Traditional Numerical Integration Research Articles

Parallel Solution of Nonlinear Projection Equations in a Multitask Learning Framework.

Nonlinear projection equations (NPEs) provide a unified framework for addressing various constrained nonlinear optimization and engineering problems. However, when it comes to solving multiple NPEs, traditional numerical integration methods are not efficient enough. This is because traditional methods solve each NPE iteratively and independently. In this article, we propose a novel approach based on multitask learning (MTL) for solving multiple NPEs. The solution procedure is outlined as follows. First, we model each NPE as a system of ordinary differential equations (ODEs) using neurodynamic optimization. Second, for each ODE system, we use a physics-informed neural network (PINN) as the solution. Third, we use a multibranch MTL framework, where each branch corresponds to a PINN model. This allows us to solve multiple NPEs in parallel by training a single neural network model. Experimental results show that our approach has superior computational performance, especially when the number of NPEs to be solved is large.

Fast Power System Transient Stability Simulation

Power system transient stability simulation is of critical importance for utilities to assess dynamic security. Most of the commercially available tools use the traditional numerical integration method to simulate power system transient stability, which is computationally intensive and has low simulation speed. This makes it difficult to identify any insecure contingency before it happens. It is already proven that power system transient stability simulation achieved using the differential transformation method (DTM) requires less computational effort and has improved simulation speed, but it still requires further improvement regarding its accuracy and performance efficiency. This paper introduces a novel power system transient stability simulation method based on the adaptive step-size differential transformation method. Using the proposed method, the step size is varied based on the estimated local solution error at each time step. The accuracy and speed of the proposed simulation approach are investigated in comparison with the classical differential transformation method and the traditional numerical integration method using the IEEE 9 bus and 39 bus test systems. The simulation results reveal that the proposed method increases the simulation speed by 20–44.57% and 83–92% when compared with the classical DTM and traditional numerical-integration-based simulation methods, respectively. It is also proved that compared with the DTM-based simulation, the proposed method provides 45.27% to 58.85% and more than 90% accurate simulation results for IEEE 9 and IEEE 39 test systems, respectively. Therefore the proposed power system transient stability simulation method is faster and relatively more accurate and can be applied for online transient stability monitoring of power system networks.

A consistent projection integration for Galerkin meshfree methods

An efficient and inherently consistent integration method, named projection integration(PI), is proposed for Galerkin meshfree methods with arbitrary order. In contrast to traditional numerical integration methods, PI defines a projection operator based on a weighted norm to systematically approximate meshfree shape functions in an appropriate and consistent function space. Then a substitute function space is established by utilizing projected shape functions as a basis, and Galerkin weak form is approximately calculated by substituting the projected shape functions for the original meshfree shape functions. Owing to the projection consistency of the projection operator, the projected shape functions can still meet consistent conditions. For convenient implementations, the triangular finite element space is selected to approximate meshfree shape functions and the weight function in the weighted norm is carefully chosen to simplify the projection operator in this paper. Arbitrary order integration constraint conditions can be easily met by utilizing the corresponding order triangular finite element shape functions with low order Gauss quadrature rules, which means the optimal convergence rate can be obtained. A series of numerical examples are presented to demonstrate the proposed method’s efficiency and superior convergence performance.

On the Exact Analytical Formulas of Leakage Current-Based Supercapacitor Model Operating in Industrial Applications

The resistance–capacitance (RC) model is one of the most applicable circuits for modeling the charging and discharging processes of supercapacitors (SCs). Although this circuit is usually used in the electric and thermal investigation of the performance of SCs, it does not include leakage currents. This paper presents exact analytical formulas of leakage-current-based supercapacitor models that can be used in industrial applications, i.e., constant-power-based applications. In the proposed model, current and voltage are represented as a solution of nonlinear equations that are solved using the standard Newton method. The proposed expressions’ accuracy is compared with the results obtained using traditional numerical integration methods with leakage current formulation and other methods, found in the literature, with no leakage current formulation. The results confirm that including leakage current represents a more accurate and realistic manner of modeling SCs. The results show that the derived expressions are precise, allowing the generation of results that closely match those obtained using traditional numerical-based methods. The derived expressions can be used to investigate SCs further and achieve more accurate and efficient regulation and control of SCs in different applications.

