Adaptive boundary element methods in industrial applications

Transcription

1 Adaptive boundary element methods in industrial applications Günther Of, Olaf Steinbach, Wolfgang Wendland Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.1/24

2 Outline Mixed boundary value problems of potential theory Symmetric Galerkin boundary integral equation formulation Realization of the boundary integral operators by the Fast Multipole Method Fast Multipole Method for Gadaptive meshes Preconditioning of the matrices of the hypersingular operator and the single layer potential Example: spray painting Evaluation of representation formula on demand Experimental setup and errors Error estimator Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.2/24

3 Mixed boundary value problem Laplace equation: u(x) = 0 for x Ω IR 3, u(x) = gd(x) for x ΓD, t(x) := (Txu)(x) = ( nu)(x) = gn (x) for x ΓN. Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.3/24

4 Mixed boundary value problem Laplace equation: u(x) = 0 for x Ω IR 3, u(x) = g D (x) for x Γ D, t(x) := (T x u)(x) = ( n u)(x) = g N (x) for x Γ N. Representation formula: u(x) = Γ [U (x, y)] t(y)ds y Γ [T y U (x, y)] u(y)ds y for x Ω. Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.3/24

5 Mixed boundary value problem Laplace equation: u(x) = 0 for x Ω IR 3, u(x) = g D (x) for x Γ D, t(x) := (T x u)(x) = ( n u)(x) = g N (x) for x Γ N. Representation formula: u(x) = Γ [U (x, y)] t(y)ds y Calderon projector for the Cauchy data u(x) and t(x): ( ) ( ) ( ) u 1 = 2 I K V u 1 t D 2 I + K t Boundary integral operators: Γ Γ [T y U (x, y)] u(y)ds y for x Ω. on Γ (V t)(x) = [U (x, y)] t(y)ds y, (Du)(x) = T x [T y U (x, y)] u(y)ds y, Γ Γ (Ku)(x) = [T y U (x, y)] u(y)ds y, (K t)(x) = [T x U (x, y)] t(y)ds y. Γ Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.3/24

6 Symmetric boundary integral formulation Symmetric boundary integral formulation (Sirtori 79, Costabel 87): (V t )(x) (Kũ)(x) = ( 1 2 I + K) g D(x) (V g N )(x) for x Γ D, (K t )(x) + (Dũ)(x) = ( 1 2 I K ) g N (x) (D g D )(x) for x Γ N. Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.4/24

7 Symmetric boundary integral formulation Symmetric boundary integral formulation (Sirtori 79, Costabel 87): (V t )(x) (Kũ)(x) = ( 1 2 I + K) g D(x) (V g N )(x) for x Γ D, (K t )(x) + (Dũ)(x) = ( 1 2 I K ) g N (x) (D g D )(x) for x Γ N. Galerkin discretization with piecewise constant (ϕ l ) and piecewise linear (ψ i ) trial and test functions leads to a system of linear equations: ( V h K h K h D h ) ( t h ũ h ) = ( ) f N f D Single Galerkin blocks for k, l = 1,..., m and i, j = 1,..., m V h [l, k] = V ϕ k, ϕ l ΓD, K h [l, i] = Kψ i, ϕ l ΓD, K h[j, k] = K ϕ k, ψ j ΓN, D h [j, i] = Dψ i, ψ j ΓN.. Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.4/24

8 Potential and Fast Multipole Method Double layer potential: Single layer potential: (Ku)(x) = 1 4π (V t)(x) = 1 4π Γ 1 n y x y u(y)ds y for x Γ Γ 1 x y t(y)ds y for x Γ Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.5/24

9 Potential and Fast Multipole Method Double layer potential: Single layer potential: (Ku)(x) = 1 4π (V t)(x) = 1 4π In the farfield numerical quadrature: 1 4π N k=1 τ k t(y) Γ 1 n y x y u(y)ds y for x Γ Γ 1 x y ds y 1 4π 1 x y t(y)ds y for x Γ N N g k=1 s=1 k ω k,s t(y k,s ) }{{} =q k,s 1 x y k,s. Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.5/24

