Uncertainty Quantification of Stresses in a Cracked Pipe Elbow Weldment Using a Logistic Function Fit, a Nonlinear Least Square Algorithm, and a Super-parametric Method1

In a 3-part series of papers, of which this paper is Part II, we investigate the applicability of the fully quadratic hexa-27 element (see Part I) to four problems of interest to the pressure vessels and piping community: (1) The solid-element-based analysis of a welded pipe elbow with a longitudinal surface crack in one of its weldments. (2) The solid-element-based analysis of the elastic bending of a simple cantilever beam, of which the exact solution is known. (3) The tetra-04 element-based analysis of the deformation of a wrench. (4) The shell-element-based analysis of a barrel vault. In this paper, we develop a two-step method first to estimate the uncertainty of a converging series of finite-element-mesh-density-parametric solutions using a 4-parameter logistic function, and then to extrapolate the results of a specific quantity (e.g., a stress component) to an extremely fine mesh density approaching the infinite degrees of freedom. The estimated parameter of the upper bound of the logistic function serves as the “best” estimate of the chosen quantity such as a specific stress component. Using a super-parametric approach, as described in Part III of this series, we show that the hexa-27 element is superior to tetra-04, hexa-08, and hexa-20.