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 daniel April 1, 2005 03:20

finite volume

Hi I want undersand the avantages and disavantages of method of finite volume in comparison of finite element.

regards Daniel

 debs April 1, 2005 06:41

Re: finite volume

look through the CFD book by malalasekara and verstaag .You will find some information.

 scar April 1, 2005 07:27

Re: finite volume

Finite Volume and Finite Element can be considered quite similar for simple problems, i.e. constant parameters, geometry, few variables, few dimensions and so on... (at least in my personal opinion). When the problem grows in complexity, Finite Element presents a formal mathematical framework that helps to treat the problem in a general way. On the other hand, in the case of FVM when the complexity grows, the numerical issues grow in a similar way as well.

In general, if the problem is simple enough, you can apply FVM or FEM, the result will be the same (around a given error). One reason to apply FVM is that you will need one day to understand the method and code your solver, while if you want to apply FEM, you will need more than one day to understand the method, and more than one day to code your solver. But, when the learning process was over, your next problem could be implemented using FEM in less that one day(generally speaking of course).

So, my suggestion is to try both methods and get your own conclusions.

 diaw April 2, 2005 00:49

Re: finite volume

I would, in the main, agree with 'scar'...

FV is an extension of the Finite Difference Method, and so is reasonably simple to implement. Is does offer advantages in its approach to coping with non-linearities. It presents a line-by-line build-up of the 'system matrix equations'. The typical 3d element will relate 1 principle node to 6 surrounding nodes. The techniques of stabilising the system matrix are well documented. I would agree that as the complexity grows, that the numerical issues begin to grow. FV solutions are generally rather fast.

FEM is a more elegant mathematically-structured approach, which, in my understanding, has the potential to offer far more information for interogation. The method itself is a little more complex in terms of the mathematical formulations with associated 'tricks' for non-linearities. Elements can have many terminal nodes. The nodal interpolation functions provide a linkage between the nodes. FEM in fluids is really still developing in terms of speed. Currently the solution time is rather slow.

In my personal opinion, the FEM approach has the potential to upstage FV in the longer-term in terms of accuracy. FEM also is well couched in Mathematics, whereas I have gained the impression that FV is a little more 'hack-&-go'. Engineers seem to favour FV, whereas Applied Mathematicians seem to favour FEM - certainly as far as for CFD problems.

To obtain further insight, take a look at www.featflow.de . Prof. Turek & the FeatFlow team have done excellent work.

diaw...

 Mani April 6, 2005 20:55

Re: finite volume

just to add a little counter-bias for balance...

>FV is an extension of the Finite Difference Method...

I don't see it that way. Conceptually, the FV method has little in common with the FD method. It seems more natural to see FVM as a (lower-order) subset of FEM, as many people do.

In my opinion, the disadvantage of the FVM is that it is difficult to construct high-order (higher than second order) schemes. In situations where high-order approximation is necessary (e.g. acoustics), the FDM is still in use. The Finite Element Method also allows for approximations of any order. A lot of people may not see this as a real disadvantage of the FVM, though. After all, the lower-order approximation is part of the reason why the method is so efficient: There are no huge matrix operations as in the FEM. Spatial accuracy can always be increased by grid refinement.

>FEM is a more elegant mathematically-structured
:approach..

Elegance and beauty are in the eye of the beholder. Mathematical elegance does not always amount to usefulness. The beauty of the Finite Volume Method is that, unlike the other methods, it is inspired by physics, not mathematics. With the Finite Volume Method, the integral conservation laws are discretized directly. I think this partly explains why the FVM is popular among physicist and engineers, and is often perceived as more natural and easier to understand than other methods. Apart from that, the direct implementation of the conservation laws provides an important feature, which is another major advantage of the FVM: A Finite Volume solution is exactly (!) conservative, at any stage in the computation, regardless of the accuracy (of course, then there may be the issue of artificial dissipation...). Exact conservation may not always be necessary, depending on your problem. However, it is an essential feature in compressible flow computations, and other cases with discontinuous solutions. Since Jameson and colleages developed the Finite Volume Method for flow problems, which enabled them to efficiently compute transonic flow, many researchers have published major improvements and extensions. So, even though the FVM is "young" compared to the FDM and FEM, it has become a standard method in many research areas... for very good reasons. Which method you prefer will mainly depend on the problem that you are trying to solve. However, it makes no sense to just "jump in" and try all of them. It would take you quite some time to do so. You may rather spend more time gathering other people's experience and then decide which is best for you.

 diaw April 6, 2005 23:44

Re: finite volume

diaw>>FV is an extension of the Finite Difference Method...

