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LA-UR August 27, 2011 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 should be subtitled: Gudelines for cell-centerd hydro acknowledge coauthors sponsor people who have shared thoughts with us

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**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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We are interested in cell-centered hydro (CCH) as a possible alternative/complement to staggered-grid hydro (SGH) 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. 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 since our areas of interest have not been widely explored in a CCH context we used a mimetic approach to guide the formulation the mimetic approach considers not only the usual finite volume equations but also a spectrum of ancillary relationships that are satisfied analytically, but not necesarily * Hyman & Shashkov 1997

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**Linear construction from cell center to cell surface**

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 Integrals are replaced with sums of fluxes about the cell Integration of fluxes Cell CV Cell CV Riemann-like solution at the node we will discussing the algorithm in 3 parts: since the fv equations are in terms of fluxes, we begin there with the objective of learning how the ancillary equations constrain the fluxes the algorithm starts with a cell gradient that is used to construct the fluxes on the surface of the nodal control volume denoted in tan coupling the entropy condition with the conservation laws in this control volume results in a Riemann like solution at the cell surface this is used to drive the final integration of the cell evolution equations we will examine these pieces Nodal CV = Dissipation region Nodal CV

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In the finite volume method, the integrals are replaced by sums of fluxes about the perimeter of the cell 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 Surface “o” used in 2nd order scheme 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 polytopal grids are described by data structures that collapse naturally from 3d to 2d & 1d so that the same coding is executed in all 3 dimensions the basic element is called an “iota” all variables are identified relative to the iota so that … the data structures permit a compact mathematical notation in particular, surface integrals are replaced by sums… The data structures generalize to 3D and collapse to 1D - so that the same code is executed in all dimensions

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**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 The two will be consistent if we require and This is the discrete version of the geometric conservation law! The GCL is simply a statement that the numerical operators should mimic the analytical expressions Surface fluxes 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! Point fluxes There seems to be an evolving consensus that CCH must obey the so-called geometric conservation law (GCL), advocated by Despres, Maire GCL had origins with Trulio & Trigger … This is an explanation of why it is important A key notion in second-order Godunov schemes is that there should be no dissipation if the flow is linear that is: a linear velocity gradient gives rise to a uniform stress rate a linear stress gradient gives rise to a uniform acceleration this means that we must be very careful about the behavior of our gradient operator in the case of a linear function Let phi be some linear function in the second-order scheme we will use a gradient operator to project values to the cell surface [equation] if the flow is linear, the Riemann solution will return the same values these in turn will be used in the SAME operator in the evolution equations this will be true only if which is the GCL In particular, the trace must be [eqn] This is uniquely true only if the coordinates and the function evaluation are at the corners of the cell not at the faces NOTE that this argues against using least squares for one and finite volume for the other THE WINNER * Trulio & Trigger 1961 Despres 2010

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**Mimetic equation summary: The scheme must consider more than the evolution equations**

Finite volume Ancillary relationships Equilibrium Curl & divergence identities Geometric conservation law (GCL) Angular momentum Rotational equilibrium Second law of thermodynamics Mass Strain Momentum Total energy Evolution Conservation to determine the surface fluxes, we must consider more than the evolution equations the principal finite volume equations are on the left notice that the evolution equations are expressed as a sum about a cell while the flux conservation equations are a sum about a node in additions to these are a number of ancillary relations we would like to satisfy curl & divergence identities volumetric compatibility angular momentum second law Limiting cases Adiabatic compression Symmetry preservation Etc CCH challenge is to determine the point fluxes given the cell values

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**This would not be true for surface stresses**

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 because 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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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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**Alternative variable (not a linearization)**

