A geometric multigrid preconditioner for solving time-harmonic elastodynamics on unbounded domains with perfectly matched layers

Abstract eng:
A scalable geometric multigrid preconditioner applicable to solving large-scale time-harmonic elastodynamic systems on unbounded domains modeled by Perfectly Matched Layers (PMLs) is presented. To solve for the forced motion of the elastodynamic system under time-harmonic excitation, one must solve a linear system of equations where the coefficient matrix is the stiffness matrix shifted by the mass matrix. Application of PMLs to model the radiation boundary condition renders these mass and stiffness matrices complex-valued symmetric and large-scale for accurate solutions. Large-scale matrices require the use of iterative methods to solve the linear systems for tractable solution time and computational memory. Complexvalued symmetric linear systems can be in general extremely difficult to solve iteratively, due to the lack of standard efficient techniques. To solve this linear system, a geometric multigrid preconditioner which can be combined with iterative methods such as GMRES is developed. The prolongation operator is constructed geometrically by constructing a nested hierarchy of meshes within superblocks and evaluating fine grid nodes with coarse grid shape functions. The smoothing operator is chosen as a Chebyshev smoother or Gauss-Seidel smoother. For a desired accuracy, to obtain satisfactory convergence rates we observe a mild restriction on the selectable PML component parameters. Heuristics for selecting PML parameters given a desired error in approximating the radiation boundary condition are presented to complement this solvability requirement our method poses on the range of selectable PML parameters. Under these restrictions, we see superior convergence of the method. A microelectromechanical disk resonator device is used to display the effectiveness of our method.

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