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The Vofire method for multi-component flows onunstructured meshes
Bruno Despres, Emmanuel Labourasse1
Frederic Lagoutiere2
Resume Nous proposons une nouvelle methode numerique pour le calculapproche de solutions d’equations de transport, en dimension quelconque etsur maillages quelconques. Cet algorithme, de Volumes Finis, repose sur unemethode de reconstruction de l’inconnue dans chaque maille, ou l’inconnue re-construite ne varie que dans les directions orthogonales a la vitesse de transport.Ceci permet de generer un schema non diffusif et stable. Utilise pour le transportde fonctions indicatrices, il est ensuite incorpore a un algorithme Lagrangienavec remaillage pour le calcul d’ecoulements multi-materiaux compressibles.
Abstract A new numerical method for the approximation of transport equa-tions in any dimension on general meshes is here described. The algorithm relieson a reconstruction of the unknown in each cell, where the recontructed func-tion varies only in the directions orthogonal to the velocity. This allows toelaborate a non-dissipative and stable scheme for the transport of characteristicfunctions. It is then incorporated in a Lagrangian scheme with remeshing forthe computation of multi-component compressible flows.
1 Introduction
Advection of discontinuous profiles by means of Finite Volume schemes is stillchallenging despite constant works. This issue arises for instance for any Eu-lerian CFD code dealing with multi-phase or multimaterial flows. This is alsothe case when ALE (Arbitrary Lagrangian Eulerian) simulations are performed.First introduced by Hirt et al. [12], these calculations consist in allowing themesh to move with an arbitrary velocity u independent on the flow velocity uf .This method degenerates naturally to the Euler framework when u = 0, and tothe Lagrangian framework when u = uf . The reader is invited to refer to thepaper of Benson [2] and to the book of George et al. [11] for an insight on ALE.
In the case of Eulerian calculations using a Cartesian mesh, several interfacereconstruction or tracking methods have been proposed (a review of the possi-bilities is available in [23]). These methods can be roughly grouped into threefamilies: Front or Interface Tracking (see for instance [28, 21]), Level-Set [24],and Simple Line Interface Calculation (SLIC, see [20]) or Volume Of Fluid (VoF,
1Commissariat a l’Energie Atomique DIF/DSSI 91680 Bruyeres-le-Chatel BP 12, France.Emails: {bruno.despres,emmanuel.labourasse}@cea.fr
2Universite Paris-Diderot and Laboratoire Jacques-Louis Lions, UMR CNRS 7598, 75252Paris Cedex 05 France, e-mail: [email protected]. This author was supported byCEA/DIF.
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see [29, 19]). Dealing with high-speed compressible flows requires the methodto be conservative, robust and able to bear three materials or more.
From now, VoF seems to be the best candidate to calculate this kind of flows(see for instance [3] or [22] for a review on VoF methods). Unfortunately, VoF isquite difficult to extend to 3D unstructured meshes, in particular for more thantwo materials. Some promising attempts have been proposed (see for instance[10, 25, 15]) but very few realistic simulations using these methods have beenpublished.
We propose here a different way to avoid the numerical diffusion of theinterfaces. This method is based on the previous work of Despres and Lagoutiere[7], further extended by one of the authors to non-linear equations in [16, 17]and [4]. See also [30] for the use of this approach in the context of Wenoschemes. In [7] a flux-based anti-dissipative scheme is defined which has therequired properties for high-speed compressible flow calculations, but which is byconstruction restricted to Cartesian meshes. This paper describes an extensionof this scheme to 3D unstructured meshes.
A basic model, convenient for expository purposes, is the transport equationwith velocity u {
∂tc + (u,∇c) = 0, t > 0, x ∈ R2,
c(0, x) = c0(x), x ∈ R2.
(1)
The velocity u = u(x) is a given smooth velocity field and the initial data c0 isbounded. The velocity field can also be a function of time, but, for the sake ofsimplicity, we drop this dependency. The equation is equivalent to
∂tc + div (cu) = cdiv (u) . (2)
The hypothesis that div (u) = 0 will be done for equation (2). This hypothesissimplifies the analysis, but shall be relaxed in a second part for realistic com-pressible flows. An important issue for the numerical treatment of this PDEproblem is the numerical diffusion. This phenomenon, easily understandable indimension one, is more complex in higher dimension. It shall be decomposedin two different types of diffusion. The diffusion of the first type, which shallbe called longitudinal diffusion, is the one that occurs in the direction of thevelocity. It is the diffusion which is present in one-dimensional computations.The second type, transverse diffusion, is typically multidimensional and is dueto the fact that the mesh may be non-aligned with the velocity. This distinctionbetween the two phenomena could appear arbitrary, but is in accordance withbasic numerical tests. For example consider the initial condition of figure 1, thecharacteristic function of a square.
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Figure 1: Initial condition.
This profile is advected with the upwind scheme. The orientation of thevelocity u has a great influence on the result. It is illustrated on figure 2 whichdisplays only longitudinal diffusion and figure 3 which displays both types of dif-fusion, transverse and longitudinal. The transport velocity is u = (1, 0) (alignedwith the mesh) for figure 2 and u = (1, 1) (diagonal, not aligned with the mesh)for figure 3. The boundary conditions on [0, 1]2 are periodic. The results aredisplayed for time t = 1 (after one period). The difference between longitudinaland transverse diffusions appears clearly in [6] which was a previous attempt toelaborate non-dissipative schemes on non-Cartesian grids. In what follows weconstruct new transport algorithms which are based on this distinction betweentransverse diffusion and longitudinal diffusion.
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Figure 2: Upwind scheme. u = (1, 0), aligned with the mesh. Time t = 1.
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Figure 3: Upwind scheme. u = (1, 1), not aligned with the mesh. Time t = 1.Diffusion acts in every direction.
An original feature of the schemes that we propose is the following. Ona Cartesian grid directional splitting methods use the natural definition of thegrid. But on a general grid this is no more possible, mainly because no particulardirection exists in the grid. So in order to reduce the scheme to a series of localone-dimensional problems (that one could solve with an efficient state of the artmethod) one needs to figure out what could be the local main direction, usingsome additional information. Classically one considers some gradient line of thesolution. What we propose to use in this work is simply the velocity direction,because this velocity direction is the support of the diffusion behavior that wewant to suppress or at least to control. The method is based on the constructionof anti-dissipative flux. So it is different from the MUSCL approach for which werefer the reader to the very complete review [1] (see also references therein). Inthe MUSCL approach the reconstructed profiles are linear functions, while in ourmethod the reconstructed profiles are discontinuous in the cell. Since the coreof the method is the idea of reconstruction, we propose to call it Vofire, whichstands for Volume fini avec reconstruction in french. As MUSCL schemes,Vofire satisfies a local maximum principle in the form
min
(cnj , min
k∈N−(j)cnk
)≤ cn+1
j ≤ max
(cnj , max
k∈N−(j)cnk
)
where N−(j) is the set of all cells such that the flow is incoming from cellk ∈ N−(j) into cell j. This inequality is slightly more restrictive than the onestudied in [1] where the minimum (resp. maximum) is taken over all neighboringcells. This inequality seems natural in the context of transport problems.
