High-accuracy Solution Research Articles

Approximate Dynamic Programming of Long Term Power Generation Schedule

With respect to the features in optimal scheduling of long-term power generation such as long cycle, large scale and strong randomness, the paper proposed a multi-stage optimized decision model based on approximate dynamic programming. In this model, the influence of random variables including load, fuel price, and water was tested by means of stochastic simulations. Staged solution was used to reduce the size and difficulty of solving the problem. The proposed value function approximation strategy of pondage and coal inventory solved the "curse of dimensionality" in staged recursion and maintained the overall optimization characteristics after staged decomposition. Through the iterative update between the decision and the approximation function, the approximate optimal decision sequence was solved to obtain the optimization scheme of power generation scheduling. The calculation and analysis results in a real system in some province showed that the modeling by approximate dynamic programming is an effective way to solve such large-scale optimization and has a promising future by virtue of its conciseness, effectiveness, convenience in dealing with random factors, high calculation accuracy and rapidity of solution.

Numerical solution of mathematical physics problems by the collocation method

A modified collocation method for the numerical solving boundary value problems of mathematical physics is proposed. The irregular arrangement of collocation nodes in the problem solving domain can sharply increase the accuracy of the numerical solution by improving the quality of the linear algebraic equations system, to which the solved boundary value problem leads. Various basis functions systems are considered. The proposed method allows one to obtain an approximate solution of boundary value problems for a wide range of linear and nonlinear elliptic, parabolic and wave equations in an analytical form. This numerical method makes it possible to significantly expand the application field of traditional numerical methods when solving applied problems for modelling fields of various physical natures, described by linear and nonlinear equations of mathematical physics. The developed method is used to solve a quantum-mechanical problem for a hydrogen molecule ion. The results obtained in this work show the high potentialities of the complete collocation method, which are based on the universality of the method and high accuracy of numerical solutions. The energy of the ion ground state calculated with the minimum number of collocation nodes differs from the experimentally obtained value by 13%.

Approximate bright-soliton solution of the higher-order nonlinear Schrödinger equation

In this paper, approximate bright-soliton solutions of the higher-order nonlinear Schrödinger equation are constructed by treating the higher-order terms as small perturbations. The first-, second-, and third-order asymptotic solutions are obtained. The errors between the asymptotic solutions and the numerical/analytical solutions are discussed, which gives a high accuracy of the approximate solutions. It is pointed that the asymptotic solutions can be used as the initial value to improve the accuracy of the numerical solutions. This paper may be helpful for undergraduate and graduate students in mathematics and physics to understand the approximate soliton solutions of the higher-order nonlinear Schrödinger equation.

Triple Archives Particle Swarm Optimization.

There are two common challenges in particle swarm optimization (PSO) research, that is, selecting proper exemplars and designing an efficient learning model for a particle. In this article, we propose a triple archives PSO (TAPSO), in which particles in three archives are used to deal with the above two challenges. First, particles who have better fitness (i.e., elites) are recorded in one archive while other particles who offer faster progress, called profiteers in this article, are saved in another archive. Second, when breeding each dimension of a potential exemplar for a particle, we choose a pair of elite and profiteer from corresponding archives as two parents to generate the dimension value by ordinary genetic operators. Third, each particle carries out a specific learning model according to the fitness of its potential exemplars. Furthermore, there is no acceleration coefficient in TAPSO aiming to simplify the learning models. Finally, if an exemplar has excellent performance, it will be regarded as an outstanding exemplar and saved in the third archive, which can be reused by inferior particles aiming to enhance the exploitation and to save computing resources. The experimental results and comparisons between TAPSO and other eight PSOs on 30 benchmark functions and four real applications suggest that TAPSO attains very promising performance in different types of functions, contributing to both higher solution accuracy and faster convergence speed. Furthermore, the effectiveness and efficiency of these new proposed strategies are discussed based on extensive experiments.

