The polarization process of ferroelectric materials analyzed in the framework of variational inequalities

TL;DR

This paper develops a mathematical framework using variational inequalities to model and analyze the polarization process in ferroelectric materials, including existence, uniqueness, and numerical solutions with finite element methods.

Contribution

It introduces a variational inequality formulation for ferroelectric polarization, proving well-posedness and developing a numerical scheme with regularization and Newton iteration.

Findings

Existence and uniqueness of solutions for the discrete update equation.

Successful numerical implementation using mixed finite elements.

Validation of the model with numerical examples in open source software.

Abstract

We are concerned with the mathematical modeling of the polarization process in ferroelectric media. We assume that this dissipative process is governed by two constitutive functions, which are the free energy function and the dissipation function. The dissipation function, which is closely connected to the dissipated energy, is usually non-differentiable. Thus, a minimization condition for the overall energy includes the subdifferential of the dissipation function. This condition can also be formulated by way of a variational inequality in the unknown fields strain, dielectric displacement, remanent polarization and remanent strain. We analyze the mathematical well-posedness of this problem. We provide an existence and uniqueness result for the time-discrete update equation. Under stronger assumptions, we can prove existence of a solution to the time-dependent variational inequality. To solve the discretized variational inequality, we use mixed finite elements, where mechanical displacement and dielectric displacement are unknowns, as well as polarization (and, if included in the model, remanent strain). It is then possible to satisfy Gauss' law of zero free charges exactly. We propose to regularize the dissipation function and solve for all unknowns at once in a single Newton iteration. We present numerical examples gained in the open source software package Netgen/NGSolve.