Lecture 3: Function Spaces I Finite Elements Modeling. Bruno Lévy

Transcription

1 Lecture 3: Function Spaces I Finite Elements Modeling Bruno Lévy

2 Overview 1. Motivations 2. Function Spaces 3. Discretizing a PDE 4. Example: Discretizing the Laplacian 5. Eigenfunctions Spectral Mesh Processing(Richard) Epilogue Continuous or Discrete?

3 1. Motivations Computing with meshes

4 1. Motivations Computing with meshes

5 2. Function spaces Vector spaces e 3 e 2 e 1

6 2. Function spaces Vector spaces - coordinates V = x e 1 + y e 2 + z e 3 e 3 V e 2 e 1

7 2. Function spaces Vector spaces - coordinates V = x e 1 + y e 2 + z e 3 e 3 V e 2 x = V. e 1 y = V. e 2 z = V. e 3 e 1

8 2. Function spaces Vector spaces - coordinates V = x e 1 + y e 2 + z e 3 e 3 V e 2 x = V. e 1 y = V. e 2 z = V. e 3 Dot product: e 1 V. W = V x W x + V y W y + V z W z

9 2. Function spaces Vector spaces - projection V

10 2. Function spaces Vector spaces - projections V W

11 2. Function spaces Vector spaces - projections V W = (V.e 1 )e 1 + (V.e 2 )e 2 W

12 2. Function spaces Vector spaces importance of the dot product The dot product can compute Coordinates of a vector in a basis Projection of a vector onto a subspace v.v

13 2. Function spaces Vector spaces importance of the dot product The dot product can compute Coordinates of a vector in a basis Projection of a vector onto a subspace v.v Hilbert space

14 2. Function spaces Dot product for vectors u i i

15 2. Function spaces Dot product for vectors u i u v = u i v i i

16 2. Function spaces Dot product for vectors u i u v = u i v i i Dot product for functions f(x) x

17 2. Function spaces Dot product for vectors u i u v = u i v i i Dot product for functions f(x) <f,g> = f(t)g(t)dt x

18 2. Function spaces Dot product for vectors u i u v = u i v i i Dot product for functions f(x) <f,g> = f(t)g(t)dt x The inner product (dot product for functions)

19 2. Function spaces Vector spaces importance of the dot product Coordinates of a function in a function basis Projection of a function onto a subspace <f,f>

20 2. Function spaces Examples of function basis The canonical polynomial basis ( i) 0(x) = 1 (x) = x 2(x) = x 2 3(x) = x 3 f(x) = i i(x)

21 2. Function spaces Examples of function basis The Fourier basis ( i) 0(x) = 1 2k(x) = sin(2k x) 2k+1(x) = cos(2k x) f(x) = i i(x)

22 2. Function spaces Meshes The P1 Function Basis (linear FEM) i(p i ) = 1 i(p j ) = 0 Courtesy of Bruno Vallet

23 2. Function spaces Meshes The P2 Function Basis (quadratic FEM) Courtesy of Bruno Vallet

24 2. Function spaces Meshes The P3 Function Basis (cubic FEM) Courtesy of Bruno Vallet

25 3. Discretizing a PDE General form of a linear PDE: Lf = g

26 3. Discretizing a PDE General form of a linear PDE: Lf = g Linear operator e.g.,

27 3. Discretizing a PDE General form of a linear PDE: Lf = g Unknown function (the one we want to compute)

28 3. Discretizing a PDE General form of a linear PDE: Lf = g Known function (right hand side)

29 3. Discretizing a PDE General form of a linear PDE: Lf = g Weak form: h, <Lf, h> = <g, h>

30 3. Discretizing a PDE General form of a linear PDE: Lf = g Weak form: h, <Lf, h> = <g, h> Test function

31 3. Discretizing a PDE General form of a linear PDE: Lf = g Weak form: h, <Lf, h> = <g, h> Test function

32 3. Discretizing a PDE Lf = g Discretization Step 1: project PDE onto function basis ( i) i, <Lf, i> = <g, i> (1)

33 3. Discretizing a PDE Lf = g Discretization Step 1: project PDE onto function basis ( i) i, <Lf, i> = <g, i> (1) Step 2: express f in ( i): f= j j and inject in (1) i, <L j j, i> = <g, i>

34 3. Discretizing a PDE Lf = g Discretization i, <L j j, i> = <g, i>

35 3. Discretizing a PDE Lf = g Discretization i, <L j j, i> = <g, i> i, j <L j, i> = <g, i> (by linearlity of <.,.> and L)

36 3. Discretizing a PDE Lf = g Discretization i, <L j j, i> = <g, i> i, j <L j, i> = <g, i> (by linearlity of <.,.> and L) i, j <L j, i> = j < j, i> (rhs g = j j)

37 3. Discretizing a PDE Lf = g Discretization i, j <L j, i> = j < j, i> Galerkin

38 3. Discretizing a PDE Lf = g Discretization i, j <L j, i> = j < j, i> Galerkin 1 1 <L i, j> = < i, j> n n

