Discrete Formula Research Articles

Recurrent neural dynamics for handling linear equation system with rank-deficient coefficient and disturbance existence

In this paper, for handling discrete-form time-variant linear equation system (DF-TV-LES) with rank-deficient coefficient and disturbance existence, a luminous discrete-time recurrent neural dynamics (DTRND) method is presented. Firstly, the continuous-time recurrent neural dynamics (CTRND) method can be discretized to the DTRND method by using recently-developed 5-instant discretization formula. Secondly, aiming at the situations of rank-deficient coefficient and disturbance existence, corresponding handling methods are presented, respectively. Specifically, on the one hand, under the situation of rank-deficient coefficient, we present an effective method to compute the least-squares solution of DF-TV-LES; on the other hand, under the situation of disturbance existence, integral state of error function is introduced, and then the presented DTRND method possesses a certain performance for restraining different types of disturbances. Finally, comparative numerical experiment substantiates the superiority of the presented DTRND method for handling DF-TV-LES.

Discrete-time ZNN-based noise-handling ten-instant algorithm solving Yang-Baxter-like matrix equation with disturbances

Time-variant Yang-Baxter-like matrix equation (YBLME) with the disturbances of noises is the hotspot in various scientific disciplines. The existing research, either can not handle the noise, or rely on the build-in numerical algorithm provided by MATLAB. Therefore, it is necessary to further study the digital-computer discrete solution of this problem. In this paper, the authors propose a noise-handling ten-instant discrete solution for the time-variant YBLME. By defining the matrix-form error function, Zhang neural network (ZNN) is exploited to design the continuous-time noise-handling ZNN model. For potential digital hardware (i.e., digital computer) realization, a discrete-time ZNN-based noise-handling ten-instant (DTZNH10i) algorithm, which is capable of handling three types of noises, is proposed on the basis of a Zhang time discretization (ZTD) formula. The convergence and precision of the proposed DTZNH10i algorithm are investigated and discussed. Theoretical analyses show the correctness and effectiveness of the proposed algorithm. The paper offers three examples of time-variant YBLME with disturbances of constant, linear, and bounded random noises to illustrate the efficacy and convergence of the proposed algorithm.

Accurate small major hysteresis loops calculation by the Preisach model with improved parametric distribution function

The inversed Preisach hysteresis model is widely used to realize the rapid coupling of hysteresis characteristics of core materials in finite element analysis. However, the small major hysteresis loop is difficult to be accurately simulated. In this paper, an improved inversed Preisach model based on an improved parametric distribution function is proposed. By analyzing the magnetic remanence, the waveform factor is coupled with the parametric distribution function, and the identification of the small major hysteresis loop is strengthened. The distribution function parameters can be found from the limit hysteresis loop through the optimization algorithm. In the numerical calculation process of the model, the amplitude correction coefficient in the discrete formula is extracted by the basic magnetization curve. Compared with the previous method, this method simulates the limit hysteresis loop well and has higher simulation accuracy for the small major hysteresis loops.

A Quantum Weak Signal Detection Method for Strengthening Target Signal Features under Strong White Gaussian Noise

As the noise power increases, the target signal features become less obvious, which leads to the failure of weak signal detection methods. To address this problem, a quantum weak signal detection method, Local Semi-Classical Signal Analysis-Singular Value Decomposition (LSCSA-SVD), for strengthening target signal features under strong white Gaussian noise is proposed. Firstly, the time domain weak signal is quantized by the Schrodinger operator and its discrete spectrum formula. Then, in the quantum domain, the later eigenvalues are used to reconstruct the time domain signal, which can protect and enhance the target signal features. Finally, the difference between signal and noise in the singular value vector is used to further extract the reconstruction signal features. In simulation, the LSCSA-SVD can accurately extract target signals from white Gaussian noise signals with a signal-to-noise ratio (SNR) of −30 dB, which is better than the comparison methods. In the experiment, the weak acceleration sensor signal and the weak signal of the test circuit are successfully extracted. The results show that the LSCSA-SVD can suppress strong noise and improve the SNR.

Block Trigonometrically Fitted Backward Differentiation Formula for the Initial Value Problem with Oscillating Solutions

This paper presents a fitted backward differentiation formula (BTBDF) whose coefficients are functions of a fixed fitting frequency and step length especially designed for the numerical integration of first-order Initial Value Problems (IVPs) with oscillatory results. The BTBDF is a product of three discrete formulas which are obtained from a continuous second derivative trigonometrically fitted method (CSDTFM). The BTBDF is applied in a block-wise form that makes it enjoys the advantages of the self-starting formula. The convergence of the BTBDF is discussed and its superiority is demonstrated in some numerical experiments to illustrate accuracy advantage.

