Rieben-DNT 1

Presented at:Czech Technical University in Prague

September 10 th, 2007

Rob Rieben , DNT / Scientific B-DivisionTzanio Kolev , Center for Applied Scientific Computing

Lawrence Livermore National Laboratory

Mixed Finite Element Methods for LagrangianHydrodynamics (U)

Lawrence Livermore National Laboratory, P.O. Box 80 8, Livermore, CA 94551

This work was performed under the auspices of the U .S. Department of Energy by the University of Calif ornia Lawrence Livermore National Laboratory under Contract No. W- 7405-Eng-48, UCRL-PRES-231264

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Introduction and MotivationIntroduction and Motivation

We would like to improve the Lagrangian hydrodynamic

algorithms to prevent spurious grid distortions as well as

eliminate artificial symmetry breaking on distorted grids.

In a given calculation, such distortions can come from a

variety of sources, including:

• Inaccuracies in the grid acceleration operator

• Inaccuracies in the internal energy update

• Grid related errors in artificial viscosity

• Hourglass modes inherent in the discretization

Consider the Noh implosion problem on a box mesh.

What is the source of the spurious grid distortion?

Ideally, we would like to remain as faithful to the continuum conservation

equations as possible. This is the notion of “compatibility”

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The Steps of a Simple Lagrangian CalculationThe Steps of a Simple Lagrangian Calculation

1. Calculate accelerations of nodes by computing the gradient of the zonal pressure and dividing by the “nodal mass”

2. Apply boundary constraints to accelerations

3. Integrate accelerations to obtain nodal velocities

4. Integrate velocities to obtain nodal coordinates of new Lagrangian grid

5. Calculate new zonal volumes from new coordinates

6. Calculate new zone densities from old zone masses and the change in zone volumes

7. Calculate internal energies from conservation equation

8. Evaluate EOS to obtain new pressure as a function of new energy and density

9. Repeat!

Consider a hydrodynamic problem with no shocks, in a material with no strength. We have 4 unknowns (velocity, density, internal energy and pressure) and 4 equations:

Vt

rr⋅∇−=

∂

∂ρ

ρ

1Mass Conservation

),( ρEEOSP =Equation of State

VPt

E rr⋅∇−=

∂

∂ρEnergy

Conservation

Pt

V∇−=

∂

∂ rr

ρMomentum Conservation

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The HEMP Spatial Finite Difference Scheme*The HEMP Spatial Finite Difference Scheme*

To compute the acceleration of a node, we calculate the gradient of the pressure by using a control volume discretization about the 4 zones which share the node:

* M. L. Wilkins, “Calculations of elastic-plastic f low,” In Methods of Computational Physics , 1964

2P1P

4P 3P

On a well behaved grid, the control volume center of mass is coincident with the node we wish to accelerate…

2P

3P4P

1P

…However, as the grid is distorted the control volume center of mass and the nodal position are no longer the same.

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Mixed Finite Element Methods A Promising Alternative

Mixed Finite Element Methods A Promising Alternative

• Control volume finite difference methods ultimately compute an averagenodal velocity that is strongly dependent on the local regularity of the grid

• Internal energy updates are typically based on changes in zone volumes, rather than direct discretization of the divergence operator

• Mixed finite element methods offer a means of computing velocities and internal energy changes in a much more accurate manner that is much less dependent on the underlying grid geometry. Furthermore, total energy conservation is possible

• In simple terms, a mixed finite element method uses different sets of basis functions to discretize the field variables in a coupled partial differential equation

• For the case of hydrodynamics, the relevant field variables are the pressureand the velocity

• The downside to this approach is the requirement of a global linear solve to compute the velocity unknowns

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Variational Formulation of the Hydro EquationsVariational Formulation of the Hydro Equations

Consider the momentum conservation equation in variational form:

∫∫∫Ω∂ΩΩ

⋅′−′⋅∇=′⋅∂

∂)ˆ()( )( nwPwPw

t

V rrrrr

ρPt

V∇−=

∂

∂ rr

ρ ∫∫ΩΩ

′⋅∇−=′⋅∂

∂wPw

t

V rrrr

)( )(ρ

Multiply by vector valued test function, integrate over problem domain

Perform integration by parts

Now assume a piecewise polynomial representation for the velocity and pressure:

Applying a standard Galerkin procedure will reduce the continuum equation to a set of ODE’s. However , we cannot choose the basis functions independently!

They must be chosen to satisfy the Babuska-Brezzi* stability condition:

∑≈i

ii xwtvtxV )( )( ),(rrrr

Velocity:

)( )( ),( xftptxP ii

i

rr∑≈Pressure:

)(Ω∈Wiwr

)(Ω∈Pif

The basis functions belong to and span a particular polynomial

function space

)ˆ()ˆ( Ω=Ω⋅∇ PWr

*See for example: D. Arnold et. al., “Differential Complexes and stability of Finite Element Methods” , 2005.

