ae_722_unstructured_meshes.ppt

UNSTRUCTURED MESH GENERATION
Issues in Unstructured Mesh Generation
by
Gopal Shevare
ppts are based on Lecture notes by T J Barth (1994)

• Un-structured Grid
• Unstructured Grids and Application of Graph Theory
• Delaunay Triangulation and Voronoi diagrams
• Methods of Delaunay Triangulations
• Method of Non- Triangulation
CONTENTS

Grid Generation is the gateway for CAE. Grids connects various simulations for
designing a products
Make grid generation easy and user friendly. If possible make it redundant
Grid generation now encompasses each and everything from creating / importing
geometry to export a single file that any solver may need irrespective what simulation
code is simulating
(Boundary Fitted Coordinate) BFC system is not grid generation! Grid generation has
become interdisciplinary area where in CAGD, Computational Geometry, CFD,
Computer Graphics experts contribute to make a package
User may require execution of around 60 to 100 tasks before grid gets ready
Present day simulations use millions of grid points
Grids are a commodity. There is an ISO format for grids
Grid Generation Present Status

Classification of Grids
Cartesian Structured Unstructured
Mono-block
BFC
Multi-Block Triangulation
/Tetrahedral
Quadrilaterals
Hexahedral
Hybrid
Quad tree /
oct -tree
Patched Grids Overlapping Grids BFC+
Triangulation
Triangulation+
Quadrilaterals
Triangulation+
Cartesian
Adaptive Grids more points where solution changes rapidly
Surface grid generation grids on curved surface
Elliptic Grid Generation systems grids produced by solving elliptic PDEs
Orthogonal systems grids lines are orthogonal to each other
Moving grids grid points move with time
Composite overset grid multiple grids overlapping
Cartesian Adaptive Grids grid lines along x, y and z axes
Hybrid grids more than one the of grids coexisting

Transform a given two-
dimensional domain in to
a rectangle
(and three-dimensional
domain in to a box) by a
suitable affine
transformation.
Structured Grid
Mould arbitrary shaped physical domain (x, y, z) in to a rectangular shaped
computational domain (, , ζ) ?
Find three functions f, g and h such that  = f (x, y, z),  = g (x, y, z), and ζ = h(x,y,z).
Lines corresponding to constant values of  = i  = i,  = j  = j, and ζ = k ζ = k
for i = 1 to i max, j = 1 to j max and k = 1, k max in (x, y, z) plane are grid lines in
(x,y, z) plane and intersection of these points gives (a) grid points (i, j) and also (b)
quadrilateral cells.
 = f (x, y) and
 = g (x, y).

Why Triangulation ?
Triangles only
quadrilaterals only
polygons
Mixture of triangles
And quadrilaterals
Theory of triagulations is
highly developed
Higher dimensions pose
no problems
Triangle is the polytope
in 2D
Simple to deal with in 2D
And also on surfaces
Easy to use in numerical
calculations
Why Triangulation & Tetrahedrons

View #2
Fill a given domain with simple shapes such as
triangles, quadrilaterals, etc. so that the given
domain is fully covered.
Emphasis is on cells (triangles in the present case);
there are grid points but no continuous lines
or what can be called as grid lines.
Unstructured Grids
t v1 v2 v3 Remarks
1 2 7 5 Good triangle
2 2 1 5 area < 0
3 2 3 7 overlap
4 5 8 4 degenerate edge
5 2 6 7 duplicate edge?
6 4 9 10 continuum?
1
2
3
4
5,8
6
7
9
10
Structure imposed on grid points

• Graphs and Meshes
• Graph theory directly affects the algorithms in unstructured grids
• However they do not depend on the method of generating triangles
• Planar graph
– graph without self loop or parallel edges
– Most important graph theory result is Euler formula which relates no of vertices, edges
and faces of polyhedron (closed surface)
• N(f) = n(e) – n(v) + 2 Euler formula
• The polyhedron is flattened to 2D space – one face is at infinity
3
1
1
2
2
3
4
4
© Zeus Numerix 25th Jan -2006
Unstructured Meshes and Graphs

