# MFS with RBF for Thin Plate Bending Problems on Elastic Foundation

## Abstract

In this chapter a meshless method, based on the method of fundamental solutions (MFS) and radial basis functions (RBF), is developed to solve thin plate bending on an elastic foundation. In the presented algorithm, the analog equation method (AEM) is firstly used to convert the original governing equation to an equivalent thin plate bending equation without elastic foundations, which can be solved by the MFS and RBF interpolation, and then the satisfaction of the original governing equation and boundary conditions can determine all unknown coefficients. In order to fully reflect the practical boundary conditions of plate problems, the fundamental solution of biharmonic operator with augmented fundamental solution of Laplace operator are employed in the computation. Finally, several numerical examples are considered to investigate the accuracy and convergence of the proposed method.

## Keywords

Radial Basis Function Thin Plate Boundary Element Method Elastic Foundation Meshless Method
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. Bittnar Z, Sejnoha J (1996) Numerical Methods in Structural Mechanics, ASCE Press, New YorkGoogle Scholar
2. Fairweather G, Karageorghis A (1998) The method of fundamental solutions for elliptic boundary value problems. Adv Comput Math 9:69–95
3. Ferreira AJM (2003) A formulation of the multiquadric radial basis function method for the analysis of laminated composite plates. Comput Struct 59:385–392
4. Golberg MA, Chen CS, Bowman H (1999) Some recent results and proposals for the use of radial basis functions in the BEM. Eng Anal Bound Elem 23:285–296
5. Karthik B, Palghat AR (1999) A particular solution Trefftz method for non-linear Poisson problems in heat and mass transfer. J Comput Phys 150:239–267
6. Katsurada M (1994) Charge simulation method using exterior mapping functions. Japan J Indust Appl Math 11: 47–61
7. Kupradze VD, Aleksidze MA (1964) The method of functional equations for the approximate solution of certain boundary value problems. USSR Comput Math Phys 4:82–126
8. Leitao VMA (2001) A meshless method for Kirchhoff plate bending problems. Int J Numer Meth Engng 52:1107–1130
9. Liu Y, Hon YC, Liew KM (2006) A meshfree Hermite-type radial point interpolation method for Kirchhoff plate problems. Int J Numer Meth Engng 66:1153–1178
10. Long SY, Zhang Q (2002) Analysis of thin plates by the local boundary integral equation (LBIE) method. Eng Anal Bound Elem 26:707–718
11. Martin HC, Carey GF (1989) The Finite Element method (4th Edition). McGraw-Hill, LondonGoogle Scholar
12. Misra RK, Sandeepb K, Misra A (2007) Analysis of anisotropic plate using multiquadric radial basis function. Eng Anal Bound Elem 31:28–34
13. Nerantzaki MS, Katsikadelis JT (1996) An analog equation solution to dynamic analysis of plates with variable thickness. Eng Anal Bound Elem 17:145–152
14. Nitsche LC, Brenner H (1990) Hydrodynamics of particulate motion in sinusoidal pores via a singularity method. AIChE J 36:1403–1419
15. Qin QH (2000) The Trefftz Finite and Boundary Element Method. WIT Press, Southampton
16. Rajamohan C, Raamachandran J (1999) Bending of anisotropic plates by charge simulation method. Adv Eng Softw 30:369–373
17. Sun HC, Yao WA (1997) Virtual boundary element-linear complementary equations for solving the elastic obstacle problems of thin plate. Finite Elem Anal Des. 27:153–161
18. Sun HC, Zhang LZ, Xu Q et al. (1999) Nonsingularity Boundary Element Methods. Dalian University of Technology Press, Dalian (in Chinese)Google Scholar
19. Wang H, Qin QH (2006) A meshless method for generalized linear or nonlinear Poisson-type problems. Eng Anal Bound Elem 30:515–521
20. Wang H, Qin QH (2007) Some problems with the method of fundamental solution using radial basis functions. Acta Mechanica Solida Sin 20:21–29Google Scholar
21. Wang H, Qin QH, Kang YL (2005) A new meshless method for steady-state heat conduction problems in anisotropic and inhomogeneous media. Arch Appl Mech 74:563-579
22. Wang H, Qin QH, Kang YL (2006) A meshless model for transient heat conduction in functionally graded materials. Comput Mech 38:51–60
23. Yao WA, Wang H (2005) Virtual boundary element integral method for 2-D piezoelectric media. Finite Elem Anal Des 41:9–10
24. Young DL, Jane SJ, Fan CM, et al. (2006) The method of fundamental solutions for 2D and 3D Stokes problems. J Comput Phys 211:1–8
25. Zhang FF (1984) The elastic thin plates. Science Press of China, Beijing (in Chinese)Google Scholar

## Authors and Affiliations

1. 1.Department of EngineeringAustralian National UniversityCanberraAustralia
2. 2.College of Civil Engineering and ArchitectureHenan University of TechnologyZhengzhouChina
3. 3.Department of Mechanical EngineeringAcademy of Armed Forces of General M. R. ŠtefánikLiptovský MikulášSlovakia