High-accuracy Solution Research Articles

High-Accuracy Solutions to the Time-Fractional KdV–Burgers Equation Using Rational Non-Polynomial Splines

A non-polynomial spline is a technique that utilizes information from symmetric functions to solve mathematical or physical models numerically. This paper introduces a novel non-polynomial spline construct incorporating a rational function term to develop an efficient numerical scheme for solving time-fractional differential equations. The proposed method is specifically applied to the time-fractional KdV–Burgers (TFKdV) equation. and time-fractional differential equations are crucial in physics as they provide a more accurate description of various complex processes, such as anomalous diffusion and wave propagation, by capturing memory effects and non-local interactions. Using Taylor expansion and truncation error analysis, the convergence order of the numerical scheme is derived. Stability is analyzed through the Fourier stability criterion, confirming its conditional stability. The accuracy and efficiency of the rational non-polynomial spline (RNPS) method are validated by comparing numerical results from a test example with analytical and previous solutions, using norm errors. Results are presented in 2D and 3D graphical formats, accompanied by tables highlighting performance metrics. Furthermore, the influences of time and the fractional derivative are examined through graphical analysis. Overall, the RNPS method has demonstrated to be a reliable and effective approach for solving time-fractional differential equations.

Error Modeling of Fiber Optic Gyroscope Universal Time Measurement

Since the fiber optic gyroscope (FOG) is rigidly strapped down to the earth’s crust, there are various errors that affect the universal time (UT1) measurements. In this paper, the errors caused by various physical factors and mechanisms are analyzed in detail, with precession and nutation errors being taken into account, and modeling of the observation equations based on precession and nutation error correction is proposed. The mapping relationship with UT1 is established based on this observation equation; after the corresponding error correction and VLBI calibration, the high-accuracy solution of UT1 is finally completed. Through 14-day measurement experiments under a room temperature environment without any vibration isolation and magnetic shielding devices, the error variation of UT1 solution compared with the earth orientation parameter (EOP) 14 C04 data is calculated at less than 3.57 ms, with UT1 solution accuracy improved by 56% compared with the traditional method. These results indicate that this work facilitates the study of giant FOG error modeling and correction, advancing our understanding of errors in giant FOG measurements and improving the accuracy of UT1 solution.

Tilt noise extraction method based on fourier transform and fitting of 2D images

Tilt control is necessary for high-power coherent beam combining systems with large array elements. A tilt noise extraction method based on 2D images has been developed, and the analysis of abnormal situations has been executed. The method achieves high accuracy solution of tilt after two-step operation of Fourier transform and fitting of interference fringes at the detecting module, and then feeds back the compensated amount to the controlling module to achieve tilt control. Monte Carlo simulation is used to solve the control accuracy of this method. The tilt residuals of the method were calculated at various intrinsic shears and data lengths. Results of the effect of the magnitude of the intrinsic shear and the amount of data on the tilt residuals verified the accuracy and stability of the method.

High-accuracy solution for fractional solitary wave dynamics in finite water depth with linear shear flow, wind, and dissipation effects

In this paper, a fractional nonlinear Schrödinger equation has been initially derived for capturing the dynamics of gravity waves in finite water depth, accounting for factors such as wind, dissipation, and shear currents. A comprehensive framework is established to enhance the model's representation of gravity wave behavior. We employ a high-order iterative method, specifically the homotopy iterative technique, along with a non-uniform collocation approach integrated into the Haar wavelet method, resulting in a novel computational method characterized by high precision and efficiency. The robustness and reliability of the proposed approach are validated through convergence analysis and comparisons with analytical solutions. Furthermore, the results indicate that the nonlinear and dispersive effects caused by the fractional orders lead to changes in the propagation characteristics of gravity waves. The impacts of the damping coefficient related to wind action and dissipative effects on the temporal evolution of solitary waves are also discussed. The construction of the fractional model holds far-reaching significance for researching the nonlinear propagation of gravity waves in actual ocean water waves. Additionally, an outstanding computational technique for solving fractional nonlinear evolution equations in diverse applications has been developed.

