麻省理工学院：《偏微分方程式数字方法》（英文版）Lecture 20 toupload

Numerical Methods for PDEs Integral Equation Methods, Lecture 1 Discretization of Boundary Integral equations Notes by Suvranu De and J. White Apri23,2003

Outline for this Module Overview of Integral Equation Methods Important for many exterior problems (Fluids, Electromagnetics, Acoustics) Quadrature and cubature for computing integrals One and Two dimensional basics Dealing with Singularities 18 and 2a Kind Integral Equations Collocation, Galerkin and Nystrom theory Alternative Integral Formulations Ansatz approach and Green's theorem SMA+HPC⊙2003M Discretization of Boundary Integral Equations 1

Outline for this Module Fast solvers Fast Multipole and FFT-based methods SMA+HPC⊙2003M Discretization of Boundary Integral Equations 1

Outline Integral Equation Methods Exterior versus interior problems Start with using point sources Standard Solution Methods Collocation Method Galerkin Method Some issues in 3D Singular integrals SMA+HPC⊙2003M Discretization of Boundary Integral Equations 2

Interior vs Exterior Problems Interior Exterior V2T=0 outside V2T=0 Inside Temperature Temperature known on surface known on surface Temperature in a tank""lce cube in a bath What is the heat flow? Heat flow =Thermal conductivity / sur face an SMA+HPC⊙2003M Discretization of Boundary Integral Equations 3

Computation of Capacitance Examples otential V4乎y=0 Outside y is given on Surface What is the capacitance? Capacitance = Dielectric Permittivity on SMA+HPC⊙2003M Discretization of Boundary Integral Equations 4

Drag force in a microresonator Examples 口 Resonator Discretized structure E Computed Forces Computed Forces Bottom view Top view SMA-HPC(C)2003 MIT Discretization of Boundary Integral Equations 5

Electromagnetic Coupling in a Package Examples J000 Picture Thanks to coventor SMA-HPC⊙2003MT Discretization of Boundary Integral Equations 6

Capacitance of Microprocessor Signal Lines Examples E coyotesystems 」 SMA-HPC⊙2003MT Discretization of Boundary Integral Equations 7

What is common about these problems? Exterior Problems MEMS device fluid (air) creates drag ackage -EXterior fields create coupling Signal Line -EXterior fields Quantities of interest are on surface MEMS device Just want surface traction force Package -Just want coupling between conductors Signal Line- Just want surface charge Exterior problem is linear and space-invariant MEMS device- EXterior Stoke's flow equation (inear) Package -Maxwell's equations in free space(linear) Signal line -Laplace's equation in free spce(inear) But problems are geometrically very complex SMA+HPC⊙2003M Discretization of Boundary Integral Equations 8