Error Estimation in Nonlinear Finite Element Analysis in Two and Three Dimensions

Examination of flux projection-type error estimators in nonlinear finite element analysis

Alternative stress-integration schemes for large-deformation problems of solid mechanics

Advances in Adaptive Computational Methods in Mechanics

A posteriori error estimator and AMR for discrete ordinates nodal transport methods

An algorithm for large-scale computational finite-deformation plasticity

A stochastic computational method based on goal-oriented error estimation for heterogeneous geological materials

We present three new a posteriori error estimators in the energy norm for finite element solutions to elliptic partial differential equations.The estimators are based on solving local Neumann problems in each element.The estimators differ in how they enforce consistency of the Neumann problems.We prove that as the mesh size decreases, under suitable assumptions, two of the error estimators approach upper bounds on the norm of the true error, and all three error estimators are within multiplicative constants of the norm of the true error.We present numerical results in which one of the error estimators appears to converge to the norm of the true error.

Patients and physicians expect accurate whole blood glucose monitoring even when patients are anemic, are undergoing peritoneal dialysis, or have slightly elevated ascorbate levels. The objective of this study was to estimate analytical error in two consumer and two hospital glucose meters contributed by variations in hematocrit, maltose, ascorbate, and imprecision. The influence of hematocrit (20-60%), maltose, and ascorbate were tested alone and in combination with each glucose meter and with a reference plasma glucose method at three concentrations of glucose. Precision was determined by consecutive analysis (n=20) at three levels of glucose. Multivariate regression analysis was used to estimate the bias associated with the interferences, alone and in combination. Total analytical error was estimated as |% bias|+1.96 (% imprecision). Three meters demonstrated hematocrit bias that was dependent upon glucose concentration. Maltose had profound concentration-dependent positive bias on the consumer meters, and the extent of maltose bias was dependent on hematocrit. Ascorbate produced small but statistically significant biases on three meters. Coincident low hematocrit, presence of maltose, and presence of ascorbate increased the observed bias and was summarized by estimation of total analytical error. Among the four glucose meter devices assessed, estimates of total analytical error in glucose measurement ranged from 6 to 68% under the conditions tested. The susceptibility of glucose meters to clinically significant analytical biases is highly device-dependent, and low hematocrit exacerbated the observed analytical error.

In this second part of the paper, the issue of a posteriori error estimation is discussed. In particular, we derive a theorem showing the dependence of the effectivity index for the Zienkiewicz–Zhu error estimator on the convergence rate of the recovered solution. This shows that with superconvergent recovery the effectivity index tends asymptotically to unity. The superconvergent recovery technique developed in the first part of the paper1 is the used in the computation of the Zienkiewicz–Zhu error estimator to demonstrate accurate estimation of the exact error attainable. Numerical tests are shown for various element types illustrating the excellent effectivity of the error estimator in the energy norm and pointwise gradient (stress) error estimation. Several examples of the performance of the error estimator in adaptive mesh refinement are also presented.

Filtered back projection (FBP) methods are the most widely used reconstruction algorithms in computerized tomography (CT). The ill-posedness of this inverse problem allows only an approximate reconstruction for given noisy data. Studying the resulting reconstruction error has been a most active field of research in the 1990s and has recently been revived in terms of optimal filter design and estimating the FBP approximation errors in general Sobolev spaces. However, the choice of Sobolev spaces is suboptimal for characterizing typical CT reconstructions. A widely used model are sums of characteristic functions, which are better modelled in terms of Besov spaces . In particular with α ≈ 1 is a preferred model in image analysis for describing natural images. In case of noisy Radon data the total FBP reconstruction error splits into an approximation error and a data error, where L serves as regularization parameter. In this paper, we study the approximation error of FBP reconstructions for target functions with positive and 1 ⩽ p, q ⩽ ∞. We prove that the L p -norm of the inherent FBP approximation error f − f L can be bounded above by under suitable assumptions on the utilized low-pass filter’s window function W. This then extends by classical methods to estimates for the total reconstruction error.

The accuracy of the diagnosis obtained from a nuclear power plant fault-diagnostic advisor using neural networks is addressed in this paper in order to ensure the credibility of the diagnosis. A new error estimation scheme called error estimation by series association provides a measure of the accuracy associated with the advisor's diagnoses. This error estimation is performed by a secondary neural network that is fed both the input features for and the outputs of the advisor. Our error estimation by series association outperforms previous error estimation techniques in providing more accurate confidence information with considerably reduced computational requirements. We demonstrate the extensive usability of our method by applying it to a complicated transient recognition problem of 33 transient scenarios. The simulated transient data at different severities consists of 25 distinct transients for the Duane Arnold Energy Center nuclear power station ranging from a main steam line break to anticipated transient without scram (ATWS) conditions. The fault-diagnostic advisor system with the secondary error prediction network is tested on the transients at various severity levels and degraded noise conditions. The results show that our error estimation scheme provides a useful measure of the validity of the advisor's output or diagnosis with considerable reduction in computational requirements over previous error estimation schemes.

An adaptive mesh refinement algorithm has been developed for non-linear computations in geomechanics, based on a smoothed stress–strain finite element formulation. This uses estimates of error in the incremental shear strain invariant to guide the regeneration of unstructured meshes at regular intervals during loading. Following each mesh-update, no re-analysis of previous increments with the new mesh is necessary. Algorithm performance has been investigated by analysing a passive earth pressure problem using a linear elastic-perfectly plastic Mohr–Coulomb soil model. Perfectly drained behaviour has been considered, as have partially drained situations using hydromechanical coupling, while undrained behaviour has been approximated using time steps close to zero. In all cases, mesh adaptivity has been successful in capturing regions of high strain gradient. The results have been compared with analytical solutions. Accurate computations of limit load and shear band orientation have been obtained for a wide range of material dilation angles. Copyright © 2000 John Wiley & Sons, Ltd.

A statistically constrained minimum mean squares error estimator (MiMSEE) has been shown to be useful in estimating internal resistivity distribution by the use of simulated data. In this study, the performance of the MiMSEE algorithm is tested by using measured data from resistor phantoms. The MiMSEE uses a priori information on body geometry, electrode position, statistical properties of tissue resistivities, instrumentation noise and linearization error to calculate the optimum inverse matrix which maps the surface potentials to unknown regional resistivities. In this study, the MiMSEE is also constrained with the variance-covariance of the modelling error to improve the estimation accuracy. The data are obtained from two different phantom geometries, namely five-region and thorax. Using the measured data, the estimations are realized and errors are calculated. Then, the results are compared with the results obtained by using a conventional least squares error estimator (LSEE). The five-region model results show similarity with the simulation study results of Baysal and Eyüboglu. On the thorax model, the total estimation error is 34.2% with the MiMSEE compared with 856% with the LSEE. It is concluded that the MiMSEE is more robust than the LSEE and applicable to measured data.

We construct a hierarchical a posteriori error estimator for a stabilized finite element discretization of convection‐diffusion equations with height Péclet number. The error estimator is derived without the saturation assumption and without any comparison with the classical residual estimator. Besides, it is robust, such that the equivalence between the norm of the exact error and the error estimator is independent of the meshsize or the diffusivity parameter. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012

Adaptive h- p procedures for high accuracy finite element analysis of two-dimensional linear elastic problems