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Group-theoretic applications in solid and structural mechanics: a review
Alphose Zingoni
- 01 Jan 2002
- pp 283-317
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TL;DR: A review of applications of symmetry groups and associated representation theory in the analysis and study of problems involving symmetry within the fields of solid and structural mechanics is given in this article, where it is shown that through the characteristic vector-space decomposition, group-theoretic methods afford considerable simplifications and reductions in computational effort in comparison with conventional methods.
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Abstract: This paper is a review of applications of symmetry groups and associated representation theory in the analysis and study of problems involving symmetry within the fields of solid and structural mechanics. Such techniques are very well-established in various branches of physics and chemistry, but the need for a more systematic and thorough exploitation of symmetry in tackling problems within solid and structural mechanics has provided the impetus over the past 30 years for the development of group-theoretic methods. This paper traces the advances made in the exploitation of group theory in areas such as bifurcation analysis, vibration analysis and finite-element analysis, and outlines the various schemes of implementation currently available. In all cases, it is shown that through the characteristic vector-space decomposition, group-theoretic methods afford considerable simplifications and reductions in computational effort in comparison with conventional methods, and render the computations amenable to the use of parallel processors.
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Citations
Group-theoretic exploitations of symmetry in computational solid and structural mechanics
TL;DR: The use of group theory in simplifying the study of problems involving symmetry is a well-established approach in various branches of physics and chemistry, and major applications in these areas date back more than 70 years as discussed by the authors.
Generalized Eigenvalue Analysis of Symmetric Prestressed Structures Using Group Theory
TL;DR: In this article, a simplified technique for analyzing dynamic characteristics of symmetric prestressed structures is described using group theory, where the generalized eigenvalue equation of a prestressed structure based on tangent stiffness matrix and lumped mass matrix is built to get natural frequencies and corresponding vibration shapes in which the contribution of initial prestresses is considered.
78
Intrinsic non-flat-foldability of two-tile DDC surfaces composed of glide-reflected irregular quadrilaterals
Pooya Sareh,Yao Chen +1 more
TL;DR: In this article, it was shown that two-tile DDC surfaces composed of glide-reflected irregular quadrilaterals are intrinsically not flat-foldable, due to geometric incompatibilities between the properties of certain unit cells and the local flatfoldability condition, regardless of the geometric specifications of their constituting quadrilateral facets.
64
Computational methods for bifurcation problems with symmetries―with special attention to steady state and Hopf bifurcation points
Michael Dellnitz,Bodo Werner +1 more
- 01 Jan 1990
TL;DR: In this paper, the authors show how group theoretical methods can be employed to utilize the symmetry of a bifurcation problem in numerical computations, where the essential numerical point is the utilization of certain reduced instead of full systems involving appropriate subgroups of the underlying symmetry group Γ.
40
A group-theoretic formulation for symmetric finite elements
TL;DR: In this article, an efficient formulation for the computation of matrices for symmetric finite elements is presented for the problem of consistent mass matrices with respect to truss, beam, plane-stress, plate bending, and solid elements.
40
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