Pseudospectral and High-Order Time-Domain Forward Solvers Qing H. Liu G. Zhao, T. Xiao, Y. Zeng Department of Electrical and Computer Engineering Duke University DARPA/ARO MURI Review, August 15, 2003
Limitation of the Conventional Finite-Difference Time- Domain (FDTD) Method for Maxwell s Equations Staggered Grid in the FDTD Method (1D) E E Material 1 Material 2 Ei 2 H i 1 Ei 1 H i Ei H i 1 E i 1 H i 2 E i 2 H i 3 E i 3 H H E and H staggered in Space and Time. Central difference approximation 2 nd -order accurate in homogeneous media 1 st -oder accurate in discontinuous media
A Simple 1D Example to Show the Low-Order Accuracy
Outline Staggered Upwind Embeded Boundary method Pseudospectral time-domain method Spectral element method Summary
Illustration of SUEB Method in 1D Staggered Grids with Embedded Boundary E E Ei 2 H i 1 Ei 1 H i Ei H i 1 E i 1 H i 2 E i 2 H i 3 E i 3 H H At each new time step Regular E, H points are updated by Yee scheme Irregular E,H near the boundary are updated by one-sided difference Match E and H across the boundary by upwind condition
Upwind Conditions P - E - & H - w 1- & w 2 - P E & H w 1 & w 2 Maintain w 1- & w 2 Compute w 2- & w 1 by B.C. (P - ) -1 w 1- & w 2 - E - & H - (P ) -1 w 1 & w 2 E & H W 1 - W 2 - W 1 W 2
Accuracy and Results (1D Case) Dielectric Boundary
PEC Boundary and PMC Boundary
SUEB in 2D
Accuracy and Results (2D Case) Geometry Outgoing Wave PML Layer ZERO Reflection 0.2 m Interior Region
Result Comparison (Source at Center)
n Discontinuity of Hy
Accuracy for Material Boundary and PEC
Outline Staggered Upwind Embeded Boundary method Pseudospectral time-domain method Spectral element method Summary
Pseudospectral Methods Fast methods for solver partial differential equations by approximating spatial derivatives using 1. Fast Fourier transform on a uniform grid (Periodic, smooth case) 2. Lagrange interpolation polynomials at quadrature points (e.g., Chebyshev points)
Example: Chebyshev Polynomial for Derivatives Collocation points: = cos( iπ / N ), i = 1,2,..., N xi -1 1 Spatial derivative: Matrix D can be found in closed form
Patching conditions are imposed between adjacent subdomains
Multidomain Pseudospectral Time-Domain (PSTD) Method Chebyshev polynomials for spatial derivatives 2 nd -order of 4-th order Runge-Kutta method for time integration Physical boundary conditions and upwind conditions are used to patch subdomains Well-posed PML ABC (Fan & Liu, 2000)
Subdomain Patching 2 1 * The field solutions are obtained independently in each subdomain. * To ensure that these solution are correct in the global solution, we apply the patching condions between adjacent subdomains. Both boundary conditions and upwind conditions have been used for the subdomain patching.
Well-Posed Perfectly Matched Layer (PML) in the PSTD Method PML absorbs outgoing waves without reflection PML highly effective for layered media PML Ref: G. Fan and Q. H. Liu, 2000.
f=60 MHz Observation point (1,-1.2) m Rx Sampling density: 3.18 points per minimum wavelength
Outline Staggered Upwind Embeded Boundary method Pseudospectral time-domain method Spectral element method Summary
Spectral Element Method
Equations 3D Maxwell s Eqs with Well-Posed PML ~ ( µ? ) ~ ~ (1) = E AH BH t ~ ( εe) ~ ~ (1) = H CE DE GE t (2 ) J (1) H ~ = H WH t (1) E ~ E WE (1) = t (2) E = E (1) t Here A, B, C, D, G and W are diagonal matrices dependent on PML coefficients and its associated terms construct lower order terms. ~ H = H WH (1) E ~ = E WE (1) ~ ~ H = H E = E. In regular region,, (1)
x E x BH AH E? L d n L d t D i i D = * (1) ] ~ [ ˆ ~ ~ ) ~ (µ x H x J GE DE CE H E L d n L d t D i i D = * (2) (1) ] ~ [ ˆ ~ ~ ) ~ (ε = Y Y n Y n n ] ~ [ ˆ ] ~ [ ˆ ] ~ [ ˆ * H E E = Z Z n Z n n ] ~ [ ˆ ] ~ [ ˆ ] ~ [ ˆ * E H H Flux jump: Simplified Galerkin formulation Special boundary: PEC: Y -> infinity PMC: Z -> infinity
Curvilinear elements have been used for curved geometries
Geometry of a buried UXO Without rock P M L Air, K=1 Ground, K=2 With rock P M L Air, K=1 Ground, K=2 ε /ε K: dielectric constant P M L P M L P M L Rock, K=4 P M L : source (0.12, 0, 0.06) : 21 receivers (0.18:0.03:0.78, 0, 0.06) UXO UXO Max Freq: 2.5G (Hz) Problem Size: 30*15*10 (min wavelengths) P M L P M L
A 3D Spectral Element Solver Air Enclosed by PML 65cm 30cm Rock 60cm UXO 20cm 30 o 14249 tetrahedron 5 th order elements (~4,300,000 degrees of freedom) Surface Mesh
Without rock Movie of Wave Propagation With rock
Difference of E z at receivers in air between with rock and without rock
Summary The FDTD method is first-order accurate for discontinuous media The Staggered Upwind Embedded Boundary (SUEB) improves the FDTD to 2 nd -order accurate The pseudospectral time-domain (PSTD) method provides spectral accuracy, and is flexible in geometry The curvilinear spectral element method is flexible and highly accurate. Easy in mesh generation PSTD has been developed for elastic waves An FDTD method has been developed for poroelastic waves (Biot theory)
Future Directions Hybrid scheme taking advantage of the best features of the SUEB, PSTD and SEM Simulation of rough surfaces and cluttered environments Incorporate the fast solver in the inversion