Proposed Error Estimator Research Articles

A strong adaptive piecewise model order reduction method for large-scale dynamical systems with viscoelastic damping

This paper develops a novel and strong adaptive piecewise model order reduction (PMOR) method for large-scale dynamical systems with viscoelastic damping. Based on polynomial least squares approximations, the system is piecewise approximated by several kth-order (k≥2) dynamical systems in a wide target frequency band, in which the orders are adaptively determined with a curvature-based method. Then the convergent reduced-order models (ROMs) of the approximate systems are obtained gradually. In the above process, for each approximate system, the orthonormal basis is constructed iteratively via the kth-order Arnoldi method to span a projection subspace. To accelerate the convergence, an influence coefficient method and an order-dependent method are proposed to automatically determine the initial order of the ROM and the order increments, respectively. More importantly, a proposed error estimation strategy can predict all forms of estimated relative errors. According to the study of these forms, a comparison-selection method is presented to determine the final ROM for the whole target band interval by interval. Four examples comprehensively validate the strong adaptive ability, high efficiency and wide applicability of the PMOR method.

Adaptive direct discontinuous Galerkin method for elliptic equations

In this paper, adaptive direct discontinuous Galerkin method for second order elliptic equations is studied in two dimensional. A numerical flux with general weighted averages and proper weights for interface problems is introduced. In addition to the optimal a priori error estimator, we propose a residual-type a posteriori error estimator. The global upper bound and local lower bound for the error in DG norm are established. Several numerical examples are carried out to confirm the reliability and efficiency of the proposed error estimator and the corresponding adaptive direct discontinuous Galerkin method.

A three-time-level a posteriori error estimator for GS4-2 framework: Adaptive time stepping for second-order transient systems

The most commonly used adaptive time stepping technique is mainly based on the simple notion of a local error estimation. Accordingly, a variety of error estimators have mostly been designed for particular algorithms on an algorithm by algorithm selection basis. In this paper, a novel design of a three-time-level general purpose a posteriori error estimator is demonstrated for second-order transient systems, providing features of: (a) being universal for the most popular and entire class of second-order accurate unconditionally stable implicit LMS methods with/without numerical dissipation developed over the past few decades and on newer and optimal designs (the focus is on unconditionally stable implicit LMS methods and not explicit methods); (b) being simple for numerical implementation in the general purpose GS4-2 framework encompassing the entire class of LMS methods that are second order time accurate such that only the acceleration at three time levels is required; (c) preserving the third-order convergence in time for the local error; (d) predicting accurate local error such that the effectivity index is close to unity. In contrast to the existing a posteriori error estimators which are designed for certain algorithms and on algorithm by algorithm basis, the newly proposed approach can be directly applied to all the known and existing and new second-order accurate time integration algorithms in the entire class of the unconditionally stable implicit LMS family under the umbrella of the GS4-2 framework; thereby, the cumbersome work of designing different error estimators for different algorithms is avoided. In particular, the proposed error estimator can be directly applied also to the novel V0 family of algorithms without any limitation. Besides the limited selection of estimators existing within the U0 family of algorithms where most of the traditional methods belong to, the remainder of the algorithms within the U0 family and several others such as the V0 family do not have estimators that can be readily designed as the V0 family of algorithms only exist in the GS4-2 framework, which are entirely novel and new and are highly competitive. The numerical examples of stiff nonlinear spring pendulum, van der Pol equation, wave propagation, and phase-field modeling are described here to show the benefits to the computational cost whilst preserving generality, accuracy, and the like when the adaptive time stepping approach coupled with the proposed error estimator is applied to the transient simulations.

