# An Optimization Strategy to Position CHIEF Points in Boundary-Element Acoustic Problems
## 2 Materials and Methods  
### The Boundary-Element Method  
Consider an infinite 2D acoustic domain \(\Omega\), containing an inclusion with a regular and smooth boundary defined as \(\Gamma\). Externally to this inclusion, there is the domain where the acoustic field exists, defined as \(\Omega^{+}\), while the interior of the inclusion is defined as \(\Omega^{-}\). This scenario is represented in Figure 1.  
Assuming that in \(\Omega^{+}\) there is a compressible and homogeneous medium with sound velocity \(c\), the linearized wave equation in the time domain in terms of acoustic pressure \(p\) is written through:  
\[\
abla^{2}p(\mathbf{x},t)=\frac{1}{c^{2}}\frac{\partial^{2}p}{\partial t^{2}}(\bm {x},t),\text{ in }\Omega^{+}\cup\Gamma. \tag{1}\]  
Considering that the waves are produced by harmonic periodic vibrations in time, the response will also be periodic through the propagation domain of the fluid; thus, the acoustic pressure can be written as:  
\[p(\mathbf{x},t)=p(\mathbf{x})e^{i\omega t}, \tag{2}\]  
where \(\omega\) is the angular frequency of vibration and response, \(i\) is the representation of the imaginary part of the complex number and \(t\) is the time of analysis. Coupling Equation (1) to Equation (2), the differential equation that governs the steady state of linear acoustics can be obtained. This is defined by the Helmholtz equation, which may be written as:  
\[\
abla^{2}p(\mathbf{x})+k^{2}p(\mathbf{x})=0,\text{ in }\Omega^{+}\cup\Gamma, \tag{3}\]  
Figure 1: General scheme of the analyzed problem with a cavity inside an infinite homogeneous acoustic medium.  
where \(p(x)\) is the wave pressure at point x in the exterior acoustic field, \(\
abla^{2}\) is the Laplacian operator, and \(k=\omega/c\) is the wave number. The boundary value problem (BVP) is defined by Dirichlet and Neumann conditions, respectively:  
\[p(x)=\overline{p}(x),\text{ in }\Gamma_{1}, \tag{4}\]  
\[\frac{\partial p}{\partial n}(x)=\overline{q}(x),\text{ in }\Gamma_{2}, \tag{5}\]  
where \(n\) is the normal unitary vector at \(\Gamma\), defined as \(\Gamma_{1}+\Gamma_{2}=\Gamma\), pointing outwards of the domain, \(\overline{q}\) is the acoustic flow, and \(\overline{p}\) is the acoustic pressure, both prescribed at \(\Gamma\).  
Rewriting Equation (3) as the Kirchhoff-Helmholtz boundary integral equation, the following is obtained:  
\[c(x)p(x)+\int_{\Gamma}q^{*}(x,\xi)p(\xi)d\Gamma(\xi)-\int_{\Gamma}p^{*}(x,\xi )q(\xi)d\Gamma(\xi)=0, \tag{6}\]  
where \(p^{*}(x,\xi)\) is the fundamental solution for the Helmholtz equation and \(q^{*}(x,\xi)\) stands for its normal derivative, defined as:  
\[p^{*}(x,\xi)=-\frac{\mathrm{i}}{4}H_{0}^{(1)}(kr), \tag{7}\]  
\[q^{*}(x,\xi)=\frac{\partial p^{*}(x,\xi)}{\partial n(\xi)}=\frac{\mathrm{i}k} {4}H_{1}^{(2)}(kr)\frac{\partial r}{\partial n(\xi)}, \tag{8}\]  
where \(r=|x-\xi|\), and \(H_{0}^{(2)}\) e \(H_{1}^{(2)}\) are the Hankel functions of the second type and order zero and one, respectively.  
This work uses constant elements, so in Equation (6), \(c(x)\) is a constant that can be defined as:  
\[c(x)=\left\{\begin{array}{ll}1\;for\;x\in\Omega^{+}\\ 0.5\;for\;x\in\Gamma\\ 0\;for\;x\in\Omega^{-}\end{array}\right.. \tag{9}\]  
By discretizing Equation (6) through boundary collocation points, it is possible to generate a system of algebraic equations, where, for each collocation point, the integration of all the boundary elements (NE) is carried out, generating NE linearly independent equations:  
\[\sum_{j=1}^{NE}H_{ij}p_{j}=\sum_{j=1}^{NE}G_{ij}q_{j}, \tag{10}\]  
where:  
\[H_{ij}=\left\{\begin{array}{ll}\int_{\Gamma}q^{*}(x,\xi)d\Gamma(\xi),\;if\;i \
eq j\\ \int_{\Gamma}q^{*}(x,\xi)d\Gamma(\xi)+\frac{1}{2}\;,\;if\;i=j\end{array}\right\}, \tag{11}\]  
\[G_{ij}=\int_{\Gamma}p^{*}(x,\xi)d\Gamma(\xi). \tag{12}\]  
The solution of this system of equations, for each wave number (\(k\)), presents, in general, a unique acoustic pressure and flow response.