Solving integral equations on non-smooth domains

Transcription

1 Solving integral equations on non-smooth domains Johan Helsing Lund University Talk at the IMA Hot Topics workshop, August 2-5, 2010 σ yy A B C σ yy

2 Acknowledgement The work presented in this talk has been supported by, inspired by, or carried out in cooperation with, among others: Michael Benedicks Björn Dahlberg Jonas Englund Leslie Greengard Bertil Gustafsson Robert V. Kohn Graeme Milton Nikoloz Muskhelishvili Rikard Ojala Gunnar Peters the Swedish Research Council the Knut and Alice Wallenberg Foundation

3 Just for fun: Stagnation in the GMRES Fredholm second kind integral equation with compact operator (I + K)x = b.

4 Just for fun: Stagnation in the GMRES Fredholm second kind integral equation with compact operator (I + K)x = b. After discretization (I + K)x = b.

5 Just for fun: Stagnation in the GMRES Fredholm second kind integral equation with compact operator After discretization (I + K)x = b. (I + K)x = b. Solve using GMRES with modified Gram-Schmidt. Feed Arnoldi process with {b,(i + K)q 1,(I + K)q 2,(I + K)q 3,...} to form orthonormal bases for Krylov subspaces. Textbook practice.

6 The result is: 10 0 GMRES for (I+K)x=b, Modified GS 10 5 Internal estimator True residual Orthogonality Relative residual Number of iterations

7 The result is: 10 0 GMRES for (I+K)x=b, Modified GS 10 5 Internal estimator True residual Orthogonality Relative residual Number of iterations Well known behavior. Is it a problem?

8 The result is: 10 0 GMRES for (I+K)x=b, Modified GS 10 5 Internal estimator True residual Orthogonality Relative residual Number of iterations Well known behavior. Is it a problem? If so, many fixes: Replace Gram-Schmidt with Householder, reverse order...

9 Empirical result: Simplest and best fix is to feed Arnoldi with rather than {b,kq 1,Kq 2,Kq 3,...} {b,(i + K)q 1,(I + K)q 2,(I + K)q 3,...}. Same Krylov subspaces. Less loss of precision. Even saves a few FLOPs!

10 The result is: 10 0 GMRES for (I+K)x=b, Modified GS, 2nd kind fix 10 5 Internal estimator True residual Orthogonality Relative residual Number of iterations

11 Comparison: 10 0 GMRES for (I+K)x=b, Modified GS Internal estimator True residual Orthogonality 10 0 GMRES for (I+K)x=b, Modified GS, 2nd kind fix Internal estimator True residual Orthogonality Relative residual Relative residual Number of iterations Number of iterations Advantage with 2nd kind fix Slightly better accuracy Slightly less work Need not worry about stopping criterion threshold Section 8 in JCP 227(5) pp (2008)

12 Interesting problems in planar elasticity Which material has the more interesting stress field? or

13 Interesting problems in planar elasticity Which material has the more interesting stress field? C ty B pr D a E y x 2h β F w or G A ty pr Figure: Smooth: JMPS (1998). Non-smooth: IJNME (2002)

14 Interesting problems in planar elasticity Which material has the more interesting stress field? C ty B pr D a E y x 2h β F w or G A ty pr Figure: Smooth: JMPS (1998). Non-smooth: IJNME (2002) If G-A is clamped and with a crack emanating from E, it is even more interesting: Corners. Mixed conditions. Triple-junctions.

15 Fredholm second kind integral equation with compact operator (I + K)x = b. on a smooth boundary open arcs or closed contours Convergence of stress field 80 GMRES iterations for full convergence (Estimated) relative error Number of iterations Number N of discretization points Number N of discretization points Figure: Convergence with mesh refinement for the elastic field in specimens with smooth interfaces and cracks. Better and better.

16 Essentially the same equation on a non-smooth boundary (I + K)x = b. The operator K is not compact. A fundamental difference Convergence of stress field 80 GMRES iterations for full convergence (Estimated) relative error Number of iterations Number N of discretization points Number N of discretization points Figure: Convergence with mesh refinement for the elastic field in specimens with non-smooth interfaces. First better then worse.

17 It this fundamental difference a problem? 10 0 Convergence of stress field 10 0 Convergence of stress field 80 GMRES iterations for full convergence 80 GMRES iterations for full convergence (Estimated) relative error (Estimated) relative error Number of iterations Number of iterations Number N of discretization points Number N of discretization points Number N of discretization points Number N of discretization points Yes, it is. Aggravated behavior in more complicated settings. Destroys the simulation of damage evolution.

18 Corners-B-gone A numerical method for corners, triple-junctions, quadruple-junctions, mixed problems... (I + K)x = b. (1)

19 Corners-B-gone A numerical method for corners, triple-junctions, quadruple-junctions, mixed problems... (I + K)x = b. (1) Assumptions: K is compact away from a finite number of boundary points γ j. b is piecewise smooth.

