Meshless FD method over FEM?
 

Hi all,

Recent meshless methods are proving to be a better way of solving nonlinear equations in CFD. I have used Least Square based Finite Difference and it proved to be a great tool.

I have read in many discussions that FD has a disadvantage that it cant be applied to complex geometries. Now, I have a doubt with this. If FD can solve complex geometries easily, does it mean that meshless FD is better than FEM?

(In Least Square Based FD, the derivatives can be approximated to a higher order easily)

Please clarify,

Regards, Shyam

Re: Meshless FD method over FEM?
 

Hi, Shyam:

Could you introduce some reference on meshless FD?

Thanks.

- Ma

Re: Meshless FD method over FEM?
 

Here is a list of references on meshless methods in CFD. There is a growing body of literature in structural mechanics, see IJNME, JNMF, and Comp. Meth. Appl. Mech. Engg.

<h3> Some references on Meshless Methods </h3>

1. T Liszka, "An interpolation method for an irregular net of nodes", Int. Jl. for Num. Meth. in Engg., 20:1599-1612, 1984.
2. T Liszka and J Orkisz, "The finite difference method at arbitrary irregular grids and its application in applied mechanics'', Comp. Struct., 11:83-95, 1980.
3. KC Chung, "A generalised finite difference method for heat transfer problems of irregular geometries'', Num. Heat Transfer, 4:345-357, 1981.
4. EJ Kansa (1990): "Multiquadrics- A scattered data approximation scheme with applications to computational fluid dynamics: II. Solutions to parabolic, hyperbolic and elliptic partial differential equations", Comput. Math. Appl., 19(6-8):147-161.
5. JT Batina, "A gridless Euler/Navier-Stokes solution algorithm for complex aircraft applications'', AIAA Paper 93-0333, 1993.
6. T Belytschko, YY Lu, and L Gu, "Element-free Galerkin methods", Int. Jl. Num. Meth. Engg., 37:229-256, 1994.
7. AK Ghosh, SM Deshpande, "Least squares kinetic upwind method for inviscid compressible flows'', AIAA paper 95-1735,1995.
8. Yagawa G and Yamada T, "Free mesh method: A new meshless finite element method'', Comp. Mech., 18:383-386, 1996.
9. T Belytschko, Y Krongauz, D Organ, M Flemming and P Krysl, "Meshless methods: An overview and recent developments", Comput. Methods Appl. Mech. Engg., Vol. 139, pp. 3-47, 1996. (This contains a comprehensive review of many meshless techniques.)
10. K Morinishi (1999): "An implicit gridless type solver for the Navier-Stokes equations'', Int. Symp. on CFD, Bremen.
11. H Wendland (1999): "Meshless Galerkin methods using radial basis functions", Math. Comp., 68:1521-1531.
12. M Junk, "Do finite volume methods really need a mesh ?", Int. Workshop on Meshfree Methods for PDE, Bonn, September, 2001.
13. R Lohner, C Sacco, E Onate and S Idelsohn (2002): "A finite point method for compressible flows'', Int. Jl. Num. Meth. Engg., Vol. 53, pp. 1765-1779.
14. Michael Griebel and Marc Schweitzer, ed. (2002): Meshfree methods for Partial Differential Equations, LNCSE, Springer, 2002.
15. G.R. Liu: Meshfree Methods - Moving beyond the Finite Element Method , 712 pages, 2002, CRC Press. ISBN: 0849312388. (This is the first textbook on meshless methods, and till now the only one that I am aware of.)