress at the structure edges is never going to reach ORVM

Nmaxσ . This

allow the optimization continues indefinitely and allows designers to choose the appropriate end point rather than simply relying on experience to prescribe a terminating criterion for each optimization. The NESO algorithm combines both shape and topology optimizations in the following way: for topology optimization, a cavity formation algorithm is used within the NESO algorithm to create circular cavities in the internal design domain during the optimization. In addition, after cavities are created within the domain, the cavity edge nodes are treated as shape design variables and qualify for the node-shifting process. This allows freedom to these nodes to explore the surrounding low stress areas and to evolve into a better cavity shape. Therefore, within the NESO algorithm both the external boundary nodes and the cavity nodes are treated as shape design variables. The NESO algorithm is depicted in fig. 1. The cavity formation algorithm and thickness optimisation algorithm within the NESO algorithm is used for topology optimization and is discussed separately in the subsequent sections.

2.1 Cavity formation algorithm

The cavity formation algorithm lies at the heart of the NESO method in topology optimization. This algorithm is used to determine the cavity locations within the unstructured meshed design domain during the evolutionary optimization. During the optimization, circular cavities are inserted into the domain. The initial radius of each new circular cavity is a user defined variable and needs to be small enough relative to the internal ineffective (low stress) regions within the domain. Here, the ineffective point is defined to be a point inside the design domain that has a nodal stress that is less than the minimum nodal stress along the external boundary nodes. The ineffective regions (groups of ineffective points) within the domain are identified by searching through the internal domain with dense grid points in both X- and Y- directions. Decreasing the point step size increases the resolution of the ineffective regions. When the domain has been searched with sufficiently small search step points, ineffective points are often found just inside the external boundaries and near existing cavities within the domain. The areas covered by these low stress points are generally very small and they usually disappear after the next few iterations of structure edge and/or cavity edge movement. Hence, we can state that cavities should be introduced within the domain only as long as their locations are neither near existing cavities or adjacent to the structure boundary.



Figure 2: Cavity formation algorithm.

Therefore, it is better to allow the existing external edges or cavities to explore these nearby ineffective regions in later iterations rather than to introduce a cavity at these points. A cavity formation criterion is therefore used to decide where cavities should be introduced within the domain during the evolutionary design process. To avoid cavities being introduced near edges or cavities during an evolutionary process, we use two control parameters: edgeDmin

For each ineffective point Delete ineffective points

no edgeedge DD min>

cavitycavity DD min> no

yes

Record all qualified ineffective point coordinates

Delete ineffective points

While the ineffective point list is not empty

Find the lowest nodal stress ineffective point

For each remaining ineffective point:

cavitycavity DD min> yes

no

Group all points within a radius R from the lowest stress ineffective points Remove all points in group

from ineffective point list

Determine the cavity centre for this group of ineffective points

Create a cavity



and cavityDmin . edgeDmin is the distance measured from the external boundary edge. Any ineffective points lie at a point within the design domain with a distance to the closest external boundary less than the edgeDmin is automatically deleted. The retained ineffective points then need to be checked for the distance to any existing cavities within the domain. To do this, cavityDmin is used to define a distance measured from the cavity edges. For each retained ineffective point, if the distance to the nearest cavity node is greater than cavityDmin then such a point is retained for cavity formation; otherwise it is deleted. When the domain is searched with small step sizes, ineffective points that are not near any existing cavities or external boundaries often appear in groups and the center of each cavity is then taken as the mean of the X- and Y- coordinates of the ineffective points for each group.

2.2 Thickness algorithm

Following the typical evolutionary process where the starting point for each optimisation is to create an initial oversized design domain, the thickness optimisation also starts by assigning each mesh point with a large thickness value. Initially, every mesh point has the same thickness value, i.e., a flat geometry. During optimisation nodal thickness is gradually decreased (representing the trimming off surface materials). With re-meshing constantly invoked during optimisation, a potential difficulty arises in accumulating the nodal thickness iteratively. One way to overcome such a difficulty is via interpolation: the nodal thickness distribution is firstly recorded before initiating re-meshing. After re-meshing, and based on the new mesh point locations (X and Y coordinates) within the design domain, new thickness values for each node are interpolated from the recorded nodal thickness distribution. The major advantage of interpolating thickness values for previous mesh data is that smooth surface variation can be maintained during iterative optimisation. In addition, nodal thickness interpolation is independent of nodal movement or cavity insertion in the plane since every node in the mesh always has a thickness value. The thickness algorithm is discussed next. The thickness optimisation algorithm starts by identifying the low stress nodes within the design domain, any nodes with nodal stress iσ less or equal to the mean nodal stress ( VMmean

Nσ ) of the design domain being treated as thickness design variables. The nodal thickness of these nodes is then reduced. For each qualifying thickness design variable, the nodal thickness is changed by an amount calculated using eqn. (1):

,....2,1,1 =

−−= iTtt dVMmean

N

iii

σ

σ (1)

where it is the nodal thickness at node i and dT is a user prescribed unit magnitude for thickness reduction. eqn. (1) reduces nodal thickness based on the stress ratio. If the nodal stress of node i is relatively small compared with the



mean stress of the design domain, then eqn. (1) tends to decrease the nodal thickness with a larger magnitude. Conversely, if the nodal stress of node i is closer to VMmean

Nσ , the nodal thickness is reduced by a smaller amount.

}max{% tntLB = (2)

Figure 3: Thickness optimisation algorithm.

