mesh generation techniqure of structured gridpdf

mesh generation method

1. Lecture 3.2
Methods for Structured Mesh
Generation
1

2. •There are several methods to develop the structured meshes:
Algebraic methods, Interpolation methods, and methods based
on solving partial differential equations.

•In algebraic methods, the transformation is done analytically
when the boundaries are rather simple and regular.

• For irregular geometries, the coordinates of the mesh (interior)
nodes are obtained by numerical interpolation between the
prescribed boundary data. One such method is transfinite
interpolation.

•PDE methods are classified as elliptic, parabolic or hyperbolic,
depending on the characteristics of the grid generation equations,
which are generally transformed onto a rectangular domain and
solved along with the governing equations of the problem.
2

3. Algebraic Mapping

•Consider the simply connected domain shown in Fig. 3.2.1
whose sides AB, BC, CD and DA are given by the equations f
1(x,
y) = 0, f
2 (x, y) = 0, f
3 (x, y) = 0 and f
4 (x, y) = 0, respectively.

•Without loss of generality, one can map the curves AB and CD
onto the lines = 0 and = 1.0, using a transformation of the
form
η = f
1 (x, y)/{f
1 (x, y) – f
3 (x, y)}

•Similarly it can be assumed that

ξ = f
4 (x, y)/{f
4 (x, y) – f
2 (x, y)}
3

4. A
B
=1
A‘
B‘
=0
C
=1
C‘
D‘
=1
D
=0
=0
=0
=1
Algebraic Mapping


Fig. 3.2.1 Algebraic mapping (a) Physical domain in x-y system with
regular boundaries (b) Rectangular body in the (ξ-η) transformed plane.
(a) (b)
4

5. D
B
C A
=0
=1
=1
=0
•In order to apply the algebraic mapping technique for a domain
between the suction surface AB and pressure surface CD of a
turbomachine blade cascade (Fig. 3.2.2), one may first
approximate the lines AB and CD in terms of analytical
functions: f
1 and f
3 and follow the other steps as suggested
earlier.
Fig. 3.2.2 Algebraic mapping of a simple blade cascade 5

6. Transfinite Interpolation

•Apply unidirectional interpolation in – direction (or –
direction) between the boundary grid data given on the curves
= 0 and = 1 (or = 0 and = 1) and obtain the coordinates
x’
p y’
p for every interior and bondary point.

•Calculate the mismatch between the interpolated and the actual
coordinates on the = 0 and = 1 (or = 0 and = 1)
boundaries.

•Linearly interpolate the difference in the boundary point
coordinates in (or direction and find the correction to be
applied to the coordinates of every interior point.
6

7. •For applying the transfinite interpolation to a cascade geometry,
consider the four – sided geometry (ABCD) (Fig. 3.2.3).

•It is desired to generate ξ – constant and η – constant lines within
ABCD which upon transformation would become equi–spaced
orthogonal grid lines inside a rectangular domain of size 1 x 1.

•The first task to be completed is the placement of grid points on
the boundary (Fig. 3.2.3). Here, in order to get a rectangular
grid, the number of points on opposite sides should be equal.

•Also, if some idea is available regarding the nature of gradients
in the problem, the boundary points can be located so as to
resolve the high gradient regions, i.e. the regions where the inter
row spacing domain merges with the blade surfaces.
7

8. Fig. 3.2.3 Transfinite interpolation for an aerofoil cascade identification of
boundaries
8

9. Fig. 3.2.4 Transfinite interpolation: Identification of boundary nodes
9

10. •Now, let us apply linear interpolation in the ξ – direction
between two grid points which lie on the sane η = constant
line. Since the total range of ξ variation is from 0 top 1, the
linear interpolation formulae for the coordinates of an
interior point P are written as

x’
p = (1 – ξ) x
E + ξx
F; y’
p = (1 – ξ) y
E + ξy
F

where E and F are the two boundary grid points.

•Carrying out the same operation for all the η – constant lines
including the boundaries (η = 0 and η = 1), the interpolated
points (marked by x) will appear as in Figure 3.2.4.

