11/1/2011 1 1 Partial Differential Equations Almost all the elementary and numerous advanced parts of theoretical physics are formulated in terms of differential equations (DE). Newton’s Laws Maxwell equations Schrodinger and Dirac equations etc. 22dtxdmmaF Since the dynamics of many physical systems involve just two derivatives, DE of second order occur most frequently in physics. e.g., the steady state distribution of heat acceleration in classical mechanics 2Ordinary differential equation (ODE) Partial differential equation (PDE) 2 Examples of PDEs • Laplace's eq.: occurs in a. electromagnetic phenomena, b. hydrodynamics, c. heat flow, d. gravitation. • Poisson's eq., In contrast to the homogeneous Laplace eq., Poisson's eq. is non-homogeneous with a source term 220k  2020/    0/The wave (Helmholtz) and time-independent diffusion eqs These eqs. appear in such diverse phenomena as a. elastic waves in solids, b. sound or acoustics, c. electromagnetic waves, d. nuclear reactors.11/1/2011 2 3 The time-dependent wave eq., Subject to initial and boundary conditions Take Fourier Transform Boundary value problem per freq • Helmholtz equation . dtetzyxpwzyxti ),,,('),,,(The time-dependent heat equation Subject to initial and boundary conditions Biharmonic Equation . Numerical Solution of PDEs • Finite Difference Methods – Approximate the action of the operators – Result in a set of sparse matrix vector equations – In 3D a discretization will have Nx× Ny× Nz points – Iterative methods such as multigrid give good performance • Finite Element Methods – Approximate weighted integral of the equation over element – Also Result in a set of sparse matrix vector equations – In 3D a discretization will have Nx× Ny× Nz elements – Iterative methods well advanced • Meshing the domain is a big problem with both – Creating a “good mesh” takes longer than solving problem • Cannot handle infinite domains well 411/1/2011 3 Boundary Element Methods • Very commonly used with FMM • Based on a “Boundary Integral” forumation of PDE • Applicable to equations with known Green’s functions • Lead to a boundary only formulation – O(N2) unknowns as opposed to O(N3) unknowns • Lead to dense matrices – Preconditioning theory not well developed • Setting up equations requires analytical and computational work • Handle infinite domains well • Method of choice in scattering problems (EM, Acoustic), potential flow, Stokes flow, Cracks, etc. 5 Outline • Review • Vector analysis (Divergence & Gradient of potential) • 3-D Cartesian coordinates & Spherical coordinates • Laplace’s equation and Helmholtz’ equation • Green's function & Green's theorem • Boundary element method • FMM11/1/2011 4 Gauss Divergence theorem • In practice we can write  Integral Definitions of div, grad and curl Elemental volume  with surface S τ0ΔSτ0ΔSτ0ΔS1lim dSτ1lim . dSτ1lim dSτ     nD D nD D nn dS D=D(r), = (r)11/1/2011 5 Green’s formula Laplace’s equation11/1/2011 6 Helmholtz equation • • Discretize surface S into triangles • Discretize`11/1/2011 7 Green’s formula • Recall that the impulse-response is sufficient to characterize a linear system • Solution to arbitrary forcing constructed via convolution • For a linear boundary value problem we can likewise use the solution to a delta-function forcing to solve it. • Fluid flow, steady-state heat transfer, gravitational potential, etc. can be expressed in terms of Laplace’s equation • Solution to delta function forcing, without boundaries, is called free-space Green’s function Boundary Element Methods • Boundary conditions provide value of j or qj • Becomes a linear system to solve for the other11/1/2011 8 Accelerate via FMM11/1/2011 9 Example 2: Boundary element method V S n 2111/1/2011 10 Boundary element method(2) Surface discretization: System to solve: Use iterative methods with fast matrix-vector multiplier: Non-FMM’able, but sparse FMM’able, dense Helmholtz equation Performance tests Mesh: 249856 vertices/497664 elements kD=29, Neumann problem kD=144, Robin problem (impedance, sigma=1) (some other scattering problems were solved)11/1/2011 11 FMM & Fluid Mechanics • Basic Equations 1 2 n Helmholtz Decomposition • Key to integral equation and particle methods11/1/2011 12 Potential Flow • Knowledge of the potential is sufficient to compute velocity and pressure • Need a fast solver for the Laplace equation • Applications – panel methods for subsonic flow, water waves, bubble dynamics, … Crum, 1979 Boschitsch et al, 1999 © Gumerov & Duraiswami, 2003 BEM/FMM Solution Laplace’s Equation • Jaswon/Symm (60s) Hess & Smith (70s), • Korsmeyer et al 1993, Epton & Dembart 1998, Boschitsch & Epstein 1999 Lohse, 200211/1/2011 13 Stokes Flow • Green’s function (Ladyzhenskaya 1969, Pozrikidis 1992) • Integral equation formulation • Stokes flow simulations remain a very important area of research • MEMS, bio-fluids, emulsions, etc. • BEM formulations (Tran-Cong & Phan-Thien 1989, Pozrikidis 1992) • FMM (Kropinski 2000 (2D), Power 2000 (3D)) Motion of spermatozoa Cummins et al 1988 Cherax quadricarinatus. MEMS force calculations (Aluru & White, 1998 Rotational Flows and VEM • For rotational flows Vorticity released at boundary layer or trailing edge and advected with the flow  Simulated with vortex particles Especially useful where flow is mostly irrotational Fast calculation of Biot-Savart integrals (x1,y1,z1) (x2,y2,z2) G nlEvaluation point x(where y is the mid point of the filament) (circulation strength) (Far field)11/1/2011 14 Vorticity formulations of NSE • Problems with boundary conditions for this equation (see e.g., Gresho, 1991)  Divergence free and curl-free components are linked only by boundary conditions  Splitting is invalid unless potentials are consistent on boundary • Recently resolved by using the generalized Helmholtz decomposition (Kempka et al, 1997; Ingber & Kempka, 2001) • This