By reducing the nodal thickness iteratively using equ. (1) during optimisation the nodal thickness may reach zero. To overcome this problem, a lower bound tLB (equ. (2)) is imposed. The thickness lower bound (tLB) is treated as the minimum acceptable nodal thickness value during optimisation and is a user prescribed value. The lower bound tLB can be a prescribed value, say 10% of the initial thickness value. Here, the lower bound is set to be a percentage of the maximum thickness in the thickness distribution list. Equ. (2) has the advantage of automatically calculating the lower bound based on a prescribed percentage (n) of the maximum thickness value found among the mesh points during optimisation. During iterative nodal thickness reduction, if any nodal thickness reaches the lower bound tLB the nodal thickness distribution is then scaled according to equ. (3).

LBLBnew tttt

tt +−= )}(max{}max{

}{}{ (3)

Equ. (3) makes sure the nodal thickness distribution is within the range between the nodal thickness lower bound tLB and the maximum nodal thickness value

Identify nodes with nodal stress maxVMNi σσ <

Apply the thickness reduction equation

Any nodal thickness less than the thickness lower bound LBt ?

yes

Apply thickness scaling equation no



found within an evolving geometry during optimisation. Noted if we substitute equ. (2) into equ. (3) we get

}max{%%)%100(}{ tnnttnew +−= (4) So if we used 10% we would get

}max{%10%90}{ tttnew += (5) Now, as max{t}>t we get tnew is greater than t. In this way, the lowest nodal thickness value is always greater than the prescribed thickness lower bound. In other words, during iterative cycles of thickness reduction, a relative thickness distribution can be obtained within a prescribed range. The thickness optimisation algorithm is summarized in fig. 3.

3 Optimisation

In this section, three commonly used examples (examples A, B & C, see figures 4,5,6, respectively) are presented to demonstrate the proposed NESO algorithm in handling both topology (examples A & B – 10mx5mx1m) and thickness optimisations (example C-10mx5mx2m). In both cases A and B, plane stress problems are been considered but in example C, the element thickness is taken as the mean of three nodal thickness for each planar triangular element during stress calculation. In addition, in example C, it is assumed that a point force is applied at the centre plane (i.e., at the plane z = 1m) and due to symmetry; optimisation of only half of the initial design domain is needed (i.e., dimensions 10mx5mx1m).

Figure 4: Initial starting point of example A.

Figure 5: Initial starting point of example B.

Figure 6: Initial starting point of example C.

For all three examples, the magnitude of the applied force is 100 N. The design domains are meshed with unstructured meshes and each element has the following standard properties: E=210 GPa, ν=0.3, ρ=7800 kg/m3. The nodal thickness lower bound tLB required in example C in set to 10% of the maximum thickness and a brief optimisation history for each example is included for



illustration purposes: The full optimised geometry of example C is visualized by producing a mirror reflection about the centre plane. A brief optimisation histories for examples A, B & C are shown through fig. 7~9, fig. 11~13 & fig. 15~17, respectively. The theoretical truss solution for these examples are shown in figures 10, 14 & 18 for comparison.

Figure 7: Optimisation history of example A: stage 1.

Figure 11: Optimisation history of example B: stage 1.

Figure 15: Optimisation history of example C: stage 1.







Figure 10: Theoretical truss solution of example A.

Figure 14: Theoretical truss solution of example B.

Figure 18: Theoretical truss solution of example C.



4 Concluding remarks

The nodal based evolutionary structural optimisation (NESO) algorithm has been presented. The cavity formation algorithm and thickness optimisation strategy that are used for topology and thickness optimisation have been discussed. A few benchmark problems have been used to demonstrate the proposed algorithm in producing optimum geometry from an initial oversized blank.

References

[1] Zhou, M. & Rozvany, G.I.N., An optimality criteria method for large systems Part I: theory, Structural Optimisation, 5, pp. 12 25, 1992.

[2] Zhou, M. & Rozvany, G.I.N., An optimality criteria method for large systems, Part II: algorithm, Structural Optimisation, 6, pp. 250 262, 1993.

[3] Rozvany, G.I.N. & Zhou, M. & Birker, T., Generalised shape optimisation without homogenization, Structural Optimisation, 4, pp 250 252, 1992.

[4] Sigmund, O., A 99 line topology optimisation code, Structural Optimisation, 21(2), pp 120 127, 2001.

[5] Bendsoe, M.P & Sigmund, O., Material interpolation schemes in topology optimisation, Archive of Applied Mechanics, 69, pp 635 654, 1999.

[6] Xie, Y.M. & Steven, G.P., Evolutionary Structural Optimisation, Springer Verlag, London, 1997.

[7] Zhao, C & Steven, G.P. & Xie, Y.M., A generalized evolutionary method for numerical topology optimisation of structures under static loading conditions, Structural Optimisation, 15, pp 251 260, 1998.

[8] Zhou, M. & Rozvany, G.I.N., On the validity of ESO type methods in topology optimisation, Structural Optimisation, 21, pp 80 83, 2001.

[9] Sigmund, O . & Petersson, J., Numerical instabilities in topology optimisation: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima, Structural Optimisation, 16, pp 68 75, 1998.

[10] Chen, Y.M & Bhaskar, A. & Keane, A., A parallel nodal based evolutionary structural optimisation algorithm, Structural and Multidisciplinary Optimisation, 23(3), pp 241 251, 2002.



a nodal based evolutionary structural … nodal based evolutionary structural optimisation algorithm...

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