A fast multipole boundary element method based on higher order elements for analyzing 2-D potential problems

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Abstract

A novel fast multipole boundary element method (FM-BEM) is proposed to analyze 2-D potential problems by using linear and three-node quadratic elements. In FM-BEM, fast multipole expansions are used for the integrals on elements which are far away from the source point, whereas direct evaluations are used for the integrals on elements which are close to the source point. The use of higher-order elements results in more complex forms of the integrands, which increases the burden in direct evaluations, especially for singular and nearly singular integrals. Herein, the complex notation is introduced to simplify the computational formulations in boundary integral equations for 2-D potential problems. The singular and nearly singular integrals on linear elements are calculated by the analytic formulas, and those on three-node quadratic elements are evaluated by a robust semi-analytical algorithm. Numerical examples show that the proposed FM-BEM possesses higher accuracy than the conventional FM-BEM. Besides, the present method can analyze thin structures and evaluate accurately the physical quantities at interior points much closer to the boundary.

Introduction

Boundary element method (BEM) is a fascinating numerical method which has been widely applied in many research fields. Different from the finite element method (FEM), BEM only discretizes the surface of the computational domain. Usually, the required numbers of the elements and the unknown quantities of BEM are less than those of FEM. However, it is difficult to deal with large-scale problems by using the conventional BEM, because the matrices produced by the conventional BEM are dense and non-symmetric. Although the number (N) of equations of the linear system is small, it requires O(N2) operations to compute the coefficients and O(N3) operations to solve the linear system using direct solvers. Once N becomes larger, the sizes of memory required for storing and the consumed CPU time can be enormous. While FEM has been used routinely to analyze the model with millions of degree of freedoms (DOFs), the conventional BEM is still limited to solve the model with several thousands of DOFs. In the late 1980s, Rokhlin [1] and Greengard and Rokhlin [2] proposed the original fast multipole method (FMM) to accelerate the evaluation of the potential and force fields in systems involving large numbers of particles. With the help of FMM, the solution time and memory required in BEM can be reduced from O(N2) to O(N), which makes reality for BEM to solve large-scale problems. Over the past three decades, the BEM embedded with FMM has been applied to analyze various scientific problems, such as elastostatic problems [3], [4], [5], [6], crack problems [7], [8], [9], acoustic problems [10], [11], [12] and many others.

In the fast multipole BEM (FM-BEM), the evaluations of the integrals on boundary elements related to a source point are distinguished into two cases. One is the far-field integrals on elements which are far away from the source point. The other is the near-field integrals on elements which are close to the source point. For the former one, the node-to-node interactions in the conventional BEM are converted to cell-to-cell interactions of far-field translations by using the fast multipole expansions. For the latter one, a direct evaluation is still needed. Of course, the singular integrals which are the main issue of the conventional BEM, still arise in FM-BEM. For boundary discretization using constant elements, all near-field integrals including both singular and nearly singular integrals can be analytically calculated. However, accurate evaluations of the singular and nearly singular integrals on higher order elements require some special strategies. Takahashi and Matsumoto [13] proposed an isogeometric BEM with the FMM for the 2-D Laplace equation, in which the singular integrals on NURBS-based elements or isogeometric elements were implemented with Taylor series. Liu et al. [14] developed an isogeometric BEM with the FMM for the 2-D acoustic problems, which has been further applied to the sensitivity analysis and shape optimization of 2-D acoustic structures. In their work, the singular integrals on NURBS elements were evaluated by using a subtraction technique of singularity.

It is known that the direct Gaussian quadrature fails to calculate the nearly singular integrals in the conventional BEM. In fact, the evaluation of the nearly singular integrals is more complicated than that of the singular integrals. To deal with the nearly singular integrals, the researchers have developed various numerical techniques, such as regularization methods [15], [16], [17], adaptive element subdivision method [18], [19], analytic and semi-analytical algorithms [20], [21], [22], [23], [24], distance transformation [25], [26], sinh transformation [27], [28], [29], exponential transformation [30], [31], [32], [33], quadrature by expansion for 2-D cases [34] and spectrally-accurate quadratures for 2-D Stokes and Laplace equations [35]. It is noteworthy that Granados and Gallego [17] and Dehghan and Hosseinzadeh [23], [24] have treated the singular and nearly singular integrals in the conventional BEM based on the complex space. In the complex space, the formulations of computing the nearly singular integrals were simple so that less CPU time of implementing the conventional BEM with higher order elements was required. All techniques reviewed above were developed to address the nearly singular integrals in the conventional BEM. To the author’s knowledge, very few available literatures reported the details about the nearly singular integrals in FM-BEM with higher order elements. One of the main reasons is that the complicated computational processing of weakly singular integrals on higher order elements may reduce the efficiency of FM-BEM.

