Scaled boundary spectral element approaches for modeling wave propagation in unbounded domains

The scaled boundary finite element method is a fundamental solution less boundary element approach, which can accurately model bounded and unbounded domains. In dealing with unbounded domains, especially for wave propagation problems, the scaled boundary methods (SBMs) can model radiation damping effect directly without needing to any extra effort. Like the finite element method, coefficient matrices in the SBMs can be lumped if properly shape functions and appropriate quadrature rule are used. Spectral element method (SEM) is a conventional way to lump these matrices. In the SEMs, quadrature points and elemental nodes are coincided together and shape functions with the Kronecker delta function property are used. Scaled boundary method has four coefficient matrices. By using spectral element method, two of them can be lumped. In this paper, two different approaches are used to construct scaled boundary spectral element method (SBSEM) and lump its coefficient matrices. The Gauss- Lobatto- Legendre (GLL) quadrature with the Lagrangian shape functions and the Clenshaw-Curtis quadrature with the Chebyshev polynomial based shape functions are used. A comparison between the integration rules and shape functions is presented then some benchmark examples are solved using the presented methods. It is shown that the GLL quadrature with the Lagrangian shape functions has better behavior in comparison with the Clenshaw-Curtis quadrature and the Chebyshev polynomial based shape functions.

Keywords:

wave propagation , scaled boundary spectral element , Chebyshev polynomial