Discrete Equations Research Articles

Dynamical behavior of liquid-filled rotors with axial motion

In the present study, the dynamical behavior of liquid-filled rotors with axial motion is investigated. Based on the differential equations of motion for ideal fluids, the fluid forces exerted on the rotor are obtained. Then, the Euler–Bernoulli beam theory is utilized to establish the governing equations of motion for the rotor system. By applying Hamilton's principle, the governing equation and the corresponding boundary conditions are derived. Furthermore, in order to solve the eigenvalue problem of the system, the extended Galerkin method is applied to discrete equations of motion for the rotor system. As a result, the system stability, divergence, and flutter instability are defined. Moreover, a comparative study is presented to verify the accuracy of the proposed model. Finally, the effects of liquid parameters on the dynamic characteristics of the rotor system are investigated in detail. The results show that for liquid-filled rotors with axial motion, the stability of the system depends on the liquid parameters, axial velocity, and spinning velocity.

Discrete energy balance equation via a symplectic second-order method for two-phase flow in porous media

We propose and analyze a second-order partitioned time-stepping method for a two-phase flow problem in porous media. The algorithm is a refactorization of Cauchy's one-leg θ-method: the implicit backward Euler method on [tn,tn+θ], and a linear extrapolation on [tn+θ,tn+1]. In the backward Euler step, the decoupled equations are solved iteratively, with the iterations converging linearly. In the absence of the chain rule for time-discrete setting, we approximate the change in the free energy by the product of a second-order accurate discrete gradient (chemical potential) and the one-step increment of the state variables. Similar to the continuous case, we also prove a discrete Helmholtz free energy balance equation, without numerical dissipation. In the numerical tests we compare this symplectic implicit midpoint method (θ=1/2) with the classic backward Euler method, and two implicit-explicit time-lagging schemes. The midpoint method outperforms the other schemes in terms of rates of convergence, long-time behavior and energy approximation, for both small and large values of the time step.

Infiltration and permeability of porous ceramics simulated by random networks

A network computational model for a 3D ceramic structure is developed. The model is applied to study the impact of geometric and material parameters of structure on the liquid metal flow through random porous ceramic medium in pressure infiltration processes. The characteristic geometric features of the ceramic structure favorable for liquid metal flow during the infiltration process are determined. The method of structural approximation and constructive homogenization are applied, and the discrete stationary Stokes equations on random graphs are considered. This approach gives a robust algorithm to determine the macroscopic permeability K of interpenetrating phases. The dependencies of K on the distribution of connections (windows) between the cells (inclusions) are derived. The numerical simulations demonstrate that the permeability K does not depend on the scaled distribution sizes of windows. This implies that K is proportional to the mean value of the window areas. The considered model takes into account a random complex structure of 3D ceramic. Hence, it complements the previous study on the local transport properties of tubes (windows) connecting the cells.

Wave propagation on hexagonal lattices

Abstract We consider propagation of two-dimensional waves on the infinite hexagonal (honeycomb) lattice. Namely, we study the discrete Helmholtz equation in hexagonal lattices without and with a boundary. It is shown that for some configurations these problems can be equivalently reduced to similar problems for the triangular lattice. Based on this fact, new results are obtained for the existence and uniqueness of the solution in the case of the real wave number k ∈ ( 0 , 6 ) ∖ { 2 , 3 , 2 } {k\in(0,\sqrt{6})\setminus\{\sqrt{2},\sqrt{3},2\}} for the non-homogeneous Helmholtz equation in hexagonal lattices with no boundaries and the real wave number k ∈ ( 0 , 2 ) ∪ ( 2 , 6 ) {k\in(0,\sqrt{2})\cup(2,\sqrt{6})} for the exterior Dirichlet problem.

Space-Time FEM for the Vectorial Wave Equation under Consideration of Ohm’s Law

Abstract The ability to deal with complex geometries and to go to higher orders is the main advantage of space-time finite element methods. Therefore, we want to develop a solid background from which we can construct appropriate space-time methods. In this paper, we will treat time as another space direction, which is the main idea of space-time methods. First, we will briefly discuss how exactly the vectorial wave equation is derived from Maxwell’s equations in a space-time structure, taking into account Ohm’s law. Then we will derive a space-time variational formulation for the vectorial wave equation using different trial and test spaces. This paper has two main goals. First, we prove unique solvability for the resulting Galerkin–Petrov variational formulation. Second, we analyze the discrete equivalent of the equation in a tensor product and show conditional stability, i.e., under a CFL condition. Understanding the vectorial wave equation and the corresponding space-time finite element methods is crucial for improving the existing theory of Maxwell’s equations and paves the way to computations of more complicated electromagnetic problems.

