LibMesh Experience and Usage - University of Texas at Austinroystgnr/libmeshflows1.pdfLibMesh Experience and Usage John W. Peterson [email protected] and Roy H. Stogner

Outline Introduction Weighted Residuals Poisson Equation Other Examples Essential BCs Some Extensions Reference

LibMesh Experience and Usage

John W. [email protected]

and Roy H. [email protected]

Univ. of Texas at Austin

September 9, 2008


1 Introduction

2 Weighted Residuals

3 Poisson Equation

4 Other Examples

5 Essential BCs

6 Some Extensions


1 Introduction


3 Poisson Equation

4 Other Examples

5 Essential BCs

6 Some Extensions


Library Description

PetscMetis

Mpich

Laspack

LibMesh

STL

RBM

TumorAngiogenesis

CompressibleNS

DD

Library Structure• Basic libraries

are LibMesh’s“roots”

• Application“branches” builtoff the library“trunk”


A Generic BVP

• We assume there is aBoundary ValueProblem of the form

M∂u∂t

= F(u) ∈ Ω

G(u) = 0 ∈ Ω

u = uD ∈ ∂ΩD

N(u) = 0 ∈ ∂ΩN

u(x, 0) = u0(x)

Ω

∂ΩD

∂ΩN


A Generic BVP

• 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, file inputs, and interfaces toTriangle and TetGen.


A Generic BVP

• 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, file inputs, and interfaces toTriangle and TetGen.


Non-Trivial Applications

• LibMesh provides several of the tools necessary toconstruct these systems, but it is not specificallywritten to solve any one problem.

• First, a few of the non-trivial applications which havebeen built on top of the library.


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.


Phase Separation

• Directed pattern self-assembly in spinodaldecomposition of binary mixture


Compressible Flow

Compressible Shocked Flow

• Original compressible flow code written by Ben Kirkutilizing libMesh.• Solves both Compressible Navier Stokes and Inviscid

Euler.• Includes both SUPG and a shock capturing scheme.

• Original redistribution code written by Larisa Branets.• Simultaneous optimization of element shape and size.• Directable via user supplied error estimate.

• Integration work done by Derek Gaston.• Combination of redistribution, h refinement.• Applicable to other problem classes.


Compressible Flow







Compressible Flow







Compressible Flow







Compressible Flow







Compressible Flow







Compressible Flow







Compressible Flow







Compressible Flow







Compressible Flow

Problem Specification

• The problem studied is that of an oblique shockgenerated by a 10o wedge angle.• This problem has an exact solution for density which

is a step function.• Utilizing libmesh’s exact solution capability the exact

L2 error can be solved for.• The exact solution is shown below:


Compressible Flow






Compressible Flow






Compressible Flow






Compressible Flow

Uniformly Refined Solutions

• For comparison purposes, here is a mesh and asolution after 1 uniform refinement with 10890 DOFs.

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

(a) Mesh after 1 uniform re-finement.

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

r

1.451.41.351.31.251.21.151.11.051

(b) Solution after 1 uniformrefinement.


Compressible Flow

H-Adapted Solutions

• A flux jump indicator was employed as the errorindcator along with a statistical flagging scheme.

• Here is a mesh and solution after 2 adaptiverefinements containing 10800 DOFs:

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

(c) Mesh, 2 refinements

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

r

1.451.41.351.31.251.21.151.11.051

(d) Solution


Compressible Flow

H-Adapted Solutions

• A flux jump indicator was employed as the errorindcator along with a statistical flagging scheme.

• Here is a mesh and solution after 2 adaptiverefinements containing 10800 DOFs:

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

(e) Mesh, 2 refinements

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

r

1.451.41.351.31.251.21.151.11.051

(f) Solution


Compressible Flow

Redistributed Solutions

• Redistribution utilizing the same flux jump indicator.

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

(g) Mesh, 8 redistributionsteps

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

r

1.451.41.351.31.251.21.151.11.051

(h) Solution


Compressible Flow

Redistributed and Adapted

• Now combining the two, here are the mesh andsolution after 2 adaptations beyond the previousredistribution containing 10190 DOFs.

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

(i) Mesh, 2 refinements

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

r

1.451.41.351.31.251.21.151.11.051

(j) Solution


Compressible Flow

Solution Comparison

• For a better comparison here are 3 of the solutions,each with around 11000 DOFs:

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

r

1.451.41.351.31.251.21.151.11.051

(k) Uniform.

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

r

1.451.41.351.31.251.21.151.11.051

(l) Adaptive.

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

r

1.451.41.351.31.251.21.151.11.051

(m) R + H.


Compressible Flow

Error Plot

• libmesh provides capability for computing error normsagainst an exact solution.