Об одном обобщении дискретной формулы Родрига для многочленов Мейкснера

Изучается обобщение многочленов Мейкснера, приводящее к новой конструкции приближений Апери. В терминах алгебраических функций получено предельное распределение нулей масштабированных многочленов. Это распределение является решением некоторой векторной задачи равновесия теории логарифмического потенциала. Библиография: 21 название.

A generalization of the discrete Rodrigues formula for Meixner polynomials

A generalization of Meixner polynomials leading to a new construction of Apéry approximations is put forward. The limiting distribution of the zeros of scaled polynomials is described in terms of algebraic functions. The resulting distribution is shown to be a solution of some vector equilibrium problem in the theory of logarithmic potential. Bibliography: 21 titles.

Higher-order deep solver of non-linear PDEs implied by a non-linear discrete Clark-Ocone formula

Higher-order deep solver of non-linear PDEs implied by a non-linear discrete Clark-Ocone formula

Theoretical Analyses on Discrete Formulae of Directional Differentials in the Finite Point Method

Theoretical Analyses on Discrete Formulae of Directional Differentials in the Finite Point Method

An efficient alternating direction implicit finite difference scheme for the three-dimensional time-fractional telegraph equation

In this work, an efficient alternating direction implicit (ADI) finite difference scheme is proposed to solve the three-dimensional time-fractional telegraph equation. The fully-discrete scheme is established via the L1 discrete formula in temporal direction and finite difference method in spatial direction. In order to reduce the computational cost of solving three-dimensional problems, an ADI algorithm is designed and applied. Then, via the energy method, we prove the stability and convergence of the scheme in L2 and H1 norms, respectively. Eventually, several numerical examples are given to validate the theoretical results.

Extending a new two-grid waveform relaxation on a spatial finite element discretization

In this work, a new two-grid method presented for the elliptic partial differential equations is generalized to the time-dependent linear parabolic partial differential equations. The new two-grid waveform relaxation method uses the numerical method of lines, replacing any spatial derivative by a discrete formula, obtained here by the finite element method. A convergence analysis in terms of the spectral radius of the corresponding two-grid waveform relaxation operator is also developed. Moreover, the efficiency of the presented method and its analysis are tested, applying the twodimensional heat equation.

A virtual-electric-field-based method for distributing sampling points on the free-form surfaces of parts

The sampling process used for on-machine inspections is one of the key factors affecting the efficiency and quality of such inspections. For high-precision contact measurements in particular, a good distribution of sampling points can reduce the time spent on the inspection process and improve the reliability of the test results. However, in practice, most production and test equipment can only provide simple measurement methods and insufficient test functions, resulting in reduced automation of the test process and poor-quality test results when testing complex part surfaces. In this paper, we propose a sampling point distribution method for on-machine inspection based on a virtual electric field, to solve the common problem of unreasonable inspection methods and the need to improve efficiency and quality when inspecting free-form surfaces during on-machine inspection. First, the discrete curvature of each node in the grid model is calculated according to Gauss–Bonnet’s discrete curvature formula, which is used as the original data for the subsequent analysis process. The corresponding spatial search sphere is then constructed in the space where the model is located, and based on this, we propose a search method for extreme curvatures on the model surface. A virtual-electric-field model of the part surface is then constructed, and the searched curvature extremes are used to determine the positive charge of an electrostatic field on the model’s surface. The sampling points are regarded as charged particles, and the repulsion equation between sampling points is given. We then analyze the motion law of the sampling points in the electronic potential field within the sampling region, and also give the step length of each motion iteration process. Finally, the adaptive distribution of the sampling points is experimentally verified using several mechanical models with complex surfaces. The results show that the sampling distribution method proposed in this paper outperforms some traditional or commonly used distribution methods in terms of fitting accuracy.

