High-order mixed finite elements for an energy-based model of the\n polarization process in ferroelectric materials

Abstract

An energy-based model of the ferroelectric polarization process is presented\nin the current contribution. In an energy-based setting, dielectric\ndisplacement and strain (or displacement) are the primary independent unknowns.\nAs an internal variable, the remanent polarization vector is chosen. The model\nis then governed by two constitutive functions: the free energy function and\nthe dissipation function. Choices for both functions are given. As the\ndissipation function for rate-independent response is non-differentiable, it is\nproposed to regularize the problem. Then, a variational equation can be posed,\nwhich is subsequently discretized using conforming finite elements for each\nquantity. We point out which kind of continuity is needed for each field\n(displacement, dielectric displacement and remanent polarization) is necessary\nto obtain a conforming method, and provide corresponding finite elements. The\nelements are chosen such that Gauss' law of zero charges is satisfied exactly.\nThe discretized variational equations are solved for all unknowns at once in a\nsingle Newton iteration. We present numerical examples gained in the open\nsource software package Netgen/NGSolve.\n