Semi-discretization Method Research Articles

A Semi-Discretization Method Based on Finite Difference and Differential Transform Methods to Solve the Time-Fractional Telegraph Equation

The telegraph equation is a hyperbolic partial differential equation that has many applications in symmetric and asymmetric problems. In this paper, the solution of the time-fractional telegraph equation is obtained using a hybrid method. The numerical simulation is performed based on a combination of the finite difference and differential transform methods, such that at first, the equation is semi-discretized along the spatial ordinate, and then the resulting system of ordinary differential equations is solved using the fractional differential transform method. This hybrid technique is tested for some prominent linear and nonlinear examples. It is very simple and has a very small computation time; also, the obtained results demonstrate that the exact solutions are exactly symmetric with approximate solutions. The results of our scheme are compared with the two-dimensional differential transform method. The numerical results show that the proposed method is more accurate and effective than the two-dimensional fractional differential transform technique. Also, the implementation process of this method is very simple, so its computer programming is very fast.

An efficient numerical method for the distributed order time-fractional diffusion equation with error analysis and stability

This article proposes a numerical method to find the numerical solutions of the time-fractional diffusion equations involving fractional distributed order operator of Caputo type. Using the finite difference approach, we solve these equations by applying the semi-discrete method regarding the time variable and the fully-discrete method regarding the spatial variable. For the distributed integral part with respect to time, the Gauss–Legendre quadrature formula is applied and to estimate the multi-term time-fractional operator, including the Caputo fractional derivative, the L2 -1 approach is utilized. In addition, the error analysis and stability of the proposed numerical method are studied in this work. Finally, some numerical examples are provided to demonstrate the accuracy and efficiency of the suggested method. These examples are compared to several numerical previous methods stated in the articles, and the results show that the accuracy of our method is superior to these methods.

Prediction of Chatter Stability in Bull-Nose End Milling of Thin-Walled Cylindrical Parts Using Layered Cutting Force Coefficients

During the bull-nose end milling operations of thin-walled structures, chatter usually occurs and adversely affects cutter performance, finished surface quality, and production efficiency. To accurately predict chatter stability, a suitable dynamic model with effective system parameters is required. In this article, a three-degree-of-freedom (3-DOF) dynamic model is developed to analyze the milling stability of the thin-walled cylinders, in which the dynamics of the bull-nose end mill along the x-axis and y-axis directions and the dynamic of the workpiece along the z-axis direction are taken into account. Then, the cutter-workpiece engagement (CWE) is extracted by employing a slice-intersection-based approach. And the layered cutting force coefficients are identified by considering the influences of varying cutter diameters on the cutting speed. Thereafter, the semi-discretization method (SDM) is adopted to compute the stability lobe diagram (SLD). In the end, a group of milling tests are carried out on a thin-walled cylinder to validate the accuracy and reliability of the proposed model, and the results show that the model predictions agree well with the experimental data.

Lyapunov-based robust optimal control for time-delay systems with application in milling process

In this paper, we propose an optimal delay-independent robust controller based on the Lyapunov–Krakovskii theorem for the milling process as a time-delay system in the presence of varying axial depth of cut and different parametric uncertainty, such as stiffness, damping, etc. Milling is widely used in the manufacturing processes for the production of complex-shaped workpieces with high accuracy. However, chatter is a self-excited vibration that may cause adverse effects, such as tool damage, poor surface quality, excessive noise, etc. The dynamic model of the milling process is considered a two-dimensional time-delay system. A nonlinear programming with linear matrix inequality constraints is solved in order to obtain the controller gain where the objective function is corresponding to the norm-2 of controller gain, and the constraints guarantee robust stability. Using the semi-discretization method, stability lobes are shown in both uncontrolled and controlled plants to illustrate the improvement of the stable region via the proposed controller. Bifurcation phenomena have been improved with this controller by postponing the adverse effects to the higher values of the axial depth of cut, reducing the amplitude of limit cycles, and changing the type of bifurcation. Finally, we will compare the proposed controller with an intelligent controller in order to show the efficiency of the proposed method. It is shown that the proposed controller has improved the integral absolute error index by about 3.65 times compared with the intelligent controller.

