Numerical Integration Method Research Articles

Improving the Variational Quantum Eigensolver Using Variational Adiabatic Quantum Computing

The variational quantum eigensolver (VQE) is a hybrid quantum-classical algorithm for finding the minimum eigenvalue of a Hamiltonian that involves the optimization of a parameterized quantum circuit. Since the resulting optimization problem is in general nonconvex, the method can converge to suboptimal parameter values that do not yield the minimum eigenvalue. In this work, we address this shortcoming by adopting the concept of variational adiabatic quantum computing (VAQC) as a procedure to improve VQE. In VAQC, the ground state of a continuously parameterized Hamiltonian is approximated via a parameterized quantum circuit. We discuss some basic theory of VAQC to motivate the development of a hybrid quantum-classical homotopy continuation method. The proposed method has parallels with a predictor-corrector method for numerical integration of differential equations. While there are theoretical limitations to the procedure, we see in practice that VAQC can successfully find good initial circuit parameters to initialize VQE. We demonstrate this with two examples from quantum chemistry. Through these examples, we provide empirical evidence that VAQC, combined with other techniques (an adaptive termination criteria for the classical optimizer and a variance-based resampling method for the expectation evaluation), can provide more accurate solutions than “plain” VQE, for the same amount of effort.

Numerical integration algorithms and constraint formulations for an ALE-ANCF cable element

In this study, the Absolute Nodal Coordinate Formulation (ANCF) is examined in the framework of the Arbitrary Lagrangian–Eulerian (ALE) formulation for cable simulation. The objective is to determine the stability, efficiency, and accuracy in different frequency domains for two different numerical time integration methods. The methods being compared are the Runge–Kutta method RK4, and the Generalized-α method. For simulations where the lower frequency domain is of interest, the Generalized-α method is stable, accurate and can be used with large time-steps; while where the high frequency domains is of interest, the RK4 procedure can be a more efficient method.The penalty method, the Lagrange multiplier method and linear coupling are tested as constraint methods for connecting ALE-ANCF cable elements. For RK4, the penalty constraint method highly affects the maximum time-step, which again affects the total solution time. For the Generalized-α method, the constraint methods all provide accurate results, and they have similar solution time. ALE-ANCF cable elements can be connected by different constraint techniques without impeding the ability to exhibit large deformations.

A Unified Singular Solution of Laplace’s Equation with Neumann and Dirichlet Boundary Conditions

Laplace’s equation is one of the important equations in studying applied mathematics and engineering problems including the study of temperature distribution of steady-state heat conduction or the concentration distribution of steady-state diffusion problems. In this study, the analytical method has been applied to solve the Laplace's equation in a two-dimensional domain. For the specified Neumann or Dirichlet boundary conditions, the analytical solution of temperature distribution in the quarter-plane can be found by several methods including the Fourier transform method, similarity method, and the method of Green’s function with images. For different boundary conditions, the solution of temperature distribution of the Laplace’s equation will be in a totally different form. Nevertheless, the merit of this research is that the solutions of steady-state temperature distribution in the quarter plane with Neumann and Dirichlet boundary conditions are unified under the singular similarity solution with source type singularity. With the typical benchmarked examples for finding the temperature distribution by the numerical integral method, it is shown that Gibbs phenomenon behaves at a jump discontinuity, where serious oscillation result was found especially near the singular points of the boundary. In addition, the temperature distribution in the domain can be easily calculated without oscillation phenomenon near the singular points from the similarity solutions.

Estimation of the volume of production of turbine vapor of a fuel boiler with stochastic exogenous factors

The paper is based on the analysis and review of the conceptual principles, methodology and practice of modelling water-steam processes of a fuel boiler of power plants; implementation of algorithms and models of the theory of system dynamics, white and pink noise; using computational technologies of difference-scheme equations and numerical integration methods, the values of the input parameters of the production of turbine vapor of the fuel boiler are determined), which, regardless of stochastic exogenous factors, provide stable states of the necessary and model volumes, norms of water intake and output of turbine vapor of the fuel boiler.* Corresponding author: kerimkhulle@gmail.com

Software for Calculating the Location Coordinates and Weighting Coefficients of Acoustic Paths of Ultrasonic Flow Meters