A third (fourth)-order computational scheme for 2D Burgers-type nonlinear parabolic PDEs on a nonuniformly spaced grid network

An implicit compact scheme is proposed to approximate the solution of parabolic partial differential equations (PDEs) of Burgers’ type in two dimensions. These nonlinear PDEs are essential because they describe various mechanisms in engineering and physics. The nonlinear convective and advective processes are discretized with high-order accuracy on an arbitrary grid, which results in a family of high-resolution discrete replacements of given PDEs. The essence of the new scheme lies in its compact character and two-level single-cell discretization, so that one discrete equation leads to the accuracy of orders three or four, depending upon the choice of the grid network. The scheme is used for solving celebrated nonlinear PDEs, such as the nondegenerate convection–diffusion equation, the generalized Burgers–Huxley equation, the Buckley–Leverett equation, and the Burgers–Fisher equation. Many computational results are presented to demonstrate the high-resolution character of the newly proposed scheme.

Існування надзвукових періодичних біжучих хвиль в дискретних рівняннях типу Клейна-Ґордона з нелокальною взаємодією

The article is devoted to discrete Klein-Gordon type equations that describe infinite chains of nonlinear oscillators with nonlocal interactions. This implies that each oscillator interacts with several of its neighbors on both sides. The main result of the article concerns the existence of periodic traveling waves in such equations. Sufficient conditions for the existence of such waves were established using the variational method and the mountain pass theorem.

Linearly implicit time integration scheme of Lagrangian systems via quadratization of a nonlinear kinetic energy. Application to a rotating flexible piano hammer shank.

This paper presents a general and practical approach for nonlinear energy quadratization based on the Euler-Lagrange formulation of the physical equations. A Scalar Auxiliary Variable -like method based on a phase formulation of the equations is applied. The proposed scheme is linearly implicit, reproduces a discrete equivalent of the power balance. It is applied to a rotating and flexible piano hammer shank. An efficient solving strategy leads to a quasi explicit algorithm which shows quadratic space/time convergence.

A fresh perspective on solution to fragmentation problems

The population balances offer a continuum framework to model fragmentation of discrete size drops, bubbles, particles, crystals, polymer chains, etc. We have examined the available approaches to solve the resulting population balances and the older discrete kinetic equations. The analytical solutions have enhanced our understanding of breakup processes. We offer in this work new analytical solutions using a fresh and counter-intuitive approach. The approximate numerical solutions approach analytical solutions as the fineness of discretization increases. We looked for scaling patterns in coarse numerical solutions and harnessed the observed scaling to obtain closed-form analytical solutions in the limit of infinitely fine discretization. The approach provides analytical solutions for a family of breakup functions, including evolving populations of infinite number of particles. The success of the new approach is contingent on the discretization step having internal consistency concerning particle number and mass. The approach holds promise for empirical breakage functions.

Strength-induced Peridynamic model for the dynamic failure of porous materials

Predicting the dynamic failure process of porous materials is a challenging task due to their complex structure. To minimize the use of time-consuming peridynamic (PD) models and to avoid surface effect issues in the dynamic failure of complex porous materials, this paper proposes a strength-induced PD model. The paper first presents the dynamic formula and relevant finite element discrete equation of the coupled PD and classical continuum mechanics (PD-CCM) model based on the Morphing method. The Morphing function is implemented to control the material parameters and enable the free transformation of PD and CCM models. Based on the coupled PD-CCM model, the strength-induced PD model is established to adaptively expand the PD subdomain in porous materials by controlling the Morphing function value through the strength state of the structure. This model enables the PD subdomain to appear automatically when the porous materials reach the critical stress state. The proposed model accurately predicts the location of crack initiation and path while minimizing computational costs and improving efficiency. Three two-dimensional numerical examples are used to verify the effectiveness, efficiency, and accuracy of the model. The results of the simulation suggest that the location where the crack initiates in the porous materials is strongly influenced by the amplitude of the dynamic load. Cracking is dependent on the pores and typically occurs through them.

