Abstract
Wave-based methods for acoustic simulations within enclosures suffer the numerical dispersion and then usually have evident dispersion error for problems with high wave numbers. To improve the upper limit of calculating frequency for 3D problems, a hybrid smoothed finite element method (hybrid SFEM) is proposed in this paper. This method employs the smoothing technique to realize the reduction of the numerical dispersion. By constructing a type of mixed smoothing domain, the traditional node-based and face-based smoothing techniques are mixed in the hybrid SFEM to give a more accurate stiffness matrix, which is widely believed to be the ultimate cause for the numerical dispersion error. The numerical examples demonstrate that the hybrid SFEM has better accuracy than the standard FEM and traditional smoothed FEMs under the condition of the same basic elements. Moreover, the hybrid SFEM also has good performance on the computational efficiency. A convergence experiment shows that it costs less time than other comparison methods to achieve the same computational accuracy.
Highlights
Small enclosures, such as small studios and aircraft cabins, are typical environments that require high sound quality and low noise level in people’s daily life
E smoothing technique in SFEM usually can be performed based on different types of smoothing domains which are created from the nodes, edges, cells, or faces of the elements. erefore, the SFEM can be generally categorized into the following types, the node-based smoothed finite element method (NS-FEM) [17], the edge-based smoothed finite element method (ES-FEM) [18,19,20,21], the cell-based smoothed finite element method (CS-FEM) [22, 23], and the face-based smoothed finite element method (FS-FEM) [24,25,26]
Based on the truth that the number of faces in a tetrahedron element is always less than that of the edges, the face-based smoothing domains are much easier to be constructed than the edge-based smoothing domains. erefore, the proposed hybrid SFEM has high efficiency on constructing the mixed smoothing domains. e numerical verifications have demonstrated that the proposed method is capable of significantly reducing the numerical dispersion and giving more accurate results
Summary
Small enclosures, such as small studios and aircraft cabins, are typical environments that require high sound quality and low noise level in people’s daily life. E smoothing technique in SFEM usually can be performed based on different types of smoothing domains which are created from the nodes, edges, cells, or faces of the elements. Erefore, lower bound solutions can be obtained with respect to the exact solution by using these SFEMs. To further improve the performance of the smoothed methods, the α-FEM [27,28,29,30] was proposed by mixing the node-based technique and the standard FEM. By using the FEM shape function for field variable interpolation in the form of equation (3), the acoustic pressure at a position x on the boundary of the smoothing domain is calculated by p(x) NI(x)pI,. After the smoothing domains are formed in each element, the local smoothing domains can be subsequently constructed by combining the ones in neighboring elements as mentioned in the last subsection
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