Numerical Integration Techniques Research Articles

Numerical implementation of solving a boundary value problem with parameter for Fredholm integro-differential equation

The boundary value problem with parameter for Fredholm integro-differential equation with degenerate kernel is investigated in this paper. The aim of the paper is to establish the solvability conditions, to construct analytical and numerical solutions of the considered problem. The basis for achieving the goal is the ideas of Dzhumabayev parameterization method, classical numerical methods of solving Cauchy problems and numerical integration techniques. A problem with parameters is obtained by introducing an additional parameter and a new unknown function. A system of equations with respect to parameters is compiled according to the initial data of the considered equation and boundary conditions. The unknown function is found as a solution of the Cauchy problem for the ordinary differential equation. The equivalence of the original problem and the problem with parameters, the conditions of unique solvability are established and the formula for finding an analytical solution is obtained. Test examples of finding analytical and approximate solutions of the original problem are given.

High-order finite volume method for solving compressible multicomponent flows with Mie–Grüneisen equation of state

In this paper, we propose a new high-order finite volume method for solving the multicomponent fluids problem with Mie–Grüneisen EOS. Firstly, based on the cell averages of conservative variables, we develop a procedure to reconstruct the cell averages of the primitive variables in a high-order manner. Secondly, the high-order reconstructions employed in computing numerical fluxes are implemented in a characteristic-wise manner to reduce numerical oscillations as much as possible and obtain high-resolution results. Thirdly, advection equation within the governing system is rewritten in a conservative form with a source term to enhance the scheme’s performance. We utilize integration by parts and high-order numerical integration techniques to handle the source terms. Finally, all variables are evolved by using Runge–Kutta time discretization. All steps are carefully designed to maintain the equilibrium of pressure and velocity for the interface-only problem, which is crucial in designing a high-resolution scheme and adapting to more complex multicomponent problems. We have performed extensive numerical tests for both one- and two-dimensional problems to verify our scheme’s high resolution and accuracy.

Multibody Dynamics and Terramechanics-based modeling and simulation of Quarter-Car with Suspension

Abstract While tires only compress when acted upon by a load in on-road scenarios, in off-road conditions, on the other hand, the tires not only undergo compression but also sinkage into the ground, in a manner that depends on the terrain. The tire-ground contact, in turn, affects the forces acting on the suspension system. Hence, it is necessary to study the vertical dynamic response of a system in off-road conditions by accounting for the terramechanics phenomenon (tire-terrain interface). The forces acting at the tire and terrain interface in off-road conditions have been studied in terramechanics separately; however, their integration with multibody systems is relatively less explored. In this paper, a multibody dynamic-based modeling and simulation of a quarter car with a linear spring-damper suspension system for off-road vehicles is presented. The generalized external forces are obtained using the Bekker-Wong based soil pressure-sinkage approach for two different kind of multibody systems interacting with sandy loam terrain. This model is integrated with the Tangent Space Ordinary Differential Equations of the quarter car and suspension system and solved using numerical integration techniques. Thus, by accounting for the terramechanics in the multibody system, the technique provides a more realistic response of the system in off-road scenarios.

Harmonic Response of a Highly Flexible Thin Long Cantilever Beam: A Semi-Analytical Approach in Time-Domain With ANCF Modeling and Experimental Validation

Abstract The main objective of this work is to use the time variational method (TVM), a semi-analytical approach to evaluate steady-state responses in the time-domain for absolute nodal coordinate formulation (ANCF) modeled systems. The gradient-deficient ANCF beam element's performance is demonstrated for a highly flexible cantilever beam under gravity and impulse loading, with comparisons to experiments. The damping behavior is compared for the Rayleigh proportional and the Navier–Stokes (NS) damping model for a gradient-deficient ANCF beam element. Classical finite element method (FEM) beam formulation's shortcomings in predicting large deflections of thin, flexible cantilever beams are highlighted. Unlike the harmonic balance method (HBM), TVM reduces the computational time for harmonic response evaluation compared to numerical integration techniques and handles nonlinear forces in the time-domain. The harmonic response is evaluated by exciting the cantilever beam in the linear region for both experiments and TVM computations.

