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<title>Finite-Integral-Methode</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Finite-Integral-Methode</span></h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Die <b>Finite-Integral-Methode</b> basiert auf der <b>Finite Integration Theorie</b> (FIT) und ist ein <a href="Numerik" class="mw-redirect" title="Numerik">numerisches</a> <a href="Simulation" title="Simulation">Simulationsverfahren</a> zur näherungsfreien Lösung der <a href="Maxwellsche_Gleichungen" class="mw-redirect" title="Maxwellsche Gleichungen">elektromagnetischen Grundgleichungen</a> nach <a href="James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell</a>. Sie bildet die mathematische Grundlage von Simulationsprogrammen für elektromagnetische Probleme wie z. B. MAFIA und <i>CST MICROWAVE STUDIO®</i>.

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<div class="mw-heading mw-heading2"><h2 id="Grundlagen">Grundlagen</h2></div>

<p>Die Finite-Integral-Methode, zuerst 1976 von <a href="Thomas_Weiland" title="Thomas Weiland">Thomas Weiland</a> vorgestellt, löst die nach Maxwell benannten elektromagnetischen Grundgleichungen

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<annotation encoding="application/x-tex">{\displaystyle \oint _{C(A)}{\vec {E}}\cdot d{\vec {s}}\;=\;-\int _{A}{\frac {\partial }{\partial t}}{\vec {B}}\cdot d{\vec {A}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8545bc2adcf22c4c6b0e3f7ea4ef2c283eec7d6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:30.691ex; height:6.176ex;" alt="{\displaystyle \oint _{C(A)}{\vec {E}}\cdot d{\vec {s}}\;=\;-\int _{A}{\frac {\partial }{\partial t}}{\vec {B}}\cdot d{\vec {A}}}" loading="lazy"></span></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \oint _{C(A)}{\vec {H}}\cdot d{\vec {s}}\;=\;\int _{A}\left({\frac {\partial }{\partial t}}{\vec {D}}+{\vec {J}}\right)\cdot d{\vec {A}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f37d67509b32f202890f8f0f8a93d1fcdc00f70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.994ex; height:6.343ex;" alt="{\displaystyle \oint _{C(A)}{\vec {H}}\cdot d{\vec {s}}\;=\;\int _{A}\left({\frac {\partial }{\partial t}}{\vec {D}}+{\vec {J}}\right)\cdot d{\vec {A}}}" loading="lazy"></span></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \oint _{A(V)}{\vec {D}}\cdot d{\vec {A}}\;=\;\int _{V}\rho \cdot dV}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb945ee531df37d41ae12df7ac68976c473be77d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.698ex; height:6.009ex;" alt="{\displaystyle \oint _{A(V)}{\vec {D}}\cdot d{\vec {A}}\;=\;\int _{V}\rho \cdot dV}" loading="lazy"></span></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \oint _{A(V)}{\vec {B}}\cdot d{\vec {A}}\;=\;0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f07f4452f1be4ce1dfc51ef16d3809ca9aeb878f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.641ex; height:6.009ex;" alt="{\displaystyle \oint _{A(V)}{\vec {B}}\cdot d{\vec {A}}\;=\;0}" loading="lazy"></span></dd></dl>

<p>näherungsfrei in deren Integralform und mittels Integralapproximation die Materialgleichungen

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<annotation encoding="application/x-tex">{\displaystyle {\vec {D}}\;=\;\varepsilon _{0}\varepsilon _{r}{\vec {E}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/122872d87fcbaa77e7dabf66694b9baeca811d22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.284ex; height:3.176ex;" alt="{\displaystyle {\vec {D}}\;=\;\varepsilon _{0}\varepsilon _{r}{\vec {E}}}" loading="lazy"></span></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle {\vec {B}}\;=\;\mu _{0}\mu _{r}{\vec {H}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89e8691da9d44bf5107b51c1151da7bd5322cf39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.048ex; height:3.343ex;" alt="{\displaystyle {\vec {B}}\;=\;\mu _{0}\mu _{r}{\vec {H}}}" loading="lazy"></span></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle {\vec {J}}\;=\;\kappa {\vec {E}}+{\vec {J_{s}}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d1566f31fdb3cd521bd143ed82bf8d42aad2b216.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.456ex; height:4.009ex;" alt="{\displaystyle {\vec {J}}\;=\;\kappa {\vec {E}}+{\vec {J_{s}}}}" loading="lazy"></span></dd></dl>

<p>in <a href="Diskretisierung" title="Diskretisierung">diskretisierter</a> Form.