We also show how to extend this method for multimaterial simulations. Weprove the theoretical properties are essentially preserved in the remap step ofa Lagrange+remap scheme. Indeed, the remap step of such a scheme can berecast in a numerical resolution of (1). In general the equivalent transportvelocity does not satisfy div (u) = 0, but the transport of a concentration c
∂t(ρc) + div (ρcu) = 0,
where ρ is the fluid density that satisfies ∂tρ + div (ρu) = 0, is equivalent to
∂tc + (u,∇c) = 0,
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which will allow to use the same analysis in the remap stage. Furthermore, thistransport or remap step is independent on the Lagrange step (staggered, cell-centered. . . ), thus the present paper does not focus on the Lagrange resolution.
The plan is as follows. In section 2 we give the notations. The principles ofthe geometric reconstruction are developed in section 3. The Vofire Algorithmis constructed in section 4. This is followed by pure transport numerical resultsin section 5. Then we show in section 6 a simple strategy to adapt this methodin the remap phase of a multimaterial algorithm. The results of 3D numericaltests are given in section 7. Some conclusions are sketched in our final section.
2 Notations in 2D
We shall develop the schemes in the context of Finite Volume methods. We beginwith notations (see figure 4). We consider a mesh in dimension two composed
k, l, m ∈ K(j)
k, l ∈ K+(j)
m ∈ K−(j)u
Tk
Tl
Tm
Tj
nj,k
lj,k
lj,m
nj,m
nj,l
lj,l
Figure 4: Mesh and notations (with u constant for simplicity).
of open cells (Tj)j∈Zsuch that Tj ∩ Tk = ∅ for every j, every k 6= j. For all
j, k ∈ Z, one denotes lj,k the one-dimensional Lebesgue measure of Tj ∩ Tk:
lj,k = length(Tj ∩ Tk
),
thus lj,k = lk,j . For each cell Tj (for j ∈ Z), one denotes K(j) the set of indicesof neighboring cells of Tj (that is to say cells having a common edge with Tj)
K(j) ={k ∈ Z \ {j} s.t. length(Tj ∩ Tk) 6= 0
}.
For j ∈ Z and for k ∈ K(j) (Tj and Tk have an edge in common), one denotesnj,k the unit normal vector to the common edge outward from Tj. We thus havenj,k = −nk,j for every j ∈ Z and every k ∈ K(j). Let uj,k be some consistentmean value of the velocity on the edge Tj ∩ Tk. We then denote by N+(j) the
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set of indices of the downwind neighbors of Tj and by N−(j) the set of indicesof the upwind neighbors of Tj:
N+(j) = {k ∈ K(j) s.t. (uj,k,nj,k) > 0} ,
N−(j) = {k ∈ K(j) s.t. (uj,k,nj,k) < 0}
where (·, ·) denotes the Euclidean dot product. Let sj denote the two-dimensionalLebesgue measure of the cell Tj ,
sj = area (Tj) .
The general form of the schemes is obtained by choosing a time step ∆t > 0 andby integrating the transport equation (2) on [n∆t, (n + 1)∆t] × Tj . One gets
sj
cn+1j − cn
j
∆t+
∑
k∈K(j)
lj,k(uj,k,nj,k)cnj,k =
∑
k∈K(j)
lj,k(uj,k,nj,k)
cn
j . (3)
In this formula, the quantities uj,k are approximate values of the given velocityu(t, x) on the edges. For simplicity we shall consider that
uj,k =1
lj,k
∫
Tj∩Tk
(u,nj)dσ (4)
and the cnj,k are approximate values of the solution on the edges between times
n∆t and (n + 1)∆t. This equation rewrites
sj
cn+1j − cn
j
∆t+
∑
k∈K(j)
lj,k(uj,k,nj,k)(cnj,k − cn
j ) = 0. (5)
This formula defines the class of finite volumes schemes. For instance the upwindscheme is obtained taking cn
j,k = cnj for k ∈ N+(j) and cn
j,k = cnk for k ∈ N−(j).
In this case we have the simplification
cn+1j − cn
j +∆t
sj
∑
k∈N−(j)
lj,k(uj,k,nj,k)(cnk − cn
j ) = 0 (6)
where the sum is taken only on the edges which correspond to upwind neigh-bors. Note that this scheme satisfies the maximum principle under the Courant-Friedrichs-Lewy condition (CFL)
νj = −∆t
sj
∑
k∈N−(j)
lj,k(uj,k,nj,k) ∈ [0, 1]. (7)
This upwind scheme is very diffusive, so useless for many computations.
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3 Geometric reconstruction in dimension 2
We propose to use some geometric reconstruction to introduce some anti-dissipativemechanism in the scheme. We consider a triangular mesh in dimension 2, asin figure 5. In order to separate the problems of longitudinal and transversediffusions we propose to perform the reconstruction in 2 steps. For the sake ofsimplicity, we assume in this section that u is constant. So
∑
k∈K(j)
lj,k(u,nj,k) =
∫
∂Tj
(u,n) dσ =
∫
Tj
div(u) dx = 0. (8)
This assumption will be removed in section 6.
3.1 First step: the transverse reconstruction
Assume for simplicity that the mesh is made with triangles in 2D (we shall relaxthis hypothesis in section 4). The transverse reconstruction boils down to “cut”cells in two parts by a segment parallel to the velocity and to modify the valueof the datum in each of these two sub-cells. Each triangle Tj has at least onedownwind neighbor and at most two. If it has only one downwind neighbor,we do not perform the transverse reconstruction (we do not cut the cell). Letus now assume that Tj has two downwind neighbors, Tk and Tl. It has thenone upwind neighbor, Tm. We consider the intersection point of the two edgesrelative to the downwind neighbors and cut Tj along the line passing on thisintersection point and parallel to u. The two sub-cells are denoted Tj,k andTj,l: Tj,k has Tk as (unique) downwind neighbor, and Tj,l has Tl as (unique)downwind neighbor. This cutting is illustrated on figure 5.
Tk
Tl
Tm
Tj,k
Tj,l
u
Tj
Figure 5: Transverse reconstruction.
We denote sj,k and sj,l the surfaces of sub-cells Tj,k and Tj,l. Of courseone has sj,k + sj,l = sj and sj,k > 0 and sj,l > 0. We now have to assign
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to each of the sub-cells a value of the reconstructed solution. The aim is todefine a reconstructed value cR
j,k in Tj,k and a reconstructed value cRj,l in Tj,l
by maximizing∣∣∣cR
j,l − cRj,k
∣∣∣ (to guarantee the anti-dissipativity). (We omit the
superscript n for the reconstructed values, for the clarity of the presentation).In this reconstruction stage, we will impose
sj,kcRj,k + sj,lc
Rj,l = sjc
nj (9)
to guarantee the local conservativity. We impose that the triplet {cnl , cR
j,l, cRj,k}
has the same monotony as the pair {cnl , cn
j } and that the triplet {cRj,l, c
Rj,k, cn
k} hasthe same monotony as the pair {cn
j , cnk}. These constraints imply in particular
that the datum in Tj would not be reconstructed if cnj was a local extremum in
the transverse direction. In the following, for any a, b ∈ R, ⌊a, b⌉ denotes [a, b]if a ≤ b, [b, a] else. The geometric algorithm on triangles is (10-12).
