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| <html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Introduction · SauterSchwabQuadrature.jl</title><meta name="title" content="Introduction · SauterSchwabQuadrature.jl"/><meta property="og:title" content="Introduction · SauterSchwabQuadrature.jl"/><meta property="twitter:title" content="Introduction · SauterSchwabQuadrature.jl"/><meta name="description" content="Documentation for SauterSchwabQuadrature.jl."/><meta property="og:description" content="Documentation for SauterSchwabQuadrature.jl."/><meta property="twitter:description" content="Documentation for SauterSchwabQuadrature.jl."/><meta property="og:url" content="https://ga96tik.github.io/SauterSchwabQuadrature.jl/"/><meta property="twitter:url" content="https://ga96tik.github.io/SauterSchwabQuadrature.jl/"/><link rel="canonical" href="https://ga96tik.github.io/SauterSchwabQuadrature.jl/"/><script data-outdated-warner src="assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="assets/documenter.js"></script><script src="search_index.js"></script><script src="siteinfo.js"></script><script src="../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href>SauterSchwabQuadrature.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li class="is-active"><a class="tocitem" href>Introduction</a><ul class="internal"><li><a class="tocitem" href="#References"><span>References</span></a></li></ul></li><li><a class="tocitem" href="details/">Details</a></li><li><a class="tocitem" href="manual/">Manual</a></li><li><a class="tocitem" href="apiref/">API Reference</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Introduction</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Introduction</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/ga96tik/SauterSchwabQuadrature.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/ga96tik/SauterSchwabQuadrature.jl/blob/master/docs/src/index.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Introduction"><a class="docs-heading-anchor" href="#Introduction">Introduction</a><a id="Introduction-1"></a><a class="docs-heading-anchor-permalink" href="#Introduction" title="Permalink"></a></h1><p>This package provides the Sauter-Schwab regularizing coordinate transformations [1] such that 4D integrals of the form</p><p class="math-container">\[\int_{\Gamma}\int_{\Gamma'}b_i(\bm{x})\,k(\bm{x},\bm{y})\, b_j(\bm{y})\,\mathrm{d}S(\bm{y})\,\mathrm{d}S(\bm{x})\]</p><p>with Cauchy-singular integral kernels <span>$k(\bm{x},\bm{y})$</span> can be integrated via numerical quadrature. The integrals denote double surface integrals over </p><ul><li>triangles (curved or flat) or </li><li>quadrilaterals (curved or flat) </li></ul><p><span>$\Gamma$</span> and <span>$\Gamma'$</span> in 3D Space. The functions <span>$b_i(\bm{x})$</span> and <span>$b_i(\bm{y})$</span> are assumed to be real valued and non-singular.</p><p>These kind of integrals occur in the area of boundary element methods (BEM) for solving elliptic partial differential equations. It can be interpreted as the interaction of the two basisfunctions <span>$b_i(\bm{x})$</span> and <span>$b_i(\bm{y})$</span>, with respect to their domains <span>$\Gamma$</span> and <span>$\Gamma'$</span>, which, for instance, correspond to the cells of a meshed surface.</p><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>The triangles or quadrilaterals must be either equal, have two vertices in common, have one vertex in common or do not touch at all. A partial overlap is forbidden.</p><p>In the current implementation <span>$\Gamma$</span> and <span>$\Gamma'$</span> have to be both either triangles or quadrilatersls. However, mixed cases can be implemented, too.</p></div></div><h2 id="References"><a class="docs-heading-anchor" href="#References">References</a><a id="References-1"></a><a class="docs-heading-anchor-permalink" href="#References" title="Permalink"></a></h2><p>[1] Sauter S. Schwab C., "Boundary Element Methods (Springer Series in Computational Mathematics)", Chapter 5, Springer, 2010.</p></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="details/">Details »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.4.1 on <span class="colophon-date" title="Monday 13 May 2024 11:36">Monday 13 May 2024</span>. Using Julia version 1.10.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html> | ||