A Plate Bending Kirchhoff Element Based on Assumed Strain Functions

To investigate static and free vibration for thin plate bending structures, a four-node quadrilateral finite element is proposed in this research paper. This element has been formulated by using both the assumptions of thin plates theory (Kirchhoff plate theory) and strain approach. The suggested element which possesses only three degrees of freedom (one transverse displacement and two normal rotations) at each of four corner nodes is based on assumed higher-order functions for the various components of strain field that satisfies the compatibility equation. The displacement functions of the developed element are obtained by integrating the assumed strains functions and satisfy the exact representation of the rigid body modes. Several numerical tests in both static and free vibration analysis are presented to assess the performance of the new element. The obtained results show high solution accuracy, especially for coarse meshes, of the developed element compared with analytical and other numerical solutions available in the literature.

Chatter stability prediction of milling considering nonlinearities

Nonlinearities have been evidenced during the chatter vibration of milling. Machinability of the thin-walled part is feed rate and position-dependent, and is subject to process damping at low cutting speed. Therefore, chatter stability prediction of milling considering nonlinear cutting force, nonlinear structural stiffness and process damping is investigated. The cutting force and stiffness are established based on the polynomial model and the process damping is investigated based on the dissipated energy. The dynamic cutting force and stability lobes are solved in the time domain with coefficients updated at each iteration. By formulating the displacement as an expanded form via the perturbation method, the time-consuming solution of delay differential equations is avoided. After formulating the identification of the nonlinear model via cutting tests and modal tests, numerical simulations considering nonlinearities are carried out and compared with the analytical method. The proposed method attains high accuracy of classic time-domain solution, but with an improved computational efficiency. Finally, cutting tests are conducted to verify the prediction of cutting force and stability lobes.

A hybrid asymptotic and augmented compact finite volume method for nonlinear singular two point boundary value problems

An accurate and efficient numerical method is proposed for nonlinear singular two point boundary value problems. The scheme combines Puiseux series asymptotic technique with augmented compact finite volume method. The main motivation is to get high order accurate solution for nonlinear singular problems. The key of the new method is to express the solution as the Puiseux series expansion on a small subinterval involving the singular point. The expansion contains an undetermined parameter which is an augmented variable in the numerical method. In this way, a nonlinear system is constructed in other subinterval and the parameter related with the semi-analytic solution near the singular point and the numerical solution can be simultaneously obtained. A rigorous error estimate for the solution of the singular differential equation is conducted and fourth order accuracy is obtained. Numerical examples confirm the theoretical analysis and efficiency of the new method.

Simple algorithm for L 1‐norm regularisation‐based compressed sensing and image restoration

L1-norm regularisation plays an important role in compressed sensing reconstruction and image restoration. However, the discontinuity of L1-norm function makes solving the involved optimisation problem very challenging with traditional optimisation methods. In this article, a simple but efficient algorithm is proposed for the L1-norm regularised compressed sensing and image restoration problem. In the proposed algorithm, the L1-norm regularised optimisation problem is converted to a non-linear optimisation problem with L1-norm approximation by a smoothening function, which then can be solved by existing powerful non-linear optimisation methods. The simulation results show that the proposed algorithm is more efficient and results in a higher accurate solution. Compared to existing methods, the proposed algorithm is very easy to implement and promising for applications in medical and biological imaging.

An improved cuckoo search with reverse learning and invasive weed operators for suppressing sidelobe level of antenna arrays

AbstractIn order to overcome some shortcomings including the premature convergence and slow convergence speed at later evolution stage of conventional cuckoo search (CS) algorithm for the sidelobe suppressions of antenna arrays, an improved CS with reverse learning and invasive weed operators (ICSRLIWO) is proposed. First, ICSRLIWO algorithm generates reverse populations through opposition‐based learning method, then selects better individuals in the mixed population consists of the original and reversed population to form a high‐quality initial population. Second, ICSRLIWO introduces the reproducing, space diffusion and survival competition operators of invasive weed optimization to improve the population diversity and global search ability of conventional CS algorithm. Simulations are conducted based on 30 test functions in the CEC 2014 test suit to verify the performance of the proposed approach, and the results show that ICSRLIWO has higher solution accuracy and faster convergence speed than other comparison algorithms. In addition, ICSRLIWO also has advantages in solving the array antenna beam pattern optimization problems.

