SFP-a new parameterization based on shape functions for optimal material selection: application to conventional composite plies

In this paper, a new parameterization of the mechanical properties is proposed for the optimal selection of materials. Recent parameterization schemes from multi-phase topology optimization (i.e. Discrete Material Optimization-DMO) are compared to this novel approach in the selection of conventional laminates including only 0 degrees., -45 degrees, 45.degrees and 90 degrees plies. In the new parameterization the material stiffness is computed as a weighted sum of the candidate material properties, and the weights are based on the shape functions of a quadrangular first order finite element. Each vertex of the reference quadrangle then represents a candidate ply. Compared to DMO, this method requires fewer design variables, since the four pseudo-densities representing the presence or the absence of a given candidate ply in DMO are now replaced, in the weights, by two design variables, which are the two natural coordinates of the reference quadrangular element sufficient to identify each of the four vertices. Another advantage of the new parameterization scheme is to penalize, in a more convenient way, the intermediate values of the design variables, possibly avoiding any blending of materials at the solution. Three simple numerical applications with in-plane loadings are proposed and solved in order to demonstrate that the new approach is an interesting alternative to DMO, able to select the optimal orientations and to combine the material distribution with optimal orientation problems.