Keywords: fixed grid, finite element analysis, meshless methods.

The discretisation of a medium by the finite element analysis (FEA) method results in the generation of a set of elements that are formed from the boundary of the structure. Therefore, a direct relationship is created between the boundary and the mesh, such that any changes in the boundary are also produced in the mesh. Significant boundary changes can necessitate the adaptation or regeneration of the mesh, as it no longer provides a correct representation of the medium. In addition to this, the discretisation duration is significantly increased as the geometrical complexity of the medium is increased. These factors incorporate a considerable amount of inefficiency in the FEA process, particularly when multiple regenerations are required, which is a problem in the field of structural optimisation.

An alternative method to FEA is the fixed grid FEA (FGFEA) method [1], where the boundary of the medium is disassociated from the mesh. 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 structure, the boundary is free to deform without inducing any element degradation. The material properties of the elements are defined as a function of the element location, and are modified as required, to reflect any such geometrical changes.

The method has been successfully implemented with a genetic algorithm (GA) [2] and evolutionary structural optimisation (ESO) [3] design methods. Initially, the only element used in all FGFEA was the two dimensional four-node bilinear rectangular plane strain element, with eight degrees of freedom. This has been extended by the incorporation of three new 4 node element types: 2D axisymmetric and plain strain [4] and a 20 DOF shell [5]. Example structures were analysed at several increasingly refined mesh densities and a comparison of the results was drawn with traditional FEA. A high correlation of results was achieved across the density range. Therefore, the necessity for this method has been proven in 2D structural optimisation where it is increasingly used, but there is still great scope for the advancement of the method, particularly in 3D analysis.

In this paper the FGFEA method was extended to 3D analysis by the incorporation of a four node brick element with three degrees of freedom at each node. The ESDU rod with a fillet example [6] was analysed using a fixed grid (FG) and traditional FEA to determine the maximum stress concentration factor. Error analysis was undertaken and the resulting correlation defined a high degree of agreement between the two methods, where FGFEA was only 5.79% less accurate than traditional FEA. The error, in comparison to the theoretical value, was -28.40% for FEA and -34.19% for FGFEA. This is at an acceptable level as less than 335 elements were used in both models. The theoretical location of the concentration factor is stated as occurring close to the fillet and minimum diameter junction [6] and the results determined occurred locally to this for both methods. Therefore, considering the premise of FGFEA and the low level of error that was achieved at the correct stress location, the initial extension of FGFEA to 3D analysis is substantiated. A further outcome of this research was the derivation of a relationship between 3D FGFEA error and the intersection angle of the feature made with the boundary element. The process defined can be used to assist in the calculation of the required FG mesh sizes.

1

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

2

Kim H, García MJ, Querin O.M, Steven GP and Xie YM. "Introduction of fixed grid in evolutionary structural optimisation", Engineering Computations, 17(4), 2000, p 427-439. doi:10.1108/02644400010334838

3

Woon SY, Querin OM, Steven GP. "Application of the Fixed-Grid FEA Method to Step-Wise GA Shape Optimisation", 2nd ASMO UK / ISSMO conference on Engineering Design Optimization, Swansea, Wales, UK, July 10-11, 2000.

4

Maan, FS, Querin OM, Barton DC, "Incorporation of Plain Strain and Axisymmetric Elements into Fixed Grid Finite Element Analysis", The 4th Australasian Congress on Applied Mechanics, Melbourne, Australia, February 16-18, 2005.

5

Maan, F.S., Querin O.M., Barton D.C., "Extension of the Fixed Grid Finite Element Method to Eigenvalue Problems", The Seventh International Conference on Computational Structures Technology, Civil-Comp Press, Stirling, UK, 2004, pp. 281-282. doi:10.4203/ccp.79.128

6

ESDU, "890048 Elastic Stress Concentration Factors, Geometric Discontinuities in Rods and Tubes of Isotropic materials", ESDU, 1989, pp. 8.

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