A wideband FMBEM for 2D acoustic design sensitivity analysis based on direct differentiation method

Abstract

This paper presents a wideband fast multipole boundary element method (FMBEM) for two dimensional acoustic design sensitivity analysis based on the direct differentiation method. The wideband fast multipole method (FMM) formed by combining the original FMM and the diagonal form FMM is used to accelerate the matrix-vector products in the boundary element analysis. The Burton–Miller formulation is used to overcome the fictitious frequency problem when using a single Helmholtz boundary integral equation for exterior boundary-value problems. The strongly singular and hypersingular integrals in the sensitivity equations can be evaluated explicitly and directly by using the piecewise constant discretization. The iterative solver GMRES is applied to accelerate the solution of the linear system of equations. A set of optimal parameters for the wideband FMBEM design sensitivity analysis are obtained by observing the performances of the wideband FMM algorithm in terms of computing time and memory usage. Numerical examples are presented to demonstrate the efficiency and validity of the proposed algorithm.

Appendices

Appendix A: evaluation of the integrals on $S_{\epsilon }$ and $\Gamma _{\epsilon }$

$S_{\epsilon }$ and $\Gamma _{\epsilon }$ are depicted in Fig. 1. The radius and shape of the semi-circle $S_{\epsilon }$ do not vary with respect to the design variable. First one can obtain easily the following formulations on $S_{\epsilon }$:

$$\begin{aligned} n_j(x)n_j(y)=\cos \theta , \end{aligned}$$

(99)

and

$$\begin{aligned} y_j-x_j=\epsilon n_j(y). \end{aligned}$$

(100)

By differentiating the above two equations with respect to the design variable, one can derive the following formulations:

$$\begin{aligned} \dot{\overline{n_j(x)n_j(y)}}=0, \end{aligned}$$

(101)

and

$$\begin{aligned} \dot{\overline{y_j-x_j}}=\epsilon \dot{\overline{n_j(y)}}. \end{aligned}$$

(102)

Moreover,

$$\begin{aligned} d\dot{S}(y)=0. \end{aligned}$$

(103)

By substituting the above equations into the singular integrals on boundary $S_{\epsilon }$ in Eqs. (33) and (34), one can obtain the following formulations:

$$\begin{aligned} \int _{S_\epsilon }F(x,y)\dot{\phi }(y)dS(y)&= -\frac{1}{2}\dot{\phi }(x), \end{aligned}$$

(104)

$$\begin{aligned} \int _{S_\epsilon }F^1(x,y)\dot{\phi }(y)dS(y)&= -\frac{1}{\pi \epsilon }\dot{\phi }(x)-\frac{1}{4}\dot{q}(x), \end{aligned}$$

(105)

and

$$\begin{aligned} \int _{S_\epsilon }G^1(x,y)\dot{q}(y)dS(y)=\frac{1}{4}\dot{q}(x). \end{aligned}$$

(106)

The results of the other integrals on boundary $S_{\epsilon }$ in Eqs. (33) and (34) are all zero.

The element $\Gamma _{\epsilon }$ will vary with respect to the design variable and the distribution of $\dot{y}_j$ varies linearly on $\Gamma _{\epsilon }$, one can obtain the following formulations on $\Gamma _{\epsilon }$:

$$\begin{aligned} \dot{y}_{l,m}&= \dot{x}_{l,m}, \end{aligned}$$

(107)

$$\begin{aligned} \dot{y}_l-\dot{x}_l&= \dot{x}_{l,j}(y_j-x_j), \end{aligned}$$

(108)

and

$$\begin{aligned} d\dot{S}(y)=[\dot{y}_{l,l}-\dot{y}_{l,j}n_l(y)n_j(y)]dS(y)+r_{,l}r_{,j}\dot{x}_{l,j}\epsilon . \end{aligned}$$

(109)

By substituting the above equations into the singular integrals on boundary $\Gamma _{\epsilon }$ in Eqs. (33) and (34), one can obtain the following equations:

$$\begin{aligned}&\quad \int _{\Gamma _\epsilon } \dot{G}(x,y)q(y)ds(y) \nonumber \\&\quad =-\frac{ik}{4}\lim _{\varepsilon \rightarrow 0}\int _{\Gamma _\varepsilon } H_1^{(1)}(kr)\dot{r}dS(y) q(x), \end{aligned}$$

(110)

$$\begin{aligned}&\quad \int _{\Gamma _\epsilon } G(x,y)\dot{q}(y)dS(y) \nonumber \\&\quad =\frac{i}{4}\lim _{\varepsilon \rightarrow 0}\int _{\Gamma _\varepsilon } H_0^{(1)}(kr)dS(y)\dot{q}(x), \end{aligned}$$

