A novel time-marching scheme using numerical Green’s functions: A comparative study for the scalar wave equation

The present paper presents the formulation of a novel time-marching method based on the Explicit Green’s Approach (ExGA) to solve scalar wave propagation problems. By means of the weighted residual method in both time and space, the time integral expression concerning the ExGA is readily established. The arising ExGA time integral expression is spatially discretized in a finite element sense and a recursive scheme that employs time-domain numerical Green’s function matrices is adopted to evaluate the displacement and the velocity vectors. These Green’s matrices are computed by the time discontinuous Galerkin finite element method only at the first time step. The system of coupled equations originated from the time discontinuous Galerkin method is then solved by an iterative predictor–multicorrector algorithm. Once the Green’s matrices are computed, no iterative process is required to obtain the displacement and the velocity vectors at any time level. At the end of the paper, numerical examples are presented in order to compare the proposed approach with other approaches.     

Multi space reduced basis preconditioners for parametrized Stokes equations

We introduce a two-level preconditioner for the efficient solution of large scale saddle-point linear systems arising from the finite element (FE) discretization of parametrized Stokes equations. This preconditioner extends the Multi Space Reduced Basis (MSRB) preconditioning method proposed in Dal Santo et al. (2018); it combines an approximated block (fine grid) preconditioner with a reduced basis (RB) solver which plays the role of coarse component. A sequence of RB spaces, constructed either with an enriched velocity formulation or a Petrov–Galerkin projection, is built. Each RB coarse component is defined to perform a single iteration of the iterative method at hand. The flexible GMRES (FGMRES) algorithm is employed to solve the resulting preconditioned system and targets small tolerances with a very small iteration count and in a very short time. Numerical test cases for Stokes flows in three dimensional parameter-dependent geometries are considered to assess the numerical properties of the proposed technique in different large scale computational settings.     

Convergence analysis of a parareal-in-time algorithm for the incompressible non-isothermal flows

This paper presents and analyzes a parareal-in-time scheme for the incompressible non-isothermal Navier–Stokes equations with Boussinesq approximation. Standard finite element method is adopted for the spatial discretization.The proposed algorithm is proved to be unconditional stability. The convergence factor of iteration error for the velocity and temperature is given at time-continuous case. It theoretically demonstrates the superlinearly convergence of the parareal iteration combined with finite element method for incompressible non-isothermal flows. Finally, several numerical experiments that confirm feasibility and applicability of the algorithm perform well as expected.     

Policy Iteration for Optimal Control of Discrete-Time Time-Varying Nonlinear Systems

Aimed at infinite horizon optimal control problems of discrete time-varying nonlinear systems, in this paper, a new iterative adaptive dynamic programming algorithm, which is the discrete-time time-varying policy iteration (DTTV) algorithm, is developed. The iterative control law is designed to update the iterative value function which approximates the index function of optimal performance. The admissibility of the iterative control law is analyzed. The results show that the iterative value function is non-increasingly convergent to the Bellman-equation optimal solution. To implement the algorithm, neural networks are employed and a new implementation structure is established, which avoids solving the generalized Bellman equation in each iteration. Finally, the optimal control laws for torsional pendulum and inverted pendulum systems are obtained by using the DTTV policy iteration algorithm, where the mass and pendulum bar length are permitted to be time-varying parameters. The effectiveness of the developed method is illustrated by numerical results and comparisons.      

基于折扣广义值迭代的智能最优跟踪及应用验证

设计了一种基于折扣广义值迭代的智能算法,用于解决一类复杂非线性系统的最优跟踪控制问题.通过选取合适的初始值,值迭代过程中的代价函数将以单调递减的形式收敛到最优代价函数.基于单调递减的值迭代算法,在不同折扣因子的作用下,讨论了迭代跟踪控制律的可容许性和误差系统的渐近稳定性.为了促进算法的实现,建立一个数据驱动的模型网络用...     

Stable iterative adaptive dynamic programming algorithm with approximation errors for discrete-time nonlinear systems

In this paper, a novel iterative adaptive dynamic programming (ADP) algorithm is developed to solve infinite horizon optimal control problems for discrete-time nonlinear systems. When the iterative control law and iterative performance index function in each iteration cannot be accurately obtained, it is shown that the iterative controls can make the performance index function converge to within a finite error bound of the optimal performance index function. Stability properties are presented to show that the system can be stabilized under the iterative control law which makes the present iterative ADP algorithm feasible for implementation both on-line and off-line. Neural networks are used to approximate the iterative performance index function and compute the iterative control policy, respectively, to implement the iterative ADP algorithm. Finally, two simulation examples are given to illustrate the performance of the present method.     