If cnj is a local extremum in the transverse direction , we do not recon-
struct c in the cell Tj, that is
cRj,l = cR
j,k = cnj if cn
j /∈ ⌊cnk , cn
l ⌉. (10)
If cnj is not a local extremum in the transverse direction , we consider
two sub-cases.
• We take{
cRj,l = cn
l ,
cRj,k = (sjc
nj − sj,lc
nl )/sj,k
if (sjcnj − sj,lc
nl )/sj,k ∈ ⌊cn
j , cnk⌉.
(11)
• And we take{
cRj,l = (sjc
nj − sj,kcn
k )/sj,l,
cRj,k = cn
k
if (sjcnj − sj,kcn
k )/sj,l ∈ ⌊cnl , cn
j ⌉.
(12)
Lemma 1 Consider the finite volume scheme (5). The CFL condition (7) onthe time step is the same for the upwind fluxes (6) and for the reconstructedflux defined by (10-12).
Proof Since u is constant one has∑
k∈N+(j)
lj,k(u,nj,k) = −∑
k∈N−(j)
lj,k(u,nj,k).
The standard condition (7) for the upwind scheme for the cell Tj thus takes theform
∆t
sj
∑
k∈N+(j)
lj,k(u,nj,k) ≤ 1,
8
that is∆t
sj(lj,k(u,nj,k) + lj,l(u,nj,l)) ≤ 1. (13)
The CFL condition for the sub-cells Tj,k and Tj,l are
∆t
sj,klj,k(u,nj,k) ≤ 1 and
∆t
sj,l(u,nj,l) ≤ 1. (14)
Let us denote lj = length(Tj,k ∩ Tj,l
)the length of the segment separating Tj,k
and Tj,l. One has sj,k =lj×lj,k(u,nj,k)
2|u| and sj,l =lj×lj,l(u,nj,l)
2|u| and
sj = sj,k + sj,l =lj × (lj,k(u,nj,k) + lj,l(u,nj,l))
2 |u|
The two inequalities of (14) and inequality (13) thus rewrite |u| 2∆tlj
≤ 1. They
are equivalent. It ends the proof.Remark. In case the triangle Tj has only one downwind neighbor, we do
not perform any reconstruction. The first reason is that in a triangle cell havingtwo upwind neighbors, the basic upwind scheme “concentrates” by itself thenumerical information. The second reason is that even if we would reconstructthe solution in this cell, the two numerical values would be then merged in thedownwind neighbor cell, making the reconstruction useless. These two reasonsare illustrated on figure 6.
u
concentration
merging
Figure 6: The case of a cell with only one downwind neighbor.
Anyway, even a case without reconstruction can be considered as a recon-struction with two values that are equal. This will be implicitly understood inthe following in order to treat the general situation.
3.2 Second step
The second part of the algorithm is more classical. Ideally we just transport thereconstructed profile. We detail the idea in the case of figure 5 only, i.e. in thecase of a cell having one upwind neighbor and two downwind neighbors. We do
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not present the other case of figure 6 because the scheme will be reformulated(in an algebraic form) in section 4. Anyway this case could be treated in anequivalent way.
What is remarkable in figure 5 is that one has locally, e.g. in the cut cell Tj
of figure 5, only one-dimensional problems. More precisely, (Tm, Tj,k, Tk) and(Tm, Tj,l, Tl) are one-dimensional triplet in the sense that
• Tj,k has one upwind neighbor: Tm, and one downwind neighbor: Tk,
• Tj,l has one upwind neighbor: Tm, and one downwind neighbor: Tl.
To perform the numerical advection of the (transversally) reconstructed data,one thus can think of any classical transport 3-point scheme.
The numerical transport can be done with a first order transport scheme (e.g.the upwind scheme. Note nevertheless that the resulting scheme would not bethe upwind scheme because of the transverse reconstruction). One can also useany algorithm of ones own choice: in the following, we pitch on a one-order anti-dissipative method, namely the limited downwind scheme (or Ultra-Bee limiter),which can be reinterpreted as a discontinuous reconstruction scheme (as well asthe first step, see [17]). Essentially one can use any linear or non-linear schemein 1D, provided it respects some maximum principle in the situation describedin figure 7.
Let us go back to one-dimensional notations. We replace (cm, cj,k, ck) or(cm, cj,l, cl) by the triplet (ci−1, ci, ci+1), see table 1.
ci−1 ci ci+1
(Tm, Tj,k, Tk) cm cRj,k ck
(Tm, Tj,l, Tl) cm cRj,l cl
Table 1: Transformation of figure 5 into two 1D problems
The one-dimensional velocity is |u|. The one-dimensional length associatedwith sub-cells Tj,k and Tj,l, that is, to the one-dimensional cell number i, is lj/2.This is suggested by the fact that the CFL condition is 2|u|∆t/lj ≤ 1 (see theproof of lemma 1). In order to perform the update of the value cn
j (to compute
cn+1j ), one needs to compute the update of cR
j,k and cRj,l, both of them being
denoted as cni in the following and on figure 7.
10
ci−1
ci
ci+1
i i + 1i − 1
∆xi = lj/2
Figure 7: Reduction to a one-dimensional problem.
It means we have to solve twice the same one-dimensional transport equation∂tc + |u|∂xc = 0. The associated discrete scheme is
cn+1i − cn
i
∆t+ |u|
cni+1/2 − cn
i−1/2
∆xi= 0 with ∆xi =
lj2
.
The fluxes are the cni+1/2. Noting νi = |u|∆t/∆xi, standard fluxes with TVD
limitation are
cni+1/2 = cn
i +1
2(1 − νi)µ
ni+1/2
(cni+1 − cn
i
).
The µni+1/2 ∈ [0, 1] coefficient is the limiter coefficient. In the sequel we focus
on the Ultra-Bee limiter
µni+1/2 = 2 Minmod
(1
νi,
cni − cn
i−1
(cni+1 − cn
i )(1 − νi)
). (15)
The Ultra-Bee limiter (equivalent to a limited downwind scheme, see [7]) isconvenient for computations where one seeks strongly optimal non-linear anti-dissipation.
3.3 Extension on general meshes and for general velocities
The generalization of the previous algorithm for general meshes and velocities ispossible in theory, resting upon geometrical methods for cutting cells in varioussituations. Such algorithms exist in particular in the meshing community.