• The exact solution is not in H1 therefore we onlyobtain the L2 convergence plot:

3.5 4.0 4.5 5.001 sfoD )N(gol

-2.2

-2.0

-1.8

-1.6

-1.4

-1.2

01)r

orrE2

L(g

ol

UniformAdaptivityRedist + Adapt

(n) LogLog plot of L2 vs DOFs.


Compressible Flow

Error Plot

• libmesh provides capability for computing error normsagainst an exact solution.

• The exact solution is not in H1 therefore we onlyobtain the L2 convergence plot:

3.5 4.0 4.5 5.001 sfoD )N(gol

-2.2

-2.0

-1.8

-1.6

-1.4

-1.2

01)r

orrE2

L(g

ol

UniformAdaptivityRedist + Adapt

(o) LogLog plot of L2 vs DOFs.


Compressible Flow


1 Introduction


3 Poisson Equation

4 Other Examples

5 Essential BCs

6 Some Extensions


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

(F(u), v) = 0 ∀v ∈ V

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

(F(uh), vh) = 0 ∀vh ∈ Vh


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

(F(u), v) = 0 ∀v ∈ V

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

(F(uh), vh) = 0 ∀vh ∈ Vh


Some Examples

Poisson Equation

−∆u = f ∈ Ω


Some Examples

Poisson Equation

−∆u = f ∈ Ω

Weighted Residual Statement

(F(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


(F(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]

(F(u), v) :=

∫Ω

[−p (∇ · v) + ν∇u :∇v− f · v

+ (∇ · u) q] dx +

∫∂ΩN

(ν∇u− pI) n · v ds


Some Examples

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


1 Introduction


3 Poisson Equation

4 Other Examples

5 Essential BCs

6 Some Extensions



• For simplicity we start with the weighted residualstatement arising from the Poisson equation, with∂ΩN = ∅,

(F(uh), vh) :=∫Ωh

[∇uh · ∇vh − fvh] dx = 0 ∀vh ∈ Vh


Element Integrals

• 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



Element Integrals

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

0 =

∫Ωh


=Ne∑

e=1

∫Ωe



Finite Element Basis Functions

• 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


Finite Element Basis Functions

• 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


Element Matrix and Load Vector

• 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

fφi dx , i = 1, . . . ,Ns

• This can be expressed naturally in matrix notation as

KeUe − Fe






Ns∑j=1

uj

∫Ωe


Ωe

fφi dx , i = 1, . . . ,Ns


KeUe − Fe






Ns∑j=1

uj

∫Ωe


Ωe

fφi dx , i = 1, . . . ,Ns


KeUe − Fe


Global Linear System

• 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

fφi dx

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

KU = F




Keij :=

∫Ωe

∇φj · ∇φi dx


Fei :=

∫Ωe

fφi dx


KU = F




Keij :=

∫Ωe

∇φj · ∇φi dx


Fei :=

∫Ωe

fφi dx


KU = F


Reference Element Map

• The integrals are performed on a “reference” elementΩe

eΩx (ξ)x ξ

Ωe




eΩx (ξ)x

Ωe

ξ

• The Jacobian of the map x(ξ) is J.

Fei =

∫Ωe

fφidx =

∫Ωe

f (x(ξ))φi|J|dξ




eΩx (ξ)x

Ωe

ξ

• Chain rule: ∇ = J−1∇ξ := ∇ξ

Keij =

∫Ωe

∇φj · ∇φi dx =

∫Ωe

∇ξφj · ∇ξφi |J|dξ


Element Quadrature

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

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


Element Quadrature




Element Quadrature



Fei =

∫Ωe

fφi|J|dξ

≈Nq∑

q=1

f (x(ξq))φi(ξq)|J(ξq)|wq


Element Quadrature



Keij =

∫Ωe

∇ξφj · ∇ξφi |J|dξ

≈Nq∑

q=1

∇ξφj(ξq) · ∇ξφi(ξq)|J(ξq)|wq


LibMesh Quadrature Point Data

• 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


Matrix Assembly Loops

• 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







Fei =

Nq∑q=1







Fei =

Nq∑q=1







Fei =

Nq∑q=1







Keij =

Nq∑q=1







Keij =

Nq∑q=1







Keij =

Nq∑q=1



1 Introduction


3 Poisson Equation

4 Other Examples

5 Essential BCs

6 Some Extensions


Convection-Diffusion Equation

• The matrix assembly routine for the linearconvection-diffusion equation,

−k∆u + b · ∇u = f


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]);




−k∆u + b · ∇u = f







−k∆u + b · ∇u = f





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

u1pKe

u2u1Ke

u2u2Ke

u2p

Kepu1

Kepu2

Kepp

Ueu1

Ueu2

Uep

− Fe

u1

Feu2

Fep

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


Stokes Flow


∇p− ν∆u = f∇ · u = 0 ∈ Ω ⊂ R2


u1u1Ke

u1u2Ke

u1pKe

u2u1Ke

u2u2Ke

u2p

Kepu1

Kepu2

Kepp

Ueu1

Ueu2

Uep

− Fe

u1

Feu2

Fep

• We have an array of submatrices: Ke[0][0]


