Computer simulation of multiscale problems

Transcription

1 Progress in the SSF project CutFEM, Geometry, and Optimal design Computer simulation of multiscale problems Axel Målqvist and Daniel Elfverson University of Gothenburg and Uppsala University Umeå Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

2 Research group and Outline My research group consists of: Daniel Elfverson, Uppsala, Multiscale and UQ, Fredrik Hellman, Uppsala, Multiscale and UQ, Anna Persson, Göteborg, Time dep. ms problems, Gustav Kettil, Göteborg, Simulation of paper, Tony Stillfjord, Göteborg, Time dep. PDE and splitting, Support from SSF project Introduction to Localized Orthogonal Decomposition techniques for solving multiscale problems (Axel) Application to eigenvalue problems (Axel) Application to problems posed on complex geometry (Daniel) Recent development and openings for collaboration (Axel) Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

3 Multiscale problems Applications such as flow in a porous medium composite materials require numerical solution of partial differential equations with rough data (e.g. permeability or module of elasticity). Major challenge: Features on multiple scales in space and time. Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

4 Motivation for multiscale techniques Poisson s equation A u = f in Ω u = 0 on Ω where A has rapid oscillations. Example (periodic coefficient): A(x) = 2 + sin(2πx/ε), ε = 2 6, f = 1 A(x) x oscillatory coefficient Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

5 Motivation for multiscale techniques Poisson s equation A u = f in Ω u = 0 on Ω where A has rapid oscillations. Example (periodic coefficient): A(x) = 2 + sin(2πx/ε), ε = 2 6, f = 1 u(x) h/ε= x solution and P1-FEM-approximation log 2 (relative error) ε log 2 (h) log 2 (H 1 (Ω) error) vs. log 2 (h) Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

6 Motivation for multiscale techniques Poisson s equation A u = f in Ω u = 0 on Ω where A has rapid oscillations. Example (periodic coefficient): A(x) = 2 + sin(2πx/ε), ε = 2 6, f = 1 u(x) h/ε= x solution and P1-FEM-approximation log 2 (relative error) ε log 2 (h) log 2 (H 1 (Ω) error) vs. log 2 (h) Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

7 Motivation for multiscale techniques Poisson s equation A u = f in Ω u = 0 on Ω where A has rapid oscillations. Example (periodic coefficient): A(x) = 2 + sin(2πx/ε), ε = 2 6, f = 1 u(x) h/ε= x solution and P1-FEM-approximation log 2 (relative error) ε log 2 (h) log 2 (H 1 (Ω) error) vs. log 2 (h) Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

8 Motivation for multiscale techniques Poisson s equation A u = f in Ω u = 0 on Ω where A has rapid oscillations. Example (periodic coefficient): A(x) = 2 + sin(2πx/ε), ε = 2 6, f = 1 u(x) h/ε= x solution and P1-FEM-approximation log 2 (relative error) ε log 2 (h) log 2 (H 1 (Ω) error) vs. log 2 (h) Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

9 Motivation for multiscale techniques Poisson s equation A u = f in Ω u = 0 on Ω where A has rapid oscillations. Example (periodic coefficient): A(x) = 2 + sin(2πx/ε), ε = 2 6, f = 1 u(x) h/ε= x solution and P1-FEM-approximation log 2 (relative error) ε log 2 (h) log 2 (H 1 (Ω) error) vs. log 2 (h) Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

10 Motivation for multiscale techniques Poisson s equation A u = f in Ω u = 0 on Ω where A has rapid oscillations. Example (periodic coefficient): A(x) = 2 + sin(2πx/ε), ε = 2 6, f = 1 u(x) h/ε= x solution and P1-FEM-approximation log 2 (relative error) ε log 2 (h) log 2 (H 1 (Ω) error) vs. log 2 (h) Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

