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Beyond Resistive MHDDuncan McGregor

Computational MultiphysicsOctober 2017, SIAMPNWS/PNWNAS

SAND2017-11628C

Acknowledgements

A. C. Robinson for his collaboration and development of ct remap

E. Love for consultation on predictor corrector

T. R. Mattsson for supporting this activity

Number of time steps

time

Fast Speed Limiting is Evident

Fast speed impacts calculation

time step

“Eddy” experiment on stagnation: the current flow in low density regions affects the dynamics.

Source: Peterson & Mattson

Resistive MHD

3

MHD simulation offer near predictive computation for a wide variety of classical warm dense matter

computations .

MHD codes neglect physics and are numerically brittle in low density regimes

but …

How do we overcome low density brittleness?

time

Source: Allen Robinson

ALEGRA MHD simulation of a dynamic materials experiment on Z-Machine.

Source: Ray Lemke

Practical Constraints

Any changes we make to solve this problem should be attainable on a short time frame 1-2 years tops

We need to solve real problems today rather than hypothetical problems in the future.

Changes will be made to the indirect ALE code ALEGRA.

Indirect ALE: Lagrange Step + (Swept) Remap

Compatible Finite Elements on unstructured hexs

Mixed material elements

Interface reconstruction is available

Notation from Electromagnetic Theory by Kovetz, a rational mechanics approach to EM in the Lagrangian frame.

Can we perturb an MHD code to fix the problem rather than having to start from scratch?

Preconditioners?

Constrained Transport Remap?

Lets try not to mess too much with hydro.

4

Part IModels

5

6

Arbitrarily large magnetic diffusion time

Maxwellian particle distribution

Indistinguishable ionization states

Negligible ratio of electron to atomic mass

Negligible electron & ion inertial length scales

Negligible electric field relaxation time

Boltzmann Equation

n-Fluid Continuum Plasma

Heavy/Light Fluid Plasma

Generalized Ohm’s Law

Full Maxwell Hydrodynamics

Magnetohydrodynamics

Ideal Magnetohydrodynamics

A Hierarchy of Plasma Models

PIC

MHD Codes

Research

Computational Problems Issues with MHD

Classical operator splitting uses an ideal MHD step

Ideal MHD step requires a positive density

�, � ± �� +� �

��

Magnetic diffusion step requires a positive conductivity even in “void”

We care about resolving physics in low density regions.

We have an explicit Lagrangianstep which depends on fast magnetosonic speeds!

x

t

F A S FAS

up

E

left state right state

To push beyond the warm dense region we will require more physics!

Maxwell-Ampere and Generalized Ohm’s Law

The Physicist’s Answer: Add More Physics

Stagnation experiments for MagLif do not capture torsion at stagnation caused by the Hall Effect.

MHD assumptions don’t hold in low density regions so maybe we shouldn’t assume them.

Do not neglect displacement currents

• Generalized Ohm’s Law starts from change of variables of the from the Two Fluid system

• Without additional assumptions the system is equivalent to the two fluid system.

• People often assume at least

and quasi neutrality

The Eulerian DG code PERSEUS (Seyler of Cornell, Martin of SNL, et al.) has very promising results using these physics.

One Step at a Time

9

Attracted to the system because of desirable characteristic and dispersive properties for the linearization

Instead of considering the full generalized Ohm’s law to begin with we started slightly simpler.

Some people refer to the system as a single fluid plasma.

We call it Full Maxwell Hydrodynamics

Does not neglect displacement currents

Assumes classical Ohm’s law.

Lagrangian Energy Balance

10

Kinetic Energy:Kinetic energy can be found by dotting the momentum equation with v, integrating, and add scaling of mass equation.

Electromagnetic Energy:Start with the frame invariant Poynting theorem

Integrate and expand terms very carefully.This is not as easy as it looks.

Internal Energy:We assume its balance law cancels all remaining internal contributions (Joule heating and mechanical work)

Part IIDiscretization

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How should we discretize?

