FVM,FDM AND FEM
Hi CFD Friends,
Can some one explain me the basic difference among these three numerical technique FVM,FDM AND FEM? why one is good for CFD, another is good for structural analysis? If some one know any link for this info then plz tell me.
Re: FVM,FDM AND FEM
when talking about "good" numerical methods, we shall stand with end users, either engineers or researchers. because that's our purpose to develope numerical methods.
three words are better to be use to describe "good" numerical methods: reliability, efficiency, and flexibility.
accuracy, stability, roubustness, simplicity, ... all are included in the above three words, and themselves are inappropriate to stand out independently, for the sake of end users.( the explnation could be a long story).
an example, saying that high-order method is more accurate does not tell anything. because the cost might be high. that is, one may refine mesh in low-order method to gain the same accuracy with less cost. so, efficiency is the right word. accuracy is not.
as a matter of fact, the two best numerical methods are: (continuous) FEM (what very known to people) and discontinuous FEM (so called Discontinuous Galerkin, which is an improper name).
the former is the best for low-Re flow and structure and heat, because all those problems are elliptic or heat type PDEs, and where solutions are usually smooth.
the latter is the best for typical flow, which is hyperbolic type PDEs, and where solutions are typically not smooth, due to convective nonlinearity and also due to pressure in the case of incompressible flow.
in structural mechanics, there is only one thing passing around: stress. but in fluid mechanics, there are two: stress and mass. in a computational view, linear elasticity is a subset of Navier-Stokes, which can display as any of three types of PDEs under different conditions. a method works for subset does not have to work for the superset.
when a method is in weak form which means the solution generated by computers may NOT be THE solution, when there are two things trying to kicking away responsibility to each other, when the solution is not smooth which means sharp changes are ubiquitous; when all three stand, a non locally conservative method is not something we shall trust.
the continuous FEM is not locally conservative, therefore it is a poor method in typical flows. FEM is not reliable for typcial flows, whatever so called stabilized methods or not. the Control-Volume based FEM, which is locally conservative in mass, is the only exception.
the discontinuous FEM is locally conservative both in mass and momentum, but for smooth problems it becomes inefficient because it introduces much more unknowns.
both methods are flexible, better than FDM, which still can handle complex geometry but in an awkward manner.
FVM is a simplified type of Discontinuous FEM, therefore it is very popular in fluids. but, FVM is NOT the best.
so called spectral element method is a high-order continuous FEM; and so called subdomain methods should be named as discontinuous Spectral Element Method, and it belongs to high-order Discontinous FEM.
so called spectral method is the one-element continous/discontinous spectral element method, and it can't handle complex geometry.
for problems where fundamental solutions from the government equations (Green's functions) can be found, consequently only BC are left to be satisfied, the Boundary Element Method/Boundary Integral Method is the most efficient one.
and there are a bunch of Lagrangian based methods like those tracking particles. Chorin's vortex method belongs to this category, I guess.
the above is mainly on 'reliability'. flexibility is mentioned but only when starting to implement a method or use a method to solve a varity of problems, one can feel the meaning of flexibility. efficiency is easy to measure in case by case base. just run the code, to see which spends less time but gains the same accuracy. but efficiency is hard to measure in general. for example, for a simple geomerty problem, nither the continous nor discontinuous FEM is the best, simply because their powerful features, which takes time for computer to run, are wasted.
overall speaking, the Discontinuous Spectral Element Method is the best of the best for fluid mechanics.
difference among three in a nature
the most apparent different among three, in a structure sense, is explained below.
FDM uses the "exact fit" found in any textbook on numerical analysis. Taylor expansion is used for this method.
FEM uses the "best fit". Polynomimal (typically lagrangian) employed in this method.
FEM has a relative, called Discontinuous Galerkin (DG). and this relative has a son, called FVM.
when in low-order polynomials, FVM looks like FDM.
Re: difference among three in a nature
Read also previous discussions:
Re: FVM,FDM AND FEM
FVM is most recent method for the CFD. Because the conservative nature of this schemes. That means the control volumes does not overlap and summation of these control volumes gives the hole domain. FVM is the natural extension of FDM and FEM. In the present time researchers encourage to find the convergence and stability of this scheme.
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