SNIF-DFA: A Signal Processing and Information Fusion Method for Smart Gua Sha Device

Due to the COVID-19 global pandemic, there is more need for remote patient care, especially in rehabilitation requiring direct contact. However, traditional Chinese rehabilitation technologies, such as gua sha, often need to be implemented by well-trained professionals. To automate and professionalize gua sha, it is necessary to record the nursing and rehabilitation process and reproduce the process in developing smart gua sha equipment. This article proposes a new signal processing and sensor fusion method for developing a piece of smart gua sha equipment. A novel stabilized numerical integration method based on information fusion and detrended fluctuation analysis (SNIF-DFA) is performed to obtain the velocity and displacement information during the gua sha operation. The experimental results show that the proposed method outperforms the traditional numerical integration method with respect to information accuracy and realizes accurate position calculations. This is of great significance in developing robots or automated machines that reproduce the nursing and rehabilitation operations of medical professionals.

The impact of negative interest rates on the pricing of options written on equity: a technical study for a suitable estimate of early termination

This work aims to investigate the main problems that impact the pricing models and the sensitivity measures of American options written on shares without a pay-out, in the presence of negative interest rates with a specific focus on the Monte Carlo method. The first paragraph carries out a review of the anomalies caused by such an odd condition and focuses thereafter on the core topic of the research by treating a wide range of numerical models suitable for unbiased evaluation of the early exercise, thus expanding the existing literature. The two following paragraphs are dedicated to describing the models used for the correct estimation of fair value: binomial lattice models (Cox-Ross-Rubinstein - CRR Tree, Leisen Reimer - LR Tree, Jarrow-Rudd - JR Tree and Tian Tree), trinomial stochastic trees, Finite Difference Method (FDM) scheme and the Longstaff-Schwartz Monte Carlo. Particular attention is paid to this last approach which allows to combine the flexibility of traditional numerical integration schemes for stochastic processes on equity with the estimation of the convenience of exercising the American option ahead of time. After conducting quantitative tests both on pricing and on the estimation of sensitivity measures, the LR Tree was selected as the most performing deterministic algorithm to be compared with the Monte Carlo stochastic technique. The final part of the work focuses on quantifying the valuation gap introduced by negative interest rates in the valuation of American options written on an unprofitable underlying comparing the traditional valuation approach and the deterministic Leisen Reimer model and the Longstaff-Schwartz stochastic model.

A Numerical Algorithm for Arbitrary Real-Order Hankel Transform

The Hankel transform is widely used to solve various engineering and physics problems, such as the representation of electromagnetic field components in the medium, the representation of dynamic stress intensity factors, vibration of axisymmetric infinite membrane and displacement intensity factors which all involve this type of integration. However, traditional numerical integration algorithms cannot be used due to the high oscillation characteristics of the Bessel function, so it is particularly important to propose a high precision and efficient numerical algorithm for calculating the integral of high oscillation. In this paper, the improved Gaver-Stehfest (G-S) inverse Laplace transform method for arbitrary real-order Bessel function integration is presented by using the asymptotic characteristics of the Bessel function and the accumulation of integration, and the optimized G-S coefficients are given. The effectiveness of the algorithm is verified by numerical examples. Compared with the linear transformation accelerated convergence algorithm, it shows that the G-S inverse Laplace transform method is suitable for arbitrary real order Hankel transform, and the time consumption is relatively stable and short, which provides a reliable calculation method for the study of electromagnetic mechanics, wave propagation, and fracture dynamics.

Angular orientation determination in SINS: traditional algorithms comparison

The principle of organization of strap-down inertial navigation systems is based on numerical integration of angular velocities and accelerations. The purpose of numerical integration algorithms is to approximate the behavior of a dynamic system (unmanned aerial vehicle – UAV) with continuous time using a digital computer. The efficiency of numerical integration is determined by the accuracy and stability of the computational process. The integration algorithm may have a small integration error, but at the same time be inefficient due to the instability of the numerical method when the step or conditions of integration change. The standard way to test integration algorithms for stability is to test them under control operating conditions (when performing a typical UAV flight along the route and canonical movement). The article presents the results of simulation modeling of traditional numerical integration algorithms in the conditions of rectilinear and conical UAV motion, when calculating the values of angular velocities by various methods. The analysis of the obtained research results is carried out, which allows us to choose an algorithm that has an advantage with respect to accuracy and computational simplicity, depending on the flight conditions. For a UAV that has no or minimal undampened angular harmonic oscillations of its body, when performing a typical flight along the route, the best, in terms of accuracy and volume of calculations, is a second-order accuracy algorithm implementing the average speed method. Its average error in calculating angles ranges from 3.6 to 43%, which is approximately equal to the errors values when using the considered algorithms (an algorithm implementing a second approximation to the average speed method, a one-step algorithm of the thirdorder of accuracy), with a three-fold smaller amount of mathematical calculations.

