Perturbation-based sensitivity analysis of optimal solutions in uncertain mechanical and structural systems

Abstract

It is generally the case that optimal designs or response configurations for mechanical and structural systems are based on the nominal values of parameters change when environmental variation is considered, while the repeated re-optimization needed to ascertain these changes becomes computationally infeasible for realistic models with many parameters. In this study come up with a sensitivity analysis framework based on perturbation, which expands the state, adjoint, and optimal multiplier variables of a constrained structural optimization problem in an asymptotic power series with respect to the small perturbation parameter. The study performs order-by-order substitution in the equilibrium and optimality conditions to derive the recursive system of linear equations, which share a nominal stiffness operator, where the first order sensitivities, as well as higher order sensitivities can be found without the need of repeated Jacobian inversion. The introduction of an adjoint formulation allows us to find gradient information quickly with respect to a multitude of uncertain parameters with one solution. The study states and proves an implicit function theorem for the perturbed solution branch showing that the branch exists, is unique, and is Frechet differentiable around the nominal solution. Moreover, provide the error bounds for the obtained series and describe the radius of convergence dependent on the behaviour of structural bifurcation and turning point. Finally, numerical validation on benchmarks of truss framework and plates considering uncertain material moduli and eccentricity of loading shows that predicted order of accuracy is achieved and that the sensitivity map for the structural members is obtained.