Variable-step Method Research Articles

Energy dissipation laws of time filtered BDF methods up to fourth-order for the molecular beam epitaxial equation

This report presents the time filtered BDF-k (FiBDF-k) methods up to fourth-order time accuracy for the molecular beam epitaxial equation with no-slope selection. The new (k+1)-order methods are developed by introducing an inexpensive post-filtering step to the variable-step BDF-k(k=1,2,3) methods. We show that the FiBDF-k methods are uniquely solvable and volume conservative. Some novel discrete gradient structures of the FiBDF-k formulas are derived such that we can build up the discrete energy dissipation laws for the associated time-steppings. Numerical examples are included to show the mesh-robustness and effectiveness of the variable-step methods especially in the long-time simulations.

Suppression of voltage gain fluctuation in LLC Resonant Converter Based on chaotic spread spectrum

In order to realize the voltage stable output of the LLC resonant converter under different loads, a voltage gain fluctuation suppression technology of the LLC resonant converter based on a chaotic spread spectrum is proposed. The gain fluctuation electromagnetic interference signal is eliminated by the Chua’s circuit; The rated voltage value is discretized, and the target network and the evaluation network are used to form a deep Q network, with the multi-threshold judgment method and the variable step method respectively completed the gain fluctuation suppression target under different working conditions. The experimental results show that the proposed technology can achieve stable voltage output in both dynamic and steady-state conditions, with good gain fluctuation suppression effect and strong timeliness. It provides data reference for the rational application of the LLC resonant converter.

A variable step size least mean p-power adaptive filtering algorithm based on multi-moment error fusion

For the channel estimation problem under α-stable distributed impulse interference, the traditional fixed-step adaptive filtering cannot satisfy the fast convergence speed and low steady-state error at the same time, whereas the variable-step method is able to effectively solve this contradiction. This paper proposes an improved variable step-size least mean p-power adaptive algorithm that offers good robustness against impulsive noise. The proposed algorithm takes into account the linkage between the errors and uses the adjustment of the step size based on the errors of the current moment and the previous k moments, thus overcoming the problems of poor anti-noise performance and large steady-state fluctuations of the fixed-step size algorithm. This algorithm ensures that the step size does not change abruptly when the system is disturbed by impulse noise and can achieve a lower steady-state error. The simulation results show that the algorithm has better convergence than the traditional fixed step-size algorithm and the existing variable step-size algorithm under the interference of impulsive noise.

Encountering singularities of a serial robot along continuous paths at high precision

A new variable step (VS) method rapidly calculates continuous kinematic paths that encounter singularities of a serial robot. Mitigating adverse effects of such encounters by applying small deviations to the intended path achieves high precision. The method is motivated by a simplified geometric model representing the arm extension, overhead and wrist singularities found in a wrist-separable 6-axis robot. A variable step length limits the kinematic closure error in a high-order combined predictor-corrector. Squaring the determinant of the robot Jacobian matrix gives its Taylor series favorable convergence. The resulting estimate of the distance to the nearest singularity affecting convergence of other kinematic variables bounds the step length, avoiding jumps between branches of path bifurcations associated with robot singularities. Finally, a 2nd order bivariate expansion of the squared determinant targets a minimum determinant magnitude by controlling deviation from an intended path. This process obviates the need for matrix regularization giving unintended switches between solution branches. The VS method demonstrates the ability to deviate from an intended path through dead center of any of the above singularities in isolation, pairwise and three-fold combination.

Application of the Complex Variable Step Method and the Boundary Element Method for Sensitivity Analysis of Steady Heat Conduction Problems

The paper concerns the problems related to applying the complex variable step method for the sensitivity analysis of the steady temperature field in the solid body domain due to the perturbations of the geometrical and physical parameters. The optimization problem using the approach proposed is also discussed. At the stage of numerical modelling, the boundary element method is used. The first part of the paper is devoted to the shape sensitivity. The results obtained are compared with the solution resulting from the implicit approach of sensitivity analysis. In the second part, the practical problem concerning optimizing the geometry of continuous casting mould cross-section is considered. The project variable vector contains the cooling pipes' radius and the volume flow rate of the cooling water. The numerical results and the conclusions are presented in the final part of the paper.