Mani> I don't see it that way. Conceptually, the FV method has little in common with the FD method. It seems more natural to see FVM as a (lower-order) subset of FEM, as many people do.

My reasoning for linking the FV method to the FD method lies in that fact that once the governing eqn has been discretised, the 'weight components' generated are lumped to their respective nodes. Thereafter, there is no direct inter-node relationship, other than these weights.

In addition, under typical FV 3D formulations, the problems are solved at each node, in a similar fashion to the FD method. The line-by-line layup of the system matrices is very similar.

This is contrasted to the FE Method, where element matrices are used to build the system matrices, providing additional 'information linkage' between the nodes... primarily due to the activity of the nodal interpolation functions. Information is contained 'between' nodes.

------------- Mani> In my opinion, the disadvantage of the FVM is that it is difficult to construct high-order (higher than second order) schemes. In situations where high-order approximation is necessary (e.g. acoustics), the FDM is still in use.

Good point.

Mani> The Finite Element Method also allows for approximations of any order. A lot of people may not see this as a real disadvantage of the FVM, though. After all, the lower-order approximation is part of the reason why the method is so efficient: There are no huge matrix operations as in the FEM. Spatial accuracy can always be increased by grid refinement.

This is definitely the strong point of FVM. I have found some anomalies when the mesh size for FVM becomes very small, in terms of error increases... The work on FeatFlow in Germany discovered this during early comparative benchmarking tests. Not too sure why...

DA>>FEM is a more elegant mathematically-structured
:approach..

Mani> Elegance and beauty are in the eye of the beholder. Mathematical elegance does not always amount to usefulness.

To be sure... :) But, the sound mathematical basis for the FEM goes to great pains to ensure unique solutions.

Mani> The beauty of the Finite Volume Method is that, unlike the other methods, it is inspired by physics, not mathematics. With the Finite Volume Method, the integral conservation laws are discretized directly. I think this partly explains why the FVM is popular among physicist and engineers, and is often perceived as more natural and easier to understand than other methods. Apart from that, the direct implementation of the conservation laws provides an important feature, which is another major advantage of the FVM: A Finite Volume solution is exactly (!) conservative, at any stage in the computation, regardless of the accuracy (of course, then there may be the issue of artificial dissipation...). Exact conservation may not always be necessary, depending on your problem. However, it is an essential feature in compressible flow computations, and other cases with discontinuous solutions. Since Jameson and colleages developed the Finite Volume Method for flow problems, which enabled them to efficiently compute transonic flow, many researchers have published major improvements and extensions.

Good points...

Mani> So, even though the FVM is "young" compared to the FDM and FEM, it has become a standard method in many research areas... for very good reasons. Which method you prefer will mainly depend on the problem that you are trying to solve. However, it makes no sense to just "jump in" and try all of them. It would take you quite some time to do so. You may rather spend more time gathering other people's experience and then decide which is best for you.

For a wonderful review of FEM at the 'cutting edge', take a look at FeatFlow at <www.featflow.de>, the graphical results are stunning... I have yet to see FD, or FVM produce this level of information.

diaw...

 scar April 7, 2005 06:10

Re: finite volume

I am in agreement with the argument exposed by Diaw, but I would also say that solving a mathematical equation should be based on mathematics (more than physics) that satisfies the requirement that the right solution is found and not what you want to get. Physics models a given process in terms of mathematical expressions, and mathematics gives tools for treating these expressions. Once solved, physics can help you to understand if the solution is acceptable or not, and if not maybe the modeling process should be repeated, more than tuning the solution process.