Effect of frequency of minute-amplitude oscillatory shear loadings on ultimate yield stress of cohesive sediments

The ultimate yield stress of cohesive sediments changes with oscillatory loading or vibration. Given a strong enough vibration, the sediment may be fluidized. In this study, the effect of the frequency of minute-amplitude oscillatory shear loadings on the ultimate yield stress of cohesive sediments was experimentally and analytically investigated in a laboratory. The test samples were prepared with cohesive sediments with a median particle size of 31 μm and at four water content (31.8% to 37.03%). The sediments were subjected to continuous oscillatory shear loadings with a constant frequency (0–131.6 Hz). The findings of the study indicated that the ultimate yield stress of cohesive sediments under minute-amplitude oscillatory shear loadings was dependent on the oscillatory shear frequency, water content, and the ultimate yield stress of the sediments prior to the application of the oscillatory shear loading. This study revealed that there exists a critical shear loading frequency fcr. When the oscillatory shear frequency (f) was less than the critical shear frequency (fcr), the ultimate yield stress decreased as f increased. However, when f was greater than fcr, the ultimate yield stress was constant, irrespective of further increase of the oscillatory shear frequency. The former condition is identified as the non-equilibrium fluidization stage and the latter, the equilibrium fluidization stage. The fcr value decreased as the water content increased. In the equilibrium fluidization stage, the dimensionless yield stress Su/Su0 did not vary with the water content or oscillatory shear frequency. Two discrete formulae were proposed to estimate the ultimate yield stress of cohesive sediments subjected to minute-amplitude oscillatory shear loadings for the non-equilibrium and equilibrium fluidization stages. To explore the engineering applications of the formula for the equilibrium fluidization stage, a series of vertical pull out tests were performed using a dynamic torpedo anchor. This study provides an important method for distinguishing the degree of fluidization and sheds lights on potential applications in coastal and marine engineering, such as pile or anchor installation and extraction and scour protection in cohesive bed.

Inverse-Free Discrete ZNN Models Solving for Future Matrix Pseudoinverse via Combination of Extrapolation and ZeaD Formulas.

Time-varying matrix pseudoinverse (TVMP) problem has been investigated by many researchers in recent years, but a new class of matrix termed Zhang matrix has been found and not been handled by some conventional models, e.g., Getz-Marsden dynamic model. On the other way, future matrix pseudoinverse (FMP), as a more challenging and intractable discrete-time problem, deserves more attention due to its significant role-playing on some engineering applications, such as redundant manipulator. Based on the zeroing neural network (ZNN), this article concentrates on designing new discrete ZNN models appropriately for computing the FMPs of all matrices of full rank, including the Zhang matrix. First, an inverse-free continuous ZNN model for computing TVMP is derived. Subsequently, Zhang et al. discretization (ZeaD) formulas and equidistant extrapolation formulas are used to discretize the continuous ZNN model to two discrete ZNN models for computing FMPs with different truncation errors. The numerical experiments are conducted for the five conventional discrete models and two new discrete ZNN models. Distinct numerical results substantiate the effectiveness and choiceness of newly proposed models. Finally, one of the newly proposed models is implemented on simulating and physical instances of robot manipulators, respectively, to show its practicability.

Constitutive model for solid-like, liquid-like, and gas-like phases of granular media and their numerical implementation

Granular media, a macroscopic amorphous matter composed of mesoscopic discrete particles, exhibits different behaviour characteristics (e.g., solid-, liquid-, and gas-like phases) depending on the volume fraction and loading conditions. Therefore, it poses a significant challenge to three-phase modelling and analysis. In this study, a description of multiple phases in granular media is realised by establishing the transition criteria and parameter conversion relations between different phases, using the already established theories of the elastic-plastic constitutive relation in solid mechanics, the viscoplastic constitutive relation in rheology, and the kinetic theory of granular flow. The new model realises the coexistence of different phases and the conversions therebetween. Smoothed discrete particle hydrodynamics are introduced to discretise the new theoretical model, and the correspondence between the smoothed and actual particles for different phases and discrete formulas under different models are described in detail. Two typical cases are used for numerical tests: the collapse of one side of a three-dimensional granular column and the collapse and flow of a granular column on an inclined surface. The calculated morphology of granular flow, distributions of velocity vectors, scaling law of spreading range under different height-width ratios, and the transient characteristics of particle spreading all accord well with the experimental results. Furthermore, quasi-static, slow, fast, and other phases of particle motions are well captured. Thus, we verify the feasibility of the new model and method for the multiple-phase simulation of granular media.