Level‐wise strain recovery and error estimation for natural element hierarchical plate models

AbstractDiffering from the finite element method, most of mesh‐free methods are restricted to two‐dimensional (2D) problems owing to the difficulty in constructing three‐dimensional (3D) grids. But, for 3D elastic structures, this limitation could be effectively overcome when hierarchical models are employed. In the hierarchical models of elastic structures, only the in‐plane displacement field is needed to be approximated because the thickness‐wise displacement field is assumed by polynomials. In this context, this paper is concerned with the reliable numerical implementation of hierarchical models for plate‐like structures using 2D natural element method (NEM). However, one critical problem in implementing the hierarchical model is to secure the continuity in strain and stress fields. To resolve this problem, this paper presents a level‐wise strain recovery method. In addition, a recovery‐based error estimation is introduced to predict the modeling error and the numerical approximation error. The proposed methods are illustrated and validated through the numerical experiments. It is confirmed that the level‐wise strain recovery successfully and CPU‐time effectively provides the continuous stress resultants and the proposed error estimator accurately predicts the modeling and approximation errors.

On error estimation for reduced-order modeling of linear non-parametric and parametric systems

Motivated by a recently proposed error estimator for the transfer function of the reduced-order model of a given linear dynamical system, we further develop more theoretical results in this work. Moreover, we propose several variants of the error estimator, and compare those variants with the existing ones both theoretically and numerically. It is shown that some of the proposed error estimators perform better than or equally well as the existing ones. All the error estimators considered can be easily extended to estimate the output error of reduced-order modeling for steady linear parametric systems.

A Hierarchical Error Estimator for the MSFEM for the Eddy Current Problem in 3-D

The aim of this article is to develop an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a posteriori</i> hierarchical error estimator for the multiscale finite element method (MSFEM). The MSFEM is used to solve the linear eddy current problem in a stack of iron sheets. It allows for a calculation of the solution without having to resolve each sheet in the finite element mesh. A hierarchical local error estimator for nodal elements and edge elements is adapted to the multiscale setting. The estimator allows for adaptive p-refinement on the coarse multiscale mesh. Numerical examples show an increased order of convergence compared to uniform refinement. The proposed error estimator increases the efficiency of the MSFEM, requiring even fewer degrees of freedom. It is the first error estimator presented for the 3-D MSFEM for the eddy current problem.

Some Contributions for Antenna 3D Far Field Characterization at Terahertz.

The three-dimensional (3D) characterization of antenna far field patterns at terahertz frequencies is addressed. This task is challenging, because the phase of the electric field is difficult to measure accurately and reliably. From the sub-millimeter wave range, the small wavelength indeed significantly increases the impact of mechanical and electrical errors. Models and procedures to estimate these errors are proposed to mitigate their effects. The 3D far field patterns of a circularly polarized horn measured at 300 GHz and a multibeam pillbox antenna at 270 GHz are shown. The agreement between the 3D measurements and the two-dimensional (2D) patterns of reference as well as the radiated pattern before and after correction demonstrates the interest of the proposed approach and experimentally validate the proposed error estimation procedures. The methodology can be applied to direct far field measurement facilities as well as compact antenna test ranges.

Explicit a posteriori and a priori error estimation for the finite element solution of Stokes equations

For the Stokes equation over 2D and 3D domains, explicit a posteriori and a priori error estimation are novelly developed for the finite element solution. The difficulty in handling the divergence-free condition of the Stokes equation is solved by utilizing the extended hypercircle method along with the Scott-Vogelius finite element scheme. Since all terms in the error estimation have explicit values, by further applying the interval arithmetic and verified computing algorithms, the computed results provide rigorous estimation for the approximation error. As an application of the proposed error estimation, the eigenvalue problem of the Stokes operator is considered and rigorous bounds for the eigenvalues are obtained. The efficiency of proposed error estimation is demonstrated by solving the Stokes equation on both convex and non-convex 3D domains.