(111)

$$\begin{aligned}&\quad \int _{\Gamma _\epsilon } G(x,y)q(y)d\dot{S}(y) \nonumber \\&\quad =\frac{i}{4}\lim _{\varepsilon \rightarrow 0}\int _{\Gamma _\varepsilon } H_0^{(1)}(kr)dS(y)q(x)[\dot{x}_{l,l}-\dot{x}_{l,j}n_l(x)n_j(x)],\nonumber \\ \end{aligned}$$

(112)

$$\begin{aligned}&\quad \int _{\Gamma _\epsilon } F^1(x,y)\dot{\phi }(y)dS(y) \nonumber \\&\quad =\frac{ik^2}{4}\lim _{\varepsilon \rightarrow 0}\int _{\Gamma _\varepsilon }\left[\frac{ H_1^{(1)}(kr)}{kr}+\frac{2i}{\pi k^2 r^2}-\frac{i}{\pi }\ln (kr)\right]dS(y)\dot{\phi }(x) \nonumber \\&\quad -\frac{k^2L}{4\pi }\left[\ln (\frac{kL}{2})+\frac{8}{k^2L^2}-1\right]\dot{\phi }(x)+\frac{1}{\pi \epsilon }\dot{\phi }(x), \end{aligned}$$

(113)

$$\begin{aligned}&\quad \int _{\Gamma _\epsilon } \dot{F^1}(x,y)\phi (y)dS(y) \nonumber \\&\quad =\left(\frac{4}{\pi L}-\frac{2}{\pi \epsilon }\right)r_{,l}r_{,j}\dot{x}_{l,j}\phi (x)-\frac{ik^2}{4} r_{,l}r_{,j}\dot{x}_{l,j}\phi (x) \nonumber \\&\quad \cdot \lim _{\varepsilon \rightarrow 0}\int _{\Gamma _\varepsilon }\left[ H_2^{(1)}(kr)+\frac{4i}{\pi k^2 r^2}\right]dS(y), \end{aligned}$$

(114)

and

$$\begin{aligned} \int _{\Gamma _\epsilon }&F^1(x,y)\phi (y)d\dot{S}(y) \nonumber \\&=\frac{2}{\pi \epsilon }r_{,l}r_{,j}\dot{x}_{l,j}\phi (x)+\frac{ik^2}{4}r_{,l}r_{,j}\dot{x}_{l,j}\phi (x) \nonumber \\&\cdot \lim _{\varepsilon \rightarrow 0}\int _{\Gamma _\varepsilon }\left[\frac{ H_1^{(1)}(kr)}{kr}+\frac{2i}{\pi k^2 r^2}-\frac{i}{\pi }\ln (kr)\right]dS(y) \nonumber \\&-\frac{k^2L}{4\pi }\left[\ln \left(\frac{kL}{2}\right)+\frac{8}{k^2L^2}-1\right]r_{,l}r_{,j}\dot{x}_{l,j}\phi (x). \end{aligned}$$

(115)

The results of the other integrals on boundary $\Gamma _{\epsilon }$ in Eqs. (33) and (34) are all zero.

Appendix B: the direct differentiation wtih respect the design variable

By differentiating Eq. (14) directly with respect to the design variable, one can obtain the following equation:

$$\begin{aligned} \frac{1}{2}\dot{\phi }(x) =&\dot{C}_1q(x)+C_1\dot{q}(x) \nonumber \\&+\int _{S\setminus S_x} [\dot{G}(x,y)q(y)-\dot{F}(x,y)\phi (y)]dS(y) \nonumber \\&+ \int _{S\setminus S_x}[G(x,y)\dot{q}(y)-F(x,y)\dot{\phi }(y)]dS(y) \nonumber \\&+\int _{S\setminus S_x}[G(x,y)q(y)-F(x,y)\phi (y)]d\dot{S}(y), \end{aligned}$$

(116)

where

$$\begin{aligned} \dot{C}_1 =&- \frac{ik}{4}\lim _{\varepsilon \rightarrow 0}\int _{\Gamma _\varepsilon } H_1^{(1)}(kr)\dot{r} dS(y) \nonumber \\&+ C_1[\dot{x}_{l,l}-\dot{x}_{l,j}n_l(x)n_j(x)]. \end{aligned}$$

(117)

Finally, one can obtain the following equation:

$$\begin{aligned} \dot{C}_1q(x)+C_1\dot{q}(x)=A_1. \end{aligned}$$

(118)

By substituting the above equation into Eq. (116), one can obtain the same equation as Eq. (35).

