Higher Order Sensitivity Analysis for Linear and Nonlinear Problems Using the Multidual Finite Element Method

Date
2022
Authors
Avila, David
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Abstract

This research work evaluated the accuracy and efficiency of the multidual complex-valued finite element method (ZFEM) in computing higher order derivatives of output variables with respect to input variables of elastic, elastic-plastic, and hyperelastic problems, up to the third order. Multidual ZFEM accuracy in elastic problems was evaluated by comparing multi-order derivatives of the displacement and stress fields with respect to material, geometric, and loading parameters in a thick-walled cylinder problem under internal pressure. A mesh sensitivity analysis with standard mesh refinement techniques was performed to determine the convergence rate of the results produced by Multidual ZFEM. The L2-norm error between the evaluated analytical solution and the solution produced by Multidual ZFEM revealed that the higher order derivative terms of the output variable of interest converge at a slower rate than the traditional output of the analysis. In addition, the results indicated that higher order derivatives with respect to material parameters are generally more accurate than those relating to geometry, regardless of the material constitutive behavior, as the geometric derivatives are heavily dependent on the perturbation region. This trend in the results was also observed for mixed derivatives between a variety of different types of input parameters, i.e., mixed derivatives between shape and material parameters. Overall, multidual ZFEM is capable of calculating accurate derivatives in elastic problems regardless of the type of input parameter in a single analysis in comparison to conventional numerical techniques, such as Finite Difference, which require multiple analyses to determine the appropriate step size of the perturbation of the input variable. For elastic-plastic problems, the results suggested that adjustments to the solver technique might be required to obtain convergence of higher order derivatives. The higher order derivatives provided by Multidual ZFEM can be useful in many branches of engineering, including uncertainty quantification.

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Keywords
Higher order sensitivity, Multidual finite element, Nonlinear problems
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Department
Mechanical Engineering