Efficient generation of large-scale pareto-optimal topologies

Structural and Multidisciplinary Optimization - Tập 47 - Trang 49-61 - 2012
Krishnan Suresh1
1University of Wisconsin, Madison, USA

Tóm tắt

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.

Tài liệu tham khảo

Allaire G (2002) A level-set method for shape optimization. Comptes Rendus Math 334(12):1125–1130 Allaire G (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194(1):363–393 Allaire G, Jouve F (2005) A level-set method for vibration and multiple loads structural optimization. Comput Methods Appl Mech Eng 194(30–33):3269–3290 Almeida SRM, Paulino G, Silva ECN (2009) A simple and effective inverse projection scheme for void distribution control in topology optimization. Struct Multidisc Optim 39(4):359–371 Amir O, Sigmund O (2011) On reducing computational effort in topology optimization: how far can we go? Struct Multidisc Optim 44(1):25–29 Amir O, Bendsøe M, Sigmund O (2009) Approximate reanalysis in topology optimization. Int J Numer Methods Eng 78:1474–1491 Amir O, Stolpe M, Sigmund O (2010) Efficient use of iterative solvers in nested topology optimization. Struct Multidisc Optim 42(1):55–72 Augarde CE, Ramage A, Staudacher J (2006) An element-based displacement preconditioner for linear elasticity problems. Comput Struct 84(31–32):2306–2315 Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71:197–224 Bendsoe MP, Sigmund O (2003) Topology optimization: theory, methods and application, 2nd edn. Springer Borisov A, Dickinson M, Hastings S (2010) A congruence problem for polyhedra. Am Math Mon 117(3):232–249 Borrvall T, Petersson J (2001) Large-scale topology optimization in 3-D using parallel computing. Comput Methods Appl Mech Eng 190:6201–6229 Bruns TE, Tortorelli DA (2001) Topology optimization of non-linear elastic structures and compliant mechanisms. Comput Methods Appl Mech Eng 190(26–27):3443–3459 Céa J, Garreau S, Guillaume P, Masmoudi M (2000) The shape and topological optimization connection. Comput Methods Appl Mech Eng 188(4):713–726 Eschenauer HA (1994) Bubble method for topology and shape optimization of structures. Struct Optim 8:42–51 Evgrafov A, Rupp CJ, Maute K, Dunn ML (2008) Large-scale parallel topology optimization using a dual-primal substructuring solver. Struct Multidisc Optim 36:329–345 Feijóo RA (2005) The topological-shape sensitivity method in two-dimensional linear elasticity topology design. In: Idelsohn VSSR (ed) Applications of computational mechanics in structures and fluids. CIMNE Goddeke D, Strzodka R, Turek S (2007) Performance and accuracy of hardware-oriented native-emulated- and mixed-precision solvers in FEM simulations. IJPEDS 22(4):221–256 Gopalakrishnan SH, Suresh K (2008) Feature sensitivity: a generalization of topological sensitivity**. Finite Elem Anal Des 44(11):696–704 Hassani B, Khanzadi M, Tavakkoli SM (2012) An isogeometrical approach to structural topology optimization by optimality criteria. Struct Multidisc Optim 45(2):223–233 He L (2007) Incorporating topological derivatives into shape derivatives based level set methods. J Comput Phys 225(1):891–909 Kim YI, Kwak BM (2002) Design space optimization using a numerical design continuation method. Int J Numer Methods Eng 53:1979–2002 Kirsch U (2000) Combined approximations—a general reanalysis approach for structural optimization. Struct Multidisc Optim 20(2):97–106 Lorensen WE, Cline HE (1987) Marching cubes: a high resolution 3D surface reconstruction algorithm. Comput Graph (Proc SIGGRAPH) 21(4):163–169 Monroe BF (1962) Aircraft seat structure. U.S. Patent 30378121962 Nguyen T, Paulino G, Song J, Le C (2010) A computational paradigm for multiresolution topology optimization (MTOP). Struct Multidisc Optim 41:525–539 Norato JA (2007) A topological derivative method for topology optimization. Struct Multidisc Optim 33:375–386 Novotny AA (2005a) Topological derivative for linear elastic plate bending problems. Control Cybern 34(1):339–361 Novotny AA (2005b) Topological sensitivity analysis for three-dimensional linear elasticity problem. Comput Methods Appl Mech Eng 196(41–44):4354–4364 Novotny AA (2006) Topological-shape sensitivity method: theory and applications. Solid Mech Appl 137:469–478 Press WH et al (2007) Numerical recipies: the art of scientific computing, 3rd edn. Cambridge University Press Rozvany G (2009) A critical review of established methods of structural topology optimization. Struct Multidisc Optim 37(3):217–237 Saad T (2003) Iterative methods for sparse linear systems (2nd ed.). SIAM, Philadelphia SchmidtBeghini A, Baker WF, Paulino G S (2009) A 2589 Line topology optimization code written for the graphics card. Univeritat Trier; www.am.uni-erlangen.de, Technical report. Preprint SPP1253-068 Sigmund O (2001) A 99 line topology optimization code written in Matlab. Struct Multidiscip Optim 21(2):120–127 Sokolowski J (1999) On topological derivative in shape optimization. SIAM J Control Optim 37(4):1251–1272 Stromberg LL, Beghini A, Baker WF, Paulino G (2012) Topology optimization for braced frames: combining continuum and discrete elements. Eng Struct 37:106–124 Suresh K (2010) A 199-line Matlab code for Pareto-optimal tracing in topology optimization. Struct Multidisc Optim 42(5):665–679 Talischi C, Paulino G, Pereira A, Menezes FM (2010) Polygonal finite elements for topology optimization: a unifying paradigm. Int J Numer Methods Eng 82(6):671–698 Talischi C, Paulino G, Pereira A, Menezes FM (2012) PolyTop: a Matlab implementation of a general topology optimization framework using unstructured polygonal finite element meshes. Struct Multidisc Optim. doi:10.1007/s00158-011-0696-x Turevsky I, Suresh K (2007) Generalization of topological sensitivity and its application to defeaturing**. In: ASME IDETC conference, Las Vegas Turevsky I, Suresh K (2011a) Efficient generation of pareto-optimal topologies for compliance**. Int J Numer Methods Eng 87(12):1207–1228 Turevsky I, Suresh K (2011b) Tracing the envelope of the objective-space in multi-objective topology optimization**. Presented at the ASME IDETC/CIE conference, Washington, DC Turevsky I, Gopalakrishnan SH, Suresh K (2009) An efficient numerical method for computing the topological sensitivity of arbitrary shaped features in plate bending**. Int J Numer Methods Eng 79:1683–1702 Wang MY (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192:227–246 Wang S, Sturler ED, Paulino G (2007) Large-scale topology optimization using preconditioned Krylov subspace methods with recycling. Int J Numer Methods Eng 69(12):2441–2468 Yoon GH (2010) Maximizing the fundamental eigenfrequency of geometrically nonlinear structures by topology optimization based on element connectivity parameterization. Comput Struct 88(1–2):120–133 Zienkiewicz OC (2005) The finite element method for solid and structural mechanics. Elsevier