•It may be noted that on the boundaries AB and CD, the grid
points obtained by unidirectional interpolation between the
coordinated of the corner nodes (A, B) or (C, D), do not
coincide with actual grid points shown as dots (.).
10

11. Fig. 3.2.5 Transfinite interpolation: Calculation of interior nodes
11

12. •In order to remove this anomaly, the difference between the
actual points and the interpolated points should be subtracted on
the η = 0 and η = 1 boundary curves.

•Moreover, some corrections need to be applied to the
coordinates of point P which have been obtained by
unidirectional interpolation in the ξ – direction, along an η =
constant line.

•Introducing such corrections brings in the influence of the
boundary grid point data in the η – direction also, for
determining the coordinates of point P. considering two grid
points G and H corresponding to the ξ = constant line on which
P lies, the corrections for the coordinates of point P are

∆x
P = (1 – η) ∆x
G + η∆x
H; ∆y
P = (1 – η) ∆y
G + η∆y
H (3.2.1)
12

13. •Here ∆x
G, ∆x
H, ∆y
G, ∆y
H, are the corrections for the boundary
points given by
∆x
G = x’
G – x
G; ∆y
G = y’
G – y
G;
∆x
H = x’
H – x
H; ∆y
H = y’
H – y
H; (3.2.2)

•In Eq. 3.2.2, the coordinates indicated with prime are those
obtained by unidirectional interpolation in ξ – direction and
those without prime are the actual boundary grid point data. The
final values of the coordinates of P (after interpolation in both ξ
and η directions)are obtained as

x
p = x’
p – ∆xp y
p = y’
p – ∆y
p

•Performing the above sequence of operations for every interior
point gives the mesh in the physical domain as shown in Fig.
3.2.6. The corresponding transformed mesh is as usual a
rectangular grid.
13

14. Fig. 3.2.6 Transfinite interpolation: Final mesh.
14

15. •It is important to note that unidirectional interpolation can be
done first in the η direction along each ξ = constant curve. In that
case, corrections will have to be applied for matching actual grid
points and the interpolated points along the boundaries AD (ξ =
0) and BC (ξ = 1). The final grid obtained by both the above
approaches will be exactly the same.
15

16. Domain Vertex Method
•Domain vertex method is also an interpolation method,
preferably used for generating multi-block structured grids. They
make use of tensor products of unidirectional interpolation
functions for two or three dimensions.

•Relation between physical (x, y) and transformed (ξ, η)
coordinates, in two dimensions, is given by:




•Here, suffix i indicates the physical coordinate directions. N and
M represent node numbers in the direction of the coordinates.

• are the unidirectional functions and
denotes tensor product. , 1,2, , 1,2
, , 1,2, 1,2,3,4
NM
i iNM
i N iN
x x i N M
x x i N NMand ,
N
16

17. •The following functions are known as Blending functions:





•An example of this transformation from the physical to
transformed domain in two dimensions is shown in Fig. 3.2.7. 1 1 1
2 2 1
3 2 2
4 1 2
ˆˆ11
ˆˆ
1
,
ˆˆ
ˆˆ
1
N
17
Fig. 3.2.7 Two dimensional domain

18. 1
2
3
4
5
6
7
8
111
11
1
11
,,
11
1
1
N 18
•Similarly in three dimensions, relation between physical and
transformed coordinates is given by




•Blending functions: ˆ ˆ ˆ , 1,2,3 , , 1,2
or
, , , 1,2,3 1,....,8
i N M P iNMP
i N iN
x x i N M P
x x i N

19. 19
Fig. 3.2.8. Transformation of Three Dimensional Domain using Domain
Vertex Method
•An example of this transformation from the physical to
transformed domain in three dimensions is shown in Fig. 3.2.8.

20. Exercise Problems

•Using transfinite interpolation method generate grids for

The region bounded by





•Using transfinite interpolation method generate grid for any
triangular region.
20 22
22
2 2 2 2
33
1, 1
4 9 4 9
33
1, 1
9 4 9 4
xx yy
x y x y

21. Summary of Lecture 3.2

Methods for algebraic mesh generation and with transfinite
interpolation are illustrated for structured grids. A domain vertex
method popular for finite element methods is also presented.
21
END OF LECTURE 3.2

22. Module 3
Mesh Generation
1

23. Lecture 3.1
Introduction
2

24. Mesh Generation Strategy
•Mesh generation is an important pre-processing step in CFD of
turbomachinery, quite analogous to the development of solid
modeling that has been discussed in the earlier module for
building the physical model of the computational domain.

