aerospace engineering seminar series

High Order Finite Elements for LagrangianComputational Fluid Dynamics

Truman E. EllisAdvisor: Dr. Faisal Kolkailah

LLNL Mentors: Dr. Robert Rieben and Dr. Tzanio Kolev

Department of Aerospace EngineeringCalifornia Polytechnic State University

San Luis Obispo, CA

[email protected]

November 18, 2009

This work performed under the auspicesof the U.S. Department of Energy by

Lawrence Livermore National Laboratoryunder Contract DE-AC52-07NA27344

LLNL-PRES-416883, -416822

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Introduction

We are developing advanced finite elementdiscretization methods for Lagrangianhydrodynamics.Our goal is to improve the currentstaggered grid hydro (SGH) algorithms inmulti-material ALE codes with respect to:

symmetry preservationenergy conservationartificial viscosity discretizationhourglass-mode instabilities

We consider a general framework forsolving the Euler equations with thefollowing features:

allows for high order field representationsallows for curved element geometryexact energy conservation by constructionreduces to classical SGH under simplifying assumptions

Truman Ellis (Cal Poly) High Order Finite Elements for Lagrangian CFD November 2009 2 / 27

Glossary

We consider several different hydro methods which make use of various finite element spaces.The specific finite element spaces considered in this talk are:

Q0: constant function on quadrilaterals

Q1: continuous bi linear functions on quadrilaterals

Q2: continuous bi-quadratic functions on quadrilaterals

Q1d: discontinuous bi-linear functions on quadrilaterals

Q2d: discontinuous bi-quadratic function on quadrilaterals

Each hydro method is designated by a pair of finite element spaces:

A continuous kinematic space

A discontinuous thermodynamic space

For example: Q2-Q1d uses continuous quadratic velocities and positions with discontinuous bi-linear pressures and densities.


Euler Equations in a Lagrangian Frame

The Euler equations of gas dynamics in a Lagrangianreference frame can be written in differential form as:

Momentum Conservation: ρ∂v∂ t

=−~∇p+ ...

Mass Conservation:1ρ

∂ρ

∂ t=−~∇ ·~v

Energy Conservation: ρ∂e∂ t

=−p~∇ ·~v+ ...

Equation of State: p = EOS(e,ρ)

Typically, these equations aresolved on a staggered spatial grid[1,2] where thermodynamicvariables are approximated aspiece-wise constants defined onzone centers and kinematicvariables are defined on thenodes.

Spatial gradients are computedusing finite volume and/or finitedifference methods:

[1] R. Tipton, CALE Lagrange Step, unpublished LLNL report, 1990[2] M. Wilkins, Calculations of Elastic-Plastic Flow, In Methods of Computational Physics, 1964


Semi-Discrete Finite Element Approximation: Computational Mesh

We propose a general semi-discrete FEM approach to solving the Euler equations in a Lagrangianframe. A semi-discrete method is only concerned with the spatial approximation of the continuumequations.

To begin, we decompose the continuum domain at an initial time into a set of non-overlappingdiscrete volumes (zones). The union of these volumes forms the computational mesh:

Ω(t0)

−→

Ω(t0) = ∪ZΩZ(t0)

The motion of the continuous medium will therefore be described by the motion of only a finitenumber of particles (mesh vertices, edge midpoints, etc )


Semi-Discrete Finite Element Approximation: Lagrangian Mesh MotionAfter deformation in time, zones are reconstructed based onparticle locations, thus defining the moved mesh.

Q1 (Bi-Linear) Approximation Q2 (Bi-Quadratic)Approximation

This reconstruction processhas an inherent geometricerror.

Initial Cartesian meshdeformed with the exactsolution of the Sedov blastwave problem.

This built in geometric error motivates the use of (higher order) finite elements which can use theadditional particle degrees of freedom to more accurately represent continuous deformations.


Lagrangian Mesh Motion: Animation


Semi-Discrete Finite Element Approximation: Curved Geometries

Finite element basis functions are defined on a reference (unit) zone. To compute quantities on anarbitrary mesh zone, we use the Jacobian matrix which defines the geometric transformation (ormapping) from reference space:

Reference Space:

φ(~x)

~∇φ(~x)

Physical Space:

φ(~x) = φ(Φ−1(~x))

~∇φ = J−1Z

~∇φ

A standard Q1 mapping gives zones with straight edges, while Q2 mappings produce curved zones.Note that we can have Q1 elements defined with a Q2 mapping and vice versa. The finite elementreference degrees of freedom are in general different from the nodes of the mapping.


Semi-Discrete Finite Element Approximation: Basis Functions

We approximate each of the continuum fields with a basis function expansion, where theexpansion coefficients are independent of space:

Velocity:~v(~x, t)≈Nv

∑i

vi(t)wi(~x)

Internal Energy: e(~x, t)≈Ne

∑i

ei(t)θi(~x)

Pressure: p(~x, t)≈Np

∑i

pi(t)φi(~x)

Density: ρ(~x, t)≈Nρ

∑i

ρi(t)ψi(~x)

Our formulation is general; we are free to choose various types of basis functions. However,stability requires that certain choices are not independent[1] (e.g. pressure and velocity.)

