Approximate Integral Method Research Articles

Unified framework for open quantum dynamics with memory

The dynamics of quantum systems coupled to baths are typically studied using the Nakajima-Zwanzig memory kernel (K\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{{\\bf{{{{\\mathcal{K}}}}}}}}$$\\end{document}) or the influence functions (I), particularly when memory effects are present. Despite their significance, formal connections between the two have not been explicitly known. We establish their connections by examining the system propagator for a N-level system linearly coupled to Gaussian baths with various types of system-bath coupling. For a certain class of problems, we devised a non-perturbative, diagrammatic approach to construct K\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{{\\bf{{{{\\mathcal{K}}}}}}}}$$\\end{document} from I for (driven) systems interacting with Gaussian baths, bypassing conventional projection-free dynamics inputs. Our work provides a way to interpret approximate path integral methods in terms of approximate memory kernels. Moreover, it offers a Hamiltonian learning procedure to extract the bath spectral density from reduced system trajectories, opening new avenues in quantum sensing and engineering. The insights we provide advance our understanding of non-Markovian dynamics and will serve as a stepping stone for future theoretical and experimental developments in this area.

An improved approximate integral method for nonlinear reliability analysis

In order to evaluate the failure probability corresponding to the Limit State Function (LSF) in structural reliability, the First Order Reliability Method (FORM) linearizes the LSF and directly calculates the failure probability based on the Most Probable Point (MPP). But this method is unable to effectively handle nonlinear problems. The Second Order Reliability Method (SORM), on the other hand, utilizes the Hessian matrix to obtain curvature information at the MPP and computes the failure probability with Breitung's second-order approximation formula. However, SORM is unable to maintain satisfactory computational accuracy in highly nonlinear problems due to its lack of detailed analysis on the shape of the limit state surface. To address this issue, this paper proposes an Improved Approximation Integration Method (IAIM). The initial approximation of the limit state surface is based on the intersection region between the hypersphere and the n-dimensional paraboloid. Subsequently, a statistical analysis of the deviation between the aforementioned region and its projected area is conducted to construct a corresponding fitting function, which is then utilized for dimensionality reduction in the multidimensional integration of the approximation area. Finally, several examples are presented to demonstrate the accuracy and feasibility of the proposed IAIM.

Energy-saving design of underground metro vertical alignment with deep reinforcement learning

The design of metro vertical alignment greatly affects train traction energy consumption. This study develops a deep reinforcement learning (DRL) algorithm for an energy-saving design of underground metro vertical alignment and analyzes the influence of train traction energy consumption calculation factors (CFs) on energy-saving design. First, a metro train operation simulation model is adopted using the approximate integral method, and the value range of CFs as input parameters is determined. Second, the 'agent', 'state', 'action', and 'reward' of the metro vertical alignment environment and the corresponding alignment constraints are determined. Third, a Dueling Double Deep Q Network (D3QN) is constructed to design the energy-saving metro vertical alignment. The proposed model is applied to a real-world metro case study, and the influence of CFs (design speed, limiting gradient in reverse slopes, and passenger capacity) on energy-saving design is analyzed. Finally, compared to manual design, the D3QN-based design outperforms GA-based design and DQN-based design, and demonstrates a remarkable energy savings of 6.17%.

Approximate Integral Method for Nonlinear Reliability Analysis

In the realm of reliability analysis methods, the First-Order Reliability Method (FORM) exhibits excellent computational accuracy and efficiency in linear problems. However, it fails to deliver satisfactory performance in nonlinear ones. Therefore, this paper proposes an Approximate Integral Method (AIM) to calculate the failure probability of nonlinear problems. Firstly, based on the Most Probable Point (MPP) of failure and the reliability index β obtained from the FORM, the Limit State Function (LSF) can be equivalent to an Approximate Parabola (AP) which divides the hypersphere space into feasible and failure domains. Secondly, through the ratio of the approximate region occupied by a parabolic curve to the entire hypersphere region, the failure probability can be calculated by integration. To avoid the computational complexity in the parabolic approximate area due to high dimensionality, this paper employs a hyper-rectangle, constructed from chord lengths corresponding to different curvatures, as a substitute for the parabolic approximate area. Additionally, a function is utilized to adjust this substitution, ensuring accuracy in the calculation. Finally, compared with the calculated result of the Monte Carlo simulation (MCS) and the FORM, the feasibility of this method can be demonstrated through five numerical examples.