11
u
u
u
u
Figure 8: Cutting of a pentagon. Left: with constant velocity (unique cut-ting direction). Right: the cutting may be pathological for some non-constantvelocity.
u
u
Figure 9: Left: cutting of a tetrahedron. Right: cutting the hexaedron in sub-cells is not immediate (even the computation of the intersection point betweenthe line and a non-coplanar face requires specific methods).
Nevertheless, in order to avoid some difficulties illustrated in figures 8 and9, we do not pursue in this direction, preferring to develop the algebraic formu-lation in the next section.
4 The Vofire scheme
The algebraic extension of the previous algorithm intends to be useful in anydimension and for any mesh. In this method, we shall forget about the geom-etry of the problem. However we state as a design principle that the algebraicalgorithm shall be equal to the previous geometrical transverse reconstruction-longitudinal limitation method on a triangle mesh and with a constant velocity(figure 5). We shall first define the reconstructed values cR
j,k and then the final
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fluxes cnj,k. We assume in this section that u is divergence free. So
∑
k∈K(j)
lj,k(u,nj,k) =
∫
∂Tj
(u,n) dσ =
∫
Tj
div(u) dx = 0. (16)
4.1 First step
The starting point is the general scheme (5) with the simplification (16) andreplacing the fluxes cn
j,k with the reconstructed values cRj,k (to be defined)
sj
cn+1j − cn
j
∆t+
∑
k∈K(j)
lj,k(uj,k,nj,k)cRj,k = 0. (17)
The final flux = cnj,k will be defined as the reconstructed cR
j,k flux plus a correc-tion. This scheme is obtained with the standard upwind “limitation” after thereconstruction in the triangle case above (taking µn
i+1/2 = 0 instead of (15)).
Remark 1 In the following we construct the reconstructed values cRj,k as well
as the fluxes cnj,k for every j ∈ Z and every k ∈ N+(j). For k ∈ N−(j), we
define cRj,k = cR
k,j and cnj,k = cn
k,j : there one has j ∈ N+(k) and the values can bedefined following the construction below. Note that imposing cn
j,k = cnk,j ensures
the conservativity of the scheme when div(u) = 0.
The analysis of the transverse reconstruction of section 3.1 indicates that
∑
k∈K+(j)
lj,k(uj,k,nj,k)(cRj,k − cn
j ) = 0, ∀j (18)
is satisfied in the case described in figure 5: this is a consequence of the conser-vativity equation (9). This constraint is defined on outgoing boundaries sincethe sum is taken over k ∈ N+(j). We state it as a necessary constraint for thetransverse reconstruction in the general case. It is convenient to define
pj,k =lj,k(uj,k,nj,k)∑
i∈N+(j) lj,i(uj,i,nj,i)∈ [0, 1], k ∈ N+(j)
and
pj,k =lj,k(uj,k,nj,k)∑
i∈N−(j) lj,i(uj,i,nj,i)∈ [0, 1], k ∈ N−(j).
One has ∑
k∈N−(j)
pj,k =∑
k∈N+(j)
pj,k = 1.
With these notations, the constraint (18) rewrites
∑
k∈N+(j)
pj,kcRj,k = cn
j . (19)
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Then we add a second ingredient, which consists in taking the reconstructedvalue cR
j,k as close as possible to the downwind value cnk . There exist many ways
to do that. We propose to minimize the function J
J =∑
k∈N+(j)
pj,k|cRj,k − cn
k | (20)
with respect to the reconstructed value cRj,k. Define for convenience λj,k ∈ R
such that cRj,k = cn
j +λj,k
(cnk − cn
j
). Then J =
∑k∈N+(j) pj,k|cn
k − cnj | |1 − λj,k|.
It seems natural for consistency to add another constraint
λj,k ∈ [0, 1] (21)
so that cRj,k ∈ ⌊cn
j , cnk⌉ and
J =∑
k∈N+(j)
pj,k|cnk − cn
j | (1 − λj,k) . (22)
Constraint (19) rewrites
∑
k∈N+(j)
pj,k
(cnk − cn
j
)λj,k = 0. (23)
Definition 1 The algebraic reconstruction that we consider is defined by theminimization of J given by (22). The minimization is taken over all λj,k’ssatisfying the constraints (21) and (23).
Let us quote some properties of this method.• The minimization problem has at least one solution. In practice, we use
the following algorithm for the computation of a solution.We compute
Aj =∑
k∈N+(j), ck−cj>0
pj,k(ck − cj), Bj =∑
k∈N+(j), cj−ck>0
pj,k(cj − ck).
There are two sets of indices k: those contributing to Aj and those to Bj . Thesolution we construct uses this distinction. There are two cases.First case: Aj < Bj . We first set
λj,k = 1 for k ∈ N+(j) s.t. ck − cj > 0.
We then take the other λj,k (for k ∈ N+(j) s.t. cj − ck > 0) as large as pos-sible such that the equality Aj =
∑k∈N+(j), cj−ck>0 pj,kλj,k(cj − ck) holds (to
ensure (23)). For the sake of simplicity, we decide to retain the formula
λj,k = λj =Aj
Bjfor k ∈ N+(j) s.t. cj − ck > 0.
14
Other choices are possible.The second case, Bj ≥ Aj , is symmetric and leads to
λj,k = 1 for k ∈ N+(j) s.t. cj − ck > 0
and
λj,k = λj =Bj
Ajfor k ∈ N+(j) s.t. ck − cj > 0.
• In the case the geometry is the one of figure 5 (triangle mesh, constantvelocity), the solution is unique and is the one described in section 3.1. Theverification is left to the reader.
• The scheme (17) preserves the maximum principle under CFL conditionfor free divergence velocities, i.e such that
∑k∈K(j) lj,k (uj,k,nj,k) = 0; Indeed
the constraint (23) is equivalent to (18). Therefore the scheme (17) rewrites
sj
cn+1j − cn
j
∆t−
∑
k∈N−(j)
lj,k(uj,k,nj,k)(cnj − cR
j,k) = 0. (24)
Since, by construction, cRj,k ∈ ⌊cn
j , cnk⌉, the maximum principle is a consequence
of the CFL condition ∆t∑
k∈N−(j) lj,k |(uj,k,nj,k)| ≤ sj.
• The scheme (17) is conservative∑
j cn+1j =
∑j cn
j (see remark 1).
4.2 Second step
The final scheme is also based on (5) with the simplification (16)
sj
cn+1j − cn
j
∆t+
∑
k∈K(j)
lj,k(uj,k,nj,k)cnj,k = 0, (25)
but with a different flux. The flux cnj,k for k ∈ N+(j) shall be defined as
a modification of the reconstructed value cRj,k by the way of some coefficients
µj,k,r ∈ [0, 1], r ∈ N−(j),
cnj,k = cR
j,k+∑
r∈N−(j)
(µj,k,rpj,r(c
nk − cR
j,k))
= cRj,k+
∑
r∈N−(j)
µj,k,rpj,r
(cn
k−cRj,k).