The article focuses on the development of software that allows you to calculate the location coordinates (x) and weighting coefficients (w) of acoustic paths of multipath chordal ultrasonic flow meters. The input data for the calculation are three (and in one case – four) parameters – the number of acoustic paths (N) ultrasonic flow meter, the type of numerical integration method, the internal diameter of the pipeline and degree (k) of weight function of Jacobi polynomial (if the Gauss-Jacobi numerical integration method was chosen). At the same time, the calculation is based on both table data (from mathematical directories) and on the analytical dependencies developed by the authors x = f(k) and w = f(k). The presence of such calculator software will simplify the work of developers, designers, scientists, and students who theoretically investigate the future influence of the geometric characteristics of ultrasonic flow meters on their metrological accuracy in various conditions of their operation.

Estimation of track lateral alignment based on function approximation by Sinc function and numerical Fourier transform.

This paper proposes a numerical method to estimate track lateral alignment. The estimation of the track lateral alignmentisformulatedastheproblemoffindinganapproximatesolutiontoafunctionalequation, andtheSincfunction approximation, which has been studied in the field of numerical analysis, and the Fourier transform approximation based on numerical integration methods are utilized to construct the approximate solution. In particular, our newly proposed method directly deals with the singularity in the frequency transfer function involved in the estimation.

Analysis of the application of various numerical integration methods and their practical implementation in MATLAB CAD

The article presents an analysis of numerical integration methods. These methods are frequently used in solving various technical problems in production. Iterative numerical methods in MATLAB CAD are considered and implemented, including the method of integration by central rectangles, the trapezoidal method, and Simpson’s rule. The practical significance of the work is familiarization with basic numerical methods that are widely used in computing, as well as with a powerful computing tool – MATLAB. In the course of the work, the algorithms of each method are described, program codes with illustrations for evaluating the results of calculating by a particular method are given. The conclusions are drawn regarding the accuracy of each method. The software and mathematical excerpts used in the work are publicly available on the Internet, which allows anyone to carry out similar work and make sure that the written codes and the obtained conclusions are correct. This work can be used not only to train students in the field of electronic device development in terms of their algorithmization and for the organization of laboratory work, but also in the creation and design of real devices both in production and within a higher educational institution, for example, for the development of laboratory work using specialized CAD. Familiarization and study of this software are conducted at the departments of one of the leading engineering universities of the Russian Federation.

Комп’ютерне рішення для знаходження поперечних динамічних зміщень товстих пружних плит в умовах невизначеності

The article poses the task of finding the function of dynamic transverse displacements of thick elastic plates under the action of axisymmetric external dynamic disturbances acting on the surface of such plates. When setting the problem, it is indicated that some initial-boundary observations are known for the state of the plate, but their number is insufficient to solve this problem by classical methods of mathematical physics or numerical integration. Therefore, it is possible to apply to this problem the method of mathematical modeling of externally distributed dynamic processes under conditions of uncertainty, i.e. under insufficient number of initial and boundary conditions. For such a method, it is necessary to have an integral mathematical model and its main components - kernels, that is, integral functions for which there is a finding method. Or they can be calculated using numerical methods of computer algebra systems. With the built integral model of the dynamics of thick elastic plates, taking into account its kernels, the method of mathematical modeling of the dynamics of spatially distributed processes leads to the result - a set of solutions that accurately satisfy the integral and differential models and agree with the initial boundary conditions according to a certain criterion. The article selects one of the many solutions to the problem of finding the function of transverse dynamic displacements, which is found according to the methodology of mathematical modeling of the dynamics of spatially distributed processes and thanks to the calculation of the integral function of the mathematical integral model. For the problem, the elastic characteristics and density of the slab corresponding to some material are fixed, some specific external dynamic disturbances and initial-at-the-edge observations are determined, which are represented by certain conditions at specific points. Under such assumptions, graphs of the functions of transverse dynamic displacements are constructed at different values of the transverse coordinate z and at the value 0 of the radial coordinate r of the cylindrical coordinate system.

