finite elements libmesh

Introduction Object Models System Assembly Examples Summary

libMesh Finite Element Library

Roy [email protected]

Derek [email protected]

1Univ. of Texas at Austin

2Sandia National LaboratoriesAlbuquerque,NM

August 22, 2007

Outline1 Introduction2 Object Models

Core ClassesBVP Framework

3 System AssemblyBasic ExampleCoupled VariablesEssential Boundary Conditions

4 ExamplesFluid DynamicsBiologyMaterial Science

5 Summary

libMesh is not

A physics implementation.

A stand-alone application.

libMesh is

A software library and toolkit.

Classes and functions for writing parallel adaptivefinite element applications.

An interface to linear algebra, meshing, partitioning,etc. libraries.

libMesh is notA physics implementation.

libMesh is

libMesh isA software library and toolkit.

For most applications weassume there is a BoundaryValue Problem to beapproximated in a FiniteElement function space

M∂u∂t

= F(u) ∈ Ω

G(u) = 0 ∈ Ω

u = uD ∈ ∂ΩD

N(u) = 0 ∈ ∂ΩN

u(x, 0) = u0(x)

Associated to Ω is theMesh data structure

A Mesh is basically acollection of geometricelements and nodes

Ωh :=⋃

Object Oriented Programming

Abstract Base Classes define user interfaces.

Concrete Subclasses implement functionality.

One physics code can work with manydiscretizations.

Core Classes

Geometric Element Classes

Elem#_nodes: Node **#_neighbors: Elem **#_parent: Elem *#_children: Elem **#_*flag: RefinementState#_p_level: unsigned char#_subdomain_id: unsigned char

+n_faces,sides,vertices,edges,children(): unsigned int+centroid(): Point+hmin,hmax(): Real

NodeElem FaceEdge Cell InfQuad InfCell

Prism Hex Pyramid Tet

Hex8 Hex20 Hex27

DofObject-_n_systems: unsigned char-_n_vars: unsigned char *-_n_comp: unsigned char **-_dof_ids: unsigned int **-_id: unsigned int-_processor_id: unsigned short int

NodeAbstract interfacegives mesh topology

Concreteinstantiations ofmesh geometry

Hides element typefrom mostapplications

Core Classes

Finite Element Classes

FEBase

+phi, dphi, d2phi

+quadrature_rule, JxW

+reinit(Elem)

+reinit(Elem,side)

Lagrange

Hermi te

Hierarchic

Monomial

Finite Element objectbuilds data for eachGeometric object

User only deals withshape function,quadrature data

Core Classes

Core Features

Mixed element geometries in unstructured grids

Adaptive mesh h refinement with hanging nodes, prefinement

Integration w/ PETSc, LASPack, METIS, ParMETIS,Triangle, TetGen

Support for UNV, ExodusII, Tecplot, GMV, UCD files

Mesh creation, modification utilities

Core Classes

Degree of Freedom Handling

DofObject subclasses store global Degree ofFreedom indices

DofMap class assigns indices to a partitioned mesh

FE classes calculate hanging node, periodicconstraints

DofMap class applies constraints

System class handles AMR/C projections

Core Classes

Generic Constraint Calculations

To maintain function space continuity, constrain “hangingnode” Degrees of Freedom coefficients on fine elementsin terms of DoFs on coarse neighbors.

uF = uC∑i

uFi φF

i =∑

uCj φC

Akiui = Bkjuj

ui = A−1ki Bkjuj

Integrated values (and fluxes, for C1 continuity) giveelement-independent matrices:

Aki ≡ (φFi , φF

Bkj ≡ (φCj , φC

Core Classes

Generic Projection Calculations

Upon element coarsening (or refinement in non-nestedspaces):

Copy nodal Degree of Freedom coefficients

Project edge DoFs, holding nodal DoFs constant

Project face DoFs, holding nodes/edges constant

Project interior DoFs, holding boundaries constant

Advantages / Disadvantages

Requires only local solves

Consistent in parallel

May violate physical conservation laws

Core Classes

libMesh Parallelization

Parallel code

PetscVector, DistributedVector classes for LinearAlgebra

Parallel assembly, error indicators, etc.

Mesh partitioning with ParMETIS, etc.