10 Potential and Fast Multipole Method Double layer potential: Single layer potential: (Ku)(x) = 1 4π (V t)(x) = 1 4π In the farfield numerical quadrature: 1 4π N k=1 τ k t(y) Γ 1 n y x y u(y)ds y for x Γ Γ 1 x y ds y 1 4π 1 x y t(y)ds y for x Γ N N g k=1 s=1 k ω k,s t(y k,s ) }{{} =q k,s Fast Multipole Method: [ Rohklin 1984; Greengard, Rohklin 1987;...] Fast evaluation of potentials in many particle systems: Φ(x j ) = N i=1 q i x j y i for j = 1,..., M. 1 x y k,s. Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.5/24

11 Idea of the Fast Multipole Method Starting point: evaluation of sums in many points x j N Φ(x j ) = q i k(x j, y i ) j = 1,..., M. i=1 effort: O(N M); for N = M : O(N 2 ) Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.6/24

12 Idea of the Fast Multipole Method Starting point: evaluation of sums in many points x j N Φ(x j ) = q i k(x j, y i ) j = 1,..., M. i=1 effort: O(N M); for N = M : O(N 2 ) Idea: splitting of the function k(x,y) k(x, y) = f(x)g(y) Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.6/24

13 Idea of the Fast Multipole Method Starting point: evaluation of sums in many points x j N Φ(x j ) = q i k(x j, y i ) j = 1,..., M. i=1 effort: O(N M); for N = M : O(N 2 ) Idea: splitting of the function k(x,y) Effect on the sums: reduced effort: O(N + M). k(x, y) = f(x)g(y) N Φ(x j ) = f(x j ) q i g(y i ) j = 1,..., M. i=1 Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.6/24

14 Realization of the Fast Multipole Method Actual realization in the Fast Multipole Method: k(x, y) = 1 p x y with spherical harmonics for m 0 Y ±m n (ˆx) = n n=0 m= n (n m)! dm ( 1)m (n + m)! dˆx m 3 x n Yn m (ˆx) Y n m (ŷ) y n+1 P n (ˆx 3 )(ˆx 1 ± iˆx 2 ) m. Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.7/24

15 Realization of the Fast Multipole Method Actual realization in the Fast Multipole Method: k(x, y) = 1 p x y with spherical harmonics for m 0 Y ±m n (ˆx) = n n=0 m= n (n m)! dm ( 1)m (n + m)! dˆx m 3 finite expansion degree = approximation; x n Yn m (ˆx) Y n m (ŷ) y n+1 P n (ˆx 3 )(ˆx 1 ± iˆx 2 ) m. Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.7/24

16 Realization of the Fast Multipole Method Actual realization in the Fast Multipole Method: k(x, y) = 1 p x y with spherical harmonics for m 0 Y ±m n (ˆx) = n n=0 m= n (n m)! dm ( 1)m (n + m)! dˆx m 3 finite expansion degree = approximation; convergence of the expansion only for x y < 1 d x n Yn m (ˆx) Y n m (ŷ) y n+1 P n (ˆx 3 )(ˆx 1 ± iˆx 2 ) m. with d > 1. Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.7/24

17 Realization of the Fast Multipole Method Actual realization in the Fast Multipole Method: k(x, y) = 1 p x y with spherical harmonics for m 0 Y ±m n (ˆx) = n n=0 m= n (n m)! dm ( 1)m (n + m)! dˆx m 3 finite expansion degree = approximation; convergence of the expansion only for x y < 1 d Effect on the sums: Φ(x j ) 1,N i NF (j) q i k(x j, y i ) + p n n=0 m= n x n Yn m (ˆx) Y n m (ŷ) y n+1 P n (ˆx 3 )(ˆx 1 ± iˆx 2 ) m. with d > 1. x j n Y m n (ˆx j ) 1,N Yn m (ŷ i ) q i y i n+1 i F F (j) }{{} =L m n. Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.7/24

