LibMesh Experience and Usage

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1 LibMesh Experience and Usage John W. Peterson Univ. of Texas at Austin January 12, 2007

2 1 Introduction 2 Weighted Residuals 3 Poisson Equation 4 Other Examples 5 Essential BCs 6 Some Extensions

3 1 Introduction 2 Weighted Residuals 3 Poisson Equation 4 Other Examples 5 Essential BCs 6 Some Extensions

4 Library Description Tumor Angiogenesis Compressible RBM NS Petsc Laspack DD LibMesh STL Metis Library Structure Basic libraries are LibMesh s roots Application branches built off the library trunk Mpich

5 A Generic BVP We assume there is a Boundary Value Problem of the form M u t = F(u) Ω G(u) = 0 Ω u = u D Ω D Ω Ω D N(u) = 0 Ω N u(x, 0) = u 0 (x) Ω N

6 A Generic BVP Associated to the problem domain Ω is a LibMesh data structure called a Mesh A Mesh is essentially a collection of finite elements Ω h Ω h := e Ω e

7 A Generic BVP Associated to the problem domain Ω is a LibMesh data structure called a Mesh A Mesh is essentially a collection of finite elements Ω h Ω h := e Ω e LibMesh provides some simple structured mesh generation routines as well as an interface to Triangle.

8 1 Introduction 2 Weighted Residuals 3 Poisson Equation 4 Other Examples 5 Essential BCs 6 Some Extensions

9 The point of departure in any FE analysis which uses LibMesh is the weighted residual statement (F(u), v) = 0 v V Or, more precisely, the weighted residual statement associated with the finite-dimensional space V h V (F(u h ), v h ) = 0 v h V h

10 The point of departure in any FE analysis which uses LibMesh is the weighted residual statement (F(u), v) = 0 v V Or, more precisely, the weighted residual statement associated with the finite-dimensional space V h V (F(u h ), v h ) = 0 v h V h

11 Some Examples Poisson Equation u = f Ω

12 Some Examples Poisson Equation u = f Ω Weighted Residual Statement (F(u), v) := [ u v fv] dx Ω + ( u n) v ds Ω N

13 Some Examples Linear Convection-Diffusion k u + b u = f Ω

14 Some Examples Linear Convection-Diffusion k u + b u = f Ω Weighted Residual Statement (F(u), v) := [k u v + (b u)v fv] dx Ω + k ( u n) v ds Ω N

15 Some Examples Stokes Flow p ν u = f u = 0 Ω

16 Some Examples Stokes Flow p ν u = f u = 0 Ω Weighted Residual Statement u := [u, p], v := [v, q] (F(u), v) := [ p ( v) + ν u: v f v Ω + ( u) q] dx + n (ν u pi) v ds Ω N

17 Some Examples To obtain the approximate problem, we simply replace u u h, v v h, and Ω Ω h in the weighted residual statement.

18 1 Introduction 2 Weighted Residuals 3 Poisson Equation 4 Other Examples 5 Essential BCs 6 Some Extensions

19 Weighted Residual Statement For simplicity we will focus on the weighted residual statement arising from the Poisson equation, with Ω N =, (F(u h ), v h ) := Ω h [ uh v h fv h] dx = 0 v h V h

20 Element Integrals The integral over Ω h... 0 = Ω h [ uh v h fv h] dx v h V h

21 Element Integrals The integral over Ω h... is written as a sum of integrals over the N e finite elements: 0 = = [ uh v h fv h] dx Ω h N e [ uh v h fv h] dx Ω e e=1 v h V h v h V h

22 Finite Element Basis Functions An element integral will have contributions only from the global basis functions corresponding to its nodes. We call these local basis functions φ i, 0 i N s. v h N s Ωe = c i φ i i= Ω e

23 Finite Element Basis Functions An element integral will have contributions only from the global basis functions corresponding to its nodes. We call these local basis functions φ i, 0 i N s. v h N s Ωe = c i φ i i=1 N s v h dx = c i φ i dx Ω e Ω e i= Ω e

