EE750 Advanced Engineering Electromagnetics Lecture 17

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1 EE75 Avan Engnrng Eltromagnt Ltur 7

2 D EM W onr a D ffrntal quaton of th form α α β f ut to p on Γ α α. n γ q on Γ whr Γ Γ Γ th ontour nlong th oman an n th unt outwar normal ot that th ounar onton ma a Drhlt uman or m Drhlt an uman. α α an β ar funton aoat wth th phal paramtr an f th taton EE75 Dr. Moham Bakr

3 EE75 Dr. Moham Bakr D EM Cont Γ Γ.5 q f - γ β α α Th funtonal aoat wth th prolm Prov t!

4 D EM Anal Th omputatonal oman v nto trangl lmnt Eah no ha oth a loal an a gloal n 5 5 A onntvt arra n an M tor th gloal n of th no EE75 Dr. Moham Bakr

5 D EM Anal Cont Γ W aum that thr ar M ln gmnt on Γ W tor th n arra n an M of gloal n of no on Γ EE75 Dr. Moham Bakr 5

6 Input Data to th D EM Anal Th oornat of th no r whr th total numr of no Th valu of α α β an f for ah lmnt Th valu of p for ah no rng on Γ Th valu of γ an q for ah gmnt wth no on Γ Th two arra n an M an n an M EE75 Dr. Moham Bakr 6

7 7 EE75 Dr. Moham Bakr Elmntal Intrpolaton Ovr th th lmnt w utlz th lnar appromaton a Th thr no of th th lmnt mut atf th lnar ntrpolaton rlaton a a a Solvng for a an w otan whr a A a a a

8 8 EE75 Dr. Moham Bakr Elmntal Intrpolaton Cont A th ara of th th lmnt an gvn A Th ntrpolaton funton atf δ

9 9 EE75 Dr. Moham Bakr Th Homognou uman BC a W frt onr th a γ q Th funtonal pr a a um of lmntal ufunton M whr f - β α α.5 Suttutng wth th lnar ntrpolaton pron w wrt

10 EE75 Dr. Moham Bakr Th Homognou uman BC a f.5 β α α f - β α α [ ] [ ] [ ]

11 Th Homognou uman BC a Cont α α β f EE75 Dr. Moham Bakr an If α α β an f ar takn a ontant ovr ah lmnt an utlzng th proprt l! m! n! l m n A l! m! n! w gt A α α δ A A f β

12 Th Pro of Aml Th pro of aml nvolv torng th loal lmntal omponnt nto thr propr loaton n th gloal tm of quaton [ ] [ ] [] Th lmnt Th lmnt a to n a to n n EE75 Dr. Moham Bakr

13 An Aml Eampl Thr ar no W ntalz an wth zro R an R Evaluat an an a thm to th propr loaton to gt EE75 Dr. Moham Bakr

14 An Aml Eampl Cont Evaluat an an a thm to th propr loaton Rpat th am tp for all lmnt EE75 Dr. Moham Bakr

15 Inorporatng a Bounar Conton of th r n In th a γ an q ar not zro γ Th tra ufuntonal q Γ a to Γ th funtonal Bau Γ ompr of M ln gmnt w ma wrt M W appromat th funton ovr th gmnt th lnar pron ξ ξ Γ ξ ξ ξ th normalz tan from no to no EE75 Dr. Moham Bakr 5

16 6 EE75 Dr. Moham Bakr Inorporatng a r n BC Cont Γ q γ U th panon Γ q - γ Dffrntat an u Γl ξ l q - l ξ ξ γ [ ] [ ] [ ] n matr form an l q l ξ ξ γ Aml thn appl to tor th offnt

17 Th Drhlt Bounar Conton Th Drhlt ounar onton ar mpo lmnatng th known no uttutng for thr valu pp up pu uu p u p u Orgnal tm uu u u up p Ru tm EE75 Dr. Moham Bakr 7

18 An Eampl: A Shl Mrotrp Ln ε o ε o /n ε r ε r Th mrotrp kpt at potntal whl th trnal hlng o kpt at potntal Smmtr ma mplo to ru th omputatonal oman on half Th govrnng BVP EE75 Dr. Moham Bakr 8

19 9 EE75 Dr. Moham Bakr An Eampl: A Shl Mrotrp Ln Cont ε ρ ε ε o r r wth on th outr onutor on th mrotrp an / nonthplanofmmtr It follow that w hav α α ε r β f Th ltr fl otan through E-. But ovr ah lmnt appromat a A A E Ovr th th lmnt

20 An Eampl: A Shl Mrotrp Ln Cont Th nt Elmnt Mtho n Eltromagnt Janmng Jn EE75 Dr. Moham Bakr

21 An Eampl: A Shl Mrotrp Ln Cont Th nt Elmnt Mtho n Eltromagnt Janmng Jn Th qu-potntal ln EE75 Dr. Moham Bakr

September 27, Introduction to Ordinary Differential Equations. ME 501A Seminar in Engineering Analysis Page 1. Outline

September 27, Introduction to Ordinary Differential Equations. ME 501A Seminar in Engineering Analysis Page 1. Outline Introucton to Ornar Dffrntal Equatons Sptmbr 7, 7 Introucton to Ornar Dffrntal Equatons Larr artto Mchancal Engnrng AB Smnar n Engnrng Analss Sptmbr 7, 7 Outln Rvw numrcal solutons Bascs of ffrntal quatons