Integral Equation Methods - MIT OpenCourseWare

Introduction to Simulation - Lecture 22Integral Equation MethodsJacob WhiteThanks to Deepak Ramaswamy, Michal Rewienski,Xin Wang and Karen Veroy

OutlineIntegral Equation MethodsExterior versus interior problemsStart with using point sourcesStandard Solution Methods in 2-DGalerkin MethodCollocation MethodIssues in 3-DPanel IntegrationSMA-HPC 2003 MIT

Interior Versus Exterior ProblemsInteriorExterior 2T 0outside 2T 0insideTemperatureknown on surface“Temperature in a Tank”Temperatureknown on surface“Ice Cube in a Bath”What is the heat flow?Heat FlowSMA-HPC 2003 MIT T nsurfaceThermal conductivity

Exterior Problem in Electrostaticspotential v- 2Ψ 0OutsideΨ is given on SurfaceWhat is the capacitance?CapacitanceSMA-HPC 2003 MITDielectric Permitivity Ψ nsurface

Drag Force in a MicroresonatorCourtesy of Werner Hemmert, Ph.D. Used with permission.ResonatorComputed ForcesBottom ViewSMA-HPC 2003 MITDiscretized StructureComputed ForcesTop View

What is common about these problems.Exterior ProblemsDrag Force in MEMS device - fluid (air) creates drag.Coupling in a Package - Fields in exterior create couplingCapacitance of a Signal Line - Fields in exterior.Quantities of Interest are on the surfaceMEMS device - Just want surface traction forcePackage - Just want coupling between conductorsSignal Line - Just want surface charge.Exterior Problem is linear and space-invariantMEMS - Exterior Stokes Flow equation (linear).Package - Maxwell’s equations in free space (linear).Signal Line - Laplace’s equation in free space (linear).But problems are geometrically very complex!

Exterior ProblemsSurfaceWhy not use Finite-Differenceor FEM methods2-D Heat Flow ExampleT 0 at But, musttruncate themesh TOnly needon the surface, but T is computed everywhere nMust truncate the mesh, T ( ) 0 becomes T ( R ) 0SMA-HPC 2003 MIT

Green’s FunctionLaplace’s Equation(In 2-D22( x x0 ) ( y y0 ))If u log 2u 2uthen 2 2 0 for all ( x, y ) ( x0 , y0 ) x yIn 3-DIf u 1( x x0 ) ( y y0 ) ( z z0 )222 2u 2u 2uthen 2 2 2 0 for all ( x, y, z ) ( x0 , y0 , z0 ) x y zProof: Just differentiate and see!SMA-HPC 2003 MIT

Laplace’s Equationin 2-DSimple Ideau is given on surfaceSurface( x0 , y0 )Let u log( 2u 2u 2 0 outside2 x y( x x0 ) ( y y0 )2 2u 2u 2 0 outside2 x y2)Problem SolvedDoes not match boundary conditions!SMA-HPC 2003 MIT

Simple IdeaLaplace’s Equationin 2-D“More Points”u is given on surface 2u 2u 2 0 outside2 x y( x2 , y2 )( x1 , y1 )nLet u ωi logi 1(( xn , yn )( x xi ) ( y yi )22) ω G ( x x , y y )ni 1iiiPick the ωi ' s to match the boundary conditions!SMA-HPC 2003 MIT

Simple IdeaLaplace’s Equationin 2-D“More Points Equations”(x , y )t1t1Source Strengths selectedto give correct potential attest points.( x2 , y2 )( x1 , y1 )( xn , yn ) G ( xt x1 , yt y1 ) L L G ( xt xn , yt yn ) ω Ψ ( xt , yt ) 111111 1 MOMM M M MOMM G x x , y y L L G x x , y y ω n Ψ x , y 11tntntnntnn tntn (SMA-HPC 2003 MIT)()()

Computational results using points approachCircle with Charges r 9.5Potentials on the Circlen 20SMA-HPC 2003 MITrR 10n 40

Laplace’s Equationin 2-DIntegral FormulationLimiting ArgumentWant to smear point charges onto surfaceResults in an Integral EquationΨ ( x ) G ( x, x′ ) σ ( x′ ) dS ′surfaceHow do we solve the integral equation?SMA-HPC 2003 MIT

