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A grouptheoretic formulation for symmetric finite elements
, 2005
"... After a brief review of the literature on applications of group theory to problems exhibiting symmetry in structural mechanics, an efficient formulation is presented for the computation of matrices for symmetric finite elements. The theory of symmetry groups and their representation allows the vecto ..."
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After a brief review of the literature on applications of group theory to problems exhibiting symmetry in structural mechanics, an efficient formulation is presented for the computation of matrices for symmetric finite elements. The theory of symmetry groups and their representation allows the vector spaces of the element nodal freedoms to be decomposed into independent subspaces spanned by symmetryadapted element nodal freedoms. Similarly, the displacement field of the elements is decomposed into components with the same symmetry types as the subspaces. In this way, element shape functions and element matrices are computed separately within each subspace, through the solution of a much smaller system of equations and the integration of a much simpler set of functions. The procedure is illustrated with reference to the computation of consistentmass matrices for truss, beam, planestress, platebending and solid elements.A numerical example has been considered in order to demonstrate the effectiveness of the approach. Overall, the formulation leads to significant reductions in computational effort in comparison with the conventional computation of element matrices. However, the proposed formulation only becomes really advantageous in the case of finite elements with a high degree of symmetry (typically solid hexahedral elements), and a large number of nodes and nodal degrees of freedom.
Application of Lie transformation group methods to classical theories of plates and rods
 Int. J. Solids Structures (submitted
"... Abstract—In the present paper, a class of partial differential equations related to various plate and rod problems is studied by Lie transformation group methods. A system of equations determining the generators of the admitted point Lie groups (symmetries) is derived. A general statement of the ass ..."
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Abstract—In the present paper, a class of partial differential equations related to various plate and rod problems is studied by Lie transformation group methods. A system of equations determining the generators of the admitted point Lie groups (symmetries) is derived. A general statement of the associated groupclassification problem is given. A simple intrinsic relation is deduced allowing to recognize easily the variational symmetries among the ”ordinary ” symmetries of a selfadjoint equation of the class examined. Explicit formulae for the conserved currents of the corresponding (via Noether’s theorem) conservation laws are suggested. Solutions of groupclassification problems are presented for subclasses of equations of the foregoing type governing stability and vibration of plates, rods and fluid conveying pipes resting on variable elastic foundations and compressed by axial forces. The obtained groupclassification results are used to derive conservation laws and groupinvariant solutions readily applicable in plate statics or rod dynamics. 1.
Symmetry Groups and Conservation Laws in Structural Mechanics
, 2000
"... this paper: Greek (Latin) indices have the range 1, 2 (1, 2, 3), unless explicitly statedoedfi'fifi2 and the usual summatio co ventio o ver a repeated index is emplo yed. The kthoh[ partial derivativeso f a dependent variable, say w, that is # k w/#x # 1 # 2 ...#x # k (k, # 1 , 2,i ..."
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this paper: Greek (Latin) indices have the range 1, 2 (1, 2, 3), unless explicitly statedoedfi'fifi2 and the usual summatio co ventio o ver a repeated index is emplo yed. The kthoh[ partial derivativeso f a dependent variable, say w, that is # k w/#x # 1 # 2 ...#x # k (k, # 1 , 2,icc# k 2,cS ), are deno=A either by w # 1 # 2 ...# k o w x # 1 x # 2 ...x # k , where x 1 2,iT are the independent variables. A similar no:A2A[ is used fo the partial derivativeso f any o2: functioO say f ,o f the independent variables but, in this case, the indices indicating the di#erentiatio are preceded by a co:G D # (# =1, 2,#j ) deno1 thetoG derivative oeratoG Fo r the basic nofi13: and statements used in thegroG analysiso f di#erentialequatioA and variatioOG proG:G3 see [1]o r [2]. P.A.Djo3A=: 2 Symmetries and conservation laws of beam equations BernoulliEE1 beams