MOVEMENT OF A PARTICLE ALONG AN INCLINED CYLINDER ROTATING AROUND ITS AXIS

It is well known that parts of agricultural machinery often have a cylindrical shape. This shape, for example, can be observed in the casing of lifting and transport machines, where the active working body rotates. Furthermore, drum grain dryers and triers use an inclined cylinder that rotates around its axis. In this case, the particles of the technological material interact with the rotating surface, which leads to their sliding, the nature of which depends on the value of the angle of inclination of the cylinder. In this study, the methods of differential geometry, vector algebra, theoretical mechanics, and numerical integration of differential equations consider the motion of a particle along the inner surface of an inclined cylinder rotating at a constant angular velocity around its axis. The axes of a fixed coordinate system are used to compose differential equations of motion. It was established that the proper initial conditions under which the particle would be stationary at a certain distance from the lower forming cylinder towards its rotation can be determined analytically. In case of movement along an inclined cylinder, the particle moves, among other things, in the axial direction, while reducing the amplitude of vibrations. Furthermore, it was found that the angle of inclination of the cylinder plays a significant role. If the latter is less than the angle of friction, then the vibrations stop, the movement of the particle stabilises, and it performs a rectilinear movement at a constant speed in the axial direction. If the angle of inclination of the cylinder is greater than or equal to the angle of friction, then the particle moves rapidly in the axial direction and its movement does not stabilise. The value of the angular velocity of rotation also plays a significant role. A certain amount of it provokes “sticking” of the particle, which does not depend on the inclination angle of the cylinder. The obtained analytical dependences can be used in the design of cylindrical working bodies of agricultural machines.

Explicit Formulas and Numerical Integral Equation of ARL for SARX(P,r)<sub>L</sub> Model Based on CUSUM Chart

The Cumulative Sum (CUSUM) chart is widely used and has many applications in different fields such as finance, medical, engineering, and other fields. In real applications, there are many situations in which the observations of random processes are serially correlated, such as a hospital admission in the medical field, a share price in the economic field, or a daily rainfall in the environmental field. The common characteristic of control charts that has been used to evaluate the performance of control charts is the Average Run Length (ARL). The primary goals of this paper are to derive the explicit formula and develop the numerical integral equation of the ARL for the CUSUM chart when observations are seasonal autoregressive models with exogenous variable, SARX(P,r)_L with exponential white noise. The Fredholm Integral Equation has been used for solving the explicit formula of ARL, and we used numerical methods including the Midpoint rule, the Trapezoidal rule, the Simpson's rule, and the Gaussian rule to approximate the numerical integral equation of ARL. The uniqueness of solutions is guaranteed by using Banach's Fixed Point Theorem. In addition, the proposed explicit formula was compared with their numerical methods in terms of the absolute percentage difference to verify the accuracy of the ARL results and the computational time (CPU). The results obtained indicate that the ARL from the explicit formula is close to the numerical integral equation with an absolute percentage difference of less than 1%. We found an excellent agreement between the explicit formulas and the numerical integral equation solutions. An important conclusion of this study was that the explicit formulas outperformed the numerical integral equation methods in terms of CPU time. Consequently, the proposed explicit formulas and the numerical integral equation have been the alternative methods for finding the ARL of the CUSUM control chart and would be of use in fields like biology, engineering, physics, medical, and social sciences, among others.

RESEARCH OF THE MOVEMENT OF A MATERIAL PARTICLE ON THE SURFACE OF AN OBLIQUE HELICOID UNDER THE ACTION OF THE FORCE OF OWN WEIGHT

The problem that describes the movement of a particle along a helical surface is used in the design of spiral separators. Modeling the motion of a material particle on helical surfaces and its investigation by modern methods of numerical integration and visualization makes it possible to obtain a real picture of motion in the absence of full-scale models of such surfaces. This makes it possible to search for helical surfaces to improve their operational characteristics. The purpose of the study was to establish the regularity of the movement of a material particle along an oblique helicoid depending on the structural parameters of the surface. The oblique helicoid has two design parameters - the pitch h and the angle b of the inclination of the rectilinear generating surfaces to the horizontal plane. With the help of these parameters, there are more opportunities to influence the nature of the movement of a particle along an oblique helicoid compared to a helical conoid and a wide helicoid. The differential equations of motion of a material particle on the surface of an oblique helicoid under the action of its own weight were formulated. The equations are solved by numerical methods. The obtained results were visualized It has been established that the movement of a material particle with a known coefficient of friction along the surface of an oblique helicoid can be ensured at a given distance from its axis by combinations of structural parameters of the surface. At the same time, the resolution of the surface during the separation of particles with different friction coefficients practically does not change. However, in the transition period (before the stabilization of the motion), there is a significant difference in the trajectories of particle motion, which increases as the angle of inclination of the rectilinear generating surfaces decreases. This gives reason to consider material separation at the stage of the transition period, which requires further theoretical and experimental research.

