A computational macroscopic model of piezomagnetoelectric materials using Generalized Multiscale Finite Element Method

Abstract

Piezomagnetoelectric materials have many applications in various areas of science and engineering. For example, such materials can be used in sensors, transducers, energy harvesters, etc. The mathematical model of piezomagnetoelectricity consists of a coupled system of partial differential equations for the electric potential, magnetic potential, and mechanical displacements.We develop multiscale algorithms based on the Generalized Multiscale Finite Element Method for solving the piezomagnetoelectricity problem. We consider two ways of constructing multiscale basis functions: split and coupled. In the split approach, we construct the basis functions separately for each field. The coupled approach constructs multiscale basis functions by solving coupled local problems. The split way requires fewer computational resources than the coupled one, but it may give a less accurate solution in the case of a strongly coupled medium. The coupled approach considers the mutual influence of the fields, which is especially important when they are strongly coupled. We use the multiscale partition of unity functions to obtain conforming basis functions for both approaches.We consider two- and three-dimensional model problems to test the effectiveness of the proposed multiscale approaches. On average, the coupled basis functions show better results than the split ones. We also consider the application of the single-step hermitian and skew-hermitian splitting for solving the discrete problem on a coarse grid. The results have demonstrated that one can successfully apply this technique to solving the piezomagnetoelectricity problem on a coarse grid.