12

Explicit• Easy to implement –linear solve/

preconditioning/etc• Time steps are cheap• Time steps are as small as your

fastest time scale• In high density regimes fast time

scales become arbitrarily large• You actually resolve the physics

you care about because you resolve all the physics

Implicit• Hard to implement• Large scale linear solvers are hard

for hero-physicists to implement• Requires preconditioners• Time steps are expensive• Time steps can step over fastest

time scales while still resolving slowest accurately

• Not cheaper than explicit at resolving the slowest scales

Traditionally these two approaches are assumed to be mutually exclusive

13

IMEX, adjPronunciation: īmig’z

A property of time integrators – usually describing RKmethods – which splits equations into slow and stiff operators. Slow operator is then discretized explicitly and the stiff operator is discretized implicitly.

u̇ = F u + R(u)

Stiff OperatorImplicit

Slow OperatorExplicit

Using IMEX we can attempt to achieve the best of both worlds but selecting the right sort of discretization in a multiphysics context.

We have to resolve the hydro : explicitWe don’t care about light waves : implicitWe still need great linear solvers and preconditioners

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IMEX Predictor Corrector for FMHD

Explicit Implicit

1D Linear Stability Analysis

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• Operator splitting a la ALEGRA MHD leads to an unstable system.

• An implicit field solve in the Lagrangian step recovers hydro stability limit!

• Requires two fields solves on the Lagrangian Mesh!

• Electric Displacement flux is the Galilean Invariant. Simplest approach requires discrete Hodge Starr.

This reproduces and extends the analysis found Love, Rider, and Scovazzi, J. Comput. Phys. 228.20 (2009): 7543-7564.

Constrained Transport Remap

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• Requires topology to not change under mesh motion – no reconnection.

• Moves dof from Lagrangianconfiguration to a more desirable mesh

• Discretized as advection (Reynold’s transport theorems) – i.e. Lie Derivatives

• If you don’t want to introduce new divergence to a field it had better be on faces

• Volume intersections computed using the D. Ibanez’s (SNL) branch of open source package R3D (LANL).

• Commutes with discrete exterior calculus

newS

1S

2S

4S

3S

oldS

Part IIIVerification

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Face Element Remap Results

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• A high order volume remap contribution for face element has been implemented.

• The volume contribution is associated with the through-face flux rather than the flux passing through the swept-edge faces in the standard div free CT algorithm.

• Below comparison of low and high order remap of charge density. Note much reduced charge diffusion for high order algorithms.

Harmonic MinmodDonor – Low Order

Discrete Maxwell’s Equations with a Hodge Star

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Electric Displacement (D) is really the frame invariant of the system for Galilean invariant Maxwell.We have preconditioners for (M + curl curl) systems:

Is this method asymptotic preserving? Does it converge to a solution of Resistive diffusion?Simple Verification test for sanity:

Verifying Maxwell + Hodge Star: Errors and rates at final time

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h Δt RL∞ Rate RL1 Rate RL2 Rate

0.1000 1.00e-1 4.96e-01 -- 3.53e-01 -- 3.82e-01 --

0.0500 0.25e-1 1.53e-01 1.69 1.08e-01 1.70 1.16e-01 1.70

0.0025 6.25e-2 4.06e-02 1.91 2.88e-02 1.91 3.10e-02 1.91

0.0125 1.56e-2 1.03e-02 1.97 7.31e-03 1.97 7.87e-03 1.97

h Δt RL∞ Rate RL1 Rate RL2 Rate

0.1000 1.00e-1 4.23e-01 -- 3.73e-01 -- 3.78e-01 --

0.0500 0.25e-1 1.33e-01 1.66 1.18e-01 1.65 1.19e-01 1.67

0.0025 6.25e-2 3.56e-02 1.90 3.17e-02 1.89 3.18e-02 1.90

0.0125 1.56e-2 9.10e-03 1.98 8.07e-03 1.97 7.87e-03 1.97

Magnetic Flux Density

Electric Displacement

Synthesis: 1D Flyer Plates

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• Prototype for a pulsed power Dynamic Materials Experiment.

• Traditionally solved with ALEGRA MHD

• Initial configuration of Aluminum (Tabular EOS and LMD conductivities) is accelerated by magnetic field push.

• Joule heating ablates the back surface and causes a stream of lower density plasma behind the flyer.

• Alfvén velocity eats your lunch and the time step crashes if you do nothing

• Previously could be circumvented by capping the Alfvén velocity and reducing forces

Fast Speed Limiting is Evident

Remap CFL control kicks in at faster velocities

Fast speed impacts calculation

time step

It works!