20 Corners-B-gone A numerical method for corners, triple-junctions, quadruple-junctions, mixed problems... (I + K)x = b. (1) Assumptions: K is compact away from a finite number of boundary points γ j. b is piecewise smooth. Let K(τ,z) denote the kernel of K. Split K(τ,z) into two functions K(τ,z) = K (τ,z) + K (τ,z), (2) where K (τ,z) is zero except for when τ and z simultaneously lie close to the same γ j. Then K (τ,z) is zero.

21 Corners-B-gone A numerical method for corners, triple-junctions, quadruple-junctions, mixed problems... (I + K)x = b. (1) Assumptions: K is compact away from a finite number of boundary points γ j. b is piecewise smooth. Let K(τ,z) denote the kernel of K. Split K(τ,z) into two functions K(τ,z) = K (τ,z) + K (τ,z), (2) where K (τ,z) is zero except for when τ and z simultaneously lie close to the same γ j. Then K (τ,z) is zero. The kernel split corresponds to an operator split K = K + K. K is a compact operator. After discretization on fine grid (I fin + K fin + K fin )x fin = b fin, (3)

22 GMRES stagnation control Interesting problems in planar elasticity Corners-B-gone Numerical Examples Figure: A coarse mesh with eight quadrature panels on a closed contour. A fine mesh of 14 panels is created by refinement near the corner nz = nz = Figure: Kfin = K fin + K fin nz =

23 Variable substitution (change of basis) x fin = (I fin + K fin ) 1 x fin (4) gives ( I fin + K fin (I fin + K fin ) 1) x fin = b fin. (5)

24 Variable substitution (change of basis) x fin = (I fin + K fin ) 1 x fin (4) gives ( I fin + K fin (I fin + K fin ) 1) x fin = b fin. (5) Here K fin (I fin + K fin ) 1 is a compact operator from a practical viewpoint x fin is piecewise smooth Neither b fin, nor x fin, nor K fin need a fine grid for resolution.

25 Introduce a prolongation operator P from the coarse grid to the fine grid and a restriction operator Q in the other direction QP = I (6)

26 Introduce a prolongation operator P from the coarse grid to the fine grid and a restriction operator Q in the other direction Then QP = I (6) b fin = Pb coa (7) x fin = P x coa (8) K fin = PK coap T W (9) Compare economy-size SVD. Here P W = W fin PW 1 coa and W contains quadrature weights on the diagonal.

27 Introduce a prolongation operator P from the coarse grid to the fine grid and a restriction operator Q in the other direction Then QP = I (6) b fin = Pb coa (7) x fin = P x coa (8) K fin = PK coap T W (9) Compare economy-size SVD. Here P W = W fin PWcoa 1 and W contains quadrature weights on the diagonal. Taken together ( ) I coa + K coap T W (I fin + K fin ) 1 P x coa = b coa. (10)

28 Introducing eq. (10) reads R = P T W (I fin + K fin ) 1 P, (11) (I coa + K coar) x coa = b coa. (12) We only need the fine grid for constructing the small (block) matrix R.

29 Introducing eq. (10) reads R = P T W (I fin + K fin ) 1 P, (11) (I coa + K coar) x coa = b coa. (12) We only need the fine grid for constructing the small (block) matrix R. The blocks of R can be constructed via an extremely fast recursion, i = 1,..., n, where step i inverts and compresses contributions to R involving the outermost quadrature panels on level i of a locally n-ply refined mesh.

30 Introducing eq. (10) reads R = P T W (I fin + K fin ) 1 P, (11) (I coa + K coar) x coa = b coa. (12) We only need the fine grid for constructing the small (block) matrix R. The blocks of R can be constructed via an extremely fast recursion, i = 1,..., n, where step i inverts and compresses contributions to R involving the outermost quadrature panels on level i of a locally n-ply refined mesh. From a practical viewpoint the corner difficulties are gone: JCP 227(20) pp (2008), IJSS 46(25/26) pp (2009), JCP 228(23) pp (2009).

31 The fast recursion R i = P T Wbc ( F{R 1 (i 1) } + I b + K b) 1 Pbc, i = 1,...,n. (13) part of coarse mesh i = 1 i = 2 refined n = 4 i = 3 Figure: Recursion on the refined mesh surrounding a corner.

32 Electrostatics: quadruple-junctions A unit cell with 10,000 squares whose conductivities vary between σ=0.001 and σ=1000. With discretization points, construction of R takes 20 minutes and iterative solution of (12) takes 14 minutes. The estimated relative error in σ eff is O(10 11 ).

33 Elastostatics: triple-junctions A perturbed honeycomb structure with 2475 grains. The coloring is based on the Young s modulus. The operator K is actually singular. Construction of R takes 40% of the total solution time. The estimated relative error in computed quantities is O(10 12 ).

34 Elastostatics: mixed boundary conditions Elasticity: Relative error of pointwise stress (max norm) Im{z} Re{z} 14 Traction and displacement prescribed at different boundary parts. The 10-logarithm of the relative error for σ xx + σ yy at 484,670 points in the computational domain is shown. The maximum pointwise error is The computing time is 24 seconds.

35 Singular integral equations on piecewise smooth curves σ yy A B C σ yy Stress intensity factors are computed for a 176-ply branched crack in an elastic plane. The estimated relative error is O(10 12 ). The computing times is less than 10 minutes.