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CivilComp Proceedings
ISSN 17593433 CCP: 83
PROCEEDINGS OF THE EIGHTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 178
Extension of the Fixed Grid Finite Element Method to ThreeDimensional Analysis F.S. Maan, O.M. Querin and D.C. Barton
School of Mechanical Engineering, University of Leeds, United Kingdom F.S. Maan, O.M. Querin, D.C. Barton, "Extension of the Fixed Grid Finite Element Method to ThreeDimensional Analysis", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Eighth International Conference on Computational Structures Technology", CivilComp Press, Stirlingshire, UK, Paper 178, 2006. doi:10.4203/ccp.83.178
Keywords: fixed grid, finite element analysis, meshless methods.
Summary
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 fournode 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. References
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