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In this paper, the shallow water wave problem is discussed in the Lagrangian description. By using the Hamilton variational principle in analytical mechanics, a displacement shallow water wave equation (DSWWE) is developed for the shallow water wave problem with a sloping water bottom and wet–dry interface. A numerical scheme based on the discretized Hamilton principle is constructed for solving the proposed displacement shallow water wave equation. The proposed numerical scheme is symplectic and explicit, and can preserve the total energy and mass of the shallow water system in the discrete sense. The correctness of the DSWWE and the effectiveness of the proposed numerical scheme are verified by using four classical numerical examples. Numerical examples show that the proposed method performs well with respect to the simulation of the shallow water problem with a sloping water bottom and wet–dry interface. [ABSTRACT FROM AUTHOR]

In this paper, the shallow water problem is discussed. By treating the incompressible condition as the constraint, a constrained Hamilton variational principle is presented for the shallow water problem. Based on the constrained Hamilton variational principle, a shallow water equation based on displacement and pressure (SWE-DP) is developed. A hybrid numerical method combining the finite element method for spatial discretization and the Zu-class method for time integration is created for the SWEDP. The correctness of the proposed SWE-DP is verified by numerical comparisons with two existing shallow water equations (SWEs). The effectiveness of the hybrid numerical method proposed for the SWE-DP is also verified by numerical experiments. Moreover, the numerical experiments demonstrate that the Zu-class method shows excellent performance with respect to simulating the long time evolution of the shallow water. [ABSTRACT FROM AUTHOR]

An algorithm based on the Paterson-Stockmeyer (PS) scheme and filtering technology is developed to compute the large matrix functions in dynamic problems accurately and efficiently. With the assistance of analysis on truncation error and error caused by filtering, the error growth law during the computation is studied, based on which an adaptive filtering threshold is proposed to help the proposed algorithm more efficiently achieve similar accuracy as the original PS scheme. Numerical examples, including 30 random matrices with different bandwidths, 10 adjacency matrices in the complex network dynamic problems, and a trampoline vibration problem, are given to verify the efficiency and accuracy of the proposed algorithm. Numerical results suggest that the proposed algorithm can achieve good accuracy and efficiency in computing the matrix function in the considered dynamic problems. [ABSTRACT FROM AUTHOR]

Purpose The purpose of this paper is to propose an accurate and efficient numerical method for determining the dynamic responses of a tensegrity structure consisting of bars, which can work under both compression and tension, and cables, which cannot work under compression.Design/methodology/approach An accurate time-domain solution is obtained by using the precise integration method when there is no cable slackening or tightening, and the Newton–Raphson scheme is used to determine the time at which the cables tighten or slacken.Findings Responses of a tensegrity structure under harmonic excitations are given to demonstrate the efficiency and accuracy of the proposed method. The validation shows that the proposed method has higher accuracy and computational efficiency than the Runge–Kutta method. Because the cables of the tensegrity structure might be tense or slack, its dynamic behaviors will exhibit stable periodicity, multi-periodicity, quasi-periodicity and chaos under different amplitudes and frequencies of excitation.Originality/value The steady state response of a tensegrity structure can be obtained efficiently and accurately by the proposed method. Based on bifurcation theory, the Poincaré section and phase space trajectory, multi-periodic vibration, quasi-periodic vibration and chaotic vibration of the tensegrity structures are predicted accurately. [ABSTRACT FROM AUTHOR]

Based on the finite element method (FEM), the parametric variational principle (PVP) is combined with a numerical time-domain integral method to simulate the dynamic behavior of the pantograph-catenary system. Based on PVP, formulations for the nonlinear droppers in the catenary and for the contact between the pantograph and the contact wire are proposed. The formulations can accurately determine the tension state or compression state of the nonlinear droppers and the contact state between the pantograph and the contact wire. Based on the periodicity of the catenary and the precise integration method (PIM), a numerical time-integration method is developed for the dynamic responses of the catenary. For this method, the matrix exponential of only one unit cell of the catenary is computed, which greatly improves the computational efficiency. Moreover, the validation shows that the formulations can compute the contact force accurately and represent the nonlinearity of the droppers, which demonstrates the accuracy and reliability of the proposed method. Finally, the dynamic behaviors of the pantograph-catenary system with different types of catenaries are simulated. [ABSTRACT FROM AUTHOR]

