Motivation Theoretical Developments Numerical Experiments
A Mimetic Tensor Artificial Viscosity Method for Arbitrary Polygonal and Polyhedral Meshes Konstantin Lipnikov1 1
Mikhail Shashkov2
Los Alamos National Laboratory Applied Mathematics and Plasma Physics Group, T-5 2 Methods and Algorithms Group, XCP-4
March 2011, Reno, NV
Konstantin Lipnikov1
Mikhail Shashkov2
A Mimetic Tensor Artificial Viscosity Method for Arbitrary Polygonal and Polyhed
Motivation Theoretical Developments Numerical Experiments
Outline
Motivation Mimetic tensor artificial viscosity Symmetry preservation on special meshes Numerical experiments Summary
Konstantin Lipnikov1
Mikhail Shashkov2
A Mimetic Tensor Artificial Viscosity Method for Arbitrary Polygonal and Polyhed
Motivation Theoretical Developments Numerical Experiments
Shock calculations Shock calculation requires an introduction of the artificial numerical viscosity. The analysis has shown that the modified equations, like the momentum equation du ∂(p + pQ ) ρ =− , dt ∂x
∂u µ2 pQ = −µ1 + ∂x ρ
∂u ∂x
2 ,
give the physically meaningful solution: Rankine-Hugoniot jump conditions are preserved ’viscous’ solution converges to the correct discontinuous solution as µi → 0.
Konstantin Lipnikov1
Mikhail Shashkov2
A Mimetic Tensor Artificial Viscosity Method for Arbitrary Polygonal and Polyhed
Motivation Theoretical Developments Numerical Experiments
Shock calculations Shock calculation requires an introduction of the artificial numerical viscosity. The analysis has shown that the modified equations, like the momentum equation du ∂(p + pQ ) ρ =− , dt ∂x
∂u µ2 pQ = −µ1 + ∂x ρ
∂u ∂x
2 ,
give the physically meaningful solution: Rankine-Hugoniot jump conditions are preserved ’viscous’ solution converges to the correct discontinuous solution as µi → 0.
The model for the artificial viscosity is given analytically. Konstantin Lipnikov1
Mikhail Shashkov2
A Mimetic Tensor Artificial Viscosity Method for Arbitrary Polygonal and Polyhed
Motivation Theoretical Developments Numerical Experiments
Edge viscosity
The Noh implosion problem on a square mesh with the edge and TTS viscosities runs to completion but results in mesh artifacts.
The strong mesh imprint indicates probably that the edge viscosity does not correspond to a continuous model.
Konstantin Lipnikov1
Mikhail Shashkov2
A Mimetic Tensor Artificial Viscosity Method for Arbitrary Polygonal and Polyhed
Motivation Theoretical Developments Numerical Experiments
Mimetic viscosity Campbell & Shashkov (JCP, 01) proposed to employ a discrete tensor calculus technique for development of artificial viscosity methods. They started with a continuous model: ρ
du = −∇p + div(σ(u)), dt
where the viscous tensor σ is either non-symmetric or symmetric: σ(u) = µ ∇u
or
σ(u) =
µ (∇u + (∇u)T ). 2
The discrete tensor calculus provides accurate approximations of the first-order operators (divergence and gradient) on arbitrary meshes.
Konstantin Lipnikov1
Mikhail Shashkov2
A Mimetic Tensor Artificial Viscosity Method for Arbitrary Polygonal and Polyhed
Motivation Theoretical Developments Numerical Experiments
Mimetic viscosity
The Noh implosion problem on a square mesh with the mimetic viscosity runs to completion without the TTS viscosity.
The developed discrete tensor calculus was limited to convex zones and zones having exactly d faces attached to each vertex, d = 2, 3.
Konstantin Lipnikov1
Mikhail Shashkov2
A Mimetic Tensor Artificial Viscosity Method for Arbitrary Polygonal and Polyhed
Motivation Theoretical Developments Numerical Experiments
Mimetic viscosity
The Noh implosion problem on a square mesh with the mimetic viscosity runs to completion without the TTS viscosity.