• Now there are two types of edges
– Internal – edges : shared by two faces
– Boundary edges : shared by only one face
• Edges for the face at infinity (boundary edges)
(you may consider that face at infinity is absent)
• Edges binding regions, (holes) embedded in the domain (not a part of the domain)
• Euler formula becomes n(f) + n(v) = n(e) + 1 - n(h) Eqn. (1)
• But no of edges and no of faces are related to each other
2 n(e)interior + n(e)boundary
max d(f)
=  i n(f)
i =3
No of pointers from
edges to triangles
No of pointers from
triangles to edges
Eqn. (2)

Subtracting 2 time Eqn (1) from Eqn (3)
n(f) = 2 n(v) - n(e)boundary -2 + 2n(h)
No of faces is approx. 2 times no of vertices
Substituting for n(f) in Eqn. (1)
n(e) = 3 n(v) - n(e)boundary - n(e) - 3 + 3n(h)
No of edges is approx. 3 time no of vertices!
What can happen if
n(f)  2 n(v) - n(e)boundary -2 + 2n(h) or
n(e)  3 n(v) - n(e)boundary - n(e) - 3 + 3n(h)

Basic Definitions
Dirichlet Tessellation
(Also called Voronoi Regions)
The Dirchlet tessellation of a set
of points is the pattern of convex
regions, each being closer to some
Point P in the point set than any
other point in the set
The edges of Voronoi polygons
comprise the diagram.
The idea of Voronoi regions
naturally extends to 3D
point edge

Basic Definitions - Duality
Duality of Graph
Let G be a planer graph of vertices, edges and cells then GDual
the Dual graph of G exhibits the following properties:
If an edge separates two faces
fi and fj in G, then the edge, GDual connects two vertices associated with faces fi and fj
GDual can be formed by (a)Medium segments, (b)Centroid segments, (c)Dirichlet
tessellation
Face in G
vertex in GDual corresponds to
Face in G
Face in GDual corresponds to
vertex in G
Vertex in G
Edge in G
Edge in GDual corresponds to
edge in G

Triangulation
 Definition : The triangulation of a point set is defined as the dual of the Voronoi diagram of
the set.
 In 2D triangulation is formed by connecting if and only if their Voronoi regions a common
border
 If no four or more points are cocircular, then vertices of the Voronoi are circumcentres of
triangles.
triangle
Voronoi diagram

Let use define
N0 = n(v), N1 = n(e), N2 = n(f), N3 = n()
Vertices edges faces volume
Generalised Euler formula is
d
(-1)k N k= 0 (Euler formula in Rd); by definition N-1 = 1
k = -1
For the surface of polyhedron in 3D,
-1 + N0 – N1 + N2 – 1 = 0 or N(f) + N(v) = N(e) +2
For polyhedral volumes
n(f) + n(v) = n(e) + n()
Like 2D case this formula does not take care of boundary effects because it is valid for
four-dimensional polytope

Geometric data : (x, y) coordinates of vertices
Connectivity data : Vertex based v-v, v-e, v-f
Edge based e-v, e-e, e-f
Face based f-v, f-e, f-f
All 9 relations are not stored because
Large RAM is required
Consistency checks must be carried out on relations
Only some relations are stored.
All the remaining relations are derived from the stored relations
The stored connectivity relations are called data structures:
Some well-known data structures: Finite element data structure.
Edge structure.
Out-degree structure.
Quad-edge structure.
Winged-edge data structure.
Data Structures

Data Structures
Storing of Structured grids in unique
Edges in 3D are always formed by (i+1, j, k) and (i, j, k) etc….
Faces are always formed by (i, j, k), (i+1, j, k), (I, j+1, k) and (i+1, j+1, k)…
How to Store Unstructured Meshes?
Collection of ensure does not ensure edges are connected
Collection of faces does not ensure faces form a 3D cell
and that faces not intersecting
.
.
.
.
.

Data Structures
Standard FE data
structure
Ti Vl, Vm, Vn
Sequence of vertices
matters
f-v connectivity
Naturally extend to 3D
Extensively used in
EFM of solids and
fluids
Edge structure
Ei  Fi, Fj
For planar meshes
Edge is list of
pair of vertices
Ideal for FVM
Out-degree structure
Vl  Vm, Vn, Vo
But this is not v-v
connectivity
Max three out degree
required
Quad-edge structure
Ei-1/2  Ei+1/2, Ej, Ek
Edges are stored as a pair
of directed edges
Stores vertex at the origin
Stores previous and next
edge

