A higher-order time-domain boundary element formulation based on isogeometric analysis and the convolution quadrature method

An isogeometric boundary element method (BEM) is presented to solve scattering problems in an isotropic, homogeneous medium. We consider wave propagation 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 variable in the framework of isogeometric analysis (IGA), entailing a patchwise smooth spline basis. While previous studies have struggled to develop higher-order discretization methods for evolutionary boundary value problems, the present work introduces a novel combination of multi-stage RK-based convolution quadratures and isogeometric spatial approximation, yielding a fully higher-order method with high convergence rates in both 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 approximate solutions in a mixed space-and-time error norm.