Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / 2006 1 Numerical Methods of Electromagnetic Field Theory I (NFT I) Numerische Methoden.

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Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / Numerical Methods of Electromagnetic Field Theory I (NFT I) Numerische Methoden der Elektromagnetischen Feldtheorie I (NFT I) / 12th Lecture / 12. Vorlesung Universität Kassel Fachbereich Elektrotechnik / Informatik (FB 16) Fachgebiet Theoretische Elektrotechnik (FG TET) Wilhelmshöher Allee 71 Büro: Raum 2113 / 2115 D Kassel Dr.-Ing. René Marklein University of Kassel Dept. Electrical Engineering / Computer Science (FB 16) Electromagnetic Field Theory (FG TET) Wilhelmshöher Allee 71 Office: Room 2113 / 2115 D Kassel

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / FIT Discretization of the 3rd and 4th Maxwell’s Equation / FIT-Diskretisierung der 3. und 4. Maxwellschen Gleichung Governing Analytic Equations Maxwell’s equations in integral form / Maxwellsche Gleichungen in Integralform FIT Grid Equations Maxwell’s grid equations / Maxwellsche Gittergleichungen

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / D FIT – Electrostatic Case / 3D-FIT – Elektrostatischer Fall Electric Gauss’ grid equation – 3rd Maxwell’s grid equation in global matrix form / Elektrische Gaußsche Gittergleichung – 3. Maxwellsche Gittergleichung in globaler Matrixform Inhomogeneous, anisotropic case / Inhomogener anisotroper Fall Homogeneous, isotropic case / Homogener isotroper Fall

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / D FIT – Electrostatic Case / 3D-FIT – Elektrostatischer Fall Inhomogeneous, anisotropic case / Inhomogener anisotroper Fall Homogeneous, isotropic case / Homogener isotroper Fall

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / FIT Discretization of Scalar Electric Potential / FIT-Diskretisierung des skalaren elektrischen Potentials Integral form / Integralform Differential form / Differentialform FIT grid equation / FIT-Gittergleichung

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / FIT Discretization of Scalar Electric Potential / FIT-Diskretisierung des skalaren elektrischen Potentials Integral form / Integralform

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / FIT Discretization of Scalar Electric Potential / FIT-Diskretisierung des skalaren elektrischen Potentials

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / D FIT – Electrostatic Case / 3D-FIT – Elektrostatischer Fall Electrostatic Poisson’s grid equation / Elektrostatische Poissonsche Gittergleichung Inhomogeneous, anisotropic case / Inhomogener anisotroper Fall Homogeneous, isotropic case / Homogener isotroper Fall with / mit

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / D FIT – Electrostatic Case / 3D-FIT – Elektrostatischer Fall Electrostatic Poisson’s grid equation / Elektrostatische Poissonsche Gittergleichung

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / D FIT – Electrostatic Case / 3D-FIT – Elektrostatischer Fall Electrostatic Poisson’s grid equation / Elektrostatische Poissonsche Gittergleichung

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / D FIT – Electrostatic Case / 3D-FIT – Elektrostatischer Fall

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / D FIT – Electrostatic Case / 3D-FIT – Elektrostatischer Fall

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / D FIT – Electrostatic Case / 3D-FIT – Elektrostatischer Fall

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / D FIT – Electrostatic Case / 3D-FIT – Elektrostatischer Fall

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / D FIT – Electrostatic Case / 3D-FIT – Elektrostatischer Fall

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / D FIT – Electrostatic Case / 3D-FIT – Elektrostatischer Fall

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / Discrete Poisson’s Grid Equation / Diskrete Poissonsche Gittergleichung

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / D FIT – Electrostatic Case / 3D-FIT – Elektrostatischer Fall Electrostatic Poisson’s grid equation / Elektrostatische Poissonsche Gittergleichung Homogeneous isotropic case for a cubic grid complex / Homogener isotroper Fall für ein kubischen Gitterkomplex Electrostatic Poisson’s grid equation / Elektrostatische Poissonsche Gittergleichung

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / Band Structure of the Div-Grad-Operator in Matrix Form / Bandstruktur des Div-Grad-Operators in Matrixform

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / Band Structure of the Div-Grad-Operator in Matrix Form / Bandstruktur des Div-Grad-Operators in Matrixform

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / Band Structure of the Div-Grad-Operator in Matrix Form / Bandstruktur des Div-Grad-Operators in Matrixform 3-D case / 3D-Fall 3-D case / 3D-Fall 2-D case in the xz plane / 2D-Fall in der xz -Ebene 2-D case in the xz plane / 2D-Fall in der xz -Ebene