Graph Operations on Unstructured Grids
Planar Graph with Minimum Out-Degree
Theorem: Every planar graph has a 3 bounded orientation; i.e. each edge of a planar
graph can be assigned a direction such that the maximum number of edges pointing
outward from any vertex is less than three
Proof.
• Find a vertex such that incident boundary edges that connected to at most
maximum two. Other boundary vertices and any number of interior edges.
• Such a vertex is called reducible vertex. It can be proved that at least two vertices
always exit
• The two edges connected to boundary vertices are directed outward
• All the remaining edges are directed inward
• The directed edges are removed from the graph
• The process is repeated on the remaining graph

Planar Graph with Minimum Out-Degree : Graphically explained

Graph Ordering Techniques
Ordering can have a large influence on CPU time and RAM requirements.
Vertices close to each other should be stored close to each other
Most of the grid generation algorithms produce poor ordering
If a Boolean matrix of size [N x N] matrix stores connectivity of vertices such that
(i,j)th entry is 1 if ith and jth vertices are connected, many 1s should be lying close to
diagonal of the matrix
1 1
1 1
1 1 1
1 1 1
1 1
1 1
1
1 1 1
1 1 1
1 1 1
1 1
1 1 1
1 1
1 1 1 1
1 1 1 1
1 1 1 1 1
1 1
1 1 1
1 1
1
Ordered to minimize bandwidth
As obtained form the algorithm

A typical unstructured mesh
Entries of Non-zero Laplacian
Laplacian of reordered mesh
Cuthill-McKee algorithm

Rosen Algorithm
Step I : Determine the band width and choose a pair of vertices (say i and j)
such that i < j
Step II : If increasing i or decreasing j reduces band width
then (increase i or decrease j and goto Step I)
Step III : If increasing i and decreasing j does not change the band width
then (increase I or decrease j and goto Step I)
Comments
Produces good ordering but is very expensive especially for large graphs

Cuthill-McKee Algorithm
Step I : Find the vertex with lowest degree
Step II : Find all the neighbouring vertices incident on root.
Step III : Order the neghbouring vertices by increasing degree
Step IV : Choose the neighbour with lowest degree if not previously not ordered.
Step V : Continue with Step II till all vertices are ordered.
Comments
Very efficient
Recommended in LD decomposition of matrices by reverse renumbering

Graph Partitioning
Graph partitioning is required for load balancing in parallel computing on unstructured grids
Requirements
Each ‘processor’ should get as per its capability
Communication between ‘processors’ should be minimum.
Communication graph
This is graph in which vertices are ‘processors’ and edge indicates that they need to
share some information between.
The colours in the graph show ‘communication cycles’
1 4
2
3
1
3
2
4
Mesh partitioned Communication graph
Communication cycles
1st cycle: 1-4 & 2-3
2nd cycle: 1-2, 3-4
3rd cycle: 1-3

Graph Partitioning
Vizing’s theorem
Any graph of maximum degree  (i.e. number of edges incident on vertex) can be
coloured with using n colours such that  <= n <= +1
Cost of Communication is A + Bm, where A – time required to initiate the message
B – rate of data transfer and m – length of the message
n
Total cost =  (A + Bm)
1
To reduce cost (i) reduce  (reduce number of messages, n), but 1 <= n < = P-1 or
(ii) reduce the length of message
Partitioning for
minimum 
Partitioning for
minimum message
length

Methods of Graph Partitioning –
Coordinate bisection
Step I : Sort the face centroids either horizontally
or vertically
Step II : Separate the triangles such that no of
faces are the same
Step III : Repeat Step II recursively
Comment
Scheme is efficient in partitioning the triangulation.
Partition is sub-optimal as the message length is not
minimum

Methods of Graph Partitioning –
CutHill-McKee Algorithm
Step I : Produce a dual of given mesh
Step II : Reorder the dual graph
Step III : Set the partition at the median or closet to
the median
Comment
Scheme is efficient in partitioning the triangulation.
Reduces number of sub-graphs
Tends to produce sub-optimal as the message length is
not minimum

