Error Estimation Technique Research Articles

An enhanced extended isogeometric analysis with strong imposition of essential boundary conditions for crack problems using B++ splines

We present the extended isogeometric analysis (XIGA) based on B++ splines for linear elastic fracture problems. The proposed approach converts the control points associated with crack elements into collocation points on the crack surface. The opening of the crack surface is characterized by double-layer collocation points whereas the tip singularity is simulated by additional collocation points. In doing so, the additional control points associated with tip elements are replaced by the additional collocation points on the crack surface, which reduces the number of unknowns. The collocation points on the entire crack surface satisfy the Kronecker Delta property, which allows directly imposing Dirichlet boundary condition on the crack surface. A posteriori error estimation technique is also presented in this paper. The values of the normalised stress intensity factors (SIFs) are calculated by interaction integral technique. Finally, the suggested approach is verified by 2D benchmark problems.

A Voronoi diagram based framework for fast and accurate evaluation of 2D free-form profile errors

Free-form objects which have vast engineering applications are inspected using coordinate measuring machines (CMM) and non-contact scanners. The output from the devices, that is the measured point-set, must be registered with the reference profile and later inspected for the profile error. Registration is accomplished in two phases, that is coarse registration and fine registration. The most crucial step in the fine-registration stage is the profile error estimation at each point of the measured point-set. Generally, this is accomplished using the point-inversion method, which has relatively high time complexity. This work proposes a two-stage framework based on the Voronoi diagram concept for profile error estimation of 2D free-form profiles. A discretized profile is prepared for the measurement process in the first stage. The profile errors are estimated from the measured point-set using the discretized profile in the second stage. Two profile error estimation technique has been proposed based on Voronoi vertices and the Voronoi edges. The effective time-complexity of the proposed framework is O(lm log n). Implementations were executed on simulated NURBS profiles and practically measured CT-scanner data of two machined airfoils modelled as NURBS profiles. The validation results illustrate that the proposed framework’s accuracy is on par with that of the point-inversion method and is at least 56% faster than the latter; thus making it a fast and accurate alternative for CMM (operated in continuous scanning mode), CNC machine-based (in-situ) and vision-based 2D profile inspection.

Robust error assessment for reduced order vibro-acoustic problems

Robust and efficient automatic model order reduction is important in the design process of vibro-acoustic structures. For correct assessment of the reduction error without requiring the solution of the full order model, reliable error estimation techniques are required. Such error estimators can be used in iterative algorithms to automatically determine a reasonable order and expansion point distribution for reduced-order models. An accurate estimation of the approximation error is important in order to design an efficient adaptive model order reduction process requiring as little a priori knowledge of the system response as possible. In this contribution, different techniques for estimating the approximation error of reduced-order models are compared and employed in adaptive algorithms aiming at generating reduced-order models of vibro-acoustic systems. For many applications, vibro-acoustic systems need to be evaluated in a specific frequency range, as certain system characteristics are only observed in this region. Therefore, the employed error estimation methods are assessed regarding their effectiveness estimating the error in a particular frequency region.

Corporate integrity, external assurance and sustainability reporting quality: evidence from the Malaysian public listed companies

PurposeThis paper aims to examine the effect of corporate integrity and external assurance on Sustainability Reporting Quality (SRQ) of Malaysian public listed companies.Design/methodology/approachThe study uses a longitudinal sample of 2,463 firm-year observations of non-financial firms listed on the main board of Bursa Malaysia from 2015 to 2019. The study employed panel regression that is, Fixed Effect (FE) Robust Standard Error estimation technique to test its hypotheses.FindingsThe panel regression results reveal that corporate integrity and external assurance positively and significantly influence the quality of sustainability reporting. Though the positive association shows an improvement in the SRQ of the sampled firms, it needs an improvement as the disclosure is more general and qualitative than quantitative. The present improvement in SRQ might result from some regulatory changes like the Sustainability Practice Note 9 Updates of Bursa Malaysia 2017 and the Revised MCCG Principle A to C within the same period.Research limitations/implicationsThe study adopts a purely quantitative approach and call for a qualitative investigation in the area in the future.Practical implicationsThe study has policy implication for the government and regulators to strengthen compliance with the sustainability reporting guide and the Practice Note 9 Updates. It also has implication for corporate integrity and external assurance for companies, to enhance SRQ and achieve sustainable development.Originality/valueThe study bridged literature gaps by offering new insights and empirical evidence on the role of corporate integrity in SRQ, which has received no empirical attention in the Malaysian context.