39 3. Discretizing a PDE Lf = g Discretization i, j <L j, i> = j < j, i> Galerkin 1 1 <L i, j> = < i, j> Discretization of the operator L (stiffness matrix) n n

40 3. Discretizing a PDE Lf = g Discretization i, j <L j, i> = j < j, i> Galerkin 1 1 <L i, j> = < i, j> n Discretization of the solution f (unknown vector) n

41 3. Discretizing a PDE Lf = g Discretization i, j <L j, i> = j < j, i> Galerkin 1 1 <L i, j> = < i, j> n Discretization of the inner product (bilinear form) n

42 3. Discretizing a PDE Lf = g Discretization i, j <L j, i> = j < j, i> Galerkin 1 1 <L i, j> = < i, j> n n Discretization of the rhs g (known vector)

43 4. Discretizing the Laplacian f = g (Poisson equation) Discretization 1 1 i, j> = < i, j> n n

44 4. Discretizing the Laplacian Discretization P1 1 1 i i, j> = < i, j> n n

45 4. Discretizing the Laplacian Discretization P1 1 1 i i, j> = < i, j> n n Coefficients of the mass matrix: < i, j share an edge < i, i> = ( t ) / 6 t in St(i) < i, j> = 0 otherwise

46 4. Discretizing the Laplacian Discretization P1 1 1 i i, j> = < i, j> n n i, j> = -< i, j > (+ boundary term)

47 4. Discretizing the Laplacian Discretization P1 1 1 i, j> = < i, j> n n i, j> = -< i, j > (+ boundary term)

48 4. Discretizing the Laplacian Discretization P1 1 1 i, j> = < i, j> n n

49 4. Discretizing the Laplacian P1 FEM Laplacian - summary 1 1 j i, j> = < i, j> i n n

50 4. Discretizing the Laplacian P1 FEM Laplacian [Pinkall and Polthier 94] [Meyer and Desbrun 99]

51 4. Discretizing the Laplacian P1 FEM Laplacian Courtesy of Rhaleb Zayer

52 4. Discretizing the Laplacian P1 FEM Laplacian [Mac Neal 1949] Courtesy of Rhaleb Zayer

53 4. Discretizing the Laplacian P1 FEM Laplacian [Mac Neal 1949] Courtesy of Rhaleb Zayer

54 5. Eigenfunctions Operator equation: Lf = f Lf = = div(grad f)) Manifold Harmonics [Vallet & L, 2007]

55 5. Eigenfunctions Operator equation: Lf = f Function basis ( i): f = i i Inner Product: <f,g dx i, <Lf, i> = <f, i>

56 5. Eigenfunctions Test function space ( i): f = i i (P1,P2,P3) i, < f, i> = <f, i> < f,g> = -< f, g> (+ boundary term) Ax = Bx a ij = -< i, j> ; b ij = < i, j> Generalized eigenvalue problem (solved by ARPAK or MATLAB)

57 5. Eigenfunctions P1 function basis P3 function basis

58 5. Eigenfunctions P1 function basis P3 function basis

59 5. Eigenfunctions

60 5. Eigenfunctions Compute the terms < i, j> and < i, j Courtesy of Bruno Vallet

61 5. Eigenfunctions Compute the terms < i, j> and < i, j Derivations: see Bruno Ph.D thesis Implementation available in Graphite / ManifoldHarmonics plugin See

62 5. Eigenfunctions Boundary conditions Dirichlet Neumann

63 5. Eigenfunctions Boundary conditions Dirichlet Neumann

64 Epilogue Discrete or continuous? How many points in a triangle?

65 Epilogue Discrete or continuous? Is this a triangle?

66 Epilogue Discrete or continuous? p 1 p 2 p 3 t = { 1 p p p 3, 0 < 1, 2, 3, =1}

67 Epilogue Discrete or continuous? I am a point of this triangle p 1 p 2 p 3 t = { 1 p p p 3, 0 < 1, 2, 3, =1}

68 Epilogue Discrete or continuous? p 1 Me too! Me too! p 2 p 3 t = { 1 p p p 3, 0 < 1, 2, 3, =1}

69 Epilogue A mesh surface is a continuous object encoded by a discrete set of parameters p 1 p 2 p 3 t = { 1 p p p 3, 0 < 1, 2, 3, =1}

70 Epilogue We can compute integrals over the triangle p 1 p 2 p 3 t = { 1 p p p 3, 0 < 1, 2, 3, =1}

71 Epilogue A Pn function is also a continuous object encoded by a discrete set of parameters

72 Epilogue Discretization of parameters (geo. or func.) Discretization of geometry

73 Epilogue

74 Epilogue (1) Smooth is beautiful

75 Epilogue (1) Smooth is beautiful

76 Epilogue (1) Smooth is beautiful (2) FEM framework is general

77 Epilogue (1) Smooth is beautiful (2) FEM framework is general (3) P1 does not always suffice (Bi-Laplacian )

78 Further reading Theory, validy, convergence Sobolev spaces Lax-Milgram Other function bases Other inner products

79 Further reading

80 Further reading