(26)This formula needs some comments and justifications. First, taking µj,k,r = 0∀j, k ∈ K(j), one recovers cn
j,k = cRj,k. The scheme is then equal to (17) and
inherits all its properties. Second, we know that this particular choice is notenough to achieve anti-dissipativity, because the definition of cR
j,k correspondsto the first step of the geometric algorithm, the second step is still missing (indimension 1, the equivalent scheme would be the upwind scheme). The numer-ical result of section 5.3 indicates the second step is necessary. The proposedformula tends to mimic the second step of the geometric algorithm.
15
In the following we propose a constructive way to achieve stability and anti-dissipativity. It is quite clear that taking large values of µj,k,r ∈ [0, 1] should leadto an anti-dissipative scheme. By stability, we mean that the scheme satisfiesthe maximum principle
min
(cnj , min
r∈N−(j)(cn
r )
)= mn
j ≤ cn+1j ≤ Mn
j = max
(cnj , max
r∈N−(j)(cn
r )
)
under a Courant-Friedrichs-Lewy condition. The satisfaction of these inequali-ties is the design principle for the definition of the µj,k,r.
On the one hand, one has
∑
k∈N+(j)
pj,kcnj,k =
∑
k∈N+(j)
pj,k
cR
j,k +∑
r∈N−(j)
µj,k,rpj,r(cnk − cR
j,k)
= cnj +
∑
k∈N+(j)
pj,k
∑
r∈N−(j)
µj,k,rpj,r(cnk − cR
j,k)
since∑
k∈N+(j) pj,kcRj,k = cn
j . On the other hand, the scheme rewrites aftersome algebra left to the reader
cn+1j =
∑
k∈N+(j)
pj,k
(1 − νj)c
nj + νj
∑
r∈N−(j)
pj,rcnj,r − νj
∑
r∈N−(j)
µj,k,rpj,r(cnk − cR
j,k)
where νj is the local Courant number given by (7). We have used the divergence-free assumption,
∑k∈N+(j) lj,k(uj,k,nj,k) = −
∑k∈N−(j) lj,k(uj,k,nj,k), and the
property∑
k∈N−(j) pj,k =∑
k∈N+(j) pj,k = 1. Since∑
k∈N+(j) pj,k = 1, it issufficient to satisfy
mnj ≤ (1−νj)c
nj +νj
∑
r∈N−(j)
pj,rcnj,r−νj
∑
r∈N−(j)
µj,k,rpj,r(cnk −cR
j,k) ≤ Mnj (27)
for all k ∈ N+(j) to get the maximum principle. This rewrites
mnj ≤
∑
r∈N−(j)
pj,r
((1 − νj)c
nj + νjc
nj,r − νjµj,k,r(c
nk − cR
j,k))≤ Mn
j , k ∈ N+(j).
Again, it is sufficient to satisfy
mnj ≤ (1 − νj)c
nj + νjc
nj,r − νjµj,k,r(c
nk − cR
j,k) ≤ Mnj ∀k ∈ N+(j), r ∈ N−(j).
(28)Let us introduce
mj,r = min(cnj , cR
j,r) ≥ mnj and Mj,r = max(cn
j , cRj,r) ≤ Mn
j . (29)
We will define the µj,k,r in such a manner that they satisfy the sufficient condi-tion
mj,r ≤ (1 − νj)cnj + νjc
nj,r − νjµj,k,r(c
nk − cR
j,k) ≤ Mj,r. (30)
16
As mj,r ≤ cnj,r ≤ Mj,r, the situation is quasi one-dimensional in the sense
that standard one-dimensional limiters do satisfy these last inequalities. Let usprovide further details. The preceding inequalities are satisfied as soon as
mj,r ≤ (1 − νj)cnj + νjmj,r − νjpj,rµj,k,r(c
nk − cR
j,k)
and(1 − νj)c
nj + νjMj,r − νjpj,rµj,k,r(c
nk − cR
j,k) ≤ Mj,r
since by construction mj,r ≤ cnj,r ≤ Mj,r. Then, either cn
k − cRj,k > 0, then the
two inequalities are equivalent to
µj,k,r ≤1 − νj
νj×
cnj − mj,r
cnk − cR
j,k
,
or, on the contrary, cnk − cR
j,k < 0, and we obtain
µj,k,r ≤1 − νj
νj×
cnj − Mj,r
cnk − cR
j,k
.
The following inequality gathers these two cases
µj,k,r ≤1 − νj
νjmax
(cnj − mj,r
cnk − cR
j,k
,cnj − Mj,r
cnk − cR
j,k
). (31)
Now the value µj,k,r is to be chosen. A priori, maximizing the anti-dissipativityof the resulting scheme is achieved by choosing the largest µj,k,r ∈ [0, 1] thatsatisfies inequality (31). This leads to
µj,k,r = min
(1 − νj
νjmax
(cnj − mj,r
cnk − cR
j,k
,cnj − Mj,r
cnk − cR
j,k
), 1
). (32)
Note the similarity with the classical one-dimensional limiter formalism [18, 27,7]. This choice is used for the following test-cases.
4.3 Interpretation of (30)
It is possible to rewrite inequality (30) in a more convenient and direct way.Indeed let us define dj,k,r, for k ∈ N+(j) and r ∈ N−(j), as the solution of
sj
dj,k,r − cnj
∆t+
∑
q∈N+(j)
lj,q(uj,q,nj,q)
d∗j,k,r+
∑
q∈N−(j)
lj,q(uj,q,nj,q)
cn
j,r = 0
(33)where the unknown flux is d∗j,k,r = cn
j + µj,k,r(cnk − cR
j,k). The quantity dj,k,r
is simply the solution of the scheme with the assumption that all incomingquantities are equal to the same value cn
j,r and that all outgoing quantities are
17
equal to the same value d∗j,k,r. What is remarkable is that after simplifications
left to the reader one recovers dj,k,r = (1 − νj)cnj + νjc
nj,r − νjµj,k,r(c
nk − cR
j,k).So (30) is equivalent to mj,r ≤ dj,k,r ≤ Mj,r. The number of such simplifiedproblems (33) is card(N+(j)) × card(N−(j)).
Once the maximal value of µj,k,r which satisfies the local maximal principlemj,r ≤ dj,k,r ≤ Mj,r has been computed, one simply combines these fluxesµj,k,r(c
nk − cR
j,k) with the pj,r coefficients. It gives (26). This method is veryconvenient for implementation considerations. It will become evident in section6.
5 Some numerical results for pure transport
We first present the result of the Zalesak test-case [31]. Then we give the re-sult of a simple convergence test. After that we compare the results of theupwind method, the Vofire method, the Vofire method without transverse re-construction but with longitudinal limiter and the Vofire method with transversereconstruction but without the upwind longitudinal scheme.
5.1 Zalesak test-case
A solid body velocity field [31] is used to rotate a complicated initial profile.The result computed on a Cartesian mesh (128× 128 cells) is given in figure 10in the left part.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
’rho.fin’ 0.8 0.6 0.4 0.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
’rho.fin’ 0.8 0.6 0.4 0.2
Figure 10: Zalesak solution computed on a Cartesian mesh (left), on a triangularmesh (left).