Serial code

Mesh storage on every node

DoF renumbering, constraint calculations

Refinement/Coarsening flagging

Mesh, solution I/O

Core Classes

Parallel codePetscVector, DistributedVector classes for LinearAlgebra

Serial code

Mesh, solution I/O

Core Classes

Serial code

Mesh, solution I/O

Core Classes

Serial code

Mesh, solution I/O

Core Classes

Serial code

Mesh, solution I/O

Core Classes

Serial codeMesh storage on every node

Mesh, solution I/O

Core Classes

Mesh, solution I/O

Core Classes

Mesh, solution I/O

Core Classes

Mesh, solution I/O

Core Classes

ParallelMesh

Parallel subclassing of MeshBase

Start from unstructured Mesh class

Add methods to delete, reconstruct non-semilocalElem and Node objects

Parallelize DofMap methods

Parallelize MeshRefinement methods

Add parallel or chunked I/O support

Add load balancing support

Also want parallel BoundaryInfo, BoundaryMesh, Data,Function, Generation, Modification, Smoother, and Toolsclasses

BVP Framework

Boundary Value Problem Framework Goals

GoalsImproved test coverage and reliability

Hiding implementation details from user code

Rapid prototyping of new formulations

Physics-dependent error estimators

MethodsObject-oriented System and Solver subclasses

Factoring common patterns into library code

Per-element Numerical Jacobian verification

BVP Framework

FEM System Classes

FEMSystem+elem_solution: DenseVector<Number>+elem_residual: DenseVector<Number>+elem_jacobian: DenseMatrix<Number>#elem_fixed_solution: DenseVector<Number>#*_fe_var: std::vector<FEBase *>#elem: Elem *

+*_time_derivative(request_jacobian)+*_constraint(request_jacobian)+*_postprocess()

NavierStokesSystem

LaplaceYoungSystem

CahnHilliardSystem

SurfactantSystem

Generalized IBVPrepresentation

FEMSystem does allinitialization, globalassembly

User code onlyneeds weighted timederivative residuals(∂u

∂t , vi) = Fi(u) and/orconstraintsGi(u, vi) = 0

BVP Framework

ODE Solver Classes

TimeSolver

+*_residual(request_jacobian)+solve()+advance_timestep()

SteadySolver EulerSolver

AdamsMoultonSolver EigenSolver

Calls user code oneach element

Assembleselement-by-elementtime derivatives,constraints, andweighted oldsolutions

BVP Framework

Nonlinear Solver Classes

NonlinearSolver+*_tolerance+*_max_iterations

+solve()

QuasiNewtonSolverLinearSolver

ContinuationSolver

Acquires residuals,jacobians fromFEMSystemassembly

Handles inner loops,inner solvers andtolerances,convergence tests,etc

Basic Example

System Assembly

For simplicity we will focus on the weighted residualstatement arising from the Poisson equation, with∂ΩN = ∅,

(R(uh), vh) :=∫Ωh

[∇uh · ∇vh − fvh

]dx = 0 ∀vh ∈ Vh

Basic Example

The element integrals . . .∫Ωe

are written in terms of the local “φi” basis functions

Ns∑j=1

∫Ωe

∇φj · ∇φi dx−∫

fφi dx , i = 1, . . . , Ns

This can be expressed naturally in matrix notation as

KeUe− Fe

Basic Example

Ns∑j=1

∫Ωe

fφi dx , i = 1, . . . , Ns

KeUe− Fe

Basic Example

Ns∑j=1

∫Ωe

fφi dx , i = 1, . . . , Ns

KeUe− Fe

Basic Example

The integrals are performed on a “reference” elementΩe, transformed via Lagrange basis to the geometricelement.

eΩx ξ)x(

Basic Example

eΩx x(ξ)

The Jacobian of the map x(ξ) is J.

∫Ωe

fφidx =

∫Ωe

f (x(ξ))φi|J|dξ

Basic Example

eΩx ξ)x(

Basic Example

The integrals on the “reference” element areapproximated via numerical quadrature.

The quadrature rule has Nq points “ξq” and weights“wq”.