Laplace’s Equationin 2-DBasis Function ApproachBasic IdeanRepresent σ ( x ) ωi ϕ i ( x ){i 1Basis FunctionsExample BasisRepresent circle with straight linesAssume σ is constant along each lineThe basis functions are “on” the surfaceCan be used to approximate the densityMay also approximate the geometrySMA-HPC 2003 MIT

Laplace’s Equationin 2-DBasis Function ApproachGeometric Approximation isnot new.Piecewise Straight surface basisTriangles for 2-D FEMFunctions approximate the circle approximate the circle too!Ψ ( x) approxsurfaceSMA-HPC 2003 MITnG ( x, x′ ) ωiϕi ( x′ ) dS ′i 1

Laplace’s Equationin 2-Dx1xnlnl1x2l2Ψ ( x) Basis Function ApproachPiecewise Constant StraightSections Example.1) Pick a set of n Points on thesurface2) Define a new surface byconnecting points with n lines.3) Define ϕ i ( x ) 1 if x is on line liotherwise, ϕ i ( x ) 0nni 1i 1G ( x, x′ ) ωiϕi ( x′ ) dS ′ ωiapproxsurfaceSMA-HPC 2003 MITG ( x, x′ ) dS ′ line liHow do we determine the ω i ' s ?

Basis Function ApproachLaplace’s Equationin 2-DR ( x) Ψ ( x) Residual Definition andminimizationnG ( x, x′ ) ωiϕi ( x′ ) dS ′approxsurfacei 1We will pick the ω i ' s to make R ( x ) small.General Approach: Pick a set of test functionsφ1 ,K , φn , and force R ( x ) to be orthogonal to the set φ ( x )R ( x ) dS 0iSMA-HPC 2003 MITfor all i.

Basis Function ApproachLaplace’s Equationin 2-DResidual minimization usingtest functions φ ( x ) R ( x ) dS φ ( x ) Ψ ( x ) dS iinφi ( x ) G ( x, x′ ) ω jϕ j ( x′ ) dS ′dS 0j 1approxsurfaceWe will generate different methods by chosing the φ1 ,K , φn ,()Collocation: φi ( x ) δ x xti (point-matching)Galerkin Method: φi ( x ) ϕ i ( x ) (basis test)SMA-HPC 2003 MIT

Basis Function ApproachLaplace’s Equationin 2-DCollocationCollocation: φi ( x ) δ ( xi ) (point-matching) δ ( x x ) R ( x ) dS R ( x ) Ψ ( x ) titi( )n Ψ xti ω jj 1(G xti , x′tiapproxsurface (n) ω ϕ ( x′) dS ′ 0j 1j)G xti , x′ ϕ j ( x′ ) dS ′approxsurface14444244443Ai , j A1,1 L L A1,n ω1 Ψ ( xt1 ) M O M M M MO M M M An ,1 L L An ,n ω n Ψ xtn ( )SMA-HPC 2003 MITj

Laplace’s Equationin 2-Dxn lnxt1l1l2x2Basis Function ApproachCentroid Collocation forPiecewise Constant Bases( )nΨ xti ω jj 1 ()G xti , x′ ϕ j ( x′ ) dS ′approxsurfaceCollocation point inline center A1,1 L L A1,n ω1 Ψ ( xt1 ) M O M M M MO M M M An ,1 L L An ,n ω n Ψ xtn ( )SMA-HPC 2003 MIT( )nΨ xti ω jj 1 G ( x , x′) dS ′tiline j1442443Ai , j

Laplace’s Equationin 2-D( )nΨ xti ω jj 1Basis Function ApproachCentroid CollocationGenerates a nonsymmetric A G ( x , x′) dS ′tiline j1442443Ai , jxt1xt2l1A1,2 l2 G ( x , x′) dS ′ G ( xt1line 2SMA-HPC 2003 MITt2line1, x′ ) dS ′ A2,1

Laplace’s Equationin 2-DBasis Function ApproachGalerkinGalerkin: φi ( x ) ϕ i ( x ) (test basis) ϕ ( x ) R ( x ) dS ϕ ( x ) Ψ ( x ) dS inϕi ( x ) G ( x, x′ ) ω jϕ j ( x′ ) dS ′dS 0i nϕi ( x ) Ψ ( x ) dS ω japproxsurfacej 1144424443bi A1,1 L L M O MO An ,1 L Lapproxsurface G ( x, x′ ) ϕi ( x ) ϕ j ( x′ ) dS ′dSapprox approxsurface surface14444444244444443Ai , jA1,n ω1 b1 M M M M M M An ,n ω n bn If G ( x, x′) G ( x′, x) then Ai , j A j ,iSMA-HPC 2003 MITj 1A is symmetric

