Numerical Methods of Electromagnetic Field Theory II (NFT II) Numerische Methoden der Elektromagnetischen Feldtheorie II (NFT II) /
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1 umerical Methods of lectromagnetic Field Theory II (FT II) umerische Methoden der lektromagnetischen Feldtheorie II (FT II) / 7th Lecture / 7. Vorlesung Dr.-Ing. René Marklein marklein@uni-kassel.de Universität Kassel Fachbereich lektrotechnik / Informatik (FB 6) Fachgebiet Theoretische lektrotechnik (FG TT) Wilhelmshöher Allee 7 Büro: Raum 23 / 25 D-342 Kassel University of Kassel Dept. lectrical ngineering / Computer Science (FB 6) lectromagnetic Field Theory (FG TT) Wilhelmshöher Allee 7 Office: Room 23 / 25 D-342 Kassel Dr. R. Marklein - FT II - SS 23
2 CG Method Conjugate Gradient Method References / KG-Methode Konjugierte Gradientenmethode Jonathan Richard Shewchuk: An Introduction to the Conjugate Gradient Method Without the Agonizing Pain, p., August 994. Dr. R. Marklein - FT II - SS
3 Penetrable Scatterer: Data quation and Lippmann-Schwinger quation 2-D TM Case/ Penetrable Streuer: Datengleichung und Lippmann-Schwinger Gleichung 2D-TM-Fall O r sc εε r, µ µ r n sc S sc = V sc ε ε ( r), µ µ r r z (, r ω) ( r, ω) z V sc r k G r r r r r sc 2 2 z (, ω) = (, ω) χ( ) (, )d r ' S z ω sc { z } = [ G][ χ ]{ z} [ G] π () j ka J ( ka)h ( k rm rn ) m n 2 Gmn = π () + j ka H ( ka) m = n 2 Dr. R. Marklein - FT II - SS
4 P Scatterer: Discretization of the Domain Integral quation Richmond Method / P-Streuer: Diskretisierung der Bereichsintegralgleichung Richmod-Methode x εε r, µ µ r S sc 2 O r r sc z (, r ω) εε ( r ), µµ, χ( r ) z r r ( r, ω) 2 x r r k G r r r r r sc 2 2 z (, ω) = (, ω) χ( ) (, )d ' S z ω sc sc { } [ ] { χ } = G = z z x y [ G] π () j ka J ( ka) H ( k rm rn ) m n n, m =, 2,, = x G 2 mn = π () + j ka H ( ka) m = n n, m =, 2,, = x 2 y y Dr. R. Marklein - FT II - SS
5 The umerical Solution of a Linear Matrix quation Requires the Storage of the Full Matrix in the Computer Main Memory. This is the Main Bottleneck in the Solution of a Large Set of Linear quations. / Die numerische Lösung einer linearen Matrixgleichung erfordert die Speicherung der vollen -Matrix im Hauptspeicher des Computers. Dies stellt den wesentlichsten Flaschenhals bei der numerischen Lösung von großen linearen Gleichungssystemen dar. CG-FFT Conjugate Gradient Fast Fourier Transform / KG-SFT Konjugierte Gradienten Schnelle Fourier-Transformation Dr. R. Marklein - FT II - SS
6 Two-Dimensional Convolution Integral / Zweidimensionales Faltungsintegral r k G r r r r r sc 2 2 z (, ω) = (, ω) χ( ) (, )d r ' S z ω sc 2 2 z (, r ω) = G( r r, ω) k χ( r ) ( r, )dr r ' S z ω = ( r, ω ) = J( r, ω ) G r ' S 2 ( r, ω ) = ( r r, ω ) J( r, ω )d r 2-D Convolution Integral in Cartesian Coordinates / 2D-Faltungsintegral in Kartesische Koordinaten ( x, y, ω) = G( x x, y y, ω)j( x, y, ω)dx dy x = y = -D Convolution Integral in Cartesian Coordinates / D-Faltungsintegral in Kartesischen Koordinaten ( x, ω) = Gx ( x, ω)j( x, ω)dx x = Dr. R. Marklein - FT II - SS
7 Infinite -D Convolution Integral in Cartesian Coordinates / Unendliches D-Faltungsintegral in Kartesischen Koordinaten G( x, ω) J e ( x, ω) ( x, ω) = Gx ( x, ω)j( x, ω)dx x = x x Finite -D Convolution Integral in Cartesian Coordinates / ndliches D-Faltungsintegral in Kartesischen Koordinaten G( x, ω) J e ( x, ω) x max ( x, ω) = Gx ( x, ω)j( x, ω)dx x = x min x x min x = x = max x Discrete Convolution / Diskrete Faltung G( x, ω ) J e ( x, ω ) = x G J m =,, 2,..., m m n n x Dr. R. Marklein - FT II - SS
8 Discrete Convolution / Diskrete Faltung = x G J m =,, 2,..., m m n n Discrete Convolution in Matrix Form / { } = x[ G] { J} Diskrete Faltung in Matrixform xample: = M = 4 / Beispiel: = M = 4 m = : m = : m = 2: m = 3: 4 = 3 m= m n n = x [ J J J J ] = xg [ J + G J + G J + G J ] = = 2= 2 2 3= = 3 G m= m n n J = xg + G + G + G = x G J [ J J J J ] = x[ G J + G J + G J + G J ] = xg + G + G + G = x 4 = 3 m= 2 m n = = 2= 2 3= [ J J J J ] = xg [ J + GJ + G J + G J ] 4 = 3 G m= 3 m n n J 2 = 2 2 = 2 2= 2 2 3= 3 J n = xg + G + G + G = x G [ J J J J ] = x[ G J + G J + G J + G J ] = xg + G + G + G 3 = 3 3 = 2 3 2= 2 3 3= Dr. R. Marklein - FT II - SS
