A low frequency elastodynamic fast multipole boundary element method in three dimensions

Abstract

This paper presents a fast multipole boundary element method (FMBEM) for the 3-D elastodynamic boundary integral equation in the ‘low frequency’ regime. New compact recursion relations for the second-order Cartesian partial derivatives of the spherical basis functions are derived for the expansion of the elastodynamic fundamental solutions. Numerical solution is achieved via a novel combination of a nested outer–inner generalized minimum residual (GMRES) solver and a sparse approximate inverse preconditioner. Additionally translation stencils are newly applied to the elastodynamic FMBEM and an implementation of the 8, 4 and 2-box stencils is presented, which is shown to reduce the number of translations per octree level by up to $60\,\%$. This combination of strategies converges 2–2.5 times faster than the standard GMRES solution of the FMBEM. Numerical examples demonstrate the algorithmic and memory complexities of the model, which are shown to be in good agreement with the theoretical predictions.

Appendix: Recurrence relations for the second-order partial derivatives of S and R

The multipole expansion of the elastodynamic fundamental solutions (Eqs. 2 and 3) requires the evaluation of first and second-order Cartesian partial derivatives of the spherical basis functions. A direct evaluation of these terms leads to cumbersome expressions [40]. Alternatively, Gumerov and Duraiswami derived compact recurrence relations for the first-order partial derivatives of the S and R spherical basis functions via the following differential operators [38]:

$$\begin{aligned} \partial _z&\equiv \frac{\partial }{\partial z} \end{aligned}$$

(41)

$$\begin{aligned} \partial _{x+iy}&\equiv \frac{\partial }{\partial x} +i\frac{\partial }{\partial y} \end{aligned}$$

(42)

$$\begin{aligned} \partial _{x-iy}&\equiv \frac{\partial }{\partial x} -i\frac{\partial }{\partial y} \end{aligned}$$

(43)

which are exactly expressible in the spherical coordinate system [22]. The x and y first-order partial derivatives are then:

$$\begin{aligned} \frac{\partial }{\partial x}&= \frac{1}{2}\left[ \partial _{x+iy} + \partial _{x-iy}\right] \end{aligned}$$

(44)

$$\begin{aligned} \frac{\partial }{\partial y}&= - \frac{i}{2}\left[ \partial _{x+iy} - \partial _{x-iy}\right] \end{aligned}$$

(45)

Applying the above differential operators to the spherical basis functions (with $F_n^m = S_n^m(\mathbf{r}), R_n^m(\mathbf{r})$ for arbitrary expansion vector $\mathbf{r}$) yields the following recurrence relations [22, 38]:

$$\begin{aligned} \partial _z F_n^m&= k\big [a_{n-1}^mF_{n-1}^m - a_{n}^mF_{n+1}^m\big ] \end{aligned}$$

(46)

$$\begin{aligned} \partial _{x+iy} F_n^m&= k\big [b_{n+1}^{-m-1}F_{n+1}^{m+1} - b_{n}^mF_{n-1}^{m+1}\big ] \end{aligned}$$

(47)

$$\begin{aligned} \partial _{x-iy} F_n^m&= k\big [b_{n+1}^{m-1}F_{n+1}^{m-1} - b_{n}^{-m}F_{n-1}^{m-1}\big ] \end{aligned}$$

(48)

with the coefficients $a_n^m$ and $b_n^m$ defined as:

$$\begin{aligned}&a_n^m = a_n^{|m|} = \left\{ \begin{array}{lll} &{} \sqrt{\frac{(n+1+|m|)(n+1-|m|)}{(2n+1)(2n+3)}}, &{} n \ge |m| \\ &{} 0, &{} n < |m| \end{array} \right. \end{aligned}$$

(49)

$$\begin{aligned}&b_n^m = \left\{ \begin{array}{lll} &{} \sqrt{\frac{(n-m+1)(n-m)}{(2n-1)(2n+1)}}, &{} m \ge 0 \\ &{} -\sqrt{\frac{(n-m+1)(n-m)}{(2n-1)(2n+1)}}, &{} m \le -1 \\ &{} 0, &{} |m| > n \end{array} \right. \end{aligned}$$

(50)

New recurrence relations for the second-order partial derivatives of S and R are derived using the following second-order differential operators, which again may be exactly expressed in the spherical coordinate system:

$$\begin{aligned} \partial _{z^2}&\equiv \frac{\partial ^2}{\partial z^2} \end{aligned}$$

(51)

$$\begin{aligned} \partial _{zx+izy}&\equiv \frac{\partial }{\partial z}\frac{\partial }{\partial x} + i \frac{\partial }{\partial z}\frac{\partial }{\partial y} \end{aligned}$$

(52)

$$\begin{aligned} \partial _{zx-izy}&\equiv \frac{\partial }{\partial z}\frac{\partial }{\partial x} - i \frac{\partial }{\partial z}\frac{\partial }{\partial y} \end{aligned}$$