By differentiating Eq. (15) directly with respect to the design variable, one can obtain the following equation:

$$\begin{aligned} \frac{1}{2}\dot{q}(x) =&\dot{C}_2\phi (x)+C_2\dot{\phi }(x) \nonumber \\&+ \int _{S\setminus S_x} [\dot{G^1}(x,y)q(y)-\dot{F^1}(x,y)\phi (y)]dS(y) \nonumber \\&+ \int _{S\setminus S_x}[G^1(x,y)\dot{q}(y)-F^1(x,y)\dot{\phi }(y)]dS(y) \nonumber \\&+ \int _{S\setminus S_x}[G^1(x,y)q(y)-F^1(x,y)\phi (y)]d\dot{S}(y), \end{aligned}$$

(119)

where

$$\begin{aligned} \dot{C}_2 =M_1+M_2, \end{aligned}$$

(120)

and

$$\begin{aligned} M_1 =-\frac{ik^2}{4} \lim _{\varepsilon \rightarrow 0}\int _{\Gamma _\varepsilon } H_2^{(1)}(kr)\dot{r}/rdS(y), \end{aligned}$$

(121)

$$\begin{aligned} M_2=\lim _{\varepsilon \rightarrow 0}\int _{\Gamma _\varepsilon } \frac{ik}{4r} H_1^{(1)}(kr)d\dot{S}(y). \end{aligned}$$

(122)

By eliminating the singular part in function $H_2^{(1)}(kr), M_1$ can be expressed as

$$\begin{aligned} M_1 =&-\frac{ik^2}{4} \lim _{\varepsilon \rightarrow 0}\int _{\Gamma _\varepsilon }\left[ H_2^{(1)}(kr)+\frac{4i}{\pi k^2 r^2}\right]\dot{r}/rdS(y) \nonumber \\&- \lim _{\varepsilon \rightarrow 0}\int _{\Gamma _\varepsilon } \frac{1}{\pi r^2} \dot{r}/rdS(y). \end{aligned}$$

(123)

From Eqs. (29) and (108), one can derived the following equation on $\Gamma _\varepsilon$:

$$\begin{aligned} \dot{r}/r=r_{,l}r_{,j}\dot{x}_{l,j}=\dot{y}_{l,l}-\dot{y}_{l,j}n_l(y)n_j(y), \end{aligned}$$

(124)

and $\dot{r}/r$ does not vary with respect to the design variable for the piecewise constant discretization. By substituting this equation into Eq. (123), one can obtain

$$\begin{aligned} M_1 =&-\frac{ik^2}{4} \lim _{\varepsilon \rightarrow 0}\int _{\Gamma _\varepsilon }\left[ H_2^{(1)}(kr)+\frac{4i}{\pi k^2 r^2}\right]dS(y)r_{,l}r_{,j}\dot{x}_{l,j} \nonumber \\&+ \left(\frac{4}{\pi L}-\frac{2}{\pi \varepsilon }\right)r_{,l}r_{,j}\dot{x}_{l,j}. \end{aligned}$$

(125)

By substituting Eq. (109) into Eq. (122), $M_2$ can be expressed as

$$\begin{aligned} M_2=\lim _{\varepsilon \rightarrow 0}\int _{\Gamma _\varepsilon } \frac{ik}{4r} H_1^{(1)}(kr)dS(y)r_{,l}r_{,j}\dot{x}_{l,j}+\frac{1}{\pi \varepsilon }r_{,l}r_{,j}\dot{x}_{l,j}. \end{aligned}$$

(126)

By substituting Eq. (17) into the above equation, one can obtain

$$\begin{aligned} M_2=C_2r_{,l}r_{,j}\dot{x}_{l,j}+\frac{2}{\pi \varepsilon }r_{,l}r_{,j}\dot{x}_{l,j}. \end{aligned}$$

(127)

By substituting Eqs. (125) and (127) into Eq. (120), $\dot{C}_2$can be expressed as

$$\begin{aligned} \dot{C}_2=&-\frac{ik^2}{4} \lim _{\varepsilon \rightarrow 0}\int _{\Gamma _\varepsilon }\left[ H_2^{(1)}(kr)+\frac{4i}{\pi k^2 r^2}\right]dS(y)r_{,l}r_{,j}\dot{x}_{l,j} \nonumber \\&+ \left(\frac{4}{\pi L}+C_2\right)r_{,l}r_{,j}\dot{x}_{l,j}. \end{aligned}$$

(128)

Finally, one can obtain the following equation:

$$\begin{aligned} \dot{C}_2\phi (x)+C_2\dot{\phi }(x)=B_1. \end{aligned}$$

(129)

By substituting the above equation into Eq. (119), one can obtain the same equation as Eq. (36).