2D Acoustic Design Sensitivity Analysis Based on Adjoint Variable Method Using Different Types of Boundary Elements

Abstract

Continuous linear and quadratic boundary elements are often applied to numerical solution. Discontinuous higher-order boundary elements are developed for 2D acoustic problems to achieve higher accuracy in this paper. The Burton–Miller formulation is used to overcome the fictitious frequency problem when using a single Helmholtz boundary integral equation for exterior boundary-value problem. The strong singular integrals in Burton–Miller formulation using different types of element discretization are evaluated explicitly and directly, respectively. An example of scattering by an infinite rigid cylinder is presented to compare the performance of different types of elements. The effect of the position of nodes on the performance of discontinuous elements is studied, and an empirical value for optimal nodal position is concluded in this paper. Adjoint variable method is applied to evaluate the sensitivity value of the objective function, and the method of moving asymptotes is used for structural optimization analysis of noise barrier.

Appendix 1: Evaluation of the Integrals on \(S_{\varepsilon }\) and \(\Gamma _{\varepsilon }\)

After piecewise discretization, the CBIE and NDBIE formulas can be rewritten as

$$\begin{aligned} \frac{1}{2}\phi (x^j)+B(x^j)&= \int _{S\setminus S_{x^j}} G(x^j,x)q(x)\mathrm{d}S(x)\nonumber \\&\quad \,-\int _{S\setminus S_{x^j}} F(x^j,x) \phi (x)\mathrm{d}S(x), \end{aligned}$$

(35)

and

$$\begin{aligned} \frac{1}{2}q(x^j)+D(x^j) =&\int _{S\setminus S_{x^j}} G^1(x^j,x)q(x)\mathrm{d}S(x)\nonumber \\&-\int _{S\setminus S_{x^j}} F^1(x^j,x) \phi (x)\mathrm{d}S(x), \end{aligned}$$

(36)

$S\setminus S_{x^j}$ denotes the boundary S except $S_{x^j}$, $S_{x^j}$ is the element containing the source point $x^j$. $S_\varepsilon$ denotes a semi-circle with a radius $\varepsilon$ centred at $x^j$ and $\Gamma _\varepsilon$ denotes $S_{x^j}\setminus S_\varepsilon$.

The function f(x) in every boundary element can be expressed as the following formulation

$$\begin{aligned} f(x)=\sum _{k=1}^{m} \Phi _k f(x^k), \end{aligned}$$

(37)

where m denotes the number of interpolation nodes in every boundary element, $\Phi$ denotes the interpolation function. Function f(x) could be chosen as $\phi (x)$, q(x) and so on.

• 1. For constant element

  $m=1$ and $\Phi =1$.

• 2. For linear element

  $$\begin{aligned} \Phi _{1}&=\frac{1}{2}\left( 1-\frac{\xi }{\beta }\right) ,\quad \Phi _{2}=\frac{1}{2}\left( 1+\frac{\xi }{\beta }\right) , \end{aligned}$$

  (38)

  where $\xi$ means the local coordinate of the point x, and $\beta$ denotes the position of interpolation nodes on the discontinuous element. When $\beta =1$, Eq. (38) denotes expression of the interpolation functions for linear continuous element.

• 3. For quadratic element.

  $$\begin{aligned}&\Phi _{1}=\dfrac{\xi }{2\beta }\left( \dfrac{\xi }{\beta }-1\right) ,\quad \Phi _{2}=1-\dfrac{\xi ^2}{\beta ^2} ,\nonumber \\&\Phi _{3}=\dfrac{\xi }{2\beta }\left( \dfrac{\xi }{\beta }+1\right) . \end{aligned}$$

  (39)

  Using Eq. (37) , one can obtain the expression of the coordinate $(x_1,x_2)$ at the point x, as follows

  $$\begin{aligned} x_1(\xi )&=\frac{1}{2}A_1 \xi ^2+A_2\xi +{x}^b_1, \end{aligned}$$

  (40)
  $$\begin{aligned} x_2(\xi )&=\frac{1}{2}A_3 \xi ^2+A_4 \xi +{x}^b_2, \end{aligned}$$

