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How to formulate the FEM for the governing equatio

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Old   March 14, 2005, 15:54
Default How to formulate the FEM for the governing equatio
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Hi there,

The governing equations are already in the integral form, say,

d/dt (S udv) = S (f dv).

where u is the velocity, f is the force and (S dv) is the volume integral.

I am wondering how I can derive the FEM formulation through the standard Galerkin residual weighting procedure?


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Old   March 15, 2005, 05:39
Default Re: How to formulate the FEM for the governing equ
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Let's take a step back. Suppose your PDE is Lu=f, with u the unknown field, f - a known RHS and L - a differential operator. Appropriate BCs should also be prescribed, but for this simple explanation they will be ignored. The domain should be meshed with nonoverlapping elements.

You start by assuming the distribution of u in any element is spanned by the shape functions, Ni, which the spatial distribution within the element and the DOFs, Ui, usually the nodal values, i.e., u = sum (Ni*Ui) (the sum is on i).

Next, you multiply your PDE by a weight function, Wk, and integrate over the domain volume to get the weak formulation. Since the volume is meshed, the integral over the entire domain may be split to sum on all elements of volume integrals over each element. Further, for the standard Galerkin method, the weight functions are identical to the shape functions, i.e., Wk=Nk.

That's more-or-less the whole process, except for the BC treatment. There are, of course, many details involved (and God is in the details) in order to end up with a successful code. For these, a good book rather than a posting in a forum is advised.

I hope this helps,

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Old   March 15, 2005, 13:32
Default Re: How to formulate the FEM for the governing equ
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Hi Rami,

Thanks kindly for your reply.

In fact, I am fully aware of the procedure you described and have been using the FEM for a number of years. However, my original question was about the governing equations that are already in the integral form before the weighting (i.e. the volume conservative form). It is no problem to span variables elementwise via shape functions. My concern is that how to do the weighting. One cannot simply move the weighting function into the integral unless using piecewise constant weighting function. However, by choosing the piecewise constant weighting function, the FEM reduces to FVM in certain sense.

My question, after rephrasing, is that can we use linear or higher-order weighting functions for the governing equations in the volume conservative form? If the answer is no, then does it mean that the FEM is not so conservative?

Thanks again.

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Old   March 16, 2005, 08:20
Default Re: How to formulate the FEM for the governing equ
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i am the student of final year mechanical. sir,we are doing project on cfd,our project details shown below. project titleEVELOPMENT OF PREPROCESSOR FOR CFD ANALYSIS ON 2D DIFFUSERS(ST.CONICAL,CURVED DIFFUSERS) sir we have problem with deriving eqation for fluid flow through these diffusers .sir i will be great full to you if you kindly help me.we want to extend our project ,so that we can call our project as complete package for diffusers.we are writing programmes in matlab.
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