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Flexural wave scattering plays a crucial role in optimizing and designingstructures for various engineering applications. Mathematically, the flexuralwave scattering problem on an infinite thin plate is described by afourth-order plate-wave equation on an unbounded domain, making it challengingto solve directly using the regular linear finite element method (FEM). In thispaper, we propose two numerical methods, the interior penalty FEM (IP-FEM) andthe boundary penalty FEM (BP-FEM) with a transparent boundary condition (TBC),to study flexural wave scattering by an arbitrary-shaped cavity on an infinitethin plate. Both methods decompose the fourth-order plate-wave equation intothe Helmholtz and modified Helmholtz equations with coupled conditions at thecavity boundary. A TBC is then constructed based on the analytical solutions ofthe Helmholtz and modified Helmholtz equations in the exterior domain,effectively truncating the unbounded domain into a bounded one. Using lineartriangular elements, the IP-FEM and BP-FEM successfully suppress theoscillation of the bending moment of the solution at the cavity boundary,demonstrating superior stability and accuracy compared to the regular linearFEM when applied to this problem.

Numerical solution of the cavity scattering problem for flexural waves on thin plates: linear finite element methods