24 Element Matrix and Load Vector The element integrals... Ω e [ uh v h fv h] dx

25 Element Matrix and Load Vector The element integrals... Ω e [ uh v h fv h] dx are written in terms of the local φ i basis functions N s j=1 u j φ j φ i dx f φ i dx, i = 1,..., N s Ω e Ω e

26 Element Matrix and Load Vector The element integrals... Ω e [ uh v h fv h] dx are written in terms of the local φ i basis functions N s j=1 u j φ j φ i dx f φ i dx, i = 1,..., N s Ω e Ω e This can be expressed naturally in matrix notation as K e U e F e

27 Global Linear System The entries of the element stiffness matrix are the integrals K e ij := φ j φ i dx Ω e

28 Global Linear System The entries of the element stiffness matrix are the integrals K e ij := φ j φ i dx Ω e While for the element right-hand side we have F e i := f φ i dx Ω e

29 Global Linear System The entries of the element stiffness matrix are the integrals K e ij := φ j φ i dx Ω e While for the element right-hand side we have F e i := f φ i dx Ω e The element stiffness matrices and right-hand sides can be assembled to obtain the global system of equations KU = F

30 Reference Element Map The integrals are performed on a reference element ˆΩ e x Ω e (ξ) x ξ Ω e

31 Reference Element Map The integrals are performed on a reference element ˆΩ e x Ω e (ξ) x ξ Ω e The Jacobian of the map x(ξ) is J. F e i = f φ i dx = Ω e f (x(ξ))φ i J dξ ˆΩ e

32 Reference Element Map The integrals are performed on a reference element ˆΩ e x Ω e (ξ) x ξ Ω e Chain rule: = J 1 ξ := ˆ ξ K e ij = φ j φ i dx = Ω e ˆΩ e ˆ ξ φ j ˆ ξ φ i J dξ

33 Element Quadrature The integrals on the reference element are approximated via numerical quadrature.

34 Element Quadrature The integrals on the reference element are approximated via numerical quadrature. The quadrature rule has N q points ξ q and weights w q.

35 Element Quadrature The integrals on the reference element are approximated via numerical quadrature. The quadrature rule has N q points ξ q and weights w q. F e i = f φ i J dξ ˆΩ e N q q=1 f (x(ξ q ))φ i (ξ q ) J(ξ q ) w q

36 Element Quadrature The integrals on the reference element are approximated via numerical quadrature. The quadrature rule has N q points ξ q and weights w q. K e ij = ˆΩ e N q q=1 ˆ ξ φ j ˆ ξ φ i J dξ ˆ ξ φ j (ξ q ) ˆ ξ φ i (ξ q ) J(ξ q ) w q

37 LibMesh Quadrature Point Data LibMesh provides the following variables at each quadrature point q Code Math Description JxW[q] J(ξ q ) w q Jacobian times weight phi[i][q] φ i (ξ q ) value of i th shape fn. dphi[i][q] ˆ ξ φ i (ξ q ) value of i th shape fn. gradient xyz[q] x(ξ q ) location of ξ q in physical space

38 Matrix Assembly Loops The LibMesh representation of the matrix and rhs assembly is similar to the mathematical statements. for (q=0; q<nq; ++q) for (i=0; i<ns; ++i) { Fe(i) += JxW[q]*f(xyz[q])*phi[i][q]; } for (j=0; j<ns; ++j) Ke(i,j) += JxW[q]*(dphi[j][q]*dphi[i][q]);

39 Matrix Assembly Loops The LibMesh representation of the matrix and rhs assembly is similar to the mathematical statements. for (q=0; q<nq; ++q) for (i=0; i<ns; ++i) { Fe(i) += JxW[q]*f(xyz[q])*phi[i][q]; } for (j=0; j<ns; ++j) Ke(i,j) += JxW[q]*(dphi[j][q]*dphi[i][q]); N q F e i = f (x(ξ q ))φ i (ξ q ) J(ξ q ) w q q=1

40 Matrix Assembly Loops The LibMesh representation of the matrix and rhs assembly is similar to the mathematical statements. for (q=0; q<nq; ++q) for (i=0; i<ns; ++i) { Fe(i) += JxW[q]*f(xyz[q])*phi[i][q]; } for (j=0; j<ns; ++j) Ke(i,j) += JxW[q]*(dphi[j][q]*dphi[i][q]); N q F e i = f (x(ξ q ))φ i (ξ q ) J(ξ q ) w q q=1

