Efficient generation of large-scale pareto-optimal topologies

The objective of this paper is to introduce an efficient algorithm and implementation for large-scale 3-D topology optimization. The proposed algorithm is an extension of a recently proposed 2-D topological-sensitivity based method that can generate numerous pareto-optimal topologies up to a desired volume fraction, in a single pass. In this paper, we show how the computational challenges in 3-D can be overcome. In particular, we consider an arbitrary 3-D domain-space that is discretized via hexahedral/brick finite elements. Exploiting congruence between elements, we propose a matrix-free implementation of the finite element method. The latter exploits modern multi-core architectures to efficiently solve topology optimization problems involving millions of degrees of freedom. The proposed methodology is illustrated through numerical experiments; comparisons are made against previously published results.

[1]  William E. Lorensen,et al.  Marching cubes: A high resolution 3D surface construction algorithm , 1987, SIGGRAPH.

[2]  M. Bendsøe,et al.  Generating optimal topologies in structural design using a homogenization method , 1988 .

[3]  V. Kobelev,et al.  Bubble method for topology and shape optimization of structures , 1994 .

[4]  Jan Sokolowski,et al.  On the Topological Derivative in Shape Optimization , 1999 .

[5]  U. Kirsch.  Combined approximations – a general reanalysis approach for structural optimization , 2000 .

[6]  J. Cea,et al.  The shape and topological optimizations connection , 2000 .

[7]  T. E. Bruns,et al.  Topology optimization of non-linear elastic structures and compliant mechanisms , 2001 .

[8]  J. Petersson,et al.  Large-scale topology optimization in 3D using parallel computing , 2001 .

[9]  Ole Sigmund,et al.  A 99 line topology optimization code written in Matlab , 2001 .

[10]  G. Allaire,et al.  A level-set method for shape optimization , 2002 .

[11]  Byung Man Kwak,et al.  Design space optimization using a numerical design continuation method , 2002 .

[12]  Xiaoming Wang,et al.  A level set method for structural topology optimization , 2003 .

[13]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[14]  G. Allaire,et al.  Structural optimization using sensitivity analysis and a level-set method , 2004 .

[15]  G. Allaire,et al.  A level-set method for vibration and multiple loads structural optimization , 2005 .

[17]  Charles E. Augarde,et al.  An element-based displacement preconditioner for linear elasticity problems , 2006 .

[18]  Edgardo Taroco,et al.  Topological-Shape Sensitivity Method: Theory and Applications , 2006 .

[19]  Raúl A. Feijóo,et al.  Topological Sensitivity Analysis for Three-dimensional Linear Elasticity Problem , 2007 .

[20]  Robert Strzodka,et al.  Performance and accuracy of hardware-oriented native-, emulated- and mixed-precision solvers in FEM simulations , 2007, Int. J. Parallel Emergent Distributed Syst..

[21]  E. Sturler,et al.  Large‐scale topology optimization using preconditioned Krylov subspace methods with recycling , 2007 .

[22]  Lin He,et al.  Incorporating topological derivatives into shape derivatives based level set methods , 2007, J. Comput. Phys..

[23]  M. Bendsøe,et al.  A topological derivative method for topology optimization , 2007 .

[24]  Krishnan Suresh,et al.  Generalization of Topological Sensitivity and Its Application to Defeaturing , 2007, DAC 2007.

[25]  Krishnan Suresh,et al.  Feature sensitivity: A generalization of topological sensitivity , 2008 .

[26]  Ole Sigmund,et al.  Approximate reanalysis in topology optimization , 2009 .

[27]  Kurt Maute,et al.  Large-scale parallel topology optimization using a dual-primal substructuring solver , 2008 .

[28]  George I. N. Rozvany,et al.  A critical review of established methods of structural topology optimization , 2009 .

[29]  Sylvia R. Almeida,et al.  A simple and effective inverse projection scheme for void distribution control in topology optimization , 2009 .

[30]  Krishnan Suresh,et al.  An efficient numerical method for computing the topological sensitivity of arbitrary‐shaped features in plate bending , 2009 .

[31]  G. Yoon.  Maximizing the fundamental eigenfrequency of geometrically nonlinear structures by topology optimization based on element connectivity parameterization , 2010 .

[32]  O. Sigmund,et al.  Efficient use of iterative solvers in nested topology optimization , 2010 .

[33]  Glaucio H. Paulino,et al.  Polygonal finite elements for topology optimization: A unifying paradigm , 2010 .

[34]  Mark Dickinson,et al.  A Congruence Problem for Polyhedra , 2010, Am. Math. Mon..

[35]  Tam H. Nguyen,et al.  A computational paradigm for multiresolution topology optimization (MTOP) , 2010 .

[36]  Krishnan Suresh,et al.  A 199-line Matlab code for Pareto-optimal tracing in topology optimization , 2010 .

[37]  Tracing the Envelope of the Objective-Space in Multi-Objective Topology Optimization , 2011, DAC 2011.

[38]  Krishnan Suresh,et al.  Efficient generation of pareto‐optimal topologies for compliance optimization , 2011 .

[39]  O. Sigmund,et al.  On reducing computational effort in topology optimization: how far can we go? , 2011 .

[40]  Volker Schulz,et al.  A 2589 line topology optimization code written for the graphics card , 2011, Comput. Vis. Sci..

[41]  Mostafa Khanzadi,et al.  An isogeometrical approach to structural topology optimization by optimality criteria , 2012 .

[42]  Glaucio H. Paulino,et al.  PolyTop: a Matlab implementation of a general topology optimization framework using unstructured polygonal finite element meshes , 2012 .

[43]  Glaucio H. Paulino,et al.  Topology optimization for braced frames: Combining continuum and beam/column elements , 2012 .

[44]  O. C. Zienkiewicz,et al.  The Finite Element Method for Solid and Structural Mechanics , 2013 .