Time-dependent solution for natural convection in a porous enclosure using the Darcy–Lapwood–Brinkman model

Natural convection (NC) in high permeable porous media is usually investigated using the Darcy–Lapwood–Brinkman model (DLB). The problem of the porous squared cavity is widely used as a common benchmark case for NC in porous media. The solutions to this problem with the DLB model are limited to steady-state conditions. In this paper, we developed a time-dependent high accurate solution based on the Fourier–Galerkin method (FG). The solution is derived considering two configurations dealing with unsteady and transient modes. The governing equations are reformulated using the stream function. The Temperature and the stream functions are expended as unknowns in space using Fourier series which are appropriately substituted in the equations. The equations are then projected to the spectral space using Fourier trigonometric trial functions. The obtained developed equations form a nonlinear differential algebraic system of equations. An appropriate technique is used to integrate the spectral system in time and to ensure high accuracy. The results of the FG method are compared to a finite element solution for different Rayleigh and Darcy numbers values. The transient and unsteady solutions are obtained with a feasible and low computational cost. The paper provides high accurate time-dependent solutions useful for benchmarking numerical models dealing with NC in porous media. The results of the developed solutions are efficient to gain physical insight into the time-dependent NC processes.

On the approximate solutions to a damped harmonic oscillator with higher-order nonlinearities and its application to plasma physics: semi-analytical solution and moving boundary method

In this paper, a strongly nonlinear oscillator equation with higher-order nonlinear (cubic) restoring force, namely damped Duffing equation with higher-order nonlinearities, has been solved and analyzed analytically and numerically using some effectiveness and high-accuracy methods. Firstly, the physical oscillator system consisting of a spring–mass moves under cubic resorting force with constant spring is considered for getting damped Duffing equation. After that a semi-analytical solution is obtained in terms of the Jacobian elliptic function cn. Also, the numerical approximate solution using Runge–Kutta-4th (RK4) procedure is carried out and compared to the semi-analytical solution. It is found that the agreement between both solutions is very good especially if the coefficient of the cubic nonlinear term becomes larger than the coefficient of the linear one. However, we made some improvements to the semi-analytical solution using the numerical moving boundary method to get higher accuracy solution. Furthermore, the error analysis is estimated for both the semi-analytical solution, improved semi-analytical solution, and RK4 solution. The method that is used for getting the semi-analytical solution is elementary, i.e., allows to obtain only integrable case without using any Lie group theory. In another meaning, our approach gives exact/analytical solution for the integrable case such as undamped Duffing equation. Moreover, the t-intercepts and amplitude points are estimated precisely. Furthermore, the plasma oscillations are discussed in the framework of damped Duffing equation. Firstly, we reduced the fluid basic equations of pair-ion plasmas with Maxwellian electrons using reductive perturbation technique to a modified korteweg de-Vries Burgers (mKdVB) equation, and by using traveling wave transformation the mKdVB equation is converted to the damped Duffing equation. After that we compared between all mentioned solutions using plasma parameters.

A CMM-Based Method of Control Point Position Calibration for Light Pen Coordinate Measuring System.

A light pen coordinate measuring system (LPCMS) is a kind of vision-based portable coordinate measuring technique. It implements coordinate measurement by analyzing the image of a light pen, which has several control points and a probe. The relative positions of control points need to be determined before measurement and serve as the measuring basis in LPCMS. How to accurately calibrate the relative positions of control points is the most important issue in system calibration. In this paper, a new method of control point position calibration based on a traditional coordinate measuring machine (CMM) is proposed. A light pen is fastened to the measuring arm of a CMM and performs accurate translational movement driven by the CMM. A camera is used to capture the images of control points at different positions, and the corresponding readings of the CMM are recorded at the same time. By establishing a separate coordinate system for each control point, the relative positions of the control points can be transformed to the differences of a series of translation vectors. Experiments show that the calibration repeatability of control point positions can reach 10 μm and the standard deviation of measurement of the whole LPCMS can reach 30 μm. A CMM is used to generate accurate translation, which provides a high accuracy basis of calibration. Through certain mathematical treatment, tremendous data acquired by moving the light pen to tens of thousands of different positions can be processed in a simple way, which can reduce the influence of random error. Therefore, the proposed method provides a high-accuracy solution of control point position calibration for LPCMS.

