Semi-discretization Method Research Articles

Dissipative spatial discretization of a phase field model of multiphase multicomponent isothermal fluid flow

This work is devoted to the development of a dissipative spatial discretization of a phase field model describing the dynamics of a multicomponent multiphase isothermal viscous compressible fluid with a resolved interface and effects associated with it (surface tension, wetting). Mass densities of mixture components are used as order parameters. The total Helmholtz free energy incorporates wall free energy providing wetting effects. The model under consideration is preliminary regularized, which allows one to increase time step in explicit time-marching methods. A finite-difference semi-discrete method is proposed and a discrete counterpart of the dissipativity theorem is proven. All the statements remain valid in the absence of regularization as well. 3D simulations of two-phase two-component and three-phase three-component fluids using explicit Euler method are performed. The simulation results confirm theoretical predictions.

Investigation on Cotes-formula-based prediction method and its experimental verification of milling stability

Abstract Regenerative chatter is the main factor affecting the stability of milling process. The reasonable prediction of milling stability lobe diagram (SLD) is the core to controlling chatter and improving production efficiency. Therefore, a milling system with consideration of the regenerative effect is formulized as the delay differential equation (DDE). By uniformly discretizing the part of the forced vibration in the tooth passing period, the state transition matrix can be achieved by sophisticatedly adopting the Cotes formula for the fit value of the DDE at every discretized point. And then, the judgment criterion of milling stability that the eigenvalues of the state transition matrix are not more than 1 is suggested by using Floquet theory. Compared with the first-order semi-discretization method (SDM), the first-order full-discretization method (FDM) and second-order FDM, the results show that the proposed method has an advantage over the first-order SDM and the FDM whether in the computational accuracy or the computational efficiency. Finally, some group of milling experiments were done to validate the proposed method. The morphology of machined surface shows the predicted SLD results are highly consistent with the experimental values. Therefore, the proposed method has a good industrial application value in determining the optimal chatter-free cutting conditions.

Stability of Milling Process with Variable Spindle Speed Using Runge–Kutta-Based Complete Method

The variable-spindle-speed (VSS) technique is effective in preventing regenerative chatter in milling processes. However, spindle-speed-modulation parameters should be deliberately selected to augment the material removal rate. Stability-prediction algorithms of stability predicting play an important role in this respect, as they allow the prediction of stability for all ranges of a given spindle speed. The increase in calculation time in variable-spindle-speed milling, which is caused by the modulation frequency, hinders its practical use in the workshop. In this paper, a Runge–Kutta-based complete discretization method (RKCDM) is presented to predict the stability of milling with variable spindle speeds, which is described by a set of delay differential equations (DDEs) with time-periodic coefficients and time-varying delay. The convergence and calculation efficiency are compared with those of the semidiscretization method (SDM) under different testing configurations and milling conditions. Results show that RKCDM is more accurate and saves at least 50% of the calculation time of SDM. The effects of modulation parameters on the stability of VSS milling are explored through stability lobe diagrams produced from RKCDM.

Stability Analysis of Milling Process with Variable Spindle Speed and Pitch Angle considering Helix Angle and Process Phase Difference

Suppression of milling chatter by disrupting regenerative effect is a well-known method to obtain higher cutting stability domain. In this paper, a dynamic model of the milling process with variable spindle speed and pitch angle considering helix angle and process phase difference is presented. Then, an updated semidiscretization method is applied to obtain the stability chart. After the effectiveness of the proposed method is confirmed by comparisons with the previously published works and the time-domain simulations, lots of analyses are conducted to deeply evaluate the influence of the helix angle, the process phase difference, and feed per tooth on milling stability. Results show that the change of helix angle can result in significant stability discrepancies for both high-speed and low-speed regions. Though the process phase difference has the randomness and immeasurability in the practical application, it has an important influence on the stability and will result in a periodic evolution of the stability with a period π. Also, its recommended values are given for the practical milling process.

Stability and Neimark–Sacker bifurcation for a discrete Nicholson's blowflies model with proportional delay

In this paper, we revisit a Nicholson's blowflies model with proportional delay, the stability and bifurcation of whose discrete version have not been studied. By using the method of semidiscretization, we dig deeply out the stability and Neimark–Sacker bifurcation of its discrete model. Especially, some results for the existence and stability of Neimark–Sacker bifurcation are derived by using the centre manifold theorem and bifurcation theory. Numerical simulations are also formulated to verify the existence of Neimark–Sacker bifurcation derived.