The iso-lines show that the general structure is preserved. The boundary ofthe characteristic function is captured on very few cells. The result computedon a triangular mesh (8648 cells) is given in figure 10 in the right part. Thesolution is plotted after projection on a Cartesian mesh. The general quality ofthe result is very similar to the left part computed on a Cartesian mesh, withperhaps little more smearing.
18
5.2 Convergence test
We consider the translation of a the characteristic function of a square witha velocity u = (1, 1). At time t = 0.4 we inverse the velocity, so that thefinal analytical solution at t = 0.8 is equal to the initial datum. The mesh isCartesian. We plot in table 2 the relative error in the L1 norm
E =
∑j |u
nj − u0
j |∑j |u
0j |
for an increasing number of cells. It shows the convergence of Vofire with respectto the size of the cells.
Cartesian mesh 60 × 60 100 × 100 150 × 150relative error 0.11 0.078 0.06
Table 2: Relative error in the L1 norm
5.3 Comparisons with the upwind scheme
We consider a solid body rotation problem, u(x, y) = (−2πy, 2πx). The initialdata is the characteristic function of a disk of radius 0.2 and center (0.5, 0.7).
We plot in figure 11 the results computed with the full upwind scheme, theVofire scheme, and also the Vofire scheme with λ ≡ 0 (but µ 6= 0 a priori) andthe Vofire scheme with µ ≡ 0 (but λ 6= 0 a priori). The best result is the onecomputed with the Vofire algorithm.
19
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
’rho.fin’ 0.8 0.6 0.4 0.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
’rho.fin’ 0.8 0.6 0.4 0.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
’rho.fin’ 0.8 0.6 0.4 0.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
’rho.fin’ 0.8 0.6 0.4 0.2
Figure 11: The Vofire method (top right) is able to capture the disk afterone turn. Top left: full upwind. Other methods incorporate some degree ofupwinding. Bottom left: λ ≡ 0, that is, no transverse reconstruction, but limiterin the longitudinal direction. Bottom right: µ ≡ 0, that is, upwind after thetransverse reconstruction. Computation done on a triangular unstructured mesh(5996 cells), the Courant number is 0.1. The iso-lines are for c = 0.2, 0.4, 0.6and c = 0.8.
We also study the function
t 7→ F (t) =
∫c(t, x)(1 − c(t, x))dx.
For the analytical solution F ≡ 0. Since the maximum principle is preservedfor the numerical solution, then F (t) ≥ 0 in the general case. Therefore this
20
function F serves as a measure of the spreading of the numerical solution. Wecompare four curves in figure 12. One is computed with the upwind scheme andshows the important diffusion of this method, the three others are computedwith the Vofire method, with the Vofire method in which the transverse recon-struction is set to zero (Mod2), and with the VOFIRE method in which thelongitudinal scheme (after the transverse reconstruction) is upwind (Mod1).
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
UPWINDVOFIRE
MOD1MOD2
Figure 12: The function t 7→ F (t) for different schemes. The best curve is theone given by Vofire. Computation done on a triangle unstructured mesh (5996cells), the Courant number is 0.1. Mod1 (resp. Mod2) corresponds bottom left(resp. bottom right) of figure 11.
6 Extension to multicomponent Euler equations
The goal is to adapt the previous algorithm to the construction of anti-dissipativefluxes for multimaterial compressible flows. We focus only on the remappingstage which follows the Lagrange stage in a Lagrange+remap algorithm, as in[2, 8]. The Lagrange step of the code presents specific difficulties and techniquesthat we do not discuss. To present the main ideas it is enough to focus on theconcentration equations of a two-fluid problem,
∂t(ρc1) + div(ρc1u) = 0 and ∂t(ρc2) + div(ρc2u) = 0. (34)
Here c1 and c2 are mass fractions and verify c1 + c2 = 1. In a Lagrange schemethe mass of each fluid is constant during the time step, therefore the massfraction of each material is also constant. However a pure Lagrange code isnot always possible in 2D and 3D, due to tangling of the mesh. So one usuallyperforms a projection on a new mesh which is close to the previous one, exceptfor the pathological cells. Our major concern is about the adaptation of the
21
method presented in the previous section to the design of an anti-dissipativemultimaterial algorithm in the remap step.
We consider that the remapping is driven by some ALE method. So theconnectivity of the mesh is preserved but the location of nodes is moving inorder to enhance the global quality of the mesh. An example is given in figure13 where a quad is remapped onto a new quad. The shadow zone gives the areathat is lost by the cell across the edge. An evaluation of the area of this shadowzone is ∆t× l× (u,n) where l is the length of the edge, n is the outward normaland u is the local edge velocity. The sign of the dot product (u,n) is importantto determine if the swept region is positive or negative (or null). A priori theedge velocity u is the half sum of the vertices velocities.
l
n u
Figure 13: The sweep method and determination of the shadow zone. Area ofthe shadow zone is ∆t × l × (u,n). The edge velocity u is the half sum of thevertices velocities.
In our simulations it is important to control the numerical diffusion of thespecies. This is why we have chosen to focus on the adaptation of the previousanti-dissipative algorithm for the mass fractions c1 and c2 and not for the totaldensity ρ. But in principle a possible evolution of the method could be toincorporate some anti-dissipation mechanisms also for the total density ρ.
The velocity field is a general one and has no reason to be divergence free.It means that div(u) 6= 0 and
∑
k∈K(j)
lj,k (uj,k,nj,k) =∑
k∈N+(j)
lj,k (uj,k,nj,k) +∑
k∈N−(j)
lj,k (uj,k,nj,k) 6= 0.
6.1 Vofire for multimaterial computations
Let us present a very simple way to incorporate the Vofire algorithm in theremapping stage based on the sweep method. An important idea is to decouplethe projection of the total density, which is done in our case with a standard
22
second order method, from the projection of the concentrations which is donewith the Vofire method. The goal is to get the simplest as possible method,unchanging the existing hydro-code.
For a two-material problem, the algorithm may be written as
1) First we use a standard second order method for the projection of the totaldensity. That is
• We consider that the displacement of the vertices of the mesh is givenby an exterior algorithm. So one computes the new area of the cellsj
n+1. A priorisj
n+1 6= snj .
Then for all interfaces, one can compute the edge velocity with thesweep algorithm. It gives uj,k for all j, k.
• For all cells, determine the incoming and outgoing faces in functionof the sign of (uj,k,nj,k). A face incoming into one cell is necessarilyoutgoing on the other side.
• Compute some second order fluxes ρ2nd
j,k for the total density with astandard second order scheme.
At the end of the first part of the method, one knows the total density ineach cell:
sjn+1ρj
n+1 − snj ρn
j + ∆t∑
k
lj,k (uj,k,nj,k) ρ2nd
j,k = 0. (35)
2) Then construct the concentration fluxes with Vofire. Operations are in order
• First step For all cells, perform a transverse reconstruction of theconcentration c = c1 using (22) with the constraints (21) and (23).The coefficients pj,k are replaced with the pj,k
pj,k =lj,kρ2nd
j,k (uj,k,nj,k)∑
i∈N+(j) lj,iρ2nd
j,i (uj,i,nj,i)∈ [0, 1], k ∈ N+(j).