A slope-based automatic identification and optimization of key time nodes method for adaptive control vector parameterization

Control vector parameterization is a mainstream numerical approach for solving dynamic programming problems. By discretizing the entire time domain into a grid of time nodes, an infinite-dimensional optimal control problem can be transformed into a finite-dimensional static optimization problem. Despite the attractive advantages of high solution accuracy and ease of implementation of control vector parameterization, the rationality of time grid partitioning significantly influences the efficiency of the solution and the approximating accuracy of the optimal control trajectory. In the traditional control vector parameterization method, the time grid is typically set beforehand and remains static in the optimization process, which has a direct impact on how closely the solution result approximates the optimal control trajectory. To address the conflict of accurate approximations and computational time, we propose a slope-based automatic identification and optimization of key time nodes method for adaptive control vector parameterization, which not only optimizes merging and inserting time nodes but also incorporates automatic identification of key time nodes. Through a simulation example, we verify that this improved method can reduce computation time, enhance approximation accuracy, and achieve higher optimization efficiency.

A novel multi-objective robust optimization method based on improved Gray Wolf optimizer and Kriging model

A novel multi-objective robust optimization method for mechanical structure products is introduced. Firstly, the method introduces Bernoulli Dynamic Adaptive Gray Wolf Optimizer (BDAGWO) to address inherent limitations within the original Gray Wolf Optimizer (GWO), such as falling into local optimal solutions and low convergence rate. Subsequently, BDAGWO is utilized to search optimal correlation coefficients of Kriging. The proposed surrogate model exhibits high fitting accuracy on different test cases, with coefficients of determination reaching above 0.99 on the test set, and mean relative error, mean absolute error, mean absolute percentage error and root mean squared error are all close to 0. To solve multi-objective optimization problems, an improved NSGA-II is introduced to amplify search capabilities. Then based on robust optimization method, combine BDAGWO-Kriging and improved NSGA-II to establish a multi-objective robust optimization framework with high accuracy and high solution efficiency. The proposed method is implemented to an EMU bogie frame, demonstrating that the robust optimization scheme reduces the maximum equivalent stress and mass, and exhibits lower fluctuation compared to deterministic optimization. This substantiates the method’s effectiveness and provides an optimization approach of large and complex mechanical products.

Multiple strategies improved spider wasp optimization for engineering optimization problem solving

The Spider Wasp Optimization (SWO) algorithm is a swarm intelligence optimization technique inspired by the collective behaviors of social animals. This algorithm, designed to address optimization challenges, emulates the unique hunting, nesting, and mating behaviors of female spider wasps. It offers several advantages, including rapid search speed and high solution accuracy. However, when tackling complex optimization problems, it can encounter issues such as getting trapped in local optima, slow early convergence, and the need for manual adjustment of the “Trade-off Rate” (TR) parameter for different problems.To improve the performance and versatility of the SWO algorithm, a Multi-strategy Improved Spider Wasp Optimizer (MISWO) is proposed. Firstly, the Grey Wolf Algorithm is integrated into the initialization phase to enhance early convergence and improve the fitness of the initial population, thereby boosting the algorithm’s global optimization capabilities.Secondly, an adaptive step size operator and Gaussian mutation are introduced during the search phase to automatically adjust the search range at different optimization stages. This enhancement increases both the optimization accuracy and the algorithm’s ability to avoid local optima. The Trade-off Rate (TR) is dynamically selected to better accommodate a variety of problems. Finally, a dynamic lens imaging reverse learning strategy is employed to update optimal individuals, further improving the algorithm’s capacity to escape local optima. To validate the effectiveness of MISWO, it was tested on 23 benchmark functions and 7 engineering optimization problems, and compared with several state-of-the-art algorithms. Experimental results show that MISWO outperforms other algorithms in terms of optimization capability, stability, and adaptability across diverse problems.