Finite Difference, Finite Volume and Finite Element can be though like derived from a similar framework (the Weighted Residual Method changing the weight function).

Finite Volume is based on the integration of the transport equation. This integration is divided in sub-volumes. In this point the problem is not close, and the only way to continue is to go to Taylor and approximates the derivatives in the same way like in Finite Difference. While, in the case of FEM the key is the use of a projection concept to expand the solution. Thus, I think that it is better to said that FVM is closer to FDM than FEM.

Finite Volume is "really" conservative because it is able to conserve whatever you introduce in your equation, whether it is the right solution or not. FVM forces the conservation of the properties by means of flux conservation, which means that it manipulates the approximation of the derivative in the faces of the volumes. In this way, what is leaving a volume is entering the next, smart trick :), but only good enough when nothing really happens inside the volume. If complex processes are going on inside the volume, FVM is not able to handle it except in the limit of the grid refinement. In this case increasing the order of the approximation can help. However, FVM is still conserving everything including the errors, of course. Thus, if it is not able to fulfill the physics what sense has to said that it is conservative?

 Mani April 7, 2005 14:11

Re: finite volume

>I am in agreement with the argument exposed by Diaw,
:but I would also say that solving a mathematical >equation should be based on mathematics (more than >physics) that satisfies the requirement that the right >solution is found and not what you want to get. >Physics models a given process in terms of >mathematical expressions, and mathematics gives tools >for treating these expressions. Once solved, physics >can help you to understand if the solution is >acceptable or not, and if not maybe the modeling >process should be repeated, more than tuning
:the solution process.

Sounds like the old question: "Which was there first, the hen or the egg?" (of course, the right answer is:"the egg"). You can see it both ways. However, experience tells us that you may get a decent mathematical description of a problem and still get the wrong solution. That's not so much because the math doesn't work out, but because any type of problem we deal with is simplified in some sense, so that the equations you are solving are not really valid.

One thing we can agree on, from physical experience rather than mathematical proof (there is none), is the conservation of mass, momentum and energy. At least in Newton's model of continuum mechanics, which has proven very accurate for many macroscopic cases of interest.

To find a mathematical model that exactly preserves this fundamental physical property is a great achievement. I'll later give you a commonly encountered example where this feature is essential.

>Finite Difference, Finite Volume and Finite Element
:can be though like derived from a similar framework
:(the Weighted Residual Method changing the weight
:function).

That's an overly simplified view, but, yes, you can see it that way, hiding the concept behind mathematical details.

>Finite Volume is based on the integration of the
:transport equation. This integration is divided in
:sub-volumes. In this point the problem is not close,
:and the only way to continue is to go to Taylor and
:approximates the derivatives in the same way like >Finite Difference.

That's not quite accurate. With the FVM,derivatives come into play only when computing viscous flow (remember, we are solving the integral equations, not the differential ones). The problem in the FVM is posed a little differently: How to approximate the fluxes on cell faces using the flow solution within the cell. Again, mathematical procedures inspired by physics of flow have been proven very useful, just to name the family of Godunov schemes as one example. When it comes to really evaluating flow gradients, I think the Green-Gauss approach is more popular and more in line with the Finite Volume paradigm, especially on unstructured FEM-type grids. Of course you can draw parallels to Taylor on cartesian grids, I agree with that. I think the reason why people commonly refer to the FVM as a subset of FEM, and less related to FDM, is rather a question of concept, not so much about mathematical details.

>While, in the case of FEM the key is the use of a
:projection concept to expand the solution. Thus, I >think that it is better to said that FVM is closer to >FDM than FEM.

Take FEM, choose the correct shape functions (usually uniform), and the correct weighting, and you will arrive at the FVM. FVM is FEM's little brother. Compare the DNA :)

>However, FVM is still conserving everything including >the errors, of course. Thus, if it is not able to
:fulfill the physics what sense has to said that it is
:conservative?

Sorry, but I really have to say that this either nonsense or an unfortunate mis-formulation of what you really meant to say.