Parameterization optimal control of an unsteady partial differential equations with convection term by an improved three-term spectrum conjugate gradient algorithm

The unsteady partial differential equations (UPDE) with convection term gives a clear descriptions for the solidification process of a slab in dynamic production of continuous casting. To give a suitable setting value of secondary cooling water flow rate for the dynamic control system, this study investigates an optimal control problem (OCP) of UPDE with convection term. Firstly, control vector discretization of OCP and the solution of UPDE are given. Secondly, due to the rapidity for gradient, this paper analyzes the expression of the gradient calculation method based on Hamiltonian function costate system by approximate treatment, matrix calculation and composite trapezoidal integral method. Thirdly, an improved three-term spectrum conjugate gradient algorithm (ITSCGA) is proposed to solve the OCP of UPDE, and the global convergence of the ITSCGA is demonstrated. Lastly, the performance of ITSCGA is demonstrated by experimental simulations. The results demonstrate that the ITSCGA provides a smaller temperature fluctuations, and improves the quality of a slab.

On optimal inequalities between three-point quadratures

We examine the family of all (at most) three-point symmetric quadratures on [-1,1] which are exact on polynomials of order 3 to find all possible inequalities between them in the class of 3-convex functions. Next we optimise them by using convex combinations of the quadratures considered. We find the optimal quadrature and use it to construct the adaptive method of approximate integration. An effective method to estimate the error of this method is also given. It needs a considerably fewer number of subdivisions of the interval of integration than the classical adaptive methods as well as the method developed by the second-named author in his recent paper (Wa̧sowicz in Aequ Math 94(5):887–898, 2020).

An experimental analog circuit realization of Matsuda’s approximate fractional-order integral operators for industrial electronics

Analog circuit realization of fractional order (FO) elements is a significant step for the industrialization of FO control systems because of enabling a low-cost, electric circuit realization by means of standard industrial electronics components. This study demonstrates an effective operational amplifier-based analog circuit realization of approximate FO integral elements for industrial electronics. To this end, approximate transfer function models of FO integral elements, which are calculated by using Matsuda’s approximation method, are decomposed into the sum of low-pass filter forms according to the partial fraction expansion. Each partial fraction term is implemented by using low-pass filters and amplifier circuits, and these circuits are combined with a summing amplifier to compose the approximate FO integral circuits. Widely used low-cost industrial electronics components, which are LF347N opamps, resistor and capacitor components, are used to achieve a discrete, easy-to-build analog realization of the approximate FO integral elements. The performance of designed circuit is compared with performance of Krishna’s FO circuit design and performance improvements are shown. The study presents design, performance validation and experimental verification of this straightforward approximate FO integral realization method.

Outage performance of UAV-assisted AF relaying with hardware impairments

The use of unmanned aerial vehicle (UAV) as aerial relay, i.e. UAV-assisted relaying, has the advantages of easy deployment, agility, and mobility compared to the traditional ground relaying. In this paper, our goal is to acquire the universal outage probability (OP) analytical expression for multi-rotor UAV-assisted AF relaying with transceiver hardware impairments (HIs) while considering a practical large-scale fading. Contrary to the existing works on the UAV-assisted relaying, where the OP was derived by assuming a specific small-scale fading, we make a more general assumption of arbitrary small-scale fading and thereafter apply an integral region shape-based approximate method (IRS-AM) to deduce a closed-form OP expression with adjustable error in a more common way. The derivation process is free of the concrete model of the small-scale fading while the final OP expression has a concise and intuitive form, which is determined by the cumulative distribution function (CDF) of the small-scale fading in both two links. Based on the derived results, we demonstrate the ceiling effect caused by HIs on the multi-rotor UAV-assisted AF relaying and obtain the end-to-end signal-to-noise-and-distortion ratio (SNDR) bound, which depends on the aggregate HIs levels and determines whether the considered relay network is always in outage or not. Computer simulations validate our studies and reveal the influence of HIs on the OP and the optimal UAV altitude.