To incorporate the Second Law into the discretization, we must first decompose the energy equation Alternative variable (not a linearization) 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 Total energy Kinetic energy Internal energy * to treat the second law, we need to decompose the total energy equation we do this by expressing the surface values as a cell center value + a difference this results in terms that can be easily identified as kinetic energy mechanical work internal energy and a dissipation term in a closed system the kinetic energy must dissipate into the internal which means the dissipation integral must be positive it is sufficient that the flux be positive the is called the “entropy condition” must be satisfied by a model such as … “Work” “Dissipation” Momentum equation Strain equation

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**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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**Algorithmic roadmap: Part 2, the “Riemann” solution– We will show results from two entropy relations**

The entropy condition and the momentum conservation law are solved on a nodal control volume to yield surface stress and velocity Linear construction from cell center to cell surface Integration of fluxes Cell CV Cell CV Riemann-like solution at the node we will discussing the algorithm in 3 parts: since the fv equations are in terms of fluxes, we begin there with the objective of learning how the ancillary equations constrain the fluxes the algorithm starts with a cell gradient that is used to construct the fluxes on the surface of the nodal control volume denoted in tan coupling the entropy condition with the conservation laws in this control volume results in a Riemann like solution at the cell surface this is used to drive the final integration of the cell evolution equations we will examine these pieces Nodal CV = Dissipation region Nodal CV

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**We first tried a conventional dissipation expression**

We first tried a conventional dissipation expression* that is mathematically sufficient to satisfy the entropy condition Substitute the dissipation expression into the flux conservation law and solve the matrix equation for velocity then go back to solve for force * Maire 2007 Carre et al 2009 we use a variation of the ingenious nodal solver of Pierre-Henri extended to tensor & multimaterial cells (not shown) substitute the entropy conditions into the momentum flux conservation law and solve for the nodal velocity then go back & evaluate the force note that this satisfies the rotational equilibrium requirement 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?

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

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

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**Chevron instability arises here with fine meshing**

The dissipation condition worked well for many problems - until we tried the Taylor Anvil* and Howell** problems (with strength) Coarse 25x50 mesh * G.I. Taylor 1948 ** Howell & Ball 2002 N12.0 L20e.0 6.808 Chevron instability arises here with fine meshing ν = 0.35, and yield stress σY= 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 = mm and ∆y = 0.064mm which results in 200 points along the axial direction and 100 points along the radial direction.

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We believe the system of equations should be closed with a physical model instead of simply a sufficient condition The conventional dissipation condition produces a force that is in the direction . 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 we use a simplified tensor impedance for our dissipation model the impedance tensor should be diagonal in the principal frame of the shock the model is in common use and assumes the tensor is diagonal in the frame of the surface normal this results in this expression relating the stress and velocity jumps in the real world, the shock is unaware of the mesh orientation, and the model needs to be replaced

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**The tensor dissipation condition does not require a matrix inversion**

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 substitute the entropy conditions into the momentum flux conservation law and solve for the nodal velocity then go back & evaluate the force note that this satisfies the rotational equilibrium requirement The 2 forces are constructed from a symmetric viscous stress tensor so that rotational equilibrium is satisfied

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**With tensor dissipation runs to completion**

With the tensor dissipation, the Taylor anvil* ran without chevrons - We also ran a coarsely zoned Howell** problem without chevrons Coarse zoning with conventional dissipation crashes due to chevron cells t = 30 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 N12 L20e.1 sym gradu t=30 With tensor dissipation runs to completion t = 150 N14.0 L20e.0 @ t=150 * G.I. Taylor 1948 ** Howell & Ball 2002

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**Preliminary testing suggests tensor dissipation does not seem to exhibit hourglass or chevron modes**

Saltzman xy t=0.75 N14.2 L20e.0 sym Q2=0 Noh xy polar grid t=0.80 N14.2 L20e.1 sym Q2=1 Noh xy box grid t=0.85 Sedov xy box grid t=0.90

