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<p>In this paper we consider 3D interior and exterior Helmholtz problems, reformulated in terms of a boundary integral equation (BIE). For their numerical solution, we rely on a collocation boundary element method (BEM) formulated in the general framework of isogeometric analysis (IgA-BEM), adopting in particular conforming multi-patch discretizations. As&#xa0;is well known, all BEM matrices are non-symmetric when collocation is adopted and in any case they are fully populated, a drawback that prevents the application of this strategy to large-scale realistic problems. As a possible remedy to reduce the global complexity of the method, we propose a numerical scheme relying on the hierarchical matrix (<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="366_2025_2144_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal {H}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation>-matrix) technique combined with an IgA-BEM approach based on function-by-function assembly and thus particularly suitable for the considered coupling. Using a suitable admissibility condition, it starts with partitioning the matrix hierarchically into full- and low-rank blocks, without requiring the preliminary computation of any of its entries. Then, only the former blocks are computed and stored in&#xa0;a conventional way, meanwhile the latter are directly approximated by the adaptive cross approximation (ACA) methodology which successfully compresses the dense matrices of the multi-patch IgA-BEM approach. Furthermore, the cost of the matrix–vector product is reduced and this allows us to increase the overall computational efficiency of the generalized minimal residual method (GMRES), adopted for the solution of the linear system. Several numerical results are given to demonstrate the accuracy and efficiency of the proposed methodology.</p>

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Hierarchical matrices for 3D Helmholtz problems in the multi-patch IgA-BEM setting

  • 期刊论文

  • 发表日期：

    2025年5月10日

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  • L. Desiderio,
  • G. A. D’Inverno,
  • M. L. Sampoli,
  • A. Sestini

摘要

In this paper we consider 3D interior and exterior Helmholtz problems, reformulated in terms of a boundary integral equation (BIE). For their numerical solution, we rely on a collocation boundary element method (BEM) formulated in the general framework of isogeometric analysis (IgA-BEM), adopting in particular conforming multi-patch discretizations. As is well known, all BEM matrices are non-symmetric when collocation is adopted and in any case they are fully populated, a drawback that prevents the application of this strategy to large-scale realistic problems. As a possible remedy to reduce the global complexity of the method, we propose a numerical scheme relying on the hierarchical matrix ( \(\mathcal {H}\) H -matrix) technique combined with an IgA-BEM approach based on function-by-function assembly and thus particularly suitable for the considered coupling. Using a suitable admissibility condition, it starts with partitioning the matrix hierarchically into full- and low-rank blocks, without requiring the preliminary computation of any of its entries. Then, only the former blocks are computed and stored in a conventional way, meanwhile the latter are directly approximated by the adaptive cross approximation (ACA) methodology which successfully compresses the dense matrices of the multi-patch IgA-BEM approach. Furthermore, the cost of the matrix–vector product is reduced and this allows us to increase the overall computational efficiency of the generalized minimal residual method (GMRES), adopted for the solution of the linear system. Several numerical results are given to demonstrate the accuracy and efficiency of the proposed methodology.