The numerical oscillation problem is a difficulty for the simulation of rapidly varying shallow water surfaces which are often caused by the unsmooth uneven bottom, the moving wet-dry interface, and so on. In this paper, an adaptive artificial viscosity (AAV) is proposed and combined with the displacement shallow water wave equation (DSWWE) to establish an effective model which can accurately predict the evolution of multiple shocks effected by the uneven bottom and the wet-dry interface. The effectiveness of the proposed AAV is first illustrated by using the steady-state solution and the small perturbation analysis. Then, the action mechanism of the AAV on the shallow water waves with the uneven bottom is explained by using the Fourier theory. It is shown that the AVV can suppress the wave with the large wave number, and can also suppress the numerical oscillations for the rapidly varying bottom. Finally, four numerical examples are given, and the numerical results show that the DSWWE combined with the AAV can effectively simulate the shock waves, accurately capture the movements of wet-dry interfaces, and precisely preserve the mass. [ABSTRACT FROM AUTHOR]

A new reliable structure-preserving algorithm for computingH∞norm of linear continuous-time periodic systems is proposed in this paper. In the computation of theH∞norm, no Riccati differential equations are needed to solve and only eigenvalues of a monodromy matrix of the associated periodic Hamiltonian system will be evaluated. First, the monodromy matrix is expressed as the product of state transition matrices of the Hamiltonian system. Second, these state transition matrices, which have been proved to be symplectic matrices, are evaluated by a structure-preserving Magnus series method. Then, in order to preserve the standard symplectic form of the monodromy matrix, the structure-preserving matrices obtained by state transition matrices are employed to compute the monodromy matrix. At last, the effectiveness and the high accuracy of the proposed structure-preserving algorithm are demonstrated by numerical examples. [ABSTRACT FROM PUBLISHER]

• The mechanical friction force during the control rod drop is treated as a time-variant uncertainty for uncertainty analysis; • The interval process model is used to describe the time-variant uncertain mechanical friction during the CR drop; • The uncertainty analysis of CR drop considering various uncertain parameters is conducted; • A novel and general computational framework for uncertainty analysis of CR drop, UA-CRD, is proposed. Due to the influence of multiple factors, the control rod drop involves some uncertain parameters. The mechanical friction on the control rod is a time-variant uncertain parameter and is related to the position of the control rod. For the first time, this paper adopts the interval process model, which accurately describes the uncertainty parameters by using small-scale samples, to describe the uncertain mechanical friction on the control rod. The numerical experimental results demonstrate that the interval process model can describe the mechanical friction with high accuracy, even when employing small-scale samples. Furthermore, it is found that the uncertainty analysis can be considered as solving the extremal problem of a black-box model. Based on it, we propose a novel and general calculation framework for uncertainty analysis of control rod drop, UA-CRD. Through the analysis of numerical examples, it is verified that UA-CRD can accurately and efficiently conduct the uncertainty analysis of control rod drop. [ABSTRACT FROM AUTHOR]

A new reliable algorithm for computing -norm of linear time-varying periodic (LTP) systems in continuous-time domain is proposed in this article. The solving of -norm is firstly transformed into the solving of the periodic Lyapunov differential equation (PLDE). Then, the key point in this article is that a new method based on the interval mixed energy matrix and the interval transition matrix computed by the structure-preserving Magnus series method is proposed for solving the PLDE. With the solutions of PLDE, the -norm of LTP systems is evaluated by a simple first-order ordinary differential equation. Finally, the effectiveness and the high accuracy of the proposed algorithms are demonstrated by two numerical examples. [ABSTRACT FROM AUTHOR]