Relationship between time-instant number and precision of ZeaD formulas with proofs

Recently, zeroing dynamics (ZD) method has shown wonderful effect in solving time-varying problems. Generally, the continuous-time ZD model should be discretized as a discrete-time algorithm for numerical computation. A good choice is to apply Zhang et al. discretization (ZeaD) formulas, which are effective time-discretization formulas. Actually, ZeaD formulas can also be applied to obtaining various discrete-time algorithms, not only the ZD algorithms. In the previous work, some piecemeal ZeaD formulas have been proposed and investigated. However, the relationship between the time-instant number and precision of ZeaD formulas is not found, which is emphatically investigated in this paper. Specifically, the ZeaD formulas from two to nine instants are investigated, and the general ZeaD formula groups are studied. Two-instant and three-instant ZeaD formula groups have linear precision at most. Four-instant ZeaD formula group has quadratic precision at most. Five-instant and six-instant ZeaD formula groups have cubic precision at most. Seven-instant and eight-instant ZeaD formula groups have quartic precision at most. Nine-instant ZeaD formula group has quintic precision at most. Theoretical analyses are presented to substantiate the relationship. Moreover, the ZeaD formulas as well as ZD method are applied to solving time-varying quadratic optimization problem, and the numerical results verify the effectiveness of ZeaD formulas.

Unified Model Solving Nine Types of Time-Varying Problems in the Frame of Zeroing Neural Network.

Many time-varying problems have been solved using the zeroing neural network proposed by Zhang et al. In this article, nine types of time-varying problems, namely time-varying nonlinear equation system, time-varying linear equation system, time-varying convex nonlinear optimization under linear equalities, unconstrained time-varying convex nonlinear optimization, time-varying convex quadratic programming under linear equalities, unconstrained time-varying convex quadratic programming, time-varying nonlinear inequality system, time-varying linear inequality system, and time-varying division, are investigated to better understand the essence of zeroing neutral network. Discrete-form time-varying problems are studied by considering the nature of unknown future and the requirement of real-time computation for time-varying problems. A unified model is proposed in the frame of zeroing neural network to uniformly solve these time-varying problems on the basis of their connections and a newly developed discretization formula. Theoretical analyses and numerical experiments, including the tracking control of PUMA560 robot manipulator, verify the effectiveness and precision of the proposed unified model.

A new well-balanced finite-volume scheme on unstructured triangular grids for two-dimensional two-layer shallow water flows with wet-dry fronts

In this study, a novel two-dimensional finite-volume method on unstructured triangular meshes for two-layer shallow water flows is developed. First, the one-dimensional relaxation approach developed in Abgrall and Karni (2009) [1] is extended to the currently studied two-dimensional two-layer shallow water equations to build an unconditionally hyperbolic system. Next, a two-dimensional finite-volume HLL scheme on triangular grids for such a non-conservative hyperbolic system is constructed. Special attention is paid to guaranteeing the well-balanced property even in the presence of the wet-dry fronts. To this end, techniques such as spatial reconstruction for the secondary variables, special cell classifications for the two-layer fluid system, and special local reconstruction of auxiliary surface variables are proposed. These special techniques are further combined with new discrete formulas for the nonconservative products and the bed-slope terms, and it thus leads to an exactly well-balanced numerical scheme for the studied two-layer system even in the presence of wet-dry fronts. The developed numerical model is second order accurate and preserves the nonnegative layer depths. The performances of the proposed numerical model are investigated by several numerical experiments.

Efficient difference method for time-space fractional diffusion equation with Robin fractional derivative boundary condition

In this paper, a numerical method is proposed to solve the time-space fractional diffusion equation with Robin fractional derivative boundary condition. Under the weak regularity assumptions of solution, we present a numerical scheme based on the L1 method for time discretization on graded mesh and the Grunwald-Letnikov formula for spatial discretization on uniform mesh. And a fast scheme for the considered problem is constructed based on the exponential summation approximation of the kernel function t−α. Meanwhile, a detailed analysis of stability and convergence is given. Then, the extrapolation method is applied to the space direction to make it reach the second-order accuracy. Finally, numerical experiments show that the proposed method is effective.

Nonlinear conjugate gradient method for identifying Young's modulus of the elasticity imaging inverse problem

Application of elasticity imaging inverse problem to identify Young's modulus in the elasticity problems in human's life is an interesting research area. In this study, we identify the modulus of elasticity for solving elasticity imaging inverse problem using a modified output least-squares method. Numerical convergence in the displacements of the direct problem for elasticity is investigated. To study the elasticity imaging inverse problem in an optimization framework, we utilize the sensitivity and adjoint problems to conceptualize a new model for computing the gradient of the minimizer. Discrete formulae in the model are then used to devise a scheme for an efficient computation gradient of the modified output least-squares objective function using the nonlinear conjugate gradient method. Numerical experiments demonstrate the effectiveness of the proposed technique.