Truss and Beam Finite Elements Revisited: A Derivation Based on Displacement-Field Decomposition

Zingoni, Alphose

doi:10.1177/026635119601100404

Cited by 24 publications

(19 citation statements)

References 2 publications

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“…First an arbitrary numbering is performed using the standard rules from group theory [30], Figure 3. Then the algorithm of Section 4 is performed.…”

Section: Group-theoretical Methodmentioning

confidence: 99%

“…It is conventional to show the 'irreps' with ( ) and the 'redreps' by ( ) . The process of reduction into irreps, would correspondingly divide the vector space V into a number of group invariant subspaces V ( ) , such that none of these subspaces can be divided into further group invariant subspaces of smaller dimension [30]. In representation theory for symmetry groups, idempotents are projection operators which nullify all vectors of a given vector space other than those which belong to a particular subspace associated with a specific symmetry type.…”

mentioning

confidence: 99%

“…In representation theory for symmetry groups, idempotents are projection operators which nullify all vectors of a given vector space other than those which belong to a particular subspace associated with a specific symmetry type. Therefore, these can be used to transform the normal variables of a problem (spanning the vector space V ) into orthogonal sets, each spanning a subspace V ( ) of the vector space V [30]. Associated with each irrep, there is a group invariant subspace and each idempotent corresponds to one of these subspaces.…”

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confidence: 99%

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Block diagonalization of Laplacian matrices of symmetric graphs via group theory

Kaveh

Nikbakht

2006

Numerical Meth Engineering

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SUMMARYIn this article, group theory is employed for block diagonalization of Laplacian matrices of symmetric graphs. The inter-relation between group diagonalization methods and algebraic-graph methods developed in recent years are established. Efficient methods are presented for calculating the eigenvalues and eigenvectors of matrices having canonical patterns. This is achieved by using concepts from group theory, linear algebra, and graph theory. These methods, which can be viewed as extensions to the previously developed approaches, are illustrated by applying to the eigensolution of the Laplacian matrices of symmetric graphs. The methods of this paper can be applied to combinatorial optimization problems such as nodal and element ordering and graph partitioning by calculating the second eigenvalue for the Laplacian matrices of the models and the formation of their Fiedler vectors. Considering the graphs as the topological models of skeletal structures, the present methods become applicable to the calculation of the buckling loads and the natural frequencies and natural modes of skeletal structures.

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“…First an arbitrary numbering is performed using the standard rules from group theory [30], Figure 3. Then the algorithm of Section 4 is performed.…”

Section: Group-theoretical Methodmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Block diagonalization of Laplacian matrices of symmetric graphs via group theory

Kaveh

Nikbakht

2006

Numerical Meth Engineering

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“…This matrix is actually a permutation of the identity matrix. The eigenvalue decomposition of this matrix is required and can be calculated analytically (see [31]); hence, matrices L and X used in Equation (15) are simply calculated…”

Section: Eigenvalues Decompositionmentioning

confidence: 99%

On the decomposition of generalized eigenproblems for the free vibration analysis of cyclically symmetric finite element models

Koohestani¹

2009

Numerical Meth Engineering

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SUMMARYFree vibration analysis is a major part of any dynamic analysis. Natural frequencies and related mode shapes may be obtained from free vibration analysis as the solutions of generalized eigenproblems. Although the eigensolutions of large-scale structures require large computational efforts, these solutions may be achieved simply for symmetric structures. We present an efficient method for the decomposition of generalized eigenproblems related to finite element models with cyclic symmetry (having nodes at the axis of symmetry) into eigensubproblems with significantly smaller dimensions. This decomposition is obtained by block diagonalization of a matrix with a special pattern known as a block circulant, using the concept of the Kronecker product and similarity transformations. The proposed method is applied to three finite element models discretized by triangular and four-node quadrilateral plate and shell elements, and its efficiency, accuracy and simplicity are evaluated.

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“…Such techniques have been widely used in chemistry and physics [13][14][15], but in structural mechanics, the applications have not been extensive as other fields of science. However, the method has been successfully utilized in different cases, some of the most recent of which are the works of Healy and Treacy [16] and Zingoni et al [4,5,17,18]. The results are compared to those of existing methods based on graph theory and linear algebra.…”

Section: Introductionmentioning

confidence: 99%

Decomposition of symmetric mass–spring vibrating systems using groups, graphs and linear algebra

Kaveh

Nikbakht

2006

Commun. Numer. Meth. Engng.

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SUMMARYThe main objective of this article is to develop a methodology for an efficient calculation of the eigenvalues for symmetric mass-spring systems in order to reduce the size of the eigenproblem involved. This is achieved using group-theoretical method, whereby the model of a symmetric mass-spring system is decomposed into appropriate submodels. The eigenvalues of the entire system is then obtained by calculating the eigenvalues of its submodels. The results are compared to those of the existing methods based on graph theory and linear algebra. Examples are provided to illustrate the simplicity and efficiency of the present method.

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Truss and Beam Finite Elements Revisited: A Derivation Based on Displacement-Field Decomposition

Cited by 24 publications

References 2 publications

Block diagonalization of Laplacian matrices of symmetric graphs via group theory

Block diagonalization of Laplacian matrices of symmetric graphs via group theory

On the decomposition of generalized eigenproblems for the free vibration analysis of cyclically symmetric finite element models

Decomposition of symmetric mass–spring vibrating systems using groups, graphs and linear algebra

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