Analysis of the variable step method of Dahlquist, Liniger and Nevanlinna for fluid flow

AbstractThe one‐leg, two‐step time discretization proposed by Dahlquist, Liniger and Nevanlinna is second order and variable step G‐stable. G‐stability for systems of ordinary differential equations (ODEs) corrresponds to unconditional, long time energy stability when applied to the Navier–Stokes equations (NSEs). In this report, we analyze the method of Dahlquist, Liniger and Nevanlinna as a variable step, time discretization of the Navier–Stokes equations. We prove that the kinetic energy is bounded for variable time‐steps, show that the method is second‐order accurate, characterize its numerical dissipation and prove error estimates. The theoretical results are illustrated by several numerical tests.

METHOD OF COMPROMISE CHOICE OF INTERVAL OF MODEL TIME CHANGE IN IMITATION MODEL

The process of simulation is considered as one of the main means for studying the dynamics of functioning of real systems, in particular, complex ones. The system can be represented by a set of components. The functioning of a component is represented by the implementation of a set of functional actions, which are represented by the corresponding activities in the form of a pair: algorithm for performing a functional action - duration of execution. The problem of displaying the simultaneous or parallel nature of the functioning of all components of a complex system in the MI is solved by introducing model or system time. The main methods for introducing model time are the fixed step method and the variable step method. In the fixed-step method, an important problem is the choice of the value of the model time variation interval. The existing recommendations for choosing the value of the interval for changing the model time are of a qualitative nature and their use makes it possible to increase the accuracy of modeling, but at the same time the consumption of the computer time resource increases. It is proposed to use quantitative estimates of the values of quality criteria - the accuracy and expenditure of the computer time resource when choosing the value of the model time change interval. The generalized criterion is represented as a weighted sum of transformations of local criteria. The values of the coefficients by which the corresponding transformations are multiplied express the preferences of the decision-maker for the local optimality criteria. A geometric interpretation of the process of determining a compromise alternative on a set of effective alternatives for various cases of the importance of local criteria is given. These estimates make it possible to substantiate the nature of the change in the quality criteria for various variants of the values of the interval of change in the model time and to use them to select a compromise option among the effective ones by minimizing the generalized criterion. The choice of the compromise value of the model time variation interval is implemented in the control simulation program.

A parallel-in-time algorithm for variable step multistep methods

This paper presents a multigrid reduction in time (MGRIT) algorithm for achieving time parallelism using multistep backward difference formula (BDF) methods on variably-spaced temporal grids. This MGRIT approach transforms the linear multistep methods into single step methods applied to groups of time steps. Stability considerations are addressed through lowering of the order on coarse grids. The methods are presented for fixed and variable time step formulations. Numerical results show moderate speedups for the heat equation as well as two IEEE power grid test problems characterized by nonlinear differential algebraic equations (DAE) of index 1. The performance of MGRIT with BDF is compared to MGRIT with Runge-Kutta for both the fixed and variable step methods of the same order.

A Predictor-Corrector Method for Power System Variable Step Numerical Simulation

This letter proposes a predictor-corrector method to strike a balance between simulation accuracy and efficiency by appropriately tuning the numerical integration step length of a power system time-domain simulation. Numerical tests indicate that, by estimating the truncation error for step length tuning based on the 2-Step Adams-Moulton method and the implicit Trapezoidal method, the proposed method can provide much more precise results at little cost of efficiency compared to a conventional variable step method based on Newton's method.

The Parameter Identification of PMSM Based on Improved Cuckoo Algorithm

In view of the multi-parameter identification problem of permanent magnet synchronous motor, a kind of parameter identification method was proposed based on an improved cuckoo search algorithm. Cuckoo search algorithm has the advantages of simple, less parameters, fast convergence etc., but it also has the defects of premature convergence and low computation accuracy. In view of the deficiency of cuckoo search algorithm, the fuzzy reasoning based on the cloud membership was designed to adjust the probability of an alien egg being discovered by host birds and adaptive variable step method was used to adjust the step size of Levy flights. The improved algorithm can accelerate the convergence speed and improve the local and global optimizing ability by increasing the diversity of the population. The multi-parameter identification results of permanent magnet synchronous motor show that the improved cuckoo algorithm can effectively identify the motor parameters, and compared with the traditional cuckoo algorithm, the effectiveness and superior performance are tested.

