Numerical Integration Solutions Research Articles

Analytical solution for the skip trajectory of hypersonic vehicles with time-varying aerodynamic coefficients

Abstract An analytical method is proposed for calculating the altitude and flight path angle of hypersonic vehicles’ skip trajectories under the influence of arbitrary time-varying aerodynamic coefficients. First, a longitudinal plane dynamics equation is established based on the inertial Earth model and the specific kinetic energy is introduced as independent variable to simplify the derivation. Subsequently, the solution is obtained through the perturbation method, where the zero-order analytical solution is based on existing literature, and the first-order linear time-varying system is decoupled through matrix linear transformation. Finally, by comparing numerical integration and analytical solutions with constant aerodynamic coefficients, the precision of this method in dealing with time-varying aerodynamic coefficients is verified, demonstrating its advantages over analytical solutions with constant aerodynamic coefficients.

AN EFFICIENT APPROACH FOR SOLVING THE FRACTAL, DAMPED CUBIC–QUINTIC DUFFING’S EQUATION

The main goal of this work is to focus on using He’s two-scale fractal dimension transform, the Caputo–Fabrizio fractional-order derivative, and the harmonic balance and the homotopy methods are applied for deriving the approximate solution of the fractal, damped cubic–quintic Duffing’s equation when the fractional derivative order of the inertia term is not twice of that of the damping term. Numerical results obtained from the derived expressions and the numerical integration solution show good agreement, especially at small values of the nonlinear parameters. Furthermore, when the fractal order of the damping term decreases, the damping oscillation frequency values increase with a decrease in the system wavelength values, which indicates a slower decay in the system oscillation amplitudes. Our solution procedures elucidate the applicability of He’s two-scale fractal dimension transform for solving nonlinear dynamic systems with inertia and damping fractal terms.

Efficient quadrature rules for numerical evaluation of singular and hyper singular integrals

In this paper Newton-Cote type quadrature rule, Haar wavelets, and hybrid functions-based quadratures are used for the numerical solution of singular and hyper singular integrals having singularity at the origin. These integrals include the Cauchy principle value (CPV) and Hadamard finite part (HFP) integrals. The proposed rules are tested numerically on some test problems to check the efficiency and accuracy of the new methods.

Comparative Analysis of Numerical Integration Solutions Pias Method and Newton Cotes Method Using Python Programming Language

Irregular areas cannot be solved by ordinary calculus formulas, so it is necessary to use numerical methods such as the Quadrature and Newton-Cotes methods. This research compares numerical integration solutions using the Quadrature method (Rectangular and Trapezoidal) and the Newton-Cotes method (Trapezoidal, Simpson 1/3, Simpson 3/8, and Weddle) with the Python programming language. Manual calculation of the first case study on integrals where the smallest error from the numerical method to the analytical method is achieved by the rectangular method of 0,017. In the second case study of tabular data for manual calculations the author only uses the Simpson 3/8 method with an absolute error for an analytical area of 1.418,583 km2. Whereas in the Python application for the first case study the smallest error was achieved by the Simpson 1/3 and Simpson 3/8 methods with an error of 0, in the sense that these two methods are very accurate to the actual analysis results. In the second case study the smallest error was achieved by the Simpson 1/3 method of 1.039,365 km2. The difference between manual calculations and application results is due to decimal rounding, and the linspace(a,b,n+1) function in the numpy library.

An eXtended finite element method implementation in COMSOL multiphysics: Thermo-hydro-mechanical modeling of fluid flow in discontinuous porous media

This paper presents the implementation of the eXtended Finite Element Method (XFEM) in the general-purpose commercial software package COMSOL Multiphysics for multi-field thermo-hydro-mechanical problems in discontinuous porous media. To this end, an exclusive enrichment strategy is proposed in compliance with the COMSOL modeling structure. COMSOL modules and physics interfaces are adopted to take account of the relevant physical processes involved in thermo-hydro-mechanical coupling analysis, namely: the mechanical deformation, fluid flow in porous media and heat transfer. Essential changes are made to the internal variables of the physics interfaces to ensure consistency in the evaluation of enriched solution fields. The model preprocessing, level-set updates, coupling of the relevant physics interfaces and postprocessing procedures are performed adopting a coherent utilization of the COMSOL’s built-in features along with the COMSOL’s LiveLink for MATLAB functions. The implementation process, remedies for the treatment of the enriched zones, XFEM framework setup, multiphysics coupling, numerical integration and numerical solution strategy are described in detail. The capabilities and performance of the proposed approach are investigated by examining several multi-field thermo-hydro-mechanical simulations involving single/multiple discontinuities in 2D/3D porous rock settings.

