Keywords: fixed grid finite element analysis, fea, eigenvalue, natural frequency, buckling.

This paper shows the development of the Fixed Grid Finite Element Analysis (FGFEA) method to the solution of eigenvalue (natural frequency and buckling) problems. An efficiency and robustness comparison between FGFEA and FEA is gained through the analysis of the plate with a central hole example. This example is analysed at various mesh densities to determine the lowest buckling load and first natural frequency. These results establish a correlation between the two methods, such that the extension of FGFEA to eigenvalue analysis is substantiated.

In order to address such problems a number of analysis methods have been developed over recent decades, where the boundary of the medium is disassociated from the mesh. These methods are termed meshless or meshfree methods, as the discretisation process is not undertaken. This is achieved by distributing nodes or analysis points over the domain of the medium and forming subdomains for the nodes, where numerical analysis is then undertaken. The Reproducing Kernel Particle (RKPM) [1], Element Free Galerkin (EFGM) [2] and Finite Spheres [3] methods are just some of the main meshfree methods. These have been effectively used in the fields of nonlinear large deformation problems [4], fracture analysis [5] and real time surgical simulations [3].

An alternative analysis method that is also considered as part of the meshfree methods, but which has not been researched in much detail, is the FGFEA method [6]. The discretisation process consists of superimposing a fixed rectangular grid of regular sized elements over the domain of the medium. The elements are then classified into three distinct types, by considering their location within the domain of the discretised medium. The three elements types are defined as: Inside (I), Outside (O) and Boundary (B) elements. The locations of these elements are inside, on the boundary, and outside of the medium, respectively.

As the grid is independent of the medium, any geometrical changes to the medium do not degrade the elements. This method has been successfully implemented with the Genetic Algorithm (GA) [7] and Evolutionary Structural Optimisation (ESO) [8] design methods. However, the only element used in all FGFEA was the two dimensional four-node bilinear rectangular plane strain element, with eight degrees of freedom. Therefore, the necessity for this method has been proven in structural optimisation where it is increasingly used, but there is still great scope for the advancement of the method.

In this paper the FGFEA method was extended by the incorporation of a three dimensional four node element, with six degrees of freedom at each node. The method was then further advanced to a natural frequency and buckling eigenvalue analysis of a structure, using the newly incorporated element. The analysed structure was a plate with a central hole, which was simply supported and had a distributed load on the vertical outer edge for the buckling analysis. The lowest buckling load and the first natural frequency were calculated, for the respective analyses, at four increasingly refined mesh densities. These results showed that convergence occurred as the mesh density was increased. The validity of the results was established by comparing the results with those from a conventional FEA of the structure. A good correlation between the two methods with a low percentage error was obtained, and FGFEA actually provided a more accurate answer at the lowest mesh density. Therefore, the use of FGFEA with a three dimensional element for eigenvalue analysis is substantiated in this paper.

1

Liu, W.K.; Jun, S.; Zhang, Y.F. "Reproducing kernel particle methods." Int.J. Numer. Meth. Fluids 20, pp.1081-1106, 1995. doi:10.1002/fld.1650200824

2

Belytschko, T., Lu, Y.Y. and Gu, L., "Crack Propagation by Element Free Galerkin Methods", Engineering Fracture Mechanics, Vol.51, No.2, 1995. doi:10.1016/0013-7944(94)00153-9

3

De, S., Hong, J.W., Bathe, K.J., "On the method of finite spheres in applications: towards the use with ADINA and a surgical simulator", Computational Mechanics, Vol. 31, pp. 27-37, 2003. doi:10.1007/s00466-002-0390-3

4

Chen, J.S., Pan, C., and Wu, C.T., "Reproducing Kernel Particle Methods for large deformation analysis of non-linear structures", Computer Methods in Applied Mechanics and Engineering, Vol. 139, pp. 195-227, 1996. doi:10.1016/S0045-7825(96)01083-3

5

Belytschko, T., Lu, Y.Y. and Gu, L., "Element Free Galerkin Methods", Int. J. Numer. Methods Engrg., Vol.37, pp. 229-256, 1994. doi:10.1002/nme.1620370205

6

Garcia, M.J., "Fixed Grid Finite Element Analysis in Structural Design and Optimisation", PhD., Dept. of Aeronautical Eng., University of Sydney, 1999.

7

Woon, S.Y., Querin, O.M., Steven, G.P., "Application of the Fixed-Grid Method to Step-Wise GA Shape Optimisation", 2nd ASMO UK / ISSMO conference on Engineering Design Optimisation, Swansea, Wales, UK, July 10-11, pp. 265-272, 2000.

8

Garcia, M.J., Ruiz, O.E., Steven, G.P., "Engineering Design Using Evolutionary Structural Optimisation Based on Iso-Stress-Driven Smooth Geometry Removal", NAFEMS (Intl Assoc. Eng. Analysis Community) World Congress, Lake Como, Italy. April 24-29, 2001.

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