MTDE: An effective multi-trial vector-based differential evolution algorithm and its applications for engineering design problems

In this article, an effective metaheuristic algorithm named multi-trial vector-based differential evolution (MTDE) is proposed. The MTDE is distinguished by introducing an adaptive movement step designed based on a new multi-trial vector approach named MTV, which combines different search strategies in the form of trial vector producers (TVPs). In the developed MTV approach, the TVPs are applied on their dedicated subpopulation, which are distributed by a winner-based distribution policy, and share their experiences efficiently by using a life-time archive. The MTV can be deployed by different types of TVPs, particularly, we use the MTV approach in the MTDE algorithm by three TVPs: representative based trial vector producer, local random based trial vector producer, and global best history based trial vector producer. Therefore, this study introduces the MTV approach to boost the performance of the MTDE and demonstrates its advantages in dealing with problems of different levels of complexity. The performance of the proposed MTDE algorithm is evaluated on CEC 2018 benchmark suite which include unimodal, multimodal, hybrid, and composition functions and four complex engineering design problems. The experimental and statistical results are compared with state-of-the-art metaheuristic algorithms: GWO, WOA, SSA, HHO, CoDE, EPSDE, QUATRE, and MKE. The results demonstrate that the MTDE algorithm shows improved performance and benefits from high accuracy of optimal solutions obtained.

Arbitrary high-order non-oscillatory scheme on hybrid unstructured grids based on multi-moment finite volume method

In this article, we propose a novel high-order multi-moment finite volume method (MMFVM) for solving linear and nonlinear hyperbolic systems on unstructured grids. Different from the previous versions of volume-average/point-value multi-moment (VPM) scheme, the present scheme is developed by using a new reconstruction procedure based on the constrained least-square method which can be naturally extended to arbitrary high-order of accuracy in two and three dimensions. The so-called VPM-CLS (VPM based on constrained least-square method) scheme substantially increases the solution accuracy on a compact stencil which largely simplifies the computations. In order to eliminate the numerical oscillations associated with high-order schemes, we propose a new limiting projection approach named AOD (adaptive order detection) to choose the optimal degree of reconstruction polynomial for trouble cells in presence of discontinuous solutions. The resulting reconstruction method, i.e. VPM-AOD that combines VPM-CLS scheme with AOD limiting projection technique is able to attain 5th-order accuracy on the stencil of all immediately adjacent cells and achieve essentially non-oscillatory and less-dissipative numerical results for discontinuous solutions. The numerical methods are extensively verified by various benchmark tests for the Euler equations of compressible gas dynamics. Numerical results demonstrate the high solution accuracy and excellent capabilities of proposed schemes to handle both complex physics and geometries.

Analyzing S-Shaped I–V characteristics of solar cells by solving three-diode lumped-parameter equivalent circuit model explicitly

In order to analyze and optimize new generation solar cells’ electrostatic performances, lumped-parameter equivalent circuit model is a common method to simulate S-shaped I–V characteristics including linear, exponential, or exponential-like current kinks. Unfortunately, three-diode lumped-parameter model is still inevitable to be solved generally in numerical iteration method. The absence of explicit solution to three-diode lumped-parameter model is actually the main bottleneck of implementing the model into photovoltaic device and circuit simulators in compact. In this paper, to overcome the problem of three-diode model’s implementation into simulators, we proposed an explicit solution to the model based on the regional approach, where high accuracy and low computation time cost are the main features of such a solution. Analytical derivation and correction for the solution to transcendent I–V equation in three-diode model leads to high computation accuracy, and avoidance of numerical iteration methods introduces low computation time cost. Finally, numerical iteration results and reconstructed experimental data of solar cells are used to validate the accuracy and applicability of our proposed explicit solution. As a result, high accuracy and efficiency of the explicit solution make it possible to implement three-diode lumped-parameter equivalent circuit model into photovoltaic device and circuit simulators and explain I–V characteristics of new generation solar cells.

Pyramidal approximation for power flow and optimal power flow

Power flow equations (PFEs) are the fundamental of power flow (PF) calculations and optimal PF (OPF) for power system analysis, but existing PFEs suffer from the trade-off among various requirements in practice. This study presents a novel pyramidal approximation (PA) for PF and OPF. A multi-sided pyramid is used to approximate the feasible region of PFEs. The PA-based PFEs not only guarantee the linearity, tightness, and no dependence on the initial guess but also have high solution accuracy. The rotation-and-fold strategy is developed to balance the computational efficiency and approximation accuracy, so that the problem for a relatively large power system can be solved in a reasonable time. Case studies in different test systems validate the tightness and accuracy of the proposed PA method. The balance of accuracy and efficiency is also discussed, and the PA has a good performance in the small or medium power systems.

