Error Estimates Research Articles

Finite element analysis for the Navier-Lamé eigenvalue problem

The present paper introduces the analysis of the eigenvalue problem for the elasticity equations when the so-called Navier-Lamé system is considered. This system incorporates the displacement, rotation, and pressure of a linear elastic structure. The analysis of the spectral problem is based on the compact operator theory. A finite element method using polynomials of degree k≥1 is employed to approximate the eigenfrequencies and eigenfunctions of the system. Convergence and error estimates are presented. An a posteriori error analysis is also performed, where the reliability and efficiency of the proposed estimator are proven. We conclude this contribution by reporting a series of numerical tests to assess the performance of the proposed numerical method for both a priori and a posteriori estimates.

Stability analysis and error estimates of implicit-explicit Runge-Kutta least squares RBF-FD method for time-dependent parabolic equation

In this paper, for the time-dependent parabolic equations defined on complex geometries domain, we develop and analyze the least-squares radial basis function finite difference method (RBF-FD) coupled with the implicit-explicit Runge-Kutta (IMEX-RK) time discretization up to third order accuracy, which improves stability and accuracy. We derive the absolute stability region and the optimal time-step constraint for four kinds of IMEX-RK schemes. Compared to the traditional explicit or implicit time discretization, these are not trivial. Under a wide time-step constraint, the stability and the error estimates in l2-norm are established. Finally, several numerical experiments on the regular domain and non-convex domain are performed to validate the theoretical analysis.

A new experiment to enable rapid systematic investigations of flux trapping dynamics for superconducting radio-frequency cavity applications.

Many modern accelerators rely on superconducting radio-frequency (SRF) cavities to accelerate particles. When these cavities are cooled to the superconducting state, a fraction of the ambient magnetic field (e.g., Earth's magnetic field) may be trapped in the superconductor. This trapped flux can significantly increase the power dissipation of the SRF cavities. It is, therefore, crucial to understand the underlying mechanism of how magnetic flux is trapped and what treatments and operating conditions can reduce the flux-trapping efficiency. A new experiment was designed that enables a systemic investigation of flux trapping. It allows for independent control of cooldown conditions, which might have an influence on flux trapping: temperature gradient across the superconductor during cooldown, cooldown rate, and ambient magnetic field. For exhaustive studies, the setup was designed for quick thermal cycling, permitting up to 300 superconducting transitions in one day. In this paper, the setup and operation is described in detail and an estimation of the measurement errors is given. Exemplary data are presented to illustrate the efficacy of the system.

Finite-Time Fault-Tolerant Control via Fully Actuated System Approaches.

In this article, a finite-time fault-tolerant controller based on the fully actuated system (FAS) theory is presented to realize system stabilization and trajectory tracking. Paralleling to first-order nonlinear state space theory, the high-order FAS (HOFAS) theory contains rich controller design approaches. The existing FAS approaches can only give general global asymptotic stability results. In order to enhance the applicability of FAS approaches in fast control systems, a parameterized FAS stabilization controller based on the homogeneity principle is established for global finite-time stability. Moreover, a finite-time FAS tracking controller based on a finite-time observer is proposed for a HOFAS model with process faults. The proposed observer can yield zero-value convergence of state estimation error and fault estimation error in a finite time, and the proposed fault-tolerant controller can yield zero-value convergence of tracking error in a finite time. The main results are proved theoretically and illustrated experimentally.

The mixed penalty method for the Signorini problem

We investigate two penalty approaches for the Signorini problem in the mixed form. The well-posedness theory has been established for the two mixed penalty problems in both the continuous and discrete sense. For the continuous case, we obtain the error of the penalty for both two approaches. We also study the error estimates of the discrete mixed penalty problems, which depend on the mesh size and the penalty parameter. Based on the two penalty approaches, we design two algorithms and show their convergence. Several numerical experiments are carried out to confirm the theoretical results.