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<div class="mw-heading mw-heading2"><h2 id="Vorgehensweise">Vorgehensweise</h2></div>

<p>Das gesamte Problemgebiet wird in ein erstes (oder primäres) dreidimensionales Netz von einzelnen, möglichst kleinen Gitterzellen (engl.: mesh cells) mit den Materialeigenschaften <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{r},\mu _{r},\kappa }">

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{r},\mu _{r},\kappa }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/326fb73953e3657c2cb307b98019d33af8d2ef5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.84ex; height:2.176ex;" alt="{\displaystyle \varepsilon _{r},\mu _{r},\kappa }" loading="lazy"></span> unterteilt, die jede bzgl. ihrer elektrischen Kantenspannung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{i}=E_{i}\cdot a}">

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<annotation encoding="application/x-tex">{\displaystyle e_{i}=E_{i}\cdot a}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/82e2e7be23b0b87d661f494f2f2aed76d8753801.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.405ex; height:2.509ex;" alt="{\displaystyle e_{i}=E_{i}\cdot a}" loading="lazy"></span> und ihres magnetischen Flusses durch die Randflächen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{j}=B_{j}\cdot A_{j}}">

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<annotation encoding="application/x-tex">{\displaystyle b_{j}=B_{j}\cdot A_{j}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b641c2a6393b0d3aaf6890a9d38aeae4a66043d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.011ex; height:2.843ex;" alt="{\displaystyle b_{j}=B_{j}\cdot A_{j}}" loading="lazy"></span> berechnet wird.

</p><p>Zusätzlich wird ein orthogonal zum ersten Gitternetz angesetztes, zweites (duales) Gitterzellennetz bzgl. der magnetischen Kantenspannung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{i}=H_{i}\cdot a}">

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<annotation encoding="application/x-tex">{\displaystyle h_{i}=H_{i}\cdot a}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/892a71a1831aed40c2880a6fcabb8ecd750f6d7c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.877ex; height:2.509ex;" alt="{\displaystyle h_{i}=H_{i}\cdot a}" loading="lazy"></span> und des elektrischen Flusses durch die Randflächen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{j}=D_{j}\cdot A_{j}}">

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<annotation encoding="application/x-tex">{\displaystyle d_{j}=D_{j}\cdot A_{j}}</annotation>

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<p>Durch die quaderförmige Form der Gitterzellen vereinfacht sich das <a href="Kurvenintegral" title="Kurvenintegral">Konturintegral</a> der elektrischen Feldstärke <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{C(A)}{{\vec {E}}\cdot d{\vec {s}}}=\int _{{C}_{1}}{{\vec {E}}\cdot d{\vec {s}}}+\int _{{C}_{2}}{{\vec {E}}\cdot d{\vec {s}}}-\int _{{C}_{3}}{{\vec {E}}\cdot d{\vec {s}}}-\int _{{C}_{4}}{{\vec {E}}\cdot d{\vec {s}}}}">

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<annotation encoding="application/x-tex">{\displaystyle \oint _{C(A)}{{\vec {E}}\cdot d{\vec {s}}}=\int _{{C}_{1}}{{\vec {E}}\cdot d{\vec {s}}}+\int _{{C}_{2}}{{\vec {E}}\cdot d{\vec {s}}}-\int _{{C}_{3}}{{\vec {E}}\cdot d{\vec {s}}}-\int _{{C}_{4}}{{\vec {E}}\cdot d{\vec {s}}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6826c95776ce711e949c5cb47c7834923803db96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:62.435ex; height:6.009ex;" alt="{\displaystyle \oint _{C(A)}{{\vec {E}}\cdot d{\vec {s}}}=\int _{{C}_{1}}{{\vec {E}}\cdot d{\vec {s}}}+\int _{{C}_{2}}{{\vec {E}}\cdot d{\vec {s}}}-\int _{{C}_{3}}{{\vec {E}}\cdot d{\vec {s}}}-\int _{{C}_{4}}{{\vec {E}}\cdot d{\vec {s}}}}" loading="lazy"></span> zur Summe <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{4}e_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{4}e_{i}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f39e1efea35af7f5cefa141603861d73bedf1315.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:5.625ex; height:7.343ex;" alt="{\displaystyle \sum _{i=1}^{4}e_{i}}" loading="lazy"></span> der Kantenspannungen einer Quaderwand der Gitterzelle.

