Hybrid absorbing scheme based on hyperelliptical layers with non-reflecting boundary conditions in scalar wave equations

Abstract

Modeling unbounded or semi-infinite media with numerical methods, such as finite elements, requires avoiding that waves pass through the boundaries of the truncated computational domain without reflection. For this reason, in areas such as Geophysics or Acoustics, absorbing layers are used to circumvent this issue. Nevertheless, sizing them is an open problem due to the necessary calibration of their parameters for an adequate performance and to prevent numerical instabilities. Absorbing layers are dependent on both wave frequency and material properties set up inside them and, affect the convergence of iterative methods in inverse problems by the inherent variation of domain properties within the process, especially in the transient regime. On the other hand, hybrid absorbing boundary conditions are the combination of absorbing layers and the application of non-reflecting boundary conditions on the layer boundary with the purpose of increasing their effectiveness. In this work, an analytical approach is presented to size the portion added to the original domain and to determine its damping parameters simultaneously. Hyperelliptical absorbing layers are introduced and, for generating the hybrid absorbing boundary conditions scheme, Sommerfeld or 1st-Higdon boundary conditions are imposed on the outer layer boundary. Therefore, this methodology is adaptive according to the intermediate solutions of the transient optimization process for reducing possible numerical instabilities and spurious information for the inverse problem. The transient equilibrium equations are implemented by using the finite element method and an implicit time integration method. Results are presented to show the potential of the proposal.