Optimum Design of Infinite and Finite Periodic Structures

This ongoing research effort research is performed by Moshe B. Fuchs and Michael Ryvkin of the School of Mechanical Engineering at Tel Aviv University. The general purpose of the research is to develop a numerical methodology for the analysis of modular structures to be used in a general design scheme of structures. The static analysis of repetitive strutures is based on the method of the representative cell, a technique developed by Michael Ryvkin and Boris Nuller:

The method of the representative cell can be construed as the static equivalent of Bloch wave decomposition or simply the discrete Fourier transform (DFT) used in the analysis of the dynamic behavior of repetitive structures. Based on the DFT, the method of the representative cell reduces the static analysis of the initial infinite modular structure under arbitrary loading to the analysis of a representative cell under appropriate transformed loading and boundary conditions. An almost typical boundary value problem is formulated for the repetitive module which can be solved by regular closed form or numerical (finite element) methods, depending on the complexity of the representative cell. Having obtained the structural response in the transformed variables the exact displacements and stresses can be computed at any station along the structure by the inverse transform.

The advantages of this approach for optimal design are evident. We now have a means of designing very long repetitive structures with relatively few variables both for the analysis and for the design. Although we are dealing with rather large structures the size of the analysis equations and of the design space is commensurate with the dimension of the representative cell. And since the structural distortions have a tendency to decay very rapidly beyond the loaded zone, the number of design constraints are also of the order of the design constraints in the representative cell. All this has far reaching implications in the context of multiple reanalysis, sensitivity calculations and many other aspects of optimum structural design.

It is instructive to mention that the present research does not deal with the design of extremal composites. Other investigators have used periodic microstructures for the topological optimization of multi-phase composite. There, the length scale of variation of the loads applied to the structure is much larger than the scale of the repeating module and consequently the stresses applied to the module are of the same period as the structure, and are given. The boundary conditions on all modules are thus known and constant throughout the material. This paves the way for the analysis of the module. In the present case, however, the length scales of the loading and of the repeating modules are of the same order and the stresses vary spatially from module to module. We are thus designing periodic structures as opposed to periodic materials.

Publications include:

Last modified by Moshe Fuchs on 28 September, 1999.