Methods of Coordinate Bi-section

Methods of CutHill-McKee
Algorithm
the message length is not minimum

Methods of Spectral Partitioning
the message length is not minimum

Properties of Triangulation
1. Uniqueness:
The triangulation is unique,
assuming that no four sites
are circular.
2. The circumcircle criteria
A triangulation for N > 2 site is
if and only if the circumcircle of
every interior triangle is point-
free
Point free
interior
Interior is
not empty
A
A
B
D
C
D
C B
Incircle test for the triangle
Fails if
∠DCB + ∠ DAB ≤π
Incircle test for the triangle
is true if
∠ DCB + ∠ DAB ≥ π
A
D
C B Since interior angles of quadrilaterals sum to 360 degrees,
swapping diagonal can be used to produce triangle

3. Edge circle property
A triangulation of sites is if and only if there exists some circle
passing through the end points of each and every edge which is point
free
4. Edge circle property for constrained triangulation
A triangulation of sites is constrained triangulation if for each
and every edge there exists some circle passing through its end points
containing other site in the triangulation which is visible to the edge
Constrained edge
Point not visible to edge

4. Equiangularity property
triangulation maximises the minimum angle of the
triangulation. (Hence Delauanay triangulation  Maxmin
triangulation.
The property is valid globally and locally. The property can
be used to produce Deluanay triangulation by
swapping of diagonal of convex quadrilateral.
Minimum
angle
Minimum
angle
Non-
triangulation
made
Edge swapping
can not be used in
concave quadrilaterals
Only one edge
is acceptable

5. Minimum Containment property
triangulation minimises the maximum containment circle
over entire triangulation
Containment circle is the smallest triangle enclosing the three
vertices of a triangle
Containment circle is (i) the circumcircle if the triangle is
acute (ii) circle with diameter equal to the largest side if
triangle is obtuse.
Containment circle 
circumcircle for acute triangles
Containment circle  circle with
diameter equal to the longest edge

6. Nearest neighbour Property
In triangulation an edge is obtained by joining the nearest
neighbour
But the nearest neighbour edge is one of the many edges
Nearest neighbour
edge
7. Minimal roughness Property
Suppose a property fi is given at all vertices Vi i = 1 to N
Any triangulation will produce piecewise linear interpolation
surface for this data
triangulation minimises the integral given by
 [(f/x)2 + (f/y)2 ] dx dy for any f
Non-nearest neighbour
edge

• Divide and conquer algorithm
• Incremental insertion algorithm
– Bowyer algorithm
– Watson algorithm
– Green and Sibson algorithm
• Global edge swapping algorithm.
Advancing front algorithm
Graded triangulation algorithm
Generation of Triangulation

Divide and conquer algorithm
Overall algorithm
• Sites are assumed to be pre-specified
• Partition the sites in to Left (L) and right (R)
half planes
• Triangulate each half recursively
• Merge each half recursively
Points to note while merging
• Only L-R or R-L edges are created
• No new L-L or R-R edges do not get generated
as L-triangulation or R-triangulation is
Delaunany. But the merging may lead to
deleting L-L and R-R edges
Merge algorithm
• Connect vertices with minimum y in the left
and right triangulation as the first guess to the
lowest common tangent (LCT) of the L and R
triangulations
contd…..
L triangulation
R
triangulation
First guess for lowest
common tangent (LCT)

Merge algorithm
• Allow a ‘bubble’ passing through LCT (edge
AB) to rise till hits a point (C) from either L or
R triangulation. From the new edge CB or CA
• Use CB as the LCT and use rising bubble to hit
a new point on either L or R triangulation
• Delete the L-L and/or R-R edges to complete
the merge either after every creation of LCT or
after all LTC are obtained.
C
A
B
rising bubble
new edge

Incremental Insertion Algorithms - Common Features
• New points are added to existing triangulation
one by one
• New points are assumed to lie in the existing
triangulation
• All incremental insertion algorithms start with
a phantom triangle large enough to enclose all
the given points
• The edges formed by these points, if given as a
part of input data is initially ignored. These
edges will be built later after all the points are
inserted.
• Every insertion of new point requires locating
(i) a triangle or (ii) circumcircles containing
this point.
• In mesh adaptation the location is known. In
initial triangulation some search is required
Phantom
triangle
Points to be
inserted
Edge ignored