The developed discrete tensor calculus was limited to convex zones and zones having exactly d faces attached to each vertex, d = 2, 3. Goal The goal of this talk is to remove these limitations by extending the discrete tensor calculus to arbitrary-shaped zones. Konstantin Lipnikov1
Mikhail Shashkov2
A Mimetic Tensor Artificial Viscosity Method for Arbitrary Polygonal and Polyhed
Motivation Theoretical Developments Numerical Experiments
Discrete Tensor Calculus Consistency Condition Symmetry Analysis
Requirements for Artificial Viscosity ρ
du = −∇p + div(µ ∇(u)) dt
Partial requirements for the artificial viscosity are Self-similar motion invariance: The viscosity should vanish for uniform contractions or rigid body rotations. Wave-front invariance: The viscosity should have no effect along a wave front of a constant phase.
They are transferred to the discrete calculus requirements: The discretization should be exact for linear u (µ = const). The coefficient µ should be a tensor. Konstantin Lipnikov1
Mikhail Shashkov2
A Mimetic Tensor Artificial Viscosity Method for Arbitrary Polygonal and Polyhed
Motivation Theoretical Developments Numerical Experiments
Discrete Tensor Calculus Consistency Condition Symmetry Analysis
Discrete Tensor Calculus div(µ ∇(u))
→
DIV µ GRAD U
where the mimetic operators are GRADe U =
U1 − U2 |x1 − x2 |
and DIV µ = (GRAD)∗ . This duality is with respect to normed discrete spaces of point-centered and edge-centered discrete fields: T DIV µ = −M−1 p (GRAD) Me .
Both mimetic operators have a compact stencil and only Me depends on µ. Konstantin Lipnikov1
Mikhail Shashkov2
A Mimetic Tensor Artificial Viscosity Method for Arbitrary Polygonal and Polyhed
Motivation Theoretical Developments Numerical Experiments
Discrete Tensor Calculus Consistency Condition Symmetry Analysis
Viscous Force
The viscous force is F = −(GRAD)T Me GRAD U ≡ −A U. A new idea is to calculate matrix A without calculating the inner product matrix Me and the discrete gradient operator.
Konstantin Lipnikov1
Mikhail Shashkov2
A Mimetic Tensor Artificial Viscosity Method for Arbitrary Polygonal and Polyhed
Motivation Theoretical Developments Numerical Experiments
Discrete Tensor Calculus Consistency Condition Symmetry Analysis
Viscous Force
Consider a weak form of the viscous force: FT V = −(A U)T V
∀V.
It can be shown that an additivity property holds: the matrix A can be assembled from small zonal matrices Az : X FT V = − (Az Uz )T Vz zones
Konstantin Lipnikov1
Mikhail Shashkov2
A Mimetic Tensor Artificial Viscosity Method for Arbitrary Polygonal and Polyhed
Motivation Theoretical Developments Numerical Experiments
Discrete Tensor Calculus Consistency Condition Symmetry Analysis
Mimicking Gauss-Green Formula Assume that the vector of nodal velocities Uz corresponds to a function uh . The matrix Az represents of the following bilinear form: Z T (Az Uz ) Vz ≈ µz ∇uh : ∇vh dx. z
The Gauss-Green formula is Z Z Z µz ∇uh : ∇vh dx = − div(µz ∇uh )vh dx + (µz ∇uh ) · n · vh dx. z
z
Konstantin Lipnikov1
Mikhail Shashkov2
∂z
A Mimetic Tensor Artificial Viscosity Method for Arbitrary Polygonal and Polyhed
Motivation Theoretical Developments Numerical Experiments
Discrete Tensor Calculus Consistency Condition Symmetry Analysis