Comparison of numerical integration methods for calculating diffraction of a plane electromagnetic wave by a rectangular aperture

We discuss features of the calculation of a Fraunhofer integral by traditional quadrature numerical integration methods and a special collocation Levin method when calculating the diffraction of a plane electromagnetic wave by a rectangular aperture. For the quadrature numerical integration methods, a criterion for the assessment of the integration step is derived depending on the screen size and required calculation accuracy. Advantages of the use of the special collocation Levin method in comparison with the traditional quadrature numerical integration methods are shown.

Integral Evaluation for a Closed-Form 2-D Potential Formulation of the Galerkin BEM

Closed-form solutions are derived for the regular and adjacent-singular integrals involving the two-dimensional Laplacian’s Green’s function and its normal derivative that arise in the Galerkin BEM. Motivation for their use is provided by comparing the accuracy and time required to compute the BEM system matrix relative to traditional numerical integration. Specifically, it is shown that the use of the closed-form solutions reduces computation time to produce the influence matrices by approximately 95% relative to Gauss quadrature with similar accuracy. All elements of these matrices are then expressed in a form convenient for implementation.

Application of Stabilized Numerical Integration Method in Acceleration Sensor Data Processing

It is necessary to perform a numerical or a quadratic integral calculation on the acceleration sensor data by rectangular integration, trapezoidal integration or Simpson integration, etc . to obtain the velocity or the displacement result in engineering practice. Since poles of transfer functions of these numerical integration systems are on the unit circle, when acceleration signals containing interference factors, such as random noise, are integrated, integration results are often divergent. Therefore, velocity and displacement results will have serious drift, which affects the accuracy of the follow-up attitude calculation and target positioning. In order to overcome acceleration integration drift, a method based on the stabilized numerical integration to obtain velocity and displacement is proposed in this paper, which makes improvements of the traditional numerical integration. The proposed method obtains accurate velocity and displacement through the acceleration integration by constructing a stable system to modify the transfer function of the integration system, and utilizes the characteristics of the stability system to effectively suppress the drift of the acceleration data multiple integration results. Experimental results show that compared with traditional acceleration integration methods, the quadratic integral displacement peak error, difference error, and absolute error of the proposed stabilized numerical integration method are 0.059, 0.0619 and 0.0084, respectively, which demonstrate that the proposed method is feasible that it can reduce the integration results drift error and improve the accuracy of the integration results.

A Modified Linear Integral Resonant Controller for suppressing jump-phenomenon and hysteresis in micro-cantilever beam structures

This paper demonstrates the effectiveness of a Modified Linear Integral Resonant Controller (MIRC) based on its original LTI cousin, known just as the ‘IRC’, for suppressing Jump-Phenomenon and hysteresis found in Micro-Cantilevers. A Method of Multiple Scales frequency response is derived, explored and compared with a Runge-Kutta based numerical integration method in order to understand any shortcomings in approximate analytical methods for the analysis of closed-loop nonlinear systems with the inclusion of a stability analysis. It is found that there exist some mild inconsistencies when comparing closed-loop Method of Multiple Scales to traditional numerical integration. A suitably designed Modified Integral Resonant Controller is implemented in closed-loop. Both analytical and numerical results agree with each other and show significant damping performance. Efficacy of the proposed control scheme is validated via frequency response plots, phase portraits and quiver plots.

Singularity cancellation method for time-domain boundary element formulation of elastodynamics: A direct approach

In the implementation of time-domain boundary element method for elasto-dynamic problems, there are two types of singularities: the wave front singularity arising when the product of wave velocity and time is equal to the distance between the source point and the field point, and the spatial singularity arising when the source point coincides with the field point. In this paper, the singularity of the first type in the integrand is eliminated by an analytical integration over time, Cauchy principal value and Hadamard finite part integral. Four types of singularities with different orders appear in the integrand after analytical time integration. In order to accurately calculate the integral, in which the integrand is piecewise continuous, the integral domain is subdivided into several patches based on the relation between the product of wave velocity and time and the distance. In singular patches, the integrands are separated into a regular part and a singular part. The regular part can be computed by traditional numerical integration method such as Gaussian integration, while the singular part can be analytically integrated. Using the proposed method, the spatial singular integrals can be calculated directly. Numerical examples using various kinds of elements are presented to verify the proposed method.