Stability Analysis of a One Degree of Freedom Robot Model With Sampled Digital Acceleration Feedback Controller in Turning and Milling

Abstract This study is interested in the stability of robots in machining. The goal is to improve the dynamic performance of robots using an additional acceleration signal fed back through the conventional built-in proportional-derivative controller provided by the manufacturer. The structure of the robot is modelled with a simple one degree-of-freedom lumped model and the control signals are fed back via a linear spring and damping. The time delays of the feedback controllers are considered as zero-order holds, which results in sawtooth-like time-periodic time delays. The resulting equation of motion is an advanced delay differential equation. The semidiscretization method is shown for such systems having multiple sampled digital delays and continuous delays. First, we establish the stable regions in the plane of the sampling delay and the gain of the acceleration signal without machining. Then, we show the possibility to improve stability in turning and milling using the additional acceleration feedback controller compared to the cases without any controller or using only the built-in proportional-derivative controller.

Stability and Neimark–Sacker Bifurcation of a Delay Difference Equation

In this paper, we revisit a delay differential equation. By using the semidiscretization method, we derive its discrete model. We mainly deeply dig out a Neimark–Sacker bifurcation of the discrete model. Namely, some results for the existence and stability of Neimark–Sacker bifurcation are derived by using the center manifold theorem and bifurcation theory. Some numerical simulations are also given to validate the existence of the Neimark–Sacker bifurcation derived.

A whole discretization method for milling stability prediction considering the discrete vibration velocities

The discretization methods have been widely used for chatter stability prediction and can be divided into two categories: semi-discretization methods (SDMs) by discretizing the time-delayed displacements, and full-discretization methods (FDMs) by discretizing both the present and time-delayed displacements. In this paper, a whole discretization method is proposed to achieve a faster and more accurate prediction for the milling stability. In addition to the discrete vibration displacements, the discrete vibration velocities are included in the solution by rebuilding the integrated matrix. The prediction error caused by the separate discretization of the cutting force coefficient matrix and vibration state variables can be avoided. The milling stability is determined with the discrete map of the state based on the Floquet theory. The proposed model is compared with the zeroth-order SDM and the first-order FDM which also approximate the solution with two neighboring discrete state variables. It is shown that the local discretization error based on the proposed method achieves o((Δt)5), which is three orders higher compared to the zeroth-order SDM and the first-order FDM. In addition, when the same prediction errors are required, the computational time of the proposed method is 11–14% of the zeroth-order SDM, and 24–27% of the first-order FDM. The results prove that considering vibration velocity can increase the efficiency and accuracy of the chatter stability prediction compared to existing SDM and FDM algorithms.

Chatter stability of robotic rotary ultrasonic countersinking

In recent years, industrial robots have received extensive attention in manufacturing field due to their high flexibility and great workspace. However, the weak stiffness of industrial robots makes it extremely easy to arouse chatter, which affects machining quality inevitably and generates noise pollution in severe cases. Compared with drilling, the chatter mechanism of robotic countersinking is more complex. The external excitation changes with cutting width and depth in countersinking. This characteristic results in time-varying and nonlinearity of robotic countersinking dynamics. Thus, it is urgent to propose a new method of chatter suppression and provide an accurate stability analysis model. As a new special machining technology, rotary ultrasonic machining has been proved to improve robotic drilling and milling stability effectively. Based on this, robotic rotary ultrasonic countersinking (RRUC) is proposed to improve the robotic countersinking stability in this paper. A three-dimensional stability domain method of RRUC is established. First, the countersinking process was divided into ρ parts. The dynamic model of every unit was constructed based on ultrasonic function angle (γ) and dynamic chip area. Then, the stability region of RRUC is obtained based on the semi-discrete method (SDM). Compared with the robotic conventional countersinking (RCC), RRUC improves the stability by 27%. Finally, the correctness and effectiveness of the stability region model are proved by robotic ultrasonic countersinking experiments.

Discontinuous Galerkin methods for stochastic Maxwell equations with multiplicative noise

In this paper we propose and analyze finite element discontinuous Galerkin methods for the one- and two-dimensional stochastic Maxwell equations with multiplicative noise. The discrete energy law of the semi-discrete DG methods were studied. Optimal error estimate of the semi-discrete method is obtained for the one-dimensional case, and the two-dimensional case on both rectangular meshes and triangular meshes under certain mesh assumptions. Strong Taylor 2.0 scheme is used as the temporal discretization. Both one- and two-dimensional numerical results are presented to validate the theoretical analysis results.