A semianalytical spectral element method (SEM) is proposed for electromagnetic simulations of 3-D layered structures. 2-D spectral elements are employed to discretize the cross section of a layered structure, and the Legendre transformation is then used to cast the semidiscretized problem from the Lagrangian system into the Hamiltonian system. A Riccati equation-based high precision integration method is utilized to perform integration along the longitudinal direction, which is the undiscretized direction, to generate the stiffness matrix of the whole layered structure. The final system of equations by the semianalytical SEM will take the form of a set of linear equations with a block tri-diagonal matrix, which can be solved efficiently by the block Thomas algorithm. Numerical examples demonstrate the high efficiency and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

Pseudo-division algorithm for matrix multivariable polynomial are given, thereby with the view of differential algebra, the sufficient and necessary conditions for transforming a class of partial differential equations into infinite dimensional Hamiltonianian system and its concrete form are obtained. Then by combining this method with Wu's method, a new method of constructing general solution of a class of mechanical equations is got, which several examples show very effective. [ABSTRACT FROM AUTHOR]

Differential‐algebraic equations (DAEs) can model constrained dynamical systems and processes from practical engineering. Therefore, research on nonlinear optimal control problems of DAEs is of theoretical significance for optimal control of constrained systems, which can generate reference trajectories and control inputs for online control strategies. In terms of the numerical solution of this type of problem, research on indirect numerical methods is still insufficient and less research focuses on symplectic‐preserving methods. In this article, a symplectic indirect approach is proposed for optimal control problems subject to index‐1 DAEs. Necessary conditions of the optimal control problem constitute a Hamiltonian boundary value problem (HBVP) and there exists a symplectic structure in the Hamiltonian system. In the proposed approach, based on specified properties of generating functions, discrete equations can preserve the symplectic structure of the Hamiltonian system. In the iterative solution, the Jacobian matrices of the discrete equations are sparse and symmetric, which are very significant to save memory and improve efficiency in practical computation. In numerical examples, the proposed approach can provide highly accurate state variables and control inputs with fewer iterations. More accurate cost functional can be obtained. Problems from the chemistry process also can be solved effectively, it verifies the problem‐solving ability of the proposed approach. [ABSTRACT FROM AUTHOR]

Variational integrators for dynamic systems with holonomic constraints are proposed based on Hamilton's principle. The variational principle is discretized by approximating the generalized coordinates and Lagrange multipliers by Lagrange polynomials, by approximating the integrals by quadrature rules. Meanwhile, constraint points are defined in order to discrete the holonomic constraints. The functional of the variational principle is divided into two parts, i.e., the action of the unconstrained term and the constrained term and the actions of the unconstrained term and the constrained term are integrated separately using different numerical quadrature rules. The influence of interpolation points, quadrature rule and constraint points on the accuracy of the algorithms is analyzed exhaustively. Properties of the proposed algorithms are investigated using examples. Numerical results show that the proposed algorithms have arbitrary high order, satisfy the holonomic constraints with high precision and provide good performance for long-time integration. [ABSTRACT FROM AUTHOR]

• A deterministic simplified model of the control rod drop and fluid flow is established. • A fifth-order adaptive collocation stochastic perturbation (ACSP) method is established to approximate the random responses. • The uncertainty analysis of the CR drop based on the ACSP method is carried out in both large and small reactor. • The research results can provide guidance for the analysis and design optimization of the control rod scram process. There are many uncertain factors to affect the control rod drop from its design and manufacture to the chain reaction in the reactor. They have to be considered in the simulation to realistically predict the dynamic behavior of control rod drop. Different from the previous method by using the nominal and extreme values of the uncertain factors, a statistical method is applied in this paper to analyze them in the process of the control rod drop. The uncertain factors obey uniform distribution, based on which a fifth-order adaptive collocation stochastic perturbation (ACSP) method is established in this paper, which is featured with simple format, high precision, and high calculation efficiency. The mean and standard deviation of the responses (such as the final time of the control rod drop) in two different types of reactors are calculated by using the ACSP method. On this basis, the response intervals and corresponding probabilities of the control rod drop are obtained. Besides, influences of each random variable on the responses are analyzed by the sensitivity analysis. In addition, the linear correlation degree and correlation between the two are discussed through the correlation coefficient of random variables and the responses. The uncertainty analysis of control rod drop in this paper not only provides a relatively strong theoretical support for scientifically dealing with uncertain factors, but also offers specific help for the optimal design of control rod assembly. [ABSTRACT FROM AUTHOR]

