Hybrid finite element – wave based method for steady-state acoustic analysis

This paper considers the extension of the frequency application range of the finite element method (FEM) applied to steady-state acoustic problems. This is achieved by the hybrid coupling of the FEM with the wave based method (WBM). The WBM is a computationally efficient alternative for the FEM, but it is restricted to acoustic problems of moderate geometrical complexity. The hybrid finite element – wave based method (HFE-WBM) benefits from the advantageous features of both methods. These are the high geometrical flexibility of the FEM and the computational efficiency of the WBM. The potential of the HFE-WBM to provide more accurate predictions for higher frequencies than the FEM is illustrated for the validation of a numerical example of a 2D acoustic car cavity. 1 Extension of frequency application range of deterministic methods The finite element method (FEM) is a commonly used deterministic numerical method for the analysis of steady-state acoustic problems [1, 2, 3]. The finite element (FE) procedure consists of subdividing an acoustic domain in a large number of small non-overlapping subdomains, i.e. the finite elements. Within each element, a linear combination of simple (polynomial) basis functions approximates the exact pressure distribution. The FEM exhibits a large geometrical flexibility since the FE subdivision is possible for most practical engineering problems. However, the FEM is restricted to the low-frequency application range due to the increasing model size and the subsequent computational efforts, which are required to keep the FE approximation errors within reasonable limits. The FE approximation errors consist of interpolation errors and dispersion errors [3]. Especially, the dispersion errors gain importance at higher frequencies. The difference between the numerical wavenumber and the physical wavenumber causes the dispersion errors. In order to control this type of error, the FE model sizes and subsequent computational costs increase drastically for increasing frequencies. A vast amount of research is currently spent on extending the frequency application range of the FEM. A common feature of these extensions is that the computational costs for increasing frequencies is reduced by controlling the dispersion errors more efficiently. Several classes of extensions exist, namely • the stabilized methods, such as the Galerkin least-squares FEM [4] and the quasi-stabilized FEM [5], • the generalized methods, such as the partition-of-unity FEM [6] and the element-free Galerkin method [7], • and the multi-scale methods, such as the residual-free bubbles method [8] and the discontinuous enrichment method [9].