Parallel Finite Element Methods with Weighted Linear B-Splines

Transkript

1 Parallel Finite Element Methods with Weighted Linear B-Splines K. Höllig, J. Hörner, and M. Pfeil Stuttgart, October 5, 2007 Cooperation partners: Prof. Dr. U. Reif, Dr. J. Wipper Supported by: TLB, HLRS K. Höllig, J. Hörner, and M. Pfeil (IMNG) PFEMLWB Stuttgart, October 5, / 16

2 Ritz Galerkin Method Approximation with finite elements u h = k K c k B k V h H Weak and variational form a(u h, v h ) = λ(v h ) v h V h Q(u h ) = 1 2 a(u h, u h ) λ(v h ) min on V h Linear System GC = F for coefficients g j,k = a(b j, B k ), f j = λ(b j ) K. Höllig, J. Hörner, and M. Pfeil (IMNG) PFEMLWB Stuttgart, October 5, / 16

3 Standard Finite Elements ART Triangulations (A. Fuchs) Element types for triangulations Lagrange Argyris Clough-Tocher Lagrange (3, 0, 10) (5, 1, 21) (3, 1, 12) (degree, smoothness, dimension) (3, 0, 20) K. Höllig, J. Hörner, and M. Pfeil (IMNG) PFEMLWB Stuttgart, October 5, / 16

4 Splines: Some Historic Steps 1947 Curry and Schoenberg (B-Splines) 1959/66 de Casteljau, Bézier (Automated Design and Manufacturing) 1967 Ahlberg, Nilson, and Walsh (first book) 197* de Boor, Schumaker (Numerical Analysis, Approximation) 1979 Dahmen, Michelli (Multivariate Splines) 1980 Oslo algorithm: Cohen, Lyche, and Riesenfeld (Computer Graphics and Image Processing) 1980 Böhm (Computer Aided Geometric Design) Λ splines, β-splines, ν-splines, ω-splines, τ-splines, A-splines, ARMA splines, B-splines, Bernoulli splines, BM-splines, Box-splines, box-splines, cardinal splines, Catmull-Rom splines, D m -splines, Dirichlet splines, discrete splines, E-splines, elliptic splines, exponential Euler splines, exponential box splines, fundamental splines, g-splines, Gibbs-Wilbraham splines, H m,p -splines, harmonic splines, Helix splines, Hermite splines, Hermite-Birkhoff splines, histosplines, hyperbolic splines, Inf-convolution splines, K-splines, L-monosplines, L-splines, Lagrange splines, LB-splines, Legendre splines, Lg splines, M-splines, metaharmonic splines, minimal-energy splines, monosplines, natural splines, NBV-splines, NURBS, ODR splines, PDL g splines, perfect splines, PL g splines, plg splines, polyharmonic splines, Powell-Sabin splines, pseudo splines, Q-splines, K. Höllig, J. Hörner, Schoenberg and M. Pfeil splines, (IMNG) simplex splines, smoothing PFEMLWBsplines, super splines, thin-plate Stuttgart, splines, October 5, triangular / 16

5 Multivariate Splines Multivariate B-splines m b k (x 1,..., x m ) = b kν (x ν ) ν=1 Linear combinations of relevant B-splines f (x) = k K K : k D c k b k K. Höllig, J. Hörner, and M. Pfeil (IMNG) PFEMLWB Stuttgart, October 5, / 16

6 B-Splines as Finite Elements homogeneous boundary condition via weight function stability via extension b k b i w b k b i + e i,j b j j J(i) K. Höllig, J. Hörner, and M. Pfeil (IMNG) PFEMLWB Stuttgart, October 5, / 16

7 Finite Element Basis domain D D h : w h (x) > 0 weighted B-splines w h bk h, k ν Z approximation ( ) ( ) u h = w k bk h c k bk h k k K. Höllig, J. Hörner, and M. Pfeil (IMNG) PFEMLWB Stuttgart, October 5, / 16

8 Numerical Integration Ritz-Galerkin integrals intersection patterns S D h ( a w h bk, h w h bk h ) ( = I w h S, k, k ) w i Poisson bilinear form t i I : rational function of w h S or t h S (automatically generated, cases) K. Höllig, J. Hörner, and M. Pfeil (IMNG) PFEMLWB Stuttgart, October 5, / 16

9 Simplification of Expressions for Ritz-Galerkin Integrals ( w0,0,0 4 2w 2 0,0,0 w1,1,0 2 w 1,1, w 0,0,0 4 w 1,0,0w1,1, (107 similar terms)) (w 0,0,0 w 1,0,0 ) 3 (w 0,0,0 w 1,1,0 ) 2 (w 0,0,0 w 1,0,1 ) 2 (w 0,0,0 w 1,1,1 ) 2 4 substitutions of type w = a/(b c) 17 substitutions of type w = a b 4 substitutions of type w = a/b simplified instruction set 20 w 5 6 w 6 w 7 10 w 8 5 w 9 10 w 10 w 11 5 w w w w 15 2 w 16 8 w w 18 2 w 19 + w w w w 23 + w w 25 K. Höllig, J. Hörner, and M. Pfeil (IMNG) PFEMLWB Stuttgart, October 5, / 16

10 Example: 3-Dimensional Random Domain Total time 48.2s (265s) Matrix Other Solve Problem type: Dirichlet Discretization: 192,100,033 (577 3 ) unknowns, 7 grids Solver: Dynamic-Multigrid-Jacobi (10 smoothings, ω = 0.88) Residual: 5.462E-09 (< 1E-8) Iterations: Machine: NEC-SX8, 8 CPUs (1 Node), 103 GB Memory K. Höllig, J. Hörner, and M. Pfeil (IMNG) PFEMLWB Stuttgart, October 5, / 16

11 Example: Multigrid Levels Grid 0 Grid 1 Grid 2 Grid 3 Grid 4 Grid 5 Grid 6 calls sopt s % hopt s % hopt + s omp % K. Höllig, J. Hörner, and M. Pfeil (IMNG) PFEMLWB Stuttgart, October 5, / 16

12 Example: Time Comparision Total Solve Matrix Total Solve Matrix Total Solve Matrix Scalar Vector (1:20 Scalar) Parallel (1:6 Vector, 1:120 Scalar) K. Höllig, J. Hörner, and M. Pfeil (IMNG) PFEMLWB Stuttgart, October 5, / 16

13 Example: Ftrace-Statistics (1 CPU) PROG UNIT TIME MFLOPS V.RATIO V.LEN total matrix mg solve matrix celltypes cell data cell data mg smooth mg smooth mg smooth K. Höllig, J. Hörner, and M. Pfeil (IMNG) PFEMLWB Stuttgart, October 5, / 16

14 Example: Ftrace-Statistics (8 CPU) PROG UNIT TIME MFLOPS V.RATIO V.LEN total matrix mg solve process matrix mg solve mg s 6 micro micro micro micro micro micro micro micro sum K. Höllig, J. Hörner, and M. Pfeil (IMNG) PFEMLWB Stuttgart, October 5, / 16

15 Advantages of the WEB-Method No grid generation Natural integration in CAD/CAM-systems based on tensor product B-splines Simple implementation and short computing times Approximations of arbitrary order of accuracy by appropriate choice of the degree of the basis functions Low dimensional approximation spaces Exact fulfillment of boundary conditions Well suited for multigrid methods and hierarchical refinement Natural parallelization of algorithms K. Höllig, J. Hörner, and M. Pfeil (IMNG) PFEMLWB Stuttgart, October 5, / 16