In this paper, a symplectic local pseudospectral (PS) method for solving nonlinear state-delayed optimal control problems with inequality constraints is proposed. We first convert the original nonlinear problem into a sequence of linear quadratic optimal control problems using quasi-linearization techniques. Then, based on local Legendre-Gauss-Lobatto PS methods and the dual variational principle, a PS method to solve these converted linear quadratic constrained optimal control problems is developed. The developed method transforms the converted problems into a coupling of a systemof linear algebraic equations and a linear complementarity problem. The coefficientmatrix involved is sparse and symmetric due to the benefit of the dual variational principle. Converged solutions can be obtained with few iterations because of the local PS method and quasi-linearization techniques are used. The proposed method can be applied to problems with fixed terminal states or free terminal states, and the boundary conditions and constraints are strictly satisfied.Numerical simulations show that the developed method is highly efficient and accurate. [ABSTRACT FROM AUTHOR]

This paper addresses the dynamic analysis of the clustered tensegrity structures via the framework of the positional formulation finite element method (FEM). The nodal coordinates are chosen as the generalized coordinates. The rod element and multi-nodes clustered cable element are first formulated. A comprehensive dynamic model for investigating the dynamic properties of the clustered tensegrity is then developed. The dynamic model is subsequently modified for the structural vibration analysis and modal analysis. The results of the modal analysis show that each order of natural frequency of a clustered tensegrity is lower than that of its corresponding classical tensegrity, indicating that the structure tends to be more flexible if a classical tensegrity is transformed into a clustered tensegrity. The results of the nonlinear deployment analysis show that the motion characteristics differ from those of the quasi-static analysis as the actuation speed increases. To pursuit a fast actuation speed to actuate the structure, mere quasi-static analysis in the existing literature is inadequate, and dynamic effects must be taken into account. The proposed method provides an effective tool to capture the dynamic properties, such as the determination of an appropriate actuation speed, and to actuate the clustered tensegrity. [ABSTRACT FROM AUTHOR]

This paper presents a sliding cable element for multibody system analysis. Unlike the existing literature on sliding cables developed using the finite element approach, the novelty of this approach is the use of the configuration of the attached rigid bodies as the generalized coordinates, rather than the traditional nodal displacements. The generalized force vector, and its related tangent stiffness and damping matrix, of the sliding cable and that of the classical cable element are analytically derived. It can also be found that the proposed sliding cable element can degenerate to the existing element formulated using the finite element approach. This allows us to use less generalized coordinates to address a system that contains few rigid or flexible body but with many pulleys. Then, this sliding cable element is employed to investigate the deployment of clustered tensegrity. Both quasi-static and dynamic analyses are carried out. Two representative examples show the effectiveness of the proposed element. The dynamic results also show that the motion characteristics of the system differ from the quasi-static solutions as the actuation speed increases. To achieve a fast actuation speed for deploying such systems, quasi-static analysis seems inadequate, and the dynamic effect must be taken into account. Under this background, the proposed element, coupled with the multibody dynamic methodology used in this work, does provide a powerful tool for analyzing the mechanical properties of such systems. [ABSTRACT FROM AUTHOR]

Control rod drop is the key part of ensuring the safety of nuclear reactor shutdown. Aiming at the control rod drop problem, we established a physical model of the control rod drive line. The proposed model considers the influence of the absorbing rod's position and hence evaluates the resultant force accurately on the control rod during the whole dropping process. Based on the physical model, a unified coupling state equation on the control rod drop and fluid flow is developed by integrating the motion variables of the control rod and the fluid into a state vector. A time step adaptive symplectic (TSAS) algorithm is proposed to solve the proposed state equation. The TSAS algorithm efficiently and stably predicts the motion states of the control rod and the fluid synchronously and accurately captures the sudden change of flow state. Numerical examples confirm the correctness of the proposed model and the TSAS algorithm. Sensitivity analysis is also conducted to discuss the effects of different parameters on the control rod drop, which provides reference and help for the design and manufacturing of the control rod drive line. [ABSTRACT FROM AUTHOR]