Mimicking Gauss-Green Formula Assume that the vector of nodal velocities Uz corresponds to a function uh . The matrix Az represents of the following bilinear form: Z T (Az Uz ) Vz ≈ µz ∇uh : ∇vh dx. z
The Gauss-Green formula is Z Z Z µz ∇uh : ∇vh dx = − div(µz ∇uh )vh dx + (µz ∇uh ) · n · vh dx. z
∂z
z
For a linear function, uh = uLh , the volume integral disappears, i.e. we need to define the function vh only on the zone boundary: Z X (Az ULz )T Vz ≈ (µz ∇uLh ) · nf · vh dx. f ∈∂z
Konstantin Lipnikov1
Mikhail Shashkov2
f
A Mimetic Tensor Artificial Viscosity Method for Arbitrary Polygonal and Polyhed
Motivation Theoretical Developments Numerical Experiments
Discrete Tensor Calculus Consistency Condition Symmetry Analysis
Mimicking Gauss-Green Formula (cont.) We apply a proper quadrature to approximate the face integral: Z vh dx = f
Konstantin Lipnikov1
Mikhail Shashkov2
X
ωf ,p Vf ,p
p∈∂f
A Mimetic Tensor Artificial Viscosity Method for Arbitrary Polygonal and Polyhed
Motivation Theoretical Developments Numerical Experiments
Discrete Tensor Calculus Consistency Condition Symmetry Analysis
Mimicking Gauss-Green Formula (cont.) We apply a proper quadrature to approximate the face integral: Z vh dx = f
X
ωf ,p Vf ,p
p∈∂f
Using the quadrature, we replace ≈ sign by equality. This gives an algebraic equation for the unknown matrix Az : (Az ULz )T Vz =
X
(µz ∇uLh ) · nf
f ∈∂z
X
ωf ,p Vf ,p
∀Vz .
p∈∂f
The equation must hold for every linear function uLh , and the corresponding vector of degrees of freedom ULz . We found a family of stable solution Az (JCP, 2010, V.229). Konstantin Lipnikov1
Mikhail Shashkov2
A Mimetic Tensor Artificial Viscosity Method for Arbitrary Polygonal and Polyhed
Motivation Theoretical Developments Numerical Experiments
Discrete Tensor Calculus Consistency Condition Symmetry Analysis
Accuracy Analysis Let µ = const. Then F = 0 for a linear velocity field on arbitrary meshes. Also, F = 0 for rigid body rotations and symmetric viscous tensor. For a variable but smooth µ, we get the following relative L2 -errors between the discrete and analytic viscous forces.
Refinement level 0 1 2 3 4 rate Konstantin Lipnikov1
9.21e-2 3.78e-2 1.15e-2 3.08e-3 7.87e-4 1.75 Mikhail Shashkov2
1.39e-1 7.85e-2 3.58e-2 1.56e-2 6.89e-3 1.10
1.61e-1 1.32e-1 9.34e-2 6.62e-2 4.65e-2 0.46
A Mimetic Tensor Artificial Viscosity Method for Arbitrary Polygonal and Polyhed
Motivation Theoretical Developments Numerical Experiments
Discrete Tensor Calculus Consistency Condition Symmetry Analysis
Symmetry Analysis on Special Meshes y
y p4
p2
∆θ
z0 p
p1
p3
(fvis )pz0
3 (fvis ref )z
1 (fvis ref )z
z
(fvis )pz
O
x O
x
We proved that the developed mimetic tensor artificial viscosity preserves symmetry in both x-y and r-z coordinate systems on ’polar’ meshes with uniform angular steps. Konstantin Lipnikov1
Mikhail Shashkov2
A Mimetic Tensor Artificial Viscosity Method for Arbitrary Polygonal and Polyhed
Motivation Theoretical Developments Numerical Experiments
Noh Implosion Problem Sedov Explosion Problem Saltzman Piston Problem
Viscosity Coefficient We use the viscosity coefficient proposed by Wilkins (JCP, 80): s 2 γ+1 γ+1 µz = ψz ρz Lz q2 |∆u|z + q22 |∆u|2z + q21 s2z 4 4 ψz = 1 for compression and 0 otherwise q1 ∼ 1 and q2 ∼ 1 according to 1D theory Velocity jump is |∆u|z = Lz DIV µ,z (Uz ) Modified Dobrev-Kolev-Rieben approach for characteristics length: 1/d Vz (t) Lz (t) = Lz (0) Vz (0) Konstantin Lipnikov1