11 Motivation for multiscale techniques Poisson s equation A u = f in Ω u = 0 on Ω where A has rapid oscillations. Example (periodic coefficient): A(x) = 2 + sin(2πx/ε), ε = 2 6, f = 1 u(x) h/ε= x solution and P1-FEM-approximation log 2 (relative error) ε log 2 (h) log 2 (H 1 (Ω) error) vs. log 2 (h) Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

12 Motivation for multiscale techniques Poisson s equation A u = f in Ω u = 0 on Ω where A has rapid oscillations. Example (periodic coefficient): A(x) = 2 + sin(2πx/ε), ε = 2 6, f = 1 u(x) h/ε= x solution and P1-FEM-approximation log 2 (relative error) ε log 2 (h) log 2 (H 1 (Ω) error) vs. log 2 (h) Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

13 Motivation for multiscale techniques Poisson s equation A u = f in Ω u = 0 on Ω where A has rapid oscillations. Example (periodic coefficient): A(x) = 2 + sin(2πx/ε), ε = 2 6, f = 1 u(x) h/ε= x solution and P1-FEM-approximation log 2 (relative error) ε log 2 (h) log 2 (H 1 (Ω) error) vs. log 2 (h) Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

14 Motivation for multiscale techniques Poisson s equation A u = f in Ω u = 0 on Ω where A has rapid oscillations. Example (periodic coefficient): A(x) = 2 + sin(2πx/ε), ε = 2 6, f = 1 u(x) h/ε= x solution and P1-FEM-approximation log 2 (relative error) ε log 2 (h) log 2 (H 1 (Ω) error) vs. log 2 (h) Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

15 Motivation for multiscale techniques Poisson s equation A u = f in Ω u = 0 on Ω where A has rapid oscillations. Example (periodic coefficient): A(x) = 2 + sin(2πx/ε), ε = 2 6, f = 1 u(x) h/ε= x solution and P1-FEM-approximation log 2 (relative error) ε log 2 (h) log 2 (H 1 (Ω) error) vs. log 2 (h) Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

16 Motivation for multiscale techniques Poisson s equation A u = f in Ω u = 0 on Ω where A has rapid oscillations. Conclusion Fine scale features have to be resolved even to get coarse solution behavior right (both H 1 and L 2 errors are large). Resolution of the fine scales by a uniform mesh is very computationally expensive. Local mesh refinement is not an option. The standard basis does not seem to be suitable for this problem. Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

17 Standard FE decomposition Coarse FE mesh with parameter H Piecewise linear continuous FE space V H I T : V V H interpolation operator Decomposition V = V H V f with V f := {v V I T v = 0} Example: rough coefficient Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

18 Standard FE decomposition Coarse FE mesh with parameter H Piecewise linear continuous FE space V H I T : V V H interpolation operator Decomposition V = V H V f with V f := {v V I T v = 0} Example: = + u V u H V H u f V f Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

19 Orthogonal multiscale decomposition Let dim(v ms H ) = dim(v H) Start from V H and add fine scale corrections in V f v ms H Decomposition V ms, v H f V f holds Ω A vms v H f dx = 0 V = V ms H V f with V f := {v V I T v = 0} Example: rough coefficient Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

20 Orthogonal multiscale decomposition Let dim(v ms H ) = dim(v H) Start from V H and add fine scale corrections in V f v ms H Decomposition V ms, v H f V f holds Ω A vms v H f dx = 0 V = V ms H V f with V f := {v V I T v = 0} Example: = + u V u ms H V ms H u f V f Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

21 Computing a basis Let V H = span {λ x x N} and further let, φ x V f solve A (λ x φ x ) w dx = 0, for all w V f. Ω Multiscale FE space V ms H = span {λ x φ x x N} Example: = λ x φ x V ms H λ x V H φ x V f Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

22 Computing a basis Let V H = span {λ x x N} and further let, φ x V f solve A (λ x φ x ) w dx = 0, for all w V f. Ω Multiscale FE space V ms H = span {λ x φ x x N} Example: = λ x φ x V ms H λ x V H φ x V f Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

