A Higher-Order Time Domain Boundary Element Formulation based on Isogeometric Analysis and the Convolution Quadrature Method

TL;DR

The paper develops a high-order time-domain boundary element method for acoustics and elastodynamics by coupling convolution quadrature with Runge-Kutta schemes to discretize time and isogeometric analysis for spatial discretization. Time integration occurs via RK-based CQM, requiring Laplace-domain solves per RK stage and employing a weighted DFT for efficiency, while space uses Bézier-extracted spline bases to achieve high-order accuracy on boundary geometries. The authors present a detailed discretization pipeline (direct and indirect BIEs), outline basis transformations for element localization, and provide convergence analyses in time and space alongside extensive numerical experiments on a scattering sphere demonstrating the expected rates. The work shows that the method attains higher-order convergence in both time and space, with the overall rate governed by the slower discrete dimension, and highlights practical considerations for accurate quadrature of singular and oscillatory integrals in boundary integral evaluations.

Abstract

An isogeometric boundary element method (BEM) is presented to solve scattering problems in an isotropic homogeneous medium. We consider wave problems governed by the scalar wave equation as in acoustics and the Lamé-Navier equations for elastodynamics considering the theory of linear elasticity. The underlying boundary integral equations imply time-dependent convolution integrals and allow us to determine the sought quantities in the bounded interior or the unbounded exterior after solving for the unknown Cauchy data. In the present work, the time-dependent convolution integrals are approximated by multi-stage Runge-Kutta (RK) based convolution quadratures that involve steady-state solutions in the Laplace domain. The proposed method discretizes the spatial variables in the framework of isogeometric analysis (IGA), entailing a patchwise smooth spline basis. Overall, it enables high convergence rates in space and time. The implementation scheme follows an element structure defined by the non-empty knot spans in the knot vectors and local, uniform Bernstein polynomials as basis functions. The algorithms to localize the basis functions on the elements are outlined and explained. The solutions of the mixed problems are approximated by the BEM based on a symmetric Galerkin variational formulation and a collocation method. We investigate convergence rates of the approximative solutions in a mixed space and time error norm.

Figures (7)

• Figure 1: The sphere constructed by six patches.
• Figure 2: Convergence plots of the Laplace domain solution in acoustics computed on the refinement levels $0,\hdots,4$.
• Figure 3: Convergence plots of the Laplace domain solution in elastodynamics computed on the refinement levels $0,\hdots,4$.
• Figure 4: Indirect formulation for the scattering sphere: Dirichlet problem. Convergence plots of the time domain solution computed on the refinement levels $0,\hdots,3$. In (b), the black dotted lines show the convergence of the CQM using the symbol of $(\frac{1}{2} \psi_1+\mathcal{K}*\psi_1)^{-1}$.
• Figure 5: Indirect formulation for the Scattering sphere: Neumann problem. Convergence plots of the time domain solution computed on the refinement levels $0,\hdots,3$. In (a), the black dotted lines show the convergence of the CQM using the symbol of $- (\frac{1}{2} \phi_2 - \mathcal{K}'*\phi_2)^{-1}$.