SAND2011-6250C
Limited Artificial Viscosity and
Hyperviscosity Based on a Nonlinear
Hybridization Method
September 7, 2011
1950’s
Bill Rider, Ed Love, and G. Scovazzi
Sandia National Laboratories
Albuquerque, NM
Multimat11 Arcachon France
2011
1
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-AC0494AL85000.
Outline of Presentation
Introduction
 Nonlinear hybridization and other limiters
 Analysis of artificial viscosity coeffients
 Filtering and hyperviscosity
 Results
 Summary and outlook

2
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.

3
*Valid for a convex EOS, nonconvex EOS’s
have valid expansion shocks.
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.
7
Overcoming Godunov’s Theorem with
nonlinear methods = High Resolution Methods
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”)
 High
Jay Boris
V. Kolgan
9
Bram van Leer
We have decided to go a different path – A
nonlinear hybridization.
 This
is Harten’s method and it is fairly simple,
(
n
∆t
u n+1
=
u
j
j
∆ x f j+1/2 - f j-1/2
)
(
)
low
high
f j+1/2 = q j+1/2 f j+1/2
+ 1- q j+1/2 f j+1/2
 The
key is defining the switch as the ratio of secondto-first derivatives.
qj =
•The
∆x
¶2 u
¶ x2
¶u
¶x
=
u j+1 - 2u j + u j-1
u j+1 - u j + u j - u j-1
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.
10
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,
u
n+1
j
=u n
j
∆t
∆x
(
v u -u
n
j
∆ j-1/2 u = u
n
n
j
) - v ( min mod (u ) - min mod (u ) );v > 0
S
= sgn (∆
u )
-u
n
j-1
∆t
∆x
n
j-1
n
n
j
j-1
n
j-1/2
j-1/2
(
)
min mod ( u ) j = S j-1/2 max é 0,min S j-1/2 ∆ j-1/2 u n , ∆ j+1/2 u n ù
ë
û
n
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* 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
min mod ( u ) j =
n

u
n+1
j
n
j
∆t
∆x
)(
n
n
n
n
+
S
∆
u
+
∆
u
∆
u
∆
u
j-1/2
j+1/2
j-1/2
j+1/2
j+1/2
j-1/2
(
v u -u
n
j
n
j-1
)-
∆t
∆x
One can show q j
 Further
12
(S
Write the nonlinear hybrid scheme as
=u -

1
4
( )
é1- q ù
j
ë
û
2
((
v 1- q j
u nj+1 -u nj-1
u nj+1 -u nj-1
2
2
=
)
u nj+1 -u nj-1
2
(
- 1- q
)
u nj -u nj-2
2
j-1
= 1- min mod ( u ) j
n
qj =
∆ j-1/2 u n ∆ j+1/2 u n +∆ j-1/2 u n ∆ j+1/2 u n
∆ j-1/2 u n + ∆ j+1/2 u n
*Graphically this is clear using the “Sweby” diagram,
a method to parametrically examine TVD limiters.
)
);v > 0
u j+1 - 2u j + u j-1
u j+1 - u j + u j - u j-1
The Van Leer or
harmonic mean
limiter (TVD)
How do we define the limiter in a FEM code?
A
linear velocity field is “smooth”, does not
represent a shocked flow,
 Computation
 Normalize
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of the velocity Laplacian in FEM
using the triangle inequality
General structure of an improved artificial
viscosity has several elements
“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
 High-order
 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).
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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.
•
•
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In this simple example situation, the limiter is one.
Details of the numerical implementation
 Use
standard single-point integration (Q1/P0) fournode 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.
17
Zero Laplacian velocity field patch test
 Linear
 Test
velocity field
on an initially distorted mesh.
 The velocity Laplacian is zero everywhere (test passes).
• Inclusion of the boundary terms is critical.
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Early enhancements of the Q
 The
linear Q was introduced by Landshoff in
1955,
¶u
*lower case “c” is
s art = -c1 rc
¶x
the sound speed
 The
concept of turning the viscosity off in
expansion by Rosenbluth also in 1955
s art = -c2 r
2 ¶u ¶u
¶x ¶x
Þ s art = -c2 r max ( 0,2
)
¶u 2
¶x
 Later
embellishments are the tensor viscosity,
“flux-limited” viscosity, and the analysis of
the coefficients of viscosity.
•
19
Noted by Wilkins in 1980, but attributed to Kurapatenko in an
earlier 1970 work in the Russian literature.
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.
r0U s
(
1
r1
)
- r1 = u0 - u1
0
r0U s (u1 - u0 ) = p1 - p0
(
e1 - e0 = - 12 ( p0 + p1 )
p = P( r, e)
20
1
r1
- r1
0
)
u p = u0 - u1 ,is the particle velocity
U s is the shock speed
e is the internal energy
p = P ( r,e ) is the EOS
Technique for Analysis
 Basically,
the equations are rewritten is a
suggestive form,
r0U s
1
1
r0U s r - r = u p
r1 =
1
0
Us - u p
p1 = p0 + r0u pU s
(
)
(
e1 - e0 = - 12 ( p0 + p1 )
1
r1
- r1
0
)
p = P( r, e) = (g -1)re
 Solve for the shock speed (for an ideal gas)
Us =
 The
g +1
4
up +
c02
+
( )(u )
g +1
4
2
p
Q comes from making the ansatz that
s art = r0Usu p Þ r0 ( c1c0 + c2 ∆ u )∆ u
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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
U s = c0 +
•
g +1
4
u p ;c2 =
g +1
u p c0 ® 0
4
Second in the limit where the velocity jump is
u c ®¥
large (relative to the sound speed!)
p
U s = c0 +

