一种新的无网格脆弱点法（FPM），在每一点上的最小未知数，在两个理论下的挠曲电分析与裂纹扩展。第一部分：理论与实践
A New Meshless Fragile Points Method (FPM) With Minimum Unknowns at Each Point, For Flexoelectric Analysis Under Two Theories with Crack Propagation. Part I: Theory and Implementation
摘要：
挠曲电性是指涉及机械应变梯度和电极化的耦合的现象。在这项研究中，无网格脆性点法（FPM），提出了分析在介电固体的挠曲电效应。局部，简单，多项式和不连续的试验和检验函数的帮助下产生的局部无网格微分求积近似的衍生物。原始和混合FPM开发，基于两个交替的挠曲电理论，有或没有电梯度效应和麦克斯韦应力。在本原始以及混合FPM中，只有位移和电势作为显式未知变量保留在最终的代数方程中的每个内部脆弱点。因此，在最终的代数方程系统中的未知数的数量保持绝对最小。提出了一种基于FPM的裂纹萌生和扩展模拟算法，该算法采用经典的基于应力的准则以及基于键合能量率（BER）的裂纹发展准则。与使用原始和混合有限元法、无单元伽辽金（EFG）法、无网格局部彼得罗夫伽辽金（MLPG）法和等几何分析（IGA）法的当前计算挠曲电分析方法相比，本原始和混合FPM方法表现出明显的优势，这是由于以下新特征：它们是使用多项式试验和检验函数的简单的Galerkin无网格方法，最小的每个点的自由度使它非常用户友好，任意多边形子域使它灵活地模拟复杂的几何形状;原始和混合FPM弱形式的数值积分非常简单，并且FPM可以很容易地用于裂纹发展模拟，而无需重新网格划分或测试函数增强。
英文摘要：
Flexoelectricity refers to a phenomenon which involves a coupling of the mechanical strain gradient and electric polarization. In this study, a meshless Fragile Points Method (FPM), is presented for analyzing flexoelectric effects in dielectric solids. Local, simple, polynomial and discontinuous trial and test functions are generated with the help of a local meshless Differential Quadrature approximation of derivatives. Both primal and mixed FPM are developed, based on two alternate flexoelectric theories, with or without the electric gradient effect and Maxwell stress. In the present primal as well as mixed FPM, only the displacements and electric potential are retained as explicit unknown variables at each internal Fragile Point in the final algebraic equations. Thus the number of unknowns in the final system of algebraic equations is kept to be absolutely minimal. An algorithm for simulating crack initiation and propagation using the present FPM is presented, with classic stress-based criterion as well as a Bonding-Energy-Rate(BER)-based criterion for crack development. The present primal and mixed FPM approaches represent clear advantages as compared to the current methods for computational flexoelectric analyses, using primal as well as mixed Finite Element Methods, Element Free Galerkin (EFG) Methods, Meshless Local Petrov Galerkin (MLPG) Methods, and Isogeometric Analysis (IGA) Methods, because of the following new features: they are simpler Galerkin meshless methods using polynomial trial and test functions; minimal DoFs per Point make it very user-friendly; arbitrary polygonal subdomains make it flexible for modeling complex geometries; the numerical integration of the primal as well as mixed FPM weak forms is trivially simple; and FPM can be easily employed in crack development simulations without remeshing or trial function enhancement.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
