Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 9523405, 8 pages

http://dx.doi.org/10.1155/2016/9523405

## High Order Projection Plane Method for Evaluation of Supersingular Curved Boundary Integrals in BEM

School of Aeronautics and Astronautics, Faculty of Vehicle Engineering and Mechanics, State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China

Received 7 December 2015; Revised 12 February 2016; Accepted 17 February 2016

Academic Editor: Ivano Benedetti

Copyright © 2016 Miao Cui et al. 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

Boundary element method (BEM) is a very promising approach for solving various engineering problems, in which accurate evaluation of boundary integrals is required. In the present work, the direct method for evaluating singular curved boundary integrals is developed by considering the third-order derivatives in the projection plane method when expanding the geometry quantities at the field point as Taylor series. New analytical formulas are derived for geometry quantities defined on the curved line/plane, and unified expressions are obtained for both two-dimensional and three-dimensional problems. For the two-dimensional boundary integrals, analytical expressions for the third-order derivatives are derived and are employed to verify the complex-variable-differentiation method (CVDM) which is used to evaluate the high order derivatives for three-dimensional problems. A few numerical examples are given to show the effectiveness and the accuracy of the present method.

#### 1. Introduction

Boundary element method (BEM) [1] is a very promising approach for solving various engineering problems, such as heat transfer problems [2–5] in energy science and engineering, in which accurate evaluation of boundary integrals [6–10] is required. Weak, strong, or hypersingularities are involved in these boundary integrals, if the source point is located on the element under integration [11]. These issues have been focused on by researchers, and many techniques, such as Gauss logarithmic quadrature rule for weak singularities [12], rigid body displacement algorithm for strong singularities [12], and regularization for hypersingularities [13], have been proposed to remove the singularities. Each method has its advantage and disadvantage [11–19]. A comprehensive review has been given in [11] on the treatment of singularities. In general, the efficient methods for singular curved line and surface integrals are those based on the operation over the intrinsic coordinate system [14–16].

Recently, Gao et al. [11] proposed another efficient direct method for numerically evaluating all kinds of singular curved boundary integrals for two-dimensional (2D) and three-dimensional (3D) BEM analyses, and numerical examples were given to demonstrate the stability of the proposed method. However, only quadratic terms were truncated when expanding the coordinate of the field point as Taylor series, which means the method will be invalid if the curved boundary is beyond the quadratic, such as the cubed curve.

In the present work, the direct method in [11] for evaluating arbitrary high order singular curved boundary integrals is developed, by considering the third-order derivatives when expanding the geometry quantities at the field point as Taylor series. Corresponding analytical expressions are derived. The first innovation is that the present method will extend the application range of the method in [11]. That is, the singular integrals over higher order elements, such as cubed curve, can be directly evaluated. Another innovation is that the complex-variable-differentiation method [20] is introduced to determine the third-order derivatives. For 2D boundary integral, the third-order derivatives are analytically derived, which are used to verify the CVDM.

The paper is organized as follows. In Section 2, review and modification of the direct method in [11] are provided. In Section 3, numerical examples are given to show the effectiveness and the accuracy of the present method. Finally, conclusions are drawn from computational results and discussions.

#### 2. Development of the Direct Method for Evaluating Arbitrary High Order Singular Curved Boundary Integrals

##### 2.1. Review of Singular Integrals [11]

Singular integrals over an element can be classified into the following form: in which and represent the source and field points, respectively; is the boundary element under integration, which is a curved line or a curved surface for 2D or 3D problems, respectively; and is the distance between the source and field points.

The term in (1) is a regular function and is the order of the singularity.

In 2D problems, the following type of singular line integrals is also frequently encountered:

It is assumed that the integrals in (1) and (2) always exist: that is, each integration has a finite value.

##### 2.2. Review of Expansions of Geometry Quantities on the Projection Line/Plane [11]

In [11], a projection line for 2D or a projection plane for 3D problems, which is the tangential line/plane of the element to the origin of the intrinsic coordinate system, was introduced. A local orthogonal coordinate system was established on the projection line/plane. The coordinate transformation between the local and global systems can be performed using the following relationships: where is the direction cosine of the local coordinate axes with respect to the global one. The repeated subscripts represent summation; is the global coordinates of the origin of the local coordinate system.

The original curved element can be projected onto the projection line or plane to form a straight or a flat projection element by (3), and then all geometry quantities can be expressed in terms of variables defined on the projection line/plane.

##### 2.3. High Order Expansion of Geometry Quantities in terms of Variables on the Projection Line for 2D Problems

For 2D and 3D problems, derivations for the modification are different, which are separately described in detail.

In 2D problems, the boundary element is a curved line as shown in Figure 1. The local orthogonal coordinate system is denoted by () and the projection line is along the axis .