Mikhail Shashkov2
A Mimetic Tensor Artificial Viscosity Method for Arbitrary Polygonal and Polyhed
Motivation Theoretical Developments Numerical Experiments
Noh Implosion Problem Sedov Explosion Problem Saltzman Piston Problem
3D Noh Implosion Problem
ideal gas with γ = 5/3 1 initial conditions are ρ = 1, p = 0, and u = − p (x, y)T 2 x + y2 a circular shock wave is generated at the origin and moves with the constant speed 1/3; the density behind shock is ρ = 64 simulation time T = 0.6
Konstantin Lipnikov1
Mikhail Shashkov2
A Mimetic Tensor Artificial Viscosity Method for Arbitrary Polygonal and Polyhed
Motivation Theoretical Developments Numerical Experiments
Noh Implosion Problem Sedov Explosion Problem Saltzman Piston Problem
Noh Problem: RZ vs 3D
initial 80 × 80 square mesh q1 = 0.5, q2 = 1 symmetric form of σ(u). Left-top picture: density as a function of distance. Konstantin Lipnikov1
Mikhail Shashkov2
A Mimetic Tensor Artificial Viscosity Method for Arbitrary Polygonal and Polyhed
Motivation Theoretical Developments Numerical Experiments
Noh Implosion Problem Sedov Explosion Problem Saltzman Piston Problem
Noh Problem: RZ vs 3D (cont.)
initial 80 × 80 × 80 cubic mesh q1 = 0.5, q2 = 1 symmetric form of σ(u). Left-bottom picture: density as a function of distance. Konstantin Lipnikov1
Mikhail Shashkov2
A Mimetic Tensor Artificial Viscosity Method for Arbitrary Polygonal and Polyhed
Motivation Theoretical Developments Numerical Experiments
Noh Implosion Problem Sedov Explosion Problem Saltzman Piston Problem
Noh Problem: Symmetry Preservation in RZ
initial 50 × 30 mesh q1 = 1, q2 = 1 non-symmetric form of σ(u). Konstantin Lipnikov1
Mikhail Shashkov2
A Mimetic Tensor Artificial Viscosity Method for Arbitrary Polygonal and Polyhed
Motivation Theoretical Developments Numerical Experiments
Noh Implosion Problem Sedov Explosion Problem Saltzman Piston Problem
Noh Problem: Impact of New Characteristic Length
smooth characteristic length is needed! Konstantin Lipnikov1
Mikhail Shashkov2
A Mimetic Tensor Artificial Viscosity Method for Arbitrary Polygonal and Polyhed
Motivation Theoretical Developments Numerical Experiments
Noh Implosion Problem Sedov Explosion Problem Saltzman Piston Problem
3D Sedov Explosion Problem
ideal gas with γ = 1.4 initial conditions ρ = 1, u = 0, and the total energy E0 is all internal and concentrated in few zones around the origin a diverging shock wave or radius rd with a peak density of ρ = 6 is generated at the origin: 1/(2+d) 2/(2+d) rd = E0 /(αd ρ0 ) t ,
α3 = 0.850937
simulation time T = 1.0
Konstantin Lipnikov1
Mikhail Shashkov2
A Mimetic Tensor Artificial Viscosity Method for Arbitrary Polygonal and Polyhed
Motivation Theoretical Developments Numerical Experiments
Noh Implosion Problem Sedov Explosion Problem Saltzman Piston Problem
Sedov Problem: RZ vs 3D
initial 80 × 80 square mesh q1 = 1, q2 = 1 symmetric form of σ(u). Left-top picture: density as a function of distance. Konstantin Lipnikov1
Mikhail Shashkov2
A Mimetic Tensor Artificial Viscosity Method for Arbitrary Polygonal and Polyhed
Motivation Theoretical Developments Numerical Experiments
Noh Implosion Problem Sedov Explosion Problem Saltzman Piston Problem
Sedov Problem: RZ vs 3D (cont.)
