ReALE - Reconnection-Based ALE

Mikhail J. Shashkov

XCP-4, Methods & Algorithms,Los Alamos National Laboratory,

Los Alamos, New Mexico, 87545, USA

e-mail:shashkov@lanl.gov; webpage: cnls.lanl.gov/∼ shashkov

Raphael Loubere - CNRS at Universite de Toulouse, France

Pierre-Henri Maire - CEA-CESTA BP 2, France

Jerome Breil - CELIA, Universite de Bordeaux, France

Stephane Galera - INRIA, Team Bacchus, France

This work was performed under the auspices of the US Department of Energy at Los Alamos National Laboratory,under contract DE-AC52-06NA25396 - LA-UR-09-02990 .

The authors acknowledge the partial support of the ASC Program at LANL and DOE Office of Science ASCR

Program. Thanks to A. Soloviev, J. Dukowicz, A. Barlow, G. Ball, M. Kucharik, T. Ringler, D. Burton, J. Fung, R.

Lowrie, M. Berndt.

1

Outline• Motivation◦ Standard ALE Methods and their Limitations◦ Reconnection-based ALE - ReALE Concept◦ Lagrangian Methods on Polygonal Meshes - Cell-centered, Staggered

• Reconnection-based Rezone Strategy◦ Voronoi Tessellation◦ Lloyd’s Algorithm◦ Initial Mesh◦ Rezone Algorithm

• Remapping• Tests and Method Comparison◦ Sedov Problem◦ Multimaterial Vortex Formation◦ Multimaterial Shock-Bubble Interaction - Cylindrical and Spherical Cases◦ RT Instability - Late Stages◦ The Rise of a Light Gas Bubble in Exponential Atmosphere◦ Two Chambers Problem

• Conclusion and Future Work

Arbitrary Lagrangian-Eulerian (ALE) Methods

• Explicit Lagrangian (solving Lagrangian equations)phase — grid is moving with fluid

• Rezone phase — changing the mesh (improvinggeometrical quality, smoothing, adaptation) — meshmovement

• Remap phase (conservative interpolation) —remapping flow parameters from Lagrangian grid torezoned mesh

Rayleigh-Taylor Instability — Limitation of ALE

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Reference-Jacobian Rezone Strategy

(Knupp, Margolin, Shashkov)

Solution: ReALE - Reconnection-based ALERezone phase — Mesh is allowed to change

connectivity

ReALE — Reconnection-based ALE

• Lagrangian phase — General polygonal meshes

• Rezone phase — Allows mesh reconnection

• Remap phase — Remapping from one polygonalmesh to another

The Devil is in the Details

Lagrangian Hydrodynamics of General PolygonalMeshes

• Staggered DiscretizationDensity, internal energy, pressure- cells(zones), velocity - nodes(points); there is also subzonalquantities

ρc εc

un

,

ρcn

∗ FLAG - LANL, KULL - LLNL

Design Principles - Burton (1994)

∗ ALE.INC

Research code T7(5)-LANL Shashkov, Campbell, Loubere (2003)

Compatible discretization, internal energy equation, subzonalmasses, edge and tensor artificial viscosity.

Lagrangian Hydrodynamics of General PolygonalMeshes• Cell-Centered Discretization

All primary quantities are cell-centered, some algorithm fordefinition of nodal velocities

∗ Free-Lagrange Efforts - Pasta, Ulam, Dyachenko, Crowley,Trease, Sahota, Ball, Shashkov, Clark, Sofronov, · · ·∗ CAVEAT - LANL - Dukowicz (1986)

∗ Despres, Mazeran - CEA (2005)

∗ CHIC - Bordeaux - Maire (2007)

Density, momentum, and total energy are defined by their mean values in the cells. The vertex velocities and the

numerical fluxes through the cell interfaces are evaluated in a consistent manner due to an original Godunov-like

solver located at the nodes (Geometric Conservation Law). There are four pressures on each edge, two for each

node on each side of the edge. Conservation of momentum and total energy is ensured. Semi-discrete entropy

inequality is provided.