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Choice of Basis FunctionsChoice of Basis Functions

• There are multiple sets of basis functions which satisfy the stability condition

• There are many things to consider when choosing a mixed finite element basis pair for hydrodynamics:

– Where are the velocity degrees of freedom defined?

– Where are the pressure degrees of freedom defined?

• For Lagrangian hydrodynamics, velocity must somehow be computed at nodes in order to move the grid

• Pressure needs to be a zone centered quantity for EOS calculations and mixed zone physics.

• We have identified a set of basis functions which is most amenable to implementation in a Lagrangian hydro code while still providing accurate representations of velocity …

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The Brezzi-Douglas-Marini Elements*The Brezzi-Douglas-Marini Elements*

α=Ω)ˆ(P

)ˆ()ˆ( Ω=Ω⋅∇ PW

−−++

++++=Ω

2222

2111

ˆˆˆ2ˆˆ

,ˆˆ2ˆˆˆ)ˆ(

ysyxryx

yxsxryx

γβα

γβαW

ˆˆˆˆ1 ),ˆˆ(2

1ˆ 2

1 yxyxxxw +−−−=r

2

1ˆ1 −=⋅∇ wrr

1rr

3rr

2rr

4rr

1n

2n

3n

4n

1v 2v

8v 3v

4v

5v6v

7vp

* F. Brezzi et. al., “Two Families of Mixed Finite E lements for Second Order Elliptic Problems”, 1985

Velocity space is piece wise quadratic vector functions :

Pressure space is piece wise constant scalars:

Together, these spaces satisfy the stability condition:

)( )(),(8

1

xwtvtxVi

i

rrrr

∑=

≈There are 8 velocity degrees of freedom per zone, which are the normal projections of the velocity on the faces.

111 ˆ)( nrVv ⋅=rr

For example, on the unit quadrilateral, we have:

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The Jacobian Matrix and Coordinate System Invariance

The Jacobian Matrix and Coordinate System Invariance

The basis functions are known on a reference (unit) quadrilateral. To map to an arbitrary quadrilateral, we need the Jacobian matrix which describes the transformation:

)ˆ(ˆ1 xwrr

Ω∈ ˆxr

Reference space:

)ˆ(ˆ||

1)( 11 xwxw T rrrr

JJ

=

Ω∈xr

Physical Space:

)ˆ(xxrr

Φ=i

jji x

x

ˆ,∂

∂=J

The BDM basis transforms covariantly to preserve normal components of

vectors on faces

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Construction of the Mass matrixConstruction of the Mass matrix

||)ˆ||

1( )ˆ

||

1(

ˆ, JJ

JJ

JM j

Ti

Tzji ww

rr

∫Ω

⋅= ρ

The variational form is discretized by numerical quadrature done at the zone level:

We transform the zone integrals to a reference zone to make computation easier and more efficient.

For every zone in the mesh, we compute the “Mass Matrix ” as:

3rd Order Gauss-Lobatto quadrature is used to ensure accurate integration of the high order polynomial integrand and to maximize sparsity of the mass matrix.

∫∫ΩΩ

′⋅∇=⋅∂

∂)( )( wPw

t

V rrrr

ρ ∑ ∫∫ΩΩ

⋅∇=⋅∂

∂

zjji

iz

zz

wPwwt

v)( )(

rrrrρ

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Semi-Discrete Formulation of the Hydrodynamic Equations

Semi-Discrete Formulation of the Hydrodynamic Equations

pDvM T

t=

∂

∂

Dvmp

e −=∂

∂

t

Given the zone based “mass ” and “derivative ” matrices, we use a standard finite element assembly procedure to obtain global sparse matrices. The resulting linear system of ODEs is given by:

p Array of length NumZones containing the zonal pressure values

e Array of length NumZones containing the zonal energy values

v Array of length 2*NumFaces containing the face based velocity degrees of freedom

M Square “mass” matrix of dimension 2*NumFaces by 2*NumFaces

D Rectangular “derivative” matrix of dimension NumZones by 2*NumFaces

For Example:

NumZones = 100

NumFaces = 220

M = 440 by 440

D =100 by 440

m Array of length NumZones containing the zonal mass values

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Discrete Energy Conservation*Discrete Energy Conservation*

ttnn

n∆

−≈≡

∂

∂ + vva

v 1Consider the case of a simple time differencing method for the MFEM velocity:

The change in kinetic and internal energy in a given time step is defined as:

+∆=− +

++ 2

2

1

2

1 111

nnnnn

Tnnn

Tn t

vvDpvMvvMv

Kinetic Energy Change Internal Energy Change

nnnnTnnE mevMv +=

2

1Then, for a given time step n, we define the total integrated energy as:

If we update the internal energy in the following manner, then discrete energy conservation follows:

+∆−= +

+ 21

1nn

n

nnn t

vvD

mp

eenn EE =+1

*P. Vassilevski, T. Kolev, “Lagrange Step in Hydro Computations”, working notes, 2007.

