A general algorithm for evaluating nearly strong-singular (and beyond) integrals in three-dimensional boundary element analysis

Abstract

One of the typical and most significant issues of almost all boundary element analyses is the accurate evaluation of nearly singular boundary element integrals. In this study, we review some numerical techniques used currently to calculate nearly singular integrals and propose an improved algorithm to calculate integrals with nearly strong (and beyond) singularities. This new method has full generally and can be easily included in any existing computer code. The method is tested in general three-dimensional boundary element analysis. Comparison of this method with some of the existing methods is also presented. It is shown that several orders of magnitude improvement in relative errors can be obtained using the proposed method when compared to a straightforward implementation of Gaussian quadrature.

Appendix: Determination of the projection point \({\varvec{s}}_{\mathbf {0}} \) and \({\varvec{t}}_{\mathbf {0}} \)

As previously mentioned, the explicit knowledge of $s_{0}$ and $t_{0}$ plays a key role in the present analysis. Their explicit expressions are now derived.

From the definitions of $(s_{0} ,t_{0} )$ and $r_{0}$ (Figs. 2, 3), the distance function $r^{2}(s,t)$ will reach the minimum value $r_{0}$ when $(s,t)\rightarrow (s_{0} ,t_{0} )$. Therefore, according to extreme value theorem, the following relations can be easily obtained

$$\begin{aligned} \left\{ {\begin{array}{l} \frac{\partial r^{2}}{\partial s}=2t_{ij} \frac{\partial x_{i} }{\partial s}\left( {x_{j} (s_{0} ,t_{0} )-y_{j} } \right) =0, \\ \frac{\partial r^{2}}{\partial t}=2t_{ij} \frac{\partial x_{i} }{\partial t}\left( {x_{j} (s_{0} ,t_{0} )-y_{j} } \right) =0, \\ \end{array}} \right. \quad i=1,2,3, \end{aligned}$$

(38)

where the derivatives, all evaluated at point $(s_{0} ,t_{0} )$, are promptly obtained from expressions (10) and (13). The problem is now reduced to find the real root $(s_{0} ,t_{0} )$ of the above nonlinear equations system with two variables. Newton’s method and the method of gradients are two most popular numerical methods that are used to solve such nonlinear equations system. Here we focused on the gradient method which solves another equivalent system of equations and obtains new approximations of roots by means of matrix computations. Now we describe the steps of gradients method.

We consider the system of nonlinear algebraic equations

$$\begin{aligned} f_{i} (x_{1} ,x_{2} ,\ldots ,x_{n} )=0,\quad i=1,2,\ldots ,n, \end{aligned}$$

(39)

where $(x_{1} ,x_{2} ,\ldots ,x_{n} )\in R^{n}$ and each $f_{i}$ is a nonlinear real function. Let $F(x_{1} ,x_{2} ,\ldots ,x_{n} )$ be a function which maps $R^{n}$ to $R^{n}$

$$\begin{aligned} F(x_{1} ,x_{2} ,\ldots ,x_{n} )=\sum \limits _{i=1}^n {f_{i}^{2} } . \end{aligned}$$

(40)

Instead of solving the system (39), we are now solving the Eq. (40).

Step 1: Let ${\varvec{x}}^{(0)}=\left( {x_{1}^{(0)} ,x_{2}^{(0)} ,\ldots ,x_{n}^{(0)} } \right)$ be a given initial vector.

Step 2: Calculate $F\left( {x_{1}^{(0)},x_{2}^{(0)} ,\ldots ,x_{n}^{(0)} } \right) =\sum \limits _{i=1}^n f_{i}^{2} \left( x_{1}^{(0)},\right. \left. x_{2}^{(0)} ,\ldots ,x_{n}^{(0)} \right) .$

Step 3: Let $\varepsilon$ be a stopping criterion. If $F\left( {x_{1}^{(0)} ,x_{2}^{(0)} ,\ldots ,x_{n}^{(0)} } \right) <\varepsilon$, it indicates $\left( {x_{1}^{(0)} ,x_{2}^{(0)} ,\ldots ,x_{n}^{(0)} } \right)$ is the ‘solution’ to the system. If not, move to Step 4.

Step 4: Calculate $\left. {\frac{\partial F}{\partial x_{i} }} \right| _{\left( {x_{1}^{(0)} ,x_{2}^{(0)} ,\ldots ,x_{n}^{(0)} } \right) } =2\sum \limits _{j=1}^n {f_{j} \frac{\partial f_{j} }{\partial x_{i} }} ,\left( {i=1,2,\ldots ,n} \right)$, and $D=\sum \limits _{j=1}^n {\left( {\left. {\frac{\partial F}{\partial x_{i} }} \right| _{\left( {x_{1}^{(0)} ,x_{2}^{(0)} ,\ldots ,x_{n}^{(0)} } \right) } } \right) ^{2}}$ .

Step 5: Using results in Step 4 to find the first iteration ${\varvec{x}}^{(1)}=\left( {x_{1}^{(1)} ,x_{2}^{(1)} ,\ldots ,x_{n}^{(1)} } \right)$ in which

$$\begin{aligned} x_{i}^{(1)}= & {} x_{i}^{(0)}\nonumber \\&-\frac{F\left( {x_{1}^{(0)} ,x_{2}^{(0)} ,\ldots ,x_{n}^{(0)} } \right) }{D}\left. {\frac{\partial F}{\partial x_{i} }} \right| _{\left( {x_{1}^{(0)} ,x_{2}^{(0)} ,\ldots ,x_{n}^{(0)} } \right) } ,\nonumber \\&i=1,2,\ldots ,n. \end{aligned}$$

Step 6: Use the results of ${\varvec{x}}^{(1)}$ to find next iteration ${\varvec{x}}^{(2)}$ by using the same procedure. Repeat the process until $F\le \varepsilon$, which indicates that we have reached the ‘solution’ of the system.

Using the procedure described above, it is now possible to give explicitly the expression of $r^{2}(s,t)$ in terms of $s_{0}$ and $t_{0}$, as shown in Eq. (22). The FORTRAN programs that were used to compute $s_{0}$ and $t_{0}$ can be found at: http://blog.sciencenet.cn/blog-425262-950615.html.