</p><p>Die zeitliche Ableitung des magnetischen Flusses durch die Randfläche der Gitterzelle wird nun dieser Summe gleichgesetzt, so dass sich folgende Gleichung ergibt:

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<annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{4}e_{i}\;=\;e_{1}+e_{2}-e_{3}-e_{4}\;=\;-{\frac {\partial }{\partial t}}b_{n}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d4d1df7e47a84b09856949b5b8cf1e55eecf3cb3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:38.493ex; height:7.343ex;" alt="{\displaystyle \sum _{i=1}^{4}e_{i}\;=\;e_{1}+e_{2}-e_{3}-e_{4}\;=\;-{\frac {\partial }{\partial t}}b_{n}}" loading="lazy"></span></dd></dl>

<p>Diese Berechnung muss für alle sechs Randflächen einer Gitterzelle wiederholt werden. In Matrixschreibweise ergibt sich das Gleichungssystem

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<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}\ldots &\ldots &\ldots &\ldots &\ldots &\ldots &\ldots \\1&\ldots &1&\ldots &-1&\ldots &-1\\\ldots &\ldots &\ldots &\ldots &\ldots &\ldots &\ldots \end{pmatrix}}\cdot {\begin{pmatrix}e_{i}\\\vdots \\e_{j}\\\vdots \\e_{k}\\\vdots \\e_{l}\end{pmatrix}}\;=\;-{\frac {\partial }{\partial t}}\;{\begin{pmatrix}\vdots \\b_{n}\\\vdots \end{pmatrix}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6a0f78ef6a3fa77d23ccbe7cb0e21d530b6b54f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.505ex; width:63.851ex; height:26.176ex;" alt="{\displaystyle {\begin{pmatrix}\ldots &\ldots &\ldots &\ldots &\ldots &\ldots &\ldots \\1&\ldots &1&\ldots &-1&\ldots &-1\\\ldots &\ldots &\ldots &\ldots &\ldots &\ldots &\ldots \end{pmatrix}}\cdot {\begin{pmatrix}e_{i}\\\vdots \\e_{j}\\\vdots \\e_{k}\\\vdots \\e_{l}\end{pmatrix}}\;=\;-{\frac {\partial }{\partial t}}\;{\begin{pmatrix}\vdots \\b_{n}\\\vdots \end{pmatrix}}}" loading="lazy"></span></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle {\textbf {C}}\cdot {\vec {e}}\;=\;-{\frac {\partial }{\partial t}}\;{\vec {b}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6012bdca08b7daa426dc5f5ccf757c1cc57c8278.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.763ex; height:5.509ex;" alt="{\displaystyle {\textbf {C}}\cdot {\vec {e}}\;=\;-{\frac {\partial }{\partial t}}\;{\vec {b}}}" loading="lazy"></span></dd></dl>

<p>Die beschreibende Matrix <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {C}}}">

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</p><p>Analog dazu werden die übrigen Maxwell’schen Gleichungen behandelt. In Matrixschreibweise ergibt sich das Gitter-Maxwell-Gleichungssystem

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<annotation encoding="application/x-tex">{\displaystyle {\textbf {C}}\cdot {\vec {e}}\;=\;-{\frac {\partial }{\partial t}}\;{\vec {b}}}</annotation>

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<mfrac>

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<mover>

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<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\textbf {C}}_{\text{Dual}}\cdot {\vec {h}}\;=\;{\frac {\partial }{\partial t}}\;{\vec {d}}+{\vec {j}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9e02243a0ab309f5ca0298d98531bc50ff067ddd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.14ex; height:5.509ex;" alt="{\displaystyle {\textbf {C}}_{\text{Dual}}\cdot {\vec {h}}\;=\;{\frac {\partial }{\partial t}}\;{\vec {d}}+{\vec {j}}}" loading="lazy"></span></dd></dl>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\textbf {S}}_{\text{Dual}}\cdot {\vec {d}}\;=\;{\vec {q}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3471ed4684b2e1ba2566f1b079e4b3da304b3af4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.09ex; height:3.176ex;" alt="{\displaystyle {\textbf {S}}_{\text{Dual}}\cdot {\vec {d}}\;=\;{\vec {q}}}" loading="lazy"></span></dd></dl>

</mrow>

<mover>

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<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\textbf {S}}\cdot {\vec {b}}\;=\;{\vec {0}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/16229b56bafc5633229b4a2e4f3fafed68b9d396.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.81ex; height:2.843ex;" alt="{\displaystyle {\textbf {S}}\cdot {\vec {b}}\;=\;{\vec {0}}}" loading="lazy"></span></dd></dl>