A posteriori choice of time-discretization step in finite difference methods for solving ill-posed Cauchy problems in Hilbert space

Abstract Finite difference semidiscretization methods for solving an ill-posed Cauchy problem in a Hilbert space are investigated. The problems involve linear positively definite selfadjoint operators. We justify an a posteriori scheme for the choice of the time-discretization step and establish accuracy estimates in terms of the error level of input data.

Higher‐order algorithms for stable solutions of fractional time‐dependent nonlinear telegraph equations in space

AbstractIn this article, three novel algorithms are developed and successfully applied to investigate the stable solutions of time‐fractional nonlinear telegraph equations. Firstly, we presented the shifted Gegenbauer polynomials through appropriate transformations. The approximation of a function u(||x||, t) is defined via shifted Gegenbauer polynomials (SGPs) and then developed the operational matrices of a positive integer and non‐integer derivatives assisted by multi‐dimensional shifted Gegenbauer polynomials vectors. Linearized fully spectral method (LFSM) is proposed by means of function approximations and developed operational matrices whilst Picard iterative scheme is used to linearize the discussed problem. To introduce a semi‐discrete method (SDM), the temporal derivative is approximated through a central difference scheme whereas spatial derivatives are approximated assisted by novel operational matrices. A fully numerical method (FNM) is proposed by introducing the central difference for temporal and spatial variables. Proposed computational methods transform the time‐fractional nonlinear telegraph equation into a system of linear algebraic equations that are easier to tackle. An inclusive comparative study is witnesses that suggested schemes are effective, accurate and well‐matched to investigate the approximate solutions of the problems udnerstudy. Convergence and error bound of the suggested methods are investigated theoretically whereas stability analysis is performed numerically. Moreover, the developed methods can be used conveniently to examine the numerical solution of other multi‐dimensional highly nonlinear fractional or variable order problems of physical nature.

Approximating the solution of the discharging process in a domestic hot water storage tank

ABSTRACT In this article a new approximation procedure for the temperature during the discharging process in a domestic hot water storage tank is developed. The main idea of this procedure is the mathematical description of the dynamic behaviour of a boundary layer that appears inside the storage tank. This leads to an approximate solution of a simple structure but a high accuracy throughout the whole discharging process for the temperature distribution inside the storage tank. For validation the approximate solution is compared with the exact solution that is constructed with help of generalized Fourier coefficients and a numerical solution that is obtained by a semi-discretization method. Adapting the method leads also to an approximation procedure for charging processes.

Convergence of a Second-order Energy-decaying Method for the Viscous Rotating Shallow Water Equation

An implicit energy-decaying modified Crank--Nicolson time-stepping method is constructed for the viscous rotating shallow water equation on the plane. Existence, uniqueness, and convergence of semidiscrete solutions are proved by using Schaefer's fixed point theorem and $H^2$ estimates of the discretized hyperbolic--parabolic system. For practical computation, the semidiscrete method is further discretized in space, resulting in a fully discrete energy-decaying finite element scheme. A fixed-point iterative method is proposed for solving the nonlinear algebraic system. The numerical results show that the proposed method requires only a few iterations to achieve the desired accuracy, with second-order convergence in time, and preserves energy decay well.

Point control of a differential-difference system with distributed parameters on the graph

The article considers the problem of point control of the differential-difference equation with distributed parameters on the graph in the class of summable functions. The differential- difference system is closely related to the evolutionary differential system and moreover the properties of the differential system are preserved. This connection is established by the universal method of semi-discretization in a time variable for a differential system, which provides an effective tool in order to find conditions for unique solvability and continuity on the initial data for the differential-difference system. For this differential-difference system, a special case of the optimal control problem is studied: the problem of point control action on the controlled differential-difference system is considered by the control, concentrated at all internal nodes of the graph. At the same time, the restrictive set of permissible controls is set by the means of conditions depending on the nature of the applied tasks. In this case, the controls are concentrated at the end points of the edges adjacent to each inner node of the graph. This is a characteristic feature of the study presented, quite often used in practice when building a mechanism for managing the processes of transportation of different kinds of masses over network media. The study essentially uses the conjugate state of the system and the conjugate system for a differential-difference system — obtained ratios that determine optimal point control. The obtained results underlie the analysis of optimal control problems for differential systems with distributed parameters on the graph, which have interesting analogies with multi-phase problems of multidimensional hydrodynamics.

