Matlab implementation of 3D topology optimization using BESO

Incorporating Sustainable Practice in Mechanics of Structures and Materials–

Fragomeni,Venkatesan,Lam&Setunge(eds)

©2011Taylor&Francis Group,London,ISBN978-0-415-61657-7 Matlab implementation of3D topology optimization using BESO

R.Huang&X.Huang

School of Civil,Environmental and Chemical Engineering,RMIT University,Melbourne,Australia ABSTRACT:This paper demonstrates a proof of concept for an open source3D finite element engine for structural topology optimization using BESO implemented in Matlab.There have been numerous attempts for the various algorithms to perform2D Finite Element Modeling for Topology Optimization using both the SIMP and BESO methods.There have also been attempts for3D modeling using the SIMP method;however,no attempts have been made to perform accurate3D topology optimization using a non commercial FEA package. Examples shown in this paper demonstrate the capability of the current Matlab code for topology optimization for3D structures using the BESO method.

1INTRODUCTION

Over the past decades,structural topology optimiza-tion has been exhaustively explored and many opti-misation methods have been developed,for example Solid Isotropic Material with Penalization(SIMP) method(Bendsoe,1989)and the evolutionary struc-tural optimisation(ESO)method(Xie&Steven, 1993)and its later version bi-directional ESO(BESO) method(Huang&Xie,2007).Sigmund(2001)has developed a highly efficient99-line algorithm using Matlab for2D topology optimization using the SIMP method.A3D structural optimization application writ-ten in C++,called Topostruct(Michalatos&Kaijima, 2008)using the SIMP method.

The importance of structural optimization is obvi-ous.The inevitable need for structures to become more optimized or efficient has become the foundation of modern structures.Topology optimization offers a mathematical modelling method to determine a more efficiently shaped structure,allowing the structure to reduce self weight,while maintaining,more or less,its original strength,which in turn allows for even higher or longer spanning structures.

The Evolutionary Structural Optimization method was first introduced by Prof.Mike Xie and Grant Steven in the early1990s(Xie&Steven,1993).The concept was rather simple,as with nature’s evolu-tion of structural elements for its own components, the process of structural optimization can be used to evolve a structural element towards optimum.This can be achieved by slowly removing material that is inefficient,through the use of FEA to guide the struc-ture to evolve along a particular evolutionary path (Xie&Steven,1993).The Bi-directional Evolution-ary Structural Optimization(BESO)method was first developed in1998as an improved methodology for structural topology optimization based on the ESO method(Querin,Steven,&Xie,1998).The current development of the BESO method has been described by Huang and Xie(2009).

The current BESO method can be easily imple-mented within Matlab for2D structures(Huang&Xie, 2009).However,there is no program for3D structures using the BESO method due to the complexity of the program.The main aim of this paper is to develop Mat-lab BESO code for optimising3D structures based on the algorithm presented in Huang&Xie(2009).

2METHOD

There are several fundamental assumptions made in the production of this algorithm.Since BESO is a post processing criterion,it is easily applicable to problems of any dimension.The BESO Matlab algorithm written by Huang&Xie(2009)uses BESO utilizing mate-rial Interpolation Scheme with Penalization(Huang &Xie,2009).For all intents and purposes,the3D algorithm is no different.Therefore,the mathematics and all derivations of Finite Element Analysis used in the algorithm presented in this paper remains the same as per Huang&Xie(2009).This paper will only present the changes and modification deemed neces-sary by the authors,to transpose the Huang&Xie (2009)algorithm into3D,for the other working parts of the algorithm,please refer to Huang&Xie(2009) and Sigmund(2001).

As previously mentioned,objective of the paper is to expand the Matlab algorithm for the soft-kill BESO method and transpose that from2D into3D.Even though it may seem simple,as the challenge is merely adding an extra dimension onto the existing code,the challenges are numerous and all but simple.The vari-ous challenges for this particular expansion are not only limited by the vast increase in computational quantity,but also to maintain consistency and avoid accumulative errors.The extra dimension of the BESO

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