To promote the computing efficiency and accuracy, a new FM-BEM with linear and 3-node quadratic elements is developed for 2-D potential problems. The remainder of this paper is organized as follows. In Section 2, the original boundary integral equations for 2-D potential problems are reviewed. The details about the evaluation of the singular and nearly singular integrals on both linear and quadratic elements are shown in Section 3 and the scheme to calculate the far-field integrals with FMM is discussed in Section 4. In Section 5, we focus on the framework of FM-BEM with the higher order elements. In Section 6, several numerical examples are provided to examine the accuracy and efficiency of FM-BEM with higher order elements. Section 7 gives the conclusions of the present study.

Section snippets

Boundary integral equations for 2-D potential problems

Consider the 2-D potential problems shown in Fig. 1. The integral equation for the potential at any interior point in Ω can be written as ϕy=ΓGx,yqx dΓΓFx,yϕx dΓ,yΩ,xΓ,where y(y1,y2) is the source point and x(x1,x2) is the field point. Γ is the boundary of Ω. ϕx and qx are the potential and the potential gradient at x on Γ, respectively. Gx,y and Fx,y are foundational solutions of 2-D potential problems as follows: Gx,y=12πlnr,Fx,y=Gx,ynx=12π1rrn, where n is the outward normal unit

The calculation of the near-field integrals on both linear and quadratic elements

In the complex plane, the field point x(x1,x2) is represented by z=x1+ix2 and the source point y(y1,y2) is represented by zs=y1+iy2, where i=1, as shown in Fig. 2.

Using the complex notation, we have G(x,y)= Re[G(z,zs)],F(x,y)= Re[F(z,zs)], where Gz,zs=12πlnzzs,Fz,zs=Gz,zsnz=Gz,zsznz=12πnzzzs. nz is a complex-valued function of the outward unit normal at z on Γe. The integrals (7)–(10) can be written as ΓeG(x,y)Nιξqιe dΓ= ReΓeGz,zsNιξqιe dΓ,ΓeFx,yNιξϕιe dΓ= ReΓeFz,zsNιξϕιe dΓ,ΓeGx

The calculation of the far-field integrals using FMM

In fast multipole BEM, the far-field integrals are calculated by the fast multipole expansions. For facilitating discussion, we rewrite the integrals in BIE (6) as below ΓeG(x,y)Nιξqιe dΓ= ReΓeGz,zsNιξqιe dΓ,ΓeFx,yNιξϕιe dΓ= ReΓeFz,zsNιξϕιe dΓ, where Gz,zs=12πlnzsz,Fz,zs=Gz,zsnz=Gz,zsznz. An expansion point zc (Fig. 6) close to the field point z is introduced, which means that |zzc||zszc|, then we have Gz,zs=12πlnzsz=12πlnzszc+ln1zzczszc.

Apply the following Taylor series

The framework of FM-BEM with higher order elements

The main computational procedures of FM-BEM with higher order elements are outlined as follows:

Step 1 Discretization with linear and three-node quadratic elements. For a given 2-D problem, the boundary Γ is discretized with linear and three-node quadratic elements same as those employed in the conventional BEM.

Step 2: Determination of the quad-tree structure of the mesh. For 2-D problems, a square called the cell of level 0 covering the entire boundary Γ is considered firstly. Secondly, this

Numerical examples

In this section, three numerical examples are given to investigate the performance of FM-BEM with linear and three-node quadratic elements. Herein, the near-field integrals on linear elements are calculated by the analytical formulas, which we referred to as FM-BEM-LA. The near-field integrals on three-node quadratic elements are evaluated by a semi-analytical algorithm, which we referred to as FM-BEM-QSA. Unknown boundary variables and physical quantities inside the domain are calculated and

Conclusions

In this paper, a novel fast multipole boundary element method (FM-BEM) with higher order elements for 2-D potential problems is proposed. The present method has several important advantages. First, linear and three-node quadratic elements are employed for discretization, which can provide more exact geometric representations. Numerical examples show that the present method possesses higher accuracy than the conventional FM-BEM with constant elements. Second, based on the complex space, the

Acknowledgment

We wish to acknowledge the support by the National Natural Science Foundation of China [Grant No.: 11702076].

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