A critical review of established methods of structural topology optimization

The aim of this article is to evaluate and compare established numerical methods of structural topology optimization that have reached the stage of application in industrial software. It is hoped that our text will spark off a fruitful and constructive debate on this important topic.

[1]  Claus Mattheck,et al.  Engineering Optimization in Design Processes , 1991 .

[2]  M. Bendsøe.  Optimal shape design as a material distribution problem , 1989 .

[3]  George I. N. Rozvany,et al.  Structural Design via Optimality Criteria , 1989 .

[4]  Pasi Tanskanen,et al.  A multiobjective and fixed elements based modification of the evolutionary structural optimization method , 2006 .

[5]  George I. N. Rozvany,et al.  Some basic issues of topology optimization , 2006 .

[6]  G. Rozvany.  Aims, scope, methods, history and unified terminology of computer-aided topology optimization in structural mechanics , 2001 .

[7]  C. S. Jog,et al.  A new approach to variable-topology shape design using a constraint on perimeter , 1996 .

[8]  N. Olhoff,et al.  Topological optimization of continuum structures with design-dependent surface loading – Part I: new computational approach for 2D problems , 2004 .

[9]  Claude Fleury.  Reconciliation of Mathematical Programming and Optimality Criteria Approaches to Structural Optimization , 1982 .

[10]  M. Fuchs,et al.  The SRV constraint for 0/1 topological design , 2005 .

[11]  N. Olhoff,et al.  Topology Optimization of Vibrating Bi-Material Structures with Respect to Sound Radiation , 2006 .

[12]  George I. N. Rozvany,et al.  DCOC: An optimality criteria method for large systems Part I: theory , 1992 .

[13]  Claus B. W. Pedersen,et al.  Industrial Implementation and Applications of Topology Optimization and Future Needs , 2006 .

[14]  George I. N. Rozvany.  Layout Theory for Grid-Type Structures , 1993 .

[15]  G. Rozvany,et al.  Analytical benchmarks for topological optimization IV: square-shaped line support , 2008 .

[16]  Jakob S. Jensen,et al.  Topology optimization of photonic crystal structures: a high-bandwidth low-loss T-junction waveguide , 2005 .

[17]  Krister Svanberg,et al.  A hierarchical neighbourhood search method for topology optimization , 2004 .

[18]  G. Rozvany.  Exact analytical solutions for some popular benchmark problems in topology optimization , 1998 .

[19]  Z. Gaspar,et al.  Addenda and corrigenda to: (1) “Aims, scope, methods, history and unified terminology of computer-aided topology optimization in structural mechanics" and (2) “On design- dependent constraints and singular topologies" (Vol. 21, No. 2, 2001, pp. 90–108; 164–172) , 2002 .

[20]  G. Allaire,et al.  Shape optimization by the homogenization method , 1997 .

[21]  N. Kikuchi,et al.  Solutions to shape and topology eigenvalue optimization problems using a homogenization method , 1992 .

[22]  George I. N. Rozvany,et al.  Combining ESO with rigorous optimality criteria , 2002 .

[23]  C. S. Jog,et al.  Topology design with optimized, self‐adaptive materials , 1994 .

[24]  R. Lipton,et al.  Optimal material layout for 3D elastic structures , 1997 .

[25]  Hojjat Adeli,et al.  Advances in Design Optimization , 1994 .

[26]  H. A. Kim,et al.  An evaluative study on ESO and SIMP for optimising a cantilever tie—beam , 2007 .

[27]  Martin P. Bendsøe,et al.  Optimization of Structural Topology, Shape, And Material , 1995 .

[28]  George I. N. Rozvany.  Optimal load transmission by flexure , 1972 .

[29]  M. Zhou,et al.  Checkerboard and minimum member size control in topology optimization , 2001 .

[30]  George I. N. Rozvany,et al.  Quality Control in Topology Optimization Using Analytically Derived Benchmarks , 2006 .

[31]  Zhan Yong-zhao,et al.  Optimization of topology and routing of WSN based on SoRCA , 2008 .

[32]  Thomas A. Poulsen.  A simple scheme to prevent checkerboard patterns and one-node connected hinges in topology optimization , 2002 .

[33]  George I. N. Rozvany,et al.  Grillages of maximum strength and maximum stiffness , 1972 .

[34]  J. Petersson,et al.  Slope constrained topology optimization , 1998 .

[35]  A. Michell.  LVIII. The limits of economy of material in frame-structures , 1904 .

[36]  N. Olhoff,et al.  Reliability-based topology optimization , 2004 .

[37]  G.I.N. Rozvany,et al.  OPTIMIZATION OF STRUCTURAL GEOMETRY , 1977 .

[38]  George I. N. Rozvany,et al.  THEORETICAL FOUNDATIONS OF SEQUENTIAL ELEMENT REJECTIONS AND ADMISSIONS (SERA) METHODS AND THEIR COMPUTATIONAL IMPLEMENTATION IN TOPOLOGY OPTIMIZATION , 2002 .