Adaptive mesh refinement for finite element analysis of the free vibration disturbance of cylindrical shells due to circumferential micro-crack damage

PurposeModerately thick circular cylindrical shells are widely used as supporting structures or storage cavities in structural engineering, rock engineering, and aerospace engineering. In practical engineering, shells often work with micro-cracks or defects. The existence of micro-crack damage may result in the disturbance of dynamic behaviours and even induce accidental dynamic disasters. The free vibration frequency and mode are important parameters for the dynamic performance and damage identification analysis. In particular, stiffness weakening of the local damage region leads to significant changes in the vibration mode, which makes it difficult for the mesh generated in the conventional finite element method to capture a high-precision solution of the local oscillation.Design/methodology/approachIn response to the above problems, this study developed an adaptive finite element method and a crack damage characterisation method for moderately thick circular cylindrical shells. By introducing the inverse power iteration method, error estimation, and mesh subdivision refinement technique for the analysis of finite element eigenvalue problems, an adaptive computation scheme was constructed for the free vibration problem of moderately thick circular cylindrical shells with circumferential crack damage.FindingsBased on typical numerical examples, the established adaptive finite element solution for the free vibration of moderately thick circular cylindrical shells demonstrated its suitability for solving the high-precision free vibration frequency and mode of cylindrical shell structures. The any order frequency and mode shape of cracked cylindrical shells under the conditions of different ring wave numbers, crack locations, crack depths, and multiple cracks were successfully solved. The influences of the location, depth, and number of cracks on the disturbance of dynamic behaviours were analysed.Originality/valueThis study can be used as a reference for the adaptive finite element solution of free vibration of moderately thick circular cylindrical shells with cracks and lays the foundation for further development of a high-performance computation method suitable for the dynamic disturbance and damage identification analysis of general cracked structures.

Design of Iterative Fuzzy Radial Basis Function Neural Networks Based on Iterative Weighted Fuzzy C-Means Clustering and Weighted LSE Estimation

In this article, a reinforced iterative fuzzy radial basis function neural networks (IFRBFNN) is introduced as an augmented FRBFNN architecture generated through the iterative refinement process of weighted fuzzy C-means (WFCM) clustering and weighted least square error estimation. It is well known that the location of the clusters have an effect on the classification performance of the FRBFNN. The underlying idea behind this article is how to define the center points of clusters based on the data distribution analysis as well as the improvement of classification performance. In a nutshell, while the fuzzy C-means clustering to define the positions of radial basis functions (RBFs) based in the unsupervised learning manner is usually used in a FRBFNN, the parameters (cluster centers and their ensuing polynomial coefficients) refinement of FRBFNN through the proposed iterative method lead to superb classification performance by relocating the positions of RBFs over the input space implied by supervised learning. The idea of the proposed approach is to relocate the centers (prototypes) of the fuzzy clusters by using WFCM clustering algorithm and to re-estimate the coefficients by weighted least square estimation with the aid of the cross-entropy loss values of data so that the classification performance is improved. The weight related to each data is determined by some auxiliary information (i.e., the modified version of the cross-entropy loss value of each data). When it comes to the estimation of the coefficients of the consequent polynomial in IFRBFNN, they are estimated by the weighted least square error estimation technique, where the weights of the coefficients are defined based on the cross-entropy loss values. The auxiliary information such as the weights for relocation of the centers of RBFs and the weights for the re-estimation of the polynomial coefficients should be defined from the viewpoint of enhancement of the classification performance. The cross-entropy loss value of each data is able to meet the necessity of the required supervision signal. Several numerical experiments are provided to demonstrate the usefulness of the proposed design method for the FRBFNN for classification problems.

