Globally convergent nonlinear 3D inverse model for smart materials with Hessian-based optimization

Abstract

A globally convergent and fully coupled 3D inverse model for smart materials is presented. In practice, stress and field (electric, magnetic, or temperature) are applied to smart materials whereas strain and flux density (electric, magnetic, or temperature) are measured. We refer to constitutive models that follow this scheme as direct models. In certain design and control situations, however, inverse models are necessary in which the field and stress are found from specified flux density and strain. This inversion typically involves an iterative procedure, which may be prone to convergence issues. An inverse model approach is proposed for arbitrary smart materials. The inversion requirement is a continuous and second order differentiable direct model for any chosen smart material. The approach is globally convergent, which makes it ideal for use in finite element frameworks. The premise of the proposed iterative system model is to constitute a recursive correction formula based on second order approximations of a novel scalar error function which offers a faster convergence rate. A continuation approach is then used to achieve global convergence for arbitrary input parameters. Magnetostrictive Galfenol is chosen to illustrate the effectiveness of the inverse model, and compact analytical derivations of the Jacobian and Hessian matrices are presented. The convergence rate of the proposed approach is superior to that of an existing inverse model. Finally, the inverse model’s robustness is demonstrated through integration of the model into a finite-element framework to simulate a magnetostrictive composite plate actuator in full 3D.