任意初态下的工业机器人迭代学习控制

对迭代初值为任意值的工业机器人轨迹跟踪控制系统,提出了一种基于滑模面的非线性迭代学习控制算法,使机器人轨迹能快速、精确跟踪上期望轨迹。基于有限时间收敛原理,构建了关于机器人轨迹跟踪误差的迭代滑模面,在滑模面内,机器人轨迹跟踪误差在预定时间内收敛到零。设计了基于滑模面的迭代学习控制算法,理论证明了随着迭代次数的增加,处于任意初态的轨迹将一致收敛到滑模面内,解决了迭代学习中的任意初值问题。数值仿真验证了该算法的有效性和抗干扰能力。     

Finite iterative algorithms for the generalized Sylvester-conjugate matrix equation {AX+BY=E\overline{X}F+S}

This paper investigates the generalized Sylvester-conjugate matrix equation, which includes the normal Sylvester-conjugate, Kalman–Yakubovich-conjugate and generalized Sylvester matrix equations as its special cases. An iterative algorithm is presented for solving such a kind of matrix equations. This iterative method can give an exact solution within finite iteration steps for any initial values in the absence of round-off errors. Another feature of the proposed algorithm is that it is implemented by original coefficient matrices. By specifying the proposed algorithm, iterative algorithms for some special matrix equations are also developed. Two numerical examples are given to illustrate the effectiveness of the proposed methods.     

Numerical simulation of rapid crack propagation in viscoplastic materials

A finite element model for the simulation of fast crack propagation under dynamic Mode I load conditions is developed. The principal tools are an enhanced strain formulation and a quasi-contact algorithm. Crack propagation is controlled by the implemented fracture criterion. The model accounts for viscoplastic material behavior, dynamic effects and contact in the crack zone. The numerical results demonstrate that these physical effects are of major importance when load speed is increased to a high level and therefore they cannot be neglected.     

Identification of a constitutive model for the simulation of time-dependent interlaminar debonding processes in composites

Time-dependent interlaminar debonding processes in polymer-based laminates are studied in this paper. A viscoplastic constitutive law is introduced to model the behaviour of the resin-enriched interfaces between adjacent plies; details concerning the formulation and the finite element implementation of the interface law are given. To calibrate the model, namely to identify model parameters, a fully numerical approach is proposed: this approach is centred on the extended Kalman filter (EKF) and on an approximate methodology to compute the sensitivity of the structural response to model parameters. First, results are presented concerning parameter identification at the constitutive level; at this stage numerical results are compared with pseudo-experimental data in order to assess the capability of the proposed approach to identify each model parameter that affects the nonlinear branch of the interface behaviour. Hence, results of the identification procedure for finite element simulations of delamination growth in standard tests are discussed.     

Stability of controllers with on-line computations

The dynamical systems theory developed by Zufiria [1], Zufiria and Guttalu [2, 3], and Guttalu and Zufiria [4] is applied to the stability analysis of control systems in which the feedback control law requires in real time the solution of a set of nonlinear algebraic equations. Since a small sampling period is assumed, the stability and performance of the controlled process can be studied with a continuous-time formulation. A singularly perturbed system is used to model both the dynamics of the system being controlled and a numerical iterative algorithm required to compute the control law. An updating control procedure has been proposed based on the iterative nature of the control algorithm. The results obtained by Zufiria [1] regarding the behavior of a dynamical system that models the numerical algorithms lead to a considerable simplification in the analysis. For the case of a control problem involving inverse kinematics, the numerical algorithm that solves for inverse kinematics can be considered as an observer (or an estimator) of the state-space variables. The study provides an estimate of the required speed of computations to preserve the stability of the controller.Recommended by E .P. Ryan     

An elasto-plastic rate-dependent finite element analysis of the metal forming process

An elasto-plastic rate-dependent finite element formulation is developed into the solution of the large strain and deformation problem. The formation is based on the power form constitutive equation for the stress-strain-strain rate relation of the material. A simple one-step Euler's time integration scheme is adopted to automatically control the time increment. After incorporating a force rate term which is due to the effect of the strain rate, the simulation is completed by modifying the updated Lagrangian formulation. The numerical results can be used as options in the selection of the adequate tool speed, die geometry, and die material.     

Convergence Analysis of Iterative Laplace Transform Methods for the Coupled PDEs from Regime-Switching Option Pricing

This paper aims to analyze the convergence rates of the iterative Laplace transform methods for solving the coupled PDEs arising in the regime-switching option pricing. The so-called iterative Laplace transform methods are described as follows. The semi-discretization of the coupled PDEs with respect to the space variable using the finite difference methods (FDMs) gives the coupled ODE systems. The coupled ODE systems are solved by the Laplace transform methods among which an iteration algorithm is used in the computational process. Finally, the numerical contour integral method is used as the Laplace inversion to restore the solutions to the original coupled PDEs from the Laplace space. This Laplace approach is regarded as a better alternative to the traditional time-stepping method. The errors of the approach are caused by the FDM semi-discretization, the iteration algorithm and the Laplace inversion using the numerical contour integral. This paper provides the rigorous error analysis for the iterative Laplace transform methods by proving that the method has a second-order convergence rate in space and exponential-order convergence rate with respect to the number of the quadrature nodes for the Laplace inversion.     

