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Solving 2D Laplace equation - Infinite boundary

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Old   August 30, 2006, 19:27
Default Solving 2D Laplace equation - Infinite boundary
  #1
Guillermo Marraco
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Hello. I write here with hope that somebody can help me with a problem that I can't solve correctly.

I want to solve the 2D Laplace equation with iterative methods like jacobi/Gauss-Seidel/etc.

The sinçmplest way is iteratively averaging each cell with neighbors. I believe that anybody in CFD know this.

I have trouble with boundary conditions.

I know the trick of mirror ghost cells to solve Dirichlet and Von Neumann booundary conditions. They are easy.

But I can't find a convergent and stable way to manage infinite boundary conditions (non bounded cells).

I thing that if exist a way, then iterative solvers will not been needed. You simply will solve the unbounded nearest cells to known cells, and then you can expand the limits to solve all the net. (I don`t want to solve the network this way).

One option is to take a really big net, much more big than nedeed, and impose far boundary nonpermeable (impervious?)conditions. But this is expensive, and is not really accurate.

I can`t find it in google. ¿somebody can help me?

(please, write to marraco at arnet.com.ar)
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Old   August 30, 2006, 20:55
Default Re: Solving 2D Laplace equation - Infinite boundar
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Adrin Gharakhani
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One approach is to use the boundary element method (BEM) at the boundaries of the outermost cells. The potential solution due to BEM will implicitly satisfy the far-field boundary condition. So, you can use BEM to solve for the potential field at the outercell boundaries and use the latter as the boundary condition for your cell-based method for the area that uses grids. Alternatively, you can just solve the BEM problem using the actual walls; i.e., solve the whole potential flow. Then you can "post-process" the BEM solution to obtain the potential field values at the cell locations. This can be done fast by classical methods if the problem size is small; for large problems there will be significant work to make BEM fast

Adrin Gharakhani
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Old   September 9, 2006, 11:05
Default Re: Solving 2D Laplace equation - Infinite boundar
  #3
Guillermo Marraco
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I thanks you for your answer.

I are looking information for the boundary element method.

I have also get a tip: if starting for any point, and following a random path until a boundary (with fixed value) is reached, one get, averaging boundary values reached, the correct value por the starting point (this is a montecarlo method for solving the laplace equation on one unique point), then this must equal a finite difference codition for unbounded boundary.

But I are not sure that I can reach a solution in this tread of thinking.
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