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

Koohestani, K.

doi:10.1002/nme.2770

Cited by 7 publications

(4 citation statements)

References 29 publications

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“…depending on the expanding generator being a circle or a path, respectively (Kaveh, 2006). Repetitive structures have been widely studied by Kangwai et al (1999), Kangwai and Guest (2000), Kaveh and Rahami (2004, 2007, 2008, Kaveh et al (2012Kaveh et al ( , 2013, , Koohestani (2010), and Zingoni (2009Zingoni ( , 2012aZingoni ( , 2012bZingoni ( , 2014. Kaveh et al (2013) analyzed the near-regular structure by using the force method.…”

Section: Equivalent Circulant Structures 343mentioning

confidence: 99%

“…Repetitive structures can be divided into two categories, namely, circulant and regular structures, depending on the expanding generator being a circle or a path, respectively (Kaveh, 2006). Repetitive structures have been widely studied by Kangwai et al (1999), Kangwai and Guest (2000), Kaveh and Rahami (2004, 2007, 2008), Kaveh et al (2012, 2013), Rahami (2012), Koohestani (2010), and Zingoni (2009, 2012a, 2012b, 2014).…”

Section: Introductionmentioning

confidence: 99%

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Analysis of repetitive and near-repetitive structures by transformation to equivalent circulant structures

Kaveh

Rahami

Jodaki

2017

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Purpose There are many structures that have a repetitive pattern. If a relationship can be established between a repetitive structure and a circulant structure, then the repetitive structure can be analyzed by using the properties of the corresponding circulant structure. The purpose of this paper is to develop such a transformation. Design/methodology/approach A circulant matrix has certain properties that can be used to reduce the complexity of the analysis. In this paper, repetitive and near-repetitive structures are transformed to circulant structures by adding and/or eliminating some elements of the structure. Numerical examples are provided to show the efficiency of the present method. Findings A transformation is established between a repetitive structure and a circulant structure, and the analysis of the repetitive structure is performed by using the properties of the corresponding circulant structure. Originality/value Repetitive and near-repetitive structures are transformed to circulant structures, and the complexity of the analysis of the former structures is reduced by analyzing the latter structures.

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Section: Equivalent Circulant Structures 343mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Analysis of repetitive and near-repetitive structures by transformation to equivalent circulant structures

Kaveh

Rahami

Jodaki

2017

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“…Koohestani [33] applied the properties of symmetry in graph theory to finite and boundary elements. For the free vibration analysis of cyclic symmetry, Koohestani [34] implemented the decomposition of extended Eigen problems. This research aimed to optimize the design of real-scale symmetric structures under frequency constraints using the GRO metaheuristic algorithm.…”

Section: Introductionmentioning

confidence: 99%

Frequency-Constrained Optimization of a Real-Scale Symmetric Structural Using Gold Rush Algorithm

2022

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The optimal design of real-scale structures under frequency constraints is a crucial problem for engineers. In this paper, linear analysis, as well as optimization by considering natural frequency constraints, have been used for real-scale symmetric structures. These structures require a lot of time to minimize weight and displacement. The cyclically symmetric properties have been used for decreasing time. The structure has been decomposed into smaller repeated portions termed substructures. Only the substructure elements are needed when analyzing and designing with the concept of cyclic symmetries. The frequency constrained design of real-scale structures is a complex optimization problem that has many local optimal answers. In this research, the Gold Rush Optimization (GRO) algorithm has been used to optimize weight and displacement performances due to its effectiveness and robustness against uncertainties. The efficacy of the concept of cyclic symmetry to minimize the time calculated is assessed by three examples, including Disk, Silo, and Cooling Tower. Numerical results indicate that the proposed method can effectively reduce time consumption, and that the GRO algorithm results in a 14–20% weight reduction of the problems.

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“…Group theory was also used for efficient topology optimization of truss structures by Bai et al [26]. Methods based on the decomposition of matrices with special patterns were also employed effectively in the analysis of cyclically symmetric structures [27] and a combinatorial optimization [28]. In summary, there are a considerable number of studies of symmetric systems in science and engineering mainly using group theory; however, for the sake of brevity, only some of them are highlighted here.…”

Section: Introductionmentioning

confidence: 99%

Exploitation of symmetry in graphs with applications to finite and boundary elements analysis

Koohestani

2012

Numerical Meth Engineering

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We present explicit and parametric forms of transformation matrices for three well-known and widely used symmetry groups: S 2 , C 2v and C 4v . Group representation theory is the most powerful method for exploiting symmetry. We propose an efficient algorithm for systematic generation of reducible representations that can be combined linearly to obtain the projection operators. The exact column spaces of these projection operators are calculated and integrated through special orderings, leading to exact explicit and parametric forms of transformation matrices. The transformation matrices could be used directly for block diagonalization of single-variable scalar field problems. Another algorithm is proposed to extend the application of the method to nonscalar and multivariable field problems. Finally, the generality and efficiency of the proposed method in relation to computation times and the accuracy of results are illustrated through examples from spectral decomposition, free vibration, buckling of FEMs and boundary element analysis of a symmetric field. One of the earliest applications of group representation theory to the stability of structures was developed by Renton [5]. The theory has since been used extensively for the study of local and global bifurcation analyses. Among the outstanding studies in bifurcation analysis are Sattinger [6], Golubitsky and Schaeffer [7,8], Ikeda and Murota [9, 10], Ikeda et al. [11,12], Healey [13] and Wohlever and Healey [14]. Group representation theory has also been employed effectively in the vibration analysis of symmetric structures by Healey and Treacy [15], Zlokovic [16], Zingoni [17, 18] and Mohan and Pratap [19]. Eigenvalue problems related to the vibration of massspring systems were studied by Kaveh and Nikbakht [20], Kaveh and Jahanmohammadi [21] and Zingoni [22] using group theory. Zingoni [23,24] used group theory to simplify the generation of Node adjacency matrix: Let G be a graph with N nodes. The adjacency matrix A is an N N matrix in which the entry in row i and column j is 1 if node i is adjacent toj , and is 0 otherwise. Note that, for simple graphs (graphs without loops and multiple edges) the diagonal entries of the adjacency matrix are always zero; however, the existence of k loops at node i implies that node i is connected to itself by k edges. In this case, the i-th diagonal entry in the adjacency matrix should equal k.Laplacian matrix: The Laplacian matrix of graph G (denoted by L) is defined as Symmetry subspacesBy defining a coordinate system for a configuration (the configuration could be a graph, a structural model, etc.) we can represent all symmetry operations as matrices. These matrices will change according to different forms of coordinate systems. This form of representation is called the reducible matrix representation and denoted here by P(see [29] for more details). However, there is always a special coordinate system (a symmetry-adapted coordinate system) such that each symmetry operation is represented on it as a unique matrix. These f...

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

Cited by 7 publications

References 29 publications

Analysis of repetitive and near-repetitive structures by transformation to equivalent circulant structures

Analysis of repetitive and near-repetitive structures by transformation to equivalent circulant structures

Frequency-Constrained Optimization of a Real-Scale Symmetric Structural Using Gold Rush Algorithm

Exploitation of symmetry in graphs with applications to finite and boundary elements analysis

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