An accurate and simple universal a posteriori error estimator for GS4-1 framework: Adaptive time stepping in first-order transient systems

The “Generalized Single Step Single Solve” framework for first-order transient systems (GS4-1) is considered to develop adaptive time stepping via a newly proposed a posteriori error estimator in this paper. Traditional approaches design and employ particular error estimators for particular algorithms on algorithm by algorithm selection basis. In sharp contrast, in this paper, an accurate and simple universal a posteriori error estimator that is agnostic is newly designed to foster features that enable the adaptive time stepping for the entire subset of second order time accurate linear multi-step (LMS) algorithms in the GS4-1 family. The core contributions are: (a) The estimated local error has the same order of convergence and error constant as that of the exact local error; thereby the underlying issues that under-/over-estimate the error for optimizing the time step are addressed with rigour. The validation tests for linear/nonlinear problems show the third-order convergence rate in time for the local error, and the overlapped profiles of estimated local error and exact local error demonstrate the accuracy of the proposed error estimator. (b) Unlike several previously reported error estimators designed for some specific existing time integration algorithms in the literature, the proposed error estimator is universal, and is independent of the algorithmic parameters; thereby it can be applied to all the numerically dissipative/non-dissipative schemes without any limitations in the GS4-1 family. (c) The adaptive time stepping computational strategy emanating from the proposed error estimator achieves an excellent balance between the solution accuracy and CPU cost. In addition, several linear/nonlinear heat transfer problems with multi-degree of freedom are solved via the proposed method to further demonstrate the effectiveness of the numerical implementation.

Form error on-line estimation and compensation for non-circular turning process

Parts with non-circular profiles are widely used in the automobile, navigation and aerospace areas, such as camshafts and convex ellipse pistons etc. The form accuracy of non-circular profiles directly affects the transmission accuracy and efficiency. However, existing non-circular turning studies mainly focused on improving the positioning accuracy of fast radial servo drives but neglected the form accuracy caused by the mismatch motions among all feed drives. In order to improve the form accuracy of non-circular turning, this paper proposes a novel control structure which treats the form error as a direct objective. The form error is estimated in real time at first within preset tolerance of calculation error. Then an on-line control strategy is designed to reduce form error through compensating the motion commands of the fast radial servo drive. The control loops of individual drives are decoupled and only position commands of the fast radial servo drive are compensated without touching the inside velocity and current loops. Comparisons with traditional contouring error control structure based on cross-coupled compensations of all axes show that the proposed error compensation strategy has better dynamic performance. Experimental results on the in-house developed non-circular turning prototype show that the proposed form error estimation and compensation methodology significantly improves the contouring accuracy than traditional controllers which only focused on tracking error control of individual drives.

A New Error Estimator for Reduced-Order Modeling of Linear Parametric Systems

Efficient error estimation is important for reliable reduced-order modeling with guaranteed accuracy. We propose an error estimator for reduced-order modeling of linear parametric dynamical systems. The error estimator estimates the error of the reduced transfer function in the frequency domain and can be easily extended to the output error estimation of the reduced-order models (ROMs) for linear steady parametric systems. It is tight and cheap to compute. Using the error estimator, the ROM can be adaptively obtained with high reliability. Numerical results show that the error estimator can accurately estimate the true error even for transfer functions with many resonances. Compared with an existing error bound, the proposed error estimator can be orders of magnitudes sharper and needs much less computational time.

Droplet Sizing in the Spray of a Fuel Injector Using Wavelet Analysis

The paper discusses the applied methods for studying the structure of the two-phase flow. Measurements of the size of liquid droplets in a spray of fuel injector were carried out using two methods: Interferometric Particle Imaging (IPI) and Glare Point Technique (GPT). The combined use of these methods allows to significantly expand the range of measured droplet sizes, and also provides an opportunity for cross-validation of methods and determining the level of errors. In this paper, we tested the IPI method using wavelet processing of interference drop images. As a result, it was shown that IPI images can have a highly inhomogeneous structure. This leads to a considerable variation in the period of the IPI-image and, as a consequence, to an increase of the error in determining the drop size. The proposed error estimation algorithm allows one to introduce an additional selection criterion and obtain statistically comparable results obtained by the IPI and GPT methods.