A novel meshfree method based on spatio-temporal homogenization functions for one-dimensional fourth-order fractional diffusion-wave equations

In this work, a spatio-temporal homogenization function method is proposed for resolving one-dimensional fourth-order fractional diffusion-wave equations (FDWEs). A homogenization function of fourth-order FDWEs satisfying the initial and boundary conditions is first constructed, and then a family of two-parameter homogenization functions is further generated via introducing shape functions with free indices. By superimposing the homogenization functions, a numerical model is established and utilized for dealing with fourth-order FDWEs. The developed approach is free of mesh generation, numerical integration, and fundamental solutions, and it is easy to program and implement on the existing software. Numerical experiments are performed to demonstrate the effectiveness and convergence of the developed approach.

Nonlinear coupled dynamics of suspended cables due to crossover points shifting and symmetry breaking

This work deals with the nonlinear coupled dynamics of suspended cables considering two unavoidable factors: temperature changes and damaged situations. Particularly, the thermal-induced crossover shifting and damage-induced symmetry-breaking are considered and investigated. A kinematically condensed mathematical model with thermal and damage effects simultaneously is employed for the transverse displacements. A weighted-residual technique based on the Galerkin method is adopted to reduce the continuous model from the partial differential equations to ordinary ones. Following the multiple scales procedure for a weakly nonlinear system, both the polar and Cartesian forms of modulation equations are achieved. Based on the eigenvalue analyses, natural frequencies and mode shapes with thermal and damage effects are presented, and the crossover and veering points between natural frequencies are observed. Resonant behaviors of suspended cables after crossover points shifting and symmetry-breaking are revealed by using the numerical continuation and bifurcation analysis. Quantitative and qualitative differences in dynamic behaviors induced by temperature and damage factors are exhibited and illustrated through some parametric studies. Direct numerical integration solutions are adopted to confirm the perturbation ones, and a good agreement is observed.

A WEIGHTED POWER-FORM FORMULATION FOR THE FRACTAL WARNER–GENT VISCOHYPERLASTIC MODEL

This paper elucidates how the two-scale fractal dimension transform, and a transformation method can be applied to replace the Warner–Gent equation that models the fractal dynamic response of porous viscohyperelastic materials with an equivalent power-form equation. Furthermore, this research work elucidates the advantages of modeling viscohyperlastic materials using the fractal Warner–Gent’s model since the values of the fractal dimension parameter unveil how the global molecular structure of viscohyperelastic materials varies as a function of the vibration frequency wavelength. Compared to the original one, the accuracy attained from the Warner–Gent power-form equivalent equation is examined by plotting the frequency–amplitude and time–amplitude curves obtained from the corresponding numerical integration solutions. It is found that both numerical integration solutions agree well since the root-mean-square-error (RMSE) values remain small.

On the Choice of Methods for Numerically Integrating the Equations of Transients in Electric Power Systems

Algorithms for solving the system of differential-algebraic equations describing electromechanical transients in electric power systems are considered along with matters concerned with ensuring the reliability and required accuracy of the solution results. Classical explicit methods, methods for implicit numerical integration and simultaneous solution of differential-algebraic systems of equations are used to model dynamic processes in power systems. On the basis of the methods used, difference models of electric power system components are obtained. The equations of transients in electric power systems are written in a homogeneous coordinate basis with the use of nodal voltage equations. A comparative analysis of algorithms based on the explicit Runge-Kutta method and implicit methods of Euler, trapezoids, Euler with fitting coefficients is carried out. The analysis results have shown that in terms of stability and accuracy, the trapezoid method has the greatest advantages, which is not inferior to the 4th order Runge-Kutta method and allows calculations with a large integration step to be carried out. The effectiveness of the combined method is shown, in which the Euler method with fitting is used to solve the differential equations describing electromagnetic processes, and the trapezoid method is used to solve the equations of electromechanical motion.

