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Introduction Applications Weighted Residuals Finite Elements Essential BCs Complex Problems Adaptivity

LibMesh

Roy H. Stogner John W. [email protected] [email protected]

Univ. of Texas at Austin

June 6, 2007


Outline

In this talk we will discuss:Goals and example applications of the libMeshlibrary.Some basic steps in writing a simple libMeshapplication.Design frameworks for more complex applications.Adaptive Mesh Refinement.


Goals

libMesh is not:A physics implementation.A stand-alone application.

libMesh is:A software library and toolkit.Objects and functions for writing parallel adaptivefinite element applications.An interface to linear algebra, meshing, partitioning,etc. libraries.


For this talk we will assumethere is a mathematicalmodel (Partial DifferentialEquation) to be solved in anengineering analysis:

∂u∂t

= R(u) ∈ Ω

u = uD ∈ ∂ΩD

∇u · n = uN ∈ ∂ΩN

δΩD

δΩN

Ω


Associated to theproblem domain Ω is alibMesh data structurecalled a Mesh

A Mesh is essentially acollection of finiteelements

Ωh :=⋃

e

Ωe

Ωh

libMesh provides some simple structured meshgeneration routines as well as an interface toTriangle.


Compressible Gas Flow

Mach 3 flow over a forward facing step.


Shallow Water Flow

Wave propagation from depth-averaged equations


Natural Convection

Tetrahedral mesh of “pipe” geometry. Stream ribbonscolored by temperature.


Surface-Tension-Driven Flow

Adaptive grid solution shown with temperaturecontours and velocity vectors.


Double-Diffusive Convection

Solute contours: a plume of warm, low-salinity fluid isconvected upward through a porous medium.


Tumor Angiogenesis

The tumor secretes a chemical which stimulatesblood vessel formation.


Cahn-Hilliard Phase Decomposition

Quenching separates fluid or alloy mixtures intomultiple material phases.


The point of departure in any FE analysis which usesLibMesh is the weighted residual statement

(R(u), v) = 0 ∀v ∈ V

Or, more precisely, the weighted residual statementassociated with the finite-dimensional space Vh ⊂ V

(R(uh), vh) = 0 ∀vh ∈ Vh


Some Examples

Poisson Equation

−∆u = f ∈ Ω


Some Examples

Poisson Equation

−∆u = f ∈ Ω

Weighted Residual Statement

(R(u), v) :=

∫Ω

[∇u · ∇v− fv] dx

+

∫∂ΩN

(∇u · n) v ds


Some Examples

Linear Convection-Diffusion

−k∆u + b · ∇u = f ∈ Ω


Some Examples

Linear Convection-Diffusion

−k∆u + b · ∇u = f ∈ Ω


(R(u), v) :=

∫Ω

[k∇u · ∇v + (b · ∇u)v− fv] dx

+

∫∂ΩN

k (∇u · n) v ds


Some Examples

Stokes Flow

∇p− ν∆u = f∇ · u = 0 ∈ Ω


Some Examples

Stokes Flow

∇p− ν∆u = f∇ · u = 0 ∈ Ω


u := [u, p] , v := [v, q]

(R(u), v) :=

∫Ω

[−p (∇ · v) + ν∇u :∇v− f · v

+ (∇ · u) q] dx +

∫∂ΩN

(ν∇u− pI) · n · v ds


To obtain the approximate problem, we simplyreplace u← uh, v← vh, and Ω← Ωh in the weightedresidual statement.


Poisson Equation

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


The integral over Ωh . . .

is written as a sum ofintegrals over the Ne finite elements:

0 =

∫Ωh

[∇uh · ∇vh − fvh] dx ∀vh ∈ Vh

=Ne∑

e=1

∫Ωe



The integral over Ωh . . . is written as a sum ofintegrals over the Ne finite elements:

0 =

∫Ωh


=Ne∑

e=1

∫Ωe



An element integral willhave contributions onlyfrom the global basisfunctions correspondingto its nodes.We call these local basisfunctions φi, 0 ≤ i ≤ Ns.