These calculations are performed on a single mesh. When moving to the newly updated mesh, a transfer operation is defined. For zonal quantities, this transfer is a trivial copy to the new mesh. For faced based velocities, a more complicated transfer is required.

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Nodal Reconstruction of Vector FieldsNodal Reconstruction of Vector Fields

∑ ∫

∑ ∫

Ω

Ω=

i

ixii

x

i

i

V

V

φ

φ ,~

∑ ∫

∑ ∫

Ω

Ω=

i

iyii

y

i

i

V

V

φ

φ ,~

xV~

xV ,1

xV ,4

For Lagrangian hydrodynamics, velocity must somehow be computed at nodes in order to move the grid.

We can use the BDM representation to evaluate a vector field at a vertex in a given zone. However, by construction, this value will not necessarily be unique for a collection of zones which share a vertex. Therefore, a type of nodal averaging is required:

*P. Vassilevski, T. Kolev, Personal Communication, 2007.

We can construct a local “quasi-interpolant” by computing the following inner products with the standard H1 nodal shape functions* and the computed BDM value.

Using simple 4 point quadrature, this quasi-interpolant reduces to area weighted averaging.

xV ,2

xV ,3

A more simple average is obtained by using equal we ighting for each zone. In practice, this simple option works be st.

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On The Issue of Hourglass ModesOn The Issue of Hourglass Modes

constP ≠

1− 1+ 1−

1+ 1− 1+

1− 1+ 1−

Unfortunately, nodal averaging can introduce hourglass modes to nodal vector fields:

Consider a “checkerboard”pressure field on a simple grid:

The MFEM solution evaluated at multiple points in a given zone is clearly non-zero as it should be …

…however, by performing nodal averaging, the nodal fields are now zero at the interior nodes.

0=∇Pr

0≠∇Pr

The accelerations computed from mixed finite element BDM spaces are naturally free of hourglass modes (i.e. they are orthogonal to the set of BDM divergence free velocity modes). However, the process by which nodal fields are computed is not constrained in this way. Therefore, some form of hourglass projection / damping is still required.

It may be possible to construct a nodal averaging m ethod which preserves the kernel of the MFEM divergence operator. This is currently being resear ched.

!ZERO

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Comparison of Acceleration Operator on a Distorted Grid – No Time Dependence

Comparison of Acceleration Operator on a Distorted Grid – No Time Dependence

pDMa T1−=

As a first test, we can use the mixed FEM to compute the nodal accelerations on a fixed, distorted grid given an analytic pressure and density:

First, we solve for the acceleration Degrees of Freedom (DOF):

Next, we use the acceleration DOF along with the BDM basis functions and the simple nodal averaging method to obtain nodal accelerations:

ii

iwaArr

∑=

=8

1

Pressure field

rxyExpPyxA

xyExpyx

yxyxP

ˆ)(1

),(

)(

1),(

),( 22

−=∇−=

=

+=

rr

ρ

ρ

Density field

Consider the following analytic functions for pressure, density and the resulting acceleration on a randomly distorted grid:

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Acceleration Operator on Distorted Polar MeshAcceleration Operator on Distorted Polar Mesh

Close-up view of computed acceleration vectors using the HEMP algorithm.

Radial symmetry has been broken:

Close-up view of computed acceleration vectors using the MFEM algorithm.

Radial symmetry is preserved:

Error in nodal acceleration along a constant “j”-line in the theta direction. MFEM is up to 2 orders of magnitude more accurate .

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Acceleration Operator on the Kershaw Z-MeshAcceleration Operator on the Kershaw Z-Mesh

The HEMP result deviates from a constant vector field along regions of the mesh with the greatest jump in mesh density and deviation from unit aspect ratios. The MFEM result preserves the constant nature of the field in all regions of the mesh.

