Why not use FDM in solid mechanics?
Finite Difference Method (FDM) and the related techniques such as FVM, are often found put to great use in fluid mechanics. See any simulation showing not only streamlines but also vortex shedding, turbulent mixing, etc.
Yet, when it comes to solid mechanics, Finite Element Method (FEM) is most often the method of choice. Actually, FEM is probably the *only* computational method used in solid mechanics. Most books on solid mechanics and structural analysis do not even mention FDM. A few that do, restrict FDM only to the Laplace's equation and the bi-harmonic equations--not to the general stress analysis problem in 3D.
Why is this so?
If you really think hard about it, you can see that the usual arguments forwarded in favor of FEM and against FDM really do not hold--not at least to the extent this is routinely supposed.
For example, consider the one big advantage usually ascribed to FEM, namely, its ability to handle complex geometries and non-rectangular meshes.
But if the first reason (complex geometries) really is the cause, then why does FEM formulation of contact stresses require a special treatment?
And what is the true relevance of the second reason (non-rect. meshes of FEM) in this day and age of abundance of memory? You can always use a little finer mesh, esp. near curved surfaces, isn't it? In any case, shouldn't it be just a matter of a little more theoretical development of FDM? (Think FVM and the related developments here.)
So, it should be possible to use FDM in Solid Mechanics--especially because FDM is so simple. Yet, it is not. Why not? Is it just a matter of predominant culture in research since the 1960's?
You are welcome to post all your thoughts. However, please let them be really well thought out. Above all, please do not write under the impression that I do not like or use FEM. I do. Also, do not write to advice me where to really pick up FEM from. I think I know that too...
It's just that I can see acertain clear conceptual advantage with FDM that is absolutely not present with FEM.
Thanks in advance for your *well thought out* replies. In particular, thanks in advance for *not* just reproducing the same opinions you heard or read in your graduate courses on FEM, numerical analysis, or solid mechanics. Thanks for *that*, really!