vh∣∣Ωe

=Ns∑

i=1

ciφi

∫Ωe

vh dx =Ns∑

i=1

ci

∫Ωe

φi dx

Ωe


The element integrals . . .∫Ωe

[∇uh · ∇vh − fvh] dx

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

Ns∑j=1

uj

∫Ωe

∇φj · ∇φi dx−∫

Ωe

f φi dx , i = 1, . . . , Ns

This can be expressed naturally in matrix notation as

KeUe − Fe


The entries of the element stiffness matrix are theintegrals

Keij :=

∫Ωe

∇φj · ∇φi dx

While for the element right-hand side we have

Fei :=

∫Ωe

f φi dx

The element stiffness matrices and right-hand sidescan be “assembled” to obtain the global system ofequations

KU = F


The integrals are performed on a “reference” elementΩe

eΩx ξ)x(

ξ

Ωe



eΩx x(ξ)

ξ

Ωe

The Jacobian of the map x(ξ) is J.

Fei =

∫Ωe

f φidx =

∫Ωe

f (x(ξ))φi|J|dξ



eΩx ξ)x(

ξ

Ωe

Chain rule: ∇ = J−1∇ξ := ∇ξ.

Keij =

∫Ωe

∇φj · ∇φi dx =

∫Ωe

∇ξφj · ∇ξφi |J|dξ


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

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


The integrals on the “reference” element areapproximated via numerical quadrature.The quadrature rule has Nq points “ξq” and weights“wq”.



Fei =

∫Ωe

f φi|J|dξ

≈Nq∑

q=1

f (x(ξq))φi(ξq)|J(ξq)|wq



Keij =

∫Ωe

∇ξφj · ∇ξφi |J|dξ

≈Nq∑

q=1

∇ξφj(ξq) · ∇ξφi(ξq)|J(ξq)|wq


LibMesh provides the following variables at eachquadrature point q

Code Math Description

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

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

dphi[i][q] ∇ξφi(ξq) value of ith shape fn. gradient

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


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]);





Fei =

Nq∑q=1

f (x(ξq))φi(ξq)|J(ξq)|wq





Keij =

Nq∑q=1

∇ξφj(ξq) · ∇ξφi(ξq)|J(ξq)|wq


Object Oriented Programming

Abstract Base Classes define user interfaces.Concrete Subclasses implement functionality.One physics code can work with manydiscretizations.


Geometric Element Classes

Elem

#_nodes

#_neighbors, _parent, _children

#_refinement_flags

Quad4

Tet4

Edge3

Hex27

Abstract interfacegives mesh topologyConcreteinstantiations ofmesh geometryHides element typefrom mostapplications


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 objectUser only deals withshape function,quadrature data


For linear problems, we have already seen how theweighted residual statement leads directly to asparse linear system of equations

KU = F


For time-dependent problems,

∂u∂t

= R(u)

we also need a way to advance the solution in time,e.g. a θ-method(

un+1 − un

∆t, vh

)=

(R(uθ), vh) ∀vh ∈ Vh

uθ := θun+1 + (1− θ)un

Leads to KU = F at each timestep.


For nonlinear problems, typically a sequence of linearproblems must be solved, e.g. for Newton’s method

(R ′(uh)δuh, vh) = −(R(uh), vh)

where R ′(uh) is the linearized (Jacobian) operatorassociated with the PDE.Must solve KU = F (Inexact Newton method) at eachiteration step.


Boundary Value Problem Framework Goals

Goals:

Improving test coverage and reliabilityHiding of implementation details from user codeRapid prototyping of differential equationapproximationsImproved error estimation

Methods:

Object-oriented System and Solver classesNumerical Jacobian verification


FEM System Classes

FEMSystem

+element_solution

+element_residual

+element_jacobian

+FE_base

+*_time_derivative(request_jacobian)

+*_constraint(request_jacobian)

+*_postprocess()

NavierStokesSystem

LaplaceYoungSystem

CahnHil l iardSystem

SurfactantSystem

Generalized IBVPrepresentationFEMSystem does allinitialization, globalassemblyUser code onlyneeds weighted timederivative and/orconstraint functions


ODE Solver Classes

TimeSolver

+*_residual(request_jacobian)

+solve()

+advance_timestep()

SteadySolver EulerSolver

AdamsMoultonSolver EigenSolver

Calls user code oneach elementAssembleselement-by-elementtime derivatives,constraints, andweighted oldsolutions


