On symmetry and non-uniqueness in exact topology optimization

The aim of this article is to initiate an exchange of ideas on symmetry and non-uniqueness in topology optimization. These concepts are discussed in the context of 2D trusses and grillages, but could be extended to other structures and design constraints, including 3D problems and numerical solutions. The treatment of the subject is pitched at the background of engineering researchers, and principles of mechanics are given preference to those of pure mathematics. The author hopes to provide some new insights into fundamental properties of exact optimal topologies. Combining elements of the optimal layout theory (of Prager and the author) with those of linear programming, it is concluded that for the considered problems the optimal topology is in general unique and symmetric if the loads, domain boundaries and supports are symmetric. However, in some special cases the number of optimal solutions may be infinite, and some of these may be non-symmetric. The deeper reasons for the above findings are explained in the light of the above layout theory.

[1]  A note on the maximum number and density of distribution of members in elastic structures of minimum weight under multiple loading conditions , 1974 .

[2]  Alphose Zingoni,et al.  Group‐theoretic exploitations of symmetry in computational solid and structural mechanics , 2009 .

[3]  E. M. L. Beale,et al.  Nonlinear and Dynamic Programming , 1965 .

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

[5]  GUY BOUCHITTÉ,et al.  Optimal Design of Thin Plates by a Dimension Reduction for Linear Constrained Problems , 2007, SIAM J. Control. Optim..

[6]  G. I. N. Rozvany,et al.  Partial relaxation of the orthogonality requirement for classical Michell trusses , 1997 .

[7]  T. Lewiński,et al.  Circular and annular two-phase plates of minimal compliance , 2003 .

[8]  Zenon Mróz,et al.  Optimization in Structural Design , 1975 .

[9]  E. Allgower,et al.  Exploiting symmetry in boundary element methods , 1992 .

[10]  W. Achtziger,et al.  Topology Optimization of Discrete Structures , 1997 .

[11]  R. Melchers.  On extending the range of Michell-like optimal topology structures , 2005 .

[12]  Niels Olhoff,et al.  Regularized formulation for optimal design of axisymmetric plates , 1982 .

[13]  P. Dewhurst,et al.  A general boundary approach to the construction of Michell truss structures , 2009 .

[14]  Cezary Graczykowski,et al.  Michell cantilevers constructed within trapezoidal domains , 2006 .

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

[16]  G. Allaire,et al.  Optimal design for minimum weight and compliance in plane stress using extremal microstructures , 1993 .

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

[18]  Peter Dewhurst,et al.  Analytical solutions and numerical procedures for minimum-weight Michell structures , 2001 .

[19]  Tomasz Sokół,et al.  On the solution of the three forces problem and its application in optimal designing of a class of symmetric plane frameworks of least weight , 2010 .

[20]  Harvey J. Greenberg,et al.  Automatic design of optimal structures , 1964 .

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

[22]  Matthew Gilbert,et al.  Optimum structure to carry a uniform load between pinned supports , 2010 .

[23]  Robert E. Melchers,et al.  On the theory of optimal, constant thickness fibre-reinforced plates—II , 1973 .

[24]  G. Rozvany,et al.  Michell layouts for various combinations of line supports. I , 1990 .

[25]  Cezary Graczykowski,et al.  Michell cantilevers constructed within trapezoidal domains—Part III: force fields , 2006 .

[26]  W. S. Hemp.  MICHELL FRAMEWORK FOR UNIFORM LOAD BETWEEN FIXED SUPPORTS , 1974 .

[27]  Cezary Graczykowski,et al.  Michell cantilevers constructed within trapezoidal domains—Part I: geometry of Hencky nets , 2006 .

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

[29]  H. Chan,et al.  Symmetric Plane Frameworks of Least Weight , 1975 .

[30]  G. I. N. Rozvany,et al.  Optimal layout theory: Analytical solutions for elastic structures with several deflection constraints and load conditions , 1992 .

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

[32]  J. Maxwell,et al.  I.—On Reciprocal Figures, Frames, and Diagrams of Forces , 1870, Transactions of the Royal Society of Edinburgh.

[33]  S. Pellegrino,et al.  AN INTRODUCTION TO THE ANALYSIS OF SYMMETRIC STRUCTURES , 1999 .

[34]  T. Lewiński,et al.  Plates, Laminates and Shells: Asymptotic Analysis and Homogenization , 2000 .

[35]  E. Anderson,et al.  Linear programming in infinite-dimensional spaces : theory and applications , 1987 .

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

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

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

[39]  W. Prager.  On Ideal Locking Materials , 1957 .

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

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

[42]  George I. N. Rozvany,et al.  Optimal Layout of Grillages , 1977 .

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

[44]  Cezary Graczykowski,et al.  Michell cantilevers constructed within a half strip. Tabulation of selected benchmark results , 2010 .

[45]  A. Ben-Tal,et al.  Two Nonsmooth Approaches to Simultaneous Geometry and Topology Design of Trusses , 1993 .

[46]  N. Olhoff,et al.  An investigation concerning optimal design of solid elastic plates , 1981 .

[47]  Marc Bonnet,et al.  Exploiting partial or complete geometrical symmetry in 3D symmetric Galerkin indirect BEM formulations , 2003 .

[48]  George I. N. Rozvany,et al.  Why multi-load topology designs based on orthogonal microstructures are in general non-optimal , 1993 .

[49]  Upper limits on the number of elements in elastic structures of minimum weight , 1997 .

[50]  N. Olhoff,et al.  Topology optimization of three-dimensional structures using optimum microstructures , 1998 .

[51]  William Prager,et al.  A General Theory of Optimal Plastic Design , 1967 .

[52]  Georgios E. Stavroulakis,et al.  A variational inequality approach to optimal plastic design of structures via the Prager-Rozvany theory , 1994 .

[53]  T. Healey.  Symmetry and equivariance in nonlinear elastostatics, Part I , 1989 .

[54]  Cezary Graczykowski,et al.  Michell cantilevers constructed within trapezoidal domains—Part II: virtual displacement fields , 2006 .

[55]  Peter G. Glockner,et al.  SYMMETRY IN STRUCTURAL MECHANICS , 1973 .

[56]  Mathias Stolpe,et al.  On some fundamental properties of structural topology optimization problems , 2010 .

[57]  Robert V. Kohn,et al.  Hencky-Prandtl nets and constrained Michell trusses , 1983 .

[58]  George I. N. Rozvany,et al.  The Theory of Optimal Load Transmission by Flexure , 1976 .

[59]  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 .

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

[61]  P. Pedersen.  Topology Optimization of Three-Dimensional Trusses , 1993 .

[62]  Alain Bossavit,et al.  Boundary Value Problems with Symmetry and Their Approximation by Finite Elements , 1993, SIAM J. Appl. Math..

[63]  P. Seppecher,et al.  Locking materials and the topology of optimal shapes , 2001 .

[64]  G. Rozvany.  Some shortcomings in Michell's truss theory , 1996 .

[65]  G. I. N. Rozvany,et al.  Layout Optimization of Large FE Systems by New Optimality Criteria Methods: Applications to Beam Systems , 1993 .

[66]  M. Bendsøe,et al.  Optimization methods for truss geometry and topology design , 1994 .

[67]  M. Beckers,et al.  Topology optimization using a dual method with discrete variables , 1999 .

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