  (41)

  where ${x}^b_k (k=1,2)$ denotes the coordinate of the central point in the boundary element, and the coefficient $A_k (k=1,4)$ can be derived by

  $$\begin{aligned} A_1&={x}^c_1-2{x}^b_1+{x}^a_1, \quad A_2 =\frac{1}{2}({x}^c_1-{x}^a_1), \nonumber \\ A_3&={x}^c_2-2{x}^b_2+{x}^a_2,\quad A_4 =\frac{1}{2}({x}^c_2-{x}^a_2), \end{aligned}$$

  (42)

  where ${x}^a_k (k=1,2)$ and ${x}^c_k$ denotes the coordinate of the two extreme points in the boundary element. Similar as linear element, Eq. (39) denotes expression of the interpolation functions for quadratic continuous element when $\beta =1$.

When different types of elements are used to discretize the boundary, one can obtain the different expression of coefficient $B(x^j)$ and $D(x^j)$ in Eqs. (35) and (36), as follows

• 1. For constant element

  $$\begin{aligned} B(x^j)&=-C_1 q(x^j), \end{aligned}$$

  (43)
  $$\begin{aligned} D(x^j)&=-C_2\phi (x^j), \end{aligned}$$

  (44)

  where coefficient $C_1$ and $C_2$ can be expressed as the following formulation

  $$\begin{aligned} C_1&= -\frac{L}{2\pi }\left[ \ln (\frac{kL}{2}) -1\right] \nonumber \\&\quad \,+\frac{{\mathrm {i}}}{4}\lim _{\varepsilon \rightarrow 0}\int _{\Gamma _\varepsilon } \left[ H_0^{(1)}(kr)-\frac{2{\mathrm {i}}}{\pi }\ln (kr)\right] \mathrm{d}S(y), \end{aligned}$$

  (45)
  $$\begin{aligned} C_2&= \frac{{\mathrm {i}}k^2}{4}\lim _{\varepsilon \rightarrow 0}\int _{\Gamma _\varepsilon } \left[ \frac{ H_1^{(1)}(kr)}{kr} +\frac{2{\mathrm {i}}}{\pi k^2r^2}-\frac{{\mathrm {i}}}{\pi }\ln (kr)\right] \mathrm{d}S(y)\nonumber \\&\quad \,-\frac{k^2L}{4\pi }\left[ \ln \left( \frac{kL}{2}\right) +\frac{8}{k^2L^2}-1\right] , \end{aligned}$$

  (46)

  where L is the length of the element in which node $x^j$ locates.

• 2. For linear element

  $$\begin{aligned} B(x^j)&= - \left[ B_1q(x^j)+B_2q(x^l)\right] , \end{aligned}$$

  (47)
  $$\begin{aligned} D(x^j)&=-\left[ D_1\phi (x^j)+D_2\phi (x^l)\right] , \end{aligned}$$

  (48)

  where point $x^l$ is the another node located on a boundary element containing node $x^j$. $\alpha$ denotes the local coordinate of the node $x^j$, and $\beta =|\alpha |$. For linear discontinuous boundary element $(\beta \ne 1)$, coefficients $B_1$, $B_2$, $D_1$ and $D_2$ are expressed as following

  $$\begin{aligned} B_1&= \frac{{\mathrm {i}}L}{8}\int _{-1}^{1} \big [H_0^1(kr)-\dfrac{2{\mathrm {i}}}{\pi }\ln (kr)\big ]\phi _1 \,d\xi -\dfrac{L}{2\pi }\ln \dfrac{kL}{2}\nonumber \\&\quad \, - \dfrac{L}{8\pi \beta } II_1, \end{aligned}$$

  (49)
  $$\begin{aligned} B_2&= \frac{{\mathrm {i}}L}{8}\int _{-1}^{1} \big [H_0^1(kr)-\dfrac{2{\mathrm {i}}}{\pi }\ln (kr)\big ]\phi _2 \,\mathrm{d}\xi \nonumber \\&\quad \,-\dfrac{L}{2\pi }\ln \dfrac{kL}{2} - \dfrac{L}{8\pi \beta } II_2, \end{aligned}$$

