  We present a novel method for finite element analysis of inelastic structures
containing Shape Memory Alloys (SMAs). Phenomenological constitutive models for
SMAs lead to material nonlinearities, that require substantial computational
effort to resolve. Finite element analysis methods, which rely on Gauss
quadrature integration schemes, must solve two sets of coupled differential
equations: one at the global level and the other at the local, i.e. Gauss point
level. In contrast to the conventional return mapping algorithm, which solves
these two sets of coupled differential equations separately using a nested
Newton procedure, we propose a scheme to solve the local and global
differential equations simultaneously. In the process we also derive
closed-form expressions used to update the internal state variables, and unify
the popular closest-point and cutting plane methods with our formulas.
Numerical testing indicates that our method allows for larger thermomechanical
loading steps and provides increased computational efficiency, over the
standard return mapping algorithm.
