Sladek Jan, Sladek Vladimir, Wen Pihua H
Institute of Construction and Architecture, Slovak Academy of Sciences, 84503 Bratislava, Slovakia.
School of Engineering and Materials Sciences, Queen Mary University of London, Mile End, London E14NS, UK.
Materials (Basel). 2020 Jun 2;13(11):2527. doi: 10.3390/ma13112527.
The meshless local Petrov-Galerkin (MLPG) method was developed to analyze 2D problems for flexoelectricity and higher-grade thermoelectricity. Both problems were multiphysical and scale-dependent. The size effect was considered by the strain and electric field gradients in the flexoelectricity, and higher-grade heat flux in the thermoelectricity. The variational principle was applied to derive the governing equations within the higher-grade theory of considered continuous media. The order of derivatives in the governing equations was higher than in their counterparts in classical theory. In the numerical treatment, the coupled governing partial differential equations (PDE) were satisfied in a local weak-form on small fictitious subdomains with a simple test function. Physical fields were approximated by the moving least-squares (MLS) scheme. Applying the spatial approximations in local integral equations and to boundary conditions, a system of algebraic equations was obtained for the nodal unknowns.
无网格局部彼得罗夫-伽辽金（MLPG）方法被开发用于分析二维的挠曲电和高阶热电问题。这两个问题都是多物理场且与尺度相关的。在挠曲电中通过应变和电场梯度考虑尺寸效应，在热电中通过高阶热通量考虑尺寸效应。应用变分原理在考虑的连续介质高阶理论内推导控制方程。控制方程中的导数阶数高于经典理论中的对应方程。在数值处理中，耦合的控制偏微分方程（PDE）在具有简单测试函数的小虚拟子域上以局部弱形式得到满足。物理场通过移动最小二乘法（MLS）方案进行近似。将空间近似应用于局部积分方程和边界条件，得到了关于节点未知量的代数方程组。
