SAND2011-6250C Limited Artificial Viscosity and Hyperviscosity Based on a Nonlinear Hybridization Method September 7, 2011 Bill Rider, Ed Love, and G. Scovazzi Sandia National Laboratories Albuquerque, NM 1950’s Multimat11 Arcachon France 2011 Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.

Outline of Presentation Introduction Nonlinear hybridization and other limiters Analysis of artificial viscosity coeffients Filtering and hyperviscosity Results Summary and outlook

The development of improved viscosity has been a long time goal. Typical artificial viscosity methods render Lagrangian hydrodynamic calculations first-order accurate. The traditional shock-capturing viscosity is active in compression*, regardless of whether the compression is adiabatic or a shock. Ideally the viscosity should vanish (go to zero) if the fluid flow is smooth (adiabatic/isentropic). The goal is to construct a second-order accurate artificial viscosity method, one which can differentiate between discontinuous (shocked) and smooth flows. *Valid for a convex EOS, nonconvex EOS’s have valid expansion shocks.

Lagrangian weak form of the conservation laws Conservation of mass Conservation of linear momentum Conservation of energy

Properties of the Q (Discontinuity Capturing) The Q is dissipative, and should only be triggered at shocks* (or large gradients/discontinuities, fixed with a “modern” flux-limited Q) The Q allows the method to correctly compute the shock jumps without undue oscillations. The Q makes the solution stable in the presence of shocks. The linear Q stabilizes the propagation of simple waves on a grid. The quadratic Q stabilizes nonlinear shocks. This comes from the analysis later in the talk * More about this later in the talk!

Major developments in Q’s The linear Q was introduced by Landshoff in 1955, The concept of turning the viscosity off in expansion by Rosenbluth also in 1955 Later embellishments are tensor viscosities, “flux-limited” viscosity, and the analysis of the coefficients of viscosity noted by Wilkins in 1980, but attributed to Kurapatenko in an earlier 1970 work in the Soviet literature. *lower case “c” is the sound speed

The standard modern Q uses limiters The technique was pioneered by Randy Christenson (LLNL), and documented in the literature by David Benson (UCSD). It uses limiters from the Godunov-type methods to reduce the dissipation where the solution is resolved Based on the work of Van Leer (and Harten’s TVD). The basic idea is to measure the smoothness of the flow using the limiter, and reduce the dissipation where the flow is found to be smooth. These limiters are usually related to the ratio of successive gradients usually in a 1-D sense. Used in most “modern” Lagrangian/ALE codes.

There have been numerous previous attempts at an advanced viscosity in ALEGRA Two or three attempts to put Christenson’s limiter into the code going back to the early 90’s. Each attempt failed due to the quality of results. We have tried a least squares limiter, but ran into conceptual problems. We also began implemented a FCT based limited, but the results were poor.

Overcoming Godunov’s Theorem with nonlinear methods = High Resolution Methods High Resolution = “Modern” developed independently by four men in 1971-1972 Jay Boris (NRL retired) = FCT Bram Van Leer (U. Leiden, now U. Michigan) = Limiters Harten’s TVD is best known version Vladimir Kolgan (Russia/USSR) = largely unknown Ami Harten (Israel) = nonlinear hybridization (also introduced TVD, first attempt is “crude”) Here’s something cool historically: Boris didn’t know about Godunov - he just decided that the physics dictated certain properties of the sol’n Van leer told me he just knew that it could be done. The other three did know about Godunov’s work. We’ll talk about all of this later (Harten & Boris in Lecture 11) Van Leer in lecture 14. Harten also contributed ENO schemes…. Passed away in ‘94 Kolgan was the closest to a modern scheme with his extension of Godunov - 2nd order ENO method. All of them bypassed Godunov’s theorem by making the methods nonlinear, the differencing depends on the sol’n itself. Jay Boris V. Kolgan Bram van Leer

We have decided to go a different path – A nonlinear hybridization. This is Harten’s method and it is fairly simple, The key is defining the switch as the ratio of second-to-first derivatives. The next step is to realize that the high order method is the standard Lagrangian hydro with Q=0, the low order is found with the Q.

Introduced by Harten and Zwas (1972) Nonlinear hybridization was an early high resolution method that never caught on. Introduced by Harten and Zwas (1972) Refined with the ACM work by Harten (1977/78) Supplanted by Van Leer’s work and Harten’s own TVD theory for monotonicity preserving methods. The archetypical TVD method is the “minmod” * scheme, * The minmod function was invented by Boris with his initial FCT work.

The nonlinear hybridization can be identical to important TVD schemes.* We can prove this for an important choice for the high-order method (Fromm’s method) Rewrite minmod algebraically as Write the nonlinear hybrid scheme as One can show Further The Van Leer or harmonic mean limiter (TVD) *Graphically this is clear using the “Sweby” diagram, a method to parametrically examine TVD limiters.

How do we define the limiter in a FEM code? A linear velocity field is “smooth”, does not represent a shocked flow, Computation of the velocity Laplacian in FEM Normalize using the triangle inequality

General structure of an improved artificial viscosity has several elements High-order “flux” is “zero artificial viscosity”. The standard method is second order accurate Low-order “flux” is “standard artificial viscosity”. The linear viscosity renders the method 1st order Limited artificial viscosity if If the velocity field is linear, then the artificial viscosity is zero on both the interior and the boundary of arbitrary unstructured meshes. Important to include boundary terms (red boxed terms on previous slide).

What happens asymptotically with mesh refinement? Assume the flow is smooth. The standard non-limited artificial viscosity is O(h). The limiter itself is O(h). In one dimension, When the artificial viscosity is multiplied by the limiter, the result is O(h2). The final limited viscosity is O(h2), and goes to zero one order faster than the standard artificial viscosity. Assume the flow is shocked, with a finite jump as h→0. In this simple example situation, the limiter is one.

