高压气体淬火过程中的温度场与相变耦合问题的计算

高压气体淬火技术是一种现代的、有效的材料加工技术．在Cheng所得到的高压气体淬火过程中非线性表面换热系数的基础上,用有限单元法对钢淬火过程的温度场与相变耦合问题进行了模拟计算．在数值计算中,材料的热物性系数被处理为温度和相变体积百分比的函数．为避免数值解的震荡,采用了Norsette有理近似法．     

θ方法解滞时微分方程的动力学性质

本文研究求解滞时微分方程的θ-方法数值解的渐近性和方程真实解的关系。首先，我们把数值方法看成以步长为参数的动力系统，考察非线性滞时微分方程θ-方法的数值稳定性。并且证明了A－稳定的θ-方法是NP－稳定的。其次我们证明了θ-方法没有伪不动点，还研究了伪周期2解的存在性。最后我们给出一个例子说明了滞时微分方程θ-方法产生的伪周期2解是不稳定的。     

二维非线性对流扩散方程特征有限元的两重网络算法

针对二维非线性对流扩散方程,构造了特征有限元两重网格算法．该算法只需要在粗网格上进行非线性迭代运算,而在所需要求解的细网格上进行一次线性运算即可．对于非线性对流占优扩散方程,不仅可以消除因对流占优项引起的数值振荡现象,还可以加快收敛速度、提高计算效率．误差估计表明只要选取粗细网格步长满足一定的关系式,就可以使两重网格解与有限元解保持同样的计算精度．算例显示:两重网格算法比特征有限元算法的收敛速度明显加快．     

求解对流扩散方程的一致四阶紧致格式

目前,许多高精度差分格式,由于未成功地构造与其精度匹配的稳定的边界格式,不得不采用低精度的边界格式.本文针对对流扩散方程证明了存在一致四阶紧致格式,它的边界点的计算格式和内点的计算格式的截断误差主项保持一致,给出了具体内点和边界格式;并分析了此半离散格式的渐近稳定性.数值结果表明该格式是四阶精度;在对流占优情况下,本文边界格式的数值结果比四阶精度的显式差分格式的的数值结果的数值振荡小,取得了不错的效果,理论结果得到了数值验证;驱动方腔数值结果显示,本文对N-S方程的离散格式具有很好的可靠性,适合对复杂流体流动的数值模拟和研究.     

爆炸冲击载荷下航空铝合金平板动态响应数值分析方法

为得到适用于爆炸冲击载荷下航空铝合金平板动态响应的数值分析方法,采用LS-DYNA显式动力学分析软件对爆炸冲击载荷下的铝合金平板进行数值仿真计算.主要研究了不同的任意Lagrange-Euler(拉格朗日-欧拉)网格（ALE）输运步算法、流固耦合方式、流固耦合点数量、网格尺寸、有限元单元类型对计算结果的影响.通过计算结果与实验结果的分析对比,表明采用van Leer+HIS输运步算法、罚函数耦合方式、在流体网格与结构网格之间采用3个耦合点、结构网格尺寸与空气域网格尺寸比例设为2∶1、结构单元采用163号壳单元时可以较为准确地计算航空铝合金平板在爆炸冲击载荷下的动态响应,并且能提高计算效率,节约计算时间.     

A collocation method for generalized nonlinear Klein-Gordon equation

In this paper, we propose a collocation method for an initial-boundary value problem of the generalized nonlinear Klein-Gordon equation. It possesses the spectral accuracy in both space and time directions. The numerical results indicate the high accuracy and the stability of long-time calculation of suggested algorithm, even for moderate mode in spatial approximation and big time step sizes. The main idea and techniques developed in this work provide an efficient framework for the collocation method of various nonlinear problems.     

An efficient step size control for continuation methods

In solving a nonlinear equation by the use of a continuation method one of the crucial problems is the choice of the step sizes. We present a model for the total computational cost of a standard numerical continuation process and solve the problem of optimal step size control for this model. Using the theoretical results as a basis, we develop an adaptive step size algorithm for Newton's method. This procedure is computationally inexpensive and it gives quite satisfactory results compared to some other numerical experiments found in the literature.     

A Second Order Unconditionally Convergent Finite Element Method for the Thermal Equation with Joule Heating Problem

In this paper, we study the finite element approximation for nonlinear thermal equation. Because the nonlinearity of the equation, our theoretical analysis is based on the error of temporal and spatial discretization. We consider a fully discrete second order backward difference formula based on a finite element method to approximate the temperature and electric potential, and establish optimal $L^2$error estimates for the fully discrete finite element solution without any restriction on the time-step size. The discrete solution is bounded in infinite norm. Finally, several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.     

An adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems

An efficient and reliable a posteriori error estimate is derived for linear parabolic equations which does not depend on any regularity assumption on the underlying elliptic operator. An adaptive algorithm with variable time-step sizes and space meshes is proposed and studied which, at each time step, delays the mesh coarsening until the final iteration of the adaptive procedure, allowing only mesh and time-step size refinements before. It is proved that at each time step the adaptive algorithm is able to reduce the error indicators (and thus the error) below any given tolerance within a finite number of iteration steps. The key ingredient in the analysis is a new coarsening strategy. Numerical results are presented to show the competitive behavior of the proposed adaptive algorithm.

