Optimum laminate design by using singular value decomposition

In practice, a structure is subjected to given loads and boundary conditions, and a multitude of stress and strain states may exist in the structure; hence, optimal construction of a laminate in a structure cannot be sought by considering only a limited number of stress resultants in the existence of multiple load cases. Then, another design objective based on optimization of a laminate for the worst possible load case emerges which is formulated as a minimax problem whose solution is shown to be equivalent to singular value minimization problem. As the squares of singular values are the bounds of power, energy and power spectral density ratios between the input and output vectors, shaping the singular values of a composite material is equivalent to shaping the response of the material. As a novel approach, singular values are used for the layout optimization of laminate. In this method, the main idea is minimization of the largest singular value of the transfer function matrix between force/moment resultants and outputs stress/strain. Thus the overall optimization problem is reduced to a simple minimization problem. Numerical examples and finite element simulations are presented for several test problems. In particular, it is shown that the use of singular values and singular vectors is computationally advantageous in case of multiple load case.