(53)

$$\begin{aligned} \partial _{(x+iy)^2}&\equiv \frac{\partial ^2}{\partial x^2} + 2i\frac{\partial }{\partial x}\frac{\partial }{\partial y} -\frac{\partial ^2}{\partial y^2} \end{aligned}$$

(54)

$$\begin{aligned} \partial _{(x-iy)^2}&\equiv \frac{\partial ^2}{\partial x^2} - 2i\frac{\partial }{\partial x}\frac{\partial }{\partial y} -\frac{\partial ^2}{\partial y^2} \end{aligned}$$

(55)

$$\begin{aligned} \partial _{(x\pm iy)}&\equiv \frac{\partial ^2}{\partial x^2} + \frac{\partial ^2}{\partial y^2} \end{aligned}$$

(56)

where the unique second-order partial derivatives may be separated as:

$$\begin{aligned} \frac{\partial }{\partial z}\frac{\partial }{\partial x}&= \frac{1}{2}\big [\partial _{zx+izy} + \partial _{zx-izy} \big ] \end{aligned}$$

(57)

$$\begin{aligned} \frac{\partial }{\partial z}\frac{\partial }{\partial y}&= -\frac{i}{2}\big [\partial _{zx+izy} - \partial _{zx-izy} \big ] \end{aligned}$$

(58)

$$\begin{aligned} \frac{\partial ^2}{\partial x^2}&= \frac{1}{4}\big [\partial _{(x+iy)^2} + \partial _{(x-iy)^2} + 2\partial _{(x\pm iy)} \big ] \end{aligned}$$

(59)

$$\begin{aligned} \frac{\partial ^2}{\partial y^2}&= -\frac{1}{4}\big [\partial _{(x+iy)^2} + \partial _{(x-iy)^2} - 2\partial _{(x\pm iy)} \big ] \end{aligned}$$

(60)

$$\begin{aligned} \frac{\partial }{\partial x}\frac{\partial }{\partial y}&= -\frac{i}{4}\big [\partial _{(x+iy)^2} - \partial _{(x-iy)^2} \big ] \end{aligned}$$

(61)

The recurrence relations for the second-order differential operators (Eqs. 51–56) for both spherical basis functions ($F_n^m = S_n^m(\mathbf{r}), R_n^m(\mathbf{r})$) can then be written as:

$$\begin{aligned} \partial _{z^2} F_n^m&= k^2\big [a_{n-1}^ma_{n-2}^mF_{n-2}^m - (a_{n-1}^m)^2F_{n}^m \nonumber \\&\quad -(a_n^m)^2F_{n}^m + a_{n}^ma_{n+1}^mF_{n+2}^m \big ] \end{aligned}$$

(62)

$$\begin{aligned} \partial _{zx+izy} F_n^m&= k^2\big [-a_{n-1}^mb_{n-1}^mF_{n-2}^{m+1} + a_{n-1}^mb_n^{-m-1}F_{n}^{m+1} \nonumber \\&\quad +a_n^mb_{n+1}^mF_{n}^{m+1} - a_{n}^mb_{n+2}^{-m-1}F_{n+2}^{m+1} \big ] \end{aligned}$$

(63)

$$\begin{aligned} \partial _{zx-izy} F_n^m&= k^2\big [-a_{n-1}^{m}b_{n-1}^{-m}F_{n-2}^{m-1} + a_{n-1}^{m}b_n^{m-1}F_{n}^{m-1} \nonumber \\&\quad +a_n^{m}b_{n+1}^{-m}F_{n}^{m-1} - a_{n}^mb_{n+2}^{m-1}F_{n+2}^{m-1} \big ] \end{aligned}$$

(64)

$$\begin{aligned} \partial _{(x+iy)^2} F_n^m&= k^2\big [b_{n}^{m}b_{n-1}^{m+1}F_{n-2}^{m+2} - b_{n}^{m}b_n^{-m-2}F_{n}^{m+2} \nonumber \\&\quad -b_{n+1}^{-m-1}b_{n+1}^{m+1}F_{n}^{m+2} + b_{n+1}^{-m-1}b_{n+2}^{-m-2}F_{n+2}^{m+2} \big ] \end{aligned}$$

(65)

$$\begin{aligned} \partial _{(x-iy)^2} F_n^m&= k^2\big [b_{n}^{-m}b_{n-1}^{-m+1}F_{n-2}^{m-2} - b_{n}^{-m}b_{n-1}^{m-2}F_{n}^{m-2} \nonumber \\&\quad -b_{n+1}^{m-1}b_{n+1}^{-m+1}F_{n}^{m-2} + b_{n+1}^{m-1}b_{n+2}^{m-2}F_{n+2}^{m-2} \big ] \end{aligned}$$

(66)

$$\begin{aligned} \partial _{(x\pm iy)} F_n^m&= k^2\big [b_{n}^{m}b_{n-1}^{-m-1}F_{n-2}^{m} - (b_{n}^{m})^2F_{n}^{m} \nonumber \\&\quad -(b_{n+1}^{-m-1})^2F_{n}^{m} + b_{n+1}^{-m-1}b_{n+2}^{m}F_{n+2}^{m} \big ] \end{aligned}$$

(67)