A survey on finite volume schemes using triangular meshes
S ANDERSON L. G ONZAGA DE O LIVEIRA
UFLA - Universidade Federal de Lavras
DCC - Departmento de Ciência da Computação
P.O. Box 3037 - Campus Universitário 37200-000 - Lavras (MG)- Brazil
[email protected]
Abstract. This review attempts to place in perspective the variety of simple triangular discretizations
which are available for constructing computational meshes in order to use the Finite Volume Method.
In general, there are two main schemes for simple finite volume discretizations in triangular meshes:
cell-centered and vertex-centered schemes. The two schemes differ in the location of the flux variable in
the control volume with respect to the mesh. This review briefly describes some variations of the grid
construction and associated techniques. Specifically, the Median Dual and variations, Voronoi Diagram
and its dual Delaunay Triangulation, the Green-Gauss integration technique, and the simplified leastsquare technique are briefly introduced.
Keywords: Finite Volume Method, partial differential equations, conservation laws, Median Dual,
Voronoi Diagrams, Delaunay Triangulation.
1 Introduction
Numerical approaches seek appropriate forms to fullfil
the Finite Volume Method requirements. For example,
numerical approaches seek to adequately establish an
orthogonal segment between evaluation points and the
outward normal vector, which is used in the Divergence
Theorem [16].
The choice of the polygons that comprise the discretization is fundamental. A mesh comprised of triangular volumes can be appropriate near physical boundary regions and features of a problem with complex geometries.
A homeomorphism (or topological isomorphism) is
a bicontinuous function (continuous function with a
continuous inverse function) between two topological
spaces. Homeomorphism can be considered the mapping which preserves all the topological properties of a
given space. Two spaces with a homeomorphism between them are the same from a topological point of
view. A closed (including the boundary points) bidimensional ball is a disk, the area bounded by a circle.
A triangle is a subset of the bidimensional Euclidean
space homeomorphic to a closed disk.
Similarly to De Floriani et al. [14], let M be a connected finite set of triangles embedded in the bidimensional Euclidean space. Then, M is a bidimensional
mesh if and only if the interiors of any pair of triangles of M are disjoint, and any triangle of M bounds at
least one triangle of M.
In general, there are two main schemes for simple
finite volume discretizations in triangular meshes: cellcentered and vertex-centered schemes. The two schemes differ in the location of the flux variable in the control volume with respect to the mesh. In cell-centered
schemes, flux quantities are stored in the finite volume
centroid themselves and the mesh has simple geometry.
In vertex-centered schemes, flux variables are stored in
the mesh vertices. Consequently, control volumes are
comprised of sub-finite volumes, i.e. parts of finite volumes, which the vertex belongs [16].
Generally, there are two main triangular vertex-centered schemes: the Median Dual scheme for general triangular meshes and Voronoi Diagrams [52] (because its
dual, the Delaunay Triangulation [10]). Both create a
dual mesh for determining the required quantities.
In relation to duality, Barth [2] consider a planar
graph G of vertices, edges, and faces, i.e. polygons.
A dual graph Gd is any graph with exhibits three properties: each vertex of Gd is associated with a face of G;
each edge of G is associated with an edge of Gd ; if an
edge separates two faces fi and fj of G then the associated dual edge connects two vertices of Gd associated
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A Survey on Finite Volume Schemes using Triangular Meshes 73
with fi and fj .
In Figure 1, edges and faces around the central vertex are formed by dual median segments, centroid segments, and by the Delaunay Triangulation. In addition,
two common techniques for simplified linear reconstruction include a Green-Gauss integration technique and the simplified least-square technique.
Figure 2: Triangulation duals [2]
Figure 1: A control volume of the Median Dual scheme (adapted
from [45])
This survey extends the review presented by Gonzaga [16] and Gonzaga and Kischinhevsky [21]. After
this introduction, section 2 addresses the Median Dual
scheme, section 3 introduces the Voronoi Diagrams and
section 4 describes cell-centered control volume schemes. Subsequently, section 5 treats the Green-Gauss integration technique and section 6 deals with the simplified
least-square technique. Afterwards, Section 7 discusses
the vertex- and cell-centered schemes. Finally, Section
8 draws some considerations.
