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Operated by Los Alamos National Security, LLC for NNSA MultiMat 2011 - 1 A cell-centered Lagrangian hydrodynamics method for multi-dimensional unstructured grids in curvilinear coordinates with solid constitutive models D.E. Burton, T.C. Carney, N.R. Morgan, S.R. Runnels, S.K. Sambasivan*, M.J. Shashkov X-Computational Physics Division * T Division Los Alamos National Laboratory MultiMat 2011 International Conference on Numerical Methods for Multi-Material Fluid Flows Arcachon, France September 5-9, 2011 Acknowledgements: U.S. DOE LANL LDRD Program A. Barlow, B. Despres, M. Kenamond, P.H. Maire, P. Roe LA-UR-11-04995 August 27, 2011

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Operated by Los Alamos National Security, LLC for NNSA MultiMat 2011 - 2 Organization of the presentation Mimetic approach Why a mimetic approach Corner vs. surface fluxes Conservation & ancillary equations Curl & divergence expressions Entropy & the energy equation Nodal solvers & the entropy condition Conventional approach A new tensor approach Spurious vorticity Mimetic approach for axisymmetric (rz) geometry The notion of a centroidal control volume Axisymmetric equations Concluding remarks

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Operated by Los Alamos National Security, LLC for NNSA MultiMat 2011 - 3 We are interested in cell-centered hydro (CCH) as a possible alternative/complement to staggered-grid hydro (SGH) Since many of these areas have not been widely explored in a CCH context, we used a mimetic* approach to guide the derivation of the difference scheme The numerical model should mimic the properties of the physical system The mimetic approach considers not only the usual finite volume equations Evolution equations Flux conservation equations but also ancillary relationships that place constraints on the formulation Geometric volume conservation Curl & divergence identities Angular momentum Entropy production etc. The latter are true analytically, but not necessarily satisfied by a difference scheme To be a viable alternative to SGH, CCH must be formulated to have comparable capabilities in the areas of: Material strength Multi-material cells Unstructured polytopal grids Multi-dimensional with curvilinear geometry Advection etc. * Hyman & Shashkov 1997

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Operated by Los Alamos National Security, LLC for NNSA MultiMat 2011 - 4 Algorithmic roadmap: There are three principal parts - We skip ahead to Part 3, the finite volume equations Linear construction from cell center to cell surface Riemann-like solution at the node Integration of fluxes Cell CV Nodal CV Nodal CV = Dissipation region Integrals are replaced with sums of fluxes about the cell

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Operated by Los Alamos National Security, LLC for NNSA MultiMat 2011 - 5 The fluxes represent time and spatial averages. Consider volume/continuity equation in which the notation implies the sum of iotas about the zone or cell, and the sum about points is In the finite volume method, the integrals are replaced by sums of fluxes about the perimeter of the cell The basic connectivity structure is called an “iota” Variables are located relative to the iota; e.g., is the cell center velocity relative to iota i is the surface stress tensor for iota i is the outward surface normal Surface “o” used in 2 nd order scheme The data structures generalize to 3D and collapse to 1D - so that the same code is executed in all dimensions

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Operated by Los Alamos National Security, LLC for NNSA MultiMat 2011 - 6 The discretization should obey the so-called “geometrical conservation law” (GCL*) - this is why In the finite volume integrals, we must choose between fluxes defined at the surface and at points Surface fluxes Point fluxes THE WINNER A linear function must satisfy both the Taylor series expansion and the finite volume gradient The GCL is satisfied by evaluating the coordinates and consequently the function at the vertices Surface-centered fluxes do not satisfy this! The GCL is simply a statement that the numerical operators should mimic the analytical expressions The two will be consistent if we require and This is the discrete version of the geometric conservation law! *Trulio & Trigger 1961 Despres 2010

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Operated by Los Alamos National Security, LLC for NNSA MultiMat 2011 - 7 Geometric conservation law (GCL) Angular momentum Rotational equilibrium Second law of thermodynamics Mass Strain Momentum Total energy Mimetic equation summary: The scheme must consider more than the evolution equations Finite volume Curl & divergence identities Ancillary relationships Evolution Conservation Equilibrium Limiting cases Adiabatic compression Symmetry preservation Etc CCH challenge is to determine the point fluxes given the cell values

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Operated by Los Alamos National Security, LLC for NNSA MultiMat 2011 - 8 because Does the discrete operator for the curl of the momentum equation vanish? Yes, because of a corner stress tensor! We need to show that the difference equations satisfy The second-order operators are evaluated on a staggered grid In other words, the internal contributions to the curl integral (dotted) vanish, so that there are no internal sources of circulation The key is that both integrals must see the same stress tensor in the corner This would not be true for surface stresses