Momentum accelerates evolutionary dynamics

We combine momentum from machine learning with evolutionary dynamics, where momentum can be viewed as a simple mechanism of intergenerational memory similar to epigenetic mechanisms. Using information divergences as Lyapunov functions, we show that momentum accelerates the convergence of evolutionary dynamics including the continuous and discrete replicator equations and Euclidean gradient descent on populations. When evolutionarily stable states are present, these methods prove convergence for small learning rates or small momentum, and yield an analytic determination of the relative decrease in time to converge that agrees well with computations. The main results apply even when the evolutionary dynamic is not a gradient flow. We also show that momentum can alter the convergence properties of these dynamics, for example by breaking the cycling associated to the rock–paper–scissors landscape, leading to either convergence to the ordinarily non-absorbing equilibrium, or divergence, depending on the value and mechanism of momentum.

Gradient-Based Optimization Method for Experimental Modal Parameter Estimation with Finite Element Model

This paper presents a novel gradient-based optimization algorithm for improving the accuracy of experimentally estimated modal parameters with the assistance of finite element models. Initially, we recast the discrete vibration response equation into a matrix form and formulate the parameter estimation problem in modal analysis as an optimization problem. Then the problem is solved with a gradient-based iterative algorithm, which explicitly exhibits the closed form of gradients used in optimization. Initial values for this iteration are parameters derived from finite element models, since every important engineering structure should be analyzed with a finite element model before it is constructed. Subsequently, the performance of this algorithm is validated by both pure numerical experiments, which simulate the physical world, and experiments using real measurement data gathered by sensors in the real physical world. The algorithm’s performance is further enhanced by incorporating gradient clipping and an adaptive iteration threshold. As a comparison, a discussion on classical least-squares time-domain method for the problem is provided. For practical applications, the Shi–Tomasi corner detection and Lucas–Kanade optical flow methods are deployed to detect corner points from videos taken during the vibration of a structure and track the motion of these points in the videos.

Dynamics analysis and chatter control of a polishing and milling nonideal flexible manipulator

Some robots are designed to be lightweight and flexible, enabling them to access small and challenging paths in various applications. These features enable robots to collaborate with humans in performing specific production tasks. However,the movement of the flexible manipulator can become self-excited when handling a cutting tool on a workpiece, which can lead to a control problem. This article presents a control solution for lightweight robotic manipulators with rotating tools, such as polishing and milling, using smart actuators. The control discretization method is also introduced to facilitate integration into digital controllers. The paper starts by describing the governing equations of the non-ideal flexible manipulator for polishing and milling and analysing its dynamic behavior. Subsequently, models for the controllers of the DC motor-only actuators and the hybrid (shape memory alloy and DC motors) actuators were formulated using shape memory alloy and a suboptimal control scheme known as the discrete state-dependent Riccati equation. The coupled system was analysed dynamically, and it was observed that it exhibits chaotic behavior. The results suggest that incorporating smart actuators into the motor group of the system could decrease the positioning error of the manipulator and significantly reduce the oscillation of the robot’s end-effector.

Darboux-integrable Reductions of the Hirota–Miwa Type Discrete Equations

Darboux-integrable Reductions of the Hirota–Miwa Type Discrete Equations

Operator-splitting finite element method for solving the radiative transfer equation

AbstractAn operator-splitting finite element scheme for the time-dependent radiative transfer equation is presented in this paper. The streamline upwind Petrov-Galerkin finite element method and discontinuous Galerkin finite element method are used for the spatial-angular discretization of the radiative transfer equation, whereas the backward Euler scheme is used for temporal discretization. Error analysis of the proposed numerical scheme for the fully discrete radiative transfer equation is presented. The stability and convergence estimates for the fully discrete problem are derived. Moreover, an operator-splitting algorithm for the numerical simulation of high-dimensional equations is also presented. The validity of the derived estimates and implementation is illustrated with suitable numerical experiments.

The Dynamics of the Hubbard Model Through Stochastic Calculus and Girsanov Transformation