A hybrid finite element–finite volume method for conservation laws

We propose an arbitrarily high-order accurate numerical method for conservation laws that is based on a continuous approximation of the solution. The degrees of freedom are point values at cell interfaces and moments of the solution inside the cell. To lowest (3rd) order this method reduces to the Active Flux method. The update of the moments is achieved immediately by integrating the conservation law over the cell, integrating by parts and employing the continuity across cell interfaces. We propose two ways how the point values can be updated in time: either by first deriving a semi-discrete method that uses a finite-difference-type formula to approximate the spatial derivative, and integrating this method e.g. with a Runge-Kutta scheme, or by using a characteristics-based update, which is inspired by the original (fully discrete) Active Flux method. We analyze stability and accuracy of the resulting methods.

Existence of a Generalized Solution for the Fractional Contact Problem

In this paper, we take into consideration the mathematical analysis of time-dependent quasistatic processes involving the contact between a solid body and an extremely rigid structure, referred to as a foundation. It is assumed that the constitutive law is fractional long-memory viscoelastic. The contact is considered to be bilateral and is modeled around Tresca’s law. We establish the existence of the generalized solution’s result. The proof is supported by the surjectivity of the multivalued maximum monotone operator, Rothe’s semidiscretization method, and arguments for evolutionary variational inequality.

A Parameter Optimization Method for Chatter Stability in Five-Axis Milling

Chatter is a dynamic instability caused by the regeneration of waviness on the workpiece surface and damages the machining efficiency and product quality. For five-axis milling, the chatter stability analysis is more complicated because the cutter-workpiece engagement and the direction of the dynamic cutting force vary along the tool path. This paper presents a parameter optimization method to avoid chatter in the five-axis milling process. The dynamic milling system is modeled in each tool-path segment, and the stability is analyzed by introducing the semi-discretization method to solve the delay differential equation. In addition, the multi-frequency solution to the stability prediction for the multiple degrees of freedom spindle-cutter system and workpiece system is presented. Considering the speed and acceleration constraints of the spindle system and the feeding system, the spindle speed optimization iterative method is applied based on the chatter prediction. The verification experiment is conducted on the five-axis milling of the S-curve part to show the predictive accuracy of the chatter model and the validity of the proposed optimization method.

Stability analysis of a digital hierarchical steering controller of autonomous vehicles with multiple time delays

This study investigates the lane-keeping control of autonomous vehicles with an emphasis on the digital delayed nature of the system. The vehicle dynamics are represented using a kinematic bicycle model, and a hierarchical lane-keeping controller is introduced with multiple delays in the feedback loop. An extension of the semidiscretization method is presented, in order to perform the stability analysis of the digitally controlled vehicle with multiple discrete time delays. The differences between the continuous approximation and the exact consideration of discrete time delays are highlighted. We show that in certain cases, neglecting the effects of quantization can lead to significant inaccuracies, especially when tuning the lower-level controller. The results are verified using a series of small-scale laboratory experiments.

Period-doubling bifurcation and Neimark-Sacker bifurcation of a discrete predator-prey model with Allee effect and cannibalism

<abstract><p>In this paper, a discrete predator-prey model incorporating Allee effect and cannibalism is derived from its continuous version by semidiscretization method. Not only the existence and local stability of fixed points of the discret system are investigated, but more important, the sufficient conditions for the occurrence of its period-doubling bifurcation and Neimark-Sacker bifurcation are obtained using the center manifold theorem and local bifurcation theory. Finally some numerical simulations are given to illustrate the existence of Neimark-Sacker bifurcation. The outcome of the study reveals that this discrete system undergoes various bifurcations including period-doubling bifurcation and Neimark-Sacker bifurcation.</p></abstract>

Modeling and Stability Analysis of Robot-assisted Thin-walled Milling

Thin-walled machining has played an important role in automotive and aerospace domains. Due to physical characteristics of the flexible structure, the thin-walled machining is prone to the deformation and the onset of chatter. To solve the problem, a robot is considered to support the flexible workpiece while cutting. This paper studies finite element dynamic modeling and stability analysis of the robot-assisted thin-walled milling system. It is divided into three submodels: thin-walled workpiece, robot, and cutting process. The finite element method (FEM) is adopted to obtain a realistic dynamic model of the workpiece and allows analysis of any location. The robot dynamics is linearized to a 1-degree-of-freedom (DOF) mass-spring-damper system for model reduction. The cutting force variation and the regenerative effects involved in the cutting process establish time-periodic and time delay terms in the milling model. The semi-discretization method (SDM) is used for the stability analysis. Simulation results show a significant improvement in the milling stability from the robot support in any location, leading to higher productivity.

