Integral Equation Methods. Jacob White. Thanks to Deepak Ramaswamy, Michal Rewienski, Xin Wang and Karen Veroy

Transcription

1 Itroducto to Smulato - Lecture 22 Itegral Equato ethods Jacob Whte Thaks to Deepak Ramaswamy, chal Rewesk, X Wag ad Kare Veroy

2 Outle Itegral Equato ethods Exteror versus teror problems Start wth usg pot sources Stadard Soluto ethods 2-D Galerk ethod Collocato ethod Issues 3-D Pael Itegrato

3 Iteror Versus Exteror Problems Iteror Exteror 2 T = sde 2 T = 0 outsde 0 Temperature kow o surface Temperature a Tak What s the heat flow? T Heat Flow = Thermal coductvty Temperature kow o surface Ice Cube a Bath surface

4 Exteror Problem Electrostatcs potetal + v 2 Ψ=0 Outsde - Ψ s gve o Surface What s the capactace? Ψ Capactace = Delectrc Permtvty surface

5 Drag Force a croresoator Courtesy of Werer Hemmert, Ph.D. Used wth permsso. Resoator Dscretzed Structure Computed Forces Bottom Vew Computed Forces Top Vew

6 What s commo about these problems. Exteror Problems Drag Force ES devce - flud (ar) creates drag. Couplg a Package - Felds exteror create couplg Capactace of a Sgal Le - Felds exteror. Quattes of Iterest are o the surface ES devce - Just wat surface tracto force Package - Just wat couplg betwee coductors Sgal Le - Just wat surface charge. Exteror Problem s lear ad space-varat ES - Exteror Stokes Flow equato (lear). Package - axwell s equatos free space (lear). Sgal Le - Laplace s equato free space (lear). But problems are geometrcally very complex!

7 Exteror Problems Why ot use Fte-Dfferece or FE methods Surface 2-D Heat Flow Example T = 0 at But, must trucate the mesh T Oly eed o the surface, but T s computed everywhere ust trucate the mesh, T( ) = 0 becomes T( R) = 0

8 Laplace s Equato Gree s Fucto If 2 2 u u the + = 0 for all xy, x, y 2 2 x y If u = u u u the + + = 0 for all xyz,, x, y, z x y z I 2-D ( ( ) 2 ) ( ) u = log x x + y y I 3-D ( x x ) + ( y y ) + ( z z ) ( ) ( ) 0 0 ( ) ( ) Proof: Just dfferetate ad see! 0 0 0

9 Laplace s Equato 2-D Smple Idea u s gve o surface x Surface, y ( ) 0 0 u x u + = 0 outsde y Let ( ( ) 2 ) ( ) u = log x x + y y 2 2 u u + = 0 outsde 2 2 x y Problem Solved Does ot match boudary codtos!

10 Laplace s Equato 2-D u s gve o surface ( x, y ) 2 2 ( x, y ) ( x, y ) u x Smple Idea ore Pots u + = 0 outsde y ( ( ) 2 ( ) 2 ) ω ( ) = + = Let u ω log x x y y G x x, y y = = Pck the ω ' s to match the boudary codtos!

11 Laplace s Equato 2-D ( x, ) t yt ( x, y ) 2 2 ( x, y ) ( x, y ) Smple Idea ore Pots Equatos Source Stregths selected to gve correct potetal at test pots. ( ) ( ) L L ( ) G xt x, y t y G x t x,, yt y ω Ψ xt y t O = O G( x ) ( ) ( ) t x, y,, t y G x t x yt y ω Ψ xt y L L t

12 Computatoal results usg pots approach Crcle wth Charges r=9.5 r Potetals o the Crcle R=0 =20 =40

13 Laplace s Equato 2-D Itegral Formulato Lmtg Argumet Wat to smear pot charges oto surface Results a Itegral Equato Ψ ( x) = G( x, x ) σ ( x ) ds How do we solve the tegral equato? surface

14 Laplace s Equato 2-D Bass Fucto Approach Basc Idea Represet σ ( x) = ω ( ) ϕ x { = Bass Fuctos Example Bass Represet crcle wth straght les Assume σ s costat alog each le The bass fuctos are o the surface Ca be used to approxmate the desty ay also approxmate the geometry

15 Laplace s Equato 2-D Bass Fucto Approach Geometrc Approxmato s ot ew. Pecewse Straght surface bass Fuctos approxmate the crcle ( ) ( ) ( ) Tragles for 2-D FE approxmate the crcle too! x G x, x ωϕ x ds Ψ = approx surface =

16 Laplace s Equato 2-D x Bass Fucto Approach Pecewse Costat Straght Sectos Example. x l l l 2 x 2 ) Pck a set of Pots o the surface 2) Defe a ew surface by coectg pots wth les. 3) Defe ϕ ( ) x = f x s o le l otherwse, ϕ x = 0 ( ) approx surface Ψ ( x) = G( xx, ) ωϕ ( x ) ds = ω G( xx, ) ds = = le l How do we determe the ω ' s?

17 Laplace s Equato 2-D Bass Fucto Approach Resdual Defto ad mmzato R x x G x, x ωϕ x ds Ψ ( ) ( ) ( ) ( ) We wll pck the approx surface = ω ' s to make R( x) small. Geeral Approach: Pck a set of test fuctos φ K φ R x,,, ad force ( ) to be orthogoal to the set φ ( ) ( ) xr x ds = 0 for all.