一类基于迭代空间条块的并行有限差分Stencil 算法

高效的并行有限差分Stencil 算法对于求解大型线性方程组是十分重要的.针对并行有限差分Stencil 算法中数据局部性差、同步和通信开销大的问题.首先改进传统有限差分Stencil 算法,提出了多层对称遍历有限差分Stencil 算法.然后给出了以迭代空间条块序作为执行序的串行算法,通过沿时间轴对迭代空间进行时滞划分,在不改变迭代算法性质的同时,对迭代空间条块内部多次迭代计算,提高算法的数据局部性.最后提出一种基于迭代空间条块的并行算法,该算法利用改进的多面体模型对迭代空间网格划分,并通过网格条块重排序减少了Cache 缺失率、通信启动和同步次数.理论分析和实验结果表明,该并行模型比传统的区域分解方法和红黑排序并行算法具有更好的数据局部性,并行效率和可扩展性.     

Bézier曲线降阶的迭代算法

为提高Bézier曲线降阶的稳定性,提出以基于L_2范数的逼近误差为指导的一种迭代算法. 该算法从一条初始Bézier曲线开始逐渐地对其控制顶点进行偏移,得到具有误差最小的逼近曲线; 同时,应用线性搜索方法来优化控制顶点的偏移,使得在每次迭代后逼近误差可以达到局部最小. 实例结果表明了该算法的快速收敛性.     

求解三对角线性方程组的迭代对角占优算法

针对并行求解三对角线性方程组的对角占优(PDD)算法,在系数矩阵为弱对角占优时,近似处理引入误差较大的问题,提出了一种PDD算法的迭代方案。该方案在解的修正值计算中采用迭代方法,计算精度得到了提高;通过对算法的误差分析,导出了算法在给定误差下迭代次数的估算式;数值实验说明了算法的有效性。通过对迭代与非迭代的PDD算法的复杂性分析,迭代算法的计算复杂性增加很小,但通信复杂性随迭代次数成倍增加。     

离散时间Itˆo 型跳变系统Lyapunov 方程的有限次迭代求解算法

针对离散时间Itˆo 型马尔科夫跳变系统Lyapunov 方程的求解给出一种迭代算法. 经证明, 在误差允许的范围内, 该算法可以在确定的有限次数内收敛到系统的精确解, 收敛速度较快, 具有良好的数值稳定性, 并且该算法为显式迭代, 可避免迭代过程中求解其他矩阵方程对结果精度产生的影响. 最后通过一个数值算例对该算法的有效性进行了验证.

     

基于滑模观测器的多智能体一致性控制

本文针对速度不可测下的多智能体系统提出一种基于滑模观测器的有限时间一致性控制算法.首先, 利用滑模观测器在有限时间内估计每个智能体的速度信息. 然后,利用估计信息设计非奇异终端滑模控制律,该控制律能够保证智能体系统在有限时间内达到一致性. 最后,通过数值仿真验证了该一致性控制算法的有效性.     

A Well-balanced Finite Volume-Augmented Lagrangian Method for an Integrated Herschel-Bulkley Model

We are interested in the derivation of an integrated Herschel-Bulkley model for shallow flows, as well as in the design of a numerical algorithm to solve the resulting equations. The goal is to simulate the evolution of thin sheet of viscoplastic materials on inclined planes and, in particular, to be able to compute the evolution from dynamic to stationary states. The model involves a variational inequality and it is valid from null to moderate slopes. The proposed numerical scheme is well balanced and involves a coupling between a duality technique (to treat plasticity), a fixed point method (to handle the power law) and a finite volume discretization. Several numerical tests are done, including a comparison with an analytical solution, to confirm the well balanced property and the ability to cope with the various rheological regimes associated with the Herschel-Bulkley constitutive law.     

离散时变多智能体系统有限时间一致性迭代学习控制

针对一类离散时变多智能体系统,通过引入虚拟领导者产生期望轨迹的方法,将虚拟领导者和所有智能体组成固定的拓扑结构,在此基础上,提出一种离散时间迭代学习控制算法.该算法对多智能体系统中的每个智能体都设计一个控制器,各控制器都是利用上一次迭代时,该智能体与虚拟领导者之间的跟踪误差和该智能体与相邻智能体之间的跟踪误差,通过拓扑结构中通信权值的组合不断修正上一次的控制律,从而获得理想控制律.同时,基于范数理论严格证明所提出算法的收敛性,并给出算法在$\lambda$-范数意义下的收敛条件.该算法能够使离散时变多智能体的输出随着迭代次数的增加在有限时间区间内完全跟踪期望轨迹.理论分析和仿真结果都表明了所提出算法的有效性.