Research on three-dimensional wake model of horizontal axis wind turbine based on Weibull function

In wind turbine wake models, Gaussian models depend on multidimensional integration to ascertain the distribution of wake velocity deficits. These integrations, which often involve complex boundary conditions, significantly enhance the complexity of mathematical computations. Due to the difficulty of obtaining analytical solutions, numerical integration methods such as Monte Carlo or other numerical integration techniques are commonly employed. This study presents a three-dimensional wake model (3DJW) for horizontal axis wind turbines, utilizing the Weibull function to simplify wake deficit characterization instead of traditional Gaussian distribution methods. The 3DJW model considers wind shear effects and mass conservation laws to enhance predictions of vertical wake velocities. By integrating incoming wind conditions and turbine parameters, the model efficiently computes downstream wake velocities, improving computational efficiency. To enhance predictions in the ultra-far wake region, an improved three-dimensional Weibull wake model is proposed using the exponential fitting method. Validation through wind tunnel experiments and wind farm data demonstrates the model's accuracy in predicting wake deficits at the hub height, with relative errors in horizontal and vertical profiles mostly within 5% and 3%, respectively. The proposed model enables accurate and rapid calculation of wake velocities at any spatial location downstream, facilitating enhanced energy utilization and reduced costs.

Thermodynamics of spin-1/2 fermions on coarse temporal lattices using automated algebra.

Recent advances in automated algebra for dilute Fermi gases in the virial expansion, where coarse temporal lattices were found advantageous, motivate the study of more general computational schemes that could be applied to arbitrary densities, beyond the dilute limit where the virial expansion is physically reasonable. We propose here such an approach by developing what we call the Quantum Thermodynamics Computational Engine (QTCE). In QTCE, the imaginary-time direction is discretized and the interaction is accounted for via a quantum cumulant expansion, where the coefficients are expressed in terms of non-interacting expectation values. The aim of QTCE is to enable the systematic resolution of interaction effects at fixed temporal discretization, as in lattice Monte Carlo calculations, but here in an algebraic rather than numerical fashion. Using this approach, in combination with numerical integration techniques (both known and alternative ones proposed here), we explore the thermodynamics of spin-1/2 fermions across spatial dimensions, focusing on the unitary limit. We find that, remarkably, extremely coarse temporal lattices, when suitably renormalized using known results from the virial expansion, yield stable partial sums for QTCE's cumulant expansion that are qualitatively and quantitatively correct in wide regions (when compared with known experimental results). This article is part of the theme issue 'The liminal position of Nuclear Physics: from hadrons to neutron stars'.

Atmospheric Chemistry Surrogate Modeling With Sparse Identification of Nonlinear Dynamics

AbstractModeling atmospheric chemistry is computationally expensive and limits the widespread use of chemical transport models. This computational cost arises from solving high‐dimensional systems of stiff differential equations. Previous work has demonstrated the promise of machine learning (ML) to accelerate air quality model simulations but has suffered from numerical instability during long‐term simulations. This may be because previous ML‐based efforts have relied on explicit Euler time integration—which is known to be unstable for stiff systems—and have used neural networks which are prone to overfitting. We hypothesize that parsimonious models combined with modern numerical integration techniques can overcome this limitation. Using a small‐scale mechanism to explore the potential of these methods, we have created a machine‐learned surrogate by (a) reducing dimensionality using singular value decomposition to create an interpretably‐compressed low‐dimensional latent space and (b) using Sparse Identification of Nonlinear Dynamics (SINDy) to create a differential‐equation‐based representation of the underlying dynamics in the compressed latent space with reduced stiffness. The root mean square error of ML model prediction for ozone concentration over 9 days is 37.8% of the root mean square concentration across all simulations in our testing data set. The surrogate model is 10× faster with 12× fewer integration timesteps compared to reference model and is numerically stable in all tested simulations. Overall, we find that SINDy can be used to create fast, stable, and accurate surrogates of a simple photochemical mechanism. In future work, we will explore the application of this method to more detailed mechanisms and their use in large‐scale simulations.

Optimizing FPGA implementation of high-precision chaotic systems for improved performance.

Developing chaotic systems-on-a-chip is gaining much attention due to its great potential in securing communication, encrypting data, generating random numbers, and more. The digital implementation of chaotic systems strives to achieve high performance in terms of time, speed, complexity, and precision. In this paper, the focus is on developing high-speed Field Programmable Gate Array (FPGA) cores for chaotic systems, exemplified by the Lorenz system. The developed cores correspond to numerical integration techniques that can extend to the equations of the sixth order and at high precision. The investigation comprises a thorough analysis and evaluation of the developed cores according to the algorithm complexity and the achieved precision, hardware area, throughput, power consumption, and maximum operational frequency. Validations are done through simulations and careful comparisons with outstanding closely related work from the recent literature. The results affirm the successful creation of highly efficient sixth-order Lorenz discretizations, achieving a high throughput of 3.39 Gbps with a precision of 16 bits. Additionally, an outstanding throughput of 21.17 Gbps was achieved for the first-order implementation coupled with a high precision of 64 bits. These outcomes set our work as a benchmark for high-performance characteristics, surpassing similar investigations reported in the literature.