[39]  N. Olhoff,et al.  Topological design of freely vibrating continuum structures for maximum values of simple and multiple eigenfrequencies and frequency gaps , 2007 .

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

[41]  Yi Min Xie,et al.  Bi‐Directional Evolutionary Structural Optimization Method , 2010 .

[42]  G. I. N. Rozvany,et al.  Least-weight design of perforated elastic plates for given compliance: Nonzero Poisson's ratio , 1988 .

[43]  George I. N. Rozvany,et al.  Layout Optimization of Structures , 1995 .

[44]  Noboru Kikuchi,et al.  Topology optimization with design-dependent loads , 2001 .

[45]  C. Mattheck,et al.  A new method of structural shape optimization based on biological growth , 1990 .

[46]  N. Olhoff,et al.  Topological design of freely vibrating continuum structures for maximum values of simple and multiple eigenfrequencies and frequency gaps , 2007 .

[47]  P. Comba,et al.  Part I. Theory , 2007 .

[48]  Alejandro R. Diaz,et al.  Optimal Material Layout in Three-Dimensional Elastic Structures Subjected to Multiple Loads* , 2000 .

[49]  George I. N. Rozvany,et al.  optimization of large structural systems , 1990 .

[50]  K. Svanberg.  The method of moving asymptotes—a new method for structural optimization , 1987 .

[51]  Robert V. Kohn,et al.  Explicit optimal bounds on the elastic energy of a two-phase composite in two space dimensions , 1993 .

[52]  Krister Svanberg,et al.  Topology optimization by a neighbourhood search method based on efficient sensitivity calculations , 2006 .

[53]  Martin P. Bendsøe,et al.  A non-linear branch and cut method for solving discrete minimum compliance problems to global optimality , 2007 .

[54]  George I. N. Rozvany,et al.  Least-weight design of perforated elastic plates. II , 1987 .

[55]  J. Sethian,et al.  Structural Boundary Design via Level Set and Immersed Interface Methods , 2000 .

[56]  R. Kohn,et al.  Topology optimization and optimal shape design using homogenization , 1993 .

[57]  Jakob S. Jensen,et al.  Systematic design of photonic crystal structures using topology optimization: Low-loss waveguide bends , 2004 .

[58]  O. Querin,et al.  Extended optimality in topology design , 2002 .

[59]  Y. Xie,et al.  A new look at ESO and BESO optimization methods , 2007 .

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

[61]  E. F. Masur.  Structural Optimization. Volume 1: Optimality Criteria (M. Save and W. Prager, eds.) , 1987 .

[62]  A. J. Morris,et al.  Foundations of Structural Optimization: A Unified Approach , 1982 .

[63]  Pasi Tanskanen,et al.  The evolutionary structural optimization method: theoretical aspects , 2002 .

[64]  George I. N. Rozvany,et al.  DCOC: An optimality criteria method for large systems Part II: Algorithm , 1993 .

[65]  A. Rietz.  Sufficiency of a finite exponent in SIMP (power law) methods , 2001 .

[66]  M. Bendsøe,et al.  Shape optimization of structures for multiple loading conditions using a homogenization method , 1992 .

[67]  G. Rozvany.  Traditional vs. extended optimality in topology optimization , 2009 .

[68]  M. Bendsøe,et al.  Topology Optimization: "Theory, Methods, And Applications" , 2011 .

[69]  Claude Fleury,et al.  CONLIN: An efficient dual optimizer based on convex approximation concepts , 1989 .

[70]  W. Prager,et al.  Optimal design of trusses for alternative loads , 1972 .

[71]  O. Sigmund,et al.  Stiffness design of geometrically nonlinear structures using topology optimization , 2000 .

[72]  Ole Sigmund,et al.  Systematic design of phononic band–gap materials and structures by topology optimization , 2003, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[73]  Krister Svanberg,et al.  Topology optimization by sequential integer linear programming , 2006 .

[74]  M. Bendsøe,et al.  Material interpolation schemes in topology optimization , 1999 .

[75]  Rakesh K. Kapania,et al.  Truss Topology Optimization with Simultaneous Analysis and Design , 1992 .

[76]  O. Sigmund.  Morphology-based black and white filters for topology optimization , 2007 .

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

[78]  Yi Min Xie,et al.  Evolutionary Structural Optimization , 1997 .

[79]  G. Rozvany,et al.  Extended exact solutions for least-weight truss layouts—Part I: Cantilever with a horizontal axis of symmetry , 1994 .

[80]  N. Olhoff,et al.  Topological optimization of continuum structures with design-dependent surface loading – Part II: algorithm and examples for 3D problems , 2004 .

[81]  J. M. Martínez,et al.  A note on the theoretical convergence properties of the SIMP method , 2005 .

[82]  Ole Sigmund,et al.  IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials , 2006 .

