# The Boundary Element Method in Acoustics: A Survey
## 5 Extending the Boundary Element Method in Acoustics
### Vibro-Acoustics
#### 5.3.1 Discrete Structural or Acoustic (Finite Element) Model  
The properties of the interior acoustic problem parallel the expected response an excited elastic structure, presuming no damping, or, more simply, simple harmonic motion. In this discussion, let us consider this further in order to bring context. With a mass M and a stiffness K, the equation of (unforced) simple harmonic motion is  
\[M\,\ddot{\underline{q}}+K\,\underline{\underline{q}}=\underline{0}\,\]  
where \(\underline{\underline{q}}\) is the displacement and \(\ddot{\underline{q}}\) is the acceleration. The same equation results from a system of masses or from the finite element method solution with \(M\) and \(K\) termed the mass and stiffness matrices and \(\underline{\underline{q}}\) and \(\ddot{\underline{\underline{q}}}\) are vectors of displacement and acceleration of the individual masses or nodes in the FEM. The phasor solution \(\underline{\underline{q}}=Qe^{i\omega t}\) returns the following equation,  
\[-\omega^{2}M\underline{Q}+K\underline{Q}=0\, \tag{88}\]  
Hence, applying the finite element method to the structural or interior acoustic modal analysis problem returns a generalised algebraic eigenvalue problem (of the form of Equation (68)). The matrices are sparse and hence are amenable to more efficient methods of solution. Although the matrices are larger and the domain needs to be meshed, the BEM with its nonlinear eigenvalue problem struggles to compete with the FEM in acoustic modal analysis. Let \(\omega_{j}^{2}\) and \(\underline{Q}_{j}\) for j = 1, 2,... be the eigenvalues (natural frequencies) and corresponding eigenvectors (mode shapes) of Equation (88). It follows that \(M^{-1}K\underline{\underline{Q}}_{j}=\omega_{j}^{2}\underline{Q}_{j}\).  
Let us also now generalise the model in order to include and external excitation force \(\underline{g}\):  
\[M\,\ddot{\underline{q}}+\ K\,\underline{\underline{q}}=\underline{g}. \tag{89}\]  
Following the phasor substitution \(\underline{g}=\underline{G}e^{i\omega t}\) Equation (89) is modified as follows:  
\[-\omega^{2}M\underline{Q}+K\underline{Q}=\underline{G}\,\ \text{or}\]\[-\omega^{2}\underline{Q}+M^{-1}K\underline{Q}=M^{-1}\underline{G}\.\]  
For the homogeneous equations (\(\underline{G}=\underline{0}\)), the above may be cast as a generalised or standard algebraic eigenvalue problems, as discussed in Section 4.2.2. Let us write the response and the excitation in terms of the modal basis  
\[\underline{Q}=\sum\
olimits_{j}\gamma_{j}Q_{j}\ \text{and}\ M^{-1}\underline{G}=\sum \
olimits_{j}a_{j}Q_{j}. \tag{90}\]  
Considering each eigen-solution in turn relates the coefficients, so that  
\[\gamma_{j}=\frac{a_{j}}{\omega_{j}^{2}-\omega^{2}}.\]  
In practice, a structure or an enclosed fluid experience damping, the simplest and usual model is to relate damping to the velocity:  
\[M\,\underline{\tilde{q}}+\boldsymbol{C}\,\underline{\dot{q}}+\ K\,\underline{ q}=\underline{g}\]  
where \(\boldsymbol{C}\) is termed the damping matrix. Following the phasor substitutions, Equation (89) is modified as follows:  
\[-\omega^{2}M\underline{Q}+j\omega\underline{C}+K\underline{Q}=\underline{G}. \tag{91}\]  
The response of the system to an applied boundary condition across a frequency sweep is to have a smoothed peak at the resonance frequency and a more gradual phase change. In general, the response of the system can be modelled as a sum of weighted modes (90) with the coefficients that are relatable as follows:  
\[\gamma_{j}=d_{j}(\omega)a_{i}. \tag{92}\]