<p>Die Matrix <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {C}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\textbf {C}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dcc7aa41c652c9a95d9e1931569b0a0eb6c42efe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:2.176ex;" alt="{\displaystyle {\textbf {C}}}" loading="lazy"></span> entspricht dem analytischen <a href="Rotation_eines_Vektorfeldes" title="Rotation eines Vektorfeldes">Rotations-Operator</a>, die Matrix <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {S}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\textbf {S}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6a95321b89b7f2bfd8fe72ee2820494d8a84cb29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.176ex;" alt="{\displaystyle {\textbf {S}}}" loading="lazy"></span> entspricht dem analytischen <a href="Divergenz_eines_Vektorfeldes" title="Divergenz eines Vektorfeldes">Divergenz-Operator</a>. Der Index <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Dual}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Dual}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49deb457ea245c0f29a8d99233125ee0a161c75c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.177ex; height:2.176ex;" alt="{\displaystyle Dual}" loading="lazy"></span> weist auf die Berechnung der Kantenspannungen und Flüsse im Dualen Gitter hin.

</p><p>Die Materialgleichungen werden analog zu den Maxwell’schen Gleichungen diskretisiert.

</p>

<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\vec {d}}\;=\;{\textbf {M}}_{\varepsilon }\;\cdot \;{\vec {e}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8bba88093965166cf518ef41a905b1ca89d9f9cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.664ex; height:3.176ex;" alt="{\displaystyle {\vec {d}}\;=\;{\textbf {M}}_{\varepsilon }\;\cdot \;{\vec {e}}}" loading="lazy"></span></dd></dl>

<mover>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {b}}\;=\;{\textbf {M}}_{\mu }\;\cdot \;{\vec {h}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4239c463fd6ff4a19928d3038875333083384daa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.552ex; height:3.509ex;" alt="{\displaystyle {\vec {b}}\;=\;{\textbf {M}}_{\mu }\;\cdot \;{\vec {h}}}" loading="lazy"></span></dd></dl>

<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\vec {j}}\;=\;{\textbf {M}}_{\kappa }\;\cdot \;{\vec {e}}\;+\;{\vec {j}}_{s},}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/091fb17d41c948bb2d1052ef28e5e85a7398e98a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.267ex; height:3.343ex;" alt="{\displaystyle {\vec {j}}\;=\;{\textbf {M}}_{\kappa }\;\cdot \;{\vec {e}}\;+\;{\vec {j}}_{s},}" loading="lazy"></span></dd></dl>

<p>wobei die Materialgrößen orts-, frequenz- und richtungsabhängig sein können.

</p><p>Die FIT-Methode ist auf alle elektromagnetische Probleme im Zeit- und Frequenzbereich anwendbar, sowohl in der Elektrostatik, als auch in der Elektrodynamik. Durch den speziellen Zuschnitt der FIT-Methode auf die Maxwell’schen Gleichungen und das daraus entstehende diskrete Analogon sind die <a href="Stetige_Funktion" title="Stetige Funktion">Stetigkeitsbedingungen</a> a priori erfüllt und die analytischen Eigenschaften der Vektoroperationen werden beibehalten.

</p><p>Für elektrodynamische Probleme werden im Frequenzbereich alle zeitlichen Ableitungen durch <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j\omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j\omega }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/635692523e5a0d8187e908408819010da7f0bd09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:2.431ex; height:2.509ex;" alt="{\displaystyle j\omega }" loading="lazy"></span> ersetzt. Das Ergebnis einer Simulation im Frequenzbereich liefert die Impulsantwort auf ein monofrequentes Eingangssignal.

</p><p>Im Zeitbereich ist eine breitbandige Anregung mit freien Signalverläufen gestattet. Die Simulationsrechnung beschreibt in diesem Fall das Frequenzverhalten über einen vorab definierten Frequenzbereich.

</p>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li>T. Weiland: <i>Eine Methode zur Lösung der Maxwellschen Gleichungen für sechskomponentige Felder auf diskreter Basis</i>, AEÜ, Band 31, Heft 3, pp. 116–120, 1977</li>

<li>T. Weiland: <i>A Discretization Method for the Solution of Maxwell’s Equations for Six-Component Fields</i>, Electronics and Communications AEUE, vol. 31, no. 3, pp. 116–120, 1977.</li></ul></div><div><div class="zim-footer">

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