WaveNet: learning to predict wave height and period from accelerometer data using convolutional neural network

Inertial sensors carried by buoys, such as accelerometers, are widely used in wave characteristics measurement. Traditional methods usually employ numerical integration on the accelerate data for wave height, where the “drifting” errors are intractable. In this paper we propose a novel method to predict wave height and period using machine learning approach, specially a convolutional neural network. The end-to-end 1D convolutional neural network named WaveNet predicts wave height and period from the raw acceleration data directly. We designed a simple device to simulate the motion of the buoy in the wave, and used it to collect data for training and testing our model. The results of the proposed method were compared with traditional numerical integration method and found that the proposed model outperforms existing method in outputting more accurate wave height and period.

Recursive Calculation Model for a Special Multivariate Normal Probability of First-Order Stationary Sequence

The consecutive hit probability of antiaircraft artillery corresponds to the multivariate normal probability distribution. The computational complexity depends on the length of the firing error sequence (i.e., the integral dimension may exceed 100). The traditional numerical integration and the Monte Carlo method are too slow for this calculation. This paper established the state equation of the firing error sequence, which was the bridge between the multivariate normal probability distribution and stochastic process theory. The recursive calculation model was given after the rigorous derivation process. The accuracy and computational complexity of the model were quantified by theoretical analysis and expressed intuitively by examples. The model shows the upper bound for the absolute error, and the computational efficiency is significantly improved.

Transient dynamic analysis of cracked structures with multiple contact pairs using generalized HSNC

The development of efficient computational methods for cracked structures is critical in the fields of civil, mechanical, and aerospace engineering since the influence of cracks on structural dynamics can play an important role in design, prognosis, and health monitoring. The nonlinearity caused by the intermittent contact on the crack surfaces typically excludes the use of fast linear methods, without which the computation of the dynamics of complex cracked structures becomes very challenging. In this paper, an efficient computational scheme for predicting both the transient and steady-state responses of cracked structures with multiple contact pairs is introduced. The new algorithm is referred to as the generalized hybrid symbolic–numeric computational (HSNC) method. The generalized HSNC method extends the original HSNC method, which was recently developed for bilinear systems, to general piecewise-linear nonlinear systems. This work also combines the HSNC with the $$\hbox {X}-\hbox {X}_{r}$$ method, a reduced-order modeling technique for cracked structures, to efficiently predict the dynamics of complex structures with cracks. The generalized HSNC approach is based on the idea that the nonlinear response of a cracked structure with multiple contact pairs can be obtained by combining linear responses of the system in each of its linear states. These linear responses can be symbolically expressed as functions of the initial conditions at starting time points in each time range where the system behaves linearly. The transition time where the system switches from one linear state to another is found using a nonlinear optimization solver with the initial values provided by an incremental search process. The method is able to individually track status of each contact pair; therefore, it can be used to predict the dynamics of the system when the crack surfaces are not completely open or closed. Moreover, both the transient and steady-state responses of complex cracked structures under various forcing conditions can be captured by the new method. The generalized HSNC method provides a flexible computational framework that is several orders of magnitude faster than traditional numerical integration methods. The dynamics of a spring–mass system and cantilever beam models that contain one crack and multiple cracks are investigated using the proposed method.

Geometrically Non-linear Analysis of Elastic Structures by Petrov-Galerkin Natural Element Method

This paper is concerned with the geometrically non-linear analysis of elastic structures using the natural element method, particularly a Petrov-Galerkin Natural Element Method (PG-NEM). The geometrically non-linear problem is linearized and formulated by the Taylor series expansion and the Total Lagrangian (TL) method. The Voronoi polygon-based Laplace Interpolation (LI) functions are used to approximate the displacement increments, while the Delaunay triangle-supported Constant-Strain (CS) basis functions are adopted for the expansion of the virtual displacement. The iterative computation is performed by the Newton-Raphson method, and a set of Delaunay triangles serves as a background cell for the traditional Gauss integration rule-based numerical integration. The proposed PG-NEM is illustrated through the numerical experiments, and its numerical accuracy is compared with MSC/Marc, CS-FEM and BG-NEM.