18 Realization of the Fast Multipole Method Multipol expansion for x > y j k i=1 q i p x y i n k n=0 m= n i=1 q i y i n Yn m (ŷ i ) Y n m (ˆx) x n+1 x Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.8/24

19 Realization of the Fast Multipole Method Multipol expansion for x > y j k i=1 q i p x y i n k n=0 m= n i=1 and local expansion for x < y i k i=1 q i p x y i n k n=0 m= n i=1 q i y i n Yn m (ŷ i ) Y n m (ˆx) x n+1 q i Y m n (ŷ i ) y i n+1 Y m n (ˆx) x n x x Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.8/24

20 Realization of the Fast Multipole Method Multipol expansion for x > y j k i=1 q i p x y i n k n=0 m= n i=1 and local expansion for x < y i k i=1 q i p x y i n k n=0 m= n i=1 farfield depends on the evaluation point x j ; different farfield sums. q i y i n Yn m (ŷ i ) Y n m (ˆx) x n+1 q i Y m n (ŷ i ) y i n+1 Y m n (ˆx) x n x x Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.8/24

21 Realization of the Fast Multipole Method Multipol expansion for x > y j k i=1 q i p x y i n k n=0 m= n i=1 and local expansion for x < y i k i=1 q i p x y i n k n=0 m= n i=1 farfield depends on the evaluation point x j ; different farfield sums. q i y i n Yn m (ŷ i ) Y n m (ˆx) x n+1 q i Y m n (ŷ i ) y i n+1 Y m n (ˆx) x n y i multipole expansion x x Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.8/24

22 Realization of the Fast Multipole Method Multipol expansion for x > y j k i=1 q i p x y i n k n=0 m= n i=1 and local expansion for x < y i k i=1 q i p x y i n k n=0 m= n i=1 farfield depends on the evaluation point x j ; different farfield sums. x j local expansion q i y i n Yn m (ŷ i ) Y n m (ˆx) x n+1 q i Y m n (ŷ i ) y i n+1 Y m n (ˆx) x n y i multipole expansion x j local expansion x x Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.8/24

23 Realization of the Fast Multipole Method Multipol expansion for x > y j k i=1 q i p x y i n k n=0 m= n i=1 and local expansion for x < y i k i=1 q i p x y i n k n=0 m= n i=1 farfield depends on the evaluation point x j ; different farfield sums. x j local expansion q i y i n Yn m (ŷ i ) Y n m (ˆx) x n+1 q i Y m n (ŷ i ) y i n+1 Y m n (ˆx) x n y i multipole expansion x j local expansion efficient computation by the use of a hierarchical structure = complexity (time and memory) with error control: O(N log 2 N) for N = M. x x Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.8/24

24 Adaptive version of the Fast multipole method Adaptive versions already exist: Cheng, Greengard, Rokhlin; Nabors, Korsmeyer, Leighton, White. Why necessary? Otherwise huge nearfield or huge tree (many expansions). Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.9/24

25 Adaptive version of the Fast multipole method Adaptive versions already exist: Cheng, Greengard, Rokhlin; Nabors, Korsmeyer, Leighton, White. Why necessary? Otherwise huge nearfield or huge tree (many expansions). Aim: Maintain the the symmetry of the matrices and error control in the Fast Multipole Method. Lemma. The Fast Multipole Method is symmetric for two points P 1 and P 2 with unit charge using a finite expansion degree, if the paths of the transformations of the expansions agree with each other except of the direction. Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.9/24

26 Adaptive version of the Fast multipole method Adaptive versions already exist: Cheng, Greengard, Rokhlin; Nabors, Korsmeyer, Leighton, White. Why necessary? Otherwise huge nearfield or huge tree (many expansions). Aim: Maintain the the symmetry of the matrices and error control in the Fast Multipole Method. Lemma. The Fast Multipole Method is symmetric for two points P 1 and P 2 with unit charge using a finite expansion degree, if the paths of the transformations of the expansions agree with each other except of the direction. = Condition for the cluster tree: symmetry of the nearfields. Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.9/24