Heat transfer model analysis of fractional Jeffery-type hybrid nanofluid dripping through a poured microchannel

This study investigates the impact of coupled heat and mass transfer on the peristaltic migration of a magnetohydrodynamic (MHD) stress-strain Jeffery-type hybrid nanofluid flowing through an inclined asymmetric micro-channel with a porous medium. The fundamental two-dimensional momentum and energy transport equations are simplified under the assumptions of long wavelength and low Reynolds number. To solve the momentum and heat transfer problems, two advanced fractional derivative approaches are employed: the fractal integral and the Prabhakar fractional derivative. The pressure difference is determined using numerical integration techniques, such as the Stehfest and Tzou's algorithms. The results are presented through graphs and tables, which illustrate the effects of various parameters on the velocity, heat transfer, and trapping phenomena. As a result, we concluded that, pressure gradients grow with higher Reynolds numbers and channel-inclined angles. The heat transfer rate is observed to decrease as the Darcy number and the orientation of the electromagnetic field increase. When comparing the fractional derivative approaches, the fractal operator exhibits a more significant impact on the momentum profiles compared to the Prabhakar fractional operator. This difference is attributed to the distinct characteristics of the integral kernels associated with each fractional derivative definition. Furthermore, when comparing hybrid nanofluids, water-based (H2O + Ag + TiO2) hybrid fluids have a somewhat more significant effect than (C6H9NaO7 + Ag + TiO2) hybrid nanofluids.

A boundary point interpolation method for acoustic problems with particular emphasis on the calculation of Cauchy principal value integrals

A boundary point interpolation method (BPIM) is presented in this paper for numerical solving acoustic problems. The BPIM is a boundary-type meshless method that combines the point interpolation technique for constructing approximate functions with boundary integral equations for reformulating boundary value problems. Since effective numerical integration techniques are crucial in the BPIM and boundary integral equation methods for calculating regular and singular integrals, two Clenshaw-Curtis integration techniques (CCITs) for regular and weakly singular integrals are derived in a unified manner with the same integration points on quadratic integration cells. In addition, a unilateral CCIT, a power series expansion technique, and an improved CCIT are proposed to effectively calculate strongly singular integrals or Cauchy principal value integrals in the BPIM. These integration techniques provide direct and efficient formulas for calculating boundary integrals on quadratic integration cells, and can be directly applied to the calculation of regular and singular integrals in the boundary element method and other meshless boundary integral equation methods. Explicit expressions of integration weights in all CCITs are derived, and the values of integration points and weights are listed. Numerical results are finally provided to validate the effectiveness of the present meshless method and integration techniques.

Semi-Bayesian active learning quadrature for estimating extremely low failure probabilities

The Bayesian failure probability inference (BFPI) framework provides a sound basis for developing new Bayesian active learning reliability analysis methods. However, it is still computationally challenging to make use of the posterior variance of the failure probability. This study presents a novel method called ‘semi-Bayesian active learning quadrature’ (SBALQ) for estimating extremely low failure probabilities, which builds upon the BFPI framework. The key idea lies in only leveraging the posterior mean of the failure probability to design two crucial components for active learning — the stopping criterion and learning function. In this context, a new stopping criterion is introduced through exploring the structure of the posterior mean. Besides, we also develop a numerical integration technique named ‘hyper-shell simulation’ to estimate the analytically intractable integrals inherent in the stopping criterion. Furthermore, a new learning function is derived from the stopping criterion and by maximizing it a single point can be identified in each iteration of the active learning phase. To enable multi-point selection and facilitate parallel computing, the proposed learning function is modified by incorporating an influence function. Through five numerical examples, it is demonstrated that the proposed method can assess extremely small failure probabilities with desired efficiency and accuracy.

A Spectral Collocation Method for Solving Caputo-Liouville Fractional Order Fredholm Integro-differential Equations

In this paper, a numerical method for solving the fractional order Fredholm integro-differential equations via the Caputo-Liouville derivative is presented. The method uses the well-known shifted Chebyshev expansion and a truncated series to represent the unknown function. It also incorporates numerical integration techniques like the Trapezoidal, Simpson’s 1/3, and Simpson’s 8/3 methods. The paper also provides an approximation for the derivative of an integer. The procedure converts the provided problem into a system of algebraic equations using shifted Chebyshev coefficients and collocation points. The coefficients are found by solving this system using well-known techniques like Newton’s method. Numerical results are presented graphycally to illustrate the applicability, efficacy, and accuracy of the approach presented in this work. All calculations in this study were performed using the MATHEMATICA software program.

