Very Efficient Numerical Solutions via the 'Mehrstellen' Method in 1D, 2D, and 3D for Complex Differential Equations Demonstrated for Acoustics and Related Fields

There is a dearth of acceptable techniques to predict Noise and vibration levels pertaining to the interior of automobiles at the mid-frequency range, 500 Hz - 5000 Hz. In this range, traditional techniques like Finite Difference (FD), Finite Element Analysis (FEA), and Boundary Integral Methods (BIM) are still computationally expensive and assumptions necessary for Statistical Energy Analysis start to break down. Due to this, different Hybrid SEA (HSEA) techniques have been introduced, but these are difficult to implement and require special expertise.;This study constructs and implements a compact high order finite difference method to solve the acoustic wave equation in the frequency and time domain. The underlying technique to build the stencil weights is called the "Mehrstellen" method which was initially introduced by Albrecht. Since the "Mehrstellen" method is at least an O(h4) method, this technique requires less nodes to represent the acoustic space and therefore less computation time is required. This study generates and applies the "Mehrstellen" method to acoustic type problems in 1D, 2D, and 3D. In the 3D case, the calculations efforts are further reduced by applying differential operator approximations introduced by Albrecht. The automobile cabin consists of materials that consist of different impedances therefore this study also looks at both Dirichlet and Neumann conditions and give strategies to tackle mixed boundary conditions.;Since the interior of automobiles consist of multi-layered materials, the second part of this study constructs a method to predict the impedance of these materials by calculating the Random Incident transmission loss. The first step to achieve this is to first apply asymptotic and homogenization techniques to simplify the acoustic-structure interaction equations. The derived equations have the same form as the equations derived by Biot. Special emphasis is applied to tie the coefficients of the equations to measurable quantities such as viscous and thermal permeability, tortuosity, and porosity. The Random Incident Transmission Loss is then calculated by applying the Transfer Matrix Method (TMM). The derived Biot type equations are applied to predict noise levels for a real automobile model and in turn these results are compared to experimental and Delany-Bazley results.