Finite Element Methods with Linear B-Splines

Transcription

1 Finite Element Methods with Linear B-Splines K. Höllig, J. Hörner and M. Pfeil Universität Stuttgart Institut für Mathematische Methoden in den Ingenieurwissenschaften, Numerik und geometrische Modellierung (IMNG) FEM / B-Spline Web-Site (with U. Reif and J. Wipper): Avignon, July 5, 2006 K. Höllig et al. (Universität Stuttgart) FE Methods with Linear B-Splines Avignon, July 5, / 13

2 Principal Features uniform grid one basis function per grid point explicit vectorizing expressions for boundary integrals fast discretization / linear precision small constant band width parallel algorithms multigrid solvers adaptive refinement applications: moderately accurate models nonsmooth problems time-sensitive simulations K. Höllig et al. (Universität Stuttgart) FE Methods with Linear B-Splines Avignon, July 5, / 13

3 Finite Element Basis domain D D h : w h (x) > 0 weighted B-splines w h b h k, k ν Z K. Höllig et al. (Universität Stuttgart) FE Methods with Linear B-Splines Avignon, July 5, / 13

4 Linear Box-Splines [de Boor, Höllig, Riemenschneider, Springer 1993] Ξ = , B Ξ (x) = 1 0 χ [0,1] 3 (x t(1, 1, 1)) dt Lagrange basis b h k (x) = B Ξ (x/h k), k Z 3 K. Höllig et al. (Universität Stuttgart) FE Methods with Linear B-Splines Avignon, July 5, / 13

5 Data Structure values on grid of bounding box w h (0 : k 1, 0 : k 2, 0 : k 3 ) p h (0 : k 1, 0 : k 2, 0 : k 3 ) finite element approximation (Dirichlet boundary conditions) ( ) ( ) u h = w h p h = wk h bh k k k p h k bh k evaluation on grid tetrahedron S ws h ph S R4 R 4 product of linear functions K. Höllig et al. (Universität Stuttgart) FE Methods with Linear B-Splines Avignon, July 5, / 13

6 Stability [cf. results by Reif and Mößner] and Accuracy Condition of the basis: u h 0,D ΓP h h with Γ the diagonal matrix of normalization constants γ k = w h bk h 0,D h. 0 Approximation order: u u h 1,D D h h u 2,D for a domain D with smooth boundary and u H 2 (D) H 1 0 (D). K. Höllig et al. (Universität Stuttgart) FE Methods with Linear B-Splines Avignon, July 5, / 13

7 Numerical Integration Ritz-Galerkin integrals S D h intersection patterns ) a (w h bk h, w h bk h = I (ws h, k, k ) w i Poisson bilinear form I : rational function of ws h or th S (automatically generated, cases) t i K. Höllig et al. (Universität Stuttgart) FE Methods with Linear B-Splines Avignon, July 5, / 13

8 Expressions for Ritz-Galerkin Integrals no intersection, full box, common factor 1/120: 24 w w 1 w w w w 1 w w 1 w 6 4 w 5 w 6 8 w 1 w 5 4 w 8 w 6 4 w 3 w w w 5 w 7 +8 w 1 w w w 4 w w w 6 w w 1 w w w w 2 w 1 4 w 3 w 7 4 w 2 w 4 4 w 7 w 8 intersection, one tetrahedron, common factor w 2 1 /120: t 2 2 t t 2 1 t 2 2 t 1 t 2 t 3 4 t 1 3 t 3 2 t 1 t 3 t 2 2 t t 1 t t 1 2 t t t t 1 t t 1 3 t 2 +5 t 1 2 t t 1 2 t 3 t 3 t 2 t 2 t 3 K. Höllig et al. (Universität Stuttgart) FE Methods with Linear B-Splines Avignon, July 5, / 13

9 K-Form [J. Koch, Dissertation 1995] sparse polynomial expressions 2xy + 3x 2 y 2 z x 2 y 2 z + 5xyz +... simplifying quadratic substitutions u = xy, v = uz, w = uv linear expressions 2u + 3w w + 5v +... K. Höllig et al. (Universität Stuttgart) FE Methods with Linear B-Splines Avignon, July 5, / 13

10 Example of K-Form Substitutions 15 substitutions t2 2 t2 1 t 3 + t 2 1t 2 2 t 1 t 2 t 3 4 t3 1 t 3 t 2 2 t2 1 t2 3 t 2 5 t 1 t 3 + 3t 2 1t 3 + 4t t t 1 t t3 1 t 2 t 3 +5 t2 1 t 2 t t2 1 t 3 t 2 t 14 + t 12 2t 13 4t 17 2t 18 5t 7 + 3t 15 +4t 10 10t 6 + t 16 2t t t 9 substitutions could be used simultanously for all 36 cases of (k, k ) 27 Substitutions for 315 products for the non-intersection case K. Höllig et al. (Universität Stuttgart) FE Methods with Linear B-Splines Avignon, July 5, / 13

11 Performance Eigenvalue Problem u = λu in D, u = 0 on D Lanczos method with dynamic multigrid unknowns: 6561 CPU time (sec.): 581 number of computed eigenvalues: 1000 time (sec.): eigenvalue Computed on Pentium 4 PC, 3.20GHz, 2GB memory K. Höllig et al. (Universität Stuttgart) FE Methods with Linear B-Splines Avignon, July 5, / 13

12 Program Demo u = f in D, u = 0 on D unit disc with random holes dynamic multigrid solver FORTRAN 90 code Matlab interface for visualization holes f = 4 f = e x ((1 + x) cos y y sin y) 0 6 grids 8 grids 6 grids 8 grids 2 6 grids 8 grids 6 grids 8 grids 4 6 grids 8 grids 6 grids 8 grids K. Höllig et al. (Universität Stuttgart) FE Methods with Linear B-Splines Avignon, July 5, / 13

13 Next Steps [in cooperation with U. Küster, HLRS] porting programs to NEC SX-8 Cluster (72 Nodes with 8 CPUs each) large scale tests of the vectorization parallel processing of different intersection types completion of the 3D implementation improvement of automatic algebraic simplifications web-sites: books: Finite Element Methods with B-Splines, SIAM, 2003 Handbook of Computer Aided Geometric Design, Elsevier 2002 K. Höllig et al. (Universität Stuttgart) FE Methods with Linear B-Splines Avignon, July 5, / 13