Trapezoidal Integration Rule Research Articles

Two linearized difference schemes on graded meshes for the time-space fractional nonlinear diffusion-wave equation with an initial singularity

Abstract In this article, two linearized difference schemes are considered to solve a class of time-space fractional nonlinear diffusion-wave equations with weak singularity near the initial time based on its equivalent integro-differential equation. The Crank-Nicolson method combined with the trapezoidal product integration rule with graded meshes is implemented to approximate the Riemann-Liouville integral. The fractional central difference formula and a novel compact difference method are used to construct fully discrete difference schemes. Using the energy method, the stability and convergence of the proposed schemes are demonstrated in terms of the L 2 norm with the accuracy of O N − min { r σ , 2 } + M − 2 and O N − min { r σ , 2 } + M − 4 , respectively. The theoretical analysis is complemented by numerical results.

Evolutional behavior of viscoelastic materials: A formulation based on logarithmic approach

In this paper, a new formulation for large deformation of visco-elastometic materials is derived. The formulation is based on logarithmic strain, and the internal variable is defined as the viscous part of the logarithmic strain. A derivation of thermodynamically consistent formulation for visco-elastomeric materials is provided. The evolution equation which is related to the deviatoric part of the strain for viscoelastic formulation is solved by the implicit trapezoidal integration rule which leads to a recursive relation.

GENERAL SOLUTION OF TWO-DIMENSIONAL PROJECTILE MOTION WITH AIR RESISTANCE

In this study, two-dimensional projectile motion is considered under the effect of a general power law model of air resistance. Classically, a projectile is treated as a point mass with mass m moving in a uniform gravitational field. The projectile is launched from the ground with an angle α to horizon. the drag force is assumed to be proportional to the speed raised to the power n. The analysis of the problem is performed using Cartesian coordinates. A general exact parametrical solution (with respect to the angle of motion) is derived for any power n, following simple steps: 1) find the speed in the direction of the axis x (horizontal – no gravity) 2) find the vertical component of the speed 3) find the time 4) find the horizontal position of the projectile and finally,5) find the vertical position of the projectile. Steps 1) and 2) give explicit closed form equations and the rest are given by exact integrals which can be solved numerically. In this study spreadsheet calculation are performed using trapezoidal rule of integration. The cases of motion in avacuum and linear drag law are used to check the accuracy of the numerical calculations.The importance of the proposed study is three-fold: a) The method of the derived solution is new, and couldnt be found elsewhere b) The derived equations make it possible to use spreadsheets for presenting the subject (no programming is required), and thus, serve as a tool to enhance teaching c) The derived equations are general for any power n thus, the same procedure could be used to find the position of the projectile at any time.

Inaccuracies due to the frequency warping in simulation of electrical systems using combined state–space nodal analysis

The simulation of electromagnetic transients may suffer from inaccuracies due to a phenomenon known as frequency warping (FW). This paper presents an analysis of the effects of FW on the accuracy of digital simulations, demonstrating that the use of the trapezoidal integration rule (TR), commonly employed in many electromagnetic transients simulators, is the root cause of such inaccuracies. Although FW is considered a major problem in digital signal processing, it is often overlooked when simulating electrical transients. The analysis is carried out in a fourth-order RLC circuit, from which the analytic solution is derived. The circuit is solved using the combined state–space nodal method, considering the TR or recursive convolutions as solution methods for the state–space representation. It was observed that the FW caused a change in the natural oscillation frequency of the system, causing a pulsating behavior of absolute error. The accumulation of errors over time can result in deteriorated solutions when either the time steps are not sufficiently small or the simulation runs for a long enough duration. This paper emphasizes the significance of accounting for the FW phenomenon in digital simulations that rely on integration methods, such as the TR.

Optimal error analysis of space–time second-order difference scheme for semi-linear non-local Sobolev-type equations with weakly singular kernel

In this paper, we construct and analyze a Crank–Nicolson difference scheme for solving the semilinear Sobolev-type equation with the Riemann–Liouville fractional integral of order α∈(0,1). The proposed scheme consists of two main stages. First, to compensate for the singular behavior of the exact solution at the initial time t=0, a Crank–Nicolson scheme and the product trapezoidal integration rule are proposed on graded meshes for time discretization. Second, a classical finite difference operator on uniform meshes is used for space discretization. Under suitable assumptions of the regularity condition, the stability and convergence analysis of this scheme in a new norm are given by the energy argument. It is shown that how the mesh grading exponent γ affects the temporal convergence rate of the presented scheme in the final convergence result. This scheme can attain a second-order accuracy in both time and space by choosing an optimal grading parameter γ. Numerical results confirm that the error analysis is sharp, and comparisons with results on uniform temporal meshes are included to indicate the effectiveness of our method.

