Die Multipol Randelementmethode in industriellen Anwendungen

Transcription

1 Fast Boundary Element Methods in Industrial Applications Söllerhaus, Oktober 2003 Die Multipol Randelementmethode in industriellen Anwendungen G. Of, O. Steinbach, W. L. Wendland Institut für Angewandte Analysis und Numerische Simulation, Universität Stuttgart Outline: 1. Mixed boundary value problem of potential theory and linear elastostatics 2. Galerkin boundary integral equation formulation 3. Realization of the boundary integral operators by the Fast Multipole Method 4. Numerical examples and industrial applications page 1

2 Mixed boundary value problem Laplace operator: u(x) = 0 for x Ω, u(x) = g D (x) for x D, t(x) := (T x u)(x) = ( n u)(x) = g N (x) for x N. linear elastostatics: div (σ(u)) = 0 for x Ω IR 3, u i (x) = g D,i (x) for x D,i, i = 1,..., 3, t i (x) := (T x u) i (x) = (σ(u)n(x)) i = g N,i (x) for x N,i, i = 1,..., 3. The stress tensor σ(u) is related to the strain tensor e(u) by Hooke s law ( Eν σ(u) = tr e(u)i + E ) (1 + ν)(1 2ν) (1 + ν) e(u). E is the Young modulus and ν (0, 1 2 ) is the Poisson ratio. The strain tensor is defined by e(u) = 1 2 ( u + u). page 2

3 Representation formula: u(x) = [U (x, y)] t(y)ds y Boundary integral formulation [T y U (x, y)] u(y)ds y for x Ω. Calderon projector for the Cauchy data u(x) and t(x) on the boundary : u = 1 2 I K V u 1 t D 2 I + on K t Boundary integral operators: (V t)(x) = [U (x, y)] t(y)ds y, (Ku)(x) = [Ty U (x, y)] u(y)ds y, (K t)(x) = [T x U (x, y)] t(y)ds y, (Du)(x) = T x [Ty U (x, y)] u(y)ds y. Fundamental solution: U (x y) = 1 4π x y, U kl(x y) = 1 + ν 8πE(1 ν) [ δ kl (3 4ν) x y + (x ] k y k )(x l y l ) x y 3. page 3

4 Symmetric variational 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 D, (K t)(x) + (Dũ)(x) = ( 1 2 I K ) g N (x) (D g D )(x) for x N. Galerkin discretization with piecewise constant (ϕ l ) and piecewise linear (ψ i ) ansatz and test functions leads to a system of linear equations: V h K h t h = f N. K h Single Galerkin blocks for k, l = 1,..., m and i, j = 1,..., m V h [l, k] = V ϕ k, ϕ l L2 ( D ), K h [l, i] = Kψ i, ϕ l L2 ( D ), K h[j, k] = K ϕ k, ψ j L2 ( N ), D h [j, i] = Dψ i, ψ j L2 ( N ). D h ũ h f D page 4

5 Symmetric realization of the single layer potential Observation: The fundamental solution (Ukl ) l,k=1..3 of linear elastostatics can be written as [ Ukl(x 1 + ν 1 δ kl y) = (3 4ν) 2E(1 ν) 4π x y 1 2 x 1 l x k x y 1 2 y 1 l y k x y 1 2 x 1 k x l x y 1 ] 2 y 1 k. y l x y Lemma 1. For x y, the single layer potential V E can be written as ( V E t ) [ (1 + ν) (x) = (3 4ν) ( V ) 1 3 ( ) (V t k k (x) x l + x ) k t l (x) 2E(1 ν) 2 x k x l l=1 1 3 y l t l (y) 1 2 y k 4π x y ds y 1 ] 3 y k t l (y) 1 2 y l 4π x y ds y. l=1 This form guarantees the symmetry of the farfield part of the Galerkin matrix. l=1 page 5

6 Double layer potential Theorem 1 (Kupradze, 1979). The double layer potential K E of linear elastostatics can be written as ( K E u ) (x) = 1 u(y) 1 4π n y x y ds y 1 1 4π x y (Mu) (y)ds y +2µ ( V E (Mu) ) (x), with µ = E 2(1 + ν), M = 0 n 2 1 n 1 2 n 3 1 n 1 3 n 1 2 n n 3 2 n 2 3 ; n 3 1 n 3 1 n 2 3 n ( K E u ) (x) = ( K u ) (x) ( V (Mu) ) (x) + 2µ ( V E (Mu) ) (x) The bilinear form of adjoint double layer potential is realized accordingly. page 6