Dual-Channel NCSs Performance Error Estimation Under DoS Attacks and Intelligent Control Supervised by Machine Learning to AGV Application

Dual-Channel NCSs Performance Error Estimation Under DoS Attacks and Intelligent Control Supervised by Machine Learning to AGV Application

Unconditional error analysis of the linearized transformed [formula omitted] virtual element method for nonlinear coupled time-fractional Schrödinger equations

This paper constructs a linearized transformed L1 virtual element method for the generalized nonlinear coupled time-fractional Schrödinger equations. The solutions to such problems typically exhibit singular behavior at the beginning. To avoid this pitfall, we introduce an identical s-fractional differential system derived from a smoothing transformation of variables t=s1/α, 0<α<1. By utilizing the discrete complementary convolution kernels, we prove the boundedness and error estimates of the solution of time-discrete system. Moreover, the unconditionally optimal error bounds of the proposed fully discrete scheme are derived in L2-norm without restriction on the grid ratio. Finally, numerical tests on a set of polygonal meshes are presented to verify the theoretical results.

Reduced basis method for non-symmetric eigenvalue problems: application to the multigroup neutron diffusion equations

In this article, we propose a reduced basis method for parametrized non-symmetric eigenvalue problems arising in the loading pattern optimization of a nuclear core in neutronics. To this end, we derive a posteriori error estimates for the smallest eigenvalue which is assumed to be simple and the associated left and right eigenvectors. The practical computation of these estimators requires the estimation of a constant called prefactor, which we can express as the spectral norm of some operator. We provide some elements of theoretical analysis which illustrate the link between the expression of the prefactor we obtain here and its well-known expression in the case of symmetric eigenvalue problems, either using the notion of numerical range of the operator, or via a perturbative analysis. Lastly, we propose a practical method in order to estimate this prefactor which yields interesting numerical results on actual test cases. We provide detailed numerical simulations on two-dimensional examples including a multigroup neutron diffusion equation.

Finite element modelling of linear rolling contact problems

The present work is devoted to the finite element modelling of linear hyperbolic rolling contact problems. The main equations encountered in rolling contact mechanics are reviewed in the first part of the paper, with particular emphasis on applications from automotive and vehicle engineering. In contrast to the common hyperbolic systems found in the literature, such equations include integral and boundary terms, as well as time-varying transport velocities, that require special treatment. In this context, existence and uniqueness properties are discussed within the theoretical framework offered by the semigroup theory. The second part of the paper is then dedicated to recovering approximated solutions to the considered problems, by combining discontinuous Galerkin finite element methods (DGMs) with explicit Runge–Kutta (RK) schemes of the first and second order for time discretisation. Under opportune assumptions on the smoothness of the sought solutions, and owing to certain generalised Courant–Friedrichs–Lewy (CFL) conditions, quasi-optimal error bounds are derived for the complete discrete schemes. The proposed algorithms are then tested on simple scalar equations in one space dimension. Numerical experiments seem to suggest the theoretical error estimates to be sharp.

Adaptive momentum equation method for overcoming singularities of dispersed phases

The singularity issue arising from the phase fraction approaching zero in multiphase flow can significantly intensify the solution difficulty and lead to nonphysical results. By employing the conservative form of momentum equations in high-phase-fraction and discontinuity regions and the phase-intensive form of momentum equations in low-phase-fraction regions, computational reliability can be assured while avoiding the singularity issue. Regarding the proposed adaptive momentum equation method, the form of momentum equations for each cell is determined by a conversion bound and a phase fraction discontinuity detector. A comparative analysis is conducted on this method and other singularity-free methods. For discontinuities of dispersed phases, an error estimation method of the conversion bound is presented through theoretical analysis. Computational results demonstrate that the discontinuity detector accurately captures discontinuities in high-phase-fraction regions while disregarding pseudo-discontinuities in low-phase-fraction regions. Compared to the conservative form corrected by the terminal velocity method, the method yields higher-quality flow fields and potentially exhibits an efficiency improvement of over 10 times. Compared to the phase-intensive form, the method benefits from the physical quantity conservation, providing higher computational reliability. When encountering discontinuities, the expected error from the error estimation method aligns well with the actual error, indicating its effectiveness. When the conversion bound is below 1/10 000 of the inlet phase fraction, the errors of the adaptive method are essentially negligible.