2 The Median Dual scheme
According to Schneider and Maliska [45], one of the
first formulations using triangular meshes for the generation of control volumes was proposed by Baliga and
Patankar [1]. The discretization by elements based on
the Finite Element Method eases the modeling of the
approximated equations and normalizes the computational implementation, resulting in adequate accuracy
in the solution. In this scheme, the finite-element test
functions were extended to the finite-volume discretization [45]. Figure 1 and 2 depict control volumes according to this scheme.
The Median Dual scheme can be used with general meshes. Barth [3] explains that one can show that
using a specific numerical quadrature, the Finite Element Method with linear elements and the Finite Volume Method on median duals are equivalent. This enables convergence analysis directly from the Finite Element Method.
The geometric center of the triangle P12 in Figure
1 is joined to the mean point of its edges P1 and P2.
Thus, the five triangles, similar to the triangle P12, form
the control volume centered in P. Each polygon contributes with two integration points ip1 and ip2 in the
balance conservation of the control volume centered in
P [45]. Maliska [33] explains that the progress of this
finite-volume scheme was modeste if one compares it
to the usage of the generalized coordinates in order to
discretize a domain.
There are many publications that used the Median Dual scheme, including: Barth [2] presented aspects of unstructured grids and finite volume solvers
for the Euler and Navier-Stokes Equations; Schulz and
Kallinderis [48] applied it in unsteady flow structure
interaction for incompressible flows; Schneider and
Maliska ([44], [46]) proposed a formulation for finite
volume; Kelleners [28] simulated inviscid compressible multi-phase flow with condensation; Koubogiannis
et al. [30] proposed one and two-equation turbulence
models for the prediction of complex cascade flows using unstructured grids; Bastian and Lang [5] applied
the Median Dual scheme in parallel procedures; Hainke
[23] applied it in a convective problem; Morais [37]
verified draining solutions through finite volumes and
described the scheme; Cordazzo et al. [9] represented
reservoirs with geological failures; Estacio [12] simulated the injection molding; Lyra et al. [32] proposed
an adaptive edge-based unstructured finite volume forINFOCOMP - Special Edition, p. 72-81, jul. 2010
A Survey on Finite Volume Schemes using Triangular Meshes 74
mulation for the solution of biphasic flows in porous
media; Sv̈ard et al. [50] proposed stable artificial dissipation operators for finite volume schemes on unstructured grids.
Aiming towards maintaining the praticity in the
mesh generation, Schneider and Raw [47] proposed to
obtain control volumes similarly to Baliga and Patankar
[1]. The advance in the field given by Schneider and
Raw [47] was the coupling introduced in the equations in order to improve the convergence process. Figure 3 illustrates an elementar volume according to this
scheme. The control volumes are obtained by joining the barycenter of the quadrangle P123 to the mean
point of the edges. Quadrangular polygons similar to
the polygon P123 form the control volume centered in
P. Furthermore, a local coordinate system is needed in
order to implement the approximations. Schneider and
Maliska [45] described that it provides more options to
create the mesh. In addition, a fixed number of points
in the coordinate directions, what differs from the generalized coordinates is not needed. Each polygon also
contributes with two integration points ip and ip1 in the
balance conservation of the control volume centered in
P.
Figure 3: a) A control volume from the scheme proposed by Schneider and Raw [47] ([45])
Figure 4: Three neighbor volumes, whose centroids are P, 1 and 2,
whereas ip is the integration point between volumes P and 1 (adapted
from [45])
Schneider and Maliska [45] explain that each polygon Pa1b has exactly one integration point. The position of the integration point in each polygon is fundamental in order to minimize the numerical error introduced by the approximation.
Points a and b in Figure 4 determine the effective
area where flow is changed in polygon Pa1b, and also
is the addition of the vector areas of the segments aip
� from the integration
and ipb, indicated by the vector A
�
point ip. Vector A is not, necessarily, parallel to P 1.
Such area can be applied at an integration point ip that
is not in the intersection of the segments ab and P 1.
According to Schneider and Maliska [45], the effetive area, where flux is changed, does not depend on
the position of the integration point. Consequently, this
value can be 12 Pa1b when using source terms involving a sub-volume quantity, i.e. the computation of subvolumes is not needed. Thus, the integration point is
the mean point of the segment P 1.