27 Construction of the adaptive cluster tree Symmetry of the nearfields and distance control. Example: Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.10/24

28 Construction of the adaptive cluster tree Symmetry of the nearfields and distance control. Example: Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.10/24

29 Construction of the adaptive cluster tree Symmetry of the nearfields and distance control. Example: Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.10/24

30 Construction of the adaptive cluster tree Symmetry of the nearfields and distance control. Example: Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.10/24

31 Construction of the adaptive cluster tree Symmetry of the nearfields and distance control. Example: Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.10/24

32 Construction of the adaptive cluster tree Symmetry of the nearfields and distance control. Example: Postprocessing of the nearfields on son levels possible. Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.10/24

33 Example of an adaptive grid Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.11/24

34 Example of an adaptive Fast Multipole Method level # elements setup solving nearfield u(x ) u h (x ) % 1.676e % 1,677e % 4.101e % 4.103e % 1.025e % 1.021e % 2.699e % 2.674e % 6.414e % 6.561e % % 1.610e-6 Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.12/24

35 Preconditioning of hypersingular operator using boundary integral operators of opposite order: [Steinbach, Wendland 95,98; McLean, Steinbach 99; Of, Steinbach 03] Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.13/24

36 Preconditioning of hypersingular operator using boundary integral operators of opposite order: [Steinbach, Wendland 95,98; McLean, Steinbach 99; Of, Steinbach 03] for the Schur complement system and the hypersingular operator: c V 1 c D 1 V 1 v, v Γ 1 + c Dv, v Γ R single layer potential with piecewise linear basis functions 1 4(1 c R ) V 1 v, v Γ. Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.13/24

37 Preconditioning of hypersingular operator using boundary integral operators of opposite order: [Steinbach, Wendland 95,98; McLean, Steinbach 99; Of, Steinbach 03] for the Schur complement system and the hypersingular operator: c V 1 c D 1 V 1 v, v Γ 1 + c Dv, v Γ R single layer potential with piecewise linear basis functions spectral equivalent preconditioning matrix C ed for D h : 1 4(1 c R ) V 1 v, v Γ. C ed [j, i] = V 1 ψ i, ψ j Γ for i, j = 1,..., M. Approximation of the preconditioning matrix C ed C ed = Mh T V 1 h M h, C 1 ed = M 1 h V hm T h with V h [j, i] = V ψ i, ψ j Γ, M h [j, i] = ψ i, ψ j Γ. Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.13/24

38 Artificial Multilevel Preconditioner [Steinbach 2003] Sequence of nested boundary element spaces (globally quasiuniform) Z 0 Z 1... Z J = Z h H 1/2 (Γ) Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.14/24

39 Artificial Multilevel Preconditioner [Steinbach 2003] Sequence of nested boundary element spaces (globally quasiuniform) L 2 projection: Z 0 Z 1... Z J = Z h H 1/2 (Γ) Q j w Z j : Q j w, v j Γ = w, v j Γ Multilevel operator: [Bramble, Pasciak, Xu 1990] A s w := J j=0 h 2s j (Q j Q j 1 ) w for all v j Z j Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.14/24

40 Artificial Multilevel Preconditioner [Steinbach 2003] Sequence of nested boundary element spaces (globally quasiuniform) L 2 projection: Z 0 Z 1... Z J = Z h H 1/2 (Γ) Q j w Z j : Q j w, v j Γ = w, v j Γ Multilevel operator: [Bramble, Pasciak, Xu 1990] A s w := J j=0 h 2s j Spectral equivalence inequalities [Oswald 1998] (Q j Q j 1 ) w for all v j Z j c 1 w 2 H 1/2 (Γ) A 1/2 w, w Γ c 2 J 2 w 2 H 1/2 (Γ) for all w Z J. Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.14/24