Exponential time propagators for elastodynamics

We propose a computationally efficient and systematically convergent approach for elastodynamics simulations. We recast the second-order dynamical equation of elastodynamics into an equivalent first-order system of coupled equations, so as to express the solution in the form of a Magnus expansion. With any spatial discretization, it entails computing the exponential of a matrix acting upon a vector. We employ an adaptive Krylov subspace approach to inexpensively and accurately evaluate the action of the exponential matrix on a vector. In particular, we use an apriori error estimate to predict the optimal Krylov subspace size required for each time-step size. We show that the Magnus expansion truncated after its first term provides quadratic and superquadratic convergence in the time-step for nonlinear and linear elastodynamics, respectively. We demonstrate the accuracy and efficiency of the proposed method for one linear (linear cantilever beam) and three nonlinear (nonlinear cantilever beam, soft tissue elastomer, and hyperelastic rubber) benchmark systems. For a desired accuracy in energy, displacement, and velocity, our method allows for 10−100× larger time-steps than conventional time-marching schemes such as Newmark-β method. Computationally, it translates to a ∼1000× and ∼10−100× speed-up over conventional time-marching schemes for linear and nonlinear elastodynamics, respectively.

Numerical investigation of the mesh-free collocation approach for solving third kind VIEs with nonlinear vanishing delays

The principal purpose of this investigation is to present an approximate algorithm for solving third-kind Volterra integral equation with nonlinear vanishing delays using mesh-free techniques based on a radial basis function collocation scheme. This technique does not require background approximation cells, and the algorithm is efficient, has good stability and does not require much computer memory. A description of the approach for the proposed equations is provided. The error estimation of the proposed approach is also provided. The accuracy and reliability of the new scheme are demonstrated through numerical examples. Some of these examples have been compared with analytical solutions and moving least squares methods. The numerical results validate their agreement with theoretical error estimates.

Mobile solution and chronic diseases: development and implementation of a mobile application and digital platform for collecting, analyzing data, monitoring and managing health care

BackgroundTechnological advances based on mobile health (mHealth), the field of digital health (eHealth) aimed at managing health services and care and their concomitant transformations, have become increasingly important in the 21st century. In this context, care for individuals diagnosed with Chronic Non-Communicable Diseases (CNCDs) deserves to be highlighted. The aim of this study is to present the creation, development and implementation of the Telehealth Center of the Federal University of Viçosa (NUTELES - UFV), for the monitoring and management of health care for individuals with Arterial Hypertension (AH) and/or Diabetes Mellitus (DM).MethodsThis study, carried out in 2022 and 2023, involved 374 patients diagnosed with AH and/or DM in the health micro-region of Viçosa/MG, Brazil, comprising nine municipalities. The research aims to describe the creation, development and implementation of a software (NutelesApp) for the collection, storage, management and analysis of epidemiological research in public health, focusing specifically, on the monitoring and management of health care for individuals with AH and/or DM monitored by Primary Health Care (PHC). The parameters collected and analyzed were obtained through a questionnaire applied to patients, consisting of 70 questions, subdivided into 12 classes of questions. The epidemiological survey data was collected using mobile devices and analyzed using computer techniques based on statistical analysis. Once the field teams had completed their work, the files were transferred to servers for general analysis processing, using estimates of means, prevalence and respective standard errors, calculated using the Statistical Package for Social Science (SPSS) program, which takes into account the planning variables and includes the basic weights resulting from the sampling process. This study was approved by the Human Research Ethics Committee and registered, prior to recruitment, by the Brazilian Registry of Clinical Trials (ReBEC), ID: RBR-45hqzmf (Last approval date: 11/30/2022).ResultsThe information obtained through data collection with subsequent exploratory analysis of epidemiological data using the NutelesApp software suggests that mobile applications intended for the purpose of monitoring and managing healthcare for people with AH and/or DM should address the variables necessary to support a process of understanding the health conditions and/or disease of the individual as a whole and provide short- and long-term learning. Regarding the results of the survey using the software, the variables of 374 people were analyzed. Majority are female (73.2%) and white (43.5%). Most patients are elderly (average 64 years), with blood pressure levels within the normal range for this population, BMI indicates overweight, AC increased risk for cardiovascular events and CC within the recommendation value. All biochemical parameters analyzed were above normal limits.ConclusionsThe description of the creation and development of the software includes practical examples of its implementation, the results collected and its applicability in real scenarios, presenting determining criteria that can provide assertive and timely interventions for monitoring and managing the health and/or disease parameters of patients with HA and/or DM. In addition, the beneficial consequences of using this application will extend to health units and their respective management, improving the services provided by PHC and enhancing strategies and actions for health promotion and disease prevention.