An Improved Method for Total Radiated Power Tests in Anechoic Chambers

This paper proposes an improved method for Total Radiated Power (TRP) tests based on a numerical integration method called Clenshaw-Curtis quadrature. It is known that TRP tests in an anechoic chamber need integral calculation of the measured samples. And the samples are discretized in the angular domain in practical over-the-air (OTA) testing. By adopting the Clenshaw-Curtis quadrature method to the TRP tests, we can achieve higher accuracy without increasing the number of sampling points or measurement time. Note that quadrature method must be selected judiciously by considering practical OTA setting. A numerically accurate quadrature method may not necessarily ensure accurate OTA tests. To illustrate this point, another measurement approach based on a modified Clenshaw-Curtis quadrature method (with even better numerical accuracy) is used for comparison. The proposed method based on the Clenshaw-Curtis quadrature shows superior performance to the conventional and modified approaches. The above findings are verified by both simulation and measurement experiments.

Semi-Contactless Power Measurement Method for Single-Phase Enclosed Two-Wire Residential Entrance Lines

This paper presents a semi-contactless power measurement method aimed at offering power measurements of residential homes, which are power supplied by the single-phase two-wire entrance lines, for studying non-intrusive load monitoring (NILM) technique. The active power is calculated using the waveforms of the line current and voltage. To measure the line current, a magnetic sensor array consisting of seven tunnel magnetoresistance (TMR) sensors are utilized to detect the magnetic fields around the entrance lines. The single-phase time-domain current is then reversely calculated on real time by matrix multiplication using the time-domain output signals of the TMR sensor array once the entrance line positions are identified through solving the formulated nonlinear least square (NLLS) problem. In terms of the line voltage, a capacitively coupling voltage sensor is proposed to achieve almost in-phase tracking with the line voltage waveform. The voltage ratio of the voltage senor is calibrated using the measured RMS value of line voltage from the outlet sockets or power strips in-situ by a multimeter (thus called semi-contactless) due to the main disadvantage for capacitively coupling voltage sensing. That is, accuracy in voltage amplitude measurement is low because of the great difficulty in accurately determining the coupling capacitances. Numerical integration method for calculating the active power based on quasi-synchronous algorithm is proposed to eliminate the errors caused by the non-integer period sampling. Prototype instrument is designed and fabricated. Frequency response characterization shows that the TMR sensor (voltage sensor) has an almost constant gain factor (voltage ratio) and an approximate zero-phase shift (< 1°) over a frequency range of 50 Hz - 3.05 kHz. Laboratory tests for both fundamental current phasor and active power measurement along with their measurement uncertainty evaluations are first conducted to verify the feasibility of the proposed method. Field tests in a residential home are subsequently performed to further demonstrate the effectiveness of the proposed method. A power measurement error of less than 5% can be obtained, which is sufficient for home energy tracking.

Evaluating statistical significance in a meta-analysis by using numerical integration

Meta-analysis is a method for enhancing statistical power through the integration of information from multiple studies. Various methods for integrating p-values (i.e., statistical significance), including Fisher’s method under an independence assumption, the permutation method, and the decorrelation method, have been broadly used in bioinformatics and computational biotechnology studies. However, these methods have limitations related to statistical assumption, computing efficiency, and accuracy of statistical significance estimation. In this study, we proposed a numerical integration method and examined its theoretical properties. Simulation studies were conducted to evaluate its Type I error, statistical power, computational efficiency, and estimation accuracy, and the results were compared with those of other methods. The results demonstrate that our proposed method performs well in terms of Type I error, statistical power, computing efficiency (regardless of sample size), and statistical significance estimation accuracy. P-value data from multiple large-scale genome-wide association studies (GWASs) and transcriptome-wise association studies (TWASs) were analyzed. The results demonstrate that our proposed method can be used to identify critical genomic regions associated with rheumatoid arthritis and asthma, increase statistical significance in individual GWASs and TWASs, and control for false-positives more effectively than can Fisher’s method under an independence assumption. We created the software package Pbine, available at GitHub (https://github.com/Yinchun-Lin/Pbine).