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Operated by Los Alamos National Security, LLC for NNSA MultiMat 2011 - 9 The same methodology yields similar results for other relationships for both the cell and the nodal control volumes Cell Nodal CV

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Operated by Los Alamos National Security, LLC for NNSA MultiMat 2011 - 10 To incorporate the Second Law into the discretization, we must first decompose the energy equation The Second Law In a closed system, the kinetic energy must dissipate into the internal, suggesting It is sufficient (but not necessary) that which is the “entropy condition” Dissipation models similar to are invoked to satisfy the entropy condition Alternative variable (not a linearization) “Work” Internal energy Total energy Kinetic energy Momentum equation “Dissipation” Strain equation

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Operated by Los Alamos National Security, LLC for NNSA MultiMat 2011 - 11 Organization of the presentation Mimetic approach Why a mimetic approach Corner vs. surface fluxes Conservation & ancillary equations Curl & divergence expressions Entropy & the energy equation Nodal solvers & the entropy condition Conventional approach A new tensor approach Spurious vorticity Mimetic approach for axisymmetric (rz) geometry The notion of a centroidal control volume Axisymmetric equations Concluding remarks

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Operated by Los Alamos National Security, LLC for NNSA MultiMat 2011 - 12 The entropy condition and the momentum conservation law are solved on a nodal control volume to yield surface stress and velocity Algorithmic roadmap: Part 2, the “Riemann” solution– We will show results from two entropy relations Linear construction from cell center to cell surface Riemann-like solution at the node Integration of fluxes Cell CV Nodal CV = Dissipation region Nodal CV

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Operated by Los Alamos National Security, LLC for NNSA MultiMat 2011 - 13 Substitute the dissipation expression into the flux conservation law and solve the matrix equation for velocity then go back to solve for force We first tried a conventional dissipation expression* that is mathematically sufficient to satisfy the entropy condition It is not necessary to explicitly evaluate the corner stress tensor since only the forces are actually used We could solve for the stress tensor but it would be non-symmetric (4 unknowns) since there are now 4 equations To satisfy rotational equilibrium, the stress tensor must be symmetric. Could this be related to observed chevron modes? * Maire 2007 Carre et al 2009

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Operated by Los Alamos National Security, LLC for NNSA The Noh* rz problem on a polar grid – relative to SGH, CCH shows reduced wall heating & reduced Gibbs oscillations 9x100 L20s experimental 9x100 L20s acoustic Substantial wall heating Gibbs phenomena Significantly reduced wall heating Density should reach 64 Density Distance SGH tensor viscosity SGH standard CCH Dissipation pushes the shock ahead SGH tensor viscosity SGH standard CCH We CCH compare with SGH using: Standard settings – as normally used Nonstandard options – tensor viscosity *Noh 1987 ** Lipnikov Campbell & Shashkov 2001

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Operated by Los Alamos National Security, LLC for NNSA Sedov* rz is a sensitive test of energy conservation and symmetry – CCH result is clearly superior to SGH Colors correspond to pressure CCH 1D spherical SGH does not preserve symmetry Cavity volumes suggest dissipation in tensor viscosity L20s rz quadratic SGH tensor viscosity SGH standard CCH Density Distance Very noisy SGH tensor viscosity SGH standard CCH * Sedov 1959 Note smooth mesh

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Operated by Los Alamos National Security, LLC for NNSA MultiMat 2011 - 16 N12.0 L20e.0 6.808 The dissipation condition worked well for many problems - until we tried the Taylor Anvil* and Howell** problems (with strength) ν = 0.35, and yield stress σ Y = 400 MPa. The material is assumed to harden linearly with a plastic modulus of 100 MPa. The calculations are carried out up to a time of 80μs (at which point nearly all the initial kinetic energy has been dissipated as plastic work). Standard mesh size ∆x = 0.1296mm and ∆y = 0.064mm which results in 200 points along the axial direction and 100 points along the radial direction. Coarse 25x50 mesh Chevron instability arises here with fine meshing *G.I. Taylor 1948 ** Howell & Ball 2002

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Operated by Los Alamos National Security, LLC for NNSA MultiMat 2011 - 17 The conventional dissipation condition produces a force that is in the direction. We believe the system of equations should be closed with a physical model instead of simply a sufficient condition The stress jump at the discontinuity should be proportional to the strain rate This can be expressed in an impedance form For a planar shock & in the principal frame of the strain rate tensor with a basis vector and signal velocity then so the stress jump reduces to Note that this is independent of the grid! Note that in the dissipation expression is a component of the strain rate tensor not a vector A corresponding vector exists and it is only when the force is calculated that the grid becomes involved This force is in the direction not