•Two contrasting methodologies are developed for mesh
generation: one, the multi-block structured mesh and the other,
fully unstructured mesh using tetrahedra, hexahedra, prisms and
pyramids.

•The former method of structured mesh generation produces the
highest quality meshes from the point of view of solver accuracy
but does not scale well on PC clusters.

•By contrast, fully unstructured meshes are fast to generate and
automate the scale well on clusters, but do not allow solvers to
deliver their highest quality solutions.
3

25. •Further, numerical tolerancing issues arise within the CAD
system and are often exacerbated while imported from the
modeler to the mesh generating tool. In the process, due to
greatly differing scales within the geometry and lack of
numerical compatibility between various geometrical
representations, the model looses “water-tightness” and
necessitates substantial “cleaning”.

•The CSG and BREP paradigms discussed in the previous
module are also applicable while developing mesh generation
algorithms and provide the required water-tightness to the
geometry.

•Most CFD analysis codes, whether commercially available or
developed in-house, follow the same (BREP) paradigm.
4

26. •In order to solve the differential equations numerically, the
continuous physical domain needs to be identified with a large
set of discrete locations called nodes.

•The number of these discrete data points should be so large that
the characteristic variations in the flow properties, determined
after solving the differential equations by the numerical method,
should be as close to the “exact solution” or “bench-mark
solution” as possible.

•A method should be developed to mark the nodes in a fashion
that is demanded by the numerical method that is to be used for
solving the differential equation.

•The popular mesh generation methods are: structured,
unstructured and hybrid.
5

27. Structured Mesh Generation
•For the implementation of numerical methods such as the finite
difference, each node in the computational domain must have
easily identifiable neighboring nodes. A grid or mesh that
satisfies this demand is the structured mesh.

•Implementation of numerical methods on structured meshes
using Cartesian or cylindrical polar grid system is possible only
for simple rectangular or axi-symmetric geometries.

•In general, the generation of structured mesh for a complex flow
domain involves automatic discretization methodology with
boundary fitting coordinates and with coordinate transformations
as discussed.

6

28. •The basic steps in the methods of generating structured meshes
for complex geometries are:

–mapping of the complex physical domain on to a simple
computational domain;

–usage of body fitting coordinates

–transformation of lengths, areas, volumes and all vector
quantities(e.g. velocity).
7

29. •The mapping transformations should preferably be

–smooth

–conformal and

–controlled for grid spacing.

•Iso-parametric mapping of sub-domains enables creation of
multi-block structured grids. The sequence of mapping
determines whether the final mesh is a pseudo rectangular, O-
type, C-type or H-type.
8

30. FIG. 3.1.1 Pseudo rectangular mesh
•Figure 3.1.1. demonstrates the method of generating the pseudo
rectangular mesh for a physical region ABCD, bounded by lines
x = 0.5 and y = √(1-x
2
).

•The co-ordinates ξ and η are body conforming.
9

31. •Using the transformation,


the domain ABCD in x-y plane (3.1.1 (a)) is mapped on to ξ-η
plane as a unit square. Note that this transformation is not unique
and we may have used suitable alternative transformations as
well.

•The grid formed by the intersection of ξ = constant and
η=constant lines in the physical domain shows the body
conforming nature of these coordinates (Fig. 3.1.1 (b)).

•As we noticed in Lecture 1.2 (refer Fig. 1.2.5), the
turbomachinery flow geometries are multiply connected
domains, for which three basic grid configurations: O-type, C-
type and H-type are widely used. For a given geometry, any
one of these configurations can be obtained by suitable mapping.
2
0.5
(1 )
y
and x
x
  

10

32. •Consider the multiply connected domain shown in Fig. 3.1.2.
For the same geometry, different grid configurations (O, C or H)
are generated by adopting slightly different methodologies. This
is described in the following.