  (50)
  $$\begin{aligned} D_1&=\dfrac{{\mathrm {i}}k^2L}{8}\int _{-1}^{1}\Big [ \dfrac{H_1^1(kr)}{kr}+\dfrac{8{\mathrm {i}}}{\pi k^2L^2}\dfrac{1}{(\xi -\alpha )^2}\nonumber \\&\quad \,-\dfrac{{\mathrm {i}}}{\pi } \ln (kr) \Big ]\,\mathrm{d}\xi +\dfrac{\ln (1+\beta ) - \ln (1-\beta )}{2\beta \pi L} \nonumber \\&\quad \,-\dfrac{k^2L}{4\pi }\ln \dfrac{kL}{2} -\dfrac{k^2L}{8\pi }II_1 \end{aligned}$$

  (51)
  $$\begin{aligned} D_2&=\dfrac{{\mathrm {i}}k^2L}{8}\int _{-1}^{1}\Big [ \dfrac{H_1^1(kr)}{kr}+\dfrac{8{\mathrm {i}}}{\pi k^2L^2}\dfrac{1}{(\xi -\alpha )^2}\nonumber \\ {}&-\dfrac{{\mathrm {i}}}{\pi } \ln (kr) \Big ]\,\mathrm{d}\xi \nonumber \\&\quad \, +\dfrac{\ln (1+\beta ) - \ln (1-\beta )}{2\beta \pi L} - \dfrac{2}{\pi L(1-\beta ^2)} \nonumber \\&\quad \,-\dfrac{k^2L}{4\pi }\ln \dfrac{kL}{2} -\dfrac{k^2L}{8\pi }II_2, \end{aligned}$$

  (52)

  where

  $$\begin{aligned} r(\xi )&= \dfrac{L}{2}|\xi -\alpha |, \end{aligned}$$

  (53)
  $$\begin{aligned} II_1&= \int _0^{1+\beta }x\ln x\,\mathrm{d}x - \int _0^{1-\beta }x\ln x\,\mathrm{d}x, \end{aligned}$$

  (54)
  $$\begin{aligned} II_2&= \int _0^{1-\beta }\ln x\,\mathrm{d}x + \int _0^{1+\beta }\ln x\,\mathrm{d}x \nonumber \\&\quad \,- \dfrac{1}{2\beta } \left( \int _0^{1+\beta }x\ln x\,dx - \int _0^{1-\beta }x\ln x\,\mathrm{d}x\right) . \end{aligned}$$

  (55)

  For linear continuous boundary element $(\beta = 1)$, coefficients $B_1$, $B_2$, $D_1$ and $D_2$ are expressed as following

  $$\begin{aligned} B_1&=\frac{{\mathrm {i}}L}{8}\int _{-1}^{1} \big [H_0^1(kr)-\dfrac{2{\mathrm {i}}}{\pi }\ln (kr)\big ]\phi _1 \,\mathrm{d}\xi \nonumber \\&\quad \,-\dfrac{L}{8\pi }(2\ln (kL)-3), \end{aligned}$$

  (56)
  $$\begin{aligned} B_2&= \frac{{\mathrm {i}}L}{8}\int _{-1}^{1} \big [H_0^1(kr)-\dfrac{2{\mathrm {i}}}{\pi }\ln (kr)\big ]\phi _2 \,\mathrm{d}\xi \nonumber \\&\quad \,-\dfrac{L}{8\pi }(2\ln (kL)-1), \end{aligned}$$

  (57)
  $$\begin{aligned} D_1&=\dfrac{{\mathrm {i}}k^2L}{8}\int _{-1}^{1}\Big [ \dfrac{H_1^1(kr)}{kr}+\dfrac{2{\mathrm {i}}}{\pi k^2r^2}-\dfrac{{\mathrm {i}}}{\pi } \ln (kr) \Big ]\phi _1\,\mathrm{d}\xi \nonumber \\&\quad \,-\dfrac{1}{2\pi L}(1+\ln 2)-\dfrac{k^2L}{16\pi }(2\ln (kL)-3), \end{aligned}$$

