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Aiming at the problem of error estimation of smart meters in distribution network, a method of error estimation of smart meters based on particle swarm optimization convolutional neural network is proposed. This method establishes an intelligent energy meter error estimation model through data collection, data prediction, and preprocessing. To address the convergence issue in training, the interlayer distribution of weights is adjusted to improve training quality. This method fully utilizes template calibration information to transform indicator detection under complex conditions into simple and effective isometric segmentation, transforming label recognition from complex text detection and recognition tasks to simple and efficient binary detection tasks, with better robustness. The effectiveness and high robustness of the proposed method have been demonstrated through experimental verification.

Particle-based meshless methods are commonly used in fluid dynamics and solid mechanics involving finite deformations, owing to their ability to break through the limitations imposed by mesh topology. Particle distribution usually plays a crucial role in determining the accuracy of simulation results for them. In this study, an error estimation was first conducted to ascertain the requirements for particle distribution necessary for accurate simulations. It was revealed that the sawtooth and chaotic particle distributions can significantly reduce numerical accuracy and even cause numerical instability in simulations. To address this issue, we propose a particle rearrangement approach including a pseudo hydrostatic pressure treatment to achieve body-fitted particles and a geometric smoothing method based on the metric of a point cloud unit to optimize chaotic particle distributions for improved regularity. The particles are iteratively moved according to their proximity to neighboring particles. Notably, the number of particles remains constant throughout the smoothing procedure, neither particles inserted or removed, nor particles overflow or volume expansion. This new approach facilitates the generation of body-fitted and relatively regular planar and surface particle distributions, meeting the requirements for arbitrarily complex shape particle distributions. The effectiveness of this method has been demonstrated through two-dimensional SPH simulations for problems of the flow around a cylinder, dam break, and Taylor–Green vortex.

A residual type a posteriori error estimator for finite elements is analyzed using a new technique. In this case, the error estimate is the result of two consecutive projections of the exact error on two finite-dimensional subspaces. The analysis introduced in this paper is based on a probabilistic approach, that is, the idea is to assess the average value of the effectivity index (the ratio estimated error over exact error) by assuming the randomness of the exact error. The average value characterizes the mean behavior of the estimator and it is found to be related with some geometric properties of the subspaces. These geometric properties are obtained from the standard matrices of the linear systems arising in the formulation of the finite element method.

We consider a class of abstract evolutionary variational inequalities arising in the study of contact problems for viscoelastic materials. We prove an existence and uniqueness result, using arguments of nonlinear evolutionary equations and Banach's fixed point theorem. We then consider the numerical approximation of the problem. By introducing a fully discrete scheme we show the existence of a unique solution and derive error estimates on the approximate solutions, which represent the main results of the work. We apply the abstract results in the study of two contact problems involving viscoelastic materials. The first one is a Signorini frictionless problem for Kelvin–Voigt materials and the second one is a frictional contact problem for linear Maxwell materials. We provide the variational and numerical analysis of the first problem including error estimates and numerical simulations and we prove the unique weak solvability of the second problem.

The implementation of finite element methods (FEMs) for nonlocal models with a finite range of interaction poses challenges not faced in the partial differential equations (PDEs) setting. For example, one has to deal with weak forms involving double integrals which lead to discrete systems having higher assembly and solving costs due to possibly much lower sparsity compared to that of FEMs for PDEs. In addition, one may encounter nonsmooth integrands. In many nonlocal models, nonlocal interactions are limited to bounded neighborhoods that are ubiquitously chosen to be Euclidean balls, resulting in the challenge of dealing with intersections of such balls with the finite elements. We focus on developing recipes for the efficient assembly of FEM stiffness matrices and on the choice of quadrature rules for the double integrals that contribute to the assembly efficiency and also posses sufficient accuracy. A major feature of our recipes is the use of approximate balls, e.g. several polygonal approximations of Euclidean balls, that, among other advantages, mitigate the challenge of dealing with ball-element intersections. We provide numerical illustrations of the relative accuracy and efficiency of the several approaches we develop.