•O-type Meshing

Introduce a branch cut and identify points (A,B,C,D) on
either side of the branch cut as shown in Fig. 3.1.2 (a). Then,
by mapping AB on A’B’, BC onto B’C’, CD on to C’D’ and
DA onto D’A’, O-type grid is obtained.

The object boundary (AB ) and the external boundary (CD)
become opposite sides of the transformed domain. The two
sides of the branch-cut (BC and AD) are also mapped onto
two opposite sides of the rectangular domain. Now, a grid
constructed by ξ = constant and η=constant lines in the
physical domain is O-type, as shown in Fig. 3.1.2(c).

The O-type meshes generated by this method for NACA
airfoil and a turbomachinery blade are given in Figs. 3.1.3
and 3.1.4 respectively.
11

33. (a) (b)
Fig. 3.1.2 O-type Grid Generation, (a) basic branch-cut scheme, (b) Cartesian
grid in ξ-η plane (c) O-grid in the physical plane
12

34. •An O-type mesh for symmetric NACA aerofoil is shown in Fig.
3.1.3.











Fig. 3.1.3 O-type mesh for symmetric NACA aerofoil
13

35. Fig. 3.1.4: O type mesh for a turbomachinery blade
14

36. •C-type Meshing

For C-type meshing of the same multiply connected domain,
a branch-cut, as shown in Fig. 3.1.5 (a) is introduced and two
points A and B are identified where the branch-cut meets the
outer boundary. Points C and D are suitably selected on the
external boundary and mapping is carried out with AB onto
A’B’, BC onto B’C’, CD onto C’D’ and DA onto D’A’.

Note that the forward sweep of the branch-cut (AP), the
object surface (PQ) and the reverse sweep of the branch-cut
(QB) comprise one side A’B’ of the transformed region. The
object surface is mapped onto the patch P’Q’ on this side. It
can be seen that in this transformation, the η-constant lines in
the interior envelop the object and the branch-cut, thus
forming a C-type configuration.

Figure 3.1.6 shows a C-type mesh for a turbomachinery
blade.
15

37. Fig. 3.1.5 C-type Mesh Generation, (a) basic branch-cut scheme, (b) Cartesian
grid in ξ-η plane (c) C-grid in the physical plane
16

38. Fig. 3.1.6: C-type mesh for turbomachinery blade
17

39. •H-type Meshing

For H-type configuration, two branch-cuts are introduced on
either side of the object and the upper and lower portions
(ABCD and EFGH) are separately mapped onto A’B’C’D’ and
E’F’G’H’ in the proper sequence (Fig. 3.1.7). Here, the object
reduces to a line P’Q’ in the middle of the transformed
domain.

It is evident form the examples that by choosing the mapping
configuration, different types of grids can be generated for the
same geometry. The appropriate choice depends on the nature
of the problem to be solved. For complex domains with many
objects, it may be necessary to map different regions
separately, using local transformations. A variety of grid
layouts such as the overlaid grids and embedded grids can be
achieved through such procedures (Fig. 3.1.7).

Figure 3.1.8 shows a H-type mesh for a turbomachinery
blade.
18

40. Fig. 3.1.7 H-type Mesh Generation, (a) basic branch-cut scheme, (b) Cartesian
grid in ξ-η plane (c) H-grid in the physical plane
19

41. 20
Fig. 3.1.8: H type mesh for turbomachinery blade

42. Summary of Lecture 3.1

Mesh generation strategies for structured mesh are discussed.
The methods for different types of meshes such as O, C and H
type grids are presented.
21
END OF LECTURE 3.1

43. Lecture 3.3
Successive Layer Grid Generation
1

44. Boundary Discretization

•All types of grid generation methods (structured and
unstructured) involve boundary discretization as its first step.
The distribution of grid density largely depends on the
distribution of the nodes on the boundaries.

•In Module 2, Fig. 2.2.6 (d) shows the discretization of the
boundary of the airfoil for various segments. A large number of
boundary nodes are obtained in this process. This boundary
curve is regarded as the base curve for the generation of the first
grid layer.