Sliding mode observers for set-valued Lur’e systems with uncertainties beyond observational range

In this paper, we introduce a new sliding mode observer for Lur’e set-valued dynamical systems, particularly addressing challenges posed by uncertainties not within the standard range of observation. Traditionally, most ofLuenberger-like observers and sliding mode observer have been designed only for uncertainties in the range of observation. Central to our approach is the treatment of the uncertainty term which we decompose into two components: the first part in the observation subspace and the second part in its complemented subspace. We establish that when the second part converges to zero, an exact sliding mode observer for the system can be obtained In scenarios where this convergence does not occur, our methodology allows for the estimation of errors between the actual state and the observer state. This leads to a practical interval estimation technique, valuable in situations where part of the uncertainty lies outside the observable range. Finally, we show that our observer is also a T-observer as well as a strong H∞ observer.

Adaptive least-squares methods for convection-dominated diffusion-reaction problems

This paper studies adaptive least-squares finite element methods for convection-dominated diffusion-reaction problems. The least-squares methods are based on the first-order system of the primal and dual variables with various ways of imposing outflow boundary conditions. The coercivity of the homogeneous least-squares functionals are established, and the a priori error estimates of the least-squares methods are obtained in a norm that incorporates the streamline derivative. All methods have the same convergence rate provided that meshes in the layer regions are fine enough. To increase computational accuracy and reduce computational cost, adaptive least-squares methods are implemented and numerical results are presented for some test problems.

Computational and numerical analysis of the fractional three-components nonlinear Schrödinger equation with singular and non-singular kernels

This work offer a computational approach to the coupled nonlinear Shrödinger equation (CNLSE) by utilizing fractional operators and double Laplace Adomian decomposition method (DLTADM). Three various fractional operators i.e Caputo, Caputo–Fabrizio (CF) and Atangana–Baleanu Caputo (ABC) are used for studying the approximate solution of the CNLSE. Dynamics of solutions are provided via graphical examples that depict system behavior. Graphs of the deduced results portray dark soliton dynamics for various fractional values. The robustness of the applied approach is studied via comparative analysis of error estimations between exact and derived results via tables and 2D graphs.

Thin layer drying and effect of temperature on the drying characteristics of bushbuck (Gongronema latifolium) leaves

Determination of the drying properties of agricultural products holds several benefits, such as nutrient preservation, processing and storage. The effect of air temperature on the drying properties and kinetics of bushbuck leaves at 30, 40, 50, 60 and 70 °C temperature was investigated. The drying time declined as the temperature rose, indicating a falling rate drying phenomenon. Drying models such as Henderson and Pabis, Newton, Modified Page, Page, Midilli and others, Logarithmic, and Two-term were applied and fitted to the experimental data's moisture ratio. The non-linear regression fitting method results showed that the Midilli and other models gave the best-fit description for the drying process. The values obtained for the coefficient of determination (R2) were observed to have varied from 0.9948 to 0.9995 for the various drying models evaluated. In contrast, the sum of square error (SSE) and root mean square (RMSE) estimate was observed to vary from 0.0056 to 0.0007 and 0.0310 to 0.0163, respectively. On the estimate for the effective moisture diffusivity (Deff), it was observed that it depends on temperature and extends from 1.33×10−9 to 4.83×10−9 m2/s. The value of the activation energy was 26.05 kJ/mol. From this study, it can be deduced that the Arrhenius model could predict the temperature effect on the drying process of bushbuck leaves.