Some Properties of a Discrete Lorenz System Obtained by Variable Midpoint Method and Its Application to Chaotic Signal Modulation

Various discretization effects caused by applying numerical integration techniques to continuous chaotic systems are broadly studied in nonlinear science. Along with the negative impact on the precision of the various finite-difference schemes, such effects may have surprisingly fruitful practical applications, e.g. pseudo-random number generation, image encryption with improved diffusion and confusion properties, chaotic path planning, and many others. One such application is chaos-based communication systems which gained attention in recent decades due to their high covertness and broadband transmission capability. A crucial problem in the design of chaotic communication systems is the modulation of carrier signals. Due to the noise-like properties of chaotic signals, they can barely be modulated using the same methods as conventional harmonic signals. Thus, developing new modulation techniques is of great interest in the field of chaotic communications. In this study, we investigate the discrete model of the Lorenz oscillator obtained using controllable midpoint numerical integration and develop a novel modulation technique for chaos-based communication systems. We discover and analyze the multistability phenomenon in the dynamics of the investigated finite-difference Lorenz model through bifurcation, the basin of attraction, and Lyapunov spectrum analysis procedures. Using a specially designed testbench, we explicitly show that the proposed modulation method outperforms commonly used parametric modulation and is nearly equal to the state-of-the-art symmetry-based modulation in terms of covertness and noise resistivity. In addition, the proposed modulation technique is much easier to implement using computer arithmetics, especially in fixed-point hardware. The reported results may be efficiently applied to designing advanced chaos-based communications systems or improving the characteristics of existing communication system architectures.

Understanding step selection analysis through numerical integration

Abstract Step selection functions (SSFs) are flexible statistical models used to jointly describe animals' movement and habitat preferences. The popularity of SSFs has grown rapidly, and various extensions have been developed to increase their utility, including the ability to use multiple statistical distributions to describe movement constraints, interactions to allow movements to depend on local environmental features, and random effects and latent states to account for within‐ and among‐individual variability. Although the SSF is a relatively simple statistical model, its presentation has not been consistent in the literature, leading to confusion about model flexibility and interpretation. We believe that part of the confusion has arisen from the conflation of the SSF model with the methods used for statistical inference, and in particular, parameter estimation. Notably, conditional logistic regression (CLR) can be used to fit SSFs in exponential form, and this model fitting approach is often presented interchangeably with the actual model (the SSF itself). However, reliance on CLR reduces model flexibility, and suggests a misleading interpretation of step selection analysis as being equivalent to a case–control study. In this review, we explicitly distinguish between model formulation and inference technique, presenting a coherent framework to fit SSFs based on numerical integration and maximum likelihood estimation. We provide an overview of common numerical integration techniques (including Monte Carlo integration, importance sampling and quadrature), and explain how they relate to popular methods used in step selection analyses. This general framework unifies different model fitting techniques for SSFs, and opens the way for improved inferential methods. In this approach, it is straightforward to model movement with distributions outside the exponential family, and to apply different SSF model formulations to the same data set and compare them with AIC. By separating the model formulation from the inference technique, we hope to clarify many important concepts in step selection analysis.

Symplectic numerical methods in optics and imaging: ray tracing in spherical gradient-index lenses and computer-generated image rendering.

This paper provides an introduction to symplectic numerical integration techniques and examines various optical applications. We first outline the fundamentals of Hamiltonian optics and detail the construction of a symplectic method via the splitting technique. Numerical experiments involving a selection of spherically symmetric gradient-index lenses compare the accuracy of various first-, second-, and fourth-order symplectic methods with equivalent nonsymplectic methods. The best-performing methods are then further tested as part of an image rendering task involving nonlinear ray tracing, comparing the trace time required by each method. Future improvements, recommendations, and uses for symplectic ray tracing are also considered.

A novel Romberg integration method for neutrosophic valued functions

This study proposes a numerical integration technique to determine the approximate integral of the neutrosophic valued function. Newton Cot’s method with a positive coefficient has been used for neutrosophic integration, and then the suggested technique is used to find the approximate value of the integral. This method is called the Romberg integration method. In this method, the composite trapezoidal rule is applied initially, and Richardson’s extrapolation technique is used next to improve the solution. Richardson’s extrapolation is the basis of Romberg’s integration. Formulas with high-order truncation errors are created using Richardson extrapolation, which employs an averaging method. Richardson’s extrapolation helps the proposed method to achieve high computational efficiency. The convergence of the numerical approximation is provided in terms of theorems. A comparative analysis of the proposed method with other existing methods is demonstrated to show the efficiency and reliability of the proposed method.

