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

2025-10-08

AI Summary

This work addresses time-domain scattering problems for acoustics and linear elastodynamics in unbounded, isotropic, homogeneous media. We propose a high-order boundary element method (BEM) that integrates isogeometric analysis (IGA) with a multistage Runge–Kutta–based convolution quadrature (RK-CQ). The method constructs a space–time coupled formulation from Laplace-domain steady-state solutions, employing smooth spline basis functions, a symmetric Galerkin variational framework, and a local Bernstein polynomial-driven inter-element non-empty-node discretization strategy. This constitutes the first systematic integration of RK-CQ with IGA-BEM, achieving simultaneous high-order convergence in both spatial and temporal dimensions. Theoretical analysis and numerical experiments confirm optimal convergence rates under a mixed space–time error norm, significantly enhancing the accuracy of Cauchy data reconstruction and the reliability of unbounded wavefield simulations.

Technology Category

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• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

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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.

Problem

• Solving wave scattering problems using isogeometric boundary elements
• Approximating time-dependent convolution integrals via Runge-Kutta methods
• Analyzing convergence rates for mixed space-time error norms

Innovation

• Isogeometric BEM for wave scattering problems
• Runge-Kutta convolution quadrature for time integrals
• Symmetric Galerkin formulation with collocation method