Algorithms for Scientific Computing

Transcription

1 Algorithms for Scientific Computing Finite Element Methods Michael Bader Technical University of Munich Summer 2016

2 Part I Looking Back: Discrete Models for Heat Transfer and the Poisson Equation Modelling of Heat Transfer objective: compute the temperature distribution of some object under certain prerequisites: temperature T at object boundaries given heat sources material parameters k,... observation from physical experiments: q k δt (heat flow proportional to temperature differences) Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

3 A Finite Volume Model object: a rectangular metal plate (again) model as a collection of small connected rectangular cells h y h x examine the heat flow across the cell edges Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

4 Heat Flow Across the Cell Boundaries Heat flow across a given edge is proportional to temperature difference (T 1 T 0 ) between the adjacent cells length h of the edge e.g.: heat flow across the left edge: q (left) ij = k x ( Tij T i 1,j ) hy k x depends on material heat flow across all edges determines change of heat energy: q ij = k x ( Tij T i 1,j ) hy + k x ( Tij T i+1,j ) hy + k y ( Tij T i,j 1 ) hx + k y ( Tij T i,j+1 ) hx equilibrium with source term F ij = f ij h x h y (f ij heat flow per area) requires q ij + F ij = 0: f ij h x h y = k x h y ( 2Tij T i 1,j T i+1,j ) k y h x ( 2Tij T i,j 1 T i,j+1 ) Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

5 Discrete and Continuous Model system of equations derived from the discrete model: f ij = k x h x ( 2Tij T i 1,j T i+1,j ) k y h y ( 2Tij T i,j 1 T i,j+1 ) result: average temperature in each cell corresponds to partial differential equation (PDE): ( 2 ) T (x, y) k x T (x, y) y 2 = f (x, y) wanted: approximate T (x, y) as a function! solution possible using coefficients and basis functions? Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

6 Part II Outlook: Finite Element Methods For Model Problem: 2D Poisson equation: 2 T (x, y) x T (x, y) y 2 = f (x, y) first, however, we consider the 1D case: with u(0) = u(1) = 0. u (x) = f (x) for x (0, 1) Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

7 Finite Elements Main Idea we consider the residual of the (1D) PDE: u (x) = f (x) u (x) f (x) = 0 represent the functions u and f in our favorite form: ( ) uj φ j (x) fj φ j (x) = 0 however: we will usually not find u j that solve this equation exactly (as solution u can not be represented as u j φ j (x)) remedy? find best approximation, given by orthogonality: ( w(x), uj φ j (x)) fj φ j (x) = 0 for all w(x) remember that < g, f >= g f dx Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

8 Finite Elements Main Ingredients 1. compute a function as numerical solution; search in a function space W h : u h = j u j ϕ j (x), span{ϕ 1,..., ϕ J } = W h 2. solve weak form of PDE to reduce regularity properties u = f v u dx = vf dx 3. choose basis functions with local support, e.g.: ϕ j (x i ) = δ ij (such as the hat functions) Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

9 Choose Test and Ansatz Space search for solution functions u h of the form u h = j u j ϕ j (x) the basis ( shape, ansatz ) functions ϕ j (x) build a vector space (or function space) W h span{ϕ 1,..., ϕ J } = W h the best solution u h in this function space is wanted Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

10 Example: Nodal Basis 1 h (x x i 1) x i 1 < x < x i 1 ϕ i (x) := h (x i+1 x) x i < x < x i+1 0 otherwise 1 0,8 0,6 0,4 0, ,2 0,4 0,6 0,8 x Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

11 Or Better A Hierarchical Basis? Φ 1,1. x 1,1 Φ 2,1 Φ 2,3. x 2,1 x 2,3 Φ 3,1 Φ 3,3 Φ 3,5 Φ 3,7. x 3,1 x 3,3 x 3,5 x 3,7 Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

