Abstract
AbstractWe 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.
Highlights
A thermodynamical framework for the description of ferroelectric materials based on the Helmholtz free energy was originally provided in the series of papers [1, 2, 3, 4] by Bassiouny, Ghaleb and Maugin
We present a variational inequality that describes the problem of polarization of ferroelectric media
The variational inequality can be extended to hold for all z ∈ H, i.e. for dielectric displacement updates with non-zero divergence
Summary
A thermodynamical framework for the description of ferroelectric materials based on the Helmholtz free energy was originally provided in the series of papers [1, 2, 3, 4] by Bassiouny, Ghaleb and Maugin. While most finite element formulations use enthalpy-based models discretizing the electric potential, we provide theory for an energy-based setting with an independent dielectric displacement. We proceed in a similar manner as in the first reference [9] and see that, under the standard assumption of convexity of the free energy function, existence and uniqueness of a solution to the time-discrete update problem can be shown. This paper is organized as follows: in Section 2 the underlying energy-based consitutive models are described, and remanent quantities, dissipation function and dissipative driving forces are introduced on a physical level. These quantities are embedded into an abstract mathematical framework of variational inequalities, where all assumptions are stated.
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