41 Matrix Assembly Loops The LibMesh representation of the matrix and rhs assembly is similar to the mathematical statements. for (q=0; q<nq; ++q) for (i=0; i<ns; ++i) { Fe(i) += JxW[q]*f(xyz[q])*phi[i][q]; } for (j=0; j<ns; ++j) Ke(i,j) += JxW[q]*(dphi[j][q]*dphi[i][q]); N q F e i = f (x(ξ q ))φ i (ξ q ) J(ξ q ) w q q=1

42 Matrix Assembly Loops The LibMesh representation of the matrix and rhs assembly is similar to the mathematical statements. for (q=0; q<nq; ++q) for (i=0; i<ns; ++i) { Fe(i) += JxW[q]*f(xyz[q])*phi[i][q]; } for (j=0; j<ns; ++j) Ke(i,j) += JxW[q]*(dphi[j][q]*dphi[i][q]); N q F e i = f (x(ξ q ))φ i (ξ q ) J(ξ q ) w q q=1

43 Matrix Assembly Loops The LibMesh representation of the matrix and rhs assembly is similar to the mathematical statements. for (q=0; q<nq; ++q) for (i=0; i<ns; ++i) { Fe(i) += JxW[q]*f(xyz[q])*phi[i][q]; } for (j=0; j<ns; ++j) Ke(i,j) += JxW[q]*(dphi[j][q]*dphi[i][q]); K e ij = N q q=1 ˆ ξ φ j (ξ q ) ˆ ξ φ i (ξ q ) J(ξ q ) w q

44 Matrix Assembly Loops The LibMesh representation of the matrix and rhs assembly is similar to the mathematical statements. for (q=0; q<nq; ++q) for (i=0; i<ns; ++i) { Fe(i) += JxW[q]*f(xyz[q])*phi[i][q]; } for (j=0; j<ns; ++j) Ke(i,j) += JxW[q]*(dphi[j][q]*dphi[i][q]); K e ij = N q q=1 ˆ ξ φ j (ξ q ) ˆ ξ φ i (ξ q ) J(ξ q ) w q

45 Matrix Assembly Loops The LibMesh representation of the matrix and rhs assembly is similar to the mathematical statements. for (q=0; q<nq; ++q) for (i=0; i<ns; ++i) { Fe(i) += JxW[q]*f(xyz[q])*phi[i][q]; } for (j=0; j<ns; ++j) Ke(i,j) += JxW[q]*(dphi[j][q]*dphi[i][q]); K e ij = N q q=1 ˆ ξ φ j (ξ q ) ˆ ξ φ i (ξ q ) J(ξ q ) w q

46 1 Introduction 2 Weighted Residuals 3 Poisson Equation 4 Other Examples 5 Essential BCs 6 Some Extensions

47 Convection-Diffusion Equation The matrix assembly routine for the linear convection-diffusion equation, k u + b u = f for (q=0; q<nq; ++q) for (i=0; i<ns; ++i) { Fe(i) += JxW[q]*f(xyz[q])*phi[i][q]; } for (j=0; j<ns; ++j) Ke(i,j) += JxW[q]*(k*(dphi[j][q]*dphi[i][q]) +(b*dphi[j][q])*phi[i][q]);

48 Convection-Diffusion Equation The matrix assembly routine for the linear convection-diffusion equation, k u + b u = f for (q=0; q<nq; ++q) for (i=0; i<ns; ++i) { Fe(i) += JxW[q]*f(xyz[q])*phi[i][q]; } for (j=0; j<ns; ++j) Ke(i,j) += JxW[q]*(k*(dphi[j][q]*dphi[i][q]) +(b*dphi[j][q])*phi[i][q]);

49 Convection-Diffusion Equation The matrix assembly routine for the linear convection-diffusion equation, k u + b u = f for (q=0; q<nq; ++q) for (i=0; i<ns; ++i) { Fe(i) += JxW[q]*f(xyz[q])*phi[i][q]; } for (j=0; j<ns; ++j) Ke(i,j) += JxW[q]*(k*(dphi[j][q]*dphi[i][q]) +(b*dphi[j][q])*phi[i][q]);