The weights ρ2nd
j,k are natural so that the functional is a measure ofthe mass through the interfaces. This functional, to be minimized, is
J(λj,k) =∑
k∈N+(j)
pj,k|cnk − cn
j | (1 − λj,k) (36)
with the constraint 0 ≤ λj,k ≤ 1 for all k and the linear constraint∑
k∈N+(j)
pj,k
(cnk − cn
j
)λj,k = 0. (37)
This is solved for all j with the method presented in section 4. Atthe end of this stage one knows the reconstructed concentration
cRj,k = cn
j + λj,k(cnk − cn
j ) ∀k ∈ N+(j).
23
• Second step For all cells perform a second limitation of the con-centration c = c1 using formulation (33). That is we consider for allk ∈ N+(j) and all r ∈ N−(j) a prediction dj,k,r
sn+1j ρn+1
j dj,k,r − snj ρn
j cnj + ∆t
∑
q∈N+(j)
lj,q(uj,q,nj,q)ρ2nd
j,q
d∗j,k,r
+∆t
∑
q∈N−(j)
lj,q(uj,q,nj,q)ρ2nd
j,q
cn
j,r = 0.
This prediction dj,k,r corresponds to a simplified problem. The out-going flux is
d∗j,k,r = cnj + µj,k,r
(cnk − cR
j,k
).
We compute the coefficient µj,k,r ∈ [0, 1] such that the predictionsatisfies dj,k,r ∈ [mj,r, Mj,r] where mj,r = min
(cnj , cn
r
)and Mj,r =
max(cnj , cn
r
). We shall see there is always at least the trivial solution
µj,k,r = 0.
After that we compute the real flux
cnj,k = cR
j,k +
∑
r∈N−(j)
µj,k,rpj,r
(cn
k − cRj,k),
• Perform the two previous steps for c = c2.
3) Once this has been done, remap the concentrations c = c1 and c = c2 with
sjn+1ρj
n+1cn+1j − sn
j ρcnj + ∆t
∑
k
lj,k (uj,k,nj,k) ρ2nd
j,k cnj,k = 0. (38)
The equations for the definition of the Vofire flux are symmetric for thetransformation c 7→ 1 − c. Therefore for a two-material problem one hasby construction
(c1)nj,k + (c2)
nj,k = (c1)
nj + (c2)
nj = 1.
So equations (35)-(38) are compatible since
(c1)n+1j + (c2)
n+1j = 1.
6.2 Stability analysis for 2 materials
Despite the apparent complexity of this method, its stability analysis is easy toconduct following the method used in section 4. In practice it is sufficient toprove that the partial masses are non-negative. Since the total mass is definedto be the sum of the partial densities, the concentration shall be in between 0and 1 provided all partial masses are non-negative. For simplicity we first detailthe analysis for the two-material case.
24
6.2.1 Only transverse reconstruction
In order prove the stability, we begin with the case where the flux is cRj,k, (c = c1
or c2) which means that the longitudinal flux (after reconstruction) is upwind:µj,k,r ≡ 0. Equations (35)-(38) imply
cn+1j =
sn
j ρnj
sjn+1ρj
n+1−
∆t
sjn+1ρj
n+1
∑
k∈N+(j)
lj,k (uj,k,nj,k) ρ2nd
j,k
cn
j (39)
+∑
k∈N−(j)
(∆t
sjn+1ρj
n+1lj,k (−uj,k,nj,k) ρ2nd
j,k
)cRj,k.
Assuming the CFL condition
snj ρn
j − ∆t∑
k∈N+(j)
lj,k (uj,k,nj,k) ρ2nd
j,k ≥ 0,
all coefficients are non-negative. The non-negativity of the mass fraction cn+1j
for the transverse reconstruction follows. More precisely one has the maximumprinciple
min(cnj , min
r∈N−(j)(cn
r )) = mnj ≤ cj
n+1 ≤ Mnj = max(cn
j , maxr∈N−(j)
(cnr )). (40)
The right and left bounds involve only incoming boundaries.
6.2.2 Complete algorithm
Now we add the second step of the method which means the µj,k,r are notnecessarily set to zero. By construction each prediction dj,k,r is such that
mj,r ≤ dj,k,r ≤ Mj,r.
Define dj,k =∑
r∈N−(j) pj,rdj,k,r such that
sn+1j ρn+1
j dj,k − snj ρn
j cnj + ∆t
∑
q∈N+(j)
lj,q(uj,q,nj,q)ρ2nd
j,q
d∗j,k,r
+∆t
∑
q∈N−(j)
lj,q(uj,q,nj,q)ρ2nd
j,q cnj,r
= 0.
Then it is easy to check that cn+1j =
∑k∈N+(j) pj,kdj,k since
sn+1j ρn+1
j cn+1j − sn
j ρnj cn
j + ∆t
∑
q∈N+(j)
lj,q(uj,q,nj,q)ρ2nd
j,q d∗j,k,r
25
+∆t
∑
q∈N−(j)
lj,q(uj,q,nj,q)ρ2nd
j,q cnj,r
= 0
and ∑
q∈N+(j)
lj,q(uj,q,nj,q)ρ2nd
j,q d∗j,k,r =∑
q∈N+(j)
lj,q(uj,q,nj,q)ρ2nd
j,q cj,k
due to∑
k∈N+(j) pj,k(cRj,k − cn
j ) = 0 which is one of the constraints of the trans-verse reconstruction. Therefore the maximum principle holds.
6.3 Three materials and more
For three materials and more the analysis is a little more tricky. Assume forsimplicity that there are three materials in the simulation. Then one can have
(c1)nj,k + (c2)
nj,k + (c3)
nj,k 6= 1
for the Vofire fluxes. It has two consequences. First in equation (39), the sum ofthe coefficients in parenthesis may be different from 1. Second the equation (35)is not the sum of the three equations (38) for c = c1, c = c2 and c = c3. Thereis an incompatibility, which means that we have to reconsider the algorithm.
Fortunately the total density computed in (35) is not really important in thedefinition of the new concentrations (38). It is more the product ρn+1
j cn+1j which
has a physical meaning: it is the new partial density in the cell. So equation(35) is now just a prediction of the total density in the cell. Let us analyze theconsequences.