Multimaterial Lagrangian Hydrodynamics

• Method of Concentrations

* Transport equations for evolving concentrations, rescaling.

* Mixture assumption - iso-pressure, iso-temperature, leads toeffective γ for mixture of ideal gasses

• Volume of fluid

◦ Interface Reconstruction◦ Closure models (constant volume fraction)

Reconnection-based rezone strategy

• Initial mesh at t = 0 is Voronoi mesh• Voronoi mesh correspond to some generators - one generator per Voronoi

cell• Location of generators control the mesh• We use weighted-Voronoi meshes - accuracy, smoothness, adaptation• On rezone stage we define new (rezoned) positions of generators which

gives us new rezoned mesh - Voronoi mesh corresponding to new positionsof generators.

• Rezone strategy is to how move generators• Connectivity of the mesh correspond to Voronoi mesh• At each Lagrangian step we start with ”almost” Voronoi mesh - ”small”

edges are removed

Voronoi Tessellation - Definition

Set of generators: gi = (xi, yi)Voronoi cell: Vi = {r = (x, y) : |r− gi| < |r− gj| , for all j 6= i}Mass centroid of the cell (ρ(r) > 0 - given function)

cρi =∫Vi

r ρ(r) dx dy/∫Viρ(r) dx dy,

If gi = cρi - weighted-centroidal Voronoi tessellation

Left - generators, Voronoi cells, centroids; Right - centroidal Voronoi Tessellation (ρ = 1)

Voronoi Tessellation - Nature

Left - from T. Ringler, Voronoi Tessellation for Ocean Modeling: Methods, Modes and Conservation. Right - fromG.W. Barlow, Hexagonal Territories, Animal Behavior, v. 22, 1974 (The territories of the male Tilapia)

Somewhere in Argentine Republic Fruit (Surface Mesh) - Maui

Voronoi Tessellation - Lloyd’s Algorithm1) Given ρ(r > 0 and positive number n2) Set positions for n generators: g0

i

3) Construct Voronoi cells V 0i corresponding to g0

i

4) Define new positions of generators to be weighted-centroids of V 0i :

g1i = cρ(V 0

i )

5) Repeat starting with #2 till some convergence criterion is satisfied

Mesh and Volume

Left - initial, central - five iterations, right - ”final”

Calculations by M. Kucharik

Initial Mesh - Example - Shock-Bubble Interaction

0.295 0.3050.05

piston

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Air

He

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Generators on the similar

distance perpendicular to

material interface. Inside the

bubble - uniform with respect

to arc-length.

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"circle/fixed/Vor_mesh.dat"

Generators on the similar

distance perpendicular to

material interface are fixed -

Centroidal Voronoi

Initial Mesh - Examples

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"RTI/b_Vor_mesh.dat"

Generators — g0i

Voronoi mesh generation code - A. Soloviev

Rezone Strategy

Requirements - Close to Lagrangian, Smooth Mesh, Adaptation capability

• At the beginning of Lagrangian step mesh is Voronoi mesh, V nicorresponding to generators gni

• At the end of Lagrangian step vertices of the cells are movedaccordingly to Lagrangian method - Lagrangian cell at time tn+1

is V n+1i - not a Voronoi mesh

• Lagrangian Phase - There is no equation for movement of thegenerators

Algorithm for Movement of Generators• Compute weighted-centroid (cρ)n+1

i of V n+1i as follows`

cψ´n+1

i=RV n+1i

rψn+1i (r) dx dy/

RV n+1i

ψn+1i (r) dx dy

ψn+1i (r) - piece-wise linear reconstruction based on mean values of some

monitor function at tn+1.• ”Lagrangian” Movement of Generators

gn+1,Lagi = gi + ∆t ui , ui =

1

|P(cni )|X

p∈P(cni

)

up

◦ gn+1,Lagi - inside Lagrangian cell at tn+1

• Final position of generators

gn+1i = gn+1,Lag

i + ωih(cρi )n+1 − gn+1,Lag

i

iωi ∈ [0, 1]