  (58)
  $$\begin{aligned} D_2&= \dfrac{{\mathrm {i}}k^2L}{8}\int _{-1}^{1}\Big [ \dfrac{H_1^1(kr)}{kr}+\dfrac{2{\mathrm {i}}}{\pi k^2r^2}-\dfrac{{\mathrm {i}}}{\pi } \ln (kr) \Big ]\phi _2\,\mathrm{d}\xi \nonumber \\&\quad \,+\dfrac{\ln 2}{2\pi L}-\dfrac{k^2L}{16\pi }(2\ln (kL)-1). \end{aligned}$$

  (59)

• 3. For quadratic element

  $$\begin{aligned} B(x^j)&= - \left[ B_1q(x^j)+B_2q(x^l)+ B_3q(x^m)\right] , \end{aligned}$$

  (60)
  $$\begin{aligned} D(x^j)&=- \left[ D_1\phi (x^j)+D_2\phi (x^l)+D_3\phi (x^m)\right] , \end{aligned}$$

  (61)

  where points $x^l$ and $x^m$ are the other nodes that are located on a boundary element containing node $x^j$. For quadratic discontinuous boundary element $(\beta \ne 1)$, coefficients $B_1$, $B_2$, $B_3$, $D_1$, $D_2$, $D_3$ are expressed as following

  $$\begin{aligned} B_m&= \frac{{\mathrm {i}}}{4}\int _{-1}^{1} \phi _m \big [H_0^1(kr)J_1-\dfrac{2{\mathrm {i}}}{\pi }\ln (kr_1)J_2\big ] \,\mathrm{d}\xi \nonumber \\&\quad \, -\dfrac{J_2}{2\pi }(\ln k+\ln J_2)\int _{-1}^{1}\phi _m\,\mathrm{d}\xi \nonumber \\&\quad \, -\dfrac{J_2}{2\pi }\int _{-1}^{1}\phi _m \ln \vert \xi -\alpha \vert \,\mathrm{d}\xi ,\quad m=1,2,3, \end{aligned}$$

  (62)
  $$\begin{aligned} D_m&= \dfrac{{\mathrm {i}}k^2}{4}\int _{-1}^{1} \phi _m\Big [ \dfrac{H_1^1(kr)}{kr}n_j(x)n_j(y)J_1(\xi )\nonumber \\&\quad \,+\dfrac{2{\mathrm {i}}}{\pi k^2}\dfrac{1}{(\xi -\alpha )^2 J_2}-\dfrac{{\mathrm {i}}}{\pi }\ln (kr_1)J_2 \Big ]\,\mathrm{d}\xi \nonumber \\&\quad \, -\dfrac{k^2J_2}{4\pi }(\ln k+ \ln J_2)\int _{-1}^{1}\phi _m \,\mathrm{d}\xi \nonumber \\&\quad \, +\dfrac{1}{2\pi J_2}\int _{-1}^{1}\dfrac{\phi _m }{(\xi -\alpha )^2}\mathrm{d}\xi \nonumber \\&\quad \, -\dfrac{k^2J_2}{4\pi }\int _{-1}^{1}\phi _m \ln \vert \xi -\alpha \vert \,\mathrm{d}\xi ,\quad m=1,2,3, \end{aligned}$$

  (63)

  where

  $$\begin{aligned} r^2&=(\xi -\alpha )^2\left\{ \left[ \frac{1}{2}A_1(\xi +\alpha )+A_2 \right] ^2\right. \nonumber \\&\quad \left. +\left[ \frac{1}{2}A_3(\xi +\alpha )+A_4 \right] ^2 \right\} , \end{aligned}$$

  (64)
  $$\begin{aligned} r_1^2&= (\xi -\alpha )^2J_2^2, \end{aligned}$$

  (65)
  $$\begin{aligned} J_1^2&=(A_1\xi +A_2)^2+(A_3\xi +A_4)^2, \end{aligned}$$

  (66)
  $$\begin{aligned} J_2^2&=(A_1\alpha +A_2)^2+(A_3\alpha +A_4)^2. \end{aligned}$$

  (67)

  When $\beta =1$, the coefficients $B_m$ and $D_m$ for quadratic continuous element can be obtained using Eqs. (62) and (63).