Numerical calculation for coupling vibration system by Piecewise-Laplace method.

To improve the reliability and accuracy of dynamic machine in design process, high precision and efficiency of numerical computation is essential means to identify dynamic characteristics of mechanical system. In this paper, a new computation approach is introduced to improve accuracy and efficiency of computation for coupling vibrating system. The proposed method is a combination of piecewise constant method and Laplace transformation, which is simply called as Piecewise-Laplace method. In the solving process of the proposed method, the dynamic system is first sliced by a series of continuous segments to reserve physical attribute of the original system; Laplace transformation is employed to separate coupling variables in segment system, and solutions of system in complex domain can be determined; then, considering reverse Laplace transformation and residues theorem, solution in time domain can be obtained; finally, semi-analytical solution of system is given based on continuity condition. Through comparison of numerical computation, it can be found that precision and efficiency of numerical results with the Piecewise-Laplace method is better than Runge-Kutta method within same time step. If a high-accuracy solution is required, the Piecewise-Laplace method is more suitable than Runge-Kutta method.

Solving periodic semilinear stiff PDEs in 1D, 2D and 3D with exponential integrators

Dozens of exponential integration formulas have been proposed for the high-accuracy solution of stiff PDEs such as the Allen–Cahn, Korteweg–de Vries and Ginzburg–Landau equations. We report the results of extensive comparisons in MATLAB and Chebfun of such formulas in 1D, 2D and 3D, focusing on fourth and higher order methods, and periodic semilinear stiff PDEs with constant coefficients. Our conclusion is that it is hard to do much better than one of the simplest of these formulas, the ETDRK4 scheme of Cox and Matthews.

High-order compact scheme for the two-dimensional fractional Rayleigh\u2013Stokes problem for a heated generalized second-grade fluid

In this article, an unconditionally stable compact high-order iterative finite difference scheme is developed on solving the two-dimensional fractional Rayleigh–Stokes equation. A relationship between the Riemann–Liouville (R–L) and Grunwald–Letnikov (G–L) fractional derivatives is used for the time-fractional derivative, and a fourth-order compact Crank–Nicolson approximation is applied for the space derivative to produce a high-order compact scheme. The stability and convergence for the proposed method will be proven; the proposed method will be shown to have the order of convergence O(tau + h^{4}). Finally, numerical examples are provided to show the high accuracy solutions of the proposed scheme.

An Assessment of Impact of Adaptive Notch Filters for Interference Removal on the Signal Processing Stages of a GNSS Receiver

With the fast growing diffusion of real-time high-accuracy applications based on the global navigation satellite system (GNSS), the robustness of GNSS receiver performance has become a compelling requirement. Disruptive effects can be induced to signal processing stages of GNSS receivers due to the disturbances from radio frequency interference (RFI), even leading to a complete outage of the positioning and timing service. A typical RFI threat to GNSS signals is represented by portable jammers, which transmit swept-frequency (chirp) signals in order to span the overall GNSS bandwidth. The implementation in the receivers of adaptive notch filters (ANFs) for chirp cancellation has been extensively investigated and proved to be an efficient countermeasure. However, the performance of the ANF is strongly dependent on its configuration setup. Inappropriate parameter settings of the ANF for interference removal may induce severe distortion to the correlation process. In addition, an effective mitigation will still introduce a vestigial signal distortion contributed by the residual unmitigated chirp and the ANF operation itself, being not negligible for high-accuracy solutions. This article addresses the detailed analysis for assessing the effects of interference mitigation by notch filtering. A bias compensation strategy is proposed, wherein for each pseudorandom noise, the biases due to parameter settings of the notch filter are estimated and compensated. The impact of using the ANF operation on chirp signals at the acquisition and tracking stages of GNSS receivers is analyzed. On the basis of the three proposed metrics, the effects can be quantitatively estimated to depict a complete picture of the most influential parameters of the chirp and the ANF configurations, as well as the optimal achievable performance at the acquisition and tracking stages.