Q1 (Bi-Linear) basis function4 DOF per Quad

Q2 (Bi-Quadratic) basis function9 DOF per Quad

Q2 basis function with degrees offreedom at the 9

Gauss-Legendre quadraturepoints

[1] See for example: D. Arnold et. al., Differential Complexes and stability of Finite Element Methods, 2005.


Semi-Discrete Momentum Conservation

To derive a semi-discrete momentum conservation law, we begin with a variational formulation:

ρd~vdt

=−~∇p =⇒∫

Ω(t)

(ρ

d~vdt

)· ~w′ =

∫Ω(t)

(~∇p) · ~w′ =∫

Ω(t)p(~∇ · ~w′)−

∫∂ Ω(t)

p(~w′ · n)

Multiply by vector valued testfunction, integrate over

computational mesh

Perform integrationby parts

Inserting our basis function expansions and applying Galerkins method by picking the velocitybasis as the test function yields the following linear system of equations:

∫Ω(t)

(ρ ∑

i

dvi

dt~wi

)·~wj =

∫Ω(t)

∑i

piφi(~∇ ·~wj)Write in matrix / vector form−→ M

dvdt

= DTp

The Mass and Derivative matrices are computed zone by zone and assembled to form globalsystems:

(Mz)ij =∫

Ω(t) ρ(~wi · ~wj)

The symmetric positive definite mass matrixdescribes how matter is distributed in space

(Dz)ij =∫

Ω(t) φi(~∇ · ~wj)

The rectangular derivative matrix mapsbetween pressure and velocity spaces. This

matrix is a discrete version of the Div operator.Its adjoint (transpose) is the Grad operator


Computing Mass and Derivative Matrices

In practice, we compute the mass and derivative matrices by integrating over the reference volumeusing a change of variables: ∫

Ω(t)f =

∫Ω(t)

(f Φ)|Jz|

The integrals are computed using Gaussian quadrature of a specified order. For certain integrandsand quadrature rules, this can be computed exactly:

∫Ω(t)

(f Φ)|Jz| ≈Nq

∑n=1

αn f Φ)|Jz|~x=~qn

2D Order 1 Gauss-Legendrerule (exact for Q1)





Semi-Discrete Mass Conservation

We define the zonal mass moments in a natural manner: mz,i =∫

Ωz(t)

ρψi

The Lagrangian description requires that:dmz,i

dt= 0

From the representation: ρ(~x, t)≈∑i

ρi(t)ψi(~x) it follows that: mz,i = ∑j

ρz,j

∫Ωz(t)

ψiψj

In matrix form we have: mz = Mρz ρz where (Mρ

z )i,j =∫

Ωz(t)

ψiψj

If we impose the stronger condition:ddt

∫Ω(t)

ρψ = 0 for any function ψ

then we obtain the strong mass conservation principle:

ρ(t)|J(t)|= ρ(t0)|J(t0)|

Note that, in general, the density defined by the above equation is not polynomial.

Note that both generalizations imply the standard zonal mass conservation


Mixed Finite Elements


High Order Methods Using the Q2 Basis

For each Q2 zone, we track 9independent vertices(coordinates, velocities,accelerations etc ) which definethe quadratic mapping fromreference space to physicalspace. This mapping is definedusing the Q2 Jacobian matrix,which is a function:

Jmat=[(-4*(y - 1)*(2*y - 1)*(2*x - 1)*X5 +

(y - 1)*(2*y - 1)*(4*x - 3)*X1 + (y- 1)*(2*y

- 1)*(4*x - 1)*X2 + 16*(y - 1)*(2*x - 1)*X9*y

- 4*(y - 1)*(4*x- 3)*X8*y - 4*(y - 1)*(4*x -

1)*X6*y - 4*(2*y - 1)*(2*x - 1)*X7*y + (2*y-

1)*(4*x - 3)*X4*y + (2*y - 1)*(4*x -

1)*X3*y); (-4*(2*y - 1)*(x - 1)*(2*x - 1)*X8

+ 16*(2*y - 1)*(x - 1)*X9*x - 4*(2*y -1)*(2*x

- 1)*X6*x + (4*y - 3)*(x - 1)*(2*x - 1)*X1 -

4*(4*y - 3)*(x -1)*X5*x + (4*y - 3)*(2*x -

1)*X2*x + (4*y - 1)*(x - 1)*(2*x - 1)*X4

-4*(4*y - 1)*(x - 1)*X7*x + (4*y - 1)*(2*x -

1)*X3*x)];

Points in reference space

Basis function in reference space

Points in physical space

Basis function in physical space


High Order Methods Using the Q2 Basis

For each zone in the computational mesh, we compute the following matrices usingGauss-Legendre quadrature of a specified order.:

The symmetric positive definite 9 by 9 massmatrix describes how matter is distributedwithin the Q2 zone

(Mz)i,j ≡∫

Ωzρz(wiwj)|Jz|

The rectangular derivative matrix mapsbetween the pressure and velocity spaces.This matrix is a discrete version of the Divoperator. Its adjoint (transpose) is the Gradoperator

(Dxz)i,j ≡

∫Ωz

φi(J−1z

~∇wj) · 1,0|Jz|

(Dyz)i,j ≡

∫Ωz

φi(J−1z

~∇wj) · 0,1|Jz|

The symmetric positive definite 9 by 9 stiffnessmatrix is a discrete version of the second orderDivGrad operator. It is used to compute theartificial viscosity

(Sz)i,j ≡∫

Ωzµz(J−1

z~∇wi)(J−1

z~∇wj)|Jz|

The bulk of the computational effort is in computing the high order Jacobian matrix. However, thiscan be computed once per DOF and shared between each matrix.

For the remaining examples, we apply mass lumping to the mass matrix to eliminate the need fora global linear solve. In addition, we compute the mass matrix only once as an initial condition.


Static Momentum Equation Solve Convergence Study

In this simple test, we project an analytic pressure functiononto a randomly perturbed mesh and solve for theresulting accelerations:

p(x,y) = cos(π

2x)cos(

π

2y)

ma =−∇p

Ma =−DT p

Randomly perturbed base line mesh and a sequence ofrefinements

Lumped mass matrix

Full linear solve


1D Sod Shock Tube: Velocity and Density

We consider two meshes: A coarse 60 zone mesh and fine 120 zone mesh

For each method, qlin = qquad = 1.0, no monotonic limiter is used

We acknowledge this is sub-standard for Q1 (i.e. limiters make a difference); however, this afair comparison between each method

For each zone, 5 plot points are evaluated

Each method captures contact discontinuity exactly due to discontinuous thermodynamicbasis


1D Sod Shock Tube: Velocity and Density (Zoomed View)

We consider two meshes: A coarse 60 zone mesh and fine 120 zone mesh

For each method, qlin = qquad = 1.0, no monotonic limiter is used

We acknowledge this is sub-standard for Q1 (i.e. limiters make a difference); however, this afair comparison between each method

For each zone, 5 plot points are evaluated

Each method captures contact discontinuity exactly due to discontinuous thermodynamicbasis


Sedov Blast Problem, 20 by 20 Mesh: Q1-Q0 FEM

Standard HG filterrequired

Shock is not wellresolved on thiscoarse mesh


Sedov Blast Problem, 20 by 20 Mesh: Q2-Q1d FEM

No HG filter used

Curved elements

81 plot points used per cell

Shock front is much sharperon the same mesh

Zone at origin has incorrectdeformation

Velocity oscillations areobserved in post shock region


Sedov Blast Problem, 20 by 20 Mesh: Q2-Q1d FEM

No HG filter used

81 plot points used per cell

Allowing 25% of the linear-qterm in expansion has asignificant benefit

Velocity oscillations in postshock region are eliminated

Curved deformation of zonesclosely resembles exactdeformation

We have not taken full benefitof the high order informationin each cell to compute theviscosity coefficient, this is anarea we plan to investigatemuch further


Sedov Blast Problem, 20 by 20 Mesh Comparison


Noh Implosion Problem, 30 by 30 Mesh: Q1-Q0 FEM

Standard HG filterrequired

Shock is not wellresolved on thiscoarse mesh

No overshoots orundershoots areobserved


Noh Implosion Problem, 30 by 30 Mesh: Q2-Q1d FEM

No HG filter used

36 plot points per zone

Shock is more sharplyresolved

Wall heating is diminished

Post shock density is closerto correct value

Strong undershoots andovershoots in the density areobserved in the single layer ofzones at shock front


Noh Implosion Problem, 30 by 30 Mesh: Q2-Q2d FEM

No HG filter used

36 plot points per zone

Shock is more sharplyresolved

Wall heating is diminished

Post shock density is closerto correct value

Under shoots in densitydiminished, but overshootspersist


Conclusions

We have developed a general FEM framework for Lagrangian CFD

Kinematic and thermodynamic space can be chosen independentlyFor triangles:

Linear, quadratic, cubic, etc. kinematic spaceConstant, linear, quadratic, etc. thermodynamic space

For quads:Bi-linear, bi-quadratic, bi-cubic, etc. kinematic spaceConstant, bi-linear, bi-quadratic, etc. thermodynamic space

This allows us to experiment with many different finite element pairs

We have also demonstrated that high order Lagrangian methods have significant promise

Permits curved elements

Increased accuracy

Sharper shocks

Density/pressure gradients in a single cell

Second derivatives of velocity in a single cell

Potential improvements for sub-zonal physics / multi-material ALE

Post shock ringing is still an issue and we plan to address this by improving the artificialviscosity coefficient (e.g. the Hyper Viscosity technique of Cook and Cabot)


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