Structured random forest for label distribution learning

Abstract Label distribution learning (LDL) has proven effective in many machine learning applications. Previous LDL methods have focused on learning a non-linear conditional probability mass function by maximizing entropy or minimizing the Kullback–Leibler (K-L) divergence. In order to make full use of the structural information among different classes, a method called structured random forest (StructRF) regression is used which has been applied to semantic image labeling and edge detection. It is a general LDL model that treats the distribution as an integral whole. In StructRF, all label distributions are mapped to a discrete space at each split node in a random forest. In this way, standard information gain measures can be evaluated. Then the predicted distribution can be obtained directly without calculating the probability of each class individually during the test. StructRF is proved to be fast in training and it reaches higher accuracies and lower standard deviations among different measurements. Besides, we propose an adaptive variable step method that can speed up the training process and reduce the calculations of information gain significantly. It is suitable for the most decision tree based models.

Stochastic resonance in a time-delayed feedback tristable system and its application in fault diagnosis

Stochastic resonance (SR) phenomena in a time-delayed feedback tristable system driven by Gaussian white noise are investigated by simulating the potential function, mean first-passage time (MFPT), and signal-to-noise ratio (SNR) of the system. Through the use of a short delay time, the generalized potential function and stationary probability density function (PDF) are obtained. The delay feedback term has a significant effect on both equations, and that the parameters b, c, and d have different effects on the three wells of the potential function. The MFPT is calculated, which plays an extremely important role in research on particles escape rates. We find that the delay feedback term can affect the noise enhanced stability (NES). In addition, the SR characteristics are studied by the index of SNR. The simulation demonstrates that SNR is a non-monotonic distributed and that the peak SNR value can be attained by adjusting the appropriate parameters. Finally, the proposed theory is combined with a variable step method and applied to the detection of high frequencies in experiments. The result indicates that the fault frequency can be identified, and that the energy of the fault signal can be enhanced under suitable delay feedback parameters.

New Optimization Method of the MPPT Algorithm and Balancing Voltage Control of the Three-Level Boost Converter (TLBC)

This paper is dedicated to studying the control of the three level Boost converters (TLBC) and the optimization method of maximum power point tracking (MPPT) based a variable step. The main objective of the optimization is to find a compromise between the response time and the amplitude of the oscillations around the optimal point. The nonlinear behavior of the (TLBC) is manifested by the presence of the disturbances. For reasons of simplicity of the control, a linearization based on the dynamic compensation of the disturbance is proposed. On the one hand, a cascaded MPPT algorithm and a simple linear regulator allow adjusting the inductance current and a maximum power operation of the wind system. On the other hand, a second linear regulator ensures balancing of the output voltages. The paper proposes a new approach to the optimization of the Inc-Cond MPPT. The suggested contribution consists of using an exponential function of the power derivative to develop a variable step. The adoption of the variable step size according to the dynamics of the wind system implies a compromise between the response time and the amplitude of the ripples around the optimal point. The simulation results showed that a variable step size, especially in transient conditions and during a very rapid climate change recover the optimum power point within a reasonable time and suitable amplitude of the oscillations. The results achieved in this study show the ability of the proposed approach to extract the maximum power according to the available wind speed while guaranteeing a better efficiency. The developed study is summarized by the following points: (a) modeling the wind conversion systems, (b) detailing the control approach of the (TLBC) and presenting the variable step method (c) presenting the simulations results and evaluating the perf

New Optimization Method of the MPPT Algorithm and Balancing Voltage Control of the Three-Level Boost Converter (TLBC)

This paper is dedicated to studying the control of the Three Level Boost Converters (TLBC) and the optimization method of Maximum Power Point Tracking (MPPT) based a variable step. The main objective of the optimization is to find a compromise between the response time and the amplitude of the oscillations around the optimal point. The nonlinear behavior of the TLBC is manifested by the presence of the disturbances. For reasons of simplicity of the control, a linearization based on the dynamic compensation of the disturbance is proposed. On the one hand, a cascaded MPPT algorithm and a simple linear regulator allow adjusting the inductance current and a maximum power operation of the wind system. On the other hand, a second linear regulator ensures balancing of the output voltages. The paper proposes a new approach to the optimization of the Inc-Cond MPPT. The suggested contribution consists of using an exponential function of the power derivative to develop a variable step. The adoption of the variable step size according to the dynamics of the wind system implies a compromise between the response time and the amplitude of the ripples around the optimal point. The simulation results showed that a variable step size, especially in transient conditions and during a very rapid climate change recover the optimum power point within a reasonable time and suitable amplitude of the oscillations. The results achieved in this study show the ability of the proposed approach to extract the maximum power according to the available wind speed while guaranteeing a better efficiency. The developed study is summarized by the following points: (a) modeling the wind conversion systems, (b) detailing the control approach of the TLBC and presenting the variable step method (c) presenting the simulations results and evaluating the perf.