Basic Example

∫Ωe

fφi|J|dξ

≈Nq∑

f (x(ξq))φi(ξq)|J(ξq)|wq

Basic Example

Keij =

∫Ωe

∇φj · ∇φi |J|dξ

≈Nq∑

∇φj(ξq) · ∇φi(ξq)|J(ξq)|wq

Basic Example

At the qth quadrature point, LibMesh can providevariables including:

Code Math Description

JxW[q] |J(ξq)|wq Jacobian times weight

phi[i][q] φi(ξq) value of i th shape fn.

dphi[i][q] ∇φi(ξq) value of i th shape fn. gradient

d2phi[i][q] ∇∇φi(ξq) value of i th shape fn. Hessian

xyz[q] x(ξq) location of ξq in physical space

normals[q] ~n(x(ξq)) normal vector at x on a side

Basic Example

The LibMesh representation of the matrix and rhsassembly 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] );

Basic Example

Nq∑q=1

Basic Example

Nq∑q=1

Basic Example

Nq∑q=1

Basic Example

Nq∑q=1

Basic Example

Keij =

Nq∑q=1

∇ξφj(ξq) · ∇ξφi(ξq)|J(ξq)|wq

Basic Example

Keij =

Nq∑q=1

Basic Example

Keij =

Nq∑q=1

Basic Example

Convection-Diffusion Equation

The matrix assembly routine for the linearconvection-diffusion equation,

−k∆u + b · ∇u = f

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]);

Basic Example

−k∆u + b · ∇u = f

Basic Example

−k∆u + b · ∇u = f

Coupled Variables

Stokes Flow

For multi-variable systems like Stokes flow,

∇p− ν∆u = f∇ · u = 0

∈ Ω ⊂ R2

The element stiffness matrix concept can extended toinclude sub-matrices Ke

u1u1Ke

u1u2Ke

Keu2u1

Keu2u2

− Fe

We have an array of submatrices: Ke[ ][ ]

Coupled Variables

Stokes Flow

∇p− ν∆u = f∇ · u = 0

∈ Ω ⊂ R2

u1u1Ke

u1u2Ke

Keu2u1

Keu2u2

− Fe

We have an array of submatrices: Ke[ 1][ 1]

Coupled Variables

Stokes Flow

∇p− ν∆u = f∇ · u = 0

∈ Ω ⊂ R2

u1u1Ke

u1u2Ke

Keu2u1

Keu2u2

− Fe

Coupled Variables

Stokes Flow

∇p− ν∆u = f∇ · u = 0

∈ Ω ⊂ R2

u1u1Ke

u1u2Ke

Keu2u1

Keu2u2

− Fe

Coupled Variables

Stokes Flow

∇p− ν∆u = f∇ · u = 0

∈ Ω ⊂ R2

u1u1Ke

u1u2Ke

Keu2u1

Keu2u2

− Fe

And an array of right-hand sides: Fe[] .

Coupled Variables

Stokes Flow

∇p− ν∆u = f∇ · u = 0

∈ Ω ⊂ R2

u1u1Ke

u1u2Ke

Keu2u1

Keu2u2

− Fe

And an array of right-hand sides: Fe[ 1] .

Coupled Variables

Stokes Flow

∇p− ν∆u = f∇ · u = 0

∈ Ω ⊂ R2

u1u1Ke

u1u2Ke

Keu2u1

Keu2u2

− Fe

And an array of right-hand sides: Fe[ 2] .

Coupled Variables

Stokes Flow

The matrix assembly can proceed in essentially thesame 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]);

Essential Boundary Conditions

Essential Boundary Data

Dirichlet boundary conditions can be enforced afterthe global stiffness matrix K has been assembledThis usually involves

1 placing a “1” on the main diagonal of the globalstiffness matrix

2 zeroing out the row entries3 placing the Dirichlet value in the rhs vector4 subtracting off the column entries from the rhs

k11 k12 k13 .k21 k22 k23 .k31 k32 k33 .. . . .

f1f2f3.

1 0 0 00 k22 k23 .0 k32 k33 .0 . . .

f2 − k21g1

f3 − k31g1

2 zeroing out the row entries

3 placing the Dirichlet value in the rhs vector4 subtracting off the column entries from the rhs

k11 k12 k13 .k21 k22 k23 .k31 k32 k33 .. . . .

f1f2f3.

1 0 0 00 k22 k23 .0 k32 k33 .0 . . .

f2 − k21g1

f3 − k31g1

2 zeroing out the row entries3 placing the Dirichlet value in the rhs vector

4 subtracting off the column entries from the rhs

k11 k12 k13 .k21 k22 k23 .k31 k32 k33 .. . . .

f1f2f3.

1 0 0 00 k22 k23 .0 k32 k33 .0 . . .

f2 − k21g1

f3 − k31g1

k11 k12 k13 .k21 k22 k23 .k31 k32 k33 .. . . .

f1f2f3.

