Simulation of singularity in the potential problem with semi-analytical elements

Highlights

• Two new singular boundary elements are constructed to model the singular flux density filed.

• The proposed method can analyze the singular flux density field of any material properties.

• The proposed method can obtain more accurate results without much computational cost.

• The first three singular coefficients can be determined without post processing.

Abstract

Based on the asymptotic solutions for primer and gradient fields, two novel semi-analytical elements are respectively proposed to evaluate potential and flux density in the 2D domain with singularity. By setting two semi-analytical elements on both sides of the singular point, the physical fields in the vicinity of the singular point can be simulated, while boundary element method is implemented for the rest part. Due to asymptotic solutions, the first three singular coefficients are set as the unknowns. The semi-analytical elements are firstly applied on the isotropic material. Regarding to the orthotropic and anisotropic materials, the coordinate system transformation method is applied to transform the governing equation into the isotropic format. Then, semi-analytical elements can be set in the new calculation domain to evaluate potential and flux density. At last, by carrying out an inverse coordinate system transformation, the results can be transformed into the original computational domain. Benefitted from the coordinate system transformation method, semi-analytical elements can be applied to solve the singularities in potential problem with respect to any kinds of material properties. Accurate physical fields as well as the first three singular coefficients can be arrived at the same time.

Introduction

In the engineering mechanics, the potential problem controlled by the Laplace equation has been concerned wildly. For this kind of problem, many different methods have been proposed for the purpose of investigating the heat conduction properties of some special materials [1], [2], [3], [4], [5] or accelerating the process of solving the potential problem [6], [7], [8], [9]. These methods can solve most potential problems, however once there is singularity due to the existence of discontinuities in the boundary conditions or abrupt changes in the geometric shape, these methods may lose effectiveness. To accurately predict the potential and flux density in the vicinity of the singular point, it is a very significant and challenging task. Hence, a reliable and effective numerical method would be desirable.

Actually, since the early 1960s, the potential problem in 2-D homogeneous materials with cracks has been investigated for evaluating the distribution of potential and flux density [10,11]. And then, the intensity factor of the temperature gradient [12] is defined to quantify the thermal energy in the neighborhood of a macrocrack tip. Since the flux density around the crack tip is found to possess the characteristic of inverse square root singularity in any material property, the singularity problem in dissimilar anisotropic material with the interface crack and functional graded material with an arbitrarily oriented crack is investigated [13], [14], [15]. For the singularity problem caused by other reasons, such as discontinuous boundary condition or structure with a V-notch, special treatment should be implemented as the characteristic of inverse square root singularity does not exist anymore. Fortunately, on the basis of the asymptotic solution [16], many new hybrid algorithms have been presented to handle the singularity. For example, the singular basis functions [17] in FEM, regularized functions [18] and singular elements [19,20] in BEM are all effective methods to deal with the problem. Moreover, the asymptotic solution can also be coupled with boundary integral method [21], edge flux intensity functions extraction method [22], [23], [24] and iterative algorithm [25] to solve the singularity problem. Actually, the basic theory of settling the singularity is special treatment to the part of near the singular point. So the modified boundary element method [26] with boundary approximation in the vicinity of the singular point and the extended boundary element method [27] with singularity analysis are implemented for the singularity problem. After introducing the theory of symplectic eigensolution, thermal intensity factors can be determined with Hamiltonian approach [28] and symplectic finite elements [29], [30], [31] are proposed to handle the singularity in anisotropic material and composite structures. Although these methods can deal with the singularity successfully, more improvement can be developed as some methods are formed for only dealing the singularity in the crack, focusing on only isotropic material, or cumbersome derivation in the vicinity of singular point are needed. Herein, a modified boundary element method will be proposed for simulating the singular physical field and determining singular coefficients without increasing computational cost.

In this study, the singularity problem for the Laplace equation in the isotropic material is investigated firstly. The goal of numerical methods is to obtain more accurate results with less calculation. To achieve this goal, two semi-analytical elements are proposed to simulate the potential and flux density fields in the neighborhood of the singular point, with the help of potential and flux density asymptotic solutions. Different from the conventional boundary element, the unknowns in the semi-analytical elements are the first three singular coefficients, which can accurately simulate the singular field and no more refinement is needed. Apart from considering isotropic material, we further investigate the orthotropic and anisotropic materials on the basis of coordinate system transformation method (CSTM), where the governing equation of potential problem can be transformed to isotropic counterparts and then the semi-analytical elements could be employed.

The paper is structured as follow. In Section 2, the basic theory of potential problem and boundary element method is reviewed firstly, and then general formulations for composing the semi-analytical elements and dealing with singular integrals is presented. The introduction about the coordinate system transformation method is then outlined in Section 3. The semi-analytical elements are then applied to two test examples in Section 4. Finally, some conclusions are summarized in Section 5.