A symplectic algorithm with nonuniform grids is proposed for solving the hypersensitive optimal control problem using the density function. The proposed method satisfies the first-order necessary conditions for the optimal control problem that can preserve the structure of the original Hamiltonian systems. Furthermore, the explicit Jacobi matrix with sparse symmetric character is derived to speed up the convergence rate of the resulting nonlinear equations. Numerical simulations highlight the features of the proposed method and show that the symplectic algorithm with nonuniform grids is more computationally efficient and accuracy compared with uniform grid implementations. Besides, the symplectic algorithm has obvious advantages on optimality and convergence accuracy compared with the direct collocation methods using the same density function for mesh refinement. [ABSTRACT FROM AUTHOR]

Abstract: An expectation-based approach to the statistical theorem is proposed for the one-dimensional randomly disordered phononic crystal. In the proposed approach, the expectations of the random eigenstates of randomly disordered phononic crystals are investigated. In terms of the expectations of the random eigenstates, the wave propagation and localization phenomenon in the random phononic crystal could be understood in a statistical perspective. Using the proposed approach, it is proved that for a randomly disordered phononic crystal, the Bloch theorem holds in the perspective of expectation. A one-dimensional randomly disordered binary phononic crystal consisting of two materials with the random geometry size or random physical parameter is addressed by using the proposed approach. From the result, it can be observed that with the increase of the disorder degree, the localization of the expectations of the eigenstates is strengthened. The effect of the random disorder on the eigenstates at higher frequencies is more significant than that at lower frequencies. Furthermore, after introducing the random disorder into phononic crystals, some random divergent eigenstates are changed to localized eigenstates in expectation sense. [Copyright &y& Elsevier]

Abstract: This paper addresses the design and computation of a guidance law for a transfer mission from an orbit near the Earth to a halo orbit around the libration point L 2 in the Sun–Earth system. The guidance law, which is designed based on receding horizon control and compensates for launch velocity errors that are introduced by inaccuracies of the launch vehicle, is solved using the generating function method. During the design of the closed-loop guidance law, the entire transfer mission, which is considered a nonlinear optimal control problem, is evaluated to obtain a nominal reference trajectory. Using the launch velocity errors and the uncertainty of the model, a spacecraft controlled by the proposed guidance law tracks the reference trajectory. Furthermore, the original Riccati differential equation in the receding horizon control algorithm is replaced by an equivalent convenient form of the Riccati differential equation that is based on the generating function. The high-efficiency solution of the equivalent equation avoids the online direct integration of the original Riccati differential equation, which significantly increases the computational efficiency for the receding horizon control problem. Numerical simulations using a nonlinear bicircular four-body model demonstrate the capabilities of the proposed receding horizon guidance law for the transfer mission. In addition, the generating function method improves the computational efficiency by at least one order of magnitude over the backward sweep method in solving the receding horizon control problem. [Copyright &y& Elsevier]

Abstract: An optimal periodic controller based on continuous low-thrust is proposed for the stabilization missions of spacecraft station-keeping and formation-keeping along periodic Libration point orbits of the Sun–Earth system. Additionally, a new numerical algorithm is proposed for solving the periodic Riccati differential equation in the design of the optimal periodic controller. Practical missions show that the optimal periodic controller (which is designed with the linear periodic time-varying equation of the relative dynamical model) overcomes the problems and limitations of the time-variant LQR controller. Furthermore, nonlinear numerical simulations are presented for the missions of a leader spacecraft station-keeping and three follower spacecraft formation-keeping. Numerical simulations show that the velocity increments for spacecraft control and relative position errors vary little with changes in the altitude of periodic orbits. In addition, the actual trajectories of the leader and the follower spacecraft track the periodic reference orbit with high accuracy under the perturbation of the eccentric nature of the Earth''s orbit and the initial injection errors. In particular, the relative position errors obtained by the optimal periodic controller for spacecraft formation-keeping are all in the range of millimeters. [Copyright &y& Elsevier]