Evaluation of Force-Based Frame Element Response Sensitivity Formulations

Abstract

The direct differentiation method (DDM) has been shown to be an accurate and efficient approach to computing the sensitivity of structural response to uncertain parameters of constitutive models and finite-element formulations. Although it is well-known that the DDM should be consistent with the numerical time stepping procedure at the structural level, it is possible for element-level numerical instabilities to arise when the response sensitivity equations are inconsistent with the equations that govern the element response. Two existing formulations of DDM force-based element response sensitivity are shown to be mathematically equivalent in exact arithmetic; however, only one is consistent with the force-based response equations and possesses a low condition number for finite arithmetic. On the other hand, the inconsistent formulation has a high condition number that is equal to the product of the largest singular values of the section and element stiffness matrices. Representative examples show that the high condition number of the inconsistent formulation is innocuous for sensitivity with respect to section-level parameters but can lead to round off errors for sensitivity with respect to element-level geometric parameters.