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Operated by Los Alamos National Security, LLC for NNSA MultiMat 2011 - 18 The stress field is discontinuous, so we must explicitly enforce conservation of momentum Substitute the dissipation expression into the momentum conservation law and solve for velocity directly then go back to solve for the forces The tensor dissipation condition does not require a matrix inversion The 2 forces are constructed from a symmetric viscous stress tensor so that rotational equilibrium is satisfied

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Operated by Los Alamos National Security, LLC for NNSA MultiMat 2011 - 19 N14.0 L20e.0 @ t=150 With the tensor dissipation, the Taylor anvil* ran without chevrons - We also ran a coarsely zoned Howell** problem without chevrons N12 L20e.1 sym gradu crash@ t=30 Coarse zoning with conventional dissipation crashes due to chevron cells t = 30 With tensor dissipation runs to completion t = 150 0 μs 150 μs Elastic-plastic shell coasts inward until it stops 4 cm cylindrical 3 cm spherical Initial velocity field is divergence-free Howell – 2Dxy usual setup *G.I. Taylor 1948 ** Howell & Ball 2002

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Operated by Los Alamos National Security, LLC for NNSA MultiMat 2011 - 20 N14.2 L20e.0 sym Q2=0 N14.2 L20e.1 sym Q2=1 Preliminary testing suggests tensor dissipation does not seem to exhibit hourglass or chevron modes Noh xy box grid Noh xy polar grid Sedov xy box grid Saltzman xy t=0.75 t=0.80 t=0.85 t=0.90

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Operated by Los Alamos National Security, LLC for NNSA MultiMat 2011 - 21 Algorithmic roadmap: Part 1, Reconstruction – We looked at several schemes – this is based on Maire’s work Linear reconstruction from cell center to cell surface Riemann-like solution at the node Integration of fluxes Cell CV Nodal CV Nodal CV = Dissipation region * Maire 2007 The finite volume integrals are conservative, but do not provide a distribution within the cell Conserved quantities can be redistributed linearly through the centroid, without altering the total The gradient is used to extrapolate from the cell center to the surface

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Operated by Los Alamos National Security, LLC for NNSA MultiMat 2011 - 22 N14.0 nc nc L20e.0 full gradu Note: chevron at piston face fixed With SGH & CCH, large scale spurious vorticity seems to be an issue - If we know the answer before hand, we can fix it. Should we? We can obtain excellent answers in SGH & CCH if we know there should be no vorticity The numerical scheme can damp vorticity in various ways: corner pressures 1 tensor viscosity 2 curl-Q 3 Dukowicz & Meltz 4 etc Here, the limited velocity gradient used in the second order extrapolation is simply symmetrized N14.0 nc nc L20e.1 sym gradu Almost as good as pavia!! t=0.75 t=0.90 Basic methodDamped Saltzman Problem 1Browne & Wallick 1971 Burton 1991 Caramana, Shashkkov, Whalen 1998 2Campbell & Shashkov 2001 3Burton 1992 Caramana & Loubere 2005 4Dukowicz & Meltz 1992 The gradient is first limited to identify shock discontinuities but does not specifically address vorticity

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Operated by Los Alamos National Security, LLC for NNSA MultiMat 2011 - 23 Unfortunately, practical problems do have vorticity – a stiffened algorithm can produce a bad answer! THE ULTIMATE GOAL is a single method that produces satisfactory answers to all problems with the same “knob” settings, or better yet, no knobs Basic method Foot @ 6.66 Damped Foot @ 5.65 N14.0 L20e.1 5.649 not acceptable Taylor anvil @ 80 Coarsely zoned 25x50 Contours are effective stress Saltzman Taylor anvil Both appear to have legitimate bending modes Why & how should we inhibit one and not the other? Because the Saltzman result was produced by a planar shock and should be irrotational! The numerical algorithm does not explicitly guarantee that planar shocks are irrotational, so we need a vorticity limiter to enforce the condition

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Operated by Los Alamos National Security, LLC for NNSA MultiMat 2011 - 24 A dynamic vorticity limiter performs nearly as well in both cases as a knobbed switch that is based upon prior knowledge N12.0 L20e.0 N14.2 L20e.3.1N114.2 L20e.1 sym N14.2 L20e.3.1 6.663 N14.2 L20e.1 sym 5.681 N14.2 L20e.0 full 6.843 6.6635.681 Basic method Damped Dynamic limiter Since shocks tend to be 2-3 cells wide, the dynamic limiter is determined by the minimum of the shock limiter including adjacent cells t=0.75 t=0.90

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Operated by Los Alamos National Security, LLC for NNSA MultiMat 2011 - 25 Organization of the presentation Mimetic approach Why a mimetic approach Corner vs. surface fluxes Conservation & ancillary equations Curl & divergence expressions Entropy & the energy equation Nodal solvers & the entropy condition Conventional approach A new tensor approach Spurious vorticity Mimetic approach for axisymmetric (rz) geometry The notion of a centroidal control volume Axisymmetric equations Concluding remarks