◦ New generators lies between Lagrangian position and position ofcentroid of new Lagrangian cell

◦ ωi = 0 - Lagrangian position; ωi = 1 - centroid of new Lagrangian cell -one iteration of Lloyd’s - smoothing the mesh

Computation of the ωi• The principle of material frame indifference: uniform translation or rotationωi = 0

• Deformation gradient tensor F

F =

∂Xn+1

∂Xn∂Xn+1

∂Y n

∂Y n+1

∂Xn∂Y n+1

∂Y n ,

!

Jacobian matrix of the map that connects the Lagrangian configurations ofthe flow at time tn and tn+1.

• The right Cauchy-Green strain tensor

C = FtF

◦ C is a 2× 2 symmetric positive definite tensor◦ Reduces to the unitary tensor in case of uniform translation or rotation◦ Two positive eigenvalues: λ1 ≤ λ2 - the rates of dilation

• Definition of ωiωi =

„1−

λ1,i

λ2,i

«ffi„1− min

i

λ1,i

λ2,i

«

Rezoned Mesh

• Creating Voronoi mesh corresponding to gn+1i

• Cleaning - removing small edges - CFL

A B

P

P

ACD

ABD

CD

A B

PABCD

CD

• New mesh is V n+1i is is cleaned Voronoi mesh

◦ Connectivity of the mesh - correspond to Voronoi mesh -automatic

Remapping

Basis - some cell-centered conservative remapping algorithm capable to remapfrom one polygonal mesh to another.

We use intersection (overlay based) with piece-wise linear reconstruction

Cell-centered Lagrangian discretization

Remapping applied to primary cells

N cp

p

p−

N cp

UpLcp

(ρc,U c, Pc)

Πcp

Πcp

Lcp

p+

Velocity of nodes recovered as followsup = M−1

p

Pc∈C(p)

h“LcpN

cp + LcpN

cp

”Pc +MpcUc

iMpc = ρc ac

hLcp

“Ncp ⊗Nc

p

”+ Lcp

“Ncp ⊗Nc

p

”i,

Mp =P

c∈C(p)Mpc

Staggered Lagrangian discretization

Main Stages

• Gathering stage Define mass, momentum, internal energy and kinetic energy in the subcells ensuring

that the gathering stage is conservative.

• Subcell remapping stage Conservative remapping of mass, momentum, internal and kinetic

energy from subcells of Lagrangian mesh to subcells of rezoned mesh.

• Scattering stage Conservatively recover the primary variables subcell density, nodal velocity, and

cell-centered specific internal energy on the new rezoned mesh. Require local matrix inversion.

c

n

nc

n

cn

c

Subcell Remap

Gathering Scattering

ReALE - Flowchart

Un

CleaningConstruction of associated Voronoi tesselation

In+1

Construction of initial Voronoi meshCleaning

Mesh I

Un=0

start a new time step

Initialisation

n=0

Initial distrib. of physical parameters

n := n+1

Mesh I

Mesh LEnd

Lagrangian phase

Un+1L

Rezone phase

n+1Mesh R

Definition of generator position for cell c:

Un+1

Remap phaseRemap from

Lagrangian mesh onto Rezoned mesh

Requires intersection−based remap

Solving Lag. equations on general polygonal mesh

Same connectivity meshes

Start

Rezoned mesh is a general polygonal mesh

n+1

n

{ Gcn+1}

L

Un+1L

Rn+1n+1

Resettingn+1

R

Tests and Comparison of Methods

Test Problems

• Sedov problem - Sanity check• Vortex formation• Interaction of shock with bubble• RT Instability

Methods

• Lagrangian - cell-centered and staggered• Standard ALE◦ Rezone - one iteration of Winslow (condition number)◦ Remap - swept region

• Eulerian = Lagrange + Remap• ReALE