Application of the integration method for assessing the quality of qualimetry objects

The analysis of existing scientific approaches to the quantitative assessment of the quality of qualimetry objects of different nature, which have different quality indicators with different measurement scales, is carried out. A number of modern scientific works related to the assessment of the quality of products, processes and services are considered, their shortcomings and possible boundaries of application are identified. As a result of the analysis, it was revealed that there are a number of unsolved problems in qualimetry, the solution of which could make it possible to develop new practical methods that would be quite universal and could be effectively used in assessing the quality of objects of various kinds. As a result of the analysis, the relevance of the topic was proved and the goals of the work were determined - to develop a methodology for obtaining a comprehensive assessment of the quality indicator, which will be suitable for assessing qualimetry objects of various kind. For a generalized assessment of the quality of qualimetry objects, it is proposed to determine an assessment for each of its characteristics or criteria, and then to determine a single assessment, taking into account all its characteristics. To determine the generalized quality indicator of one criterion, it is proposed to apply the integration method, that is, find the area under the broken surface, which emerged as a result of connecting points on the plane of the XOY coordinate system. For this, quadrature formulas are applied using the trapezoidal method. The trapezium method is an approximate integration method, useful in cases where it is not possible to find the original function and calculate the integral through it. A step-by-step algorithm for determining a generalized quality indicator of a qualimetry object of various natures is proposed, using the integration by the trapezium method. In this case, we get a multi-criteria assessment of the quality of the qualimetry object. For integration, you can apply other methods, which will determine the most effective one. The developed technique can be used for multi-criteria assessment of the quality of qualimetry objects if, instead of a time scale, a scale of single assessments of various criteria that characterizes the object is used.

Об интегрировании дифференциальных уравнений неустановившегося постепенно изменяющегося движения потока в открытом русле в условиях высокогорья при прорыве плотины

Цель. Выбор научно-технически обоснованного метода интегрирования уравнений неустановившегося постепенно изменяющегося движения потока в открытом русле в условиях высокогорья при прорыве плотины.
 Методы. Методы приближенного интегрирования дифференциальных уравнений.
 Результаты. Предложено использование конечно-разностного метода интегрирования дифференциальных уравнений (метода характеристических уравнений С.А. Христиановича) для неустановившегося постепенно или плавно изменяющегося движения в условиях высокогорья при прорыве плотины.
 Область применения исследований. Предложенное решение уравнений неустановившегося постепенно изменяющегося движения в условиях высокогорья при прорыве плотин можно использовать при решении практических задач по определению основных параметров волны перемещения и зон вредного воздействия на прилегающую к высокогорью территорию.

Approximate solutions to one-phase Stefan-like problems with space-dependent latent heat

The work in this paper concerns the study of different approximations for one-dimensional one-phase Stefan-like problems with a space-dependent latent heat. It is considered two different problems, which differ from each other in their boundary condition imposed at the fixed face: Dirichlet and Robin conditions. The approximate solutions are obtained by applying the heat balance integral method (HBIM), the modified HBIM and the refined integral method (RIM). Taking advantage of the exact analytical solutions, we compare and test the accuracy of the approximate solutions. The analysis is carried out using the dimensionless generalised Stefan number (Ste) and Biot number (Bi). It is also studied the case when Bi goes to infinity in the problem with a convective condition, recovering the approximate solutions when a temperature condition is imposed at the fixed face. Some numerical simulations are provided in order to assert which of the approximate integral methods turns out to be optimal. Moreover, we pose an approximate technique based on minimising the least-squares error, obtaining also approximate solutions for the classical Stefan problem.

On a certain adaptive method of approximate integration and its stopping criterion

We introduce a new quadrature rule based on Chebyshev’s and Simpson’s rules. The corresponding composite rule induces the adaptive method of approximate integration. We propose a stopping criterion for this method and we prove that if it is satisfied for a function which is either 3-convex or 3-concave, then the integral is approximated with the prescribed tolerance. Nevertheless, we give an example of a function which does satisfy our criterion, but the approximation error exceeds the assumed tolerance. The numerical experiments (performed by a computer program created by the author) show that integration of 3-convex functions with our method requires considerably fewer steps than the adaptive Simpson’s method with a classical stopping criterion. As a tool in our investigations we present a certain inequality of Hermite–Hadamard type.

