Itroducto to Smulato - Lecture 22 Itegral Equato ethods Jacob Whte Thaks to Deepak Ramaswamy, chal Rewesk, X Wag ad Kare Veroy
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
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
Exteror Problem Electrostatcs potetal + v 2 Ψ=0 Outsde - Ψ s gve o Surface What s the capactace? Ψ Capactace = Delectrc Permtvty surface
Drag Force a croresoator Courtesy of Werer Hemmert, Ph.D. Used wth permsso. Resoator Dscretzed Structure Computed Forces Bottom Vew Computed Forces Top Vew
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!
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
Laplace s Equato Gree s Fucto If 2 2 u u the + = 0 for all xy, x, y 2 2 x y If u = 2 2 2 u u u the + + = 0 for all xyz,, x, y, z 2 2 2 x y z I 2-D ( ( ) 2 ) ( ) 2 0 0 u = log x x + y y I 3-D ( x x ) + ( y y ) + ( z z ) 2 2 2 0 0 0 ( ) ( ) 0 0 ( ) ( ) Proof: Just dfferetate ad see! 0 0 0
Laplace s Equato 2-D Smple Idea u s gve o surface x Surface, y ( ) 0 0 u x u + = 0 outsde y 2 2 2 2 Let ( ( ) 2 ) ( ) 2 0 0 u = log x x + y y 2 2 u u + = 0 outsde 2 2 x y Problem Solved Does ot match boudary codtos!
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 2 2 ( ( ) 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!
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
Computatoal results usg pots approach Crcle wth Charges r=9.5 r Potetals o the Crcle R=0 =20 =40
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
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
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 =
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?
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.
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)
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 4444244443 A, = ( x ) t A, L L A, ω Ψ O = O A ( ), L L A, ω Ψ x t
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 442443 A,
Laplace s Equato 2-D Bass Fucto Approach Cetrod Collocato Geerates a osymmetrc A ( ) (, ) t ω t Ψ = x G x x ds = le 442443 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
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 44424443 444444244444443 b A, A, L L A, ω b O = O A, L L A, ω b
Laplace s Equato 2-D Bass Fucto Approach Galerk for Pecewse Costat Bases l l x x 2 l 2 ( ) ω (, ) = le le le 4243 44424443 b x ds G x x ds ds Ψ = A, A, L L A, ω b O = O A, L L A, ω b
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
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 44424443 A, ( x ) c A, L L A, ω Ψ O = O A ( ), L L A, ω Ψ x c
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, 4 0.25* Area x = c x pot
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
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
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
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 442443 = 444442444443 b A For pecewse costat Bass Ψ ( x) ds = ω dsds 4243 pael pael = x x b 4444244443 A,, A, L L A, ω b O = O A, L L A, ω b
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!
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