Efficient Implementation of Multi-Port Frequency Dependent Network Equivalents for Electromagnetic Transient Studies using Norton Equivalent Circuits

Power system electromagnetic transient studies require that a small part of the electrical network be modelled in detail. The rest of the system is represented by a network equivalent taking into account the frequency dependence of the system components (FDNE). With the purpose of getting a FDNE, this paper presents a procedure for including high-order rational models in EMTP-type simulation programs based on Norton equivalent circuits. This approach has the advantage of using few conductance branches compared to those obtained by fitting any particular circuit structure.In relation to the work carried out, this paper presents the following issues: a) a procedure for calculating the admittance matrix of the rest of the system as a function of frequency b) this matrix is synthesized in the form of multi-terminal π-equivalent circuits, whose branches are fitted by rational functions in the pole-residue form c) each partial fraction is converted into a differential equation in time domain. The trapezoidal rule of integration is applied to that differential equation, resulting in a conductance in parallel with a history term current source. Considering all the partial fractions corresponding to a branch, a Norton equivalent circuit is obtained for that branch.This paper also shows how the procedure is implemented in three different ways in an EMTP-type program, along with the validation of such approach through simulations in an electric utility 500 kV transmission system.

Numerical solution to an integral equation for the kth moment function of a geometric process

In this paper, an integral equation for the kth moment function of a geometric process is derived as a generalization of the lower-order moments of the process. We propose a general solution to solve this integral equation by using the numerical method, namely trapezoidal integration rule. The general solution is reduced to the numerical solution of the integral equations which will be given for the third and fourth moment functions to compute the skewness and kurtosis of a geometric process. To illustrate the numerical method, we assume gamma, Weibull and lognormal distributions for the first interarrival time of the geometric process.

An improved Trapezoidal rule for numerical integration

Numerical Methods have attracted of research community for solving engineering problems. This interest is due to its practicality and the improvement of highspeed calculations done on current century processors. The increase in numerical method tools in engineering software, such as Matlab, is an example of the increased interest. In this paper, we are present a new improved numerical integration method, that is based on the well-known trapezoidal rule. The proposed method gives a great enhancement to the trapezoidal rule and overcomes the issue of the error value when dealing with some higher order functions even when solving for a single interval. After literature review, the proposed system is mathematically explained along with error analysis. Few examples are illustrated to prove improved accuracy of the proposed method over traditional trapezoidal method.

Modeling of an isothermal flow of a magnetohydrodynamic, viscoplastic fluid during forward roll coating process

An isothermal and incompressible flow of a magnetohydrodynamici (MHD) viscoplastic fluid has been examined while passing through the area among two co-rotating rolls. Applying lubrication approximation theory (LAT), the basic flow equations of mass and momentum are derived and subsequently non-dimensionalized. Exact solutions of velocity and pressure gradient distributions are evaluated. To evaluate the complex integrals, Trapezoidal rule for numerical integration is used. The significant engineering parameters like, the roll separation force, maximum pressure and the power transferred to the fluid by the rolls, are also numerically computed. The results demonstrate that the velocity field and pressure gradient are significantly affected due to the presence of the viscoplastic parameter. Magnetic field is found to be the controlling parameter for power input, separating force, and distance among attachment and detachment points, being extremely beneficial for the roll coating process. It has been learnt through this study that MHD can be an effective tool in the coating industry to regulate the engineering quantities in which coating thickness is the most important.

A practical method for stability assessment of a damaged ship

Reliable onboard stability assessment for a damaged ship is of great importance to ship operators. This paper presents a practical method to calculate ship's damage stability directly from the full three-dimensional (3D) geometric models, in which the method of lost buoyancy is used. First, the computational database is established by cutting the 3D models of hull and compartments in transverse direction according to ship's frame distance table; Then, trapezoidal rule integration method is adopted to calculate ship's volumetric data; After that, the centre of gravity of liquid loads is calculated using Newton's iterative method; Finally, ship's equilibrium state of balanced trim and draft for a given heel angle is determined using a successive linear method, and the right level could be obtained. Lloyd's Register (LR) test cases for IACS UR L5 Type 1, 2 and 3 are used to verify the proposed method. In every stage of the test program, the method achieves a high-accuracy result, and thus can be used onboard to calculate intact stability and damage stability of ships by direct application of preprogrammed damage cases, especially for bulk carriers and tankers.