Search is is carried out by using either quad-
tree or split-tree data structures as they give
0(NlogN) complexity
Alternately, ‘walking’ in the direction towards
new vertex using graphs can also be used as
these algorithms are simpler. A worst case in
this gives 0(N2) complexity
Changing the connectivity of vertices to
accommodate criterion. This may require at worst
0(N2) changes to the data structure.
Remove the phantom triangle and the recreate
boundary or edges given as input
Normally the triangles thus produced require some
smoothening
Triangle containing
the new point
Circumcircles containing
the new point
new point
Incremental Insertion Algorithms

Incremental Triangulation - Bowyer’s Algorithm
Assume some triangulation exists; Insert a new point
(site) into Voronoi diagram
Find out all the vertices in the Voronoi diagram
closer to new point than the existing points (sites).
Delete these vertices (shown as v1, v2, v3 and v4)
Delete the edges connecting the vertices if both of
them have been deleted
Produce the new vertices and the edges by
calculating the vertices
Join the sites to form the triangulation by joining the
sites separated by an edge
Existing points
Existing points
new points
Voronoi diagram
v1
v2
v3
v4

Incremental Triangulation - Watson’s Algorithm
into existing triangles
Find the triangles in which the new point lies.
Staring with this triangles perform a tree search to
find all triangles with circumcircle containing the
new point
This produces a polygonal cavity around the new
point.
Connect the new point with vertices of the
polygonal cavity
Swap edges if triangles are not Delaunay

Global edge swapping by Lawson
Choose an interior
edge
Select the
quadrilateral with
the edge as a
diagonal
Swap to satisfy
Delaunay criterion
Continue
swapping edges
till there is no
swapping

Incremental Triangulation - Green and Sibson Algorithm
into existing triangles
Find the triangles in which the new point lies.
Create three additional edges by connecting the new
point to the vertices of this triangle
If the new point is on an edges, delete that and make
four edges.
The newly connected edges are always
But the old edges do not remain Delauany; swap
these edges recursively

Non- Methods of Triangulation
These methods may or may not produce triangulation
Triangulation is NOT essential from the point of view of FVM/FEM
approximations
It is essential that angles are not too close to 0 or 180 degrees are
avoided
Advancing Front Triangulation
Min-Max Triangulation
Greedy Triangulation
Data Dependent Triangulation
Steiner Triangulation
Stretched Triangle Refinement

1. This method is particularly suited for the boundary representation of domains.
2. Boundary is discretised based upon the number of triangles required in the domain.
3. Initial front is a set of line segments defining the boundary completely.
4. The edge to be deleted from the front is chosen based upon some criteria.
5. For the edge to be deleted
a) Select the trial point position.
b) Search of any already existing point within a certain proximity of the trial point. If any
such point exists then it becomes trial point.
c) Determine whether element formed with the trial point crosses any edge. If yes, select a
new trial point and try again. go to (b)
6. Add the new point, edges and triangles to the respective lists.
7. Delete base edge from the front and add new edges to the front.
8. If the front is non-empty, go to 4
Advancing Front Algorithm

• .
Boundary is discretised based upon the number of triangles required
in the domain.
• Initialize front as a set of line segments defining the boundary
completely
• Front is advanced in to the domain producing triangles as it
advances
• Front advances in a variety of ways depending on the angle between
two consecutive edges of the font: three possibilities arise
• Determine whether element formed with the trial point crosses
any edge.
• If yes, select a new trial point and try again.
• Add the new point, edges and triangles to the respective lists.
Delete base edge from the front and add new edges to the
front till front is empty
Non- methods of triangulation
Deleting
vertices
Deleting
edges
Adding
vertices
Adding
edges
Included angle
1 2 Nil 1  < 30
1 2 1 2 30 <  < 120
Nil 1 1 2  < 120
New front
Trial
point



Advancing Front Algorithm

• .
Advantages:
Method can be used for surfaces and volume though a lot of modification
is required
Anisotropic grids can be produced
Disadvantages
Grid quality is not high
Advancing fronts collide to produce poor grids
Front Advancing from
airfoils surface
Front Advancing triangulation
completed
Advancing front Algorithm

MinMax Triangulation
• Uses Lawson’s edge swapping
algorithm for convex quadrilateral
• Minimises maximum angle
• Algorithm is guaranteed to converge
in finite number of operations
• Green Sibson algorithm Maxmin
algorithm can be modified to
execute Minmax algorithm
• Minmax is useful in many CFD
simulations
Basic steps
Select an edge of a convex quadrilateral
Replace it if does not satisfy minmax test.
Use the edges recursively to complete
forward and backward propagation

