Checkerboard‐free topology optimization using non‐conforming finite elements

The objective of the present study is to show that the numerical instability characterized by checkerboard patterns can be completely controlled when non-conforming four-node finite elements are employed. Since the convergence of the non-conforming finite element is independent of the Lame parameters, the stiffness of the non-conforming element exhibits correct limiting behaviour, which is desirable in prohibiting the unwanted formation of checkerboards in topology optimization. We employ the homogenization method to show the checkerboard-free property of the non-conforming element in topology optimization problems and verify it with three typical optimization examples.

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

[2]  S. C. Brenner,et al.  Linear finite element methods for planar linear elasticity , 1992 .

[3]  Yoon Young Kim,et al.  Multiobjective Topology Optimization of a Beam Under Torsion and Distortion , 2002 .

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

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

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

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

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

[9]  Dongwoo Sheen,et al.  Nonconforming quadrilateral finite elements:¶a correction , 2000 .

[10]  R. Rannacher,et al.  Simple nonconforming quadrilateral Stokes element , 1992 .

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

[12]  Thomas A. Poulsen,et al.  Topology optimization in wavelet space , 2002 .

[13]  Y. Kim,et al.  Multi-resolution multi-scale topology optimization — a new paradigm , 2000 .

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

[15]  J. Douglas,et al.  PEERS: A new mixed finite element for plane elasticity , 1984 .

[16]  P. Raviart,et al.  Conforming and nonconforming finite element methods for solving the stationary Stokes equations I , 1973 .

[17]  Dongwoo Sheen,et al.  Nonconforming Galerkin methods based on quadrilateral elements for second order elliptic problems , 1999 .

[18]  Dongwoo Sheen,et al.  A Locking-Free Nonconforming Finite Element Method for Planar Linear Elasticity , 2003, Adv. Comput. Math..

[19]  K. Bathe.  Finite Element Procedures , 1995 .

[20]  M. Chipot.  Finite Element Methods for Elliptic Problems , 2000 .