NUMERICAL-ANALYSIS OF A NONCONVEX VARIATIONAL PROBLEM RELATED TO SOLID-SOLID PHASE-TRANSITIONS

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 (martensit ic) energy wells, the numerical approximation of the deformation gradi ent does not converge, but tends to oscillate between the two wells, a s the size of the mesh is refined. These oscillations may be interpret ed in terms of microstructures. Using a nonconforming P-1 finite eleme nt, an estimate is given for the rate of convergence of the probabilit y for the approximated deformation to have its gradient ''near'' one o f the two (martensitic) wells.