libmesh experience and usage - university of texas at austinroystgnr/libmeshflows1.pdflibmesh...

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

LibMesh Experience and Usage

John W. Petersonpeterson@cfdlab.ae.utexas.edu

and Roy H. Stognerroystgnr@cfdlab.ae.utexas.edu

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

Laspack

LibMesh

TumorAngiogenesis

CompressibleNS

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)

∂ΩD

∂ΩN

A Generic BVP

• Associated to theproblem domain Ω is aLibMesh data structurecalled a Mesh

• A Mesh is essentially acollection of finiteelements

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 :=⋃

• 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

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

Uniformly Refined Solutions

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

0 0.2 0.4 0.6 0.8 10

(a) Mesh after 1 uniform re-finement.

0 0.2 0.4 0.6 0.8 10

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:

0 0.2 0.4 0.6 0.8 10

(c) Mesh, 2 refinements

0 0.2 0.4 0.6 0.8 10

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:

0 0.2 0.4 0.6 0.8 10

(e) Mesh, 2 refinements

0 0.2 0.4 0.6 0.8 10

1.451.41.351.31.251.21.151.11.051

(f) Solution

Compressible Flow

Redistributed Solutions

• Redistribution utilizing the same flux jump indicator.

0 0.2 0.4 0.6 0.8 10

(g) Mesh, 8 redistributionsteps

0 0.2 0.4 0.6 0.8 10

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.

0 0.2 0.4 0.6 0.8 10

(i) Mesh, 2 refinements

0 0.2 0.4 0.6 0.8 10

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:

0 0.2 0.4 0.6 0.8 10

1.451.41.351.31.251.21.151.11.051

(k) Uniform.

0 0.2 0.4 0.6 0.8 10

1.451.41.351.31.251.21.151.11.051

(l) Adaptive.

0 0.2 0.4 0.6 0.8 10

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

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

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) :=

[∇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) :=

[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) :=

[−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:

∫Ωh

[∇uh · ∇vh − fvh] dx ∀vh ∈ Vh

=Ne∑

∫Ωe

Element Integrals

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

∫Ωh

=Ne∑

∫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

∫Ωe

vh dx =

Ns∑i=1

∫Ωe

φi dx

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

∫Ωe

vh dx =

Ns∑i=1

∫Ωe

φi dx

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

∫Ωe

∇φj · ∇φi dx−∫

fφi dx , i = 1, . . . ,Ns

• This can be expressed naturally in matrix notation as

KeUe − Fe

Ns∑j=1

∫Ωe

fφi dx , i = 1, . . . ,Ns

KeUe − Fe

Ns∑j=1

∫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

eΩx (ξ)x

• The Jacobian of the map x(ξ) is J.

∫Ωe

fφidx =

∫Ωe

f (x(ξ))φi|J|dξ

eΩx (ξ)x

• 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

∫Ωe

fφi|J|dξ

≈Nq∑

f (x(ξq))φi(ξq)|J(ξq)|wq

Element Quadrature

Keij =

∫Ωe

∇ξφj · ∇ξφi |J|dξ

≈Nq∑

∇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]);

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

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

u2u1Ke

u2u2Ke

− Fe

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

Stokes Flow

∇p− ν∆u = f∇ · u = 0 ∈ Ω ⊂ R2

u1u1Ke

u1u2Ke

u2u1Ke

u2u2Ke

− Fe

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

Stokes Flow

∇p− ν∆u = f∇ · u = 0 ∈ Ω ⊂ R2

u1u1Ke

u1u2Ke

u2u1Ke

u2u2Ke

− Fe

Stokes Flow

∇p− ν∆u = f∇ · u = 0 ∈ Ω ⊂ R2

u1u1Ke

u1u2Ke

u2u1Ke

u2u2Ke

− Fe

Stokes Flow

∇p− ν∆u = f∇ · u = 0 ∈ Ω ⊂ R2

u1u1Ke

u1u2Ke

u2u1Ke

u2u2Ke

− Fe

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

Stokes Flow

∇p− ν∆u = f∇ · u = 0 ∈ Ω ⊂ R2

u1u1Ke

u1u2Ke

u2u1Ke

u2u2Ke

− Fe

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

Stokes Flow

∇p− ν∆u = f∇ · u = 0 ∈ Ω ⊂ R2

u1u1Ke

u1u2Ke

u2u1Ke

u2u2Ke

− Fe

• 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