Investigating mesh‐based approximation methods for the normalization constant in the log Gaussian Cox process likelihood

The log Gaussian Cox process (LGCP) is a frequently applied method for modeling point pattern data. The normalization constant of the LGCP likelihood involves an integral over a latent field. That integral is computationally expensive, making it troublesome to perform inference with standard methods. The so‐called stochastic partial differential equation–integrated nested Laplace approximation (SPDE‐INLA) framework enables fast approximate inference for a range of hierarchical models, where a key component is to approximate the latent field by a triangulated mesh. Recent research has made it possible to fit LGCP models with this framework using an approximate integration method to compute the integral. We carefully describe several alternative variants of that approximate integration method and derive an analytical formula for the integral in question, which actually is exact under the triangular mesh assumption used by SPDE‐INLA. We compare the different integration strategies through a comprehensive simulation study and find that the analytical formula is often more accurate, but not always. Among the approximate integration methods, we recommend a simple extension to a method implemented in an R‐package for fitting LGCP models.

Numerical Methods for Transverse Vibration Analysis of Straight Euler and Timoshenko Beams

This paper presents a short review of numerical methods used for lateral vibration analysis in the case of straight Euler and Timoshenko beams. A particular integral method is described with more details. This approximate integral method is based on the use of flexibility influence functions (Green’s functions). It leads to a matrix formulation and to an eigenvalue problem for vibration analysis. The presented approach is able to estimate separately the shear effects and the rotary inertia effects and also the combined effects. Simple numerical examples are also presented for comparisons with analytical and finite elements results. The results show good agreement.

Sequential optimization and uncertainty propagation method for efficient optimization-based model calibration

The goal of model calibration is to improve the predictive capability of a computational model by estimating the unknown input variables of the model. Optimization-based model calibration (OBMC) is a probabilistic way to estimate the unknown input variables through the use of optimization techniques. Performing optimization in a probabilistic sense requires a high computational cost to obtain statistics about the outputs at every iteration of the optimization. To improve optimization efficiency, this paper proposes a sequential optimization-based model calibration approach that makes use of first an efficient, and then a highly accurate probabilistic assessment method, in sequence. At the earlier stage of the sequential optimizations, approximate integration methods are used to accelerate the probabilistic assessment process. As a calibration metric, the moment matching metric is devised to use the obtained statistics of the outputs. When the optimization reaches near-convergence, a more accurate method, such as a sampling method with an accurate surrogate model, is substituted for the probabilistic assessment. Thus, this paper provides an efficient and accurate procedure for optimization-based model calibration. Two engineering applications, model calibration of a shallow strip footing model and an automotive steering wheel-column model, are presented to demonstrate the effectiveness of the proposed method.

Advanced Statistical Model Calibration to Determine Manufacturing-Induced Variations of Effective Elastic Properties of SAC Solder Joints in Leadless Chip Resistor Assemblies

The unknown statistical distributions of two effective elastic properties of Sn-3.0Ag-0.5Cu solder joint of leadless chip resistors (LCRs), induced by an assembly condition, are determined by the advanced statistical model calibration. Two key elements are involved in the model calibration: an uncertainty propagation (UP) analysis and an optimization process using a calibration metric. In this paper, the UP analysis utilizes an advanced approximate integration method, which allows us to take into account the statistical variations of six additional known input variables, including die thickness, solder joint height, termination length, and thickness and elastic moduli of a printed circuit board. The cyclic bending test results of LCR assemblies are used in conjunction with the maximum-likelihood metric to obtain the statistical distributions of the effective properties. The cycles-to-failure distribution of the identical LCR assemblies subjected to a different loading level is predicted accurately by the calibrated model, which corroborates the validity of the proposed approach.

Integral methods of solving boundary-value problems of nonstationary heat conduction and their comparative analysis

The modern state of approximate integral methods used in applications, where the processes of heat conduction and heat and mass transfer are of first importance, is considered. Integral methods have found a wide utility in different fields of knowledge: problems of heat conduction with different heat-exchange conditions, simulation of thermal protection, Stefantype problems, microwave heating of a substance, problems on a boundary layer, simulation of a fluid flow in a channel, thermal explosion, laser and plasma treatment of materials, simulation of the formation and melting of ice, inverse heat problems, temperature and thermal definition of nanoparticles and nanoliquids, and others. Moreover, polynomial solutions are of interest because the determination of a temperature (concentration) field is an intermediate stage in the mathematical description of any other process. The following main methods were investigated on the basis of the error norms: the Tsoi and Postol’nik methods, the method of integral relations, the Gudman integral method of heat balance, the improved Volkov integral method, the matched integral method, the modified Hristov method, the Mayer integral method, the Kudinov method of additional boundary conditions, the Fedorov boundary method, the method of weighted temperature function, the integral method of boundary characteristics. It was established that the two last-mentioned methods are characterized by high convergence and frequently give solutions whose accuracy is not worse that the accuracy of numerical solutions.