Large Deformation Hermitian Finite Element Coupled Thermoelasticity Analysis of Wave Propagation and Reflection in a Finite Domain

In the present paper, a finite element nonlinear coupled thermoelasticity formulation is presented for analysis of the wave propagation, reflection, and mixing phenomena in the finite length isotropic solids. The governing equations are derived based on the second Piola-Kirchhoff stress and the full form of Green’s strain-displacement tensors to account for the large deformations and finite strains. In contrast to the available researches, the assumption of very small temperature changes compared to the reference temperature is released in the present research. Galerkin’s method, a weak formulation, and cubic elements are employed to obtain the time-dependent non-linear finite element governing equations. The proposed solution procedure to the resulting highly nonlinear and time-dependent governing equations employs an updating algorithm and Newmark’s numerical time integration method. The wave propagation and reflection phenomena are investigated for both the mechanical and thermal shocks and time variations of distributions of the resulting displacements, temperature rises, and stresses are illustrated graphically and discussed comprehensively. Furthermore, the effects of the non-linear terms are discussed comprehensively. Results reveal that in the non-linear analysis, no fixed speed of wave propagation can be defined.

Theoretical approach for uncertainty quantification in probabilistic safety assessment using sum of lognormal random variables

Probabilistic safety assessment is widely used to quantify the risks of nuclear power plants and their uncertainties. When the lognormal distribution describes the uncertainties of basic events, the uncertainty of the top event in a fault tree is approximated with the sum of lognormal random variables after minimal cutsets are obtained, and rare-event approximation is applied. As handling complicated analytic expressions for the sum of lognormal random variables is challenging, several approximation methods, especially Monte Carlo simulation, are widely used in practice for uncertainty analysis. In this study, a theoretical approach for analyzing the sum of lognormal random variables using an efficient numerical integration method is proposed for uncertainty analysis in probability safety assessments. The change of variables from correlated random variables with a complicated region of integration to independent random variables with a unit hypercube region of integration is applied to obtain an efficient numerical integration. The theoretical advantages of the proposed method over other approximation methods are shown through a benchmark problem. The proposed method provides an accurate and efficient approach to calculate the uncertainty of the top event in probabilistic safety assessment when the uncertainties of basic events are described with lognormal random variables.

The Periodicity of the Accuracy of Numerical Integration Methods for the Solution of Different Engineering Problems

Newton-Cotes integration formulae have been researched for a long time, but the topic is still of interest since the correctness of the techniques has not yet been explicitly defined in a sequence for diverse engineering situations. The purpose of this paper is to give the readers an overview of the four numerical integration methods derived from Newton-Cotes formula, namely the Trapezoidal rule, Simpson's 1/3rd rule, Simpson's 3/8th rule, and Weddle's rule, as well as to demonstrate the periodicity of the most accurate methods for solving each engineering integral equation by varying the number of sub-divisions. The exact expressions by solving the numerical integral equations have been determined by Maple program and comparisons have been done using Python version 3.8.

Replacement of force-to-motion relationship with state–space model for dynamic response analysis of floating offshore structures

