Using Normalized Parameter Perturbations to Investigate Design, Sensitivity Analysis, and Uncertainty Quantification

摘要

This effort is intended to enrich the well-established practices of sensitivity analysis and calculation of stochastic means and variances by considering normalized parameter perturbations as independent design variables. The effect of the forward normalized parameter perturbation formulation on analysis, sensitivity analysis and uncertainty quantification are explored for fundamental analytic relationships. Basic operations (multiplication and addition/subtraction) are used to develop the normalized parameter perturbation formula for a system of linear equations exemplified by the force-stiffness-displacement relationship in standard finite element analysis. Analytic relationships for means, variances, and covariances are formulated and compared for the standard and the normal parameter perturbed system. Derivatives, Taylor series expansion, and direct and reciprocal approximations are shown for normalized parameter perturbations. Structural sensitivity analysis relationships are explored for the first (mean) and second (variance) statistical moments of the stochastic response (displacement) due to uncertainties in the input parameters of the system and loads. A generic higher order equation is considered to illustrate the usefulness of a parameter perturbation approach when attempting to quantify sufficient, best value, and exquisite solutions for cost analysis. Foundational structural analysis problems are investigated using the developed formulations.