•In the method of successive layer grid generation, a fine layer of
grid is first generated in vicinity of these so generated boundary
nodes.
2

45. •A structured grid is thus generated by starting from the boundary
curve and marching in successive layers. Simple algebraic
relations are derived based on analytical geometry
considerations for ensuring the cell orthogonality and providing
cell areas.

•Next, quadrilateral cells are constructed on outward normals at
all grid points of the base curve.

•The outer surface of these cells is treated as the base curve for
generation of the next grid layer. This marching process is
continued till a prescribed number of (say, N) layers are formed.

•The procedure for obtaining the outward normals and the cell
area are explained in the following section.
3

46. Construction of Outward Normal
•Consider a base curve Γ
i as shown in Fig. 3.3.3, which has
discretized points such as a, b and c. At point b, bm and bn are
unit vectors along the outward normals of the edges ab and bc
respectively. The outward normal at the point b of the curve Γ
i is
along the ray bo, which is the bisector of angle mbn. The co-
ordinates of the points m, n and o are derived as:
(3.3.2)

(3.3.3)



(3.3.4)

where, l
i
ab and l
i
bc are lengths of the edge ab and bc on Γ
i
respectively. ,,
,,
and
,,
22
ii
b ab b a b ab b a
mm ii
ab ab
ii
b bc b c b bc b c
nn ii
bc bc
m n m n
oo
x l y y y l x x
xy
ll
x l y y y l x x
xy
ll
x x y y
xy
4

47. Cell Area

•Consider that a cell should be constructed about an edge (ab).
The cell area is a function of the length of the specific edge (l
1
ab)
on the boundary curve (Γ
1) and a distance parameter, δ
i, i = 1 to
N.

•The distance parameter δ determines the normal dimension of
the cell. It is formulated with the help of an exponential
stretching function (given in Eq. (3.3.1)):



where L = δ
N, the value of L is chosen as a function of the
characteristic dimension in question e.g., chord length for an
airfoil. 1
1
1
1
S N i
N
i
S
e
Le 5

48. •The cell formed over the edge ab on Γ
i will have an area A
abed.

•This area is obtained by giving chosen weightages to the lengths
of edges on the boundary curve (Γ
1) and the base curve (Γ
i). The
area A
abed is given by:

A
abed = (l
i
abε+ l
1
ab(1 – ε))h
i (3.3.5)

where ε is called the cell size control factor and h
i = (δ – δ
i-1). A
suitable value for ε (between 0 and 1) is selected to get an
appropriate cell size distribution within the domain.
6

49. Fig. 3.3.3: Schematic diagram illustrating the successive layer grid-
generation method.
7

50. Cell Construction
•The cell above the edge ab of Fig. 3.3.3 is formed by knowing
the co-ordinates of the points a, b, d, o and the prescribed cell
area, A
abed.

•Here e is the grid point in the new layer and it corresponds to the
point b on the base curve.

•While the co-ordinates of the point o are calculated from Eqs.
(3.3.2 to 3.3.4), the co-ordinates of point d are obtained during
the cell formation on the preceding edge of ab.

•As triangles dbe and dbo in Fig. 3.3.3 are similar, the ratio of
areas A
dbe and A
dbo is equal to the ratio of the lengths l
be and l
ba.
8

51. •Thus, the co-ordinates of the point e are estimated as
(x
e, y
e) = (fx
o + (1 – f )x
b, fy
o + (1 – f )y
b) (3.3.6)
where
f = (l
be/l
bo) = (A
dbe/A
dbe) = (A
abed – A
abd) /A
bod

•The initial cells of every layer are constructed as rectangular
cells.

Method for Damping Grid Oscillations

•The above grid generation method produces grid oscillations.
Dampening grid oscillations in the present method is done by
averaging the normal distances.

•For instance, the point e in Fig. 3.3.3 is relocated to e', such that
the distance between b and e' is the arithmetic mean of the
distances l
ad , l
ba and l
af .
9

52. •In this manner all the points in the new grid layer are relocated
repeatedly for a few iterations, typically five times.

•For situations such as sharp corners, where there is a danger of
grid line intersection, the cell (h
i) used in Eq. (3.6.1) is varied
linearly to facilitate gradual turning of grid lines. Thus while
smoothing the grid, small variations in the cell area are allowed.
10
Fig. 3.3.4: Grid oscillations in orthogonal grid without damping (∆ = 0.1, N
= 10, s = –5 and ε = 0.25).