Parameter Estimation and Error Calibration for Multi-Channel Beam-Steering SAR Systems

Multi-channel beam-steering synthetic aperture radar (multi-channel BS-SAR) can achieve high resolution and wide-swath observations by combining beam-steering technology and azimuth multi-channel technology. Various imaging algorithms have been proposed for multi-channel BS-SAR but the associated parameter estimation and error calibration have received little attention. This paper focuses on errors in the main parameters in multi-channel BS-SAR (the derotation rate and constant Doppler centroid) and phase inconsistency errors. These errors can significantly reduce image quality by causing coarser resolution, radiometric degradation, and appearance of ghost targets. Accurate derotation rate estimation is important to remove the spectrum aliasing caused by beam steering, and spectrum reconstruction for multi-channel sampling requires an accurate estimate of the constant Doppler centroid and phase inconsistency errors. The time shift and scaling effect of the derotation error on the azimuth spectrum are analyzed in this paper. A method to estimate the derotation rate is presented, based on time shifting, and integrated with estimation of the constant Doppler centroid. Since the Doppler histories of azimuth targets are space-variant in multi-channel BS-SAR, the conventional estimation methods of phase inconsistency errors do not work, and we present a novel method based on minimum entropy to estimate and correct these errors. Simulations validate the proposed error estimation methods.

Formula omitted]-adaptive discontinuous Galerkin methods for non-stationary convection–diffusion problems

An a posteriori error estimator for the error in the (L2(H1)+L∞(L2))-type norm for an interior penalty discontinuous Galerkin (dG) spatial discretisation and backward Euler temporal discretisation of linear non-stationary convection–diffusion initial/boundary value problems is derived, allowing for anisotropic elements. The proposed error estimator is used to drive an hp-space–time adaptive algorithm wherein directional mesh refinement is employed to give rise to highly anisotropic elements able to accurately capture layers. The performance of the hp-space–time adaptive algorithm is assessed via a number of standard test problems characterised by sharp and/or moving layers.

Weak Galerkin based a posteriori error estimates for second order elliptic interface problems on polygonal meshes

Abstract In this paper, we present a posteriori error estimate of weak Galerkin (WG) finite element methods based on the second order elliptic interface problems. This estimate can be applied to polygonal meshes or meshes with hanging nodes. The reliability and efficiency of the designed error estimator has been proved in this work. Extensive numerical tests are performed to validate our algorithm. These results demonstrate the effectiveness of the adaptive mesh refinement guided by the proposed error estimator.

Statistical Model of Total Target Registration Error in Image-Guided Surgery

In a paired-point rigid registration, target registration error (TRE) is deemed to be the most important quality metric. TRE usually cannot be directly measured, and thus many TRE estimation algorithms have been proposed. However, target localization errors (TLEs) in two spaces are not considered in the definition of TRE. In this paper, we propose a new type of evaluation metric that is referred to as total TRE (TTRE) at a given target point with TLE incorporated. Statistics including mean, root mean square (rms), and covariance matrix of TTRE are derived without making any assumption of the TLE magnitude. TTRE and fiducial registration error (FRE) are proved to be uncorrelated when an ideal weighting scheme is adopted in solving the registration problem. The proposed error model is validated through extensive experiments. In the first experiment with random fiducials and targets, in 90% of the test cases, there shows no difference between the predicted and simulated TTRE statistics when six fiducials are used. In the second experiment of deep-brain stimulation surgery, the mean value of CC(TTRE,FRE) being $8.9246\times 10^{-4}\pm 0.0389 $ was observed, which indicates that TTRE and FRE are uncorrelated. In the third experiment of surgical tool-tip tracking, the mean and standard deviation of percentage differences between predicted and simulated TTRE rms values are $2.22\%\pm 0.77\%$ for the planar tool and $2.62\%\pm 0.59\%$ for the textral tool. In summary, our proposed algorithm can well model the TTRE metric. Note to Practitioners —This paper was motivated by considering the existence of target localization error in two spaces to be registered in a rigid point-based registration. Given fiducial localization error (FLE) and TLE probability distributions, the proposed TTRE model gives the covariance matrix and root mean square of TTRE at a given target point. The possible applications of the proposed TTRE model in an image-guided surgery (IGS) include: 1) estimation of surgical tool-tip tracking error distribution when pivot calibration uncertainty is considered; 2) accuracy evaluation of an image-to-patient registration; 3) updating of a preoperative surgical plan on a computed tomography/MRI image to consider the registration error at a given target point; and 4) estimation of real-time tool-tip error in touching a desired “true” surgical target, such as a tumor during surgery. To use the proposed error estimation algorithm, FLE and TLE in two spaces to be registered have to be known or estimated in advance to a registration process. The proposed error model is validated through extensive simulations.