Continuum model based chatter stability prediction for highly flexible parts in turning process with accurate dynamic force modeling

Stability analysis is rather important in the turning process for achieving a high machining efficiency under the condition of without compromising the surface integrity. However, the occurrence of chatter is not simply attributed to the cutter or the part, and sometimes may be alternating. In this paper, a new chatter stability model is built for the turning process of a typical step shaft with the consideration of the 2-dimensional (2D) compliance of tool and part. Among the model, an accurate dynamic force modeling method is first presented, into which the tool nose radius, the size effect, and the effect of vibrations in chip thickness and width directions are incorporated simultaneously. Then, the position-dependent vibration equations of the 2D compliant turning system are formulated based on the continuous Timoshenko beam theory, and position-dependent modal parameters are subsequently derived by using a matrix recursion algorithm. Further, the dynamic equations for the 2D tool-workpiece compliant system are solved with an extended second-order semi-discretization method. Finally, the simulations and experiments are conducted to verify the built model. The results of stability prediction show a good agreement with the experimental results no matter the flexibility of the cutter or the part plays a dominant role in the turning process.

Chatter stability analysis of variable speed milling with helix angled cutters

The number of factors considered in the establishing of chatter model for regenerative chatter would directly affect the accuracy of numerical analysis. In this paper, considering the helix angle of cutter, the establishment method of three degrees of freedom complex model is investigated synthetically. An improved semi-discretization method is adopted for the prediction of milling stability of variable speed milling with helix angle. The influences of milling cutter helix angle, radial immersion rate and variable speed modulation coefficient on chatter suppression in milling process are studied. The results show that the selection of milling cutters with different helix angles has a certain influence on the overall chatter stability, especially the effect cannot be ignored under local special machining conditions. Excessive radial immersion rate will increase the probability of chatter. Compared with constant speed milling, the chatter can be suppressed obviously by variable speed milling, and the selection of variable speed modulation coefficient will directly affect the suppression effect of chatter.

Measurement-Based Modal Analysis and Stability Prediction on Turn-Milling of Hollow Turbine Blade

Hollow blades with honeycomb structures are increasingly used in the turbine engines for reducing weight and saving costs. The hollow blade is a typical thin-walled structural part with low stiffness, the machining system of which is often unstable and likely to chatter. The most effective solution to avoid the machining chatter is to guide the hollow blade to be machined in a stable machining zone. This paper proposes a measurement-based approach for modal analysis and stability prediction of turn-milling hollow blade. The impact test was carried out to achieve the FRF curves on the hollow blade and the milling tool. An extremum method was employed to obtain an equivalent FRF curve, from which the modal parameters involving the natural frequency, damping ratio, and stiffness were computed. Afterwards, the semidiscretization method was used to draw a stability lobe diagram to predict the stability when turn-milling hollow blades. The experimental results confirm the feasibility of the predicted stability lobe diagram.

A novel harmonic solution for chatter stability of time periodic systems

Chatter vibrations strongly limit productivity in milling. Due to the presence of rotating parts with asymmetric stiffness and stability enhancement strategies which act through a periodic variation of stiffness, there is growing interest in estimating the stability maps of systems with Linear Time Periodic dynamics together with periodic cutting excitation. Applying Exponentially Periodic Modulated test signals to the dynamic cutting force equation and representing the dynamics of the system through the Harmonic Transfer Function, the innovative Harmonic Solution (HS) and its zero-order approximation were derived in this research. HS is a frequency domain representation of a system described by the combination of two independent periodicities. It is possible to take into account these periodicities together in HS or singularly, resulting in the Zero Order HS or in the well-known Multi-Frequency Solution. This novel formulation can deal efficiently with spindle dependent and independent dynamics and is prone to industrial applications due to its flexibility and efficiency. More specifically, in this work the developed methodologies were used to assess the cutting stability of systems with a periodically modulated stiffness. The accuracy and efficiency of HS were validated by comparison with the results achieved by the use of the semi-discretization method. Results are in agreement with those obtained using semi-discretization. Moreover, admitting a slight precision loss, HS and its zero-order approximation are orders of magnitude faster than semi-discretization, giving reliable stability maps from seconds to a few minutes.

H(div) conforming methods for the rotation form of the incompressible fluid equations

New H(div) conforming finite element methods for incompressible flows are designed that involve the rotation form of the equations of motion and the Bernoulli function. With a specific choice of numerical fluxes, we recover the same velocity field as in Guzman et al. (IMA J Numer Anal 37(4):1733–1771, 2016) for the incompressible Euler equation in the convection form. Error estimates are presented for the semi-discrete method. We further study the incompressible Navier-Stokes equation with the full version of the stress tensor $$\nu \left( \nabla \varvec{u}+ \nabla \varvec{u}^T - \frac{2}{3} \left( \nabla \cdot \varvec{u}\right) \mathbb {I} \right)$$ , instead of partially enforcing the divergence free constraint at the continuous level (as is commonly done in finite element methods), we let the numerical scheme to fully control the enforcement of this constraint. Finally, we test the behavior of the proposed methods with some numerical simulations. Our results show that (1) We recover the same velocity field in Guzman et al. (2016), (2) When H(div) conforming with BDM-DG elements, we achieve less errors in the velocity compared with Schroeder et al. (SeMA J 75(4):629–653, 2018) when polynomial order $$p\in \{2,3\}$$ , (3) When H1 conforming with Taylor-Hood elements, the use of full stress tensor helps to reduce errors in both the velocity and the Bernoulli function, (4) H(div) conforming method does a better job in long time structure preservation compared with the classical mixed method even with the grad-div stabilization.