The FVM will get you a solution to the discretized integral equations, whose error is a question of consistency and convergence, not conservation. Any discretization method will give you discretization errors, that's a no-brainer. However, the FVM guarantees that this discretization error will not mess up the most important and most fundamental part of your problem: Conservation. Neither FDM nor FEM give you this guarantee. In other words, in some case that I will explain below, you can get a completely converged solution with FDM or FEM, which is not even conservative, let alone accurate!

The problem with conservation is this: The FDM, being based on the differential equations, fails whenever the differential equations fail. Example: in compressible continuum flow, a compression shock is a discontinuity. Derivatives across the shock are not mathematically defined, and it is hard (or impossible) to get a reliable solution from FDM. Physically, to get the correct jump across the shock, exact conservation is necessary! FEM will have similar problems as FDM, whenever continuous shape functions are used. There are efforts to solve this problem and make the FEM work for discontinuous flow, something the FVM does naturally.

Never forget that the true equations are the integral conservation laws. The differential equations, derived from the integral ones, are valid only in the case of continuous flow, everywhere in the flow field. That's the reason why a direct implementation of the integral equations (FVM) is valid over a wider range of problems. I am biased towards compressible flow because I happen to work on aerospace and turbomachinery applications. That's why I prefer methods that work favorably over a wide range of the Mach scale. As I said before, if you are familiar with the flows you are dealing with, it will help you decide which method will work best for you. That may or may not be FVM. It's really a question of applicability.

 scar April 8, 2005 06:12

Re: finite volume

MANI> Sounds like the old question: "Which was there first, the hen or the egg?" (of course, the right answer is:"the egg"). You can see it both ways. However, experience tells us that you may get a decent mathematical description of a problem and still get the wrong solution. That's not so much because the math doesn't work out, but because any type of problem we deal with is simplified in some sense, so that the equations you are solving are not really valid. One thing we can agree on, from physical experience rather than mathematical proof (there is none), is the conservation of mass, momentum and energy. At least in Newton's model of continuum mechanics, which has proven very accurate for many macroscopic cases of interest. To find a mathematical model that exactly preserves this fundamental physical property is a great achievement. I'll later give you a commonly encountered example where this feature is essential. I agree with you. You probably misunderstood my point or I expressed it wrongly. MANI>That's an overly simplified view, but, yes, you can see it that way, hiding the concept behind mathematical details. Take FEM, choose the correct shape functions (usually uniform), and the correct weighting, and you will arrive at the FVM. FVM is FEM's little brother. Compare the DNA :) Following your line, choose Dirac delta functions and you get FDM. Thus, you see all are brothers!!. In particular, these ideas can be expressed in a general form in framework of the Method of Weighted Residual. MANI>The problem with conservation is this: The FDM, being based on the differential equations, fails whenever the differential equations fail. Example: in compressible continuum flow, a compression shock is a discontinuity. Derivatives across the shock are not mathematically defined, and it is hard (or impossible) to get a reliable solution from FDM. Physically, to get the correct jump across the shock, exact conservation is necessary! FEM will have similar problems as FDM, whenever continuous shape functions are used. There are efforts to solve this problem and make the FEM work for discontinuous flow, something the FVM does naturally.

In a compressible flow, the occurrence of a shock wave, modeled like a discontinuity, divides the domain in two domains. In each domain the transport equations are still valid. The shock wave imposes a jump condition which can be considered like a common boundary condition between each domain. A more convenient approach is to consider properties which are continuous through the shock wave, and this case you can apply again for example FDM. In this case, physics helps you to find the correct variables to use in your problem.

MANI> The FVM will get you a solution to the discretized integral equations, whose error is a question of consistency and convergence, not conservation. Any discretization method will give you discretization errors, that's a no-brainer. However, the FVM guarantees that this discretization error will not mess up the most important and most fundamental part of your problem: Conservation. Neither FDM nor FEM give you this guarantee. In other words, in some case that I will explain below, you can get a completely converged solution with FDM or FEM, which is not even conservative, let alone accurate!

