On the efficiency of immersed boundary finite element methods for wave propagation problems
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Bypassing the tedious task of boundary-conforming mesh generation, immersed boundary methods provide a straightforward and fully automatic discretization based on Cartesian grids and customized quadrature schemes to account for complex geometric models. For wave propagation problems the combination of immersed boundary methods such as the finite cell method with Lagrange polynomials through Gauss-Lobatto-Legendre points (SCM) or B-splines (IGA-FCM) exploits the spectral convergence with respect to the interpolation based on higher order polynomials to reduce the discretization error. The disadvantage of such a discretization is the potentially small overlap between certain elements in the grid and the geometry. Since these poorly cut elements with small physical support adversely affect the eigenvalue spectrum of the corresponding global system, they pose a particular challenge in dynamic simulations by introducing unfeasibly small critical time step sizes for explicit solvers. In this presentation, we discuss the performance of various approaches, in particular material and eigenvalue stabilization methods and tailored time marching schemes, as potential remedies. While previous studies focus on the effectiveness in increasing the critical time step size and the numerical efficiency in terms of accuracy per degree of freedom, we evaluate the computational times required to achieve a given accuracy. The computational efficiency depends not only on the critical time step size and the number of degrees of freedom, but also on the chosen spatial discretization, the number of integration points, the support of the basis functions, and the time marching scheme employed. We perform runtime comparisons on various numerical examples in 2D and 3D.