[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83102-en":3,"doc-seo-83102-105":30,"detail-sidebar-cat-0-en-105":92},{"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":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83102,1099514067415,"Rowan","https://ap-avatar.wpscdn.com/avatar/100002539d78ffe74a7?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779092875211072502",8,"Research & Report","Trefftz DG Approximation of the T-Matrix for Scattering by Periodic Layered Structures","This work investigates scattering of time-harmonic electromagnetic waves by periodic layered gratings modeled with the 2D Helmholtz equation. The obstacle may include penetrable or impenetrable regions and consists of finitely many stacked layers. A single periodic cell is treated via a quasi-periodic boundary value problem with quasi-periodic boundary conditions. Radiation in vertical directions is enforced using Dirichlet-to-Neumann operators. A T-matrix framework decomposes the global problem into layer problems using quasi-periodic modal expansions. Each layer’s T-matrix is numerically approximated with a plane-wave Trefftz discontinuous Galerkin method, yielding linear complexity in the number of layers and enabling scalable coupling of local responses.","arXiv :2607 .06475v1 [math .NA] 7 Jul 2026  \nTrefftz DG Approximation of the T-Matrix for Scattering by Periodic Layered Structures  \nArmando Maria Monforte∗, Andrea Moiola†, Simone Zanotto‡  \nJuly 8, 2026  \nAbstract  \nWe study the scattering of time-harmonic electromagnetic waves by periodic layered gratings, modelled by the 2D Helmholtz equation. The periodic obstacle may include penetrable and impenetrable regions, and consists of a finite number of stacked layers. The boundary value problem is formulated on a single periodic cell using quasi-periodic boundary conditions. The radiation condition in the vertical directions is imposed through Dirichlet-to-Neumann (DtN) operators.  \nTo efficiently treat multilayer configurations, we adopt a formulation based on the T-matrix method. The global scattering problem is decomposed into boundary value problems posedon individual layers. On the layer boundaries, the field is expressed in terms of quasi-periodic modal expansions, and the layer T-matrix describes the map between incoming and outgoing wave modes. Each local T-matrix is approximated numerically using a plane-wave based Trefftz Discontinuous Galerkin (TDG) method, which provides an efficient discretization of the layer scattering response. The T-matrix technique leads to linear computational complexity in the number of layers in the grating.  \nKeywords: Diffraction grating, Quasi-periodic, Helmholtz equation, T-matrix, Discontinuous Galerkin, Trefftz method, Plane wave basis  \nMathematics Subject Classification (2020): 65N30, 35J05, 35Q60, 78A45, 78M10  \n1 Introduction  \nThe scattering of time-harmonic electromagnetic waves by periodic structures, known as diffraction gratings, arises in a wide range of applications in photonics, optics, and metamaterial design [2, 12] . In the two-dimensional transverse-electric setting, Maxwell’s equations reduce to the Helmholtz equation with piecewise-constant, periodic relative permittivity ε . Classical numerical approximations of such diffraction gratings reduce the computational domain toa single periodic cell using quasi-periodic boundary conditions and Dirichlet-to-Neumann operators [3, 5 , 14] .  \nTo curb the computational cost of discretizing large and complex cells, we borrow the Tmatrix technique from multiple-scattering theory [17, 18 , 23] . To approximate the scattering by many obstacles, each of these is associated to a so-called T-matrix, which maps the  \n∗ Department of Mathematics, University of Pavia, Via Ferrata 5, Pavia, Italy (armandomaria.monforte01 @universitadipavia.it), ORCID: 0009-0000-7687-2217  \n†Department of Mathematics, University of Pavia, Via Ferrata 5, Pavia, Italy ([andrea.moiola@unipv.it](andrea.moiola@unipv.it)), ORCID: 0000-0002-6251-4440  \n‡Istituto Nanoscienze – CNR, NEST-SNS, Piazza San Silvestro 12, Pisa, Italy ([simone.zanotto@nano.cnr.it](simone.zanotto@nano.cnr.it)), ORCID: 0000-0001-7180-3335  \nFourier coefficients of any wave impinging on the obstacle into the Fourier coefficients of the corresponding scattered field. The size of a T-matrix only depends on the wavenumber and the obstacle size, but is independent of the obstacle shape and material properties. Multiple scattering problems coupling all obstacles are solved from a linear system built from the T-matrices of each obstacle, leading to considerable savings in the computational effort.  \nWe adapt this idea to the quasi-periodic setting. First, the domain Ω is split in N layers Ωj , which may contain inhomogeneities and impenetrable regions. Second, a T-matrix Tj , mapping incoming to outgoing fields, is computed for each layer, exploiting the natural Fourier expansion dictated by quasi-periodicity. Third, these layer T-matrices are coupled through propagation matrices into a global banded block linear system.  \nA key feature of this formulation is the decoupling of scales: the global system is N × N block tridiagonal, with (4M + 2) × (4M + 2)-sized blocks, where M is the truncation","cbCaipqrhmhSWWgr","https://ap.wps.com/l/cbCaipqrhmhSWWgr","pdf",4265113,5,1,25,"English","en",105,"# Introduction\n# Helmholtz quasi-periodic boundary value problems and DtN operators\n# Layer formulation and local boundary value problems\n# T-matrix coupling for the quasi-periodic layered setting\n# DtN-TDG method to approximate layer T-matrices\n# Numerical results and comparisons","[{\"question\":\"What physical and mathematical model is used for the scattering problem?\",\"answer\":\"Time-harmonic electromagnetic wave scattering is modeled through the 2D Helmholtz equation for a periodic layered grating. The periodic cell is governed by a quasi-periodic boundary value problem with Helmholtz-compatible radiation handling via DtN operators.\"},{\"question\":\"How is the radiation condition implemented in the formulation?\",\"answer\":\"Radiation in the vertical directions is imposed using Dirichlet-to-Neumann (DtN) operators. These operators incorporate the influence of the exterior into the boundary value problem defined on the periodic cell.\"},{\"question\":\"How does the proposed T-matrix approach improve computational efficiency across layers?\",\"answer\":\"The global scattering problem is decomposed into layer-wise boundary value problems and each layer is represented by a T-matrix mapping incoming to outgoing quasi-periodic modal expansions. Coupling yields a block tridiagonal system whose size depends on the truncation order of the DtN operators and the number of layers, enabling linear complexity in the number of layers.\"}]",1784185268,63,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":28},"trefftz-dg-approximation-of-the-t-matrix-for-scattering-by-periodic-layered-structures","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/trefftz-dg-approximation-of-the-t-matrix-for-scattering-by-periodic-layered-structures/83102/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-25","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What physical and mathematical model is used for the scattering problem?","Question",{"text":76,"@type":77},"Time-harmonic electromagnetic wave scattering is modeled through the 2D Helmholtz equation for a periodic layered grating. The periodic cell is governed by a quasi-periodic boundary value problem with Helmholtz-compatible radiation handling via DtN operators.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How is the radiation condition implemented in the formulation?",{"text":81,"@type":77},"Radiation in the vertical directions is imposed using Dirichlet-to-Neumann (DtN) operators. These operators incorporate the influence of the exterior into the boundary value problem defined on the periodic cell.",{"name":83,"@type":74,"acceptedAnswer":84},"How does the proposed T-matrix approach improve computational efficiency across layers?",{"text":85,"@type":77},"The global scattering problem is decomposed into layer-wise boundary value problems and each layer is represented by a T-matrix mapping incoming to outgoing quasi-periodic modal expansions. Coupling yields a block tridiagonal system whose size depends on the truncation order of the DtN operators and the number of layers, enabling linear complexity in the number of layers.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":20,"slug":138},19,"General","general"]