A New Meshless "Fragile Points Method (FPM)" Based on A Galerkin Weak-Form for 2D Flexoelectric Analysis

TL;DR

This paper introduces a novel meshless Fragile Points Method (FPM) based on a Galerkin weak-form for efficient 2D flexoelectric analysis, offering simplicity, symmetry, and computational advantages over existing methods.

Contribution

The paper develops a new meshless FPM using local polynomial trial functions and interior penalty fluxes, simplifying 2D flexoelectric analysis and avoiding complex numerical integration.

Findings

Produces symmetric, sparse matrices

Validates with isotropic and anisotropic materials

Simpler than FEM, EFG, and MLPG methods

Abstract

A meshless Fragile Points Method (FPM) is presented for analyzing 2D flexoelectric problems. Local, simple, polynomial and discontinuous trial and test functions are generated with the help of a local meshless differential quadrature approximation of the first three derivatives. Interior Penalty Numerical Fluxes are employed to ensure the consistency of the method. Based on a Galerkin weak-form formulation, the present FPM leads to symmetric and sparse matrices, and avoids the difficulties of numerical integration in the previous meshfree methods. Numerical examples including isotropic and anisotropic materials with flexoelectric and piezoelectric effects are provided as validations. The present method is much simpler than the Finite Element Method, or the Element-Free Galerkin (EFG) and Meshless Local Petrov-Galerkin (MLPG) methods, and the numerical integration of the weak form is trivially simple.