53. •The above method of grid generation is named as successive
layer grid generation scheme.

•This method has been applied for obtaining orthogonal grid over
an ellipse having a major axis of 2 units and a minor axis of 0.2
units.

•Figures 3.3.4 and 3.3.5a show the resulted grid without and with
grid smoothing respectively.

•In Fig. 3.3.4, small grid oscillations are found from sixth layer,
which amplify from one layer to the next.

•It is found that the grid in Fig. 3.3.5 obtained using ε = 0.25 is
smooth even for large normal distances from the surface.
11

54. •Figure 3.3.5 shows the enlarged views of the grid near the
training edge obtained with two more values of ε, zero and one
respectively.

•It is seen that as ε increases, the grid lines near the rear
stagnation point tend to diverge as the cell areas increase from
inner layers to the outer.

•Figure 3.3.6 shows the grid generated around a 90
o
sharp corner
with ∆ = 1, N = 20 s = –1 and ε = 0.5, which demonstrates the
capability of the method to discretize open geometries having
sharp corners.

•The case of possible grid line intersection with controlled cell
area variations for the corner (cavity) problem, shown in Fig.
3.3.7.
12

55. Fig. 3.3.5: Grid around an ellipse with damping (∆ = 0.1, N = 10, s = –5) (a)
ε = 0.25 (b) ε = 0.0 (c) ε = 1.
13

56. Fig. 3.3.6: Grid exterior to a corner








Fig. 3.3.7: Grid inside the corner of a cavity
14

57. Exercise Problem

•Generate a grid over NACA0012 using successive layer method.

Summary of Lecture 3.3

The methodology for another algebraic method named
successive grid generation is described. Technique for damping
the oscillations is also presented with different examples.

15
END OF LECTURE 3.3

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64. Lecture 3.4
Differential Equation Based Schemes
1

65. Differential Equation Based Schemes
•As stated in the previous lecture, another important and most
widely used method of structured mesh generation is based on
solving partial differential equations. These techniques may be
based on any one of the following schemes depending on the
characteristics of the grid generation equations. They are:

–Hyperbolic PDE based schemes

–Elliptic PDE based schemes

–Parabolic PDE based schemes
2

66. Elliptic PDE Based Methods

•In this lecture, Elliptic PDE based grid generation schemes will
only be discussed. These methods are particularly useful for
confined physical domains of turbomachinery flows.

•The elliptic PDEs for grid generation describe the variation of
the body fitting coordinates (ξ, η, ζ ) in the interior of the
physical domain, with prescribed values or slopes at the
boundary.

•In the computational domain, however, the physical coordinates
(x, y, z) are treated as the unknown variables on the grid formed
by the ξ=constant, η= constant and ζ=constant lines. They are
determined by numerically solving the transformed grid
generation equations.
3

67. Elliptic Solvers

•Consider the transformation functions, which are solutions of an
elliptic Dirichlet boundary value problem. The mathematical
problem is given by


(3.4.1)




with fixed , , on the boundaries. P, Q and R are source
functions, which can be used to grid point controlling. 2 2 2
2 2 2
222
2 2 2
222
2 2 2
( , , )
( , , )
( , , )
P x y z
x y z
Q x y z
x y z
R x y z
x y z
4

68. Laplace Solvers

•The condition: P=Q=R=0, results in to uniform distribution of
the points. The system then becomes the Laplacian.

•The Laplacian grid generation system satisfies the maximum and
the minimum principles , i.e. both maximum and minimum for
, , occur only at the boundary.

