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PROCEEDINGS OF THE ELEVENTH INTERNATIONAL CONFERENCE ON CIVIL, STRUCTURAL AND ENVIRONMENTAL ENGINEERING COMPUTING
Edited by: B.H.V. Topping
Symmetric Finite Element Formulation using Linear Algebra and Canonical Forms: Truss and Frame Elements
A. Kaveh and M. Nikbakht
Iranian Academy of Sciences
A. Kaveh, M. Nikbakht, "Symmetric Finite Element Formulation using Linear Algebra and Canonical Forms: Truss and Frame Elements", in B.H.V. Topping, (Editor), "Proceedings of the Eleventh International Conference on Civil, Structural and Environmental Engineering Computing", Civil-Comp Press, Stirlingshire, UK, Paper 130, 2007. doi:10.4203/ccp.86.130
Keywords: finite elements, symmetry, truss element, frame element, canonical forms, decomposition.
In this paper a new computational approach is presented for finding the matrices of elements in the finite element method (FEM), using the symmetry analysis of each element. In this method, we first adapt the appearance of the element and its degrees of freedom with one of the canonical symmetry forms which are well-known in linear algebra. This is done by the means of an appropriate numbering and sign convention. Then, we use the properties of the canonical forms in order to decompose the element into a number of sub-elements. This reduces the number of DOFs which are involved in forming the matrices of the element. In other words, we decompose the vector space of the first problem into a number of independent subspaces with smaller orders. Each of the resulted subspaces is physically associated with a symmetry type of the structure (this is the meaning of the symmetry analysis through which we decouple different symmetry modes of a symmetrical system). We use the concept of the symmetry type of each subspace and the decomposition of the overall shape function of the element into a number of sub-functions, each of which corresponds to the symmetry type of one of the subspaces (e.g. symmetric and anti-symmetric terms). When such a decomposition is valid and each sub-element has its own shape function, it will be very easy to form the matrices of each sub-element by means of one of the conventional methods - such as potential energy method - using its own shape function. Finally, we combine the matrices of different sub-elements, based on the properties of the canonical forms, and construct the matrix of the original element.
The method is originally inspired by group theoretical methods which are presented in the literature, but the present approach involves less computational time and effort and relatively less judgment is needed in this method, compared to the pure group theoretical approach. Combination of matrices of sub-elements and forming the matrix of the main element is much easier and more direct in this method, and in the case of elements with odd number of nodes, this approach seems to be more adaptable.
The present method can be more helpful in the case of complex elements having a number of nodes, where usually one of the canonical forms of symmetry exists, however, in this paper only the formulation for simple truss and beam elements are derived, since the focus of the paper is on the general concepts. It should be noted that in the case of more complex elements, the same steps are involved. As an example this idea can easily be applied to three-node and five-node line elements, where the symmetry of the element has the canonical Form III symmetry.
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