Regrettably, although FVM is based on an integral form of the equation, this does not imply that FVM will conserve the properties dictated by the differential equation. In any stage of the method (and of any method) you need to introduce approximations, that is life!!. After performing the integration of the differential equation, dividing this integral in sub-volumes, and applying Gauss, the derivatives must be approximated (using physically arguments or not). In general, in the case of FVM the flux through a face is written like a function of the values of the function in the cell of the neighbour volumes. That physically implies that "what is going out from one volume is going to the next", i.e. forcing the conservation, and this is an approximation. So although you start from the integral conservative form, you need to introduce approximations that can affect the conservation statements. However, due to the nature of the approximation (low order or high order) used to force the conservation the problem is modified. An example is the introduction of numerical diffusion in the case of an upwind scheme. Thus, you end up with a conservative solver of a different problem. Of course, this is the general picture of all the low order methods.

 Mani April 8, 2005 13:11

Re: finite volume

>In a compressible flow, the occurrence of a shock >wave, modeled like a discontinuity, divides the domain >in two domains. In each domain the transport equations >are still valid.

Sure, but that doesn't help you much if you don't get the shock location and jump conditions right. And for that you need conservation. There's no way around it.

>The shock wave imposes a jump condition which can be >considered like a common boundary condition between >each domain. A more convenient approach is to consider >properties which are continuous through the shock >wave, and this case you can apply again for example >FDM.

The trouble is, you don't know the shock position a priori. A non-conservative method will neither give you the right shock position nor the right shock strength. So, knowing that there must be a shock somewhere, and knowing that away from the shock, the differential equations are valid doesn't help you. You would need a method to predict the shock location. Of course this could be done in some heuristic way, but these complications can be avoided altogether by using a conservative method that naturally solves the integral equations. Just face it: That's the reason why the FVM has found wide-spread use in compressible flow simulation. I don't know why people tend to see different methods in competition. Every method has its use... for different problems. Methods that are of no use don't survive. Of course the discretized equations will not exactly solve the original equations. The same can be said about any numerical method. Maybe it's not clearly understood what I was trying to say: It is preferable (and sometimes essential) to have a numerical method that at least observes certain principles in form of a constraint. In the case of fluid flow, this constraint is "conservation".

Let me give another example. If you perform numerical simulations on structural deformation, it is preferable that your Finite Element Method be able to exactly reproduce rigid-body displacements. Why? Because the deformation of each finite element can be described as a superposition of rigid-body displacement and element-deformation. You will often find that the rigid-body displacement contributes more to the change of the element, than the actual deformation. It is therefore good if you can at least describe that major part exactly. Doesn't that make sense? Of course, it doesn't mean that it's a necessary condition. Of course, it also doesn't mean that you suddenly created an exact numerical method (there is no such thing). But you will see that your numerical method is more robust and more accurate than without the constraint. You'll still get only an approximate solution, but you have significantly increased the chance that it's physically meaningful. And that's the point.

 Tom April 9, 2005 06:02

Re: finite volume

Actually the position, strength and speed of the shock can be determined by adding viscosity to the finite difference formulation and not worrying about whether the scheme is conservative or not. After all the Rankine-Hugoniot conditions across the shock are the artifact of taking the limit of zero-viscosity. The reason your conservative scheme (conservation is not enough by itself here) captures the shock is that you're adding a large amount numerical viscosity (such as in a TVD scheme) near to the shock.

Some further points:-

(i) If you perform a convergence study the conservative scheme may not converge to the true solution as quickly as the nonconservative one (pointwise)

(ii) FVM can be derived from either FEM of FDM just as FDM can be considered a special case of FEM.

(iii) Whether FDM is conservative or not really depends upon which version of the equations you discetize (point ii).

(iv) FVM only tends to conserve the first moments of the quantities you are solving for. What about enstrophy? many geophysical fluid dynamicists would argue that for a long time simulation you'll need to conserve this as well? (Arakawa showed how to do this with finite differences in the 60's).