50 Stokes Flow For multi-variable systems like Stokes flow, p ν u = f u = 0 Ω R 2 The element stiffness matrix concept can extended to include sub-matrices Ku e 1 u 1 Ku e 1 u 2 Ku e 1 p U Ku e 2 u 1 Ku e 2 u 2 Ku e u e 1 F 2 p U e u e 1 Kpu e 1 Kpu e 2 Kpp e u 2 F e Up e u 2 Fp e We have an array of submatrices: Ke[ ][ ]

51 Stokes Flow For multi-variable systems like Stokes flow, p ν u = f u = 0 Ω R 2 The element stiffness matrix concept can extended to include sub-matrices Ku e 1 u 1 Ku e 1 u 2 Ku e 1 p U Ku e 2 u 1 Ku e 2 u 2 Ku e u e 1 F 2 p U e u e 1 Kpu e 1 Kpu e 2 Kpp e u 2 F e Up e u 2 Fp e We have an array of submatrices: Ke[0][0]

52 Stokes Flow For multi-variable systems like Stokes flow, p ν u = f u = 0 Ω R 2 The element stiffness matrix concept can extended to include sub-matrices Ku e 1 u 1 Ku e 1 u 2 Ku e 1 p U Ku e 2 u 1 Ku e 2 u 2 Ku e u e 1 F 2 p U e u e 1 Kpu e 1 Kpu e 2 Kpp e u 2 F e Up e u 2 Fp e We have an array of submatrices: Ke[1][1]

53 Stokes Flow For multi-variable systems like Stokes flow, p ν u = f u = 0 Ω R 2 The element stiffness matrix concept can extended to include sub-matrices Ku e 1 u 1 Ku e 1 u 2 Ku e 1 p U Ku e 2 u 1 Ku e 2 u 2 Ku e u e 1 F 2 p U e u e 1 Kpu e 1 Kpu e 2 Kpp e u 2 F e Up e u 2 Fp e We have an array of submatrices: Ke[2][1]

54 Stokes Flow For multi-variable systems like Stokes flow, p ν u = f u = 0 Ω R 2 The element stiffness matrix concept can extended to include sub-matrices Ku e 1 u 1 Ku e 1 u 2 Ku e 1 p U Ku e 2 u 1 Ku e 2 u 2 Ku e u e 1 F 2 p U e u e 1 Kpu e 1 Kpu e 2 Kpp e u 2 F e Up e u 2 Fp e And an array of right-hand sides: Fe[].

55 Stokes Flow For multi-variable systems like Stokes flow, p ν u = f u = 0 Ω R 2 The element stiffness matrix concept can extended to include sub-matrices Ku e 1 u 1 Ku e 1 u 2 Ku e 1 p U Ku e 2 u 1 Ku e 2 u 2 Ku e u e 1 F 2 p U e u e 1 Kpu e 1 Kpu e 2 Kpp e u 2 F e Up e u 2 Fp e And an array of right-hand sides: Fe[0].

56 Stokes Flow For multi-variable systems like Stokes flow, p ν u = f u = 0 Ω R 2 The element stiffness matrix concept can extended to include sub-matrices Ku e 1 u 1 Ku e 1 u 2 Ku e 1 p U Ku e 2 u 1 Ku e 2 u 2 Ku e u e 1 F 2 p U e u e 1 Kpu e 1 Kpu e 2 Kpp e u 2 F e Up e u 2 Fp e And an array of right-hand sides: Fe[1].

57 Stokes Flow The matrix assembly can proceed in essentially the same way. For the momentum equations: for (q=0; q<nq; ++q) for (d=0; d<2; ++d) for (i=0; i<ns; ++i) { Fe[d](i) += JxW[q]*f(xyz[q],d)*phi[i][q]; } for (j=0; j<ns; ++j) Ke[d][d](i,j) += JxW[q]*nu*(dphi[j][q]*dphi[i][q]);

58 1 Introduction 2 Weighted Residuals 3 Poisson Equation 4 Other Examples 5 Essential BCs 6 Some Extensions