Basis Function ApproachLaplace’s Equationin 2-Dlnl1 xnl2Galerkin for PiecewiseConstant Basesx2n Ψ ( x ) dS ω G ( x, x′) dS ′dSlinei14243bij 1jlinei line j144424443Ai , j A1,1 L L A1,n ω1 b1 M O M M M MO M M M An ,1 L L An ,n ω n bn SMA-HPC 2003 MIT

3-D Laplace’sEquationBasis Function ApproachPiecewise Constant BasisIntegral Equation: Ψ ( x ) surfaceDiscretize Surface intoPanels1σ ( x′ ) dS ′x x′nRepresent σ ( x ) ωi ϕ i ( x ){i 1Basis Functionsϕ j ( x ) 1 if x is on panel jPanel j ϕ ( x ) 0 otherwisejSMA-HPC 2003 MIT

3-D Laplace’sEquationPut collocation points atpanel centroidsBasis Function ApproachCentroid Collocation( )nΨ xci ω jxci Collocationpointj 1G(x panel jci), x′ dS ′144424443Ai , j A1,1 L L A1,n ω1 Ψ ( xc1 ) M O M M M MO M M M An ,1 L L An ,n ω n Ψ xcn ( )SMA-HPC 2003 MIT

Basis Function Approach3-D Laplace’sEquationCalculating Matrix Elementsxci CollocationzpointyAi , j x panel j1dS ′xci x′Panel jOne pointquadratureApproximationFour pointquadratureApproximationSMA-HPC 2003 MITPanel AreaAi , j xci xcentroid j40.25* Areaj 1xci x po int jAi , j

Basis Function Approach3-D Laplace’sEquationCalculating “Self-Term”xci CollocationzpointyAi ,i x panel i1dS ′xci x′Panel iOne pointquadratureApproximationAi ,i panel iSMA-HPC 2003 MITAi ,iPanel Area xci xci1424301dS ′ is an integrable singularityxci x′

Basis Function Approach3-D Laplace’sEquationCalculating “Self-Term”Tricks of the tradezxci CollocationypointPanel ix Ai ,i panel iDisk of radius Rsurroundingcollocation point Integrate in two Ai ,i diskpiecesDisk Integral hassingularity but hasanalytic formulaSMA-HPC 2003 MIT disk1dS ′xci x′11dS ′ dS ′ xci x′rest of panel xci x′R 2π1dS ′ xci x′0 01rdrdθ 2π Rr

Basis Function Approach3-D Laplace’sEquationCalculating “Self-Term”Other Tricks of the tradezxci CollocationypointxPanel iAi ,i panel i1dS ′xci x′14243Integrand is singular1) If panel is a flat polygon, analytical formulas exist2) Curve panels can be handled with projectionSMA-HPC 2003 MIT

Basis Function Approach3-D Laplace’sEquationGalerkin (test basis)nϕ ( x ) Ψ ( x ) dS ω ϕ ( x ) G ( x, x′ )ϕ ( x′ ) dS ′dS 14424431444442444443ibij 1jijAi , jFor piecewise constant Basisn1dS ′dSΨ ( x ) dS ′ ω j 14 panelipaneljx x′243 j 114444244443biAi , j A1,1 L L A1,n ω1 b1 M O M M M MO M M M An ,1 L L An ,n ω n bn SMA-HPC 2003 MIT

3-D Laplace’sEquationBasis Function ApproachProblem with dense matrixIntegral Equation Method Generate HugeDense Matrices A1,1 L L A1,n ω1 Ψ ( xc1 ) M O M M M MO M M M An ,1 L L An ,n ω n Ψ xcn ( )Gaussian Elimination Much Too Slow!SMA-HPC 2003 MIT

SummaryIntegral Equation MethodsExterior versus interior problemsStart with using point sourcesStandard Solution MethodsCollocation MethodGalerkin MethodNext Time Æ “Fast” SolversUse a Krylov-Subspace Iterative MethodCompute MV products Approximately

Integral Equation Methods Exterior versus interior problems Start with using point sources Standard Solution Methods Collocation Method Galerkin Method Next Time Æ“Fast” Solvers Use a Krylov-Subspace Iterative Method Compute MV produc