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Answers appear to be the same except for wiggles in the EM field at the material-vacuum transition.Doesn’t seem to effect hydro behavior.Could be due to Hxn boundary condition backed onto the conductor at time zero

Circularly Polarized Alfven Wave

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Lagrangian verification problem for ideal MHD on a periodic box. Planewave mode propagating at 30 degrees

Δt x rate v rate ε rate B rate

1.00E-02 3.26E-06 -- 4.19E-06 -- 6.09E-07 -- 3.06E-03 --

5.00E-03 1.47E-06 1.15E+00 1.29E-06 1.70E+00 2.64E-07 1.20E+00 1.48E-03 1.04E+00

2.50E-03 7.00E-07 1.07E+00 4.41E-07 1.54E+00 1.18E-07 1.16E+00 7.08E-04 1.07E+00

1.25E-03 3.67E-07 9.33E-01 1.63E-07 1.44E+00 5.21E-08 1.19E+00 3.21E-04 1.14E+00

6.25E-04 1.47E-07 1.32E+00 5.84E-08 1.48E+00 2.05E-08 1.34E+00 1.28E-04 1.32E+00

3.13E-04 3.66E-08 2.00E+00 1.47E-08 1.99E+00 5.13E-09 2.00E+00 3.21E-05 2.00E+00

As lim��→�

� → 0 so dynamics are

increasingly MHD like – recovers � Δ��

accuracy anticipated by predictor corrector.

Δx x rate v rate ε rate B rate

1.44E-01 7.99E-03 1.44E-03 3.95E-03 4.93E-03

7.20E-02 2.19E-03 1.86E+00 4.66E-04 1.62E+00 5.67E-04 2.80E+00 1.58E-03 1.64E+00

3.60E-02 5.72E-04 1.94E+00 1.18E-04 1.98E+00 7.03E-05 3.01E+00 4.29E-04 1.88E+00

1.80E-02 1.48E-04 1.95E+00 3.16E-05 1.90E+00 1.40E-05 2.32E+00 1.12E-04 1.93E+00

9.00E-03 3.44E-05 2.11E+00 8.01E-06 1.98E+00 4.35E-06 1.69E+00 2.87E-05 1.97E+00

Grid convergence recovers near optimal super convergence for the a single period of the planewave. Expected to degenerate after a long time.

Wave Zoo (MHD burst)

24

FMHD Ideal MHD

To recover ideal MHD behavior we really need E = 0 except where there are currents. Exn=0 on the boundary

Closure

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1. ALEGRA needs additional physics to meet the demands of next generation pulsed power systems (Z-Next)

2. Working with Maxwell Hydrodynamics we were able to eliminate Fast-Alfvén time step restrictions for more traditional Resistive MHD problems

3. Eliminating variables may make your equations harder to solve.

Next Steps:1. How many low density

kludges can be removed by using FMHD?

2. Can we modify IMEX RK SSP(2,2,2) for Lagrangianstep to gain second order accuracy in time for no additional solves?

3. Once we’re confident in FMHD’s implementation we will introduce a GOL and begin iterating again

References

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1. E. Love, W. J. Rider, and G. Scovazzi. "Stability analysis of a predictor/multi-corrector method for staggered-grid Lagrangian shock hydrodynamics." Journal of Computational Physics 228.20 (2009): 7543-7564.

2. E. Love and G. Scovazzi. "On the angular momentum conservation and incremental objectivity properties of a predictor/multi-corrector method for Lagrangian shock hydrodynamics." Computer Methods in Applied Mechanics and Engineering 198.41 (2009): 3207-3213.

3. M. Martin. "Generalized Ohm's law at the plasma-vacuum interface." (2010).4. A. C. Robinson, et al. "Arbitrary Lagrangian–Eulerian 3D ideal MHD algorithms." International Journal

for Numerical Methods in Fluids 65.11-12 (2011): 1438-1450.5. A. C. Robinson et al. "ALEGRA: An arbitrary Lagrangian-Eulerian multimaterial, multiphysics

code." AIAA Paper 1235 (2008): 2008.6. A. R. Schiemenz and A. C. Robinson. “Energy Based Magnetic Force Computation using Automatic

Differentiation.” SAND Report 2007-79777. D. A. McGregor and A. C. Robinson. “Explicit Indirect ALE Discretization of MHD Systems without a

Fast Magnetosonic Timestep Restriction.” In preparation.

Open Problem

28

Can we discretize a weak Lie derivative?

Will it commute with the codifferential?

Face Lie Derivative Edge Lie Derivative

Questions

29