Verification and Validation of the VANGARD Pinwise Nodal Code with Analyses of BEAVRS

This paper validates the graphical processing unit (GPU)–based pinwise nodal core calculation code VANGARD using the Benchmark for Evaluation And Validation of Reactor Simulations (BEAVRS), focusing on hot-zero-power (HZP) physics tests, hot-full-power (HFP) depletion, and load follow operation for both Cycles 1 and 2. Results are compared with measured data and other high-fidelity numerical solutions. In HZP physics tests, the critical boron concentration (CBC), control rod bank worth, and isothermal temperature coefficient agree well with measured data, satisfying the design review criteria for typical zero power physics tests. Pin power distributions for various rodded cases confirm the high accuracy of pin-resolved solutions compared to other numerical results. For HFP depletion, CBC closely matches measured data and other high-fidelity numerical solutions, showing differences smaller than 35 and 10 ppm, respectively, throughout the whole depletion steps. Computational performance analysis reveals that a cycle depletion of a realistic pressurized water reactor core can be completed within 3.5 min using a single gaming GPU, which affirms the feasibility of practical pinwise core designs.

The Use of Slack Variables in the Adjoint Method Handling Inequality Constraints in Optimal Control and the Application to Tumor Drug Dosage

Abstract In this article a modified gradient approach, based on the adjoint method, is introduced, to deal with optimal control problems involving inequality constraints. So far, the only way to incorporate inequality constraints in the adjoint approach, is to introduce additional penalty terms in the cost functional. However, this may distort the optimal control due to weighting factors required for these terms and raise serious concerns about the magnitude of the weighting factors. The method in this article avoids penalty functions and can be used for the iterative computation of optimal controls. In order to demonstrate the key idea, first, a static optimization problem in the Euclidean space is considered. Second, the presented approach is applied to the tumor anti-angiogenesis optimal control problem in medicine, which addresses an innovative cancer treatment approach that aims to inhibit the formation of the tumor blood supply. The tumor anti-angiogenesis optimal control problem with free final time involves inequality and final constraints for control and state variables and is solved by a modified adjoint gradient method introducing slack variables. It has to be emphasized that the novel formulation and the special use of slack variables in this article delivers high accurate solutions without distorting the optimal control.

Comparison of rigorous scattering models to accurately replicate the behaviour of scattered electromagnetic waves in optical surface metrology

Rigorous scattering models are based on Maxwell's equations and can provide high-accuracy solutions to model electromagnetic wave scattering from objects. Being able to calculate the scattered field from any surface geometry and considering the effect of the polarisation of the incident light, make rigorous models the most promising tools for complex light-matter interaction problems. The total intensity of the electric near-field scattering from a silicon cylinder illuminated by the transverse electric and transverse magnetic polarisation of the incident light is obtained using various rigorous models including, the local field Fourier modal method, boundary element method and finite element method. The intensity of the total electric near-field obtained by these rigorous models is compared using the Mie solution as a reference for both polarisation modes of the incident light. Additionally, the intensity of the total electric near-field scattered from a silicon sinusoid profile using the same rigorous models is analysed. The results are discussed in detail, and for the cylinder, the deviations in the intensity of the total electric field from the exact Mie solution are investigated.

K-means-ten clustering algorithm based on min-max criterion and region division

Aiming at the problem of unstable clustering results and low solution accuracy of -means- + algorithm, an I--means-+ clustering algorithm based on min-max criterion and region division is proposed. Firstly, the min-max criterion is proposed to calculate the distance from each data point to the nearest center, and the data point with the largest distance is preferentially selected as the new cluster center to avoid the situation where multiple initial centers are clustered in the same cluster; secondly, the data points in the split cluster are divided into different regions, and a data point is selected in each region as the candidate center to increase the diversity of the candidate centers; finally, for the clusters that fail to pair, the new split cluster is reselected by gain to pair with the original deleted cluster again to improve the pairing success rate and further reduce the objective function value. Experimental results show that compared with the I means-+ algorithm, the proposed algorithm has an average improvement in solution accuracy −k −and a more stable clustering result kunder the premise of basically equivalent operation efficiency compared with 6.47%-means and kmeans+ algorithms, the proposed algorithm has higher solution accuracy.