[83]  O. Sigmund,et al.  Checkerboard patterns in layout optimization , 1995 .

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

[85]  G. I. N. Rozvany,et al.  Exact analytical solutions for some popular benchmark problems in topology optimization III: L-shaped domains , 2008 .

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

[87]  G. Rozvany,et al.  Exact analytical solutions for some popular benchmark problems in topology optimization II: three-sided polygonal supports , 2007 .

[88]  George I. N. Rozvany,et al.  On the validity of ESO type methods in topology optimization , 2001 .

[89]  G. Steven,et al.  A performance-based optimization method for topology design of continuum structures with mean compliance constraints , 2002 .

[90]  M. Zhou,et al.  Generalized shape optimization without homogenization , 1992 .

[91]  N. Olhoff,et al.  Topology optimization of continuum structures subjected to pressure loading , 2000 .

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

[93]  Daniel A. Tortorelli,et al.  An element removal and reintroduction strategy for the topology optimization of structures and compliant mechanisms , 2003 .

[94]  Krister Svanberg,et al.  Sequential integer programming methods for stress constrained topology optimization , 2007 .

[95]  C Edwards,et al.  Investigation on the Validity of Topology Optimisation Methods , 2006 .

[96]  Noboru Kikuchi,et al.  Topology and Generalized Layout Optimization of Elastic Structures , 1993 .

[97]  Ole Sigmund,et al.  A web-based topology optimization program , 2001 .

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

[99]  G. Rozvany,et al.  Extended exact least-weight truss layouts—Part II: Unsymmetric cantilevers , 1994 .

[100]  Kurt Maute,et al.  Adaptive topology optimization of shell structures , 1996 .

[101]  P. Pedersen.  On optimal shapes in materials and structures , 2000 .

[102]  M. Zhou,et al.  Applications of the COC Algorithm in Layout Optimization , 1991 .

[103]  G. A. Hegemier,et al.  On Michell trusses , 1969 .

[104]  O. M. Querin,et al.  Erratum for the Brief Note “Extended optimality in topology design” by G.I.N. Rozvany, O.M. Querin, Z. Gaspar, V. Pomezanski, (SMO 24, 257–261, 2002) , 2005 .

[105]  E. Ramm,et al.  Adaptive topology optimization of elastoplastic structures , 1998 .

[106]  Niels Olhoff,et al.  On CAD-integrated structural topology and design optimization , 1991 .

[107]  O. M. Querin,et al.  Weight-increasing effect of topology simplification , 2003 .

[108]  J. E. Taylor,et al.  A Finite Element Method for the Optimal Design of Variable Thickness Sheets , 1973 .

[109]  M. Zhou,et al.  The COC algorithm, Part II: Topological, geometrical and generalized shape optimization , 1991 .

[110]  Ole Sigmund,et al.  On the Design of Compliant Mechanisms Using Topology Optimization , 1997 .

[111]  J. Petersson,et al.  Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima , 1998 .

[112]  R. Haftka,et al.  Elements of Structural Optimization , 1984 .

[113]  Ole Sigmund,et al.  Design of multiphysics actuators using topology optimization - Part I: One-material structures , 2001 .

[114]  Richard E. McConnel.  Least-Weight Frameworks for Loads Across Span , 1974 .

[115]  M. P. Bendsøe,et al.  The Michell layout problem as a low volume fraction limit of the perforated plate topology optimization problem: An asymptotic study , 1993 .

[116]  Andrej Cherkaev,et al.  Effective Characteristics of Composite Materials and the Optimal Design of Structural Elements , 1997 .

[117]  George I. N. Rozvany,et al.  Sequential Element Rejections and Admissions (SERA) method: Application to multiconstraint problems , 2004 .

[118]  Mats Werme.  Globally optimal benchmark solutions to some small-scale discretized continuum topology optimization problems , 2006 .

[119]  K. Svanberg,et al.  An alternative interpolation scheme for minimum compliance topology optimization , 2001 .

[120]  R. Kohn,et al.  Optimal design and relaxation of variational problems, III , 1986 .

[121]  Niels Olhoff,et al.  Minimization of sound radiation from vibrating bi-material structures using topology optimization , 2007 .

[122]  P. Hajela,et al.  Genetic algorithms in truss topological optimization , 1995 .

[123]  G. Rozvany.  Stress ratio and compliance based methods in topology optimization – a critical review , 2001 .

[124]  George I. N. Rozvany,et al.  CO-SIMP: extended SIMP algorithm with direct COrner COntact COntrol , 2005 .

[125]  Martin P. Bendsøe,et al.  Topology design of structures , 1993 .

[126]  G. I. N. Rozvany,et al.  Layout and Generalized Shape Optimization by Iterative COC Methods , 1993 .

[127]  Xiaoming Wang,et al.  Structural shape and topology optimization in a level-set-based framework of region representation , 2004 .

[128]  E. Ramm,et al.  Adaptive topology optimization , 1995 .