Affine decomposition of structural stiffness matrices and their application in metaheuristic design optimization of fiber composite structures

Abstract A novel affine decomposition strategy for the finite element linear and geometric stiffness matrices of a fiber composite plate structure is presented here. The main goal of this decomposition is to express the stiffness matrices as an affine summation of terms formed by parameter-independent matrices and parameter-dependent scalars. Such a grouping has applications in improving efficient finite element analysis and improved performance of metaheuristic optimization algorithms used in structural optimization. The linear stiffness matrix is decomposed using lamination parameters and invariant matrices from classical lamination theory. The decomposition of the geometric stiffness matrix is more complicated due to the involvement of the in-plane load-induced transverse stiffness. First, the static response of the structure is approximated using a set of projection basis vectors, and the coefficient associated with each vector has an implicit dependence on the fiber composite configuration. The design-independent components of the geometric stiffness matrix are computed in terms of the projection basis vectors of the static response, and the coefficients are computed on the fly using least squares approximation. The application of these decompositions in a parametric model order reduction-based two-step optimization framework is demonstrated. The proposed approach resulted in a significant reduction in the time and resource requirement needed to carry out this optimization.