23 Computing a basis We have proven exponential decay of λ x φ x! Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

24 Computing a basis This allows us to truncate to a patch. Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

25 Computing a basis This allows us to truncate to a patch and fine scale discretization. Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

26 Computing a basis Localized patch with refined mesh gives computable basis functions The multiscale basis functions are computed by solving the PDE on local patches. The basis functions are totally independent, leading to trivial parallelization. Exponential decay has been proven and is crucial. Remember dim(v ms) = dim(v H H). Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

27 Computing the multiscale approximation Multiscale approximation: u ms H A u ms H Ω v dx = V ms H Ω satisfies f v dx for all v V ms H We have proven error bound (using k = log(1/h) layers): (u u ms H ) CH f, where v 2 = Ω v2 dx and C is independent on variations in A. Note that for the standard FEM with A = A( x ) we have, ɛ (u u H ) C H ɛ f. Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

28 Numer. exp. (Poisson, f L 2 (H), f H 1 0 (H2 )!) H = 2 1, 2 2,..., 2 7 h = 2 9, k = log(1/h) (u h u ms,h ) vs. #dof H,k Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

29 Numer. exp. (linear eigenvalue problem, H 4 ) l λ (l) h e (l) (1/2 2) e (l) (1/4 2) e (l) (1/8 2) e (l) (1/16 2) Table: Errors e (l) (H) =: λ(l) H λ(l) h λ (l) h and h = Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

30 The quadratic eigenvalue problem Consider discretized structure with damping, Kx + λcx + λ 2 Mx = 0, where K ij = (A φ j, φ i ) is stiffness, C ij = c(φ j, φ i ) is damping, and M ij = (φ j, φ i ) is mass matrix. Linearization: (y = λx) ] ] Ax := [ K 0 0 M ] [ x y [ C M = λ M 0 ] [ x y := λbx, Note that T = A 1 B has eigenvalues λ 1. T is not symmetric (even if C is), it is compact but not necessarily in the limit h 0. The analysis is very different from symmetric case. Semi-simple eigenvalues are ok but defective leads to unknown constants. Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

31 Numerical experiment H H = 2 1, 2 2,..., 2 5 h = 2 6, k = λ h λ ms / λ H h vs. #dof A (pic), c(u, v) = (1 + sin(10x))u v dx Ω Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

32 Numerical experiment H 2 H = 2 1, 2 2,..., 2 5 h = 2 6, k = 10 3 λ h λ ms / λ H h vs. #dof πc(u, v) = (atan(10x 5) + π )A u v dx Ω 2 Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

33 Relevant papers Some papers relevant for the SSF project (work package D): Poisson P1, M. & Peterseim, Math. Comp., Poisson DG, E., Georgoulis, M., & Petersiem, SINUM, Linear Eig., M. & Peterseim, Numer. Math Gross-Pitaevskii, Henning, M., & Peterseim, SINUM, Helmholtz, Peterseim, two preprints LOD is applicable when: The diffusion is present and lower order terms are not dominating. LOD is efficient when: basis can be reused many times in the calculation (nonlinear iteration, time stepping,...). Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

34 LOD complex geometry Consider the Poisons equation u = f in Ω, ν u = κu on Γ R, u = 0 on Γ D, where Γ D and Γ R are not resolved by the mesh Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

35 Objective Construct a multiscale method for complicated geometries with the following properties: No pre-asymptotic effect Linear convergence without resolving the boundary Conditioning is independent of how the boundary cut the mesh Only enrich the coarse finite element space in the vicinity of the boundary Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

36 Poincaré-Friedrich inequality Difficulties: We need a inequality of the type where inf c R u c L 2 (ω) C(ω)diam(ω) u L 2 (ω) C(ω) is bounded independent of diam(ω) Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

37 Error analysis Let ω Γ be a vicinity of the complicated boundary Theorem (Locally enriched LOD method) Given that u H 1 Γ D H 2 (Ω \ ω Γ ) and that u Γ H V Γ,L H solution, then is the LOD u u LOD H h (u I h u) L 2 (ω Γ ) + H u H 2 (Ω\ω Γ ) + H f L 2 (ω Γ ) + H 1 (L) d/2 γ L f L 2 (Ω) holds. The condition number scale as κ H 2. I h u: interpolation operator on the fine mesh L: Number of layers 0 < γ < 1: depends on the Poinaré-Friedrich constant Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

