In this paper, we study multiscale methods for piezocomposites. We consider a model of static piezoelectric problem that consists of deformation with respect to components of displacements and a function of electric potential. This problem includes the equilibrium equations, the quasi-electrostatic equation for dielectrics, and a system of coupled constitutive relations for mechanical and electric fields. We consider a model problem that consists of coupled differential equations. The first equation describes the deformations and is given by the elasticity equation and includes the effect of the electric field. The second equation is for the electric field and has a contribution from the elasticity equation. In previous findings, numerical homogenization methods are proposed and used for piezocomposites. We consider the Generalized Multiscale Finite Element Method (GMsFEM), which is more general and is known to handle complex heterogeneities. The main idea of the GMsFEM is to develop additional degrees of freedom and can go beyond numerical homogenization. We consider both coupled and split basis functions. In the former, the multiscale basis functions are constructed by solving coupled local problems. In particular, coupled local problems are solved to generate snapshots. Furthermore, in the snapshot space, a local spectral decomposition is performed to identify multiscale basis functions. Our approaches share some common concepts with meshless methods as they solve the underlying problem on a coarse mesh, which does not conform heterogeneities and contrast. We discuss this issue in the paper. We show that with a few basis functions per coarse element, one can achieve a good approximation of the solution. Numerical results are presented.

在本文中，我们研究了压电复合材料的多尺度方法。我们考虑一个静态压电问题模型，该模型由关于位移分量的变形和电势函数组成。该问题包括平衡方程、电介质的准静电方程以及机械和电场的耦合本构关系系统。我们考虑一个由耦合微分方程组成的模型问题。第一个方程描述了变形，由弹性方程给出，包括电场的影响。第二个方程是针对电场的，它对弹性方程有贡献。在之前的研究结果中，数值均质化方法被提出并用于压电复合材料。我们考虑广义多尺度有限元方法 (GMsFEM)，它更通用，并且已知可以处理复杂的异质性。GMsFEM 的主要思想是开发额外的自由度，并且可以超越数值均匀化。我们考虑耦合和分裂基函数。前者是通过求解耦合局部问题构建多尺度基函数。特别地，耦合局部问题被解决以生成快照。此外，在快照空间中，执行局部谱分解以识别多尺度基函数。我们的方法与无网格方法共享一些共同的概念，因为它们解决了不符合异质性和对比度的粗网格上的潜在问题。我们在论文中讨论了这个问题。我们表明，每个粗元素有几个基函数，就可以很好地逼近解。给出了数值结果。