## 2 Numerical Pre-Design Regarding Positioning of Adaptronic Loudspeakers  
The following section presents a preliminary assessment of the door structure using numerical analysis to reduce the sound transmission of structure-borne sound, and avoid unwanted vibrations of the support structure and adjacent structures. A comparison of possible areas, which are suitable for the integration of the piezoceramic transducers, is demonstrated with an abstracted finite element (FE)-model. The main difference of the integration areas are their radiating surface and edge stiffness. The calculation of force-excited surface vibrations was conducted with a FE-solver implemented into the software tool wave6 [21]. It is well known that the choice of element type and size is related to the upper-frequency limit of the simulation and the resulting wavelength. In this study, the bending wavelength \(\lambda_{B}\) is the limiting factor. Considering the thickness \(h\) of 3 mm, which is determined by the mold geometry, a Young's Modulus \(E\) of 4 GPa, and a density \(\rho\) of 500 kg/m\({}^{3}\)[20], the bending stiffness \(B\) equals to 9.89 Nm and the mass per unit area \(m^{\prime\prime}=\rho\cdot h\) to 1.5 kg/m\({}^{2}\); at a frequency \(f\) of 5000 Hz, the wavenumber yields [22]:  
\[{k_{B}}^{2}=2\pi f\sqrt{\frac{m^{\prime\prime}}{B}}=2\pi\cdot 5000\;\mathrm{Hz }\sqrt{\frac{1.5\frac{\mathrm{kg}}{\mathrm{m}^{2}}}{9.89\;\mathrm{Nm}}}=110.61 \frac{1}{\mathrm{m}^{2}} \tag{1}\]The wavelength results in the direct relationship, presented in Equation (2), from wavenumber \(k_{B}\) and wavelength \(\lambda_{B}\):  
\[\lambda_{B}=\frac{2\pi}{k_{B}}=0.056\ \text{m}. \tag{2}\]  
Additionally, the loss factor was set to 2%. The model was built using CQUAD8 elements with an element length \(l_{Element}\) of 9 mm, to have at least 6 elements per bending wavelength:  
\[l_{Element}=\frac{\lambda_{B}}{6}=0.0095\ \text{m}. \tag{3}\]  
Furthermore, a convergence analysis with the h-refinement procedure was performed to estimate the error of the FE-solution based on the difference of the calculated eigenfrequencies. The number of calculated eigenmodes \(N_{f}\) is 500, to calculate the eigenfrequencies up to 5300 Hz. Different mesh sizes are considered and compared with a reference mesh of 2.5 mm element length. The error \(\left|\vec{e}_{cm}\right|\) between the eigenfrequency of the reference mesh \(f_{ref,i}\) and the eigenfrequency of a coarsed mesh \(f_{coarse,i}\) are calculated with the following Equation (4):  
\[\left|\vec{e}_{cm}\right|=\frac{1}{N_{f}}\sum_{i=1}^{N_{f}}\frac{\left|f_{ref,i }-f_{coarse,i}\right|}{\left|f_{ref,i}\right|}. \tag{4}\]  
The element type of all mesh sizes is CQUAD8. Figure 1 shows a strong correlation of mesh size and the error \(\left|\vec{e}_{cm}\right|\). By analysing the results of different mesh refinements, it can be assumed that the result for 9 mm element length has converged and causes errors of less than 1%. The whole FE model includes 25,712 CQUAD8 elements. The meshed simulation model is shown in Figure 2. The free edges of the model have been clamped in all translation and rotation directions (marked red). The piezoceramic was abstracted as point force and placed at 1/3 of the total length of each possible integration area. The possible integration areas 1 to 3 were chosen, because of their large individual surface \(A_{i}\) with good geometric decoupling to the overall structure by a geometric stiffness jump. The depth \(t_{i}\) is described by the offset of the possible integration areas to the surrounding structure (compare Figure 2). The pre-design includes the analyzes of the mean surface velocity and the individual reduction of sound transmission over the whole surface. * Area 1: Due to the pronounced recess of the edges, a good decoupling to the entire structure can be expected. However, area 1 is the smallest area and therefore the overall level will be reduced (area: \(A_{1}=445\ \text{cm}^{2}\); depth: \(t_{1}=52\ \text{mm}\)).