 Mani April 9, 2005 21:04

Re: finite volume

I am not sure if adding viscosity to a nonconservative scheme would be enough to capture the shock. It should be conservative to get the Rankine-Hugoniot conditions right. For one, the scheme has to be consistent in the sense that, if it converges, it will converge to the correct, conservative solution. But then there is also the question if you can guarantee that the nonconservative scheme will arrive at a unique (?) conservative solution.

Besides, shocks exist in inviscid flow as well, and that's where they are really discontinuities. Adding viscosity is a little like changing the subject. Of course, when viscosity has a smoothing effect, the flow derivatives may then be defined and even the differential equations are valid.

Your comment on artificial dissipation is right, but that's rather a question of getting the wave propagation right, in the supersonic and subsonic parts of the domain, regardless of natural diffusion. You could say, artificial dissipation is added in order to locally remove (reduce) the error you made in the basic discretization.

Then let's say you get your nonconservative scheme to work for transonic flow. If the convergence is faster or slower than that of a conservative scheme should very likely depend on the two schemes you are comparing and on the problem you are solving. I find it hard to believe that you can give general proof of your statement that nonconservative schemes are faster. That would certainly be worth considering.

 dalton April 10, 2005 03:59

Re: finite volume

Too much argue. But no one solve the problem. Words don't represent action. Are you using FVM to solve the practical problem? Some disadvantages of FVM: (1)too much trick for face flux treatment? (2)calculated result is not good enough for source term with large gradient or for small time step. FVM is indeed a bad approach.

 Tom April 10, 2005 05:35

Re: finite volume

The two representations (conservative and nonconservative) must converge to the same solution - if they do not then either the solution does not exist, in which case both numerical solutions are wrong, or one of the representations is mathematically inconsistent. After all they are different representations of the same pde.

For a shock to exist in an inviscid flow you must give meaning to the discontinuity. The standard way to do this is to consider the viscous solution in the limit of zero viscosity (see for example Linear and nonlinear waves by G.B. Whitham). Any "inviscid" scheme which captures the shock must be doing it through artificial/numerical viscosity.

 Mani April 10, 2005 14:25

Re: finite volume

Too much of an argument, no solution? :)

I think you are right, but that's not surprising. As I have tried to say throughout the discussion, there is no scheme that can be claimed to be the "best" in a general sense. Anyone who is trying to say so just reveals his lack of experience. Just because some schemes didn't work out well in his experience, doesn't necessarily mean they are of no use in general. If methods have found such widespread use as the FEM, FDM, and FVM, you may assume that all of them (and others) are worth considering.

Nonphysical diffusion is apparent in all grid-based Eulerian methods, regardless which method you use and regardless if it's needed for shock capturing or not. For shock capturing, you may control it by trying to get wave propagation right with the use of higher-order upwinding schemes (not the first-order TVD), or you may just add excessive artificial diffusion to stabilize your scheme. The option of a gridless method wasn't mentioned in the original question, so it's been left out of the discussion.

Back to the original question, just try to find out which method has proven applicable to your particular problem. For any of the methods mentioned you will find tons of published journal papers over decades of research. It is rare to find someone who is expert in all methods, and even if he is, it's probably just for a very particular field of fluid dynamics. It's been fun trying to kick of a discussion in a forum like this, but in the end you'll need to survey the literature for a more balanced and competent view of the subject.

 dalton April 10, 2005 21:47

Re: finite volume

> tons of published journal papers over decades of research

Agree with you. Most of them is useless, actually. Some one can claim that the problem has been solved by his paper or his method. But indeed, problem still exists. Much of time is wasted on these kinds of papers.

 diaw April 11, 2005 03:32

Re: finite volume

I believe that the best way to 'come to grips' with FVM & FEM is to do advanced study courses at a post-graduate level - preferably in parallel. This forces one to really get to the essence of how each method is constructed.

Each method will have its strong & weak points, as has been mentioned before... Most papers seem to be either application-centred, or at a very theoretical level... neither helps most folks to answer many of the fundamental questions.

Good authors help to sift through the waffle & put down the essentials in a form that can be easily assimilated.

diaw...

 dalton April 11, 2005 20:36

Re: finite volume

Good point, diaw

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