Schneider and Maliska [45] showed a formulation
applied to an evolutionary convective-diffusive problem, whose velocity field �u is known and the concentration φ evolves as
ρ
The arbitrary discretization proposed by Schneider
and Maliska [45] consists of obtaining the control volume from a set of diamond polygons limited by two
centers of volumes and one area, where the equations
are integrated. The polygon Pa1b in Figure 4 links the
volumes centered in P and 1 through the segment P 1.
Figure 4 sketches three volumes, whose centroids
are indicated by points P, 1 and 2. The edge between
volumes P and 1 is limited by points a and b. The integration point between those volumes is indicated by
� [16].
point ip, and the vector area is indicated by A
∂φ
+ �uρ∇ · φ = Γφ ∇2 φ + S φ ,
∂t
(1)
where Γφ is the diffusion coefficient, S φ is the source
term, ρ is the density and t is time.
Following the finite-volume basic formulation for
non-uniform meshes, the integration is performed over
the volume named P. Applying the Divergence Theorem, with the source term lineariazed, numerical integration in time and space yield
MPn φnP
−
MPn−1 φn−1
P
+ �t
3
�
βip = 0,
(2)
ip=1
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A Survey on Finite Volume Schemes using Triangular Meshes 75
−→ −
→
� ip φip −Γφ (−
where βip = ρ(�u · A)
∇φP . A )ip −(SP φip +
�VP a1b
SC ) 2 , MP = ρ�VP is the mass contained in the
� is the vector area of each face and
control volume, A
�VP is the area of the polygon. When the quantity of
the parcels of all polygons are added and boundary conditions applied, there is a conservative algebraic equation of volume P , connected to its neighbors.
Applying this scheme to all volumes results in a system of algebraic equations. When it is solved, the values φ in all centroids that comprise the discretization
are obtained.
Schneider and Maliska [45] proposed this scheme
in vertex-centered scheme and suggested the possibility
to extend it to a cell-centered scheme. The reader is referred to Schneider and Maliska [45] for further details.
The authors Gonzaga and Kischinhevsky [17] extended this scheme to a cell-centered approach that determinates the gradients as easy as in a vertex-centered
dual scheme without generating a dual mesh. Gonzaga
and Kischinhevsky ([18], [20]) performed simulations
with this scheme in order to study a space-filling curve.
In addition, Gonzaga and Kischinhevsky [19] applied
it in an adaptive mesh refinement approach. Moreover, Gonzaga [16] showed an approach in order to
solve partial differential equations applying the Finite
Volume Method and adaptive mesh refinement pursuing to maintain an appropriate accuracy with low computational cost. The dot product between the gradient vector of the partial differential equation dependent
variable and the vector area in Eq. 2 was simplified
in those notes. Since point 1 is on the right side of
point P, i.e. x1 > xP if they are at the same horizontal coordinate, vector �c = (x1 − xP , y1 − yP ) is
generated from segment P 1. In addition, since cos α
is given by the inner product between �c = (xc , yc ),
� = (xA , yA ) = (yb − ya , xa − xb ), and the distance of
A
a segment is given by the Euclidian norm, calculations
of square root are eliminated in
3
Voronoi Diagrams
Voronoi Diagram is another important vertex-centered
scheme for its simplicity in the formulation. Voronoi
Diagrams take advantage that edges that comprise a
control volume are orthogonal to the segment between
control volume centroids [16].
Voronoi Diagrams of a set of points S are convex
regions in the plane. Each region is the portion of the
plane closer to one of the points of S than to any other
point of S. Figure 2 shows a Voronoi Diagram.
Voronoi Diagram is also called Voronoi tessellation
or decomposition. A tessellation (or tiling) of the plane
is a collection of figures of the plane that fullfills the
plane (without overlapping or gaps). Voronoi Diagram
is still called as Dirichlet tessellation since Dirichlet
[11] used bidimensional and three-dimensional Voronoi
Diagrams in his study of quadratic forms. Other names
include Voronoi polygons and Thiessen polygons (or
polytopes).