Now cn+1j is just a prediction of the new concentration since ρn+1
j is also aprediction of the new total density in the cell. But what is physically importantis the product ρn+1
j cn+1j . In case of three materials one gets at the end of
this procedure three products in the cell, which are ρn+1j (c1)
n+1j , ρn+1
j (c2)n+1j
and ρn+1j (c3)
n+1j . Then one defines the partial masses (ρ1)
n+1j = ρn+1
j (c1)n+1j ,
(ρ2)n+1j = ρn+1
j (c2)n+1j and (ρ3)
n+1j = ρn+1
j (c3)n+1j . The new total density is
of course the sum of the partial densities(ρn+1
j
)true
= (ρ1)n+1j + (ρ2)
n+1j +
(ρ3)n+1j . The new concentrations are
((c1)
n+1j
)true
=(ρ1)
n+1
j
(ρn+1
j )true
. . . Therefore
one recovers a global maximum principle 0 ≤((c1)
n+1j
)true
≤ 1. . . Which is
enough in practice. The local maximum principle (40) could be ensured byenforcing (c1)
nj,k + (c2)
nj,k + (c3)
nj,k = 1 with specific methods such as in [13].
7 Numerical results in 3D
We have incorporated the multimaterial Vofire method in the 3D ALE multi-physics Arcane architecture. The following simulations intend to assess the goodaccuracy of the algorithm.
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7.1 Test-case 1
In the first test-case we transport a square profile in a 3D mesh made withnon-structured tetrahedra.
Figure 14: The tetrahedral mesh where the initial condition c = 0.
The mesh is represented in figure 14 and figure 15. The velocity is u =(1, 1, 0)
Figure 15: The tetrahedral mesh where the initial condition c = 1.
A planar cut of the solution at mid z and a cut in the diagonal direction ofthe solution are provided in figure 16 and 17, for both the initial condition andthe solution at t = 0.4. The overall quality of the displacement is quite good.
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Figure 16: Initial condition t = 0. Planar cut at mid z on the right, and diagonalcut of the planar cut on the left.
Figure 17: Solution at t = 0.4. Planar cut at mid z, and diagonal cut of theplanar cut. The trace of the mesh around the box of figure 15 is visible on theplanar cut
7.2 Test-case 2
Our second test case is a two-material shock tube problem. The initial data arethose of the 1D Sod shock test-case, but the 3D mesh of figure 18 is distortedfollowing the analytical law
δx(x, y, z) = 0.7 sin(πx)(1 − 2y)(0.5 − x)(1 − z).
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where the dimensions are [0, 1] × [−0.1, 0.1] × [−0.1, 0.1]. The points are firstequidistributed, then we move them along the x-axis by a distance equal toδx(x, y, z). The mesh is skewed so that the problem is non-trivial from a nu-merical point of view.
Figure 18: The mesh is made of skewed hexaedra.
The left and right initial chambers are declared as different materials. Ateach time step we perform a Lagrange step, and then a remap step. Since theremap is done on the initial mesh, then the final computation is done in anEulerian mode, that is the mesh is fixed. For this problem the concentrationis computed with three different algorithms, which are the upwind (donor cell)method, a second order advection a la Van Leer scheme and the Vofire method.This is represented in figure 19. A zoom is given in figure 20. The numericalsolution computed with the Vofire method presents all features of the resultcomputed with an interface tracking algorithm. In any case the diffusion ismore important with the donor cell algorithm and also with a standard secondorder MUSCL algorithm for the prediction of the flux of the mass fraction.The anti-dissipative feature of Vofire is visible on the contact discontinuity. Weobserve that Vofire is optimal in the sense that the number of mixed cells (thatis 2) is equal to the number of cell layers. We also plot the pressure in figure 21to check that Vofire has no effect on the global thermodynamic quantities. Theoscillations are related to the skewness of the 3D mesh.
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Figure 19: Concentrations computed with the upwind scheme, a second orderscheme with limitations and the Vofire method.
Figure 20: Zoom on the concentrations. The upwind scheme gives the smearedcurve, the second order scheme is better, Vofire is the steepest curve, nearlyoptimal.
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Figure 21: The pressure.
7.3 Test-case 3
We now consider a more challenging problem, which is representative of thedifficulties encountered by other interface reconstruction algorithms: the initialgeometry contains a T junction where the three different materials are in con-tact; the initial density and pressure gradients generate a 3D vortex; it makesimpossible the computation of such a flow with a pure Lagrangian scheme andrequires specific ALE techniques.
The dimension of the problem represented in figure 22 in 3D are [0, 1] ×[0., 0.25]× [0., 0.25]. In the left part (x ≤ 0.3), we take ρ = 1, p = 1 and c1 = 1.In the right interior part (x ≥ 0.3, y ≤ 0.125 and z ≤ 0.125), we take ρ = 1,p = 0.1 and c2 = 1. In the complementary part the initial data are ρ = 0.125,p = 0.1and c3 = 1. The initial velocity is u = (0, 0, 0) everywhere. The EOSis a perfect gas law with γ = 1.4 (the same for the three parts), so a referencesolution of this problem can easily be computed with a standard Eulerian code.For these initial data, the solution at the final time t = 0.4 contains a distortedT junction inside the vortex region.
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Figure 22: Initial mesh for test case 3.
A shock is created at the vertical interface, oriented to the right. The resultshave been computed with ALE techniques, that is the mesh moves. We usethe Lagrange scheme described in [9], but the results are quite similar withanother Lagrange scheme. A vortex is created. Therefore the mesh tanglesunless specific remeshing algorithms are used. In our case we used a weightedTipton-Jun method [14, 26] to regularize the mesh, the weights are calculatedin order to adapt the mesh around the pressure and concentration gradients.
Figure 23: Concentration of the internal block.
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Figure 24: The mesh after displacement.
The concentration of the interior zone is represented in figure 23. The meshat time t = 0.4 and y = 0.25 corresponding to the superior plane is repre-sented on figure 24, together with a map of the concentrations. One clearlysees that the initial T-junction is inside the vortex, which makes this computa-tion challenging. For this we use exactly the algorithm described in section 6.One observes that the concentration profile is steep (even if it is not perfect),especially near the vortex region, thanks to the Vofire algorithm. A similarcomputation but without any reconstruction or anti-dissipative method leadsto an excessive amount of mixing. We have also validated the results by meansof a comparison with a reference 3D Eulerian code with a fine mesh [5].
8 Final comments
We have done various numerical experiments with the Vofire algorithm. Theresults have revealed good for all test problems. Other properties are in order.1) The method is conservative by construction.2) The method is robust in 1D, 2D and 3D. The CFL condition is standard andequal to the CFL condition of the donor cell method (i.e. the upwind scheme).3) The method is reasonably cheap and easy to implement even in parallel. De-spite the apparent complexity of the formulas, it is very simple to incorporateVofire in an existing Finite Volume-based CFD code on 2D and 3D unstructuredmeshes.4) We observed that the algorithm presented in this work for general unstruc-tured meshes gives the same quality of results than many other anti-dissipativemethods which are nowadays mastered on Cartesian meshes.
The price to pay is the geometry of the interface which is approximated insome way, as in any anti-dissipative method. Few mixed cells remain, togetherwith some very low diffusion. It makes the accuracy acceptable but not perfect.Nevertheless the interface is correct to O(hs), h being the local mesh size. Thefactor s > 0 may be computed experimentally. From the results of table 2, thetheoretical guess is s ≈ 0.5. From our experience, this theoretical bound is theworst case.
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