As a typical quantum many body problem, we consider the time evolution of density matrix elements in the Bose-Hubbard model. For an arbitrary initial state, these quantities can be obtained from an SDE or stochastic differential equation system. To this SDE system, a Girsanov transformation can be applied. This has the effect that all the information from the initial state moves into the drift part, into the mean field part, of the transformed system. In the large N limit with g=UN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g=UN$$\\end{document} fixed, the diffusive part of the transformed system vanishes and as a result, the exact quantum dynamics is given by an ODE system which turns out to be the time dependent discrete Gross Pitaevskii equation. For the two site Bose-Hubbard model, the GP equation reduces to the mathematical pendulum and the difference of expected number of particles at the two lattice sites is equal to the velocity of that pendulum which is either oscillatory or it can have rollovers which then corresponds to the self trapping or insulating phase. As a by-product, we also find an equivalence of the mathematical pendulum with a quartic double well potential. Collapse and revivals are a more subtle phenomenom, in order to see these the diffusive part of the SDE system or quantum corrections have to be taken into account. This can be done with an approximation and collapse and revivals can be reproduced, numerically and also through an analytic calculation. Since expectation values of Fresnel or Wiener diffusion processes, we write the density matrix elements exactly in this way, can be obtained from parabolic second order PDEs, we also obtain various exact PDE representations. The paper has been written with the goal to come up with an efficient calculation scheme for quantum many body systems and as such the formalism is generic and applies to arbitrary dimension, arbitrary hopping matrices and, with suitable adjustments, to fermionic models.

Coupled vibration analysis of the spacecraft with the flexible shaft and solar panels assembly

Dynamical characteristics of the spacecraft with flexible components are investigated in this paper. The component consists of a flexible shaft and solar panels, where solar panels are completely fixed on the shaft. A novel power series multiplier polynomial method is proposed to describe the connecting condition between the shaft and solar panels. The deformation and force matching conditions between the one dimensional shaft and two dimensional panels are established, which is the major contribution about the multidimensional combined structure of this study. According to constraining boundary conditions of the shaft and solar panels, the bending-torsion coupled deformation of the component is expressed. The attitude maneuvering of the central platform and the elastic vibration of the component cause the rigid-flexible coupled effect. Then the discrete dynamical equation of the spacecraft is obtained by using the rigid-flexible coupled modal shapes. The correctness of the proposed method is verified by comparing the natural characteristics with that of the finite element model. The relative error of the higher order frequency is not more than 4%. Numerical results show that the degree of the bending angle polynomial should be at least quadratic to ensure the accuracy of the calculation. The proposed power series multiplier polynomial breaks through the barriers that the traditional Lagrange multiplier cannot express the continuous constraint. It is an effective method to fit the constrained boundary by the polynomial rather than by the discrete multipoint method.

SUMMABILITY AND ASYMPTOTICS OF POSITIVE SOLUTIONS OF AN EQUATION OF WOLFF TYPE

SUMMABILITY AND ASYMPTOTICS OF POSITIVE SOLUTIONS OF AN EQUATION OF WOLFF TYPE

Non-abelian discrete Toda chains and related lattices

We have derived a non-abelian analog for the two-dimensional discrete Toda lattice which possesses solutions in terms of quasideterminants and admits Lax pairs of different forms. Its connection with non-abelian analogs for several well-known (1+1) and one-dimensional lattices is discussed. In particular, we consider a non-commutative analog of the scheme: discrete Toda equations → Somos-N sequences → discrete Painlevé equations.

Development of a fully implicit ODE-solver for containment analysis code

The thermal–hydraulic dynamics in containment are governed by a system of stiff ordinary differential equations (ODEs). A fully implicit discretization scheme is adopted to discretize these ODEs in order to mitigate the effects of stiffness. In comparison with explicit or semi-implicit discretization schemes that are subject to Courant limits on time steps, the fully implicit discretization scheme is more suitable for a containment analysis code that focuses on predicting both short-term and long-term thermal–hydraulic parameters after an accident. This study introduces a general-purpose ODE solver for the containment analysis code. The outline of the solver is as follows: The fully implicit discrete equations lead to a large set of nonlinear equations that need to be solved using Newton’s iterative method. The partial derivative components in the Jacobi matrix are calculated by the perturbation method using finite difference approximation, which avoids the complicated derivation of partial derivatives. The scaling modification technique is incorporated into this ODE solver to deal with significant differences in unknown variable magnitudes, and the line search method is introduced to address the difficulty of obtaining an accurate root estimate with Newton’s method when the initial guess is far from the actual root. This proposed ODE solver was applied to two typical stiff ODE problems to test its stiffness-suppressed ability and to demonstrate that this proposed solver can perform calculations with a very large time step. Then, the CASSIA code, a containment analysis code developed by China Nuclear Power Technology Research Institute Co., Ltd (CNPRI), equipped with this ODE solver, was applied to the CSNI (Committee on the Safety of Nuclear Installations) benchmark problem and the Carolinas Virginia Tube Reactor (CVTR) test 3 problem to preliminarily demonstrate that the proposed ODE solver can perform containment thermal–hydraulic analysis correctly. This study could provide references for the development of a home-made containment analysis code.