L2-Norm Based a Posteriori Error Estimates of Compressible and Nearly-Incompressible Elastic Finite Element Solutions

The displacement and stress-based error estimates in a posteriori error recovery of compressible and nearly-incompressible elastic finite element solutions is investigated in the present study. The errors in the finite element solutions, i.e., in displacement and stress, at local and global levels are computed in L2-norm of quantity of interest, namely, displacements and gradients. The error estimation techniques are based on the least square fitting of higher order polynomials to stress and displacement in a patch comprising of node/elements surrounding and including the node/elements under consideration. The benchmark examples of compressible and incompressible elastic bodies, with known solutions employing triangular discretization schemes, are implemented to measure the finite element errors in displacements and gradients. The mixed formulation involving displacement and pressure is used for incompressible elastic analysis. The performance of error estimation is measured in terms of convergence properties, effectivity and mesh required for predefined precision. The error convergence rate, in FEM original solution, recovered solution using displacement recovery-based and stress-based error recovery technique for stresses, are obtained as (1.9714, 2.8999, and 2.5018) and (0.9818, 1.7805, and 1.4952) respectively for compressible and incompressible self-loaded elastic plate benchmark example using higher order triangular elements. It is concluded from the study that displacement fitting technique for extracting higher order derivatives shows a very effective technique for recovery of compressible and nearly-incompressible finite element analysis errors.

A matrix-collocation method for solutions of singularly perturbed differential equations via Euler polynomials

In this paper, a matrix-collocation method which uses the Euler polynomials is introduced to find the approximate solutions of singularly perturbed two-point boundary-value problems (BVPs). A system of algebraic equations is obtained by converting the boundary value problem with the aid of the collocation points. After this algebraic system, the coefficients of the approximate solution are determined. This error analysis includes two theorems which consist of an upper bound of errors and an error estimation technique. The present method and error analysis are applied to three numerical examples of singularly perturbed two-point BVPs. Numerical examples and comparisons with other methods are given. These applications and comparisons show that the method gives effective results.

Nutrient optimization for plant growth in Aquaponic irrigation using Machine Learning for small training datasets

With the recent trends in urban agriculture and climate change, there is an emerging need for alternative plant culture techniques where dependence on soil can be eliminated. Hydroponic and aquaponic growth techniques have proven to be viable alternatives, but the lack of efficient and optimal practices for irrigation and nutrient supply limits its applications on a large-scale commercial basis. The main purpose of this research was to develop statistical methods and Machine Learning algorithms to regulate nutrient concentrations in aquaponic irrigation water based on plant needs, for achieving optimal plant growth and promoting broader adoption of aquaponic culture on a commercial scale. One of the key challenges to developing these algorithms is the sparsity of data which requires the use of Bolstered error estimation approaches. In this paper, several linear and non-linear algorithms trained on relatively small datasets using Bolstered error estimation techniques were evaluated, for selecting the best method in making decisions regarding the regulation of nutrients in hydroponic environments. After repeated tests on the dataset, it was decided that Semi-Bolstered Resubstitution Error estimation technique works best in our case using Linear Support Vector Machine as the classifier with the value of penalty parameter set to one. A set of recommended rules have been prescribed as a Decision Support System, using the output of the Machine Learning algorithm, which have been tested against the results of the baseline model. Further, the positive impact of the recommended nutrient concentrationson plant growth in aquaponic environments has been elaborately discussed.

Offset-Symmetric Gaussians for Differential Privacy

The Gaussian distribution is widely used in mechanism design for differential privacy (DP). Thanks to its sub-Gaussian tail, it significantly reduces the chance of outliers when responding to queries. However, it can only provide approximate (&#x03B5;,&#x03B4;(&#x03B5;))-DP. In practice, &#x03B4;(&#x03B5;) must be much smaller than the size of the dataset, which may limit the use of the Gaussian mechanism for large datasets with strong privacy requirements. In this paper, we introduce and analyze a new distribution for use in DP that is based on the Gaussian distribution, but has improved privacy performance. The so-called offset-symmetric Gaussian tail (OSGT) distribution is obtained through using the normalized tails of two symmetric Gaussians around zero. Consequently, it can still have sub-Gaussian tail and lend itself to analytical derivations. We analytically derive the variance of the OSGT random variable and &#x03B4;(&#x03B5;) of the single-dimensional OSGT mechanism. We extend the OSGT mechanism to <i>k</i>-dimensional queries, iteratively compute its &#x03B4;_<i>k</i>(&#x03B5;), derive its R&#x00E9;nyi differential privacy, and study its composition. Numerical results show the OSGT mechanism can offer better privacy-utility performance compared to the Gaussian and Laplace mechanisms. We also derive a method for post processing the output of the OSGT mechanism to approximate the query based on the minimum mean square error (MMSE) estimation technique. The simulation results of such processing confirm the efficacy of the OSGT mechanism over the Gaussian mechanism.