1 0 0 00 k22 k23 .0 k32 k33 .0 . . .

f2 − k21g1

f3 − k31g1

k11 k12 k13 .k21 k22 k23 .k31 k32 k33 .. . . .

f1f2f3.

1 0 0 00 k22 k23 .0 k32 k33 .0 . . .

f2 − k21g1

f3 − k31g1

Cons of this approach :

Works for an interpolary finite element basis but not ingeneral.May be inefficient to change individual entries oncethe global matrix is assembled.

Need to enforce boundary conditions for a genericfinite element basis at the element stiffness matrixlevel.

Cons of this approach :Works for an interpolary finite element basis but not ingeneral.

May be inefficient to change individual entries oncethe global matrix is assembled.

Cons of this approach :Works for an interpolary finite element basis but not ingeneral.May be inefficient to change individual entries oncethe global matrix is assembled.

A “penalty” term is added to the standard weightedresidual statement

(R(u), v) +1ε

∫∂ΩD

(u− uD)v dx︸︷︷︸penalty term

= 0 ∀v ∈ V

Here ε 1 is chosen so that, in floating pointarithmetic, 1

ε+ 1 = 1

This weakly enforces u = uD on the Dirichletboundary, and works for general finite element bases.

(R(u), v) +1ε

∫∂ΩD

= 0 ∀v ∈ V

ε+ 1 = 1

(R(u), v) +1ε

∫∂ΩD

= 0 ∀v ∈ V

ε+ 1 = 1

LibMesh provides:

A quadrature rule with Nqf points and JxW f[]

A finite element coincident with the boundary facethat 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];

LibMesh provides:

Fluid Dynamics

Introduction

Laplace-Young equation model surface tension effects forenclosed liquids.

Combining surface tension, gravity and contact the energyfunctional for Laplace-Young is:∫

√1 + |∇u|2 dΩ +

12κu2 dΩ−

∫∂Ω

σu ds

Where κ is the ratio of surface energy to gravitationalenergy and u is the height of the liquid.

While the weak formulation of the stationary condition isgiven by:(

∇u√1 + |∇u|2

,∇ϕ

+ κ (u, ϕ)Ω = σ (1, ϕ)∂Ω (1)

By specifying the parameter σ = cos(γ) (where γ is thecontact angle) along the boundary, the liquid height u canbe solved for.

Fluid Dynamics

Introduction

√1 + |∇u|2 dΩ +

12κu2 dΩ−

∫∂Ω

σu ds

∇u√1 + |∇u|2

,∇ϕ

+ κ (u, ϕ)Ω = σ (1, ϕ)∂Ω (1)

Fluid Dynamics

Introduction

√1 + |∇u|2 dΩ +

12κu2 dΩ−

∫∂Ω

σu ds

∇u√1 + |∇u|2

,∇ϕ

+ κ (u, ϕ)Ω = σ (1, ϕ)∂Ω (1)

Fluid Dynamics

Introduction

√1 + |∇u|2 dΩ +

12κu2 dΩ−

∫∂Ω

σu ds

∇u√1 + |∇u|2

,∇ϕ

+ κ (u, ϕ)Ω = σ (1, ϕ)∂Ω (1)

Fluid Dynamics

Introduction

√1 + |∇u|2 dΩ +

12κu2 dΩ−

∫∂Ω

σu ds

∇u√1 + |∇u|2

,∇ϕ

+ κ (u, ϕ)Ω = σ (1, ϕ)∂Ω (1)

Fluid Dynamics

Instead of explicitly finding the Jacobian, we’ll use FEMSystem to finite difference the weak form.

element constraint()

for (unsigned int qp=0; qp != n_qpoints; qp++) Number u = interior_value(0, qp);Gradient grad_u = interior_gradient(0, qp);Number K = 1. / sqrt(1. + (grad_u * grad_u));

for (unsigned int i=0; i != n_u_dofs; i++) Fu(i) += JxW[qp] * (( _kappa * u * phi[i][qp] ) +

( K * grad_u * dphi[i][qp] ));

side constraint()

for (unsigned int qp=0; qp != n_qpoints; qp++) for (unsigned int i=0; i != n_u_dofs; i++)

Fu(i) -= JxW[qp] * _gamma * phi[i][qp] ;

0B@ ∇uq1 + |∇u|2

,∇ϕ