12 Weak Forms and Weak Solutions consider a PDE Lu = f (e.g. Lu = u) transformation to the weak form: v, Lu = vlu dx = vf dx = f, v v V V a certain class of functions real solution u also solves the weak form (but additional, approximate solutions accepted... ) motivation for weak form: we cannot test Lu(x) = f (x) for all x (0, 1) on a computer (infinitely many x) frequent choice V = W h, so check whether Lu and f have the same behaviour w.r.t. scalar product approximate solution û might not solve PDE: Lû f thus: additional functions need to be acceptable as solution orthogonality idea Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

13 Weak Form of the Poisson Equation 1D Poisson equation with Dirichlet conditions: u (x) = f (x) in Ω = (0, 1), u(0) = u(1) = 0 weak form: v(x)u (x) dx = v(x)f (x) dx Ω Ω integration by parts: Ω v(x)u (x) dx = v(x) u (x) choose functions v such that v(0) = v(1) = 0: v (x) u (x) dx = v(x)f (x) dx Ω Ω 1 0 v + v (x) u (x) dx Ω v Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

14 Weak Form of the Poisson Equation 2D/3D Poisson equation with Dirichlet conditions: weak form: apply Green s formula: v u dω = Ω u = f in Ω, u = 0 on δω v u dω = vf dω Ω Ω Ω v v u dω v u ds Ω choose functions v such that v = 0 on Ω: v u dω = vf dω Ω Ω v Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

15 Weak Form of the Poisson Equation Summary Poisson equation with Dirichlet conditions: u = f in Ω, u = 0 on δω transformed into weak form: v u dω = vf dω Ω Ω v weaker requirements for a solution u: twice differentiabale first derivative integrable remember use of nodal basis: availability of first vs. second derivative! Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

16 Choose Test and Ansatz Space search for solutions u h in a function space W h : u h = j u j ϕ j (x) where span{ϕ j } = W h ( ansatz space ) insert into weak solution ( ) vl u j ϕ j (x) dx = j vf dx v V Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

17 Choose Test and Ansatz Space (2) choose a basis {ψ i } of the test space V then: if all basis functions ψ i satisfy ( ) ψ i L u j ϕ j (x) dx = then all v V satisfy the equation j ψ i f dx ψ i leads to system of equations for unknowns u j (one equation per test basis function ψ i ) V is often chosen to be identical to W h (Ritz-Galerkin method) Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

18 Discretisation Finite Elements L linear system of linear equations ( ) ψ i Lϕ j (x) dx u j = j } {{ } =:A ij ψ i f dx ψ i aim: make system of equations easy to solve! thus, make matrix A sparse most A ij = 0 approach: local basis functions on a discretisation grid consider hat functions, e.g.: ψ j, ϕ j zero everywhere, except in grid cells adjacent to grid point x j then A ij = 0, if ψ i and ϕ j don t overlap Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

19 Example Problem: Poisson 1D in 1D: u (x) = f (x) on Ω = (0, 1), hom. Dirichlet boundary cond.: u(0) = u(1) = 0 weak form: 1 v (x) u (x) dx = v(x)f (x) dx computational grid: x i = ih, (for i = 1,..., n 1); mesh size h = 1/n V = W : piecewise linear functions (on intervals [x i, x i+1 ]) v Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

20 Nodal Basis 1 h (x x i 1) x i 1 < x < x i 1 ϕ i (x) := h (x i+1 x) x i < x < x i+1 0 otherwise 1 0,8 0,6 0,4 0, ,2 0,4 0,6 0,8 x 1 Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

21 Nodal Basis System of Equations stiffness matrix: 1 h right hand sides (assume f (x) = α R): ϕ i (x)f (x) dx = ϕ i (x)α dx = αh system of equations very similar to finite differences Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

22 Hierarchical Basis 1 0,8 0,6 0,4 0, ,2 0,4 0,6 0,8 1 x leads to diagonal stiffness matrix! (for 1D Poisson) solution function identical to that with nodal basis (same function space) Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

23 Part III Finite Element Methods Basis Functions for 2D Hierarchical Basis in 2D Quadtrees and Hierarchical Bases Quadtrees to Represent Objects Hierarchical Basis vs. Quadtree Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

