Study On The Numerical Simulation Of2D And3D Acoustic Problems Based On Smoothed Finite Element Methods

Acoustics is one of the most relevant topics that are very close to our daily life, so theresearch on acoustic problems in various engineering applications is of great importance.The acoustic problems in an inviscid medium, governed by the Helmholtz equation playsan important role in engineering practices. The analytical solutions are only available forsimple geometries and it is impossible to find analytical solutions for more realisticproblems with complicated shapes. Owing to the limitations of the analytical methods,many numerical methods have been developed to solve various acoustic problems.It is well known that the standard finite element method (FEM) are the mostwell-developed and widely-used numerical methods for the simulation of the acousticwave propagation problems, Unfortunately, it is known to all that the FEM can onlyprovide acceptable results in a lower frequency range because the so-called numericaldispersion error‘caused by the overly-stiff ‘nature of the FEM model can growsignificantly with increasing the frequency range, and the numerical results can deviatefrom the exact solution substantially. The smoothed finite element method (S-FEM)developed recently shows great efficiency in solving solid mechanics and acousticsproblems, because S-FEM can provide a proper softening effect to the―overly-stiff‖FEMmodel, significant improvements are achieved on the accuracy of solution. In this paper,muti-dimensional acoustic problems and coupled structural-acoustic problems are studiedusing the S-FEM method, the detail works were recapitulated as follows:1) An edge-based smoothed finite method (ES-FEM) is proposed to solve thetwo-dimensional acoustic problems using the three-node triangular elements and thefour-node quadrilateral elements. In the ES-FEM model, the gradient field of the problemis smoothed using gradient smoothing operations over the smoothing domain associatedwith the edges of the triangulars and quadrilaterals. Owing to this edge-based gradientsmoothing operation, the ES-FEM model behaves much softer the standard FEM modeland hence can reduce the numerical dispersion error significantly in solving the acousticproblems. 2) The three-dimensional acoustics problems are solved by a face-based smoothedfinite method (FS-FEM). The FS-FEM uses four-node tetrahedral elements that can begenerated automatically for complicated domains. In the FS-FEM, the system stiffnessmatrix is computed using strains smoothed over the smoothing domains associated withthe faces of the tetrahedral elements. Typical numerical examples have been conducted toverify the effectiveness of the FS-FEM for the three-dimensional acoustics problems.3) The static and modal analyses of the Reissner-Mindlin plate are solved by theedge-based smoothed finite method (ES-FEM) using three-node triangular elements. Thecalculation of the system stiffness matrix is performed by using the strain smoothingtechnique over the smoothing domain associated with edges of elements, the discrete sheargap was proposed to avoid the transverse shear locking and to improve the accuracy of theresults.4) The coupled structural-acoustic problems are studied using the S-FEM method,Three-node triangular elements and four-node tetrahedral elements are used to discretizethe two-dimensional and three-dimensional domains, respectively. The gradient field ofthe problem is smoothed using gradient smoothing operations over the edge-based andface-based smoothing domains in two-dimensional plate and three-dimensional fluid,respectively. Because the gradient smoothing technique can provide a proper softeningeffect to the―overly-stiff‖FEM model, significant improvements are achieved on theaccuracy of solution for the coupled systems.