7 Hypersingular operator Theorem 2 (Houde Han (1994), Kupradze (1979)). The bilinear form of the hypersingular operator D E in linear elastostatics can be written in the form ( 3 ) D E µ 1 u, v L2 () = (M k+2,k+1 v) (x) (M k+2,k+1 u) (y) ds y ds x 4π x y k=1 ( ) + (Mv) µ I (x) 2π x y 4µ2 U (x, y) (Mu) (y)ds y ds x 3 +µ (M k,j v i ) (x) 1 1 4π x y (M k,iu j ) (y)ds y ds x. i,j,k=1 All the boundary integral operators in linear elastostatics respectively their bilinear forms are reduced to those of the Laplacian. Together with integration by parts for bilinear form of the hypersingular operator of the Laplacian, it is sufficient to deal with the single and double layer potential of the Laplacian. page 7

8 Fast Multipole Method for potential theory (V t)(x) = 1 4π In the farfield computation by numerical integration: 1 4π N 1 t(y) τ k x y ds y 1 4π k=1 multipole expansion for x > y M q j x y j Φ p(x) = j=1 and local expansion for x < y M j=1 q j x y j Φ p(x) = with spherical harmonics for m 0 Y ±m n (ˆx) = 1 x y t(y)ds y for x p N N g k=1 i=1 n M n=0 m= n j=1 p n M n=0 m= n j=1 (n m)! dm ( 1)m (n + m)! dˆx m 3 k ω k,i t(y k,i ) } {{ } =q k,i 1 x y k,i. q j y j n Yn m (ŷ j ) Y n m (ˆx) x n+1 q j Y m n (ŷ j ) y j n+1 Y m n (ˆx) x n P n (ˆx 3 )(ˆx 1 ± iˆx 2 ) m. page 8

9 Post model Dividing in nearfield and farfield by a hierarchical structure. page 9

10 Properties of the FMM approximation With an appropriate choice of the parameters of the multipole approximation: positive definiteness of the approximated matrices Ṽh and D h same (optimal) convergence rate as in the standard approach All the boundary integral operators in linear elastostatics are reduced to those of the Laplacian: Single and double layer potential are sufficient. regularization of boundary integral operators less effort on integration symmetric implementation of boundary integral operators Fast boundary element method with O(N log 2 N) demand of time and memory, applicable to complex problems of industrial interest preconditioning by boundary integral operators and using hierarchical strategies page 10

11 Example: foam (H. Andra, ITWM Kaiserslautern) N generation solving It h 7.3 h 246 Gu nther Of Institut fu r Angewandte Analysis and Numerische Simulation SFB 404, Universita t Stuttgart page 11

12 Example: part of a machine (W. Volk, M. Wagner, S. Wittig, BMW) N generation solving It min 2.35 h 342 Gu nther Of Institut fu r Angewandte Analysis and Numerische Simulation SFB 404, Universita t Stuttgart page 12

13 Example: Capacitance (M. Kaltenbacher, Universität Erlangen) minimal distance distance between the fingers: size of elements number of elements page 13

14 Example: spraying (R. Sonnenschein, Daimler Chrysler, Dornier) boundary elements. page 14

15 Needles and adaptivity mesh ratio 1454,5. Gu nther Of Institut fu r Angewandte Analysis and Numerische Simulation SFB 404, Universita t Stuttgart page 15

16 Field evaluation in points or better interactive on demand. = Fast Multipole Methode Gu nther Of Institut fu r Angewandte Analysis and Numerische Simulation SFB 404, Universita t Stuttgart page 16

17 Thickness of the wall 0.8 mm, size of the wall about 1 m. data range: Gu nther Of Institut fu r Angewandte Analysis and Numerische Simulation SFB 404, Universita t Stuttgart page 17

18 create domain decomposition Domain Decomposition automatic domain decomposition page 18

19 automatic domain decomposition preconditioners BETI parallel solvers Domain Decomposition page 19