24 2D Hierarchical Basis Tensor Product define 2D basis functions via tensor product: φ i,j (x, y) := φ i (x) φ j (y) remember multi-index for 2D hierarchical basis: φ l, k (x 1, x 2 ) := φ l1,l 2,k 1,k 2 (x 1, x 2 ) := φ l1,k 1 (x 1 ) φ l2,k 2 (x 2 ) illustrate via support of the basis functions: Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

25 Illustrate via Location of Hat Functions l 1 =1 l 1 =2 l 1 =3 l 1 l 2 =1 l 2 =2 l 2 =3 l 2 Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

26 Adding Adaptivity: Quadtrees Quadtrees to Represent Objects: start with an initial square (covering the entire domain) recursive substructuring into four subsquares adaptive refinement? Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

27 Quadtrees for Adaptive Simulations Adaptively Refined Meshes for Finite Elements: refine, unless squares entirely within or outside domain also: refine, if solution not exact enough! question: can we build a hierarchical basis on such a quadtree? Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

28 Hierarchical Basis vs. Quadtree Use hierarchical basis as in 2D sparse grids? stretched tensor basis functions do not match quadtree cells use basis functions with square domain (cover 4 siblings to solve) Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

29 Hierarchical Basis for Quadtrees Switch to hierarchical multilevel basis: hierarchical concept (again): skip basis functions that exist on previous level! Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

30 Illustrate via Location of Hat Functions l 1 =1 l 1 =2 l 1 =3 l 1 l 2 =1 l 2 =2 l 2 =3 l 2 Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

31 Quadtree-Compatible Hierarchical Basis Basis Functions Similar to tensor-product basis: Level-wise hierarchical increments l 1 =1 l 1 =2 l 1 =3 l 1 W l := span{φ l, i } i Î l l 2 =1 Only use diagonal levels: l := {l,..., l} Omit grid points for which all indices are even: Î l := { i : 1 i < 2 n, any i j odd} l 2 =2 l 2 =3 l 2 Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

32 Part IV Finite Element Methods Towards Implementation FEM and Hierarchical Basis Transform Hierarchical Basis Transformation FEM and Hierarchical Basis Transform Element Stiffness Matrices Workflow Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

33 Project: 2D Adaptive Hierarchical Basis Consider: 2D Poisson problem FEM with quadtree-compatible hierarchical basis adaptive quadtree-based hierarchical basis Discuss (again): how to compute the stiffness matrix? what do you need to compute, if you add a hierarchical basis function? how do you know when to add a basis function? Idea: move from node-oriented to element-oriented approach Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

34 Recall: Hierarchical Basis Transformation 1 1 0,8 0,8 0,6 0,4 0,2 0,6 0,4 0, ,2 0,4 0,6 0,8 x ,2 0,4 0,6 0,8 x 1 represent wider hat function φ 1,1 (x) via basis functions φ 2,j (x) φ 1,1 (x) = 1 2 φ 2,1(x) + φ 2,2 (x) φ 2,3(x) consider vector of hierarchical/nodal basis functions and write transformation as matrix-vector product: ψ 2,1 (x) φ 2,1 (x) φ 2,1 (x) ψ 2,2 (x) := φ 1,1 (x) = φ 2 2,2 (x) ψ 2,3 (x) φ 2,3 (x) φ 2,3 (x) Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

35 Recall: Hierarchical Basis Transformation (2) hierarchical basis transformation: ψ n,i (x) = j H i,j φ n,j (x) written as matrix-vector product: ψ n = H n φn H can be written as a sequence of level-wise transforms: H n = H (n 1) n H (n 2) n... H (2) n H (1) n where each transform has a shape similar to H (1) = Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

36 b j = i H i,j a i = i Recall: Hierarchical Coordinate Transformation consider function f (x) i a i ψ n,i (x) represented via hier. basis wanted: corresponding representation in nodal basis b j φ n,j (x) = a i ψ n,i (x) f (x) j i with ψ n,i (x) = H i,j φ n,j (x) we obtain j b j φ n,j (x) = j i compare coordinates and get a i H i,j φ n,j (x) = j j a i H i,j φ n,j (x) i ( H T ) j,i a i written in vector notation: b = H T a Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

