Computational Structures Technology
Edited by: B.H.V. Topping and Z. Bittnar

Chapter 13

Topological Transformations in Structural Mechanics

A. Kaveh
Department of Civil Engineering
Iran University of Science and Technology, Narmak, Tehran, Iran

Keywords: transformation, graph theory, topology, rigidity, force method, ordering, decomposition, finite element, meshless, configuration processing, analysis. In this paper some topological transformations are designed for simplifying certain problems involved in mechanics of structures. For each case, the main problem is stated and the proposed transformation is established. Once the required topological analysis is completed, a back transformation results the solution for the main problem.

Introduction

For optimal analysis of structures, three conditions should be fulfilled. The structural matrices (stiffness or flexibility) should be sparse, properly structured (e.g. banded) and well-conditioned. The latter property is not purely topological and is treated elsewhere, Kaveh [1,2]. Only the problems relevant to sparsity and proper structuring are studied in this paper.

Pattern equivalence of structural matrices and those of graph theory simplifies structural problems and allows advances made in this field to be to be transferred to structural mechanics. As an example, for rigid-jointed frames the sparsity of flexibility matrices can be provided by construction of sparse cycle basis adjacency matrices. Similarly using sparse cut set basis matrices, the formation of sparse stiffness matrices become feasible. Proper structuring of the flexibility and stiffness matrices of a structure can also be provided by structuring the pattern of cycle and cut set adjacency matrices of its model, respectively.

This paper is devoted to the study of some structural problems in which topological graph theory plays an important role. Topological graph theory is primarily concerned with representing graphs on surfaces. An embedding or a drawing of a graph can be considered as identification of the graph with a subset of a surface in an appropriate fashion. For some problems it is beneficial to define a new graph with more simple connectivity properties than the original model.

The transformations presented in this article cover the following subjects in the field of structural mechanics:

• Degree of static indeterminacy of space structures,
• Rigidity of grid-shaped planar trusses,
• Cycle bases selection; manifold and collapsible embeddings,
• Generalized cycle bases,
• Cycle and generalized cycle basis ordering,
• Graph models of finite element meshes,
• Element ordering for frontwidth reduction,
• Element and nodal ordering,
• Duality of cycle bases and cut set bases,
• Graph models for meshless discretization,
• Configuration processing,
• Other applications.

Concluding Remarks

A collection of transformations is presented for the study of topological properties of structures. These transformations provide useful tools for optimal analysis of structures. It is hoped that other transformations can be found and better classifications can be made.

References

1

Kaveh, A., Structural Mechanics: Graph and Matrix Methods, Research Studies Press (John Wiley), 2nd Edition, UK, 1995.

2

Kaveh, A., Optimal Structural Analysis, Research Studies Press (John Wiley), UK, 1997.

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