A mass constraint formulation for structural topology optimization with multiphase materials

This work is focused on the topology optimization of lightweight structures consisting of multiphase materials. Instead of adopting the common idea of using volume constraint, a new problem formulation with mass constraint is proposed. Meanwhile, recursive multiphase materials interpolation (RMMI) and uniform multiphase materials interpolation (UMMI) schemes are discussed and compared based on numerical tests and theoretical analysis. It is indicated that the nonlinearity of the mass constraint introduced by RMMI brings numerical difficulties to attain the global optimum of the optimization problem. On the contrary, the UMMI-2 scheme makes it possible to formulate the mass constraint in a linear form with separable design variables. One such formulation favors very much the problem resolution by means of mathematical programming approaches, especially the convex programming methods. Moreover, numerical analysis indicates that fully uniform initial weighting is beneficial to seek the global optimum when UMMI-2 scheme is used. Besides, the relationship between the volume constraint and mass constraint is theoretically revealed. The filtering technique is adapted to avoid the checkerboard pattern related to the problem with multiphase materials. Numerical examples show that the UMMI-2 scheme with fully uniform initial weighting is reliable and efficient to deal with the structural topology optimization with multiphase materials and mass constraint. Meanwhile, the mass constraint formulation is evidently more significant than the volume constraint formulation. Copyright © 2011 John Wiley & Sons, Ltd.

[1]  G. K. Ananthasuresh,et al.  Topology optimization of compliant mechanisms with multiple materials using a peak function material interpolation scheme , 2001 .

[2]  C. Fleury,et al.  A modification of convex approximation methods for structural optimization , 1997 .

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

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

[5]  Michael Yu Wang,et al.  Synthesis of shape and topology of multi-material structures with a phase-field method , 2004 .

[6]  S. Torquato,et al.  Design of materials with extreme thermal expansion using a three-phase topology optimization method , 1997 .

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

[8]  V. Braibant,et al.  Structural optimization: A new dual method using mixed variables , 1986 .

[9]  James K. Guest,et al.  Achieving minimum length scale in topology optimization using nodal design variables and projection functions , 2004 .

[10]  Hae Chang Gea,et al.  Design of an energy-absorbing structure using topology optimization with a multimaterial model , 2006 .

[11]  Mei Yulin,et al.  A level set method for structural topology optimization with multi-constraints and multi-materials , 2004 .

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

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

[14]  Weihong Zhang,et al.  Topology optimization of multiphase material structures under design dependent pressure loads , 2009 .

[15]  Shengli Xu,et al.  Volume preserving nonlinear density filter based on heaviside functions , 2010 .

[16]  Erik Lund,et al.  Discrete material optimization of general composite shell structures , 2005 .

[17]  Seog-Young Han,et al.  Development of a Material Mixing Method Based on Evolutionary Structural Optimization , 2005 .

[18]  J. Thomsen,et al.  Topology optimization of structures composed of one or two materials , 1992 .

[19]  T. Borrvall.  Topology optimization of elastic continua using restriction , 2001 .

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

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

[22]  Shiwei Zhou,et al.  Multimaterial structural topology optimization with a generalized Cahn–Hilliard model of multiphase transition , 2006 .

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

[24]  Weihong Zhang,et al.  Topology optimization involving thermo-elastic stress loads , 2010 .