Sensitivity Analysis in Structural Dynamics using the ZFEM Complex Variable Finite Element Method

Abstract

SPECIAL SESSION ON THE DIGITAL TWINSensitivity anal 1 ysis of structural systems is of great importance for structural dynamic modifications. The sensitivity analysis helps identify key input parameters that influence the dynamic response of a model. The present paper demonstrates how to accurately and efficiently obtain derivatives of linear dynamic systems using the complex step method and the generalized multicomplex step method implemented within a complex variable finite element method (ZFEM). The highly accurate derivatives are computed using a single complex variable finite element analysis of the structural system. An undamped and damped simply supported beam was modeled using ZFEM and the response derivatives were computed with respect to load amplitude, load frequency, beam cross sectional dimension, and Rayleigh damping coefficients. Comparisons of the derivatives against analytical and finite difference estimates verify the accuracy of the methodology. Nomenclatureth undamped natural frequency ! " n = n th mode shape ! K e = local stiffness matrix ! M e = local consistent mass matrix ! " = mass-normalized mode shape matrix ! " = diagonal matrix of natural frequencies squared ! " = vector of nodal degrees of freedom ! I = identity matrix ! "t = time step ! " = forcing frequency ! Q = uniform load ! " = Rayleigh damping mass proportional coefficient ! " = Rayleigh damping stiffness proportional coefficient ! * = denotes a complex variable

Sensitivity Analysis for Transient Thermal Problems Using the Complex-Variable Finite Element Method

“…This technique has been widely applied in sensitivity analysis in many fields because it is easy to implement as it only requires multiple model evaluations with a small step size for the analyzed variable. Nevertheless, when the model becomes nonlinear, obtaining high precision sensitivities using FD is challenging because the method is subject to computational subtraction cancellation errors and truncation errors [55,56]. In addition, the step size for each input to the model is problem dependent and usually requires a convergence analysis to select it for each of the inputs.…”