38 Error analysis Choose L = C log(h 1 ), Theorem (Locally enriched LOD method) Given that u H 1 Γ D H 2 (Ω \ ω k 1 Γ ) and u Γ H V Γ,L, then u u LOD H h H holds. The condition number scale as κ H 2. H Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

39 Numerical verification Fractal domain Consider the Poisson equation in the domain: Correctors are computed everywhere Mixed boundary condition Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

40 Numerical verification Fractal domain The convergence in relative energy norm Conditioning Error Fractal O(H) Mesh size H Condition number 10 4 Fractal 10 3 O(H 2 ) Mesh size H Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

41 Numerical verification Local singularities Consider the Poisson equation in the domains: Correctors are computed in the red areas Dirichlet boundary condition Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

42 Numerical verification Local singularities The convergence in relative energy norm Conditioning Error L-shaped Slit O(H) Mesh size H Condition number Mesh size H L-shaped Slit O(H 2 ) Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

43 Numerical verification Saw tooth boundary Consider the Poisson equation in the domain: Correctors are computed in the red area Test both Dirichlet and Neumann boundary condition on the saw tooth Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

44 Numerical verification Saw tooth boundary The convergence in relative energy norm Conditioning Error Dirichlet Neumann O(H) Mesh size H Condition number Mesh size H Dirichlet Neumann O(H 2 ) Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

45 Numerical verification Cut domain Consider the Poisson equation in the domain: a b c 2) c b a 1) Correctors are computed everywhere Conside a fix mesh and cut the domain in different ways Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

46 Numerical verification Cut domain Cut 1 e rel (a) e rel (b) e rel (c) D D-N N-D cond(a) cond(b) cond(c) D D-N N-D Cut 2 e rel (a) e rel (b) e rel (c) D D-N N-D cond(a) cond(b) cond(c) D D-N N-D Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

47 Recent development and collaboration We consider a Poisson type model problem with a diffusion coefficient A that varies between two different values: A u = f in Ω, u = 0 on Ω. The objects may be short fibres/particles or cracks/holes the solution has low regularity at the boundary of the objects local mesh refinement work for single configuration but not when multiple configurations or moving objects are allowed Monte Carlo simulation or optimization procedure Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

48 Recent development In order to treat multiple configurations we want a regular static coarse background mesh. We only compute LOD basis functions in a region around each object and use standard P1 FEM basis functions in between. The size of the surrounding region can be controlled by H to meet tolerance, p adaptivity, DG.. If the regions overlap we compute basis functions for the union of the overlapping domains. Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

49 Recent development CutFEM allows us (we hope) to glue the LOD computations to the background mesh avoiding remeshing. The assembly of the LOD parts is independent of the orientation of the object (rotation,translation). If the objects have the same shape huge savings are possible. Compared to CutFEM using spatially adapted subgrids around the objects the LOD approach leads to much smaller systems of equations for each configuration of the objects. Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

50 Recent development Random distribution of objects: Monte-Carlo simulation can be performed on the coarse scale with fine scale accuracy. Design of materials: Optimization procedure where the objects are moved in order to minimize some output. Again repeated solves on the coarse scale still gives fine scale accuracy. Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30

51 Comments/questions/possible collaborations The problem is very similar to complex boundary, holes is a direct application. Any comments from or difficulties seen by the CutFEM experts? We got the idea from porous media flow applications, where there is a great uncertainty in the location and direction of defects. We seek applications in composite materials which fits our framework. Uncertainty quantification in material science is a possible application for this approach. What about applicability to optimal design problems? We will start this work after the summer, in collaboration with Fredrik Hellman Målqvist (Gothenburg) Computer simulation of multiscale problems Umeå / 30