Journal of Computational Acoustics

Solving Helmholtz problems for low frequency sound fields by a truncated modal basis approach is very efficient. The most time-consuming process is the calculation of the undamped modes.

Using traditional FE solvers, the user has to provide a mesh which has at least six nodes per wavelength in each spatial direction to achieve acceptable results. Because the mesh size increases with the 3rd power of the highest frequency of interest, this uniform dense mesh approach is a very expensive way of creating a modal space.

However, the number of modes and the accuracy of the modal basis directly influences the solution quality. It is well known that the representation of sound fields by modal basis functions φ_i is optimal with respect to the L² error norm. This means that having a modal basis Φ := {φ_i, i = 1⋯n}, the distance between true and approximated sound field takes its minimum in the mean square.

So, it is necessary to have a FE basis which also minimizes the discretization error when computing the modal basis. One can reach this goal by applying adaptive mesh refinements. Additionally, this yields the opportunity of using fast multigrid methods to solve discrete eigenvalue problems.

In context of this presentation we will discuss the results of our adaptive multigrid algorithms.

The process of mesh coarsening describes the reduction and simplification of a highly detailed mesh in such a way that the original geometry is best possible preserved. The earliest coarsening algorithms were developed for real time visualizations in the upcoming field of computer graphics and later on adopted for some element topologies of the finite element method (FEM). An algorithm is presented in this article which applies the mesh decimation process to the requirements of both the FEM and the Boundary Element Method (BEM) in acoustics. The capabilities of the algorithm in order to significantly reduce the CPU times for both numerical methods are shown.

Continuous interpolation of the sound pressure is favored in most applications of boundary element methods for acoustics. Few papers are known to the authors in which discontinuous elements are applied. Mostly they were used because they guarantee C¹ continuity of the geometry at element edges. The effect of superconvergence is known for boundary element collocation on discontinuous elements. This effect is observed if the collocation points are located at the zeroes of orthogonal functions, e.g. at the zeroes of the Legendre polynomials.

In this paper, we start with a review of related work. Then, formulation of discontinuous elements and position of nodes on the element are presented and discussed. One parameter controls the location of nodes on the element. The major part of the paper consists of the investigation of the computational example of a long duct. For that, the numeric solution is compared with the analytic solution of the corresponding one-dimensional problem. Error dependence in terms of frequency, element size and location of nodes on discontinuous elements is reported. It will be shown that the zeroes of the Legendre polynomials account for an optimal position of nodes. Similar results are observed for triangular elements. It can be seen that the error in the Euclidean norm changes by one or two orders of magnitude if the location of nodes is shifted over the element. It can be seen that the optimal location varies with the wave-number although remaining in the vicinity of the zeroes of orthogonal functions. The irregular mesh of a sedan cabin compartment accounts for the second example. Optimal choice of node position is confirmed for this example. One of the key results of this paper is that discontinuous boundary elements perform more efficiently than continuous ones, in particular for linear elements. This, however, further implies that nodes are located at the zeroes of orthogonal functions on the element.

Our work is devoted to the solution of large scale (kl = 10…100π) three dimensional radiation and scattering problems covered by the time harmonic Helmholtz equation. We present an application of the Regular Grid Method and Multilevel Fast Multipole Method to acoustic scattering problems. These methods lead to a memory requirement of that enables us to solve scattering or radiation problems with several ten-thousands of unknowns. In a computational examples we show the efficiency of these methods.

The calculation efficiency of the conventional single-domain acoustic BEM (SBEM) is important for analyzing large scaled or complicated acoustic cavity systems. Although the multi-domain BEM (MBEM) has been developed to deal with such complex shaped and large size acoustic systems effectively, it is generally known that the MBEM requires more computation time and computer memory space than the SBEM. However, if the proper division were applied on a single cavity, it is thought that the MBEM could be the better method than the SBEM in the viewpoint of accuracy and efficiency. This might be possible when one reminds the fact that the effort in the MBEM calculations also depends strongly on the shape of the total acoustic cavity. In this article, the general computational characteristics of the MBEM for analyzing the interior acoustic fields are investigated to provide a guideline in the division of a single cavity into several subdomains for having better computational performance than using the SBEM. A two-dimensional long duct comprised of a number of linear elements is taken as a demonstration example. It is clearly shown that the modification of MBEM model through the present guidelines achieves more accurate and efficient computation than using the SBEM.

Optimization of structures with the intention to reduce noise emission has become an efficient tool during the past decade. Various approaches and applications have been published and will be briefly reviewed in this paper. Then, the structural component model of a spare wheel well and the fluid model of a sedan cabin are described. The noise transfer function is defined as the sound pressure level in vicinity of the driver's ear due to a harmonic force excitation at engine supports. The frequency range of 0–100 Hz is considered. In a first investigation, it is tested whether stiffening of the entire structural component really decreases the noise transfer function. It can be seen that this stiffening mainly affects noise emission in the upper frequency range. In a contribution analysis, i.e. analysis of the surface contribution to the noise at the driver's ear, the original model and the stiffened model are compared. This contribution analysis includes frequency ranges by summation of contribution and/or contribution levels. Modification of the structure by design variables consists of modification of the shell geometry, i.e. curvature. Two regions are selected at the bottom of the wheel well. Optimization of 30 design variables leads to a gain of 1.15 dB in the objective function being the root mean square value of the sound pressure level at the driver's ear. Finally, we discuss the results. In most papers on structural acoustic optimization, higher decreases have been reported. An explanation is provided, why this was not possible for the structure that has been investigated here. The new shape, however, seems to be a reasonable choice.

Time dependent analysis of the dynamic damped behavior of continua are mathematically modelled by partial differential equations. One obtains uniqueness, existence and stability (well posed problems) by the implementation of the correct initial boundary conditions. However, by taking memory effects into consideration, any change in the past of the system changes the future dynamic behavior. Classical damping descriptions fail when describing the behavior of many materials, like teflon. This is because in classical theory the operators are local ones. The implementation of fractional time derivatives into the partial differential equations is an alternative technique to overcome these problems. Thereby the time derivative operator is a global one, memory effects in structure borne sound can be calculated. In this paper the theory of fractional time derivative operators and their application in continuum mechanics is presented. The main result when using this method for damping behavior is that a global operator is needed which takes the whole history into account. We call this theory the functional calculus method instead of the well-known fractional calculus with the use of initial conditions. In order to show the efficiency of this method the calculated impulse response of a viscoelastic rod is compared with measurements. It is shown that the damping behavior is described much better than by other models with comparably few parameters. Moreover, it is the only one that works in a wide frequency range and can describe the dispersion of the resonance frequencies. The implementation of this damping description in a Boundary Element Code is an application of dynamics of 3D continua in the frequency domain.

A method previously developed for asymptotic solution of systems of integro-differential equations with singular memory is applied to the determination of the time-domain asymptotic Green's function of Biot's poroelasticity. Asymptotic time-domain Green's functions are constructed in a neighborhood of the wavefronts. The general anisotropic medium as well as the isotropic case are considered.