Match-based solution of general parametric eigenvalue problems

We describe a novel algorithm for solving general parametric (nonlinear) eigenvalue problems. Our method has two steps: first, high-accuracy solutions of non-parametric versions of the problem are gathered at some values of the parameters; these are then combined to obtain global approximations of the parametric eigenvalues. To gather the non-parametric data, we use non-intrusive contour-integration-based methods, which, however, cannot track eigenvalues that migrate into/out of the contour as the parameter changes. Special strategies are described for performing the combination-over-parameter step despite having only partial information on such migrating eigenvalues. Moreover, we dedicate a special focus to the approximation of eigenvalues that undergo bifurcations. Finally, we propose an adaptive strategy that allows one to effectively apply our method even without any a priori information on the behavior of the sought-after eigenvalues. Numerical tests are performed, showing that our algorithm can achieve remarkably high approximation accuracy.

A Dini-Derivative-Aided Zeroing Neural Network for Time-Variant Quadratic Programming Involving Multi-Type Constraints With Robotic Applications.

Time-variant quadratic programming (QP) with multi-type constraints including equality, inequality, and bound constraints is ubiquitous in practice. In the literature, there exist a few zeroing neural networks (ZNNs) that are applicable to time-variant QPs with multi-type constraints. These ZNN solvers involve continuous and differentiable elements for handling inequality and/or bound constraints, and they possess their own drawbacks such as the failure in solving problems, the approximated optimal solutions, and the boring and sometimes difficult process of tuning parameters. Differing from the existing ZNN solvers, this article aims to propose a novel ZNN solver for time-variant QPs with multi-type constraints based on a continuous but not differentiable projection operator that is deemed unsuitable for designing ZNN solvers in the community, due to the lack of the required time derivative information. To achieve the aforementioned aim, the upper right-hand Dini derivative of the projection operator with respect to its input is introduced to serve as a mode switcher, leading to a novel ZNN solver, termed Dini-derivative-aided ZNN (Dini-ZNN). In theory, the convergent optimal solution of the Dini-ZNN solver is rigorously analyzed and proved. Comparative validations are performed, verifying the effectiveness of the Dini-ZNN solver that has merits such as guaranteed capability to solve problems, high solution accuracy, and no extra hyperparameter to be tuned. To illustrate potential applications, the Dini-ZNN solver is successfully applied to kinematic control of a joint-constrained robot with simulation and experimentation conducted.

Traffic assignment optimization to improve urban air quality with the unified finite-volume physics-informed neural network

Emissions by road transportation constitute the major contributor of air pollutants to threat the public health, especially on mega cities with high-rise buildings and growing population. A practicable and economical remedy is assigning the traffic volume judiciously, termed traffic assignment optimization (TAO). Incorporating the computational fluid dynamics (CFD), this method provides precisely optimized traffic volumes under versatile meteorological conditions. However, the high-dimensional degree of freedoms (DoF) involved in CFD hampers its further application toward the large-scale problem where a massive urban area is considered. With the more complicated urban traffic network, the larger decision variable dimension also impedes the time-efficient acquisition of the optimization results. To alleviate this curse-of-dimensionality, the current work proposes an optimization framework with the surrogate model based on the unified finite-volume physics-informed neural networks (UFV-PINN). The UFV-PINN plays the dual role as the partial differential equation (PDE) solver and the surrogate model for pollutant concentration prediction. It reduces the DoF within the PDE solution process and enables the gradient-based optimization. The current work optimizes the average CO concentration and accumulative travel time in Kowloon peninsula of Hong Kong, using the multiple-gradient descent algorithm (MGDA). This optimization framework is tested with various working conditions. Results indicate that the proposed method attain high accuracy solutions, comparable to the heuristic algorithms. This study is the first attempt to incorporate UFV-PINN into the TAO with air quality consideration. Endowed with the differentiability by UFV-PINN, the framework will facilitate the TAO to provide precise suggestions toward large-scale problems.

