Generation of Structured Grid by Solving Elliptic PDEs

1
Generation of Structured Grid by Solving
Elliptic PDEs

• Dzhmova Olga

2
The main problem

• For generation of structured grid we use the
  transformation of coordinates
• The main aim of this transformation is the
  construction of uniform grid at the computational
  domain where physical boundaries fit (lie in) the
  coordinate system

3
Abstract

• Curvilinear coordinates
• Transformation of coordinates
• Metric tensor
• Generation of structured grid
• Classification of techniques for structured grid
• Elliptic grid generation
• Conclusion

4
Curvilinear coordinates

• the object in the physical domain is transposed
  to the rectangular in the curvilinear
  coordinates
• it is efficient to do calculations at the
  computational domain

5
Curvilinear coordinates (an example)
At the physical domain the point with coordinates
correspond to the point (j,k)
6
Curvilinear coordinates
The mapping of the physical domain to the domain
of curvilinear coordinates can do the concentrate
of coordinate lines in the areas of the physical
domain where expected appearance of large
gradients. If the areas of large gradients are
changed with time, for e.g. moving of air-blast,
the physical grid can be reformed so, that the
local grid will be so fine for getting the
solution with required precision.
7
Curvilinear coordinates

• There are some difficulties when we use the
  curvilinear coordinates.
• to define the form of equations in the
  curvilinear coordinates
• at this questions there are some additional
  terms (parameters of transformation, for e.g.
  ) which are additional source of errors

8
Transformation of coordinates
Assume, that a one-to-one mapping can be
established between physical and curvilinear
coordinates, which can be written as
and correspondingly
Using these functional dependences
the equations can be transform to the
form, which contain partial derivations over
9
Transformation of coordinates
As an example, the first-order derivatives u, v,
and w over x, y, z.
,where
(1.1)
J the matrix transformation of Jacobi.
10
Transformation of coordinates
In practice it is more convenient to work with
the inverse matrix of Jacobi.
(1.2)
1.3
(1.4)
In case of two-dimensional
11
Transformation of coordinates
Using (1.3) and (1.4), the elements of matrix J
in (1.2) can be expressed in form
(1.5)
12
Metric tensor
For the connections between curvilinear,
orthogonal, conformal coordinates are better to
use the metric tensor , which connect to
matrix Jacobi.   Assume, that the physical
domain is in Cartesian coordinates and the
computational domain is in the curvilinear
coordinates
13
Metric tensor
(1.6)
can be connected with the
corresponding curvilinear coordinate
(1.7)
So,
(1.8)
where
(1.9)
14
Metric tensor
In the two-dimensional case (1.8) we can write
(2.0)
, where and
determinant of matrix Jacobi.
15
Metric tensor
(2.1)
(2.3)
16
Orthogonal and conformal grids

• For orthogonal system of coordinates some terms
  of transformation are vanished and the
  equation is simlified

Then, using (2.3) for the two-dimensional grid
we get
(2.4)
In the three-dimensional case the orthogonal
condition is
(2.5)

• Using conformal transformation allows to keep the
  same structure of model equations as in
  computational domain and also in the Cartesian
  space. (the parameters of transformation equals
  to 1 or 0 )

17
Generation of structured grid
Grid generation is usually based on a map between
a simple computational region and a complicated
physical region. It can be used numerically to
create a mesh for a discretization method to
solve a given system of equations posed on the
physical domain. Alternatively, it can be used to
transform equations posed on the physical region
to ones posed on the computational region, where
the transformed equations are then solved.
18
Generation of structured grid
We have to consider boundary-value problem
on
in
19
Generation of structured grid
For solving the boundary problem to find the
positions of inner points of grid there are 2
ways such as differential equations methods and
the interpolation of inner points on boundaries.
20
Generation of structured grid

• This mapping, at the least, has to satisfy the
  follow requirements
• A mapping which guarantees one-to-one
  correspondence ensuring grid lines of the same
  family does not cross each other.
• (the determinant of matrix has to be
  non-zero and finite).
• A smooth grid point distribution with minimum
  grid line skewness, orthogonality or near
  orthogonality and a concentration of grid points
  in regions where high flow gradients occurs are
  required.