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Algorithmic roadmap: Part 1, Reconstruction – We looked at several schemes – this is based on Maire’s work 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 Linear reconstruction from cell center to cell surface Integration of fluxes Cell CV Cell CV Riemann-like solution at the node we will discussing the algorithm in 3 parts: since the fv equations are in terms of fluxes, we begin there with the objective of learning how the ancillary equations constrain the fluxes the algorithm starts with a cell gradient that is used to construct the fluxes on the surface of the nodal control volume denoted in tan coupling the entropy condition with the conservation laws in this control volume results in a Riemann like solution at the cell surface this is used to drive the final integration of the cell evolution equations we will examine these pieces Nodal CV = Dissipation region Nodal CV * Maire 2007

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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? Basic method Damped 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 pressures1 tensor viscosity2 curl-Q3 Dukowicz & Meltz4 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!! N14.0 nc nc L20e.0 full gradu Note: chevron at piston face fixed t=0.75 t=0.90 Saltzman Problem 19.25 The gradient is first limited to identify shock discontinuities but does not specifically address vorticity Browne & Wallick 1971 Burton 1991 Caramana, Shashkkov, Whalen 1998 Campbell & Shashkov 2001 Burton 1992 Caramana & Loubere 2005 Dukowicz & Meltz 1992

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Unfortunately, practical problems do have vorticity – a stiffened algorithm can produce a bad answer! Taylor anvil Saltzman Taylor 80 Coarsely zoned 25x50 Contours are effective stress Both appear to have legitimate bending modes Why & how should we inhibit one and not the other? Basic method 6.66 Damped 5.65 N14.0 L20e.1 5.649 not acceptable 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 THE ULTIMATE GOAL is a single method that produces satisfactory answers to all problems with the same “knob” settings, or better yet, no knobs

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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.3.1 N114.2 L20e.1 sym t=0.75 t=0.90 Dynamic limiter Basic method Damped N14.2 L20e.0 full 6.843 N14.2 L20e.3.1 6.663 N14.2 L20e.1 sym 5.681 Since shocks tend to be 2-3 cells wide, the dynamic limiter is determined by the minimum of the shock limiter including adjacent cells 6.663 5.681

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**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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**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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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 Second order We only know the kinetic & internal energy here First order Energy is flat within the cell a common interpretation of FV methods is that the conserved quantities represent cell averages. that is, average specific volume, velocity, total energy in a first order scheme in which there are no interior gradients, then kinetic energy is known, then internal energy, finally stress if there are gradients, a different interpretation is necessary if the velocity field is linear, the kinetic energy and therefore internal is not – how can we find a stress? the conserved properties can be redistributed linearly through the centroid without altering the cell averages. this means that the ke, ie, stress that one would calculate in a first order scheme actually represents values at the centroid this clearly dictates how advection must be done. can there be some other way of looking at things? the answer is yes Let us define a centroidal subcell by scaling the geometry and linearly interpolating the fluxes

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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 Must also address curvilinear formulations: axial symmetry in particular the issue here is rotational symmetry on a polar grid, a spherically symmetric initial condition should produce symmetric results if we perform a full 3d surface integral in the limit of vanishing azimuthal angle we find that the volume and total energy equations will preserve symmetry but the momentum equation will not at this point, most people simply use an area-weighting scheme attributed to Wilkins this slide provides a formal justification for this common practice if you create a geometrically similar subcell about the cell centroid, you can show that in the limit the area weighting equations are both conservative and rotationally symmetric and reflect the motion of the centroid 1343

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**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 ref: TN38z temp, TN 38(19)

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

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**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 34

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Summary: Because we are in relatively unexplored territory, we used a mimetic approach to guide the formulation of the difference scheme 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 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 lots of words here, so let me summarize because we were in relatively unexplored territory we felt we needed a mimetic derivation to guide our work we demonstrated some of these results we also felt that the mimetic approach gave us a much better understanding as to what worked and what did not the weakest part seems to be the second-order construction we have yet to find the optimal solution FUTURE PLANS tie up loose ends improve the gradient operator re examine our face based method complete some work on wall heating & Riemann solvers curvature

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