59 Essential Boundary Data Dirichlet boundary conditions can be enforced after the global stiffness matrix K has been assembled This usually involves 1 placing a 1 on the main diagonal of the global stiffness matrix 2 zeroing out the row entries 3 placing the Dirichlet value in the rhs vector 4 subtracting off the column entries from the rhs

60 Essential Boundary Data Dirichlet boundary conditions can be enforced after the global stiffness matrix K has been assembled This usually involves 1 placing a 1 on the main diagonal of the global stiffness matrix 2 zeroing out the row entries 3 placing the Dirichlet value in the rhs vector 4 subtracting off the column entries from the rhs

61 Essential Boundary Data Dirichlet boundary conditions can be enforced after the global stiffness matrix K has been assembled This usually involves 1 placing a 1 on the main diagonal of the global stiffness matrix 2 zeroing out the row entries 3 placing the Dirichlet value in the rhs vector 4 subtracting off the column entries from the rhs

62 Essential Boundary Data Dirichlet boundary conditions can be enforced after the global stiffness matrix K has been assembled This usually involves 1 placing a 1 on the main diagonal of the global stiffness matrix 2 zeroing out the row entries 3 placing the Dirichlet value in the rhs vector 4 subtracting off the column entries from the rhs

63 Essential Boundary Data Dirichlet boundary conditions can be enforced after the global stiffness matrix K has been assembled This usually involves 1 placing a 1 on the main diagonal of the global stiffness matrix 2 zeroing out the row entries 3 placing the Dirichlet value in the rhs vector 4 subtracting off the column entries from the rhs k 11 k 12 k 13. f g 1 k 21 k 22 k 23. k 31 k 32 k 33., f 2 f 3 0 k 22 k k 32 k 33., f 2 k 21 g 1 f 3 k 31 g

64 Essential Boundary Data Cons of this approach : Works for an interpolary finite element basis but not in general. May be inefficient to change individual entries once the global matrix is assembled. Need to enforce boundary conditions for a generic finite element basis at the element stiffness matrix level.

65 Essential Boundary Data Cons of this approach : Works for an interpolary finite element basis but not in general. May be inefficient to change individual entries once the global matrix is assembled. Need to enforce boundary conditions for a generic finite element basis at the element stiffness matrix level.

66 Essential Boundary Data Cons of this approach : Works for an interpolary finite element basis but not in general. May be inefficient to change individual entries once the global matrix is assembled. Need to enforce boundary conditions for a generic finite element basis at the element stiffness matrix level.

67 Essential Boundary Data Cons of this approach : Works for an interpolary finite element basis but not in general. May be inefficient to change individual entries once the global matrix is assembled. Need to enforce boundary conditions for a generic finite element basis at the element stiffness matrix level.

68 Penalty Formulation One solution is the penalty boundary formulation A term is added to the standard weighted residual statement (F(u), v) + 1 (u u D )v dx = 0 v V ɛ Ω } D {{} penalty term Here ɛ 1 is chosen so that, in floating point arithmetic, 1 ɛ + 1 = 1 ɛ. This weakly enforces u = u D on the Dirichlet boundary, and works for general finite element bases.

69 Penalty Formulation One solution is the penalty boundary formulation A term is added to the standard weighted residual statement (F(u), v) + 1 (u u D )v dx = 0 v V ɛ Ω } D {{} penalty term Here ɛ 1 is chosen so that, in floating point arithmetic, 1 ɛ + 1 = 1 ɛ. This weakly enforces u = u D on the Dirichlet boundary, and works for general finite element bases.

70 Penalty Formulation One solution is the penalty boundary formulation A term is added to the standard weighted residual statement (F(u), v) + 1 (u u D )v dx = 0 v V ɛ Ω } D {{} penalty term Here ɛ 1 is chosen so that, in floating point arithmetic, 1 ɛ + 1 = 1 ɛ. This weakly enforces u = u D on the Dirichlet boundary, and works for general finite element bases.

71 Penalty Formulation One solution is the penalty boundary formulation A term is added to the standard weighted residual statement (F(u), v) + 1 (u u D )v dx = 0 v V ɛ Ω } D {{} penalty term Here ɛ 1 is chosen so that, in floating point arithmetic, 1 ɛ + 1 = 1 ɛ. This weakly enforces u = u D on the Dirichlet boundary, and works for general finite element bases.

