Numerical approximation of non-convex variational problems

Noel J. Walkington

doi:10.1117/12.174226

1 May 1994 Numerical approximation of non-convex variational problems

Noel J. Walkington

Proceedings Volume 2192, Smart Structures and Materials 1994: Mathematics and Control in Smart Structures; (1994) https://doi.org/10.1117/12.174226
Event: 1994 North American Conference on Smart Structures and Materials, 1994, Orlando, FL, United States

Abstract

We present some numerical results for the solution of non-convex variational problems. In general, the problems of interest do not attain a minimum energy. Functions that generate a minimizing sequence of energies develop infinitely fine oscillations, and it is believed that these oscillations model the fine scale structures that are ubiquitously observed in metallurgy, ferromagnetism, etc. Direct simulation of these variational problems on discrete meshes is plagued with practical problems. We present a simple 1-D example that exhibits problems typical of those encountered with such an approach. Many of these problems can be traced to the fundamental problem that the variational problem doesn't have a solution. An alternative is to consider the generalized solutions of L. C. Young. We present some numerical experiments using this algorithm for variational problems that involve vector valued functions.

Citation Download Citation

Noel J. Walkington "Numerical approximation of non-convex variational problems", Proc. SPIE 2192, Smart Structures and Materials 1994: Mathematics and Control in Smart Structures, (1 May 1994); https://doi.org/10.1117/12.174226

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