Project Title

MATHEON project C31: Numerical Minimization of Nonsmooth Energy Functionals in Multiphase Materials

This project is part of application area 'C' (Production) of the DFG Research Center Matheon.
Project Description

In this project we consider the classical nonlinear elasticity poblem in variational form. Of particular interest is the examination of the semi-smoothness properties of typical energy densities and the corresponding effect of semi-convexification, discretization, and potential regularization.

Moreover, suitably adapted versions of generalized Newton, BFGS or bundle-type methods for the numerical solution of the semi-convexified energy minimization problem should be analyzed and implemented. Algorithmically, the goal is to cope with the limited smoothness and pointwise degeneracies while exploiting the semi-convex structure. Theoretically we are interested in local convergence results for semi-smooth Newton and BFGS-type solvers. The mesh (in)dependence behavior of these methods will be studied.

Potential new methods will find relevant applications in material science such as in case of compressible,non-Hookean materials.

Project [website] at MATHEON.
Selection of project-related references
Author Title Journal / Publisher
J. Kristensen On the non-locality of quasiconvexity Ann. Institut Henri Poincaré - Analyse non linéaire 16(1):1-13, 1999
B. Dacorogna,
J. Haeberly
Some numerical methods for the study of the convexity notions arising in the calculus of variations RAIRO, Modélisation Math. Anal. Num. 32(2):153-175, 1998.
L. Eneya Pointwise evaluation of polyconvex envelopes Ph.D. thesis, Humboldt-Universität zu Berlin, 2010
J.M. Ball,
B. Kirchheim,
J. Kristensen
Regularity of quasiconvex envelopes SIAM J. Numer. Anal. 43(1):363-385, 2000
M. Hintermüller,
K. Ito,
K. Kunisch
The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. 17(1):159-187, 2003
K. Klatte
B. Kummer
Nonsmooth equations in optimization Kluwer Academic Publishers, Dordrecht, 2002.