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The Coupled System of Equations

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Old   May 30, 2016, 23:07
Default The Coupled System of Equations
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Can somebody explain the notation of the system of equations written in CFX theory guide ?

\sum_{nb\,i} a^{nb}_i\,\phi^{nb}_i =b_i

I don't understand the double notation, subscrit and superscript ...

Last edited by MissCFD; May 31, 2016 at 08:28.
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Old   May 30, 2016, 23:15
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Quote:
Originally Posted by MissCFD View Post
Can somebody explain the notation of the system of equations written in CFX theory guide ?
\sum_{nb\,i} a^{nb}_i\,\phi^{nb}_i =b_i

I don't understand the double notation, subscrit and superscript ...
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Old   May 31, 2016, 10:50
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nb stand for (N)eigh(B)ors

i stand for equation_i

the equation reads as "sum around the neighbors of its influence coefficients on i times its variable value is equal to b_i"

Those coefficients include the advection, diffusion and other physics interacting between "control volume_i" and its neighbors..

Note the equation may be written in multiple forms where phi can be the value of the solved variable, or the iterative correction to the variable. Based on the interpretation of phi, b can be the source or the residual of the equation.

Hope the above gives some context.
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Old   May 31, 2016, 14:47
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Quote:
Originally Posted by Opaque View Post
nb stand for (N)eigh(B)ors

i stand for equation_i

the equation reads as "sum around the neighbors of its influence coefficients on i times its variable value is equal to b_i"

Those coefficients include the advection, diffusion and other physics interacting between "control volume_i" and its neighbors..

Note the equation may be written in multiple forms where phi can be the value of the solved variable, or the iterative correction to the variable. Based on the interpretation of phi, b can be the source or the residual of the equation.

Hope the above gives some context.
Thank you for your response !

I'm not sure to visualize, so can you tell me if this notation means the same ?

a_i\,\phi_i + \sum_{nb} a^{nb}\,\phi^{nb} =b_i

Does this correspond to system equations for one node i ?
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Old   May 31, 2016, 21:04
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Assuming "i" stands for node number, and "nb" includes all the other influencing "i"s (say j) around "i", your are correct.
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Old   May 31, 2016, 22:43
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Quote:
Originally Posted by Opaque View Post
Assuming "i" stands for node number, and "nb" includes all the other influencing "i"s (say j) around "i", your are correct.
Thank you, so are these two equations equivalent ?

\sum_{nb\,i} a^{nb}_i\,\phi^{nb}_i =b_i (1) & a_i\,\phi_i + \sum_{nb} a^{nb}\,\phi^{nb} =b_i (2)

And, this equations (1 or 2) is written for all the nodes of mesh, so we obtain i times this equation and this is what forms the matricial system ?
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Old   June 1, 2016, 14:12
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I would read them as equivalent
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