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Old   November 24, 1998, 22:33
Default elliptic grid generation (orthogonal)
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i am trying to generate an orthogonal grid based on elliptic equations. i was reading several papers such as that by steger and sorenson, jcp,v33,1979.

the problem is that of a circle in a rectangular domain. the grid lines should be normal on the circle (1st boundary) and on the rectangular boundary (second). the rectangle is quite large compared to circle. (imagine a circle of radius 1 unit at the center of a rectangle of size 4x20). (it will be much better if the grid is orthogonal throughout the domain though)

i keep having problems with the method of control functions. for instance, which control functions are suitable? also, how do we calculate d2X/deta2 on the outer boundary?(eta is the vertical coordinate in computational domain and X is the horizontal coordinate in physical space)

steger and sorenson give an expression for inner boundary as: d2x/deta2 (xi,0)=(-3.5*X(xi,1)+4.*X(xi,2)-.5*X(xi,3))/deta**2-3*dxdeta@(xi,1)/deta

i am not sure how this is obtained and also, how this expression will look for d2x/deta2(xi,etamax=jm)

any suggestions? thanks in advance.
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Old   November 25, 1998, 07:32
Default Re: elliptic grid generation (orthogonal)
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The topology of your solution region precludes an orthogonal grid. Where do you want the non-orthogonalities? How many blocks can you use?

The combination of "orthogonality" and "control functions" is unusual but I am not familiar with the work sited. Do you require orthogonality or is it simply desirable? If required, does it need to be conformal? If orthogonality is required, I would be suspicious of any source terms (if that is what your "control functions" are) and stick to the basic equations and manipulate through the boundary conditions. However, if it is only desirable and you really want to use "control functions" then tap into some of the stuff from J.F.Thompson.

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Old   November 25, 1998, 13:47
Default Re: elliptic grid generation (orthogonal)
Vasu Veerapaneni
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i was having problems even without orthogonality, when the rectangular outer boundary is bigger than the circle (i.e, the small side of rectangle is comparable, but the long side is, say, 10 times larger than circle). in such a case when i solve the elliptic equations: alpha*d2X/dxi2-2*beta*d2Xdxdeta+gamma*d2Xdeta2=0, similar for Y i get most of the grid lines in almost a square around the circle. i generate initial grid by drawing rays out of the center of circle and see where they intersect the circle and outer rectangle (gives grid points on the circle as well as rectangle) and interpolate inbetween for intermediate points. solution is slightly better if i calculate the derivatives for the coefficients using cubic spline method rather than the finite difference methods!

Orthogonal: eventually i want to calculate drag force on the circle when a fluid is going through the rectangle. isn't it better to have the grid lines orthogonal to the circle. regarding outer boundary, the top will be free surface, and the bottom will be a solid surface. so, i would prefer orthogonality at these two surfaces also!

any suggestions are greatly welcome. i am new to this area! thanks a lot!!!

p.s.: i am not familiar with the blocks method. so, i am trying to generate a single grid for the whole domain.
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Old   December 1, 1998, 00:45
Default Re: elliptic grid generation (orthogonal)
Tom Wanat
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I can't answer your questions about the control functions from Steger and Sorenson, but I suggest you investigate the method for controlling orthogonality by using Poisson equations for xi and eta. An excellent description is found in "Numerical Methods for Engineers and Scientists" by Joe Hoffman. I used this method and found it relatively easy to understand and implement. Good luck.
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Old   March 18, 2011, 07:13
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hey buddy i am just new to CFD and i have a C code for one cylinder placed in rectangular domain; i need to apply orthogonality on the surface of the cylinder and in the inner surface of the rectangle .

if u have an algorithm/code for it then please mail it to me ;
email id:

thanks in advance
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Old   March 19, 2011, 16:51
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When it comes to structured grids, smoothing them with an elliptic PDE-based method, and enforcing angle and spacing constraints on the boundaries the method you cite (Steger and Sorenson) is an excellent method and probably the grandfather of them all. A similar method is by von Lavante, Hilgenstock, and White. The two methods differ in that the form (SS) enforces the angle and spacing constraints on the boundaries in an approximate manner while vLHW enforces them fairly precisely.

For any questions about elliptic grid generation methods, the primary reference should be Thompson's Numerical Grid Generation:

For SS (as embodied in the GRAPE code), here's a new-ish reference:

It's harder to find an online reference for vLHW.

You are taking the right approach to the problem and hopefully the references I've provided will help. For the record, these methods are implemented in our software for years and are at the core of our structured grid capabilities.
John Chawner / /
on Twitter: @jchawner
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Old   March 24, 2011, 07:22
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I m also generating the grid over cylinder but i want the notes of steger and sorenson. if u have this notes then plz send it at my email id-
Thanking u
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