Numerical simulation of SMA structures using interior-point methods

This Communication addresses the numerical simulation of quasi-static evolutions in monocrystalline SMA structures. Following the works of Govindjee and Miehe (2001), Anand and Gurtin (2003), the phase transformation is described locally by an internal vectorial variable representing the volume fractions of the n variants of martensite. That internal variable is physically constrained to satisfy n+1 inequalities at each point. This is a major difference with the framework of plasticity, in which the internal variable is usually subjected to equality constraints. From a structural simulation point of view, handling such local constraints in the evolution problem is not obvious. Robustness and computational time are particularly sensitive issues. Most of the strategies used so far rely on active set strategies and regularization. Here we propose a new approach, which consists in reformulating the incremental problem as a Linear Complementarity Problem (LCP). This allows one to use a Interior-point method (such as the algorithm of Ye) for the resolution. Such methods are indeed known to be very efficient for solving large-scale LCPs. This approach has been implemented in Matlab (using the finite element toolbox OpenFem) and validated by comparison with the experimental results of Shield (1995). Comparison of computational costs with other resolution techniques from the literature shows the relevancy of the proposed approach.