Numerical analysis of a nonconvex variational problem related to solid- solid phase transitions. (English) Zbl 0797.65052

Gremaud, Pierre-Alain

SIAM J. Numer. Anal. 31, No. 1, 111-127 (1994).

The description of equilibria of shape memory alloys or other ordered materials gives rise to nonconvex variational problems. In this paper, a two-dimensional model of such materials is studied. Due to the fact that the corresponding functional has two symmetry-related (martensitic) energy wells, the numerical approximation of the deformation gradient does not converge, but tends to oscillate between the two cells, as the size of the mesh is refined. These oscillations may be interpreted in terms of microstructures. Using a nonconforming \(P_ 1\) finite element, an estimate is given for the rate of convergence of the probability for the approximated deformation to have its gradient “near” one of the two (martensitic) wells.

Reviewer: M.A.Noor (Riyadh)

MSC:

65K10	Numerical optimization and variational techniques
49M15	Newton-type methods
49J40	Variational inequalities

Keywords:

nonconvex variational problem; solid-solid phase transitions; oscillations; finite element; rate of convergence