Analysis of Frequency-Dependent Network Equivalents in Dynamic Harmonic Domain

Rational function-based models have proved to be very efficient for accurate frequency-dependent modeling of power system components. These models are able to characterize the components terminal behaviours (analysing the admittance matrix) for nodal analysis. This provides a fast convergence and inherent stability to the solution routine of the model. This work presents a general framework for interfacing the dynamic phasor method to the rational models. That would be promising for the electromagnetic transient analysis (under harmonic distortion), in the frequency domain. Therefore, Y-element rational pole-residue models (employing the vector fitting method) are developed. Moreover, the pole-residue model is converted into the state-space representation. Next, the dynamic harmonic approach (in the frequency domain) is employed for harmonic analysis. It is shown that the order of state-space system can become a major concern for frequency-dependent networks analysis. Therefore, to generate a reduced-order model, the balanced realization theory is applied. Moreover, (for the sake of simplicity and efficiency) the trapezoidal integration rule is employed to discretise the state-space equations. For validation of the modelling, it is applied on three test case studies and results of these studies are compared with their time-domain analysis results.

Quasi-Isotropic Initial Triangulation of NURBS Surfaces

Isotropic triangulation of NURBS surfaces provides high quality triangular meshes, where all triangles are equilateral. This isotropy increases representation quality and analysis accuracy. We introduce a new algorithm to generate quasi-isotropic triangulation on NURBS surfaces at once, with no prior meshing. The procedure consists of one front made of vertexes that advances in a divergence manner avoiding front collision. Vertexes are calculated by arcs intersection whose radius length is estimated by trapezoidal rule integration of directional derivatives. The parameter space is discretized in partitions such that the error of trapezoidal rule is controlled. A new space, called pattern space, is used to infer the direction of the arcs intersection. Derivatives, whose analytical computation is expensive, are estimated by NURBS surface fitting procedures, which raise the speed of the process. The resultant algorithm is robust and efficient. The mesh achieved possesses most of the triangles equilateral and with high uniformity of sizes. The performance is illustrated by measuring angles, vertex valences and size uniformity in numerical examples

Formula omitted] Multi-contingency transient stability constrained AC optimal power flow with volt/var controllers

In order to guarantee the continuous operation of electric power systems (EPS), a proper planning of these networks in steady and transient state must be performed. This paper presents an N−1 multi-contingency AC optimal power flow (OPF) that embeds, in the same mathematical optimization model, a set of transient stability constraints (TSC) that guarantee rotor angle and angular velocity variation within technical limits, at given N−1 contingencies. Furthermore, the proposed model considers the operation of volt/var controllers, such as shunt elements and OLTC transformers, to further improve the operation of the EPS, under given levels of demand and generation. Taking advantage of the classical first-order transient stability model of synchronous machines and the implicit trapezoidal integration rule, the proposed model can be formulated as a stand-alone mixed-integer nonlinear programming (MINLP) model. Then, through well-established linearization techniques, the initially proposed MINLP model is transformed into a new mixed-integer linear programming (MILP) model, which can be implemented via algebraic programming languages, such as AMPL, and solved using convex optimization solvers, such as CPLEX. Three systems with dissimilar number of synchronous machines and nodes have been used for tests (i.e., the 9-Bus/3-Generator Western System Coordinating Council (WSCC) system, the 39-Bus/10-Generator New England system and the 68-Bus/16-Generator IEEE system). The efficiency of the proposed linearization techniques and the stability requirements of the solutions have been validated using an exact AC power flow and the transient stability analysis program PSAT-Matlab. Results show the ability of the proposed MILP model to provide stable operating points under N−1 contingencies, at minimum production cost.

Frequency-Dependent Resistances and Inductances in Time-Domain Transient Simulations of Power Transformers

This paper presents the methodology and validation of high-frequency (HF) transformer simulations, which take into account the frequency dependence of the winding resistances and inductances. The parameters’ frequency dependence is approximated by curve fitting of the values computed by the finite element method (FEM) simulations in the frequency domain. After inverse Fourier transformation into the time domain, the convolution integrals are solved efficiently by the trapezoidal integration rule. The simulation results are validated using measurement data.

Combined interpolation, scale change, and noise reduction in spectral analysis

The authors present a simple, convenient, and accurate noise-reduction approach for interpolating spectra, in particular, for converting those available as discrete points equally spaced in wavelength, acquired, for example, by a photodiode-array detector, to equivalent spectra equally spaced in energy, as needed for analysis. Based on continuum mathematics, the algorithm uses Gaussian kernels and capitalizes on the fact that trapezoidal-rule integration is accurate to fourth order in the ratio of point separation to width when applied to Gaussian functions. The approach can be expanded to perform differentiation and other operations. Examples include false-data verification, wavelength-to-energy conversion of near-bandgap interference oscillations of a GaN film, and calculation of the second energy derivative of monolayer MoS2 in the exciton region.