Confidence Level Estimation for Advanced Decision-Making in Transformer Short-circuit Fault Diagnosis

Frequency response analysis (FRA) is almost certainly the most efficient and well-established method for evaluating the mechanical integrity of transformer active part. FRA has a solid background in both industrial practice and academic research. Besides conventional visual analysis, a statistical analysis has been applied to the FRA data interpretation. The existing standards and recent studies have considered different statistical indicators (SIs) on an individual basis. However, the utility of each SI in its &#x201C;silo&#x201D; may lead to different and possibly contradictory decisions. Inspired by the bolstered error estimation technique used in pattern recognition, this article presents a new method that can utilize multiple SIs obtained from the FRA data to classify transformer operating conditions and estimate the level of confidence in decisions made. The probabilistic method proposed herein is an attempt to bridge the gap between these decision-making silos. The practical implementation of the proposed technique on distribution and power transformers revealed a reliable interpretation and classification results. At the same time, having an estimate for the level of confidence in the decision made by the method further helps engineers and utility operators make informed decisions and have a better understanding of the level of the transformer fault severity.

Application of Vieta–Lucas Series to Solve a Class of Multi-Pantograph Delay Differential Equations with Singularity

The main focus of this paper was to find the approximate solution of a class of second-order multi-pantograph delay differential equations with singularity. We used the shifted version of Vieta–Lucas polynomials with some symmetries as the main base to develop a collocation approach for solving the aforementioned differential equations. Moreover, an error bound of the present approach by using the maximum norm was computed and an error estimation technique based on the residual function is presented. Finally, the validity and applicability of the presented collocation scheme are shown via four numerical test examples.

Microstructure classification in the unsupervised context

Traditional microstructure classification requires human annotations provided by a subject matter expert. The requirement of human input is both costly and subjective and cannot keep up with the current volume of experimentally and computationally generated microstructure images. In this work, we develop a framework that is capable of reducing the cost of human annotation in this process by leveraging novel machine learning procedures for class discovery and label assignment. To reduce the penalty of a poor label assignment made by this automated process, labels are only assigned to high-confidence observations while ambiguous data are left unlabeled. Semi-supervised classification is then employed to leverage the high- and low-confidence label assignments, and a novel generalization of an established semi-supervised error estimation technique to the multi-class context is introduced to assess the resulting classifiers. Finally, it is shown that this framework can be used to produce highly accurate classifiers over microstructure image class taxonomies which are discovered solely through data-driven methods and which display consistent structural trends within and distinct morphological differences between classes.

An operational matrix method to solve the Lotka–Volterra predator–prey models with discrete delays

In this paper, we consider the Lotka–Volterra predator–prey model with discrete delays. We apply the operational matrix method in [20] to this model which corresponds to a problem of nonlinear differential equations. The method is based on the operational matrices of the standard bases functions. In order to find the polynomial approximations of the known functions in the equation, the operational matrices are created by the least-squares method. The method converts the problem into a system of algebraic equations. By solving this system, the coefficients are determined. The approximate solutions are obtained by writing the coefficients in form of the approximate solutions. In addition, the accuracy of the approximate solution is checked and an error estimation technique is presented. The method is applied to some examples and the numerical results are reported. Numerical applications and comparisons with other methods show the efficiency of the current one.

Advances in Error Estimation and Uncertainty Quantification for Numerical Methods in CEM

We summarize the application of duality-based techniques for error estimation and error control along with uncertainty quantification. The approach yields quality discretizations for estimating uncertain effects in a target quantity of interest (QoI) and naturally extends to the treatment of multiple quantities of interest. By adopting the adjoint approach in both error estimation and uncertainty quantification, expensive computations may be reused. The dual solution enables us to tie changes in the QoI to changes in the discretization, resulting in a significant enhancement of efficiency. We demonstrate examples of the proposed approaches to illustrate the application to scattered quantities of interest.

