Partition Of Unity Based Methods Research Articles

Smoothed FE-Meshfree method for solid mechanics problems

This paper presents a smoothed FE-Meshfree (SFE-Meshfree) method for solving solid mechanics problems. The system stiffness matrix is calculated via a strain-smoothing technique with the composite shape function, which is based on the partition of unity-based method, combing the classical isoparametric quadrilateral function and radial-polynomial basis function. The corresponding Gauss integration in the element is replaced by line integration along the edges of the smoothing cells, so no derivatives of the composite shape functions are needed during the field gradient estimation process. Several numerical examples including an automobile mechanical component are employed to examine the presented method. Calculation results indicate that SFE-Meshfree can obtain a high convergence rate and accuracy without introducing additional degrees of freedom to the system. In addition, it is also more tolerant with respect to mesh distortion. The volumetric locking problem is also explored in this paper under a selective smoothing integration scheme.

Investigation of linear dependence problem of three-dimensional partition of unity-based finite element methods

A known problem of partition of unity-based generalized finite element methods is the linear dependence of the approximation space, which leads to singular stiffness matrix. Up to now, the linear dependence problem has not been fully understood and an efficient way to alleviate it is not available. In our previous paper “Prediction of rank deficiency in partition of unity-based methods with plane triangular or quadrilateral meshes” [Comput. Methods Appl. Mech. Engrg. 200 (2011) 665–674], the origin of the linear dependence problem was first dissected and then a method was proposed to reliably predict the rank deficiency of the linearly dependent global approximations of two-dimensional partition of unity-based generalized finite element methods. This paper extends the previous work to three-dimensional cases. The linear dependence problem is first investigated at an element level and then extended to the whole mesh. Derivation of general formulations in a three-dimensional setting is undoubtedly more challenging than a two-dimensional setting because of the complicated element topology. The principle of the increase of rank deficiency is once more applied. The methodology of summing up the added rank deficiency of each element as that of the whole mesh is further proved to be valid in three-dimensional cases. This work together with the previous work is regarded as the essential step to successfully and completely solve the linear dependence problem in partition of unity-based finite element methods.

Prediction of rank deficiency in partition of unity-based methods with plane triangular or quadrilateral meshes

In the partition of unity (PU)-based methods, the global approximation is built by multiplying a partition of unity by local approximations. Within this framework, high-order approximations are achieved by directly adopting high-order polynomials as local approximations, and therefore nodes along sides or inside elements, which are usually adopted in the conventional finite element methods, are no more required. However, the PU-based approximation constructed in this way may suffer from rank deficiency due to the linear dependence of the global degrees of freedom. In this paper, the origin of the rank deficiency in the PU-based approximation space is first dissected at an element level, and then an approach to predict the rank deficiency for a mesh is proposed together with the principle of the increase of rank deficiency. Finally, examples are investigated to validate the present approach. The current work indicates such a fact that the rank deficiency is an unrelated issue to the nullity of the global matrix. It can be resolved in its own manner.

Immersed particle method for fluid–structure interaction

AbstractA method for treating fluid–structure interaction of fracturing structures under impulsive loads is described. The coupling method is simple and does not require any modifications when the structure fails and allows fluid to flow through openings between crack surfaces. Both the fluid and the structure are treated by meshfree methods. For the structure, a Kirchhoff–Love shell theory is adopted and the cracks are treated by introducing either discrete (cracking particle method) or continuous (partition of unity‐based method) discontinuities into the approximation. Coupling is realized by a master–slave scheme where the structure is slave to the fluid. The method is aimed at problems with high‐pressure and low‐velocity fluids, and is illustrated by the simulation of three problems involving fracturing cylindrical shells coupled with fluids. Copyright © 2009 John Wiley & Sons, Ltd.