Optimal design of infinite repetitive structures

Abstract

An approach for designing optimal repetitive struc- tures under arbitrary static loading is presented. It is shown that the analysis of such infinite structures can be reduced to the analy- sis of the repeating module under transformed loading and bound- ary conditions. Consequently, both the design parameters and the analysis variables constitute a relatively small set which facilitates the optimization process. The approach hinges on the representa- tive ceil method. It is based on formulating the analysis equations and the continuity conditions for a sequence of typical modules. Then, by means of the discrete Fourier transform this problem translates into a boundary value problem of a representative cell in transformed variables, which can be solved by any appropriate analytical or numerical method. The real structural response any- where in the structure is then obtained by the inverse transform. The sensitivities can also be calculated on the basis of the sensi- tivities of the representative cell. The method is illustrated by the design for minimum compliance with a volume constraint of an infinite plane truss. It is shown that by employing this analysis method within an optimal design scheme one can incorporate a reduced analysis problem in an intrinsically small design space.