P-Adaptive Refinement Based on Stress Recovery Technique Considering Ordinary Kriging Interpolation in L-Shaped Domain

Kwang S Woo,Jae S Ahn

doi:10.1155/2017/1790256

Abstract

The primary objectives of this study are twofold. Firstly, the original SPR method of stress recovery has been modified by incorporating the kriging interpolation technique to fit a polynomial to the derivatives recovered at the Gauss points. For this purpose, the p-version of finite element analysis is performed to produce the stresses at the fixed 10×10 Gauss points where the integrals of Legendre polynomials are used as a basis function. In contrast to the conventional least square method for stress recovery, the weight factor is determined by experimental and theoretical variograms for interpolation of stress data, unlike the conventional interpolation methods that use an equal weight factor. Secondly, an adaptive procedure for hierarchical p-refinement in conjunction with a posteriori error based on the modified SPR (superconvergent patch recovery) method is proposed. Thirdly, a new error estimator based on the limit value approach is proposed by predicting the exact strain energy to verify the kriging-based SPR method. The validity of the proposed approach has been tested by analyzing two-dimensional plates with a rectangular cutout in the presence of stress singularity.

Highlights

The error assessment tools used in finite element analysis are well known and usually classified into two strategies: recovery-based error estimators and residual-type estimators [1, 2]
The objective of this study is to demonstrate the applicability of OK interpolation to the padaptive refinement of L-shaped domain problem employing the modified superconvergent patch recovery (SPR) method for stress recovery
For a two-dimensional problem, under the assumption that the error in the energy norm has entered the asymptotic range where Uex and Ufe are the strain energy, the rate of convergence for the p-version of FEM can be derived by the inverse theorem [23, 27] as where Uex and Ufe are the exact strain energy estimated by the limit value and the approximate strain energy by FEM, α is the strength of singularity, and Np and k are the degrees of freedom for the polynomial order p and a constant which depends on the mesh, respectively

Summary

Introduction

The error assessment tools used in finite element analysis are well known and usually classified into two strategies: recovery-based error estimators and residual-type estimators [1, 2]. In the context of FEM model, the kriging interpolation technique has been employed as an alternative for estimating the derivative of the unknown variable at any point of interest [17] This method uses a variogram to express the spatial correlation, Mathematical Problems in Engineering and it minimizes the error of predicted values. The objective of this study is to demonstrate the applicability of OK (ordinary kriging) interpolation to the padaptive refinement of L-shaped domain problem employing the modified SPR method for stress recovery To verify this method, the limit value approach is proposed to predict the exact strain energy for nonsmooth problems based on the application of the equation of a prior error indicator in the asymptotic range to three FEMs with three successively higher levels of polynomial approximation

Ordinary Kriging Interpolation

A p-Adaptive Refinement Using Modified SPR Technique

Numerical Analysis

Conclusions