•The solutions of the Laplacian operator, , , are either
harmonic, sub-harmonic or super-harmonic. Therefore they are
very smooth functions. Further, the , , have continuous
derivatives of all orders. This makes the solution of the
transformed governing equation accurate
5

69. Two dimensional Transformation Functions

•Let us consider the body fitting coordinate transformation, of the
form = (x,y) and = (x,y). Therefore, one can write,

(3.4.2)

•Considering the reverse transformation x=x(, ) and y=y(, ),

(3.4.3)

•Combining the above two, we have

(3.4.4) ( , )
( , )
xyxy
xyxx
dx dyx y d dx
dx dyx y d dy ( , )
( , )
xxx x dx d
yyy y dy d x y
x y
xx
yy J
y x
yx
1
1
||
6

70. Two dimensional Transformation Functions

•One can therefore write the two dimensional transformation
functions as


(3.4.5)




•Note that the Jacobian, J, represents local area scaling factor and
should not become zero. ( , )
()( , )
( , )
( , )
( , )
()( , )
( , )
( , )
x
y
fy
y f y f
f
xy J
xf
x f x f
f
xy J
7

71. Extension to Three Dimensions

•Extending the arguments to three dimensions, one can derive the
relationships between the derivatives of the Cartesian
coordinates (x,y,z) and the curvilinear coordinates (, , ) in the
form:


(3.4.6)

1
x y z
x y z
x y z
x x x
y y y
z z z
8

72. Elliptic Solvers in 2D





(3.4.7) 2 2 2
22
2 2 2
22
22
22
22
22
20
20
1
1
x x x x x
PQ
y y y y y
PQ
x x y y
P
x x y y
Q
22 22
,
x y x y
x x y y
9

73. Demonstration

•Consider a planar region, as shown in Fig. 3.4.1, in which a
structured grid has to be generated












Fig. 3.4.1 Physical domain
10

74. •Generate a rectangular (ξ, η) = (0,1)x(0,1) uniformly discretized,
as shown in Fig. 3.4.2, (ξ
i = i*Δξ, η
j=Δη, Δξ, Δη are the uniform
step lengths in ξ, η directions, respectively ) plane given by











Fig. 3.4.2 Uniform grid in computation domain
11

75. Algorithm

•Map the boundaries (from physical to computational) as shown
in Fig. 3.4.3.








Fig. 3.4.3 Mapping of physical to computational domain

•Due to the mapping of the physical boundaries over the
boundaries of the computation domain, x, y are known along the
boundaries of the computational domain. 12

76. •Therefore, once x, y are computed in the interior of the
computational domain, the required grid in the physical domain
is established.

•To obtain x, y in the interior of the computational domain, solve
(using the boundary values of x and y as boundary conditions)


(3.4.8)



for x, y over the uniformly discretized computational domain 2 2 2
22
2 2 2
22
20
20
x x x x x
PQ
y y y y y
PQ
13

77. Multiply Connected Domain

•The algorithm described in the earlier slides works for simply
connected domains.

•For multiply connected domains, for example like annular
regions shown in Fig. 3.4.4, artificial boundaries can be
introduced to convert them in to simply connected regions.






Fig. 3.4.4 Introduction of artificial boundary
14

78. Boundary Conditions on the Artificial Boundaries

•Dirichlet boundary conditions over the artificial boundaries in
the multiply-connected regions may lead to non-smooth grid
lines as shown in Fig. 3.4.5.









Fig. 3.4.5 Non smooth grid lines over artificial boundary
15

79. Periodic Boundary Conditions on Artificial Cuts

•However Periodic boundary conditions over artificial cuts
generate smooth grid lines as shown in Fig. 3.4.6.









Fig. 3.4.6 Smooth grid over artificial boundaries
16

80. Grid Lines Attraction and Repulsion

•In general, grid points are attracted in the convex regions and
repulsive in the concave regions as shown in the Fig. 3.4.7.
17
Fig. 3.4.7 Grid point attraction and repulsion

81. Exercise Problems

•Repeat the exercise problems of Lecture 3.2 using PDE method.

•Generate uniform grid in square and cubical, rectangular and
cubical region.

•Generate uniform grid in cylindrical and spherical regions.

•Two-dimensional region bounded by circle of radius r = 1.

•Three-dimensional region bounded by sphere of radius r = 1.

•Annular region in 2D bounded by r = a and r = b with a < b.

•Annular region in 3D bounded by r = a and r = b with a < b.
18

82. Summary of Lecture 3.4

•Mesh generation schemes by solving hyperbolic, elliptic and
parabolic partial differential equation methods are presented.
The methods are explained through examples.

19
END OF LECTURE 3.4

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