72 Penalty Formulation LibMesh provides: A quadrature rule with Nqf points and JxW f[] A finite element coincident with the boundary face that has shape function values phi f[][] for (qf=0; qf<nqf; ++qf) { for (i=0; i<nf; ++i) { Fe(i) += JxW_f[qf]* penalty*ud(xyz[q])*phi_f[i][qf]; } } for (j=0; j<nf; ++j) Ke(i,j) += JxW_f[qf]* penalty*phi_f[j][qf]*phi_f[i][qf];

73 Penalty Formulation LibMesh provides: A quadrature rule with Nqf points and JxW f[] A finite element coincident with the boundary face that has shape function values phi f[][] for (qf=0; qf<nqf; ++qf) { for (i=0; i<nf; ++i) { Fe(i) += JxW_f[qf]* penalty*ud(xyz[q])*phi_f[i][qf]; } } for (j=0; j<nf; ++j) Ke(i,j) += JxW_f[qf]* penalty*phi_f[j][qf]*phi_f[i][qf];

74 Penalty Formulation LibMesh provides: A quadrature rule with Nqf points and JxW f[] A finite element coincident with the boundary face that has shape function values phi f[][] for (qf=0; qf<nqf; ++qf) { for (i=0; i<nf; ++i) { Fe(i) += JxW_f[qf]* penalty*ud(xyz[q])*phi_f[i][qf]; } } for (j=0; j<nf; ++j) Ke(i,j) += JxW_f[qf]* penalty*phi_f[j][qf]*phi_f[i][qf];

75 1 Introduction 2 Weighted Residuals 3 Poisson Equation 4 Other Examples 5 Essential BCs 6 Some Extensions

76 Time-Dependent Problems For linear problems, we have already seen how the weighted residual statement leads directly to a sparse linear system of equations KU = F

77 Time-Dependent Problems For time-dependent problems, u t = F(u) we also need a way to advance the solution in time, e.g. a θ-method ( ) u n+1 u n, v h = ( F(u θ ), v h) v h V h t u θ := θu n+1 + (1 θ)u n Leads to KU = F at each timestep.

78 Nonlinear Problems For nonlinear problems, typically a sequence of linear problems must be solved, e.g. for Newton s method (F (u k )δu k+1, v) = (F(u k ), v) where F (u k ) is the linearized (Jacobian) operator associated with the PDE. Must solve KU = F (Inexact Newton method) at each iteration step.

79 Non-Trivial Applications Common to each is the need to solve a linear system (or systems) of equations KU = F. LibMesh provides several of the tools necessary to construct these systems, but it is not specifically written to solve any one problem. We now briefly show a few of the non-trivial applications which have been built on top of the library.

80 Natural Convection Tetrahedral mesh of pipe geometry. Stream ribbons colored by temperature.

81 Surface-Tension-Driven Flow Adaptive grid solution shown with temperature contours and velocity vectors.

82 Double-Diffusive Convection Solute contours: a plume of warm, low-salinity fluid is convected upward through a porous medium.

83 Tumor Angiogenesis The tumor secretes a chemical which stimulates blood vessel formation.

84 Natural Convection Tetrahedral mesh of pipe geometry. Stream ribbons colored by temperature.

85 Surface-Tension-Driven Flow Adaptive grid solution shown with temperature contours and velocity vectors.

86 Double-Diffusive Convection Solute contours: a plume of warm, low-salinity fluid is convected upward through a porous medium.

87 Tumor Angiogenesis The tumor secretes a chemical which stimulates blood vessel formation.

88 B. Kirk, J. Peterson, R. Stogner and G. Carey, libmesh: a C++ library for parallel adaptive mesh refinement/coarsening simulations, Engineering with Computers, vol. 22, no. 3 4, p , 2006.

LibMesh Experience and Usage

LibMesh Experience and Usage John W. Peterson peterson@cfdlab.ae.utexas.edu and Roy H. Stogner roystgnr@cfdlab.ae.utexas.edu Univ. of Texas at Austin September 9, 2008 1 Introduction 2 Weighted Residuals