Optimally truncated WKB approximation for the 1D stationary Schrödinger equation in the highly oscillatory regime

This paper is dedicated to the efficient numerical computation of solutions to the 1D stationary Schrödinger equation in the highly oscillatory regime. We compute an approximate solution based on the well-known WKB-ansatz, which relies on an asymptotic expansion w.r.t. the small parameter ɛ. Assuming that the coefficient in the equation is analytic, we derive an explicit error estimate for the truncated WKB series, in terms of ɛ and the truncation order N. For any fixed ɛ, this allows to determine the optimal truncation order Nopt which turns out to be proportional to ɛ−1. When chosen this way, the resulting error of the optimally truncated WKB series behaves like O(exp(−r/ɛ)), with some parameter r>0. The theoretical results established in this paper are confirmed by several numerical examples.

Demand analysis of cayenne chilli pepper in Surakarta City

Background: This study aims to analyze the factors that affect the cayenne chilli pepper demand, to know its elasticity and to forecast its demand for 2019 up to 2023. Findings: By using time series data of 1993-2017, this study applies multiple regresiion analysis. Methods: The method used for demand foreasting analysis is Least Squares Method and for the error estimator is used Mean Average Deviation. Conclusion: The results show that cayenne chilli pepper price, red chilli pepper price, onion price, the population and income per influence the cayenne chilli pepper demand in Surakarta City. The price elasticity of demand is inelastic and the pruduct categorized as normal goods. Cross elasticity indicates that cayenne chilli pepper subsituted with red chilli with the value of cross elasticity 0,137 and complementered with onion which the value of cross elasticity is -0,094. The result of cayenne chilli pepper demand forecasting for 2019 up to 2023 shows that the potential demand will always increas every year. The value of demand forecast on 2019 is 852.350,24 kg; the value of demand forecast on 2020 is 854.291,99 kg; the value of demand forecast on 2021 is 856.233,74 kg; the value of demand forecast on 2022 is 858.175,49 kg and the value of demand forecast on 2023 is 860. 117,24 kg with the value of error estimator is 0,0048 kg.

Method for determining phase shift using two-phase interpolation transformation

This study focuses on a method for accurately determining the phase shift between two harmonic signals. The proposed technique compares the waveform of the combined signal, which is obtained by applying a two-phase interpolation transformation, with a set of predefined reference functions. This study aims to develop a method for accurately measuring the phase shift between two harmonic signals to reduce the measurement errors caused by phase asymmetry in signal transmission channels and minimize the costs associated with parameter control. The sources of error in the proposed measurement method are identified and addressed. The task involves setting up the measurement objective to determine the phase shift between two harmonic signals. This section presents the analyses of known methods for measuring phase shifts using of analog-to-digital conversion. The next step involves selecting indicators and criteria to determine how closely the investigated signal matches the reference function. Then, we propose the synthesis of an algorithm to find the extremum of the sum of squares deviations for the set of reference functions and the investigated signal, relying on the golden ratio method. Finally, the paper will analyze possible sources of error that could affect the measurement outcome. The methods used are the following: methodology for conducting digital signal processing and measuring error estimation, numerical methods for extremum search, and methodology for single-factor experiments. The following results were obtained. A proposed compensation method for measuring phase shift is based on comparing the shape of the normalized signal, which is obtained as a result of the summation of harmonic signals after their semiperiodical transformation, with a set of normalized reference functions synthesized by computational means. A list of measurement and auxiliary operations, which should be ideally conducted to implement this measurement method, has been determined. An analysis of the components of measurement errors was conducted. Conclusions. The scientific novelty of the obtained results lies in the following: a method for measuring the phase shift of a signal has been developed, which, in our opinion, should be classified as a compensatory measurement method. This will significantly reduce the error component caused by the phase asymmetry of signal transmission channels and reduce the costs of controlling parameters (up to 10%) without decreasing the quality of control; a list of the measuring and auxiliary operations necessary for the implementation of the proposed measurement method has been identified; sources of errors have been determined.