Nonlinear Solver Classes

NonlinearSolver

+*_tolerance

+*_max_iterations

+solve()

QuasiNewtonSolverLinearSolver

ContinuationSolver

Acquires residuals,jacobians from ODEsolverHandles inner loops,inner solvers andtolerances,convergence tests,etc


1D refinement example

Consider the 1D model convection-diffusion equationequation

−u′′ + bu′ = 0 ∈ 0 ≤ x ≤ 1u(0) = 0u(1) = 1

The convection-diffusion equation can be thought ofas a particularly simple form of the drift-diffusionequation.The exact solution is

u =1− exp(bx)1− exp(b)


For large values of b, the solution changes rapidlynear x = 1.The solution for b = 10.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1


We assume here that we have an approximatesolution uh which is the linear interpolant of u.We will measure the error between the exact solutionu and the approximate solution uh in the following (L2)norm:

‖e‖2L2

:= ‖u− uh‖2L2

=

∫ 1

0(u− uh)

2 dx

Consider a sequence of uniformly-refined meshes . . .


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

uu

h

4 elements, ‖e‖L2 = 0.09


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

uu

h

8 elements, ‖e‖L2 = 0.027


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

uu

h

16 elements, ‖e‖L2 = 0.0071


Adaptive Refinement

Q: Can we do “better” than uniform refinement?A: Yes, if we refine cells which have higher errorrelative to others.


A Simple Error Indicator

In general we don’t know the exact solution, and sowe need a way of estimating the error.Consider the following formula for estimating theerror, η, in an element defined on the interval (xi, xi+1)

η2 :=h2

i+1∑k=i

Ju′(xk)K2

where h := xi+1 − xi is the element length, andThe “jump” in u′ is

Ju′(xk)K :=∣∣u′(x+

k )− u′(x−k )∣∣


Error Indicator, Uniformly-Refined Grids

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

x

u’

1 2 3 40

0.5

1

1.5

2

Element Number

η

4 elements



0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

x

u’

1 2 3 4 5 6 7 80

0.5

1

1.5

Element Number

η

8 elements



0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

x

u’

5 10 150

0.2

0.4

0.6

0.8

1

Element Number

η

16 elements


Adaptation Strategy

A simple adaptive refinement strategy with r maxrefinement steps for this 1D example problem is:

r=0;while (r < r_max)Compute the FE solution (linear interpolant)Estimate the error (using flux-jump indicator)Refine the element with highest errorIncrement r

end


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

uu

h

2 elements


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

uu

h

3 elements


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

uu

h

4 elements


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

uu

h

5 elements


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

uu

h

6 elements


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

uu

h

7 elements


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

uu

h

8 elements


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

uu

h

9 elements


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

uu

h

10 elements


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

uu

h

11 elements


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

uu

h

Final: 13 elements


Error Plot vs. Number of Nodes

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−3

−2.5

−2

−1.5

−1

−0.5

0

log10

Nnodes

log 10

||e|

|

UniformAdaptive


h Adaptivity Constraints

Non-conforming meshes lead to “hanging nodes”, and toprovide continuous finite element function the fineelement degrees of freedom must be constrained in termsof coarse element degrees of freedom.

FC

uF = uC∑i

uFi φF

i =∑

j

uCj φC

j

uFi = CijuC

j


Adaptive p Constraints

p refinement is donewith hierarchicadaptivityHanging degree offreedom coefficientsare simply set to 0


Diffuse Interface Modeling with AMR/C

Mesh coarsening insmooth regions istraded for meshrefinement in sharplayersEquivalent accuracyis achieved here with75% fewer degreesof freedom than auniform mesh


Installing libMesh

PETSc Download, Installation instructions,Documentation:http://www-unix.mcs.anl.gov/petsc/petsc-2/

libMesh Downloads, Installation instructions,Documentation:http://libmesh.sourceforge.net/

http://www-unix.mcs.anl.gov/petsc/petsc-2/

http://libmesh.sourceforge.net/


Reference

B. Kirk, J. Peterson, R. Stogner and G. Carey,“libMesh: a C++ library for parallel adaptive meshrefinement/coarsening simulations”, Engineering withComputers, in press.

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