A family of homogeneous analysis models for the design of scalable structures

Research in optimum structural design has shown that mathematical programming techniques can be employed efficiently only in conjunction with explicit approximate constraints. In the course of time a well-established approximation for homogeneous functions (scalable structures) has emerged based on the linear Taylor expansion of the displacement functions in the compliance design space (Reciprocal approximation). It has been shown that the quality of this approximation is based on the property that homogeneity of the constraints is maintained and consequently the approximation is exact along the scaling line. The present paper presents a new family of approximations of homogenous functions which have the same properties as the Reciprocal approximation and which produce more accurate models in most of the tested cases. The approximations are obtained by mapping the direct linear Taylor expansion of the constraints unto a space spanned by intervening variables (original design variables to a power m). Taking the envelope of these constraints along the scaling line yields a new family of approximations governed by the parameter m. It is shown that the Reciprocal approximation is a particular member of this family of approximations (m = -1). The new technique is illustrated with classical examples of truss optimization. An optimal plate design is also reported. A parametric study of the results for various values of the exponent m is presented. It is shown that for special values of the exponent m the new approximations usually yield better models than those based on the Reciprocal approximation.