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / Numerical Methods for Linear Systems / Numerische Methoden für lineare Gleichungssysteme One the simplest type of iterative methods for solving the linear system / Einer der einfachsten Typen von iterativen Methoden zur Lösung des linearen Gleichungssystems Jacobi method (J method) / Jacobi-Methode (J-Methode) Gauss-Seidel method (GS method) / Gauß-Seidel-Methode (GS-Methode) Successive overrelaxation method (SOR method) / Überrelaxationsverfahren (SOR-Methode) Symmetric successive overrelaxation method (SSOR method) / Symmetrisches Überrelaxationsverfahren (SSOR-Methode) Jacobi method (J method) / Jacobi-Methode (J-Methode) Gauss-Seidel method (GS method) / Gauß-Seidel-Methode (GS-Methode) Successive overrelaxation method (SOR method) / Überrelaxationsverfahren (SOR-Methode) Symmetric successive overrelaxation method (SSOR method) / Symmetrisches Überrelaxationsverfahren (SSOR-Methode) Gaussian elimination (Gauss method) / Gaußsches Eliminationsverfahren (Gauß-Methode) Conjugate gradient method (CG method) / Konjugierte Gradientenmethode (KG-Methode) Multi grid methods (MG method) / Mehrgitterverfahren (MG-Methode) Algebraic multi grid method (AMG method) / Algebraische Mehrgitterverfahren (AMG-Methode) Conjugate gradient method (CG method) / Konjugierte Gradientenmethode (KG-Methode) Multi grid methods (MG method) / Mehrgitterverfahren (MG-Methode) Algebraic multi grid method (AMG method) / Algebraische Mehrgitterverfahren (AMG-Methode)

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / Numerical Methods for Linear Systems / Numerische Methoden für lineare Gleichungssysteme Algebraic multi grid method (AMG method) / Algebraische Mehrgitterverfahren (AMG-Methode)

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / Numerical Methods for Linear Systems / Numerische Methoden für lineare Gleichungssysteme Algebraic multi grid method (AMG method) / Algebraische Mehrgitterverfahren (AMG-Methode)

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / Iterative Methods for Linear Systems / Iterative Methoden für lineare Gleichungssysteme LU decomposition of matrix [A] / LU-Zerlegung der Matrix [A] Lower triangular matrix / Unter Dreiecksmatrix Main diagonal matrix / Hauptdiagonalmatrix Upper triangular matrix / Obere Dreiecksmatrix

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / Iterative Methods for Linear Systems / Iterative Methoden für lineare Gleichungssysteme Jacobi method (J method) / Jacobi-Methode (J-Methode) Gauss-Seidel method (GS method) / Gauß-Seidel-Methode (GS-Methode) Successive overrelaxation method (SOR method) / Überrelaxationsverfahren (SOR-Methode) Symmetric successive overrelaxation method (SSOR method) / Symmetrisches Überrelaxationsverfahren (SSOR-Methode) Jacobi method (J method) / Jacobi-Methode (J-Methode) Gauss-Seidel method (GS method) / Gauß-Seidel-Methode (GS-Methode) Successive overrelaxation method (SOR method) / Überrelaxationsverfahren (SOR-Methode) Symmetric successive overrelaxation method (SSOR method) / Symmetrisches Überrelaxationsverfahren (SSOR-Methode)

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / Jacobi Method / Jacobi-Methode It follows with the LU decomposition of matrix [A] / Mit der LU-Zerlegung der Matrix [A] folgt

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / Jacobi Method / Jacobi-Methode 2-D case in the xz plane / 2D-Fall in der xz -Ebene 2-D case in the xz plane / 2D-Fall in der xz -Ebene

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / Jacobi & Gauss-Seidel Method / Jacobi- & Gauss-Seidel-Methode

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / Successive Overrelaxation Method (SOR Method) / Überrelaxationsverfahren (SOR-Methode)

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / Successive Overrelaxation Method (SOR Method) / Überrelaxationsverfahren (SOR-Methode)

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / Successive Overrelaxation Method (SOR Method) / Überrelaxationsverfahren (SOR-Methode)

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / Symmetric Successive Overrelaxation Method (SSOR Method) / Symmetrische Überrelaxationsverfahren (SSOR-Methode) Forward SOR step / Vorwärts-SOR-Schritt Backward SOR step / Rückwärts-SOR-Schritt

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / Convergence of Point Iterative Methods / Konvergenz von punktiterativen Methoden

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / Error Vector and Error Measure / Fehlervektor und Fehlermaß Symmetric positive definite [A] matrix / Symmetrische positiv-definite [A] Matrix Approximation for a D-dimensional Dirichlet problem / Approximation für ein D-dimensionales Dirichlet-Problem

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / Optimal Relaxation / Optimaler Relaxationsparameter

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / Medium (2) Medium (1) Medium Transition Conditions / Übergangsbedingungen Boundary Conditions / Randbedingungen Electrostatic (ES) Fields / Elektrostatische (ES) Felder

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / Boundary conditions for the electrostatic potential / Randbedingungen für die elektrostatische Potential Medium Electrostatic (ES) Fields / Elektrostatische (ES) Felder Dirichlet Boundary Condition for Ф e / Dirichlet-Randbedingung für Ф e Neumann Boundary Condition for Ф e / Neumann-Randbedingung für Ф e

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / Example: 2-D Dirichlet Problem / Beispiel: 2D-Dirichlet-Problem Dirichlet boundary condition for Ф e / Dirichlet-Randbedingung für Ф e

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / Example: 2-D Dirichlet Problem / Beispiel: 2D-Dirichlet-Problem

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / Example: 2-D Dirichlet Problem / Beispiel: 2D-Dirichlet-Problem

Dr.-Ing. René Marklein - NFT I - Lecture 12 / Vorlesung 12 - WS 2005 / End of Lecture 12 / Ende der 12. Vorlesung