The Cummins equation is commonly used to express the motion behaviour of a floating offshore structure. It is a differential equation with convolution terms, which make it inconvenient to conduct a dynamic response analysis. Moreover, the calculation of the simulation response is not highly efficient because of the presence of these convolution terms, and a serious error accumulation problem occurs when calculating the dynamic response using a numerical integration method. Therefore, a method for estimating the dynamic response of a floating offshore structure by replacing the force-to-motion relationship with a state–space model has been described in literature. One theoretical development is the translation of dynamic response calculation for a floating structure into the task of obtaining the output of the state–space model system, which is easy to implement and convenient to use for designing motion control systems. Meanwhile, this method does not require the calculation of the convolution terms, which implies that the algorithm can considerably improve the calculation efficiency of a dynamic response simulation and avoid the problem of error accumulation. Three examples were used to verify the proposed algorithm. The first was a single degree of freedom system in the form of the Cummins equation, which was used to exhibit the entire process of the proposed algorithm. Studies have shown that the dynamic response computed by the proposed method is in accordance with that of the average acceleration method, and the proposed method can avoid the error accumulation caused by large time steps. The second was a semi-submersible floating offshore wind turbine, which expanded the application of the proposed algorithm to an estimation of the dynamic response of a floating offshore structure with six degrees of freedom. The results showed that the proposed algorithm, as compared with the time domain method, could obtain an accurate response and avoid the numerical dissipation problem in the numerical integration algorithm. Meanwhile, the computational efficiency of the proposed method was significantly higher than that of the average acceleration method. The last example was a physical model of a semi-submersible platform. The heave response estimated using the proposed algorithm was good match with the experimental data.

A time-stepping method for multibody systems involving frictional impacts and phases with persistent contact

This work complements and continues efforts towards developing a complete numerical integration method for multibody dynamic systems with frictional impacts. Specifically, the new method is a time-stepping scheme, which is applicable to mechanical systems subject to a set of bilateral and a single unilateral motion constraint. During an impact free phase of the motion, a suitable augmented Lagrangian methodology is applied. When an impact is detected during a time step, the post-impact state is determined by employing a special process, which addresses properly the inherent numerical stiffness associated with the impact events. The basic contribution of the present study is the extension to cover cases where the contact between the interacting bodies is possible to exist for a finite period of time after an impact. This phenomenon occurs frequently in machines and mechanisms and is known as persistent contact. First, a complete strategy is developed for establishing a general numerical procedure for capturing and treating both activation of contact and loss of contact between two bodies. Then, the accuracy and efficiency of this procedure is tested and demonstrated by applying it to a selected set of mechanical examples.

The Numerical simulation of composite sandwich cylindrical shells reinforced with circular frames under local loadings

The paper considers a woven fabric (based on fiberglass) composite structure in the form of a foam core sandwich cylindrical shell reinforced with circular frames, under the action of local loads applied to the frames. A technique is described for obtaining a numerical solution (with the confirmed reliability) to the problem of the stress-strain state of this type of structure using two alternative computational models, one of which is based on the numerical integration method, and the other one is based on the finite element method. The first model is developed by adopting a scheme, in which the frames are considered as short cylindrical shells obeying the single normal hypothesis. The foam core sandwich sections connected to them are considered within the framework of the zig-zag theory of soft core sandwich shells based on the assumption of incompressibility of the core along thickness. In this case, the corresponding calculation problem is formulated in the form of systems of algebraic and differential (in partial derivatives) equations for each of the shell sections being considered, which are supplemented by kinematic and force conditions at their joints, as well as boundary conditions at the left and right ends of the structure. The problem solving for each harmonic number is reduced to solving a set of boundary value problems for systems of 8 and 12 linear ordinary differential equations of the first order, related by conditions at the specified joints and using the procedure for expanding the parameters of the stress-strain state and applied loads into Fourier series in the circumferential direction. The solution algorithm is constructed using the numerical integration procedure in the orthogonal sweep version combined with the displacement method procedure (to satisfy the conditions at the joints). The declared finite element model is built within the ABAQUS software package using S4 shell elements (for composite layers) and C3D20 volume elements (for frames and core). The formed model, when setting deliberately overestimated values of the corresponding elastic modules, can implement a situation close to fulfilling the set of hypotheses adopted when constructing the first model. Having fixed in such a situation the consistency of the calculation results on the basis of the alternative computational models constructed in this way and thereby confirming the reliability of the obtained numerical solution, we carry out the transition to the calculation using the real values of the mentioned modules and the analysis of their influence on the stress-strain state of the investigated sandwich shell. The presented example of the calculation of a composite sandwich cylindrical structure, one of the frames of which is under the action of two local axial loads, demonstrates the possibilities of the adopted modeling method.