An inverse-free Getz-Marsden dynamic system and its eleven-instant discrete model for time-variant linear equations solving

Time-variant linear equations (TVLEs) are fundamental problems appeared in many practical applications. There is an increasing demand for algorithms that can solve TVLEs with high accuracy. This paper proposes an inverse-free continuous-time Getz–Marsden dynamic system (CTGMDS) for solving TVLEs. Additionally, a general eleven-instant finite-difference method (FDM) is proposed to discretize the CTGMDS, resulting in a general eleven-instant discrete-time GMDS (11-DTGMDS) model. Theoretically, the proposed 11-DTGMDS has truncation error of seven-order precision that delivers higher precision than the existing FDMs. This paper also compares five DTGMDS models using other FDMs to discretize the CTGMDS. The convergence of the CTGMDS and 11-DTGMDS is proven by theoretical analysis. Numerical validations are performed by comparing the 11-DTGMDS and other DTGMDSs, showing that the 11-DTGMDS provides higher solution accuracy. Finally, a path-tracking task is completed by applying the 11-DTGMDS model to a Franka Emika Panda robot, validating the practicality of the 11-DTGMDS.

An Effective Matrix Technique for the Numerical Solution of Second Order Differential Equations

In this paper, an effective technique for solving differential equations with initial conditions is presented. The method is based on the use of the Legendre matrix of derivatives defined on the close interval [-1,1]. Properties of the polynomial are outlined and further used to obtain the matrix of derivative which was used in transforming the differential equation into systems of linear and nonlinear algebraic equations. The systems of these algebraic equations were then solved using Gaussian elimination method to determine the unknown parameters required for approximating the solution of the differential equation. The advantage of this technique over other methods is that, it has less computational manipulations and complexities and also its availability for application on both linear and nonlinear second-order initial value problems is impressive. Other advantage of the algorithm is that high accurate approximate solutions are achieved by using a greater number of terms of the Legendre polynomial and once the operational matrix is obtained, it can be used to solve differential equations of higher order by introducing just a little manipulation on the operational matrix. Some existing sample problems from literature were solved and the results were compared to show the validity, simplicity and applicability of the proposed method. The results obtained validate the simplicity and applicability of the method and it also reveals that the method perform better than most existing methods.

Impact of electric vehicles and wave energy systems on OPF of power networks using hybrid Osprey-PSO approach

In this article, a novel optimization algorithm named hybrid Osprey Optimization and Particle Swarm Optimization (OOPSO) is proposed to solve the Probabilistic Optimal Power Flow (POPF) problem in power systems with high penetration of renewable energy resources (RERs), such as photovoltaic, wind, and wave energy and integration of electric vehicles (EVs). The algorithm's results are compared with established optimization algorithms like Osprey Optimization Algorithm (OOA), Particle Swarm Optimization (PSO), and Heap Optimization. Applied to the 30 and 118-bus IEEE test systems, the OOPSO algorithm efficiently deals with the uncertainties produced by RERs generation. This research study includes EV charging profiles for weekdays and weekends. It considers home and work-level charging scenarios. The RERs and EVs penetration ensure system reliability while minimizing generation costs and satisfying power flow constraints. Compared to existing algorithms, the OOPSO algorithm achieves superior performance, faster convergence, and higher solution accuracy. The simulation results show the effectiveness of OOPSO. It appears to be a robust tool for POPF, integrating RERs in power systems with EVs. This work highlights the potential of OOPSO as a feasible approach for power system optimization in the context of RERs and EV integration, paving the way for further research into unconventional grid optimization techniques.