Figure 5 represents another Voronoi volume. It is
obtained from joinning the mean point of each edge to
the adjacent triangle circumcenters. Each segment contributes with one integration point ip in the balance conservation. Point location in Voronoi Diagrams can be
performed in O(log(n)) time with O(n) storage cost for
n regions [2].
Figure 5: A Voronoi volume [45]
−→ �
X1 (ya − yb ) + Y1 (xb − xa )
Φ,
∇φ · A =
2
c xA +yc yA )
2
(X12 + Y12 )( (x(x2 +y
2 )(x2 +y 2 ) )
c
c
A
(3)
A
where X1 = (x1 − xP ), Y1 = (y1 − yP ), Φ =
(Φ1 − ΦP ), and ΦP and Φ1 are values of φ stored in
the respective vertices of the polygon Pa1b.
This approximation depends on the direction when
constructing a function in order to interpolate the
derivatives of φ. Heinemann and Brand [24] explains
that the direction used depends on the flux, that is previously unknown. Schneider and Maliska [45] wrote
� direction of the diamont
an interpolation function in A
polygon Pa1b.
Voronoi Diagrams use an elegant characteristic from the Delaunay Triangulation: Voronoi volumes
have edges orthogonal to the line segment between adjacent Voronoi volume centroids. This feature is crucial
in the Finite Volume Method because the Divergence
Theorem is based on the outward normal vector of the
edge of a control volume [16]. In addition, that intersection point is at the mean point of the edge that connects
two vertices.
Examples of the wide research in Voronoi Diagrams
are: Guibas and Stolfi [22] presented primitives for the
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A Survey on Finite Volume Schemes using Triangular Meshes 76
manipulation of general subdivisions and the computation of Voronoi Diagrams; Barth [2] presented aspects of unstructured grids and finite volume solvers
for the Euler and Navier-Stokes Equations; Shewchuk
[49] presented aspects of the Delaunay mesh generation; Ju [27] presented an algorithm for mesh generation; Iske and Kaser [26] proposed a conservative semilagrangian advection on adaptive unstructured meshes;
Galante [15] applied paralell multigrid methods in the
simulation of problems in computational fluid dynamics.
is point-free. This characterization provides a mechanism for defining a constrained Delaunay Triangulation
where certain edges are prescribed a priori. Consider
an edge with endpoints A and B. The edge is shared by
two triangles, say �ABC and �ABD. This property
states that points C and D are not interior to the circle passing through points A and B (see Figure 6). A
triangulation of points is a constrained Delaunay Triangulation if it preserves this property.
3.1 Delaunay Triangulation
Barth [2] defines the Delaunay Triangulation of a point
set as the dual of the Voronoi Diagrams of the set.
The bidimensional Delaunay Triangulation is formed
by connecting two points if and only if their Voronoi regions have a common border segment. If no more than
three points are cocircular, the vertices of the Voronoi
Diagrams are circumcenters of the Delaunay triangles.
This is because vertices of the Voronoi represent locations that are equidistant to three (or more) sites. If one
ignores boundary, from the definition of duality, each
edge of a Voronoi Diagram corresponds exactly to an
edge of the Delaunay Triangulation.
Because edges of the Voronoi Diagrams are the
loci of points equidistant to two sites, each edge of the
Voronoi Diagrams is perpendicular to the corresponding edge of the Delaunay Triangulation. According to
Barth [2], this duality extends straightforwardly to three
dimensions.
Some important properties of the bidimensional Delaunay Triangulation are described in the following.
The reader is referred to Barth [2] for further details.
1) Uniqueness: The Delaunay Triangulation is unique. This supposes that no more than three points are
cocircular. The uniqueness follows from the uniqueness
of the Voronoi Diagram [4].
2) The circumcircle criteria: A triangulation with
more than two points is Delaunay if and only if the
circumcicle of every interior triangle is point-free. If
this were not true, the Voronoi regions associated with
the dual would not be convex and the Voronoi Diagram would be invalid. One can test if any point
D is interior to the circumcircle formed by other distint points A, B, and C. One can perform this test as
α = ∠ABC + ∠CDA: if α < 180◦ then the test is
false; if α > 180◦ then the test is true; if α = 180◦ then
the points A, B, C, and D are cocircular.