Error estimate for indirect spectral approximation of optimal control problem governed by fractional diffusion equation with variable diffusivity coefficient

In this paper an indirect spectral method of an optimal control problem governed by a space-fractional diffusion equation with variable diffusivity coefficient is studied. First-order optimality conditions of the proposed model are derived and the regularity of the solutions is analyzed. Indirect spectral methods via weighted Jacobi polynomials are built up based on the “first optimize, then discretize” strategy, and a priori error estimates of the discrete optimal control problem in weighted norms are derived. As proposed indirect spectral discrete schemes are designed to accommodate the impact of the variable coefficients, which are complicated and not formulated under the variational framework, conventional error estimate techniques of the discrete space-fractional optimal control problems do not apply. Novel treatments of the discrete variational inequality are developed to resolve the aforementioned issues and to support the error estimates. Numerical examples are presented to verify the theoretical findings.

Adaptive Stepsize Control for Extrapolation Semi-Implicit Multistep ODE Solvers

Developing new and efficient numerical integration techniques is of great importance in applied mathematics and computer science. Among the variety of available methods, multistep ODE solvers are broadly used in simulation software. Recently, semi-implicit integration proved to be an efficient compromise between implicit and explicit ODE solvers, and multiple high-performance semi-implicit methods were proposed. However, the computational efficiency of any ODE solver can be significantly increased through the introduction of an adaptive integration stepsize, but it requires the estimation of local truncation error. It is known that recently proposed extrapolation semi-implicit multistep methods (ESIMM) cannot operate with existing local truncation error (LTE) estimators, e.g., embedded methods approach, due to their specific right-hand side calculation algorithm. In this paper, we propose two different techniques for local truncation error estimation and study the performance of ESIMM methods with adaptive stepsize control. The first considered approach is based on two parallel semi-implicit solutions with different commutation orders. The second estimator, called the “double extrapolation” method, is a modification of the embedded method approach. The introduction of the double extrapolation LTE estimator allowed us to additionally increase the precision of the ESIMM solver. Using several known nonlinear systems, including stiff van der Pol oscillator, as the testbench, we explicitly show that ESIMM solvers can outperform both implicit and explicit linear multistep methods when implemented with an adaptive stepsize.

A Laguerre approach for solving of the systems of linear differential equations and residual improvement

In this study, a collocation method based on Laguerre polynomials is presented to numerically solve systems of linear differential equations with variable coefficients of high order. The method contains the following steps. Firstly, we write the Laguerre polynomials, their derivatives, and the solutions in matrix form. Secondly, the system of linear differential equations is reduced to a system of linear algebraic equations by means of matrix relations and collocation points. Then, the conditions in the problem are also written in the form of matrix of Laguerre polynomials. Hence, by using the obtained algebraic system and the matrix form of the conditions, a new system of linear algebraic equations is obtained. By solving the system of the obtained new algebraic equation, the coefficients of the approximate solution of the problem are determined. For the problem, the residual error estimation technique is offered and approximate solutions are improved. Finally, the presented method and error estimation technique are demonstrated with the help of numerical examples. The results of the proposed method are compared with the results of other methods

A general adaptive finite element eigen-algorithm stemming from Wittrick-Williams algorithm

Stemming from Wittrick-Williams algorithm for transcendental eigenproblems, a general adaptive finite element (FE) procedure is proposed for general linear eigenproblems. The pivot of the proposed procedure is the counting method based on Sturm sequences, a simplified version of the Wittrick-Williams algorithm for linear matrix eigenproblems, based on which a general eigen-algorithm is developed by integrating a number of effective approaches such as a two-phased divide-and-conquer scheme, a guided-and-guarded technique for eigenvalue seeking, the inverse and subspace iterations for mode finding, etc. The adopted error estimator in the adaptive FE procedure is the recently developed one which is based on the element energy projection (EEP) technique for pointwise error estimation and hence is able to control errors by maximum norm with local mesh refinement. Representative numerical examples of plates and shells are given to show the validity, reliability and efficiency of the proposed procedure.

A computational approach with residual error analysis for the fractional-order biological population model

ABSTRACT In this study, a fractional Bernstein series solution method has been submitted to solve the fractional-order biological population model with one carrying capacity. The numerical method has been implemented by an effective algorithm written on the computer algebraic system Maple 15. An error-bound analysis is performed by using a process similar to the RK45 method. An error estimation technique relating to residual function is presented. In the numerical application, the variations in the population of prey and predator with respect to time and situations of these two species relative to each other are plotted. The outputs obtained from our method are compared with the homotopy perturbation Sumudu transform method and reproducing kernel Hilbert space method. The approximate solutions gained from the Bernstein series method are consistent with those of other methods. The advantage of our method is that it requires less computational cost compared with methods involving more complex operations.