[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84109-en":3,"doc-seo-84109-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84109,1099514067438,"River Wang","https://ap-avatar.wpscdn.com/avatar/100002539ee87300030?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780474512215547542",8,"Research & Report","Nested Volume-Surface Integral Equations for Acoustics","High-frequency acoustic wave propagation in unbounded domains with locally heterogeneous materials and sharp, high-contrast interfaces remains difficult for standard numerical methods. The volume-surface integral equation (VSIE) framework is attractive because Green’s functions enforce the radiation condition at infinity, Newton potentials treat heterogeneous volumes, and surface operators model interface scattering. This study extends VSIE to nested heterogeneous media with parameter jumps across interfaces, and validates accuracy and mesh convergence via extensive benchmarks against coupled finite-element and boundary-element methods using open-source software.","arXiv :2607 .06429v1 [math .NA] 7 Jul 2026  \nNested Volume-Surface Integral Equations for  \nAcoustics  \nDanilo Aballay∗ Elwin van’t Wout∗†  \nJuly 7, 2026  \nAbstract  \nThe simulation of high-frequency acoustic wave propagation in unbounded domains with local heterogeneous materials and high-contrast interfaces poses significant challenges to numerical methods. The volumesurface integral equation (VSIE) method is an attractive approach as it automatically satisfies the radiation condition at infinity via Green’s functions, handles heterogeneous materials via Newton potentials, and models scattering at high-contrast interfaces via surface integral operators. However, its effectiveness in practical simulations has been limited by high computational costs, sensitivity to sharp interfaces, and insufficient computational verification. This study extends the applicability of VSIE by deriving integral formulations for nested heterogeneous materials with parameter jumps at interfaces. We also develop extensive benchmarks against coupled finite-element and boundary-element methods to verify the VSIE’s accuracy and mesh convergence. The various benchmarks using open-source software demonstrate the effectiveness of VSIE for large-scale acoustic simulations.  \n1 Introduction  \nThe Helmholtz equation is an ubiquitous model for harmonic wave propagation and the mathematical backbone of many computational acoustics simulations [1, 2, 3] . This partial differential equation reads  \n−ρ∇ · 􀀒 1ρ∇p􀀓− 􀀐ωc 􀀑 2 p = f (1)  \nfor the unknown pressure p (x), known source f (x) with angular frequency ω , and a material with mass density ρ(x) and speed of sound c (x), all at the position x ∈ R3 . This study considers a domain consisting of two nested subdomains  \n∗ Institute for Mathematical and Computational Engineering, School of Engineering and Faculty of Mathematics, Pontificia Universidad Cat´olica de Chile, Santiago, Chile.  \n†Contact: [e.wout@uc.cl](e.wout@uc.cl)  \nembedded in an unbounded exterior region. The material parameters ρ and c may be heterogeneous in the interior domains and discontinuous across interfaces. Except for special choices of material parameters, the Helmholtz equation for multiple, heterogeneous materials does not have an analytical solution and efficient numerical methods must be used [4] .  \nSolving the acoustic transmission model with volumetric methods such asthe finite element method (FEM), finite volume method, and finite difference method is challenging. Handling scattering in the unbounded domain necessitates artificial boundary conditions [5] or absorbing layers [6] to limit the computational domains and avoid spurious reflections. Also, for a fixed number of grid elements per wavelength, the size of the sparse discretization matrix scales with the third power of the frequency [7] . Worse, these algorithms suffer from the pollution effect and need increasingly finer mesh resolutions or high-order discretizations at high frequencies to maintain accuracy [8, 9] .  \nAn alternative approach to solving the Helmholtz equation is to reformulate it as an integral equation via representation formulas based on Green’s functions. When the material parameters are piecewise constant, the Green’s identities can be employed to rewrite the volumetric Helmholtz equation into boundary integral equations at the material interfaces [10] . The advantage of this boundary integral approach is that the number of degrees of freedom in the surface meshes scales only quadratically, and the radiation condition at infinity is automatically satisfied [11] . However, the discretization matrix of the boundary element method (BEM) is dense and must be solved using acceleration techniques such as the fast multipole method [12] or hierarchical matrix compression [13] . Importantly, few elements per wavelength suffice to achieve accurate solutions, even at high frequencies [14, 15] . However, the BEM is fundamentally limited to piecewise-constant materia","cbCaikFAVvPzzvEo","https://ap.wps.com/l/cbCaikFAVvPzzvEo","pdf",1763461,1,35,"English","en",105,"# Introduction\n## Helmholtz equation in unbounded heterogeneous media\n## Integral equation reformulation via Green’s functions\n## Limits of FEM, BEM, and coupled FEM-BEM at high frequency\n## VSIE for heterogeneous volumes and high-contrast interfaces","[{\"question\":\"What challenges motivate using VSIE for high-frequency acoustics?\",\"answer\":\"High-frequency propagation in unbounded domains with local heterogeneity and high-contrast interfaces creates high computational cost, sensitivity to sharp interfaces, and limited computational verification for existing methods.\"},{\"question\":\"How does the VSIE approach address the radiation condition at infinity?\",\"answer\":\"It uses representation formulas based on Green’s functions for the exterior problem, which automatically satisfy the radiation condition at infinity.\"},{\"question\":\"How is VSIE validated in the study?\",\"answer\":\"The study performs extensive benchmarks comparing VSIE results with coupled finite-element and boundary-element methods, including checks for accuracy and mesh convergence using open-source software.\"}]",1784192907,88,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"nested-volume-surface-integral-equations-for-acoustics","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/nested-volume-surface-integral-equations-for-acoustics/84109/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What challenges motivate using VSIE for high-frequency acoustics?","Question",{"text":75,"@type":76},"High-frequency propagation in unbounded domains with local heterogeneity and high-contrast interfaces creates high computational cost, sensitivity to sharp interfaces, and limited computational verification for existing methods.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the VSIE approach address the radiation condition at infinity?",{"text":80,"@type":76},"It uses representation formulas based on Green’s functions for the exterior problem, which automatically satisfy the radiation condition at infinity.",{"name":82,"@type":73,"acceptedAnswer":83},"How is VSIE validated in the study?",{"text":84,"@type":76},"The study performs extensive benchmarks comparing VSIE results with coupled finite-element and boundary-element methods, including checks for accuracy and mesh convergence using open-source 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