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January 19, 2007, 06:08 
Finite volumes vs. finite elements: applications

#1 
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Hi all,
I heard that finite elements is a pretty less dissipative method. This means that numerical computations makes less artificial viscosity than finite volumes. Since, I suppose that finite volumes are used in fluids because it can happen that problems are not perfectly posed, and artificial viscosity makes it converge easier than finite elements, but the solution may be less accruate for similar meshes. That could be the reason why most structure simulation softwares works with finite elements: the problems are well posed because there is less boundary conditions types (no pressurevelocity coupling for example) and then we can get high precision solutions with no problem. What do you think about it ? Kindest regards Lionel 

January 19, 2007, 08:24 
Re: Finite volumes vs. finite elements: applicatio

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Dear Lionel,
I am not proficient in finite element method, but to my understanding the foremost reason why FVM is preferred for fluids is the conservation property bulit into the FV formulation, which lacks in the FEM. It therfore necessiates that the conservation be satisfied when FEM is used in the fluids framework. It looks to me that the primary reason for the choice of FV for fluids and FE for structures is more historic, with either being formulated to solve problems specifically governed by the constituive equations pertaining to the field. Regards, Ganesh 

January 19, 2007, 09:31 
Re: Finite volumes vs. finite elements: applicatio

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There are definitely commerciallyavailable CFD codes using the FEM method rather than FV. High accuracy and conservation, too.


January 19, 2007, 09:37 
Re: Finite volumes vs. finite elements: applicatio

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I'd say that FVM is preferred for CFD because it's computationally cheaper. At least when compared to naive FEM implementations. FEM comes from structural analysis where the equilibrium of the solution must be satisfied nodally, it's almost the same what is claimed by Finite Volumes Methods as local conservation through flux balances (I should say that conversely to Ganesh, I'm not proficient in FVM). There are some recent FEM papers with deep discussions about such historical war between FVM and FEM regarding local conservation.
Regards Renato. 

January 19, 2007, 13:07 
Re: Finite volumes vs. finite elements: applicatio

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Hello,
Adding to the earlier conservation point, FEMsuite is not good for Hyperbolic or End Parabolic type of equations[High speed Flows]. The weight or shape functions are the main simulation or solution deciding factor and it looks like this(FEM) approach is good for elliptic(dissipate to all>distance physics) type of equations rather than hyperbolic. You can get more clear picture from FEM for Fluid Dynamics vol 3 by Zienkiewiez and Taylor. you can get this ebook from www.esnips.com 

January 19, 2007, 17:15 
Re: Finite volumes vs. finite elements: applicatio

#6 
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Ok thanks you for all your ideas, I didn't know that FEM could make problems in the case of hyperbolic equations.
All other remarks are welcome... 

January 19, 2007, 20:30 
Re: Finite volumes vs. finite elements: applicatio

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Well, there's a bunch of FEM solvers to high speed flows and other related hyperbolic problems. In fact, FEM formulations can be improved with stabilizations and shock capturing terms, not to mention the newest techniques such as Discontinuous Galerking and Variationa Multiscale which have opened a broad research area in the family of FEM solvers.
[]'s Renato. 

January 20, 2007, 06:31 
Re: Finite volumes vs. finite elements: applicatio

#8 
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Is this means FVM is going to be on decline onwards due to more and more people opting for FEM


January 22, 2007, 06:53 
Re: Finite volumes vs. finite elements: applicatio

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I think (as Renato) that the main reason is the computational cost. Using Discontinuous Galerkin formulation the conservation is satified locally (element by element) but the computational cost is still higher than using finite volumes.
About the problems of FEM formulations with hyperbolic problems, I recommend reading J. Donea and A. Huerta "Finite Element Methods for Fluids problems" (2002) Ruben 

January 22, 2007, 09:36 
Re: Finite volumes vs. finite elements: applicatio

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AcuSolve uses exactly these formulations  Discontinous Gerlerkin. Very fast, very parallel, very stable, very good at conservation.


January 22, 2007, 11:51 
Re: Finite volumes vs. finite elements: applicatio

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I think that it is not correct to say computational cost of FEM is more than FVM, main portion of cost is related to solution of linear (or nonlinear) system of equations, that its cost is function of computational stencil.
If usual vertex centered FVM is used, its stencil is same as FEM with linear shape functions. If cellcentered FVM is used (such as Fluent, OpenFoam) stencil is smaller only when we have orthogonal grid (e.g. for triangular mesh line that connect center of twoneighbor cell pass from face normal vector), else people usualy use skeness correction technique that don't has essentially better performance, althouth sometimes such correction is neglected in practice (but 2nd order accuracy is not essentially acheived), so it can be said that (efficient) FVM is more sensitive to grid quality and is prefared when we have smooth grid. Finally, i think that FEM make life more easy due to its ease of implementation. 

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