On the Asymptotic Evaluation of the Physical Optics Approximation for Plane Wave Scattering by Circular Conducting Cylinders

In this paper, the scattering far-field from a circular electric conducting cylinder has been analyzed by physical optics (PO) approximation for both H and E polarizations. The evaluation of radiation integrations due to the PO current is conducted numerically and analytically. While non-uniform and uniform asymptotic solutions have been derived by the saddle point method, a separate approximation has been made for forward scattering direction. Comparisons among our approximation, direct numerical integration and exact solution results yield a good agreement for electrically large cylinders.

An Applied N C Differentiation Interpolation technique for improved random Anomalous values in Data Mining

In data mining, the word “interpolation” refers to interpolating some anonymous information from a given set of known information. The method of interpolation is extensively used as a valuable tool in science and engineering. The predicament is a classical one and dates back to the time of Newton, who needed to solve such a problem in analyzing data on the numerical computations. Numerical applications of interpolation include derivation of computational techniques for numerical differentiation, numerical integration and numerical solutions of differential equations. In this paper a closest fit Application of the formula to numerical data for recovering haphazard Anomalous values in Data Mining has been shown in the case of representing the data on the dataset global carbon dioxide emissions from fossil fuel burning by Fuel Type corresponding as a method of time. The formula is suitable in the situation where the values of the argument are at equal interval.

Recent strategy to study fractal-order viscoelastic polymer materials using an ancient Chinese algorithm and He’s formulation

This paper introduces a novel methodology to determine the frequency-amplitude relationship of fractal-order viscoelastic polymer materials using the two-scale fractal dimension transform, the equivalent power-form representation of the conservative restoring forces, and a simple coordinate transformation to eliminate viscoelastic effects. Then, the ancient Chinese algorithm Ying Bu Zu Shu and He’s formulation are used for obtaining the frequency-amplitude relationship. Simulation results obtained from the derived expressions exhibit good agreement when compared to numerical integration solutions. This article elucidates how the molecular structure of polymer chains influences the relaxation oscillations as a function of the fractal parameter values.

Analysis and Design of the Energy Storage Requirement of Hybrid Modular Multilevel Converters Using Numerical Integration and Iterative Solution

Increasing the modulation index by utilizing the negative voltage states of full-bridge submodules (FBSMs) can greatly reduce capacitor usage of modular multilevel converters (MMCs), thereby optimizing the cost and volume. The hybrid MMC is composed of half-bridge submodules (HBSMs) and FBSMs, and the capacitor voltages of the two types of submodules (SMs) have different shapes as long as negative voltage states exist. This condition greatly complicates the analysis and design of the energy storage requirement of the hybrid MMC, which utilizes the negative voltage states of FBSMs to boost the AC voltage. A numerical calculation method for solving the capacitor voltages and designing the capacitances of FBSMs and HBSMs is proposed in order to accurately determine the minimum energy storage requirement considering the difference between the energy variations in FBSMs and HBSMs. In the numerical calculation, the energy storage and voltage of the arm are decomposed into FBSM and HBSM parts. According to the physical switching process, the output voltages of FBSM and HBSM parts are determined separately. The one-cycle waveforms of the capacitor voltages are then obtained by numerical integration of the power flows in FBSM and HBSM parts. An iterative solution procedure and the termination criterion that can ensure the accuracy of the obtained one-cycle waveforms are also proposed. Using the numerical integration and iterative solution procedure as the kernel algorithm, the proposed method can accurately analyze the capacitor voltages of the FBSMs and HBSMs and determine the minimum energy storage requirement of the hybrid MMC. Furthermore, the proposed method is applicable for various operating working conditions and various proportions of FBSMs. The simulation results verify the feasibility and accuracy of the analysis and design method.

Interpretation Method of GATEM Data based on PID Controller Iteration Downward Continuation Method

The Ground-source Airborne Time-domain Electromagnetic (GATEM) system has advantages for high efficiency and complex areas such as mountainous zone. Because of ignoring the impact of flight height, the section interpretation method seriously affects the interpretation and imaging accuracy of shallow anomalies. The PID controller iteration downward continuation method is proposed. Based on the original iteration continuation method, the differential coefficient and integral coefficient are added. The result shows that the new method remarkably decreases the iteration number, and the accuracy are verified by comparison with the numerical integration solution. The PID controller iteration downward continuation method is applied to the interpretation of GATEM data. For synthetic data, the interpretation results of continued electromagnetic response are closer to the true model than the z = 30 m interpretation results. The method is also applied to GATEM field data in Yangquan City, Shanxi Province, China. The interpretation results perform reliability using PID controller iteration downward continuation method in a GATEM field.

