Computational field of solid mechanics is mostly dominated by FEM. This is a well known topic.
But is it that FDM (finite difference method) has no chance of surviving
in solid mechanics?
What are the present developments in the use of FDM in case of elasticity analysis of solid body?
I would like to mention some of the recent research works on this topic here.....
Recently a new mathematical model has been developed by Prof. S. Reaz Ahmed and his students known as the "displacement potential approach".
In the literature, FEM has sometimes been characterized as a local approach, but IMO this needs to be corrected.
The piecewise continuous trial-functions of FEM can be looked at from two different viewpoints:
(i) If FEM is seen as an expansion method making use of basis functions, then naturally the comparison is with the Fourier-theoretic approaches (and all the derived, consequent or similar ones). The basis functions for the latter are global in the sense they have supports all over the domain. This, indeed, is unlike the limited (piecewise) support of the FEM trial-functions.
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?