Operated by Los Alamos National Security, LLC for NNSA MultiMat 2011 - 1 A cell-centered Lagrangian hydrodynamics method for multi-dimensional unstructured.

Operated by Los Alamos National Security, LLC for NNSAMultiMat 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 DivisionLos 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-11-04995August 27, 2011


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


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


Algorithmic roadmap: There are three principal parts - We skip ahead to Part 3, the finite volume equations

uo

uz

∇zu

Linear construction from cell center to cell surface

uo

u

p

Riemann-like solution at the node

&vz

us

Integration of fluxes

Cell CV Cell CV

Nodal CV

Nodal CV =Dissipation

region

Integrals are replaced with sums of fluxes

about the cell

M z &vz = dnz—∫ ⋅u

→ Ni ⋅usi

i

z

∑


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

Ni

ze→ f

ptrianglei surfaces

uzi

σ si

Ni =Nini

Ni

i

p

∑ =0

Ni =Ninio

z

p

s

iota i

f

Surface “o” used in 2nd order scheme

The data structures generalize to 3D and collapse to 1D - so that the same code is executed in all dimensions

Ni

i

z

∑ =0

M z &vz = dnz—∫ ⋅u

→ Ni ⋅usi

i

z

∑


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

Ni

p

gz

∇zϕ =

1Vz

Niϕ si

i

z

∑

x si ϕ s

i

Surface fluxes

Ni

p

g z

∇zϕ =

1Vz

Niϕ pi

i

z

∑

x

pi ϕ p

i

Point fluxes

THE WINNER

A linear function must satisfy both the Taylor series expansion and the finite volume gradient

∇zϕ =1Vz

Niϕ i

i

z

∑

→1Vz

Ni ϕ z + xi −xz( ) ⋅gz⎡⎣

⎤⎦

i

z

∑

=1Vz

Nixi

i

z

∑⎡

⎣⎢

⎤

⎦⎥⋅gz

ϕ i =ϕ z + xi −xz( ) ⋅gz

The GCL is satisfied by evaluating the coordinates and consequently the function at the vertices

Surface-centered fluxes do not satisfy this!

x i → xpi

ϕ i → ϕ pi

∇x=I

∇⋅x=3

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!

1

Vz

N ix i

i

z

∑ =I

∇zϕ → gz

1

Vz

N i ⋅xi

i

z

∑ =3

* Trulio & Trigger 1961Despres 2010


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

Mz&τ z = Ni ⋅σ p

i ⋅upi

i

z

∑

0 = Ni ⋅σ pi ⋅up

i

i

p

∑

&M z =0

Finite volume Curl &

divergence identities

Ancillary relationships

&Lz= xp

i × Ni ⋅σ pi

( )i

z

∑

0 = xpi × Ni ⋅σ p

i( )

i

p

∑

∇p× Ni upi

i

z

∑ =0

∇p ⋅ Ni ×upi

i

z

∑ =0

∇p× Ni ⋅σ pi

i

z

∑ =0

∇p ⋅ Ni ×σ pi

i

z

∑ =0

etc.

ρ &γ =∇u

ρ&u=∇⋅σ

ρ&j =∇⋅σ ⋅u( )

∇× ∇u( ) =0

∇⋅∇×u( ) =0

∇× ∇⋅σ( ) =0

∇⋅∇×σ( ) =0

Evolution

Conservation

Equilibrium

N i ⋅ϕ z( )

i

z

∑ =0

Limitingcases

∇x=I

∇⋅x=3

1

Vz

N i ⋅xpi

i

z

∑ =3

• Adiabatic compression• Symmetry preservation• Etc

CCH challenge is to determine the point fluxes

given the cell values

d i = ni ⋅δpzi σ( )⋅δpz

i u≥0

σ symmetric


because

Does the discrete operator for the curl of the momentum equation vanish? Yes, because of a corner stress tensor!

∇z ⋅σ p →1Vz

N j ⋅σ pj

j

z

∑

∇p× ∇z ⋅σ( ) →1

Vp

−Ni( )

i

p

∑ × ∇z i( )⋅σ⎡

⎣⎤⎦

= 1Vp

1V

z c( )

−Ni( )× N j ⋅σ p

j

j

z i( )

∑⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

i

p

∑

= 1Vp

1V

z c( )

−Nc( )× Nc ⋅σ p

c( )⎡

⎣⎤⎦

c

p

∑

=0

Nc ×Nc =0

We need to show that the difference equations satisfy

The second-order operators are evaluated on a staggered grid

ρ&u=∇gσ∇× ρ&u( ) =∇×∇gσ =0

∇p×∇z ⋅σ p

∇z ⋅σ p

Nc

σ p

c

In other words, the internal contributions to the curl integral (dotted) vanish, so that there are no internal sources of circulation

Nj

Ni

The key is that both integrals must see the same stress tensor in the corner

This would not be true for surface stresses


The same methodology yields similar results for other relationships for both the cell and the nodal control volumes