Optimization of Compatible Meshfree Quadrature Rule for Nonlocal Problems with Applications to Peri Dynamics Study

The optimization of a compatible meshfree quadrature rule for nonlocal problems with applications to Peridynamics is investigated in this study. Peridynamics is a nonlocal continuum mechanics theory that models material behavior based on interactions between material points within a finite neighbourhood. In Peridynamics, the accurate evaluation of nonlocal integral equations is crucial for obtaining reliable and efficient solutions. Traditional numerical integration methods, such as Gaussian quadrature, are not directly applicable to nonlocal problems due to their local nature. Hence, the development of a compatible meshfree quadrature rule that can effectively handle nonlocal interactions is of great importance. The objective it to optimize a quadrature rule that accurately captures the nonlocal interactions in Peridynamics while maintaining computational efficiency. The optimization process involves the selection of appropriate quadrature points and weights that minimize the quadrature error and maximize the computational efficiency. Various optimization techniques, such as genetic algorithms, particle swarm optimization, or machine learning algorithms, are explored to search for an optimal quadrature rule. The optimized quadrature rule is then applied to several Peridynamics problems, including fracture mechanics, material failure, and dynamic response analysis. The performance of the optimized quadrature rule is evaluated by comparing the results with those obtained using traditional quadrature methods and analytical solutions, when available. The accuracy, stability, and computational efficiency of the optimized quadrature rule are analysed and discussed. The findings provide valuable insights into the development and optimization of compatible meshfree quadrature rules for nonlocal problems, particularly in the context of Peridynamics. The optimized quadrature rule offers improved accuracy in capturing nonlocal interactions and reduces the computational cost compared to traditional methods. The applications of the optimized quadrature rule to various Peridynamics problems demonstrate its effectiveness and reliability.
 The implications of it extend beyond Peridynamics and can be applicable to other nonlocal models in the field of computational mechanics. The optimized quadrature rule has the potential to enhance the accuracy and efficiency of numerical simulations involving nonlocal phenomena, enabling more reliable predictions of material behavior and structural response. The findings of this contribute to the advancement of numerical methods for nonlocal problems and provide a foundation for further research and developments in the field of compatible meshfree quadrature rules for nonlocal problems.

Computing Integrals Involved the Gaussian Function with a Small Standard Deviation

We develop efficient numerical integration methods for computing an integral whose integrand is a product of a smooth function and the Gaussian function with a small standard deviation. Traditional numerical integration methods applied to the integral normally lead to poor accuracy due to the rapid change in high order derivatives of its integrand when the standard deviation is small. The proposed quadrature schemes are based on graded meshes designed according to the standard deviation so that the quadrature errors on the resulting subintervals are approximately equal. The integral in each subinterval is then computed by considering the Gaussian function as a weight function and interpolating the smooth factor of the integrand at the Chebyshev points of the first kind. For a finite order differentiable factor, we design a quadrature scheme having accuracy of a polynomial order and for an infinitely differentiable factor of the integrand, we design a quadrature scheme having accuracy of an exponential order. Numerical results are presented to confirm the accuracy of these proposed quadrature schemes.

Treat All Integrals as Volume Integrals: A Unified, Parallel, Grid-Based Method for Evaluation of Volume, Surface, and Path Integrals on Implicitly Defined Domains.

We present a unified method for numerical evaluation of volume, surface, and path integrals of smooth, bounded functions on implicitly defined bounded domains. The method avoids both the stochastic nature (and slow convergence) of Monte Carlo methods and problem-specific domain decompositions required by most traditional numerical integration techniques. Our approach operates on a uniform grid over an axis-aligned box containing the region of interest, so we refer to it as a grid-based method. All grid-based integrals are computed as a sum of contributions from a stencil computation on the grid points. Each class of integrals (path, surface, or volume) involves a different stencil formulation, but grid-based integrals of a given class can be evaluated by applying the same stencil on the same set of grid points; only the data on the grid points changes. When functions are defined over the continuous domain so that grid refinement is possible, grid-based integration is supported by a convergence proof based on wavelet analysis. Given the foundation of function values on a uniform grid, grid-based integration methods apply directly to data produced by volumetric imaging (including computed tomography and magnetic resonance), direct numerical simulation of fluid flow, or any other method that produces data corresponding to values of a function sampled on a regular grid. Every step of a grid-based integral computation (including evaluating a function on a grid, application of stencils on a grid, and reduction of the contributions from the grid points to a single sum) is well suited for parallelization. We present results from a parallelized CUDA implementation of grid-based integrals that faithfully reproduces the output of a serial implementation but with significant reductions in computing time. We also present example grid-based integral results to quantify convergence rates associated with grid refinement and dependence of the convergence rate on the specific choice of difference stencil (corresponding to a particular genus of Daubechies wavelet).