Analysis of unsteady non-Newtonian Jeffrey blood flow and transport of magnetic nanoparticles through an inclined porous artery with stenosis using the time fractional derivative

In the present study, a Caputo–Fabrizio (C–F) time-fractional derivative is introduced to the governing equations to present the flow of blood and the transport of magnetic nanoparticles (MNPs) through an inclined porous artery with mild stenosis. The rheology of blood is defined by the non-Newtonian visco-elastic Jeffrey fluid. The transport of MNPs is used as a drug delivery application for cardiovascular disorder therapy. The momentum and transport equations are solved analytically by using the Laplace transform and the finite Hankel transform along with their inverses, and the solutions are presented in the form of Laplace convolutions. To display the solutions graphically, the Laplace convolutions are solved using the numerical integration technique. The study presents the impacts of different governing parameters on blood and MNP velocities, volumetric flow rate, flow resistance, and skin friction. The study demonstrates that blood and MNP velocities boost with an increase in the fractional order parameter, Darcy number, and Jeffrey fluid parameter. The volumetric flow rate decreases and flow resistance increases with enhancement in stenosis height. The non-symmetric shape of stenosis and the rheology of blood decrease skin friction, whereas enhancement in MNP concentration increases skin friction. A comparison of the present result with the previous work shows excellent agreement. The present study will be beneficial for the field of medical science to further study atherosclerosis therapy and other similar disorders.

An efficient Artificial Neural Network algorithm for solving boundary integral equations in elasticity

This study presents a novel numerical integration technique based on Artificial Neural Network (ANN) algorithms to overcome intrinsic limitations characterizing the Boundary Element Method (BEM). The proposed approach, taking advantage of some peculiar properties of the BEM equations, provides an effective alternative to traditional numerical techniques for evaluating the integrated kernels required to compute the displacements and stresses of a two-dimensional solid. Assuming isotropy and homogeneity, and modeling both the geometry and the mechanical parameters using quadratic shape functions, all the integrals in the classical BEM formulation can be expressed as the sum of two terms that are independent of the constitutive properties and solely dependent on four geometric parameters: three components of two distance vectors and a parameter representing the element’s curvature. This interesting property of boundary integral equations in elasticity makes them particularly amenable to numerical evaluation using artificial neural networks. Results from numerical tests, which were conducted using increasingly complex integrals, demonstrate the high precision of the proposed approach as long as the integration and collocation points are sufficiently separated to avoid issues with singularity.

Effect of an efficient numerical integration technique on the element-free Galerkin method

The reproducing kernel gradient smoothing integration (RKGSI) is an efficient technique to tackle the integration problem and optimal convergence in meshless methods. In this paper, the effect of the RKGSI on the element-free Galerkin (EFG) method is studied for elliptic boundary value problems with mixed boundary conditions. Theoretical results of smoothed gradients in the RKGSI are provided. Fundamental criteria on how to determine integration points and weights of quadrature rules are established according to necessary algebraic precision. By using the Nitsche's technique to impose Dirichlet boundary condition, the existence, uniqueness and error estimations of the solution of the EFG method with numerical integration are analyzed. Numerical results validate the theoretical analysis and the optimal convergence of the method.

Deep Ritz method with adaptive quadrature for linear elasticity

In this paper, we study the deep Ritz method for solving the linear elasticity equation from a numerical analysis perspective. A modified Ritz formulation using the H1/2(ΓD) norm is introduced and analyzed for linear elasticity equation in order to deal with the (essential) Dirichlet boundary condition. We show that the resulting deep Ritz method provides the best approximation among the set of deep neural network (DNN) functions with respect to the “energy” norm. Furthermore, we demonstrate that the total error of the deep Ritz simulation is bounded by the sum of the network approximation error and the numerical integration error, disregarding the algebraic error. To effectively control the numerical integration error, we propose an adaptive quadrature-based numerical integration technique with a residual-based local error indicator. This approach enables efficient approximation of the modified energy functional. Through numerical experiments involving smooth and singular problems, as well as problems with stress concentration, we validate the effectiveness and efficiency of the proposed deep Ritz method with adaptive quadrature.