∇z × ∇puo( ) =0

∇z × ∇p ⋅σo( ) =0

∇z ⋅∇p ×uo( ) =0

∇z ⋅∇p ×σo( ) =0

∇p× ∇zup( ) =0

∇p× ∇z ⋅σ p( ) =0

∇p ⋅∇z×up( ) =0

∇p ⋅∇z×σ p( ) =0

∇p e ∇z

∇z

u

pσ p

uoσ o

∇p

∇z e ∇p

Cell

Nodal CV


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)σ p

i = σ z +δ pzi σ

u pi = uz +δ pz

i u

d i = ni ⋅δpzi σ( )⋅δpz

i u≥0

&kz =1

Mz

Ni ⋅σ pi

i

z

∑⎡

⎣⎢

⎤

⎦⎥⋅uz

=&uz ⋅uz

&wz =1

Mz

σ z : Ni upi

i

z

∑⎡

⎣⎢

⎤

⎦⎥

=σ z : &γ

&dz =1

Mz

Ni ⋅δpzi σ ⋅δpz

i u( )i

z

∑

“Work”

Internalenergy

Totalenergy

Kineticenergy

Momentum

equation ni ⋅δpz

i σ : μδpzi u

( &τ =0)

&dz ≥0

&ez =&τ z−&kz

&wz + &d

⎧⎨⎪

⎩⎪

“Dissipation”

Mz&τ z = Ni ⋅σ p

i ⋅upi

( )i

z

∑

= Ni ⋅σ pi

i

z

∑⎡

⎣⎢

⎤

⎦⎥⋅uz +σ z : Ni up

i

i

z

∑⎡

⎣⎢

⎤

⎦⎥+ Ni ⋅δpz

i σ ⋅δpzi u( )

i

z

∑⎡

⎣⎢

⎤

⎦⎥

=Mz&kz + &wz + &dz

⎡⎣ ⎤⎦

Strain equation

uz

δpzi u = u p

i − uz

Ni

u pi

i






Concluding remarks


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

uo

uz

∇zu

Linear construction from cell center to cell surface

uo

u

p


&vz

u

p


Cell CV Cell CV


region Nodal CV

d i = ni ⋅δ iσ( )⋅δ iu≥0

N i ⋅σ p

i

i

p

∑ =0


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

0= Nifpi

i

p

∑

=up ⋅ Niμ i nini( )

i

p

∑ + Ni ni ⋅σ oi −μ iiuo

i( )

i

p

∑

= A[ ] up⎡⎣ ⎤⎦− B[ ]

u p⎡⎣ ⎤⎦= A[ ]−1 B[ ]

f

pi =ni ⋅σ o

c +μ ii upi −uo

i( ) ⋅ nini

σ o uo

σ p

u

p

F

p1

F

p1

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?

δf =μ δu⋅n( ) n

fp1 =n1 ⋅σ p

c

fp2 =n2 ⋅σ p

c

σ pc =

σ xx σ xy

σ yx σ yy

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

* Maire 2007Carre et al 2009

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The Noh* rz problem on a polar grid – relative to SGH, CCH shows reduced wall heating & reduced Gibbs oscillations

9x100L20sexperimental

9x100L20s acoustic

Substantial wall heating

Gibbs phenomena

Significantly reduced wall heating

Density should reach 64

Den

sity

Distance

SGH tensor viscosity

SGH standard

CCH

Dissipation pushes the shock

ahead

SGH tensor

viscosity

SGHstandard

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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Sedov* rz is a sensitive test of energy conservation and symmetry –CCH result is clearly superior to SGH

Colors correspond to pressure

CCH 1D sphericalSGH does not

preserve symmetry

Cavity volumes suggest dissipation in tensor viscosity

L20s rz quadratic

SGH tensor viscosity

SGH standard

CCH

Den

sity

Distance

Very noisy

SGH tensor

viscosity

SGHstandard

CCH

* Sedov 1959

Note smooth mesh


N12.0 L20e.06.808

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 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 = 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


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

δf =n⋅δσ =μn n⋅δu( )

n

L =aδt

δL=δuδt

a

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!

a

δσ =μ ⋅δw

a

n

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

δu=δua

δu

δf =n⋅δσ→ μδu n⋅a

δu n

δu


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

0= Fpi

i

p

∑

=up Niμci ni ⋅ac

i

i

p

∑ + Ni

i

p

∑ ni ⋅σ oi −μc

iuoi ni ⋅ac

i⎡⎣

⎤⎦

F

pi =Ni ⋅σ p

i =Ni ⋅σ oi +μc

i up−uoi

( ) Ni ⋅aci

u p =− Ni ni ⋅σ o

i −μciuo

i ni ⋅aci⎡

⎣⎤⎦

i

p

∑

Niμci ni ⋅ac

i

i

p

∑

σ o uo

σ p

c

u

p

The 2 forces are constructed from a symmetric viscous stress tensor

so that rotational equilibrium is satisfied

δf =μδu n⋅a

F

p1

F

p2


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

N12L20e.1 sym graducrash@ 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 – 2Dxyusual setup