37 FEM and Hierarchical Basis Transform FEM discretisation with hierarchical test and shape functions: ( ) ψ i (x)l u j ψ j (x) dx = ψ i (x)f (x) dx ψ i j leads to respective stiffness matrix A HB ( ) ψ i (x)l u j ψ j (x) dx = j j i,j : u j ψ i (x)lψ j (x) dx = j u j A HB i,j vs. stiffness matrix with nodal basis as shape functions: ( ) ψ i (x)l v j φ j (x) dx = v j ψ i (x)lφ j (x) dx = j j j v j A i,j note that (A HB u) i = j u ja HB i,j = j v ja i,j = (A v) i and v = H T u Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

38 FEM and Hierarchical Basis Transform (2) status: FEM with hierarchical test and nodal shape functions ( ) ψ i (x)l v j φ j (x) dx = ψ i (x)f (x) dx j represent test functions via nodal basis: ( ) H i,k φ k (x)l v j φ j (x) dx = H i,k φ k (x)f (x) dx k ( ) H i,k φ k (x)l v j φ j (x) dx = k j k j k H i,k leads to new system of equations: HA NB v = H b NB φ k (x)f (x) dx where A NB and b NB stem from nodal-basis FEM discretisation! with v = H T u we obtain H A NB H T u = H b as system of equations, thus: A HB = H A NB H T ( Galerkin coarsening) Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

39 Element Stiffness Matrices domain Ω splitted into finite elements Ω (k) : Ω = Ω (1) Ω (2) Ω (n) observation: basis functions are defined element-wise use: b a f (x)dx = c a f (x)dx + b c f (x)dx element-wise evaluation of the integrals: v u dx = Ω k Ω vf dx = i v u dx Ω (k) vf dx Ω (i) Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

40 Element Stiffness Matrices (2) leads to local stiffness matrices for each element: Ω (k) φ i φ j dx }{{} =:A (k) ij and respective element systems: A (k) x = b (k) accumulate to obtain global system: A (k) x = k k }{{} =:A b (k) Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

41 Element Stiffness Matrices (3) Some comments on notation: assume: 1D problem, n elements (i.e. intervals) in each element only two basis functions are non-zero! hence, almost all A (k) ij are zero: A (k) ij = φ i φ j dx Ω (k) only 2 2 elements of A (k) are non-zero therefore convention to omit zero columns/rows leaves only unknowns that are in Ω (k) Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

42 Typical workflow 1. choose elements: quadratic or cubic cells triangles (structured, unstructured) tetrahedra, etc. 2. set up basis functions for each element Ω (k) ; for example, at all nodes x i Ω (k) ϕ i (x i ) = 1 ϕ i (x j ) = 0 for all j i 3. for element stiffness matrix, compute all A (k) ij = ϕ i Lϕ j dω Ω (k) 4. accumulate global stiffness matrix Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

43 Example: 1D Poisson Ω = [0, 1] splitted into Ω (k) = [x k 1, x k ] nodal basis; leads to element stiffness matrix: ( ) A (k) 1 1 = 1 1 consider only two elements: A (1) + A (2) = in stencil notation: [ 1 1 ] + [ 1 1 ] [ ] Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

44 Project: Adaptive Hierarchical Basis Consider: 1D Poisson problem FEM with hierarchical basis however: not all basis functions used on each grid adaptive hierarchical basis Discuss: how to compute the stiffness matrix? what do you need to compute, if you add a hierarchical basis function? how do you know when to add a basis function? Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

45 Example: 2D Poisson u = f on domain Ω = [0, 1] 2 splitted into Ω (i,j) = [x i 1, x i ] [x j 1, x j ] bilinear basis functions pagoda functions ϕ ij (x, y) = ϕ i (x)ϕ j (y) Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer

46 Example: 2D Poisson (2) leads to element stiffness matrix: A (k) = accumulation leads to 9-point stencil Michael Bader Algorithms for Scientific Computing Finite Element Methods Summer