Reduced Order Modeling for Efficient Stability Analysis in Structural Optimization

Abstract

Design optimization involving complex structures can be a very resource-intensive task. Convex optimization problems could be solved using gradient-based approaches, whereas non-convex problems require heuristic methods. Over the past few decades, many optimization techniques have been presented in the literature to improve the efficiency of both these approaches. The present work focuses on the non-convex optimization problem involving eigenvalues that arises in structural design optimization. Parametric Model Order Reduction (PMOR) was identified as a potential tool for improving the efficiency of the optimization process. Its suitability was investigated by applying it to different eigenvalue optimization techniques. First, a truss topology optimization study was conducted that reformulated the weight minimization problem with a non-convex lower-bound constraint on the fundamental frequency into the standard convex optimization form of semidefinite programming. Applying PMOR to this, it was found the reduced system was able to converge to the correct final designs, given a reduced basis vector of suitable size was chosen. At the same time, it was shown that preserving the sparse nature of the mass and stiffness matrices was crucial to achieving reduced solution times. In addition, the reformulation to convex optimization form, while possible with the discretized form of vibrational governing equations, is not straightforward with the buckling problem. This is due to the non-linear dependence of the geometric stiffness matrix on the design variables. Hence, we turned to a metaheuristic approach as an alternative and explored the applicability of PMOR in improving its performance. A two-step optimization procedure was developed. In the first step, a set of projection vectors that can be used to project the solutions of the governing higher-order partial differential equations to a lower manifold was assembled. Invariant components of the system matrices that do not depend on the design variables were identified and reduced using the projection vectors. In the second (online) step, the buckling analysis problem was assembled and solved directly in the reduced form. This approach was applied to the design of variable angle tow (VAT) fiber composite structures. Affine matrix decompositions were derived for the linear and geometric stiffness matrices of VAT composites. The resulting optimization framework can rapidly assemble the reduced order matrices related to new designs encountered by the optimizer, perform the physics analysis efficiently in the reduced space, evaluate heuristics related to the objective function, and determine the search direction and convergence based on these evaluations. It was shown that the design space can be traversed efficiently by the developed PMOR-based approach by ensuring a uniform error distribution in objective values throughout the design space.