Stokes Flow


∇p− ν∆u = f∇ · u = 0 ∈ Ω ⊂ R2


u1u1Ke

u1u2Ke

u1pKe

u2u1Ke

u2u2Ke

u2p

Kepu1

Kepu2

Kepp

Ueu1

Ueu2

Uep

− Fe

u1

Feu2

Fep



Stokes Flow


∇p− ν∆u = f∇ · u = 0 ∈ Ω ⊂ R2


u1u1Ke

u1u2Ke

u1pKe

u2u1Ke

u2u2Ke

u2p

Kepu1

Kepu2

Kepp

Ueu1

Ueu2

Uep

− Fe

u1

Feu2

Fep



Stokes Flow


∇p− ν∆u = f∇ · u = 0 ∈ Ω ⊂ R2


u1u1Ke

u1u2Ke

u1pKe

u2u1Ke

u2u2Ke

u2p

Kepu1

Kepu2

Kepp

Ueu1

Ueu2

Uep

− Fe

u1

Feu2

Fep

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


Stokes Flow


∇p− ν∆u = f∇ · u = 0 ∈ Ω ⊂ R2


u1u1Ke

u1u2Ke

u1pKe

u2u1Ke

u2u2Ke

u2p

Kepu1

Kepu2

Kepp

Ueu1

Ueu2

Uep

− Fe

u1

Feu2

Fep

• And an array of right-hand sides: Fe[0].


Stokes Flow


∇p− ν∆u = f∇ · u = 0 ∈ Ω ⊂ R2


u1u1Ke

u1u2Ke

u1pKe

u2u1Ke

u2u2Ke

u2p

Kepu1

Kepu2

Kepp

Ueu1

Ueu2

Uep

− Fe

u1

Feu2

Fep

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


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]);


1 Introduction


3 Poisson Equation

4 Other Examples

5 Essential BCs

6 Some Extensions


Essential Boundary Data

• Dirichlet boundary conditions can be enforced afterthe global stiffness matrix K has been assembled

• This usually involves1 placing a “1” on the main diagonal of the global

stiffness matrix2 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 .. . . .

,

f1

f2

f3

.

→

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

,

g1

f2 − k21g1

f3 − k31g1

.






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

,

f1

f2

f3

.

→

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

,

g1

f2 − k21g1

f3 − k31g1

.






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

,

f1

f2

f3

.

→

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

,

g1

f2 − k21g1

f3 − k31g1

.






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

,

f1

f2

f3

.

→

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

,

g1

f2 − k21g1

f3 − k31g1

.






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

,

f1

f2

f3

.

→

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

,

g1

f2 − k21g1

f3 − k31g1

.



• 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 genericfinite element basis at the element stiffness matrixlevel.




















Penalty Formulation

• One solution is the “penalty” boundary formulation• A term is added to the standard weighted residual

statement

(F(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

ε.

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


Penalty Formulation


statement

(F(u), v) +1ε

∫∂ΩD


= 0 ∀v ∈ V


ε+ 1 = 1

ε.



Penalty Formulation


statement

(F(u), v) +1ε

∫∂ΩD


= 0 ∀v ∈ V


ε+ 1 = 1

ε.



Penalty Formulation


statement

(F(u), v) +1ε

∫∂ΩD


= 0 ∀v ∈ V


ε+ 1 = 1

ε.



Penalty Formulation

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];


Penalty Formulation







Penalty Formulation







1 Introduction


3 Poisson Equation

4 Other Examples

5 Essential BCs

6 Some Extensions


Time-Dependent Problems

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

KU = F


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(

un+1 − un

∆t, vh

)=

(F(uθ), vh) ∀vh ∈ Vh

uθ := θun+1 + (1− θ)un

• Leads to KU = F at each timestep.


Nonlinear Problems

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

(F′(uk)δuk+1, v) = −(F(uk), v)

where F′(uk) is the linearized (Jacobian) operatorassociated with the PDE.

• Must solve KU = F (Inexact Newton method) at eachiteration step.


• B. Kirk, J. Peterson, R. Stogner and G. Carey,“libMesh: a C++ library for parallel adaptive meshrefinement/coarsening simulations”, Engineering withComputers, vol. 22, no. 3–4, p. 237–254, 2006.

• Public site, mailing lists, SVN tree, examples, etc.:http://libmesh.sf.net/

Documents

LibMesh Experience and Usage - University of Texas at Austinroystgnr/libmeshflows1.pdfLibMesh Experience and Usage John W. Peterson [email protected] and Roy H. Stogner