Assembly yield prediction of plastically encapsulated packages with a large number of manufacturing variables by advanced approximate integration method

An advanced approximate integration scheme called eigenvector dimension reduction (EDR) method is implemented to predict the assembly yield of a plastically encapsulated package. A total of 12 manufacturing input variables are considered during the yield prediction, which is based on the JEDEC reflow flatness requirements. The method calculates the statistical moments of a system response (i.e., warpage) first through dimensional reduction and eigenvector sampling, and a probability density function (PDF) of random responses is constructed subsequently from the statistical moments by a probability estimation method. Only 25 modeling runs are needed to produce an accurate PDF for 12 input variables. The results prove that the EDR provides the numerical efficiency required for the tail-end probability prediction of manufacturing problems with a large number of input variables, while maintaining high accuracy.

Thermal explosions in spherical vessels at large Rayleigh numbers

This paper investigates effects of buoyancy-driven motion on the “slowly reacting” mode of combustion, and its thermal-explosion limits, of an initially cold gaseous mixture enclosed in a spherical vessel with a constant wall temperature. As in Frank-Kamenetskii’s seminal analysis, the strong temperature dependence of the effective overall reaction is modeled with a single irreversible reaction with an Arrhenius rate having a large activation energy. Besides the classical Damköhler number Da, measuring the ratio of the heat-release rate by chemical reaction evaluated at the wall temperature to the rate of heat removal by heat conduction to the wall, the solution is seen to depend on the Rayleigh number Ra, measuring the effect of buoyancy-induced motion on the heat-transport rate. For values of Da below a critical value Dac the system evolves in a slowly reacting mode where the heat losses to the wall limit the temperature increase associated with the chemical reaction, whereas for Da>Dac the initial stage of slow reaction ends abruptly at a well-defined ignition time, at which a thermal runaway occurs. Transient numerical integrations of the initial stage of slow reaction, formulated in the distinguished limit Da∼1 and Ra∼1 with account taken of the effects of the temporal pressure variation, are used to investigate influences of natural convection on thermal-explosion development, including changes in ignition times for Da>Dac and modified explosion curves. Our analysis reveals that Frank-Kamenetskii’s criterion for the determination of critical explosion conditions, based on the investigation of existence of steady solutions, provides values of Dac(Ra) that are identical to those extracted from the transient computations. Specific consideration is given to the structure of the steady solution in the asymptotic limit Ra≫1 in which the flow includes a thin chemically frozen near-wall boundary layer of downward moving cold gas bounding a central inviscid region of slowly rising reacting flow driven by the boundary-layer entrainment, with the critical explosion conditions predicted to occur for Dac∼Ra1/4. The mathematical structure of the resulting boundary-layer problem is fundamentally similar to that found in the unrelated problem of flow in curved pipes at large Dean numbers. As in that similar problem, the boundary layer exhibits a region of recirculating flow, so that the problem must be formulated as a boundary-value problem accounting for the self-similar local solutions that exist near the upper and lower stagnation points. The problem is solved with use made of an approximate integral method. The resulting asymptotic prediction for the critical Damköhler number Dac=0.655Ra1/4 is found to be in excellent agreement with the results obtained by integration of the complete conservation equations for Ra≫1.

The quasi-enthalpy based lattice Boltzmann model for solid-liquid phase change

The phase change material (PCM) receives more and more attention for the large amount of latent heat and little temperature variation during phase change. In this paper, a new phase change lattice Boltzmann (LB) model, which is based on the quasi-enthalpy (QE) method, has been developed. The update of temperature is divided into two step, the “prediction” and “consumption (for melting)” or “release (for solidification)”. The problems of one-region phase change, two-region phase change, phase change under constant heat flux and phase change by convection are solved by the present model. All the results are compared with that of analytical solution, integral approximate solution and numerical method. No significant differences can be obtained for all the simulations, which demonstrates that the QE based LB model is able to capture the location of solid-liquid interface and reveal the temperature distribution of PCM accurately.