This paper introduces a novel approach to approximate a broad range of reaction–convection–diffusion equations using conforming finite element methods while providing a discrete solution respecting the physical bounds given by the underlying differential equation. The main result of this work demonstrates that the numerical solution achieves an accuracy of $O(h^k)$ in the energy norm, where $k$ represents the underlying polynomial degree. To validate the approach, a series of numerical experiments had been conducted for various problem instances. Comparisons with the linear continuous interior penalty stabilised method, and the algebraic flux-correction scheme (for the piecewise linear finite element case) have been carried out, where we can observe the favorable performance of the current approach.

Knowing the roles of mathematics and computation in experimental science is important for computational biology because these roles determine to a great extent how research in this field should be pursued and how it should relate to biology in general. The present paper examines the epistemology of computational biology from the perspective of modern science, the underlying principle of which is that a scientific theory must have two parts: (1) a structural model, which is a mathematical construct that aims to represent a selected portion of physical reality and (2) a well-defined procedure for relating consequences of the model to quantifiable observations. We also explore the contingency and creative nature of a scientific theory. Among the questions considered are: Can computational biology form the theoretical core of biology? What is the basis, if any, for choosing one particular model over another? And what is the role of computation in science, and in biology in particular? We examine how this broad epistemological framework applies to important statistical methodologies pertaining to computational biology, such as expression-based phenotype classification, gene regulatory networks, and clustering. We consider classification in detail, as the epistemological issues raised by classification are related to all computational-biology topics in which statistical prediction plays a key role. We pay particular attention to classifier-model validity and its relation to estimation rules.

The upper frequency limit of vibroacoustic calculations with finite element methods is usually concluded from resolution rules for the minimal wavelength encountered in the problem. Here, we derive general resolution rules that account for the pollution effect in finite element solutions of time-harmonic equations. These rules are given for the Helmholtz equation and the Bernoulli beam equation. The latter are based on an analysis of numerical dispersion for finite difference solutions. The theoretical results are given in the broader context of industrial vibroacoustic computations in the medium-frequency range. The governing equations of deterministic vibroacoustic computations and statistical energy analysis are reviewed with the goal to indicate, respectively, upper and lower frequency bounds for the applicability of either model. From the discussion of priorities in industrial application, open questions for theoretical investigations are deduced.

This paper presents a posteriori error estimation and h-adaptive refinement techniques for transient dynamic analysis of stiffened plates/shells using the finite element method (FEM). We furnish the formulation of stiffness and mass matrices for finite element (FE) models, QL9S2 and QUAD4S2 for dynamic analysis of plates/shells with arbitrarily-located concentric/eccentric stiffeners. Procedures for computing a posteriori errors for spatial and temporal errors have been presented for transient dynamic problems. An h-adaptive refinement strategy for stiffened plate/shell panels by employing QL9S2 and QUAD4S2 FE models has been discussed. An adaptive time stepping scheme, which is to be used with the time errors for quality control of the time steps, has also been presented. Numerical studies have been conducted to evaluate the efficacy of the error estimators and the adaptive mesh refinement and time stepping algorithm. The spatial error estimator for transient dynamic analysis is found to exhibit monotonic convergence at all time steps. The temporal error estimator for transient dynamic analysis in association with the adaptive time stepping is able to compute more accurate and reliable time steps to keep the time errors within the specified tolerance limits.

The accuracy issues of on-wafer noise characterization for a linear noisy two-port are presented in this paper. It starts with the description of a microwave noise measurement system and the possible source of error due to the microwave power meter in the measurement system. With the description of noise characterization techniques, this paper reviews a couple of methods for noise parameter extraction to handle the errors in the measured noise powers, noise factors, and source admittances. It also presents the methods to extract the physical noise parameters and to take care of different source admittances in the hot and cold states for accuracy enhancement.

