Convex topology optimization for hyperelastic trusses based on the ground-structure approach

Most papers in the literature, which deal with topology optimization of trusses using the ground structure approach, are constrained to linear behavior. Here we address the problem considering material nonlinear behavior. More specifically, we concentrate on hyperelastic models, such as the ones by Hencky, Saint-Venant, Neo-Hookean and Ogden. A unified approach is adopted using the total potential energy concept, i.e., the total potential is used both in the objective function of the optimization problem and also to obtain the equilibrium solution. We proof that the optimization formulation is convex provided that the specific strain energy is strictly convex. Some representative examples are given to demonstrate the features of each model. We conclude by exploring the role of nonlinearities in the overall topology design problem.

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

[2]  Krister Svanberg,et al.  On Local and Global Minima in Structural Optimization , 1981 .

[3]  Wolfgang Achtziger.  Truss topology optimization including bar properties different for tension and compression , 1996 .

[4]  R. Ogden.  Non-Linear Elastic Deformations , 1984 .

[5]  E. Ramm,et al.  Large deformations and stability in topology optimization , 2005 .

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

[7]  Ulf Ringertz,et al.  OPTIMIZATION OF STRUCTURES WITH NONLINEAR RESPONSE , 1989 .

[8]  R. Haftka,et al.  Simultaneous nonlinear structural analysis and design , 1989 .

[9]  E. Hinton,et al.  A review of homogenization and topology optimization III—topology optimization using optimality criteria , 1998 .

[10]  Wolfgang Achtziger,et al.  Local stability of trusses in the context of topology optimization Part I: Exact modelling , 1999 .

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

[12]  N. S. Khot,et al.  Minimum weight design of truss structures with geometric nonlinear behavior , 1985 .

[13]  Tomasz Sokół,et al.  A 99 line code for discretized Michell truss optimization written in Mathematica , 2011 .

[14]  Anders Klarbring,et al.  An Introduction to Structural Optimization , 2008 .

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

[16]  J. Barthelemy,et al.  Two point exponential approximation method for structural optimization , 1990 .

[17]  Ole Sigmund,et al.  Extensions and applications , 2004 .

[18]  A. Klarbring,et al.  Topology optimization of hyperelastic bodies including non-zero prescribed displacements , 2013 .

[19]  Yusuf Cengiz Toklu,et al.  Nonlinear analysis of trusses through energy minimization , 2004 .

[20]  J. A. Stricklin,et al.  Formulations and solution procedures for nonlinear structural analysis , 1977 .

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

[22]  Makoto Ohsaki.  Optimization of Finite Dimensional Structures , 2010 .

[23]  P. Wriggers.  Nonlinear Finite Element Methods , 2008 .

[24]  Uri Kirsch,et al.  Optimal topologies of truss structures , 1989 .

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

[26]  A. Klarbring,et al.  A note on the min-max formulation of stiffness optimization including non-zero prescribed displacements , 2012 .

[27]  Hirohisa Noguchi,et al.  Homologous topology optimization in large displacement and buckling problems , 2001 .

[28]  G. Rozvany.  Topology optimization in structural mechanics , 2001 .

[29]  Samy Missoum,et al.  Displacement-Based Optimization for Truss Structures Subjected to Static and Dynamic Constraints , 2002 .

[30]  A. Groenwold,et al.  On the equivalence of optimality criterion and sequential approximate optimization methods in the classical topology layout problem , 2008 .