The flexoelectricity describes an electromechanical coupling effect between the electric polarization and mechanical strain gradient. As a result of the recent miniaturized trend of electromechanical systems, which leads a tremendous growth in the scale of strain gradient, there has been an increasing demand for a reliable and accurate numerical approach for flexoelectric analysis. The Fragile Points Method (FPM) is an elementarily simple Galerkin meshless method using point stiffness matrices. It has shown great accuracy and efficiency in solving 2D thermal conductivity and elasticity problems. In this study, we formulate and apply the FPM for analyzing linear flexoelectric solids.

 

In contrast to the element-based shape function in the Finite Element Method (FEM) or the complex, global continuous Moving Least Squares (MLS) approximated shape function used in the Element-Free Galerkin (EFG) method and Meshless Local Petrov-Galerkin (MLPG) method, the proposed FPM employs local, simple, polynomial and discontinuous trial and test functions. In flexoelectric analysis, the point-based trial function is written in the same form of a third order Taylor expansion, where the first to third derivatives are approximated by the displacement (or electric potential) values at a set of neighboring points. Since the FPM is purely meshless, a differential quadrature method based on local radial basis function (RBF) is applied in the approximation. The number of the neighboring points (known as the support of a point) is related to the bandwidth of the stiffness matrix. Finally, the FPM leads to sparse and symmetric matrices, which is an advantage for modeling complex problems with a large number of DoFs. The Galerkin functional in the FPM can also be integrated quite simply, comparing with the previous EFG and MLPG methods.

 

The Interior Penalty (IP) Numerical Flux Corrections are employed to ensure the consistency and stability of the method. However, unlike the Discontinuous Galerkin methods, the trial functions in neighboring subdomains in the present FPM are not completely independent. Thus, a small penalty parameter is enough to stabilize the FPM and ensure the accuracy. The essential boundary conditions, including the normal derivatives, are imposed by a collocation method and a penalty formulation. As a result of the discontinuous trial functions, the FPM also has a great potential in analyzing systems involving cracks, ruptures and fragmentations.

 

Numerical examples are carried out for validation. A benchmark infinitely long hollow cylinder subject to mechanical displacement and electric potential at the inner and outer surface is considered. The present FPM exhibits great accuracy comparing with analytical solutions. Some other numerical examples with significant strain gradient and electric field are also tested and compared with mixed FEM solution to show the applicability of the FPM. The preprocessing module of ABAQUS can be connected with the FPM solver, helping to generate the points and subdomains (just for numerical integration) in the FPM. In general, the present FPM is accurate, efficient, and has a great potential in analyzing flexoelectric problems in complex geometries.


A New Meshless \u201cFragile Points Method (FPM)\u201d Based on a Galerkin Weak-Form for 2D Flexoelectric Analysis

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Paper Proceedings: https://asmedigitalcollection.asme.org/IMECE/IMECE2020/volume/84478

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