41 Artificial Multilevel Preconditioner [Steinbach 2003] Sequence of nested boundary element spaces (globally quasiuniform) L 2 projection: Z 0 Z 1... Z J = Z h H 1/2 (Γ) Q j w Z j : Q j w, v j Γ = w, v j Γ Multilevel operator: [Bramble, Pasciak, Xu 1990] A s w := J j=0 h 2s j Spectral equivalence inequalities [Oswald 1998] (Q j Q j 1 ) w for all v j Z j c 1 w 2 H 1/2 (Γ) A 1/2 w, w Γ c 2 J 2 w 2 H 1/2 (Γ) for all w Z J. The sequence of nested boundary element spaces is build from the clustering of the boundary elements by a hierarchical structure as used in the Fast BEM. Extendable to adaptive grids. Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.14/24

42 Institut fu r Angewandte Analysis und Numerische Simulation Universita t Stuttgart Simulation of spray painting (with R. Sonnenschein, Daimler Chrysler, Dornier) boundary elements mesh ratio hmax hmin 1454, 5 thickness of the wall: 0.8 mm evaluation points size of the wall: about 1 m 150 iteration steps flux: Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.15/24

43 Field evaluation in points or better interactive on demand. = Fast Multipole Method Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.16/24

44 Evaluation on demand Evaluation of the representation formula at many points one by one. exact evaluation instead of interpolation on a grid (FEM) no grid to construct (FEM) build cluster tree without information on the evaluation points build separate trees for the geometry and the evaluation. new admissibility condition: number of nearfield panels limited extra feature: prediction of the point of intersection of the boundary and a straight line (point and direction (electric field)) Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.17/24

45 Evaluation on demand Evaluation of the representation formula at many points one by one. exact evaluation instead of interpolation on a grid (FEM) no grid to construct (FEM) build cluster tree without information on the evaluation points build separate trees for the geometry and the evaluation. new admissibility condition: number of nearfield panels limited extra feature: prediction of the point of intersection of the boundary and a straight line (point and direction (electric field)) example: evaluation points on demand at once 254 sec 333 sec sec Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.17/24

46 Experimental setup (with R. Sonnenschein, Daimler Chrysler, Dornier) boundary elements evaluation points 88 iteration steps (1 refined 92 iteration steps) Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.18/24

47 Isolines of the potential Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.19/24

48 Isolines of the potential Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.20/24

49 Adaptive meshes triangles triangles triangles Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.21/24

50 Cauchy data and representation formula unrefined mesh potential u e h distance x 10 6 uniform refined mesh potential u e h 780 once adaptively refined mesh potential u e h distance x 10 6 twice adaptively refined mesh potential u e h distance x distance x 10 6 Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.22/24

51 A posteriori error estimator Neumann problem: u(x) = 0 for x Ω IR 3, t(x) := (T x u)(x) = ( n u)(x) = g(x) for x Γ. compatibility condition: Γ g(x) 1 ds x = 0 Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.23/24

52 A posteriori error estimator Neumann problem: u(x) = 0 for x Ω IR 3, t(x) := (T x u)(x) = ( n u)(x) = g(x) for x Γ. compatibility condition: Γ g(x) 1 ds x = 0 Define ũ for a discrete solution u h : ũ(x) = V g(y) + ((1 σ(x))i K)u h (x) for x Γ. Lemma (Schulz,Steinbach). The error u u h of the approximation u h is a solution of the boundary integral equation (σ(x)i + K)(u u h )(x) = (ũ u h )(x) for x Γ. A simple error estimator e h : c K ũ u h D u u h D 1 1 c K ũ u h D Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.23/24

53 Current and future work domain decomposition methods for spray painting geometry = parallel solvers Boundary Element Tearing and Interconnecting methods (BETI) automatic generation of domain decompositions adaptive meshes based on the error estimator industrial applications... Adaptive Fast Boundary Element Methods in Industrial Applications, Hirschegg, September 30th, 2004 p.24/24