* G.I. Taylor 1948** Howell & Ball 2002

δf =μδu n⋅a


N14.2L20e.0 symQ2=0

N14.2L20e.1 symQ2=1

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

Noh xybox grid

Noh xypolar grid

Sedov xybox grid

Saltzman xyt=0.75

t=0.80

t=0.85

t=0.90


Algorithmic roadmap: Part 1, Reconstruction – We looked at several schemes – this is based on Maire’s work

uo

uz

∇zu

Linear reconstruction from cell center to cell surface

uo

u

p


&vz

u

p


Cell CV Cell CV

Nodal CV


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


N14.0 nc ncL20e.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 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 ncL20e.1 sym gradu

Almost as good as pavia!!

t=0.75

t=0.90

Basic method Damped

SaltzmanProblem

1 Browne & Wallick 1971Burton 1991Caramana, Shashkkov, Whalen 1998

2 Campbell & Shashkov 20013 Burton 1992

Caramana & Loubere 20054 Dukowicz & Meltz 1992

∇zu→ ε +ψω

ε =12∇zu+∇z

Tu( )

ω =12∇zu−∇z

Tu( )

ψ =0 ψ =1

The gradient is first limited to identify shock

discontinuities

but does not specifically address vorticity


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 methodFoot @ 6.66

DampedFoot @ 5.65

N14.0 L20e.15.649 not acceptable

Taylor anvil @ 80Coarsely zoned 25x50

Contours are effective stress

SaltzmanTaylor 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


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.1sym

N14.2 L20e.3.16.663

N14.2 L20e.1 sym5.681

N14.2 L20e.0 full6.843

6.843 6.663 5.681

ψ =1 ψ =0

Basic method

Damped Dynamic

limiter

ψ =dynamic

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






Concluding remarks


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


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/strainmomentumtotal energy

and derived quantities:kineticinternal energystress

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

vz,γz

uz

τ z

kz=

uz2

2ez =τ z−kz

σ z =σ z γz,ez( )

Can there be a fundamental principal we are missing?

Yes, a “centroidal” control volume

uz

u

kkz

uz

u

kkz

First orderEnergy is flat within the cell

Second orderWe only know the kinetic & internal energy here


In planar geometry, construct a geometrically similar subcell centered at the centroid and scaled by a factor α

u p

%u p

N %N

uz

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.,

%ρz = ρ z

%Vz = α 3Vz

%M z = α 3M z

%Ni = α 2Ni

%u p

i =uz +α upi −uz( )

%&vz=1%Mz

%Ni ⋅%upi

i

z

∑ →1

α3Mz

α3Ni ⋅upi

i

z

∑=&vz

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

Operated by Los Alamos National Security, LLC for NNSAMultiMat 2011 - 29ref: 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

N+ +N−=− Nc

c

z

∑ =−Azr

N+ ⋅uz+ +N−⋅uz

−=0

N+uz+ +N−uz

−=Az −ruz +ϕϕ uz ⋅r( )⎡⎣ ⎤⎦

N+ ⋅σ z+ +N−⋅σ z

−=−σ zϕϕAzr

N+ ⋅σ z+ ⋅uz

+( ) +N−⋅σ z

−⋅uz−

( ) =0

M z &vz = Ni ⋅upi

i

z

∑

Mz&γz = Niupi

i

z

∑ +Az −ruzc +ϕϕ uz ⋅r( )⎡⎣ ⎤⎦

Mz&uz = Ni ⋅σ pi

i

z

∑ −Azσ zϕϕ r

Mz&τ z = Ni ⋅σ pc ⋅up

c( )

i

z

∑

The usual mimetic momentum & strain equations do NOT preserve symmetry

but the volume & total energy do

Δϕ → 1

Vz = Rz Az

M z = constant

ρ z ≡M z

Vz

A

z

RΔϕ

x

y

r

N

N−

N+


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

M z %&vz = Nc ⋅upc

c

z

∑

Mz%&τ z = Nc ⋅σ pc ⋅up

c( )

c

z

∑

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

ρzAz %&uz = Ai ⋅σ pi

i

z

∑ +1

Rz

r ⋅σ z −σ zϕϕ r( )

ρ zAz %&γ z = Aiu pi

i

z

∑ +1

Rz

ϕϕ uz ⋅ r( )

The point: The canonical area weighted equations* are in fact

• mimetic• conservative• preserve symmetry• appropriate for an infinitesimal

region about the centroid

M z %&uz = Ni ⋅σ pi

i

z

∑ −Azσ zϕϕ r

=Rz A i ⋅σ pi

i

z

∑ +Az r ⋅σ z −σ zϕϕ r( ) +O α( )

Mz%&γz = Niupi

i

z

∑ +Az −ruz +ϕϕ uz ⋅r( )⎡⎣ ⎤⎦

=Rz A iupi

i

z

∑ +Azϕϕ uz ⋅r( ) +O α( )

%Ni ⋅%ψpi

i

z

∑ → α 3 Rz A i ⋅ψpi

i

z

∑ +Azr ⋅ψz +O α( )⎧⎨⎩

⎫⎬⎭

One can show

* Wilkins, 1964

See Burton 1994 for a proof of symmetry preservation






Concluding remarks


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


END

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