Fast Algorithms for \(\ell_p\)-Regression

The ℓ p -norm regression problem is a classic problem in optimization with wide ranging applications in machine learning and theoretical computer science. The goal is to compute \(\boldsymbol {\mathit {x}}^{\star } =\arg \min _{\boldsymbol {\mathit {A}}\boldsymbol {\mathit {x}}=\boldsymbol {\mathit {b}}}\Vert \boldsymbol {\mathit {x}}\Vert _p^p \) , where \(\boldsymbol {\mathit {x}}^{\star }\in \mathbb {R}^n,\boldsymbol {\mathit {A}}\in \mathbb {R}^{d\times n},\boldsymbol {\mathit {b}} \in \mathbb {R}^d \) and d ≤ n . Efficient high-accuracy algorithms for the problem have been challenging both in theory and practice and the state-of-the-art algorithms require \(poly(p)\cdot n^{\frac{1}{2}-\frac{1}{p}} \) linear system solves for p ≥ 2. In this paper, we provide new algorithms for ℓ p -regression (and a more general formulation of the problem) that obtain a high-accuracy solution in O ( pn ( p − 2)/(3 p − 2) ) linear system solves. We further propose a new inverse maintenance procedure that speeds-up our algorithm to \(\widetilde{O}(n^{\omega }) \) total runtime, where O ( n ω ) denotes the running time for multiplying n × n matrices. Additionally, we give the first Iteratively Reweighted Least Squares (IRLS) algorithm that is guaranteed to converge to an optimum in a few iterations. Our IRLS algorithm has shown exceptional practical performance, beating the currently available implementations in MATLAB/CVX by 10-50x.

Semi-analytical vibration modeling of complex axisymmetric shells using shifted Legendre series

This paper presents a semi-analytical approach based on the shifted Legendre series for vibration modeling of complex axisymmetric shells. This method divides the structure along its generator into complex axisymmetric shells comprised solely of conical and spherical shell segments. A global cylindrical coordinate system is established to unify the coordinate system of complex axisymmetric shells. Coupling between segments and the simulation of arbitrary boundary conditions are achieved using artificial spring technology. Based on the first-order shear deformation theory (FSDT), the energy functional of each segment is derived. The circumferential displacement is expanded using Fourier series, while the meridional displacement is expanded using the shifted Legendre series at shifted Gauss-Lobatto nodes. Utilizing the inner product matrix and differential matrix of the shifted Legendre series, the integration and differentiation processes in the vibration characteristic equation are simplified into a product form, significantly enhancing computational efficiency. Through convergence analysis and comparison of the computational results of the present method with numerical simulation results, experimental results and literature results demonstrate the advantages of high convergence, high accuracy and fast solution speed of this solution method. On this basis, the vibration characteristics of ring-stiffened double-layer hemispherical shells are investigated. The results demonstrate the great applicability of this method for vibration modeling of complex axisymmetric shells in engineering.

Dynamic analysis of planetary gear transmission based on Lagrange interpolation polynomials

Abstract This paper proposes a novel dynamic model considering manufacturing errors and eccentricity errors to analyze the dynamics of planetary gear transmission (PGT). The dynamic model is established based on the fractional-order calculus (FOC) and solved by an enhanced fourth-order Lagrange interpolation polynomials (LIP) method. Three numerical examples and the vibration experiments of planetary gear transmission are employed for verification. The comparison results indicate that the proposed solution method has higher solution accuracy and efficient than the existing algorithms in solving fractional equations, and the relative errors of the proposed solution method in three examples are 0.32%, 0.78% and 0.16%, respectively. The proposed dynamic model of PGT has better agreement with the experimentally measured signal compared with the integer-order dynamic model, and the maximum error and average error of the characteristic frequency amplitude between the proposed dynamic model and the measured signal are 4.76% and 3.57%, respectively. The proposed method contributes to the theoretical foundation for the signal monitoring of PGT.