9 Discrete Convolution / Diskrete Faltung = x G J m =,, 2,..., m m n n xample: = M = 4 / Beispiel: = M = 4 Discrete Convolution in Matrix Form / Diskrete Faltung in Matrixform G J + G J + G J + G J = x G J + G J + G J + G J = x G J + G J + G J + G J = x G J + G J + G J + G J = x G G G 2 G J G G G G 2 J G2 G G G3 J2 = 2 x G G G G J 2 3 = x [ G]{ J} { } Dr. R. Marklein - FT II - SS
10 Discrete Convolution in Matrix Form / Diskrete Faltung in Matrixform G G G 2 G J G G G G 2 J G2 G G G3 J2 = 2 x G G G G J 2 3 = x [ G]{ J} { } The Matrix [G] is a Matrix and is a General Toeplitz Matrix / Die Matrix [G] ist eine -Matrix und ist eine allgemeine Toeplitz-Matrix All Different lements of the Matrix [G] are given by the 2 ntries of the st Row and st Column / Alle unterschiedlichen lemente der Matrix [G] sind durch die 2 inträge der. Zeile und. Splate gegeben G,, G, G, G, G, G,, G 2 2 lements / 2 lemente Dr. R. Marklein - FT II - SS 23
11 Discrete Convolution in Matrix Form / Diskrete Faltung in Matrixform G G G G J 2 G G G G 2 J G2 G G G3 J2 = 2 x G G G G J 2 3 = x [ G]{ J} { } The Sequenz of the lements of the Matrix [G] are Periodic after lements / Die Sequenz der lemente der Matrix [G] sind periodisch nach lementen Gn = Gn n =,, 2,..., Then, the Discrete Convolution is a Circular Discrete Convolution of the Length 2 and the Matrix [G] is a Circular Matrix. Otherwise the Discrete Convolution is a Linear Discrete Convolution. / Dann ist die diskrete Faltung eine zirkulare diskrete Faltung der Länge 2 und die Matrix [G] ist eine zirkulierende Matrix. Anderenfalls ist die diskrete Faltung eine lineare diskrete Faltung. Dr. R. Marklein - FT II - SS 23
12 G G G 2 G J G G G G 2 J G2 G G G3 J2 = 2 x G G G G J 2 3 G = G n =,, 2,..., n n Periodic / Periodisch G G2 G G G G 2 G G G G2 G G G G 2 G2 G G G2 G G G3 = J x G2 G G J G G2 G G G 2 G 3 G 2 2 J 2 J 2 2 Dr. R. Marklein - FT II - SS
13 G G2 G G G G 2 G G G G2 G G G G 2 G2 G G G2 G G G3 = J x G2 G G J G G2 G G G 2 G 3 G 2 2 J 2 J J J J2 G G G 2 G G G2 G G G G G2 G G G 2 J G2 G G G3 G2 G G = x G2 G G G G 2 G 3 G G G2 G 2 2 Dr. R. Marklein - FT II - SS 23 Periodic / Periodisch
14 Discrete Convolution / Diskrete Faltung = x G J m =,, 2,..., m m n n Comment: very Linear Discrete Convolution of the Length can be formulated in a Circular Discrete Convolution of the Length 2 by xpanding the Sequence G to a Periodic Sequence of the Length 2. And the Sequence J will be Filled with Zeros up to a Length of 2 : Zero Padding / Anmerkung: Jede lineare diskrete Faltung der Länge kann in eine zirkulierende diskrete Faltung der Länge 2 gebracht werden. Dazu wird die Sequenz G in eine periodische Sequenz der Länge 2 umgewandelt. Und die Sequenz J wird mit ullen aufgefüllt: ullen-auffüllung { J} J J J2 J = 2 Original Sequence / Originalsequenz Zero Padding / ullen-auffüllung Dr. R. Marklein - FT II - SS
15 m m n n 2 ( ) = x G J m =,, 2,..., O The Fast Fourier Transform (FFT) is an fficient Way of Implementation the Discrete Fourier Transform (DFT) / Die schnelle Fourier-Transformation (SFT) ist ein effizienter Weg der Implementierung der diskreten Fourier-Transformation (DFT) k k = j2 π nk / G = G e n =,, 2,..., n j2 π nk / Gk = Gn e k =,, 2,..., is a Power of / p 2 = 2, p =,,2,3, ist eine Potenz von { G } = FFT { G} G = FFT G { } { } { } = x { G} { J} n =,, 2,..., FFT FFT FFT { } { G} { J} { },, 2,..., ( log ) = xfft FFT FFT n = O Dr. R. Marklein - FT II - SS
16 nd of 7th Lecture / nde der 7. Vorlesung Dr. R. Marklein - FT II - SS
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