Interpolation Type Stress Recovery Technique Based Error Estimator for Elasticity Problems

Mohd. AHMED; Devender SINGH; M. Noor DESMUKH

摘要

Finite Element Method (FEM) is a powerful technique for simulating real life problems, but it is only able to provide an approximated solution that necessitates reliable error control of computed solutions. The various error estimation techniques have been developing for last several years. A critical review of different error estimation techniques to get the practical finite element solution of linear, non-linear and transient problems is given in [1]. Error estimators can either be classified based on its convergence through the global effectivity index, which is defined as ratio of the estimated error to the true error or can be classified based on the procedures to obtain the estimates. The classification of error estimators based on convergence are asymptotically exact, asymptotically (upper) bounded and asymptotically not bounded. The classification of error estimators as per the estimation procedure are the residual-based error estimators [2], constitutive relation error (CRE) based error estimators [3] and the recovery-type error estimators [4], The explicit-type recovery error estimator in energy norm is proposed in [5] for the linear elasticity problem using smooth solution. In [6], a posteriori error estimate has been developed and improved convergence is shown using non-coinciding meshes for problems in linear elasticity. An effective error estimator presented in [7] is based on continuous estimated stress field that is achieved by interpolating from nodal stresses over the element with the shape functions. In [8], it is shown that the technique of duality error majorants can serve as an effective local indicator of modeling errors of nonlinear problems. A posteriori error estimate based on an equilibrated stress reconstruction that is obtained from mixed finite element solutions of local Neumann linear elasticity problems is presented in [9]. The quality of recovery based error estimation depends on the way to obtain the smoothed or post- processed continuous stress field. The recovery techniques are based on the least square fitting of velocity (or the displacement) field or their derivatives (stress field) by a higher order polynomial over a patch of elements or nodes. The various authors have proposed procedures for recovery of postprocessed stress field. The super-convergent patch recovery scheme is proposed in [10] in which post- processed stresses are obtained by interpolating from a stress surface fitted to the superconvergent stress points surrounding the node of interest. In [11], a patch recovery scheme is employed in which recovery is performed for all components simultaneously. The coupling of the stress components is made through the equations of equilibrium. In [12], computational aspect of Goal-oriented error estimates (GOEE) based on enhanced Superconvergent Patch Recovery is presented in controlling the local error in quantities of interest (Qol). A method of extrapolation based on patch recovery is proposed in [13] for obtaining continuous stress field in a local manner. A recovery technique based upon the least square fitting of velocity field over an element patch is proposed in [14]. In [15], a moving least squares (MLS) recovery-based procedure to obtain postprocessed smoothed stresses field is presented in which the continuity of the recovered field is provided by the shape functions of the underlying mesh. Investigations are reported in [16] for getting improved recovery of stress field using domain decomposition method in heterogeneous structures. In [17], an improvement of the SPR technique, called SPR-C technique (Constrained SPR), is presented and uses the appropriate constraint equations in order to obtain stress interpolation polynomials in the patch.

机译：有限元方法（FEM）是一种用于模拟现实生活中问题的强大技术，但是它只能提供一种近似的解决方案，而该解决方案需要对计算出的解决方案进行可靠的错误控制。近几年来已经开发了各种误差估计技术。在[1]中给出了对不同误差估计技术的批判性综述，以获得线性，非线性和瞬态问题的实用有限元解。误差估计器可以基于其通过全局有效性指数的收敛性进行分类，全局有效性指数定义为估计误差与真实误差之比，或者可以基于获取估计值的过程进行分类。基于收敛的误差估计量的分类是渐近精确的，渐近（上）有界的和渐近无界的。根据估计程序，误差估计器的分类为：基于残差的误差估计器[2]，基于本构关系误差（CRE）的误差估计器[3]和恢复型误差估计器[4]，显式恢复误差能量范数的估计是在[5]中提出的，用于使用平滑解的线性弹性问题。在[6]中，已经开发了后验误差估计，并且对于线性弹性问题，使用非重合网格显示了改进的收敛性。 [7]中提出的有效误差估计器基于连续估计应力场，该连续应力场是通过将具有形状函数的单元上的节点应力插值而获得的。在文献[8]中，证明了对偶误差主观分析技术可以作为非线性问题建模误差的有效局部指标。在[9]中提出了基于平衡应力重构的后验误差估计，该应力重构是从局部诺伊曼线性弹性问题的混合有限元解获得的。基于恢复的误差估计的质量取决于获得平滑或后处理的连续应力场的方式。恢复技术基于速度（或位移）场或其导数（应力场）的最小二乘拟合，该最小二乘拟合是通过元素或节点补丁上的高阶多项式实现的。各种作者都提出了恢复后应力场的程序。在[10]中提出了超收敛补丁恢复方案，其中后处理应力是通过从拟合到围绕感兴趣节点的超收敛应力点的应力表面进行插值获得的。在[11]中，采用了补丁恢复方案，其中对所有组件同时执行恢复。应力分量的耦合通过平衡方程进行。在[12]中，提出了基于增强的超收敛补丁恢复的面向目标误差估计（GOEE）的计算方面，用于控制感兴趣数量（Qol）中的局部误差。在[13]中提出了一种基于补丁恢复的外推方法，以局部方式获得连续应力场。文献[14]中提出了一种基于速度场的最小二乘拟合的恢复技术。在[15]中，提出了一种基于移动最小二乘（MLS）恢复的过程来获取后处理的平滑应力场，其中恢复场的连续性由基础网格的形状函数提供。文献[16]报道了在异质结构中使用区域分解法改善应力场恢复的研究。在[17]中，提出了对SPR技术的一种改进，称为SPR-C技术（Constrained SPR），并使用适当的约束方程式来获得补丁中的应力插值多项式。

Interpolation Type Stress Recovery Technique Based Error Estimator for Elasticity Problems

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