A finite element method (FEM)-based solution of an industry-grade problem with complex geometry, partially-validated material property databases, incomplete knowledge of prior loading histories, and an increasingly user-friendly human-computer interface, is extremely difficult to verify because of at least five major sources of errors or solution uncertainty (SU), namely, (SU-1) numerical algorithm of approximation for solving a system of partial differential equations with initial and boundary conditions; (SU-2) the choice of the element type in the design of a finite element mesh; (SU-3) the choice of a mesh density; (SU-4) the quality measures of a finite element mesh such as the mean aspect ratio.; and (SU-5) the uncertainty in the geometric parameters, the physical and material property parameters, the loading parameters, and the boundary constraints. To address this problem, a super-parametric approach to FEM is developed, where the uncertainties in all of the known factors are estimated using three classical tools, namely, (a) a nonlinear least squares logistic fit algorithm, (b) a relative error convergence plot, and (c) a sensitivity analysis based on a fractional factorial orthogonal design of experiments approach. To illustrate our approach, with emphasis on addressing the mesh quality issue, we present a numerical example on the elastic deformation of a cylindrical pipe with a surface crack and subjected to a uniform load along the axis of the pipe.
A Super-Parametric Approach to Estimating Accuracy and Uncertainty of the Finite Element Method (*)
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Rainsberger, RB, Fong, JT, & Marcal, PV. "A Super-Parametric Approach to Estimating Accuracy and Uncertainty of the Finite Element Method (*)." Proceedings of the ASME 2016 Pressure Vessels and Piping Conference. Volume 1B: Codes and Standards. Vancouver, British Columbia, Canada. July 17–21, 2016. V01BT01A059. ASME. https://doi.org/10.1115/PVP2016-63890
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