Advances in Mathematical Physics

Volume 2015, Article ID 916029, 38 pages

http://dx.doi.org/10.1155/2015/916029

## Hybrid Fundamental Solution Based Finite Element Method: Theory and Applications

Research School of Engineering, The Australian National University, Acton, ACT 2601, Australia

Received 13 October 2014; Revised 23 December 2014; Accepted 24 December 2014

Academic Editor: Luigi C. Berselli

Copyright © 2015 Changyong Cao and Qing-Hua Qin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

An overview on the development of hybrid fundamental solution based finite element method (HFS-FEM) and its application in engineering problems is presented in this paper. The framework and formulations of HFS-FEM for potential problem, plane elasticity, three-dimensional elasticity, thermoelasticity, anisotropic elasticity, and plane piezoelectricity are presented. In this method, two independent assumed fields (intraelement filed and auxiliary frame field) are employed. The formulations for all cases are derived from the modified variational functionals and the fundamental solutions to a given problem. Generation of elemental stiffness equations from the modified variational principle is also described. Typical numerical examples are given to demonstrate the validity and performance of the HFS-FEM. Finally, a brief summary of the approach is provided and future trends in this field are identified.

#### 1. Introduction

A novel hybrid finite element formulation, called the hybrid fundamental solution based FEM (HFS-FEM), was recently developed based on the framework of hybrid Trefftz finite element method (HT-FEM) and the idea of the method of fundamental solution (MFS) [1–5]. In this method, two independent assumed fields (intraelement filed and auxiliary frame field) are employed and the domain integrals in the variational functional can be directly converted to boundary integrals without any appreciable increase in computational effort as in HT-FEM [6–8]. It should be mentioned that the intraelement field of HFS-FEM is approximated by the linear combination of fundamental solutions analytically satisfying the related governing equation, instead of -complete functions as in HT-FEM. The resulting system of equations from the modified variational functional is expressed in terms of symmetric stiffness matrix and nodal displacements only, which is easy to implement into the standard FEM. It is noted that no singular integrals are involved in the HFS-FEM by locating the source point outside the element of interest and do not overlap with field point during the computation [9].

The HFS-FEM mentioned above inherits all the advantages of HT-FEM over the traditional FEM and the boundary element method (BEM), namely, domain decomposition and boundary integral expressions, while avoiding the major weaknesses of BEM [10–12], that is, singular element boundary integral and loss of symmetry and sparsity [13]. The employment of two independent fields also makes the HFS-FEM easier to generate arbitrary polygonal or even curve-sided elements. It also obviates the difficulties that occur in HT-FEM [14, 15] in deriving -complete functions for certain complex or new physical problems [16]. The HFS-FEM has simpler expressions of interpolation functions for intraelement fields (fundamental solutions) and avoids the coordinate transformation procedure required in the HT-FEM to keep the matrix inversion stable. Moreover, this approach also has the potential to achieve high accuracy using coarse meshes of high-degree elements, to enhance insensitivity to mesh distortion, to give great liberty in element shape, and to accurately represent various local effects (such as hole, crack, and inclusions) without troublesome mesh adjustment [17–20]. Additionally, HFS-FEM makes it possible for a more flexible element material definition which is important in dealing with multimaterial problems, rather than the material definition being the same in the entire domain in BEM. However, we noticed that there are also some limitations of HFS-FEM, for example, determining the positions of source points used for approximation interpolations. It is also known that fundamental solution based approximations can perform remarkably well in smooth problems but tend to deteriorate when high-gradient stress fields are presented.

This paper is organized as follows: in Section 2, the basic idea and formulations of the HFS-FEM are presented through a simple potential problem. Then, plane elasticity problems are described in Section 3. Section 4 extends the 2D formulations of the HFS-FEM to general three-dimensional (3D) elasticity problems. The method of particular solution and radial basis function approximation are shown to deal with body force in this Section. In Section 5 we extend the HFS-FEM to thermoelastic problems with arbitrary body force and temperature change. In Section 6, the HFS-FEM for 2D anisotropic elastic materials is described based on the powerful Stroh formalism. Plane piezoelectric problem is discussed in Section 7. Finally, typical numerical examples are presented in Section 8 to illustrate applications and performance of the HFS-FEM. Concluding remarks and future development are discussed at the end of this paper.

#### 2. Potential Problems

##### 2.1. Basic Equations of Potential Problems

The Laplace equation of a well-posed potential problem (e.g., heat conduction) in a general plane domain can be expressed as [21, 22]with the boundary conditionswhere is the unknown field variable and represents the boundary flux, is the component of outward normal vector to the boundary , and and are specified functions on the related boundaries, respectively. The space derivatives are indicated by a subscript comma, that is, , and the subscript index takes values for two-dimensional and for three-dimensional problems. Additionally, the repeated subscript indices imply summation convention.

For convenience, (3) can be rewritten in matrix form aswith .

##### 2.2. Assumed Independent Fields

In this section, the procedure for developing a hybrid finite element model with fundamental solution as interior trial function is described based on the boundary value problem defined by (1)–(3). Similar to the conventional FEM and HT-FEM, the domain under consideration is divided into a series of elements [15, 16, 21, 23–30]. In each element, two independent fields are assumed in the way as described in [31] and are given in Section 2.2.

###### 2.2.1. Intraelement Field

Similar to the method of fundamental solution (MFS) in removing singularities of fundamental solution, for a particular element occupying subdomain , we assume that the field variable defined in the element domain is extracted from a linear combination of fundamental solutions centered at different source points located outside the element (see Figure 1):where is undetermined coefficients, is the number of virtual sources, and is the fundamental solution to the partial differential equation:asEvidently, (5) analytically satisfies (1) due to the solution property of .