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Operated by Los Alamos National Security, LLC for NNSA MultiMat 2011 - 26 It is widely believed the mimetic equations in rz are conservative, but do not preserve 1D symmetry* Conversely, it is also widely believed that the area weighted equations are flawed because they preserve symmetry at the cost of rigorous conservation By considering a second order extension of the mimetic notion, we show that the axisymmetric (rz) equations: are mimetic are conservative do preserve symmetry are appropriate for an infinitesimal region about the centroid are the canonical area weighted ones That is, the canonical area weighted momentum & strain equations ARE the correct ones Symmetry preservation: on an equiangular polar grid, spherical loading should produce spherical results

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Operated by Los Alamos National Security, LLC for NNSA MultiMat 2011 - 27 Second order requires a shift in our thinking from “cell averages” to solutions at an integration point, the centroid In first order schemes, the solution consists of cell averages of conserved quantities: volume/strain momentum total energy and derived quantities: kinetic internal energy stress In second order, conserved quantities are redistributed linearly through the centroid without altering the totals However, the kinetic & internal energies become nonlinear functions and cannot be cell averages So the solution consists of values at the centroid, NOT cell averages Can there be a fundamental principal we are missing? Yes, a “centroidal” control volume First order Energy is flat within the cell Second order We only know the kinetic & internal energy here

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Operated by Los Alamos National Security, LLC for NNSA MultiMat 2011 - 28 In planar geometry, construct a geometrically similar subcell centered at the centroid and scaled by a factor α Geometrically, we have We linearly interpolate coordinates & all fluxes between the surface and the centroid by the same factor, e.g., Flux is conserved by detailed balance between the “doughnut” and the “hole” for all values of α In particular, the resulting equations for quantities at the centroid are identical to those for cell averages, e.g., The point: The extended mimetic finite volume equations at the centroid are the same as those for the cell as a whole (at least in xy geometry) This is important for understanding rz geometry

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Operated by Los Alamos National Security, LLC for NNSA MultiMat 2011 - 29 ref: TN38z temp, TN 38(19) The usual mimetic methodology is conservative in rz, but does not preserve symmetry Consider a section of a thin “orange slice” The conservation relations must now include contributions from the lateral surfaces so that the mimetic evolution equations reduce to The usual mimetic momentum & strain equations do NOT preserve symmetry but the volume & total energy do

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Operated by Los Alamos National Security, LLC for NNSA MultiMat 2011 - 30 However, we can write the mimetic rz equations for the centroid subcell - They do preserve symmetry and are conservative The volume & total energy are the same as those for the entire cell and also preserve symmetry Then the subcell momentum & strain equations are conservative by detailed balance for all α, including the limit After taking the limit α 0 and substituting for mass, the mimetic method yields symmetry preserving, area weighted expressions The point: The canonical area weighted equations* are in fact mimetic conservative preserve symmetry appropriate for an infinitesimal region about the centroid One can show * Wilkins, 1964 See Burton 1994 for a proof of symmetry preservation

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Operated by Los Alamos National Security, LLC for NNSA MultiMat 2011 - 31 Organization of the presentation Mimetic approach Why a mimetic approach Corner vs. surface fluxes Conservation & ancillary equations Curl & divergence expressions Entropy & the energy equation Nodal solvers & the entropy condition Conventional approach A new tensor approach Spurious vorticity Mimetic approach for axisymmetric (rz) geometry The notion of a centroidal control volume Axisymmetric equations Concluding remarks

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Operated by Los Alamos National Security, LLC for NNSA MultiMat 2011 - 32 Summary: Because we are in relatively unexplored territory, we used a mimetic approach to guide the formulation of the difference scheme We reached many of the same conclusions as previous investigators (Despres, Maire, …), but differ on others. All conservation and ancillary relations are satisfied The GCL is satisfied by corner, not surface, fluxes Curl-divergence relations are satisfied Both the mechanical & viscous stress tensors are symmetric & preserve rotational equilibrium The area weighted equations in rz are conservative and preserve symmetry The conventional dissipation expression gave rise to chevron modes A new tensor dissipation model leads to a simpler nodal solution and does not appear to introduce chevron or hourglass modes A physics-based vorticity limiter seems promising for addressing large scale spurious vorticity Historically, the weakest link in CCH has been the nodal motion The community seems to be converging on a solution The mimetic approach provided firm guidelines in formulating CCH for: Material strength Unstructured polytopal grids Multi-dimensional formulation with curvilinear geometry Multi-material cells (not presented) This approach greatly constrained the difference equations and reduced the introduction of inconsistencies

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Operated by Los Alamos National Security, LLC for NNSA MultiMat 2011 - 33 END

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