Simulations of magnetic, magnetostrictive, and pseudoelastic behavior exhibit hysteresis. These systems have a highly nonlinear character involving both short range anisotropy and elastic fields and, when appropriate, dispersive demagnetization fields. In this report we discuss our experience with this type of computation and the applications which it may serve. We implemented continuation based on the conjugate gradient method, although the same results were obtained by other methods as well. Nonetheless, the propensity of optimization procedures to become marooned at local extrema when applied to nonconvex situations presents a fundamental challenge to analysis. We present some computational results and diagnostics, developed using methods of nonlinear analysis. In a simple case described in the paper, the width of the hysteresis loop may be determined analytically. For a magnetic system, this analysis rests on the introduction of a shadow energy for a simplified version of the system. This simplified version suggests possible dispersive interactions that may be attributed to shape-memory or pseudoelastic body. We provide a brief illustration of this. A principal objective of this investigation is to study the magnetostrictive behavior of Terfenol-D.
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