Exponentially Convergent Sparse Discretizations and Application to - PowerPoint PPT Presentation

Exponentially Convergent Sparse Discretizations and Application to Near Surface Geophysics Murthy N. Guddati North Carolina State University November 9, 2017

Outline Part 1: Impedance Preserving Discretization  Part 2: Absorbing Boundary Conditions (1-sided DtN map)  Joint work with Tassoulas, Druksin, Lim, Zahid, Savadatti,  Thirunavukkarasu Part 3: Complex-length FEM for finite domains (2-sided DtN map)  Joint work with Druskin, Vaziri Astaneh  Part 4: Inversion for Near-surface Geophysics  Joint work with Vaziri Astaneh  2

Part1. Impedance Preserving Discretization

Model Problem     2 2 2 2 u u u 1 u      3D wave equation in free space  0     2 2 2 2 2 x y z c t     ik y ik z i t Fourier transform in t, y, z, with u Ue y z      2 2 U        2 2 2 k U 0   k k k is complex valued  2 y z 2   x c    Exact solution:  ikx ikx U Ae Be , where is the horizontal wavenumber k     . is a plane/evanescent wave    i kx k y k z ( t )  ikx U e u e y z 4

Finite Element Solution on a Uniform Grid in x  2 U FE discretization of:     2 k U 0  2 x Element contribution matrix with uniform element size of h :    2 2 1 1 k h   1 1     A           h 3 1 1 A B 1 3 6      2 k   k h     elem         h 1 1 1 1 B A 2 2 1 k h      B 1     6 3   h 6 Assembly results in the difference equation:     BU 2 AU BU 0   j 1 j j 1 5

Changing Mesh Size: Reflections A simple analysis using two uniform meshes with  different element sizes (h, H), but the same material What happens when a right propagating wave hits the interface?  Exact solution – just passes through  Finite element solution – reflections due to impedance mismatch   Z : discrete impedance of left domain Z Z  h H h R  Z Z Z : discrete impedance of right domain H H h 6

Computing Discrete Impedance (Half-space Stiffness) Basic idea: discrete half-space + finite element = discrete half-space               A B U Z U A Z B U 0       0 h 0    h 0      2 2 2     A Z B   h             B A Z U 0 B A Z U 0 h 1 h 1   2 2 1 1 k h     A   2   kh h 3      2 2 Z A B ik 1 h   2 2 1 2 1 k h      B 1   h 6 Error term . depends on element size, resulting impedance mismatch when  Z h the element size changes, resulting in reflections 7

Optimal Integration for Minimizing Reflection Error Minimize the error in impedance by using generalized integration rules              2 1 1    2 2    A 1 k h   2 kh   h  4        2 2 2 Z A B ik 1 h 4       2 1 1     2 2    B 1 k h Error term   h  4    Minimize the error term by choosing 0  The error in impedance is completely eliminated! No more reflections  Formally valid for more general 2 nd order equations (anisotropic, visco-  elasticity etc., electromagnetics etc. – G, 2006, CMAME) Linear elements + midpoint integration = Impedance Preserving Discretization 8

Part 2. Absorbing Boundary Conditions Perfectly Matched Discrete Layers

Perfectly Matched Discrete Layers …Impedance Preserving Discretization of PML Perfectly Matched Layers (PML) (Berenger, 1994; Chew et.al. 1995)  Step 1: Bend the domain into complex space  Reduced reflection into the interior e   2 ikL R PML Interior (real x) PML Region (imaginary or complex x) Step 2: discretize PMDL domain (in complex space)  Impedance is no longer preserved; perfect matching is destroyed  Requires a large number of carefully chosen PML layers  Impedance preserving discretization comes to the rescue!  Impedance is preserved/matched, irrespective of element length, small, large,  real, complex – Perfectly Matching Discrete Layers (PMDL) 2     Discretize with 3-5 complex-length linear finite elements j nlayer  1 ikL / 2     j R    PMDL No discretization error, but truncation causes  1 ikL / 2    j 1 j reflections. The reflection coefficient is derived as 10

PMDL vs PML: Effectiveness of Midpoint Integration PML with 3 layers PMDL with 3 layers 11

PMDL: Some More Old Results Impedance preservation property is valid for any equation that is linear  and second order in space (G, CMAME, 2006) Elastic and other complicated wave equations (G, Lim & Zahid, 2007)  PMDL with 5 layers PML with 5 layers Evanescent waves can be treated effectively  Padded PMDL – contains large real lengths with midpoint integration  (Zahid & G, CMAME, 2006) 12

Salient Features of PMDL 2     j nlayer k k  Exponential convergence    j  R    k k    j 1 j Near optimal discretization  Optimal: need staggered grids (with Druskin et al., 2003)  Links PML to rational ABCs  Lindman, Engquist-Majda, Higdon and variants (e.g. CRBC)  We started this from E-M/Higdon ABCs (G, Tassoulas, 2000)  Extensions to corners is straightforward  Additional advantage: Provides solutions to some difficult cases  Backpropagating waves: anisotropy  PML for discrete/periodic media  13

PMDL for Backpropagating Waves Opposing signs of phase and group velocities Backpropagating waves grow in the PML region  PML cannot work! (Bécache, Fauqueux and Joly,  2003) Reduced Reflection into the interior PML result: radiation in anisotropic media PML Region Interior A counter-intuitive idea: make the reflections in  PML region decay faster than the growth of the incident wave Works only with PMDL:  needs impedance preserving discretization! Result from PMDL after the fix Savadatti & G (2012), J Comp. Phys. 14

Anisotropic elasticity – Tilted Elliptic Case Arbitrary parameters Ideal Slowness Stable parameters Savadatti & G (2012), J Comp. Phys. 15

Anisotropic elasticity – Non-elliptic Case Traditional mesh Two different coordinated materials Savadatti & G (2012), J Comp. Phys. 16

PMDL for Periodic Media (after discretization) Periodic media has internal reflections and  transmissions Constructive interference leads to long-range  propagation PML’s complex stretching spoils this balance and  internal reflections and transmissions get mixed up! PML for Lattice Waves: 7% reflections w/20 PML Basic Ideas (Discrete/Periodic PMDL):  layers Periodic media = Discrete vector wave equation  (vector size = ndof in a cell) Discrete vector equation = impedance preserving  discretization of more complicated wave equation Apply PMDL on the complicated wave equation results  in impedance matching for periodic media Discrete PMDL: less than Open problem: stability for complex problems 1% error w/ 4 PMDL layers  G & Thirunavukkarasu, JCP (2009), Waves 2011 17

Part 3. Two-Sided DtN Map Complex-length Finite Element Method

Facilitating the Approximation of 2-Sided DtN Map  2 u Consider the equation:       2 u 0, z (0, ) L  2 z   A B Exact 2-sided DtN map:  K =   exact   B A By definition, exact DtN Map is impedance preserving:    2 2 2 A B Z exact Consider impedance preserving discretization of the interval:    A B   2 2 2   K = , A B Z exact exact   B A Error in A and B would be similar since:      2 2 2 2 2 A B Z A B exact Approximating two-sided map reduces to approximating one-sided map  Better derivation based on Crank-Nicolson discretization of the propagator  19

1D Helmholtz Equation  2 u     2 u 0  2 z z  0 exp(- i  L ) exp(+ i  L ) 1 st Order Form    v u / z           z L u 0 i i u    i        Eigenvalues           z v i / i 0 v i / Downwardwaves: z 20