| <html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Introduction · SauterSchwabQuadrature.jl</title><meta name="title" content="Introduction · SauterSchwabQuadrature.jl"/><meta property="og:title" content="Introduction · SauterSchwabQuadrature.jl"/><meta property="twitter:title" content="Introduction · SauterSchwabQuadrature.jl"/><meta name="description" content="Documentation for SauterSchwabQuadrature.jl."/><meta property="og:description" content="Documentation for SauterSchwabQuadrature.jl."/><meta property="twitter:description" content="Documentation for SauterSchwabQuadrature.jl."/><meta property="og:url" content="https://ga96tik.github.io/SauterSchwabQuadrature.jl/"/><meta property="twitter:url" content="https://ga96tik.github.io/SauterSchwabQuadrature.jl/"/><link rel="canonical" href="https://ga96tik.github.io/SauterSchwabQuadrature.jl/"/><script data-outdated-warner src="assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="assets/documenter.js"></script><script src="search_index.js"></script><script src="siteinfo.js"></script><script src="../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href>SauterSchwabQuadrature.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li class="is-active"><a class="tocitem" href>Introduction</a><ul class="internal"><li><a class="tocitem" href="#References"><span>References</span></a></li></ul></li><li><a class="tocitem" href="details/">Details</a></li><li><a class="tocitem" href="manual/">Manual</a></li><li><a class="tocitem" href="apiref/">API Reference</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Introduction</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Introduction</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/ga96tik/SauterSchwabQuadrature.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/ga96tik/SauterSchwabQuadrature.jl/blob/master/docs/src/index.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Introduction"><a class="docs-heading-anchor" href="#Introduction">Introduction</a><a id="Introduction-1"></a><a class="docs-heading-anchor-permalink" href="#Introduction" title="Permalink"></a></h1><p>This package provides the Sauter-Schwab regularizing coordinate transformations [1] such that 4D integrals of the form</p><p class="math-container">\[\int_{\Gamma}\int_{\Gamma'}b_i(\bm{x})\,k(\bm{x},\bm{y})\, b_j(\bm{y})\,\mathrm{d}S(\bm{y})\,\mathrm{d}S(\bm{x})\]</p><p>with Cauchy-singular integral kernels <span>$k(\bm{x},\bm{y})$</span> can be integrated via numerical quadrature. The integrals denote double surface integrals over </p><ul><li>triangles (curved or flat) or </li><li>quadrilaterals (curved or flat) </li></ul><p><span>$\Gamma$</span> and <span>$\Gamma'$</span> in 3D Space. The functions <span>$b_i(\bm{x})$</span> and <span>$b_i(\bm{y})$</span> are assumed to be real valued and non-singular.</p><p>These kind of integrals occur in the area of boundary element methods (BEM) for solving elliptic partial differential equations. It can be interpreted as the interaction of the two basisfunctions <span>$b_i(\bm{x})$</span> and <span>$b_i(\bm{y})$</span>, with respect to their domains <span>$\Gamma$</span> and <span>$\Gamma'$</span>, which, for instance, correspond to the cells of a meshed surface.</p><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>The triangles or quadrilaterals must be either equal, have two vertices in common, have one vertex in common or do not touch at all. A partial overlap is forbidden.</p><p>In the current implementation <span>$\Gamma$</span> and <span>$\Gamma'$</span> have to be both either triangles or quadrilatersls. However, mixed cases can be implemented, too.</p></div></div><h2 id="References"><a class="docs-heading-anchor" href="#References">References</a><a id="References-1"></a><a class="docs-heading-anchor-permalink" href="#References" title="Permalink"></a></h2><p>[1] Sauter S. Schwab C., "Boundary Element Methods (Springer Series in Computational Mathematics)", Chapter 5, Springer, 2010.</p></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="details/">Details »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.4.1 on <span class="colophon-date" title="Wednesday 12 June 2024 14:39">Wednesday 12 June 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html> |
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