Here we consider the case of a simple linear pressure field projected onto the “Kershaw Z-mesh” )ˆˆ(),(

),(

yxPyxA

yxyxP

+−=∇−=

+=rr

The exact solution is a constant vector field

Pressure fieldHEMP result

Mean Error: 3.45448*10-3

MFEM result

Mean Error : 3.09136*10-6

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A Dynamic Example – The Coggeshall* Adiabatic Compression Problem #2

A Dynamic Example – The Coggeshall* Adiabatic Compression Problem #2

Now consider a time dependent adiabatic compression on a distorted planar-polar mesh:

*S. V. Coggeshall, “Analytic Solutions of Hydrodyna mic Equations”, 1990

2/3),(

t

rtr =ρ

rt

rtrV ˆ

4

3),( −=

r

2

2

32

3),(

t

rtrE −=

ργ EtrP )1(),( −=

Initial distorted grid:

Initial velocity:

Final HEMP grid

Final HEMP velocity (close up)

Final MFEM grid

Final MFEM velocity (close up)

We compress the mesh for a total of 0.8 micro seconds. The HEMP grid eventually goes unstable while the MFEM grid preserves the radial symmetry for all time

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What about Sub-Zonal Pressures?What about Sub-Zonal Pressures?

The method of Sub-Zonal Pressures * can be used to alleviate mesh distortion by computing artificial resistive forces from the changes in subzonal pressures

However, a control volume finite difference method is still used to compute the primary acceleration of the grid, and the subsequent results still break symmetry:

HEMP result after 60 cycles

Radial Symmetry in Acceleration is Broken

Sub-Zonal Pressure(merit factor = 1)

result after 60 cycles

Radial Symmetry in Acceleration is Broken

MFEM result after 60 cycles

Radial Symmetry in Acceleration is

Preserved

*E. J. Caramana and M. Shaskov, “Elimination of Arti ficial Grid Distortion and Hourglass-Type Motions by Means of Lagrangian Subzonal Masses and Pr essures”, 1998

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Artificial Viscosities: The Noh Implosion Problem on a Box Mesh

Artificial Viscosities: The Noh Implosion Problem on a Box Mesh

• To account for shocks, we need some form of artificial viscosity

• It is well known that artificial viscosity formulations can cause spurious grid distortions

• Consider the Noh test problem* on a box mesh of an ideal gas:

.

MFEM + Scalar Monotonic QHEMP + Scalar Monotonic Q

The MFEM still suffers from jetting along the axes due to the non-uniform Q-heating, but the mesh does not become tangled along the 45 degree line – indicating that artificial viscosity is not the sole cause of symmetry breaking in this problem

The Scalar Monotonic Q produces non-uniform heating along the 45 degree line causing jets to form. These jets lead to further errors in accelerations with the HEMP operator, eventually causing extreme mesh distortion along the 45 degree line and the axes

*W. F. Noh, “Errors for Calculations of Strong Shoc ks Using Artificial Viscosity and an Artificial Hea t Flux”, 1987

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A New Variant of the Scalar Bulk-QA New Variant of the Scalar Bulk-Q

*J. C. Campbell and M. J. Shaskov, “A Tensor Artifi cial Viscosity using a Mimetic Finite Difference Al gorithm,” 2000

LVr

BVr

RVr

TVr

xdv

ydr

⋅−+⋅−=∆

yd

ydVV

xd

xdVVV BTLR r

rrr

r

rrr

)()(2

1

• The so called Bulk-Q can be used to reduce the spurious jets along the axis.

• Unfortunately, this results in under compression and spurious “ringing” at the shock front.

• Here, we formulate a simple alternative to the scalar Bulk-Q by taking advantage of the MFEM’s ability to evaluate velocity at any point in space:

)( slinquadzz CqVqVq −∆∆= ρ

• This simple approach takes the shock velocity direction into account

• This is just a first step, Tensor viscosities * are also possible

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The Noh Problem RevisitedThe Noh Problem Revisited

HEMP + Standard BulkQ

Axes jets have been reduced

• Final density is too low

• Spurious “ringing”occurs at shock front

• Energy is not conserved

MFEM + Modified BulkQ

Axes jets have been reduced

Final density is closer to true value

Amplitude of spurious ringing is greatly reduced

Density behind shock is more uniform

Energy is conserved

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The Planar Sedov ProblemThe Planar Sedov Problem

As a final example, we run the Sedov test problem in planar coordinates to verify energy conservation of the method:

Computational grid and pressure at time t = 1.0

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Remarks and ConclusionsRemarks and Conclusions

• Numerical symmetry breaking can come from a variety of sources in a given Lagrangian calculation

• One key source are errors in velocity calculations on irregular grids.

• Mixed FEM is a promising alternative to standard control volume finite difference methods for computing velocities and internal energy changes in a conservative manner

• The main drawback at this point is the computational effort required to assemble and solve a sparse linear system at every Lagrangian time step

• Velocity representation can possibly be used to construct a tensor artificial viscosity for further robustness

• We are currently adapting the method for axisymmetric problems (i.e. cylindrical coordinate systems) and are very close to making this work…

Thank you for your time!