24
g +1
2
u p ;c2 =
0
g +1
2
This recovers Richtmyer’s original result for
the quadratic coefficient!
The new analysis assumes piecewise
constant pressure, and a velocity jump
 State
the equations in a cell-centered form,
(
= r U (u
)
-u )
ps - p j = r jU s us - u j-1/2
p j - ps
j
s
j+1/2
s
 Solve
for the shock pressure and velocity and express
the shock velocity in terms of the known variables as
before, p = p + r U U
1
s
•
•
26
j
j
p
s
(
)
(
us = 2 u j-1/2 + u j+1/2
With U s = 12 c j + 14 c2 u j+1/2 - u j-1/2
The shock velocity relates to the artificial viscosity
coefficients.
)
Finishing the analysis with conclusions
 Taking
this result gives us recommended
viscosity coefficients
(
)
U s = 12 c j + 14 s j u j+1/2 - u j-1/2 ® c1 = 12 ;c2¢ = 14 c2
• 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.
27
Numerical Simulations I

Noh Implosion Test
Limited
Significant mesh distortion
28
Not Limited
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
m-1
¶2 m u
beyond standard second-order viscosity, (-1) lm ¶x 2 m
 The difference between filtered lower order dissipation
can define a hyperviscosity. 2 m
¶ u
¶2( m-1)u
¶2( m-1)u
¶x 2 m
¶x 2( m-1)
¶x 2( m-1)
 It is used in some aerospace codes using a certain
form that we will examine (2nd plus 4th order viscosity),
»
LO
s art = q js art
+ (1- q j )s hyper
•
29
-
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)
30
HyperViscosity can be developed by
filtering the second-order viscosity.
 Define
as the mean rate of deformation over a patch of
elements.
 Add
 The
additional viscosity
hyperviscosity also vanishes for a linear velocity
field since in that case
.
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Numerical Simulations I

Noh Implosion Test
Limited
Significant mesh distortion
32
Limited+Hyperviscosity
Numerical Simulations II

Saltzmann Test
Limited+Hyper
Not Limited
Limited+Hyper
Not Limited
34
Numerical Simulations II

Saltzmann Test
Not Limited
35
Limited 1st
Limited 2nd
Limited 2nd Hypervisc.
Numerical Simulations III

Sedov Blast Wave Test
Limited
36
Not Limited
Numerical Simulations III

Sedov Blast Wave Test
Limited+Hyper
37
Not Limited
Numerical Simulations III
 Sedov
Blast Wave Test
Limited+Hyper
• These results look promising…
38
Not Limited
These techniques are useful for problems
with unusual EOS’s
Expansion
Shock
Original
Limiter
time
39
pressure
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.
density
 In
Original
Limiter
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.
40