21
Structured grid
        The grid system has a restrict (or
structured) requirement for grid specification
such that in 2D domain each grid can only have
four grids connected to it and in 3D domain each
grid can only have six grids connected to it.
Therefore, the shape of grids are quadrilateral
and tetrahedral in 2D and 3D domain,
respectively. The FDM can use the structured
grid system.
Advantages with structured grids are Easy to
implement, and good efficiency With a smooth
grid transformation, discretization formulas can
be written on the same form as in the case of
rectangular grids.  Disadvantages - Difficult
to keep the structured nature of the grid, when
doing local grid refinement - Difficult
to handle complex geometry. (Mapping of an
aircraft to the unit cube).
22
Structured grid
Structured (Elliptic PDEs )
23
Classification of techniques for structured grid

• Complex variable methods 
• Algebraic techniques
• Techniques based on numerical solution of partial
  differential equations
•  

24
Elliptic grid generation
(2.6)
where - the coordinates in the
computational domain and P, Q the known
functions which using for the control of
concentration of inner grid points.
The elliptic PDEs satisfy the maximum principle
(i.e. max and min of x and h are reached on the
boundary). Tomson Thompson 1982 notes that it
guarantees single-valued transformation.
25
Elliptic grid generation
By interchanging the independent and the
dependent variables, we obtain the transformed
system in the transformed computational domain
(2.7)
26
Elliptic grid generation
This method has some advantages a) the
transformation between the grids is smooth , b)
the mapping is one-one , c) the boundaries of
the complicated form are operated easily. The
disadvantage is that the control functions may
not easy to derived. It is difficult to control
the distribution of grid nodes in the inner
part. And at the present time the grid has to
reconstruct after every step by the time. So it
takes the large costs of the computer time.  
27
Numerical solution
These equations (2.7) are approximated by e.g.,
(FDM)
etc.
where now the index space 1 i m and 1 j n
is a uniform subdivision of the (x,h)
coordinates, x (i - 1)/(m- 1), h (j- 1)/(n-
1). The number of grid points is specified as m
n.
28
Numerical solution
The approximation equations can be solved by
iterative methods.
For solving these questions it used the method
successive overrelaxation and got that the
parameter of acceleration can be gt 1
if As the optimal choice and the number of
iterations depend on the choice of the P, Q.
29
Numerical solution
In the work of Thompson, 1977a Thompson
recommends follow choice of parameters P and Q
and an analogous control function Q(?,?).
Where a, b, c and d are chosen so that to provide
the grid density at each point in the domain.
30
Elliptic questions
A1- the transformation between the grids is
smooth A2 considering of boundaries conditions
on all boundaries of physical domain A3 - one
one mapping of physical and computational domains
A4 flexible mechanism of control for the
distribution of inner points of grid
(Discontinuities at boundaries can be smooth out
at the interior regions.) A5- The elliptic grid
generation is the most extensively developed
method. It is commonly used for 2D problems and
has been extended to 3D problems. D1- More
computation time involved for the elliptic grid
system (elliptical questions are solved by
iterative methods) D2- Sometimes the control
functions may not easy to derived
31
Conclusion

•    In the computational domain the physical
  boundaries fit (lie on ) the coordinate system so
  there is no local interpolations when we set the
  boundary conditions.
• If the grid is orthogonal or conformal then some
  additional terms will be vanished at the
  equations.
• There are some problems with the discretization
  questions in the curvilinear coordinates as a
  rule it is concerning the approximation of
  parameter of transformation. Usually it
  recommends using the same difference formulas,
  which use for the discretization derivatives of
  dependent variables.
• If the discretization is going on the homogeneous
  computational grid it can be reach high accuracy.
  However it is right for computational domain,
  but for physical domain it is correct not always.
  If the stretching parameter of grid is not small
  we can get the degradation of accuracy.