Classical linear wavelet representations of images have the drawback that they are not optimally suited to represent edge information. To overcome this problem, nonlinear multiresolution decompositions have been designed to take into account the characteristics of the input signal/image. In our previous work^20,22,23 we have introduced an adaptive lifting framework, that does not require bookkeeping but has the property that it processes edges and homogeneous image regions in a different fashion. The current paper discusses the effects of quantization in such an adaptive wavelet decomposition. We provide conditions for recovering the original decisions at the synthesis and for relating the reconstruction error to the quantization error. Such an analysis is essential for the application of these adaptive decompositions in image compression.

In this paper, the interval analysis method is introduced to calculate the bounds of the structural displacement responses with small uncertain levels' parameters. This method is based on the first-order Taylor expansion and finite element method. The uncertain parameters are treated as the intervals, not necessary to know their probabilistic distributions. Through dividing the intervals of the uncertain parameters into several subintervals and applying the interval analysis to each subinterval combination, a subinterval analysis method is then suggested to deal with the structures with large uncertain levels' parameters. However, the second-order truncation error of the Taylor expansion and the linear approximation of the second derivatives with respect to the uncertain parameters, two error estimation methods are given to calculate the maximum errors of the interval analysis and subinterval analysis methods, respectively. A plane truss structure is investigated to demonstrate the efficiency of the presented method.

The objective of the present research is to apply element-free Galerkin (EFG) method to adaptive analyses. It is necessary to estimate error of numerical solutions of EFG method for adaptive analyses to evaluate accuracy of EFG method. Posteriori errors can be estimated by differences of solutions between the linear and the quadratic basis functions. But it is not economical to perform the two respective calculations in regard to the linear and the quadratic basis function. Then the linear function is used only when the stiffness matrix is calculated. Then both the linear and the quadratic basis function are used when the stress and strain are calculated. The error estimation is performed in an each background cell by using the error of energy norm. In the adaptive analysis, a node is added at the quadrature point which is the same as the center of gravity of a background cell where the error is higher than the threshold value.

The nodal relocation method is applied to smoothing the distribution of nodes in domain of an analysis model. The nodal relocation method in which nodes are automatically moved and relocated using physical interbubble forces called bubble meshing for FEM is applied to the adaptive analysis after additional nodes are generated. The nodes are relocated corresponding to the error of the background cells. The calculations of the analysis can be repeated again after the posteriori error estimation.

The adaptive EFG method and the nodal relocation method are applied to a problem of an infinite plate with a hole subjected to a uniform tension. Nodal density is increased at the vicinity of the hole where the error is large in the analysis. The nodal relocation method can be successful to relocate the nodes which are generated at the quadrature points of higher posteriori error. The difference between the calculated solution and the exact solution are smaller than that of the previous solution as the calculations are repeated.

In this paper, an error estimator for element-free Galerkin (EFG) method has been proposed. Since meshfree methods do not require a structured mesh or a sense of nodal belongingness, the methods offer the advantage of insertion, deletion, and redistribution of nodes adaptively in the problem domain. The trial function of the field variable is constructed entirely in terms of consistent basis functions and its associated coefficient. The proposed error estimator is based on the nodal coefficient-vector of the basis functions that are used to construct the trial function. After obtaining the nodal coefficient-vector from EFG solution, an attempt is made to recover the best nodal coefficient-vector based on the reduced domain of influence [Chung and Belytschko (1998)], which is sufficient enough to maintain the regularity of the EFG moment matrix and also ensuring that sufficient influencing nodes are present in all the four quadrants defined at the sample node. The vertices of the Voronoi polygon of the critical error nodes are considered as potential neighborhood and new nodes are inserted at the vertices. Numerical studies have been carried out to illustrate the performance of the proposed methodology of error estimator and adaptivity.

