TY - JOUR
T1 - Stability of finite element models for distributed-parameter optimization and topology design
AU - Jog, Chandrashekhar S.
AU - Haber, Robert B.
N1 - Funding Information:
The authorsw ish to thank Martin P. Bendsoe of the MathematicalI nstitute and Ole Sigmundo f the Department of Solid Mechanicsa t the Technical University of Denmark (DTU) for their encouragement and many helpful suggestionsT. he authorsw ould also like to thank the following organizations for their support of this work. The National Science Foundation (USA), the Danish Research Academy, the Danish Technical Research Council (Program for Research on Computer-Aided Design), Cray Research, Inc. and the Center for SupercomputingR esearch and Development. Computations were performed on the Cray Y-MP at the National Center for Supercomputing Applications.
PY - 1996/4/1
Y1 - 1996/4/1
N2 - We address a problem of numerical instability that is often encountered in finite element solutions of distributed-parameter optimization and variable-topology shape design problems. We show that the cause of this problem is numerical rather than physical in nature. We consider a two-field, distributed-parameter optimization problem involving a design field and a response field, and show that the optimization problem corresponds to a mixed variational problem. An improper selection of the discrete function spaces for these two fields leads to grid-scale anomalies in the numerical solutions to optimization problems, similar to those that are sometimes encountered in mixed formulations of the Stokes problem. We present a theoretical framework to explain the cause of these anomalies and present stability conditions for discrete models. The general theoretical framework is specialized to analyze the stability of specific optimization problems, and stability results for various mixed finite element models are presented. We propose patch tests that are useful in identifying unstable elements.
AB - We address a problem of numerical instability that is often encountered in finite element solutions of distributed-parameter optimization and variable-topology shape design problems. We show that the cause of this problem is numerical rather than physical in nature. We consider a two-field, distributed-parameter optimization problem involving a design field and a response field, and show that the optimization problem corresponds to a mixed variational problem. An improper selection of the discrete function spaces for these two fields leads to grid-scale anomalies in the numerical solutions to optimization problems, similar to those that are sometimes encountered in mixed formulations of the Stokes problem. We present a theoretical framework to explain the cause of these anomalies and present stability conditions for discrete models. The general theoretical framework is specialized to analyze the stability of specific optimization problems, and stability results for various mixed finite element models are presented. We propose patch tests that are useful in identifying unstable elements.
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U2 - 10.1016/0045-7825(95)00928-0
DO - 10.1016/0045-7825(95)00928-0
M3 - Article
AN - SCOPUS:0030123599
SN - 0045-7825
VL - 130
SP - 203
EP - 226
JO - Computer Methods in Applied Mechanics and Engineering
JF - Computer Methods in Applied Mechanics and Engineering
IS - 3-4
ER -