Generalized Numerical Differentiation Method for Stability Calculation of Periodic Delayed Differential Equation: Application for Variable Pitch Cutter in Milling

This paper proposes a generalized numerical differentiation method for stability prediction of the non-autonomous delayed differential equations (DDEs) with periodic coefficients and discrete delays. Firstly, the periodic DDE is described in state-space form and the period of a system is equally discretized. Then, the discrete first derivatives versus time are approximated by a linear combination of the state function values at multiple neighboring sampling grid points based on the finite-difference formulas. Such that, the original DDE is approximated as a series of algebraic equations and the Floquet transition matrix can be constructed on one period. At last, the system stability is determined according to the Floquet theory by checking the eigenvalues. The delayed damped Mathieu equation is regarded as a typical case to verify the effectiveness and efficiency of the presented method. The stability diagrams and rate of convergence are computed in comparison with those via the benchmark algorithm (the semi-discretization method). As an application, the presented method is used to predict the stability of milling with variable pitch cutter, and the computational result agrees well with the experimentally verified example.

Three degrees of freedom chatter stability prediction in the milling process

Chatter in the cutting process has a great influence on workpiece surface quality, machining efficiency, and service life of the machine tool. This work presents three degrees of freedom (DOF) dynamic model applied to tool chatter for thin-walled structures in milling. Combined with dynamic cutting force modelling in the radial, axial and tangential directions, the stability chart was solved by adopting an improved semi-discretization method in time domain. And the stability prediction model considering regenerative effect was validated by cutting tests with specific axial depth of cut and spindle speed. The cutting experimental results performed at different axial depth of cuts were in accordance with predicted stability lobe diagram (SLD). Moreover, the effects of cutting contact angle and radial immersion ratio on SLD were also analyzed. The study results show that the dynamic model could make a good prediction of chatter stability, and the established approach is high efficient in predicting the chatter stability lobe considering the actual situation.

Chebyshev pseudospectral method in the reconstruction of orthotropic conductivity

In this paper, we present a method to reconstruct the spatially varying conductivity tensor in isotropic and orthotropic materials, involved in a two-dimensional transient anisotropic model with Robin boundary conditions. For the reconstruction, the partial differential equation is solved by a semi-discrete method that combines a pseudospectral collocation method for spatial variables and Crank–Nicolson for time. The conductivity tensor is reconstructed through a non-linear least-squares problem solved by Levenberg–Marquardt method (LMM), along with Morozov's discrepancy principle as stopping rule to cope with noise in the data. Unlike classic LMM implementations that mitigate poor conditioning in calculating iterates using nonsingular diagonal scaling matrices, in this paper, singular regularization matrices are used. The impact of such a modification is illustrated with numerical experiments using discrete differential operators as scaling matrices. Numerical results show that accurate conductivity values can be obtained using a fairly small number of discretization points at a very low computational cost.

Stochastic semidiscretization method: Second moment stability analysis of linear stochastic periodic dynamical systems with delays

In this paper, an efficient numerical approach is presented, which allows the analysis of the moment dynamics, stability, and stationary behavior of linear periodic stochastic delay differential equations. The method leads to a high dimensional stochastic mapping with periodic statistical properties, from which the periodic first and second moment mappings are derived. The application of the method is demonstrated first through the analysis of the stochastic delay Mathieu equation. Then a practical case study, where the effect of spindle speed variation on the stability and the resulting surface quality of turning operations is investigated.

Investigation of milling stability based on a cutting force segmentation model

The chatter vibration is the major limiting factor for higher metal removal rates in milling process. This paper proposed an integral model of cutting force with helical cutter considering the sectional characteristics of cutting force. Based on the model, the variation law of the upper and lower limit angles of the cutting process under different radial cutting depths is analysed. The results show that the shape of cutting force for each tooth has a characteristic of three sections. Moreover, the process dynamic equation of thin-walled parts cutting is constructed based on the model. And, the stability lobe diagram is obtained by semi-discretization method. The time-domain simulations are carried out in different cutting parameters around the stable island to verify the correctness of the prediction method. The results show that the method has good ability to predict the cutting stability of cutters with helix angle.