Limited Artificial Viscosity and Hyperviscosity Based on a Nonlinear Hybridization Method

1. SAND2011-6250C Limited Artificial Viscosity and Hyperviscosity Based on a Nonlinear Hybridization Method 1950’s September 7, 2011 Bill Rider, Ed Love, and G. Scovazzi Sandia National Laboratories Albuquerque, NM 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.

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

3. 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.

4. 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!

5. 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.

6. 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”) Jay Boris V. Kolgan Bram van Leer

7. 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.

8. 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.

9. 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.

10. 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

11. 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).

12. 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.

13. 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.

14. 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.

15. 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

16. 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.

17. 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

18. 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!

19. 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.

20. 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.

21. Numerical Simulations I • Noh Implosion Test Limited Not Limited Significant mesh distortion

22. 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.

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

24. 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 .

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

26. Numerical Simulations II • Saltzmann Test Limited+Hyper Not Limited Limited+Hyper Not Limited

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

28. Numerical Simulations III • Sedov Blast Wave Test Limited Not Limited

29. Numerical Simulations III • Sedov Blast Wave Test Limited+Hyper Not Limited

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

31. 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

32. 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.