We present a stabilized finite volume element method for the coupled Stokes–Darcy problem with the lowest order $P_1-P_0$ element for the Stokes region and $P_1$ element for the Darcy region. Based on adding a jump term of discrete pressure to the approximation equation, a discrete inf-sup condition is established for the proposed method. The optimal error estimates in the $H^1$-norm for the velocity and piezometric head and in the $L^2$-norm for the pressure are proved. And they are also verified through some numerical experiments. Two figures are given to show the full comparison for the local mass conservation between the proposed method and the stabilized finite element method. And this method can also be computed directly in the irregular domain according to the last experiment.

In this paper, a new approaching technique is offered to unravel multi-pantograph-type delay differential equations. The suggested new method is a collocation method based on integration and Boubaker polynomials. As the main idea of the method, the process starts by approaching the first derivative function in the equation in the form of truncated Boubaker series. Then this approximating form is composed in the matrix form. The unknown function is then obtained by integrating the approximate derivative function and expressing it as a matrix. Using the approximation, the matrix forms for the proportionally delayed terms in the equation are derived. In addition, operational matrix forms are constructed for convenience in the method. By using these matrix forms and matrix operations, the problem is reduced to a system of algebraic linear equations. The method is illustrated through numerical implementations and compared with existing techniques in the literature. The results demonstrate the effectiveness and reliability of the proposed approach, highlighting its superiority over other methods.

The struggle for the existence of the biological species is a well-known Prey–Predator model study in the literature. In this study, we present an improved model of Jerri [J. Abdul, Introduction to Integral Equations with Applications, Vol. 10 (Wiley, New York, 1999)] by introducing the intra-species competition term between the same species in addition to the existing environmental changes and few other factors in the model. The demand from the existing (limited) resources and other requirements induces competition between the same species which may alter the survival tactics among themselves. This intra-species term provides strength to the model as it makes the model more realistic. The governing equations are a system of two nonlinear delay integro differential equations, which are solved using spectral collocation method. The role of intra-species coefficients denoting the logistic growth/decay of the two species and two other parameters affecting the population dynamics are analyzed with the three basis functions such as Chebyshev, Legendre and Jacobi polynomials. With the help of simple matrix analysis, the governing equations are converted into a system of nonlinear algebraic equations. Detailed error estimation is computed to compare our results with the existing methods. It is shown with the help of tables and figures that the present method is very efficient, has better accuracy and has least computational cost.

In this paper, we present a construction of hidden variable bivariate fractal interpolation functions (HVBFIFs) with function vertical scaling factors and estimate errors of HVBFIFs on perturbation of the function vertical scaling factor. We construct HVBFIFs on the basis of the iterated function system (IFS) with function vertical scaling factors. The perturbation of the function vertical scaling factors in the IFS causes a change in the HVBFIF. An upper estimation of the errors between the original HVBFIF and the perturbed HVBFIF is given.

Everyone knows about the complicated solution of the nonlinear Fredholm integro-differential equation in general. Hence, often, authors attempt to obtain the approximate solution. In this paper, a numerical method for the solutions of the nonlinear Fredholm integro-differential equation (NFIDE) of the second kind in the complex plane is presented. In fact, by using the properties of Rationalized Haar (RH) wavelet, we try to give the solution of the problem. So far, as we know, no study has yet been attempted for solving the NFIDE in the complex plane. For this purpose, we introduce the continuous integral operator and real valued function. The Banach fixed point theorem guarantees that, under certain assumptions, the integral operator has a unique solution. Furthermore, we give an upper bound for the error analysis. An algorithm is presented to compute and illustrate the solutions for some numerical examples.

Classical finite element method (FEM) has been applied to solve some fractional differential equations, but its scheme has too many degrees of freedom. In this paper, a low-dimensional FEM, whose number of basis functions is reduced by the theory of proper orthogonal decomposition (POD) technique, is proposed for the time fractional diffusion equation in two-dimensional space. The presented method has the properties of low dimensions and high accuracy so that the amount of computation is decreased and the calculation time is saved. Moreover, error estimation of the method is obtained. Numerical example is given to illustrate the feasibility and validity of the low-dimensional FEM in comparison with traditional FEM for the time fractional differential equations.