initial 80 × 80 × 80 cubic mesh q1 = 1, q2 = 1 symmetric form of σ(u). Left-bottom picture: density as a function of distance. Konstantin Lipnikov1
Mikhail Shashkov2
A Mimetic Tensor Artificial Viscosity Method for Arbitrary Polygonal and Polyhed
Motivation Theoretical Developments Numerical Experiments
Noh Implosion Problem Sedov Explosion Problem Saltzman Piston Problem
Sedov Problem: Symmetry Preservation in RZ
initial 50 × 30 mesh q1 = 1, q2 = 1 non-symmetric form of σ(u). Konstantin Lipnikov1
Mikhail Shashkov2
A Mimetic Tensor Artificial Viscosity Method for Arbitrary Polygonal and Polyhed
Motivation Theoretical Developments Numerical Experiments
Noh Implosion Problem Sedov Explosion Problem Saltzman Piston Problem
Sedov Problem: Impact of New Characteristic Length
smooth characteristic length is needed! Konstantin Lipnikov1
Mikhail Shashkov2
A Mimetic Tensor Artificial Viscosity Method for Arbitrary Polygonal and Polyhed
Motivation Theoretical Developments Numerical Experiments
Noh Implosion Problem Sedov Explosion Problem Saltzman Piston Problem
Saltzman Piston Problem
ideal gas with γ = 5/3 initial conditions are ρ = 1, u = 0, ε = 10−7 ; piston velocity is 1 a 1D shock wave propagates through a two-dimensional mesh, reflects from the opposite fixed end of the box at time t = 0.8 and hits the piston at time t = 0.9; the final density behind the shock is 20 and the density ahead of the shock is 10 simulation time T = 0.925
Konstantin Lipnikov1
Mikhail Shashkov2
A Mimetic Tensor Artificial Viscosity Method for Arbitrary Polygonal and Polyhed
Motivation Theoretical Developments Numerical Experiments
Noh Implosion Problem Sedov Explosion Problem Saltzman Piston Problem
Saltzman Problem: X-Y vs R-Z Coordinate Systems
X-Y
R-Z
Konstantin Lipnikov1
Mikhail Shashkov2
A Mimetic Tensor Artificial Viscosity Method for Arbitrary Polygonal and Polyhed
Motivation Theoretical Developments Numerical Experiments
Noh Implosion Problem Sedov Explosion Problem Saltzman Piston Problem
Conclusion
We developed a new TAV method for arbitrary polygonal and polyhedral meshes using the mimetic tensor calculus technique. Non-symmetric and symmetric viscous tensors showed robust behavior for the Noh, Sedov and Saltzman problems on variety of meshes. New selection of the characteristic length improved drastically symmetry of a discrete solution.
Konstantin Lipnikov1
Mikhail Shashkov2
A Mimetic Tensor Artificial Viscosity Method for Arbitrary Polygonal and Polyhed
Motivation Theoretical Developments Numerical Experiments
Noh Implosion Problem Sedov Explosion Problem Saltzman Piston Problem
Acknowledgments
DOE Office of Science Program in Applied Mathematics Research (ASCR) for financial support DOE Advanced Simulation & Computing (ASC) Program for financial support D.Burton, M.Bement and M.Kenamond for their help with the FLAG code.
Konstantin Lipnikov1
Mikhail Shashkov2
A Mimetic Tensor Artificial Viscosity Method for Arbitrary Polygonal and Polyhed