3) Edge circle property: A triangulation of points
is Delaunay if and only if there exist some circle passing through the endpoints of each and every edge which
Figure 6: Constrained Delaunay Triangulation (adapted from [2])
4) Equiangulatity property: Delaunay Triangulation
maximizes the minimum angle of the triangulation. Delaunay Triangulation is called MaxMin Triangulation
for this reason. This is also locally true for all pairs
of adjacent triangles which form a convex quadrilateral.
This is the basis for the local edge swapping succesful
algorithm of Lawson [31].
5) Minimum Containment Circle: Rajan [41] showed that the Delaunay triangulations minimizes the maximun containment circle over the entire triangulation.
The containment circle is the smallest circle enclosing
the three vertices of a triangle. This is the circumcircle for acute triangles and a circle with diameter equal
to the longest side of an obtuse triangle. This property
extends to n dimensions.
6) Nearest neighbor property: An edge that joins a
vertex to its nearest neighbor is an edge of the Delaunay
Triangulation. This property allows Delaunay Triangulation a powerful tool in solving the closest proximity
problem. It is worth pointing out that the nearest neighbor edges do not describe all edges of the Delaunay Triangulation.
7) Minimal roughness: The Delaunay Triangulation
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A Survey on Finite Volume Schemes using Triangular Meshes 77
is a minimal roughness triangulation for arbitrary sets of
scattered data (see Rippa [42]). Given arbitrary data fi
at all mesh points and a triangulation of theses points,
a unique piecewise linear interpolating surface can be
constructed. The Delaunay Triangulation has the property that of all triangulations it minimizes the roughness
of the surface as measured by the Sobolev semi-norm
�
∂f
∂f
( )2 + ( )2 dxdy.
(4)
∂y
T ∂x
This result does not depend on the current form of
the data. This indicates that Delaunay Triangulation approximates well those functions which minimize this
Sobolev norm.
Barth [2] still describes several algorithms for
bidimensional Delaunay Triangulations, including divide and conquer algorithm; incremental insertion algorithms (Bowyer, Watson, Green and Sibson, and randomized); and Lawson’s global edge swapping. The
reader is referred to Barth [2] for details of their description.
4 Cell-centered control volume schemes
In this strategy, the control volumes are the cell themselves. For instance, Figure 7 sketches a triangular
mesh with centroids in the proper finite volumes.
Figure 8 is not necessarily the mean point between the
volumes P and 1.
Figure 8: Example of hybrid mesh with control volumes in the proper
finite volume [45]
Many publications used this strategy. Examples are:
Turner and Ferguson [51] applied a hexagonal mesh
in the numerical simulation of mass and heat transfer in porous media; a generalization of the discretization through convex arbitrary polygons was proposed
by Mathur and Murthy [34] employing a versatile mesh;
Berger et al. [6] described their approach to compute accurate solutions for time dependent fluid flows
in complex geometry; Pascal and Ghidaglia [39] introduced an algorithm for the discretization of second
order elliptic operators in the context of finite volume
schemes on unstructured meshes.
5
Figure 7: Example of mesh with control volumes in the proper finite
volume [4]
Figure 8 sketches the discretization using arbitrary
polygons. The finite volume approximations are performed on each edge of the polygon P. Schneider and
Maliska [45] explained that the integration point ip in
The Green-Gauss integration technique
A common technique for simplified linear reconstruction is the Green-Gauss linear integration technique reconstruction, where gradients are computed in specific
integration points. Moreover, convective and diffusion
terms are evaluated on all control volume edges [16].
A disadvantage of Green-Gauss reconstruction is
one that occurs in many schemes: the angle evaluation between the segment between each control volume centroids and edges. Some correction should be
done and numerical oscillations can occur in turbulent
flows and discontinuities. Besides, special calculations,
through physical and mathematical knowledge of the
studied problem should be performed for second- or
higher-order accuracy [21]. As an example of this strategy, Sachdev et al. [43] implemented a parallel adaptive mesh refinement scheme for turbulent multi-phase
rocket motor core flows.
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A Survey on Finite Volume Schemes using Triangular Meshes 78
6 The simplified least-square technique
Bramkamp et al. [8] explains that the principle of the
least-square reconstruction is to minimize the error in
numerically approximating the integrals in the cell averages of the neighboring cells, which locally support
the higher-order method. Moreover, Mavriplis [35] explains that the least-square technique may include error
term weights, leading to different gradient approximations for non-linear functions.