Bifurcation analysis in a discrete predator–prey model with herd behaviour and group defense

<abstract><p>In this paper, we utilize the semi-discretization method to construct a discrete model from a continuous predator-prey model with herd behaviour and group defense. Specifically, some new results for the transcritical bifurcation, the period-doubling bifurcation, and the Neimark-Sacker bifurcation are derived by using the center manifold theorem and bifurcation theory. Novelty includes a smooth transition from individual behaviour (low number of prey) to herd behaviour (large number of prey). Our results not only formulate simpler forms for the existence conditions of these bifurcations, but also clearly present the conditions for the direction and stability of the bifurcated closed orbits. Numerical simulations are also given to illustrate the existence of the derived Neimark-Sacker bifurcation.</p></abstract>

Flip bifurcation and Neimark-Sacker bifurcation in a discrete predator-prey model with Michaelis-Menten functional response

<abstract><p>In this paper, we use a semi-discretization method to explore a predator-prey model with Michaelis-Menten functional response. Firstly, we investigate the local stability of fixed points. Then, by using the center manifold theorem and bifurcation theory, we demonstrate that the system experiences a flip bifurcation and a Neimark-Sacker bifurcation at a fixed point when one of the parameters goes through its critical value. To illustrate our results, numerical simulations, which include maximum Lyapunov exponents, fractal dimensions and phase portraits, are also presented.</p></abstract>

Prediction method of cutting stability of roadheader based on the Newton–Lagrange mixed discrete method

In order to study the influence of the multiple interaction of coal and rock and the velocity effect of the cutting head on the flutter stability of the cantilever roadheader, considering the physical characteristics of coal and rock, the structural parameters of the cutting head and the motion parameters of the cutting head, the functional expression of the cutting depth of the cutting head and the participating cutting teeth is fitted, the cutting state mechanical characteristic equation of the dynamic cutting of the cutting teeth is established, and based on the mapping relationship between the cutting teeth and the cutting head, a cutting dynamic model including multiple interactions between cutting head and coal and rock and the speed effect is constructed. An improved discrete method based on Newton–Lagrange mixed interpolation is proposed, and the influence law of the coupling effect of regeneration effect and velocity effect on the stability of cutting flutter under cutting state is clarified. The improved full discrete method is compared with the full discrete method and the semi-discrete method, and the superiority of the improved full discrete method based on the mixed interpolation method is proved. Based on the improved total discrete method, the influence of different cutting system dynamic parameters on the stability is studied. A cutting head coal rock system is built to simulate the flutter of the cutting system. The results show that the improved fully discrete method can reasonably predict the actual cutting state.

Complicate Dynamics of a Discrete Predator-prey Model with Competition Between Predators

We consider a discrete predator-prey model with competition between predators in this paper. By simplifying the corresponding continuous predator-prey model, and using the semidiscretization method to obtain a new discrete model, we discuss the existence and local stability of nonnegative fixed points of the new discrete model. What’s more important, by using the bifurcation theory, we derive the sufficient conditions for the occurrences of Neimark-Sacker bifurcation and the stability of closed orbit bifurcated. Finally, the numerical simulation are presented, which not only illustrate the existence of NeimarkSacker bifurcation but also reveal some new dynamic phenomena of this model.

Fluid transient analysis by the method of characteristics using an object-oriented simulation tool

Abstract This paper presents the development of an efficient, interactive and fully extendable computer code to analyse hydraulic transients – water hammer – using a generic object-oriented modelling and simulation (OOMS) tool to simplify and shorten the development process. OOMS tools provide continuous simulation features by means of ordinary differential equation (ODE) and differential algebraic equation (DAE) solvers, and discrete simulation features by means of event handling algorithms. Previous OOMS tool applications to simulate fluid flow in pipe networks primarily used semi-discrete methods with continuous solvers. A novel aspect of this work is the application of the method of characteristics (MOC) using the discrete simulation features of the OOMS tool. When compared to a code developed with an OOMS tool using a semi-discrete method and to commercial software that uses the MOC, the new code shows much higher accuracy and performance than the former and is similar to the latter in accuracy and calculation time. It has a graphical user interface, and its modularity makes it easily extendable with new components and algorithms. An academic version is available on github.