Determination of the frequency-amplitude response curves of undamped forced Duffing’s oscillators using an ancient Chinese algorithm

The ancient Chinese algorithm, also known as the method of surplus and deficiency that has been used by He to find the angular frequency of nonlinear oscillators, is adapted to derive the approximate frequency-amplitude equations of undamped forced cubic, cubic-quintic, and cubic-quintic–heptic oscillators of the Duffing-Type. The accuracy of the obtained frequency-amplitude equations is examined by comparing theoretical predictions with each Duffing-type oscillator’s numerical integration solutions. These examples show that the ancient Chinese algorithm is a powerful and easy-to-follow method to find the frequency of forced nonlinear oscillators.

EQUIVALENT POWER-FORM TRANSFORMATION FOR FRACTAL BRATU’S EQUATION

In this paper, an equivalent power-form transformation method with a weighted function is applied for solving the one-dimensional fractal Bratu’s boundary value equation. Numerical integration solutions obtained from the equivalent Bratu’s equation as well as those from its approximate Taylor’s series solution reveal that the proposed methodology yields highly accurate solutions. Therefore, it is believed that by applying the power-form transformation, various fractal differential equations in which two scales are needed because of the physical laws involved in modeling the observed phenomena, can be solved by treating the nonlinear terms as equivalent power-form terms.

Numerical Solution of Nonlinear Fredholm and Volterra Integrals by Newton–Kantorovich and Haar Wavelets Methods

The current study proposes a numerical method which solves nonlinear Fredholm and Volterra integral of the second kind using a combination of a Newton–Kantorovich and Haar wavelet. Error analysis for the Holder classes was established to ensure convergence of the Haar wavelets. Numerical examples will illustrate the accuracy and simplicity of Newton–Kantorovich and Haar wavelets. Numerical results of the current method were then compared with previous well-established methods.

Automatic Differentiation for Diffusion Operator Integral Variance Reduction

We demonstrate applications of automatic differentiation in the diffusion operator integral variance reduction framework originally proposed by Heath and Platen. Combining this estimator with automatic differentiation techniques for computing derivatives allows for a flexible implementation without trade-offs in numerical stability or accuracy. This fully mitigates a key practical shortcoming of the original estimator, as we remove the dependency on error-prone and problem-specific manual calculations. We perform a relative error analysis of the estimator and standard Monte Carlo estimation against the numerical integration solution of the European call option in the Heston model. Benchmarks show computational time savings in excess of three orders of magnitude for comparable expected relative errors and sublinear growth of mean relative errors in strikes. The implementation is further applied to the valuation of floating-strike lookback put options, demonstrating the ease of generalizing the automatic differentiation approach and the ability of the estimator to mitigate the monitoring bias for this class of options.

A homogenization function method for inverse heat source problems in 3D functionally graded materials

A simple and effective method is proposed for solving inverse heat source problems in functionally graded materials based on the homogenization function. Making use of given conditions, a homogenization function for the boundary value problem is conceived and a family of homogenization functions is further derived. Then, the superposition of homogenization functions method is developed and used for determining the heat source of the inverse problems. In this new methodology, the inverse heat source problems are directly solved by calculating a linear matrix system. Importantly, this scheme does not involve mesh generation, numerical integration, iteration, regularization and fundamental solutions, and it is easy to program and implement on the existing software. Four numerical examples defined on the cuboid domains are presented to demonstrate the accuracy and efficiency of the presented tool.

Broadening the frequency bandwidth of a finite extensibility nonlinear vibration absorber by exploiting its internal resonances

In this paper, the dynamics response of a finite extensibility nonlinear elastic (FENE) absorber attached to a nonlinear primary system is investigated by writing the modal form equations of motion as two decoupled, damped, and forced cubic–quintic expressions of the Duffing type. Then, numerical integration solutions of these decoupled expressions are used to find frequency–amplitude response plots, the largest Lyapunov exponents, suppression bandwidth, and force transmissibility that are used for investigating how the addition of the FENE absorber modifies the vibration conditions of the main system. The results suggest that the addition of the FENE absorber can lead to vibration mitigation of the primary system while enhancing its performance when the system operates in the vicinity of internal resonance conditions.