Element Free Galerkin Post-processing Technique Based Error Estimator for Elasticity Problems

The study present a Mesh Free based post-processing technique for asymptotically (upper) bounded error estimator for Finite Element Analyses of elastic problems. The proposed technique uses Galerkin Element Free procedure for recovery of the displacement derivatives over a patch of nodes in radial domains. The radial nodes patches are used to construct the trial shape functions utilizing the moving least-squares (MLS) techniques. The proposed technique has been tested on three benchmark elastic problems discretized using 4-node quadrilateral elements. The recovered nodal stresses are utilized to calculate the error in finite element solution in energy norm. The study also demonstrates adaptive analysis application of proposed error estimator. The performance of proposed error estimator based on mesh independent node patches has been compared with that of mesh dependent node patches based Zienkiewicz-Zhu (ZZ) error estimator on structured and unstructured mesh. The improved results of the proposed error estimator in terms of convergence rate and effectivity are obtained. It is shown that present study incorporates the superiority of the Mesh Free Galerkin method into finite element analysis environment.

A Posteriori Error Analysis for the Weak Galerkin Method for Solving Elliptic Problems

In this paper, we study the a posteriori error estimate for weak Galerkin finite element method solving elliptic problems. A residual type error estimator is proposed and is proven to be reliable and efficient. This estimator provides global upper and lower bounds on the exact error in a discrete [Formula: see text]-norm. Numerical experiments are given to illustrate the effectiveness of the proposed error estimator.

A priori and a posterior error estimate of new weak Galerkin finite element methods for second order elliptic interface problems on polygonal meshes

Abstract In this paper, we present a posteriori error estimate of a new weak Galerkin (WG) finite element method for the second order elliptic interface problems. Instead of introducing the local weak derivatives on each element, the bilinear form is computed through the inner product of traditional defined derivatives. The proposed numerical scheme is symmetric and positive definite, and can be applied to polygonal meshes. In the a posteriori error estimate, we construct a error indicator that can be applied to polygonal meshes or meshes with hanging nodes. The reliability and efficiency of the designed error estimator have been proved in this work. Extensive numerical tests are performed to validate our algorithm. These results demonstrate the effectiveness of the adaptive mesh refinement guided by the proposed error estimator.

On the robustness of variational multiscale error estimators for the forward propagation of uncertainty

The numerical simulation of physical phenomena and engineering problems can be affected by numerical errors and various types of uncertainties. Characterizing the former in computational frameworks involving system parameter uncertainties becomes a key issue. In this work, we study the behavior of new variational multiscale (VMS) error estimators for the propagation of parametric uncertainties in a Convection–Diffusion–Reaction (CDR) problem. A sensitivity analysis is performed to assess the performance of the error estimator with respect to the mesh discretization and physical parameters (here, the viscosity value and advection velocity). Three different manufactured analytical solutions are considered as benchmarking tests. Next, the performance of the VMS error estimators is evaluated for the CDR problem with uncertain input parameters. For this purpose, two probabilistic models are constructed for the viscosity and advection direction, and the uncertainties are propagated using a polynomial chaos expansion approach. A convergence analysis is specifically carried out for different configurations, corresponding to regimes where the CDR operator is either smooth or non-smooth. An assessment of the proposed error estimator is finally conducted for the three tests, considering both the viscous- and convection-dominated regimes.