For some simulations equilateral triangular is not the choice of triangulation
(e.g. viscous flow simulations)
produces equilateral triangulation
Stretched triangulation can be considered as a special case of Steiner triangulation
Steiner point is inserted based stretching property demanded
Algorithm
Choose one dimensional stretching function r = f(distance perpendicular to boundary)
Choose an existing triangulation
Find a point on the boundary (vboundary) closed to each vertex (v) in the triangulation
Store the distance between these two pints as S
Choose a major given by
Maximum dimension of the triangle projected along the perpendicular direction to boundary
divided by r
Stretched Triangulation

Here triangles never undoes what it has done before!
Keep adding edges, but do not check quality of triangulation
Triangulation satisfies Euler equation
Normally used for producing triangle with shortest edge length
Basic steps
Start with initial triangulation with boundary edges listed
Connect all the vertices to all other vertices to form additional edges
Order the edges according to their length
Choose the smallest edge
If it does not interest any edge triangulation add to edges of the triangulation
Keep choosing the next edge form the set of ordered edges till Euler equation is satisfied
Greedy Triangulation

Best triangulation for the given data defined
at vertices is required
Best edge is defined based either
(a) on value of the jump in normal derivative
(JND); the best edge has least jump
or
(b) Included angle between normals (ABN)
If equation for property P with equations
defining these properties as
P1 = a1x + b1y + c1 and
P2 = a2x + b2y + c2 then
JND cost function =
| nx(a1 – a2) + ny (b1 – b2) + nz (c1 – c2) | and
ABN cost function =
(a1a2 + b1b2 + 1)/[(a1
2 + b1
2 +1)(a2
2 + b2
2 + 1)]
Edges are swapped to minimise JND or ABN
summed up over all edges
Data dependent Triangulation

By definition Steiner triangulation is any triangulation that adds additional sites to an
existing triangulation to improve its measure of quality
Every Steiner triangulation requires a strategy to insert a point
Some of the strategies are:
Equilateral triangles – Good for Inviscid flows Smooth area variation
Triangles with minimum ration of circumcircle to incircle diameters – Good for viscous flows
Circumcircle Refinement
Place the new site in a position that guaranties no other site in the triangulation can lie
Closer than the radius of the circumcircle.
This produces sites as far as possible as possible form existing sites
Algorithm
Construct triangulation of boundary points
Compute the quality of each triangle such
(e.g. area / ration of area/(ava. Area of neighbouring triangles, etc.)
If quality of all triangles is acceptable stop triangulation
Otherwise choose the worst triangle and insert a point at its circumcentre and produce new
triangulation
Steiner Triangulation

Motivation:
In physical problem like high speed flows over bodies the mesh is required to be
adapted in particular direction i.e more number of triangles are required around the
boundary.
triangulation, criterion for insertion of points is not addressed.
It is triangulation scheme with a control on where to insert points
Basic Concept
Produce a mesh with only boundary points. This becomes background mesh
Each grid point on the boundary carries an averaged length of the edges incident on
it. This value is called (point distribution function) ‘pdf’
2
pdf = ( lengthi)/2
i=1
Point on
the
boundary
length1
length2
Graded Triangulation
(Automatic triangulation)

• The length of the new trial point from the edge of front is calculated as –.
L = *L1 + ß*L2 + *L3.
where (, ß, ) are called barycentric coordinates.
and  + ß +  = 1
• L1, L2, L3 are mesh densities at nodes of the triangle containing the trial
point.
• Mesh density at any node is defined as the mean of the lengths of the
edges meeting at that point.
• The barycentric coordinates (, ß, ) are the ratios of the area of the
three triangles formed by the trial point and the background triangle to
the area of the background triangle.

Every triangle is considered for inserting a point; its pdf is calculated using
barycentric coordinates of triangle : Li = (A1*L1 + A2*L2 + A3*L3), where (A1/A,
A2/A, A3/A) are called barycentric coordinates such that (A1/A + A2/A + A3/A) = 1
A point is accepted for insertion if the following two conditions are met
(i) pdfpoint > (minimum of pdf of the vertices)
(ii) pdfpoint > (minimum of pdf defined at the neighbouring points)
Use the circumcircle criteria to create triangle
A1
A3
A2

ae_722_unstructured_meshes.ppt

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