18 Laplace s Equato 2-D Bass Fucto Approach Resdual mmzato usg test fuctos ( ) ( ) ( ) ( ) ( ) ( ) ( ) φ x R x ds= φ x Ψ x ds φ x G x, x ω ϕ x dsds = 0 approx = surface We wll geerate dfferet methods by chosg the,,, φ K φ C ollocato: φ (pot-matchg) ( x) = δ ( x x ) t Galerk ethod: φ ( x) = ϕ ( x) (bass = test)

19 Laplace s Equato 2-D Bass Fucto Approach Collocato δ Collocat o: φ ( ) ( ) x = δ x (pot-matchg) ( ) ( ) ( ) ( ) ( ) t t t t ω ϕ ( ) x x R x ds= R x =Ψ x G x, x x ds = 0 approx surface ( ) ( ) t ω t, ϕ ( ) Ψ = x G x x x ds = approx surface A, = ( x ) t A, L L A, ω Ψ O = O A ( ), L L A, ω Ψ x t

20 Laplace s Equato 2-D x l l l 2 x 2 x t Bass Fucto Approach Cetrod Collocato for Pecewse Costat Bases ( ) ( ) t ω t, ϕ ( ) Ψ = x G x x x ds = approx surface Collocato pot le ceter ( x ) t A, L L A, ω Ψ O = O A ( ), L L A, ω Ψ x t ( ) (, ) t ω t Ψ = x G x x ds = le A,

21 Laplace s Equato 2-D Bass Fucto Approach Cetrod Collocato Geerates a osymmetrc A ( ) (, ) t ω t Ψ = x G x x ds = le A, x t l 2 l x t2 (, ) (, ) t t A G x x ds G x x ds A = = 2,2 2, le2 le

22 Laplace s Equato 2-D Bass Fucto Approach Galerk Galerk: φ ( x) = ϕ ( x) (test=bass) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ϕ x R x ds= ϕ x Ψ x ds ϕ x G x, x ω ϕ x dsds = 0 approx = surface approx surface If Gxx (, ) Gx (, x) the A = A = A s symmetrc,, ( ) ( ) (, ) ( ) ( ) Ψ = ϕ x x ds ω G x x ϕ x ϕ x dsds = approx approx surface surface b A, A, L L A, ω b O = O A, L L A, ω b

23 Laplace s Equato 2-D Bass Fucto Approach Galerk for Pecewse Costat Bases l l x x 2 l 2 ( ) ω (, ) = le le le b x ds G x x ds ds Ψ = A, A, L L A, ω b O = O A, L L A, ω b

24 3-D Laplace s Equato Itegral Equato: Ψ x = Bass Fucto Approach Pecewse Costat Bass ( ) ( ) surface x x σ x ds Dscretze Surface to Paels Represet σ ( x) ω ( ) ϕ x { = Bass Fuctos Pael ϕ x = x ϕ = ( ) f s o pael ( x) 0 otherwse

25 3-D Laplace s Equato Bass Fucto Approach Cetrod Collocato Put collocato pots at pael cetrods ( ) (, ) c ω c Ψ = c x Collocato pot x G x x ds = pael A, ( x ) c A, L L A, ω Ψ O = O A ( ), L L A, ω Ψ x c

26 3-D Laplace s Equato Bass Fucto Approach Calculatg atrx Elemets z y x x Collocato c pot Pael A, = pael x x c ds Oe pot quadrature Approxmato A, Pael Area x c x cetrod Four pot quadrature Approxmato A, * Area x = c x pot

27 3-D Laplace s Equato Bass Fucto Approach Calculatg Self-Term z y x x Collocato c pot Pael A, = pael x x c ds Oe pot quadrature Approxmato A, Pael Area x c 4243 A, = ds s a tegrable sgularty x x pael c 0 x c

28 3-D Laplace s Equato z y Itegrate two peces x Pael Bass Fucto Approach x Collocato c pot Dsk of radus R surroudg collocato pot Calculatg Self-Term Trcks of the trade A, = pael x x c ds A, = ds + ds x x x x dsk c rest of pael c Dsk Itegral has sgularty but has aalytc formula dsk x c x R 2π ds = rdrdθ = 2π R r 0 0

29 3-D Laplace s Equato z y x Pael Bass Fucto Approach x Collocato c pot Calculatg Self-Term Other Trcks of the trade A, = pael x c x 4243 Itegrad s sgular ds ) If pael s a flat polygo, aalytcal formulas exst 2) Curve paels ca be hadled wth proecto

30 3-D Laplace s Equato Bass Fucto Approach Galerk (test=bass) ( ) ( ) ( ) (, ) ( ) ϕ x Ψ x ds = ω ϕ x G x x ϕ x dsds = b A For pecewse costat Bass Ψ ( x) ds = ω dsds 4243 pael pael = x x b A,, A, L L A, ω b O = O A, L L A, ω b

31 3-D Laplace s Equato Bass Fucto Approach Problem wth dese matrx Itegral Equato ethod Geerate Huge Dese atrces ( x ) c A, L L A, ω Ψ O = O A ( ), L L A, ω Ψ x c Gaussa Elmato uch Too Slow!

32 Summary Itegral Equato ethods Exteror versus teror problems Start wth usg pot sources Stadard Soluto ethods Collocato ethod Galerk ethod Next Tme Fast Solvers Use a Krylov-Subspace Iteratve ethod Compute V products Approxmately