Seam-based variable-step Bresenham blending method for real-time video mosaicking

Image blending plays an important role in video mosaicking, which has a high demand for real-time performance and visual quality. This paper proposes a fast blending method based on Bresenham algorithm, which realizes blending by controlling the storing addresses of source pixels. The starting storing location is accurately computed based on the coordinates of the middle pixel of the seam instead of the first pixel’s, reducing the accumulated error along the seam significantly. The other storing addresses are acquired using a variable-step Bresenham method, which takes advantage of burst mode operation of a dynamic memory and can achieve a good trade-off between the operation convenience and memory requirement. By the proposed method, complicated calculations of storing addresses are simplified into integer additions and subtractions, which is more suitable for hardware implementation. A hardware architecture based on field programmable gate array is presented to evaluate the proposed method with clock frequency analysis and resource assessment. The experimental results show that the proposed method achieves good performance of high image quality, low computational complexity, and low memory requirement.

A Novel Fractional Order Normalized LMS Algorithm with Direction Optimization

A generalized fractional order least mean squares (LMS) algorithm with direction optimization is proposed in this paper. Firstly, the LMS algorithm with 0 < α < 2 is investigated, and it is shown that the larger iteration order, the faster convergence speed can be achieved. Besides, to improve the convergence speed further, the direction optimization method is introduced for the fractional order case, which extracts an optimal direction in a given data block as the iteration direction. In addition, for decreasing the weight noise in steady state, a more general variable step method with more design freedom is developed. Finally, the effectiveness and applicability of the proposed method are demonstrated in three numerical examples.

Complex variable step method for sensitivity analysis of effective properties in multi-field micromechanics

This paper shows an approach to computing the effective properties of multi-field composite materials and their first-order sensitivities. The approach is based on the application of the complex variable step method for the micromechanical Mori–Tanaka scheme; hence, the first-order sensitivities can be computed in the same analysis. Numerical results are presented for magnetoelectroelastic properties of piezoelectric–piezomagnetic composite materials. A comparison of the results to those obtained by other methods shows that the presented sensitivity analysis gives highly accurate and stable results, but the values of the results are dependent on the applied micromechanical model. The presented approach may be used to solve ill-posed problems of optimal design or identification in coupled fields micromechanics.

Modeling and Solution of a Nonlinear Vibration Model of the Bending-Torsion Coupled Spur Gear Transmission System Based on Lumped Mass Method

After comprehensive considerations of the influences of stiffness excitations, deviation excitations, meshing impact excitations, friction of tooth surfaces, gap changes and other kinds of nonlinear factors, it established a nonlinear vibration model of the bending-torsion coupled spur gear driven system based on Lumped Mass Method (LMM). By transforming the model to dimensionless form and using fifth order adaptive variable step (Runge-Kutta) method to solve the nonlinear vibration model of the system, it can get the time domain charts, spectrum charts, phase charts, Poincare charts, FFT charts and the system bifurcation figures. And then it discussed the influence of system parameter vibrations on its dynamic characteristics, and this could provide a foundation for system dynamic optimization design

Modeling and Solving of Non-Linear Vibration Model of the Multi-Clearance and Bend and Torsion Coupled Gear-Driven System

Comprehensively taking the effects of variable rigidities, tooth flank clearances, bearing clearances, contact rigidities, and other multiple non-linear factors into account, it built a bend and torsion coupled gear-driven system’s dynamics analysis model in use of lumped mass method. After dimensionless dealing, it solved this dynamic model by the method of fifth order adaptive variable step (Runge-Kutta) method, and then it obtained system vibratory responses’ time domain diagrams, frequency domain diagrams, phase map, Poincare diagram, fast Fourier transformation (FFT) diagrams, and system branch diagrams under different parameters. All of them analyzed the effects of parameter variations on gearing system’s dynamic characteristics, and it provided a foundation for gearing system dynamic optimum designs.

Variable Step Perturbation MPPT Method Based on Fuzzy Control for Wind Power Generation System

Traditional MPPT control strategy by hill climbing searching for wind power generation system (WPGS) dependent on wind turbines parameters, the oscillation around maximum power point (MPP) was serious. A new variable step perturbation MPPT was presented based on fuzzy control, the variable perturb step was determined with the power characteristic curves point slope, the control system was described with fuzzy language for maximum power capture, the principle of MPPT was analyzed and the variable step method was adopted with fuzzy control, the simulation result shows that this advanced control strategy could improve tracking speed, enhance the stability and the oscillation around maximum power point reduced.