The least-square technique represents a linear function for vertex and cell-centered discretizations on arbitrary meshes. The reader is referred to Barth and
Ohlberger [4] for further details. As an example of application of the simplified least-square technique, the
authors Kobayashi et al. [29] applied it in a conservative finite volume second-order accurate projection
method on hybrid unstructured grids in steady bidimensional incompressible viscous recirculating flows.
There are publications that proposed hybrid approaches using the Green-Gauss technique and
the least-square method. Examples are: Bramkamp
et al. [13] developed a flow solver employing local adaptation based on multiscale analysis on b-spline
grids; the authors Bramkamp et al. [7] described
h-adaptive multiscale schemes for the compressible
Navier-Stokes Equations with polyhedral discretization
and Data Compression; Northrup [38] implemented a
parallel adaptive mesh refinement scheme for predicting laminar diffusion flames; Bramkamp et al. [8] presented an adaptive multiscale finite volume solver for
unsteady and steady state flow computations; Iaccarino
and Ham [25] presented automatic mesh generation for
large-eddy simulations in complex geometries.
7 A discussion between cell and vertex-centered control volume schemes
Determining the dependent variable of the partial differential equation in the triangular volume centroids has an
advantage similar to that of Median dual. That is, using finite volume centroids for generating control volumes, one can assure that the mesh will not have exterior points in the mesh. In addition, general meshes can
be used [16]. This is different from control volumes created, for instance, from Voronoi Diagrams, whose centroids are circumcenters and hence, the centroids can be
exterior to the triangular volumes (because it requires
the Delaunay Triangulation).
Yousuf [53] explains that vertex-centered schemes
are first-order accurate on non-smooth grids. On Cartesian or on smooth grids, the vertex-centered schemes
are second or higher-order accurate depending on the
flux evaluation scheme. On the other hand, the discretization error of cell-centered schemes depends on
the smoothness of the grid.
Generally, a cell-centered scheme on triangular/tetrahedral grid leads to about two/six times more control volumes than a vertex-centered scheme. Hence,
cell-centered schemes have more degrees (more unknowns) of freedom than the Median Dual scheme. In
addition, control volumes in cell-centered schemes are
usually smaller than those in vertex-centered schemes.
This suggests that cell-centered schemes are more accurate than vertex-centered schemes. However, residue of
cell-centered schemes results from much smaller number of fluxes compared to the Median Dual scheme,
which is a vertex-centered scheme [53].
Boundary condition implementation in vertex-centered schemes requires additional logic in order to assure a consistent solution at boundary points. In the opposite, it is simple in cell-centered schemes [53]. As a
result, there is no clear evidence about the best scheme.
There are publications that used both cell and
vertex-centered strategies. For example, McManus et
al. [36] showed a scalable strategy for the parallelization of multiphysics unstructured mesh-iterative codes
on distributed-memory systems.
8
Conclusions
In the scheme applied by Schneider and Maliska [45],
other directions can be used in order to construct the
derivative interpolating function of the polygon. Specifically, a function aligned to the direction of the segment
P1 (see Figure 4) could be applied. A future work shall
apply a hybrid scheme, where the interpolating function
shall depend on the flux.
The approach showed by Gonzaga [16] shall be extended to Voronoi Diagrams in order to achieve accuracy maintaining low computational cost with adaptive mesh refinament. Specifically, Plaza and collaborators [40] proposed the 7-triangle Delaunay partition
(see Figure 9), and it shall be applied.
Figure 9: 7-triangle Delaunay partition [40]
The author hopes that this review serves to introduce the ideas, principles and schemes that constitute
the state-of-art in this subject. Additionally, the author
hopes that the list of references and descriptions to the
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A Survey on Finite Volume Schemes using Triangular Meshes 79
large body of work on this issue can provide a useful
starting point to one faced with the task of constructing a triangular mesh in order to numerically solve partial differential equations applying the Finite Volume
Method.
A future survey shall cover technical details of the
schemes. In addition, it shall cover second and higherorder schemes.
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