In general the limiter is a highly non-linear function of the discrete gradients More generally, Shock: Smooth:

Details of the numerical implementation Use standard single-point integration (Q1/P0) four-node finite elements to discretize the weak form. Flanagan-Belytschko viscous hourglass control (scales linearly with sound speed) with parameter 0.05 Second-order accurate (in time) predictor-corrector time integration algorithm. Artificial viscosity limiting based on Laplacian of velocity field. Gamma-law ideal gas equation-of-state. A more complex EOS used in the last example.

Zero Laplacian velocity field patch test Linear velocity field Test on an initially distorted mesh. The velocity Laplacian is zero everywhere (test passes). Inclusion of the boundary terms is critical.

Early enhancements of the Q The linear Q was introduced by Landshoff in 1955, The concept of turning the viscosity off in expansion by Rosenbluth also in 1955 Later embellishments are the tensor viscosity, “flux-limited” viscosity, and the analysis of the coefficients of viscosity. Noted by Wilkins in 1980, but attributed to Kurapatenko in an earlier 1970 work in the Russian literature. *lower case “c” is the sound speed

Relationship of EOS to the Q The key aspect of the Wilkins-Kurapatenko analysis is defining a clear relation between the Rankine-Hugoniot relations + EOS and the values of the constants in the artificial viscosity.

Technique for Analysis Basically, the equations are rewritten is a suggestive form, Solve for the shock speed (for an ideal gas) The Q comes from making the ansatz that

The classic Von Neumann-Richtmyer Q The original Q was introduced by Richtmyer in 1948 derived for an ideal gas using the shock relations, In the famous Von Neumann-Richtmyer Q removed the reference to the analytical derivation given by Richtmyer. Here the coefficient is given as The form of viscosity is motivated purely by stably placing a discontinuity on a finite number of mesh cells (order 3 or 4 cells).

Analysis of Q values, accounting for variable location. Previous analyses of the values to use with linear and quadratic coefficients have involved the Rankine-Hugoniot relations, but assuming all variables are collocated. We will now make use of the variable locations on the staggered grid,

Example: Analytical Q for ideal gas comes directly This involves doing a Taylor series expansion in two limits First in the limit where the velocity jump is small Second in the limit where the velocity jump is large (relative to the sound speed!) This recovers Richtmyer’s original result for the quadratic coefficient!

Artificial viscosity and Riemann solvers can be viewed in the same theoretical framework. Making some simple substitutions, These terms can be related to thermodynamic quantities, Since Riemann solvers are based on the R-H conditions the relation is direct. Dukowicz provided the bridge and developed a Riemann solver based on these ideas,

The new analysis assumes piecewise constant pressure, and a velocity jump State the equations in a cell-centered form, Solve for the shock pressure and velocity and express the shock velocity in terms of the known variables as before, With The shock velocity relates to the artificial viscosity coefficients.

Finishing the analysis with conclusions Taking this result gives us recommended viscosity coefficients Much smaller than traditionally used. We also note that Riemann solvers do not turn off the “viscosity” on expansion, and at the very least the linear dissipation is retained. If one looks at a “Q” as determining a dynamic pressure, retaining the Q on expansion would be called for.

Numerical Simulations I Noh Implosion Test Limited Not Limited Significant mesh distortion

Hyperviscosity is considered as a manner to improve the performance of the limiter. Hyperviscosity is a commonly used approach for stabilizing simulations Hyperviscosity is any dissipation that occurs in a form beyond standard second-order viscosity, The difference between filtered lower order dissipation can define a hyperviscosity. It is used in some aerospace codes using a certain form that we will examine (2nd plus 4th order viscosity), Note: Hourglass control is a particular form of hyperviscosity. The limiter’s use makes the hourglass modes more prevalent and difficult to control.

The Starter Results with Hyperviscosity The hypothesis is that hyperviscosity might keep hourglass modes from developing (it will not damp them)

HyperViscosity can be developed by filtering the second-order viscosity. Define as the mean rate of deformation over a patch of elements. Add additional viscosity The hyperviscosity also vanishes for a linear velocity field since in that case .

Numerical Simulations I Noh Implosion Test Limited Limited+Hyperviscosity Significant mesh distortion

Numerical Simulations II Noh Implosion Test Mesh looks better Limited+Hyper

Numerical Simulations II Saltzmann Test Limited+Hyper Not Limited Limited+Hyper Not Limited

Numerical Simulations II Saltzmann Test Not Limited Limited 1st Limited 2nd Limited 2nd Hypervisc.

Numerical Simulations III Sedov Blast Wave Test Limited Not Limited

Numerical Simulations III Sedov Blast Wave Test Limited+Hyper Not Limited

Numerical Simulations III Sedov Blast Wave Test Limited+Hyper Not Limited These results look promising…

These techniques are useful for problems with unusual EOS’s In this case there is an admissible shock on release (fundamental derivative is negative). Leaving viscosity on in expansion can handle this problem without difficulty, and sharper. Expansion Shock density pressure Original Limiter Original Limiter time time

Recommendations for Q’s in ALEGRA We must quantitatively evaluate these viscosities. The limiter allows the use of the analytical value of the linear coefficient. The limiter allows the viscosity to be on in expansion without causing undue dissipation Too little dissipation is much worse than too much! Too much dissipation is not accurate Too little dissipation is not physical “when in doubt, diffuse it out” The coefficients for the linear and quadratic should be carefully considered. The coefficients for viscosity ideally should be material and state dependent.