Index-3 divide-and-conquer algorithm for efficient multibody system dynamics simulations: theory and parallel implementation

There has been a growing attention to efficient simulations of multibody systems, which is apparently seen in many areas of computer-aided engineering and design both in academia and in industry. The need for efficient or real-time simulations requires high-fidelity techniques and formulations that should significantly minimize computational time. Parallel computing is one of the approaches to achieve this objective. This paper presents a novel index-3 divide-and-conquer algorithm for efficient multibody dynamics simulations that elegantly handles multibody systems in generalized topologies through the application of the augmented Lagrangian method. The proposed algorithm exploits a redundant set of absolute coordinates. The trapezoidal integration rule is embedded into the formulation and a set of nonlinear equations need to be solved every time instant. Consequently, the Newton–Raphson iterative scheme is applied to find the system coordinates and joint constraint loads in an efficient and highly parallelizable manner. Two divide-and-conquer-based mass-orthogonal projections are performed then to circumvent the effect of constraint violation errors at the velocity and acceleration level. Sample open- and closed-loop multibody system test cases are investigated in the paper to confirm the validity of the approach. Challenging simulations of multibody systems featuring long kinematic chains are also performed in the work to demonstrate the robustness of the algorithm. The details of OpenMP-based parallel implementation on an eight-core shared memory computer are presented in the text and the parallel performance results are extensively discussed. Significant speedups are obtained for the simulations of small- to large-scale multibody open-loop systems. The mentioned features make the proposed algorithm a good general purpose approach for high-fidelity, efficient or real-time multibody dynamics simulations.

Simple and accurate SEP approximation of hexagonal‐QAM in AWGN channel and its application in parametric , , fading, and log‐normal shadowing

In this study, the authors propose a simple yet tighter approximations for the special two-dimensional Gaussian Q functions using the Trapezoidal rule of numerical integration. This enables a simplified and accurate symbol error probability (SEP) approximation of the hexagonal-quadrature amplitude modulation (HQAM) in additive white Gaussian noise channel. The proposed approximation further simplifies the SEP calculation of HQAM in parametric α − μ , η − μ , and κ − μ fading distributions. Also, the SEP of HQAM over log-normal shadowing is calculated in this study. The accuracy of the analytical framework is verified using computer simulations.

Developing an algorithm for automated geometric analysis and classification of landslides incorporating LiDAR-derived DEM

Amending landslides inventories is immensely important to policy and decision makers alike. Sliding creates geometric shapes on the Earth’s surface. This study presents the utilization of LiDAR high-resolution digital elevation model (DEM) in the Alborz Mountains, Iran to refurbish the existing landslide inventory dataset by implementing the proposed algorithm. The method consists of the automated derivation of landslide geometry (length, width, and area) followed by classification of landslide types considering length, width and flow direction. This study has used the trapezoidal rule for numerical integration to develop the proposed algorithm. The landslides were then classified into four types (very long, long, very wide, and wide) based on slope, length, and width. This geometric classification of landslides is based on the geographical coordinates, slope angle (θ), length (L), and width (W), and further failure flow direction. A total of 95 landslides were updated from the existing inventory database. The proposed method was verified and evaluated by field observations; and 14 samples were tested to determine the relative error. The results demonstrated that the mean percentage relative error is 0.496% in length and width and 0.008% in area, related to the GIS analysis. The accuracy performance of determining the landslide’s type is 92%. The purposefulness of this algorithm is to increase the accuracy performance of landslides geometry analysis and automated measurements associated with the usual GIS platforms such as ArcGIS.

Bending of Euler–Bernoulli nanobeams based on the strain-driven and stress-driven nonlocal integral models: a numerical approach

Eringen’s nonlocal elasticity theory is extensively employed for the analysis of nanostructures because it is able to capture nanoscale effects. Previous studies have revealed that using the differential form of the strain-driven version of this theory leads to paradoxical results in some cases, such as bending analysis of cantilevers, and recourse must be made to the integral version. In this article, a novel numerical approach is developed for the bending analysis of Euler–Bernoulli nanobeams in the context of strain- and stress-driven integral nonlocal models. This numerical approach is proposed for the direct solution to bypass the difficulties related to converting the integral governing equation into a differential equation. First, the governing equation is derived based on both strain-driven and stress-driven nonlocal models by means of the minimum total potential energy. Also, in each case, the governing equation is obtained in both strong and weak forms. To solve numerically the derived equations, matrix differential and integral operators are constructed based upon the finite difference technique and trapezoidal integration rule. It is shown that the proposed numerical approach can be efficiently applied to the strain-driven nonlocal model with the aim of resolving the mentioned paradoxes. Also, it is able to solve the problem based on the strain-driven model without inconsistencies of the application of this model that are reported in the literature.

Numerical solution of Volterra partial integro-differential equations based on sinc-collocation method

We provide the numerical solution of a Volterra integro-differential equation of parabolic type with memory term subject to initial boundary value conditions. Finite difference method in combination with product trapezoidal integration rule is used to discretize the equation in time and sinc-collocation method is employed in space. A weakly singular kernel has been viewed as an important case in this study. The convergence analysis has been discussed in detail, which shows that the approach exponentially converges to the solution. Furthermore, numerical examples and illustrations are presented to prove the validity of the suggested method.