     

预测水驱油藏中油、水流动问题的快速方法

以最佳正交分解(POD)技术为基础提出了一种快速预测油藏中油、水流动问题的方法.采用POD技术建立了水驱油藏中油、水两相流动的低阶模型.通过油藏数值模拟方法获得二维水驱油藏模型在时间0～500 d内的压力和含水饱和度的100个样本, 并从样本中提取出一组压力和含水饱和度的POD基函数.当注采参数不断变化后,采用已求得的POD基函数结合低阶模型对新的物理场进行预测.研究结果表明:POD方法能够快速、准确地预测出水驱油藏的压力和含水饱和度场,文中算例给出压力和含水饱和度场的预测误差分别不超过1.2%与1.5%,且计算速度比直接进行油藏数值模拟快50倍以上.     

Time adaptivity in the diffusive wave approximation to the shallow water equations

We discuss the use of time adaptivity applied to the one dimensional diffusive wave approximation to the shallow water equations. A simple and computationally economical error estimator is discussed which enables time-step size adaptivity. This robust adaptive time discretization corrects the initial time step size to achieve a user specified bound on the discretization error and allows time step size variations of several orders of magnitude. In particular, the one dimensional results presented in this work feature a change of four orders of magnitudes for the time step over the entire simulation.     

A fast numerical approach for the simulation of highly viscous non-isothermal non-Newtonian fluids

This paper deals with the efficient simulation of polymer melts, as an example of highly viscous non-isothermal non-Newtonian fluids. In flow fields of our interest, which are characterized by small Reynolds numbers and large Prandtl numbers, steep gradients occur in thin boundary layers of the temperature distribution, whereas the boundary layers associated with the velocity field possess a considerable different length scale. In order to benefit from these properties, we introduce a physically motivated multigrid approach by computing velocity and temperature fields on different meshes. This new development is achieved by the modification of a discrete projection method. Numerical experiments are presented which confirm that the method decreases the computational effort considerably, while preserving the numerical accuracy.     

结构动力响应中急动度的计算

急动度(jerk)在工程实践中具有重要的意义.将径向基函数逼近与配点法相结合,发展了一种能够有效求解动力响应的数值算法.该方法使用径向基函数插值来逼近真实的运动规律,能够用于急动度和急动度(三阶)方程的计算,弥补了传统的数值方法无法计算急动度的不足.并针对微分方程的特点,提出了改进的多变量联合插值函数,同时添加与微分方程同阶的初值条件,可显著减小数值震荡.算例表明,该方法具有计算过程简单、精度高的特点,同时对急动度方程也有很好的适用性.     

Existence and stability of periodic solutions of high-order Hopfield neural networks with impulses and delays

By using the continuation theorem of coincidence degree theory and constructing suitable Lyapunov functions, the global exponential stability and periodicity are investigated for a class of delayed high-order Hopfield neural networks (HHNNs) with impulses, which are new and complement previously known results. Finally, an example with numerical simulation is given to show the effectiveness of the proposed method and results. The numerical simulation shows that our models can occur in many forms of complexities including periodic oscillation and the Gui chaotic strange attractor.     

The dynamic complexity of a three-species Beddington-type food chain with impulsive control strategy

In this paper, by using theories and methods of ecology and ordinary differential equation, the dynamics complexity of a prey–predator system with Beddington-type functional response and impulsive control strategy is established. Conditions for the system to be extinct are given by using the Floquet theory of impulsive equation and small amplitude perturbation skills. Furthermore, by using the method of numerical simulation with the international software Maple, the influence of the impulsive perturbations on the inherent oscillation is investigated, which shows rich dynamics, such as quasi-periodic oscillation, narrow periodic window, wide periodic window, chaotic bands, period doubling bifurcation, symmetry-breaking pitchfork bifurcation, period-halving bifurcation and crises, etc. The numerical results indicate that computer simulation is a useful method for studying the complex dynamic systems.     

Computational inelasticity for loading conditions on multiple time scales by adaptive step size control

This contribution proposes an algorithm based on adaptive step size control for the simulation of inelastic solids and structures undergoing loading conditions at multiple time scales. Adaptivity in time integration of viscoelastic constitutive laws is directed by an refinement indicator which is constructed from integrators of different order, here a fourth-order Runge-Kutta (RK) method and linear Backward-Euler. The key novel aspect is that by virtue of an recently established consistency condition the higher order methods, p ≥ 2, can achieve their full nominal order without fulfilling the weak form of balance of linear momentum in the RK stages, but only at the end of the time interval. A representative numerical example illustrates the performance of the present adaptive method and underpins the computational savings compared with uniform time step sizes. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

High resolution upwind-mixed finite element methods for advection-diffusion equations with variable time-stepping

Numerical methods for advection-diffusion equations are discussed based on approximating advection using a high-resolution upwind finite difference method, and incorporating diffusion using a mixed finite element method. In this approach, advection is approximated explicitly and diffusion implicitly. We first describe the basic procedure where each advection time-step is followed by a diffusion step. Because the explicit nature of the advective scheme requires a CFL time-step constraint, the basic procedure may be expensive, especially if the CFL constraint is severe. Two alternative time-stepping approaches are presented for improving computational efficiency while preserving accuracy. In the first approach, several advective time-steps are computed before taking a diffusion step. In the second approach, the advective time-steps are also allowed to vary spatially. Numerical results for these three procedures for a model problem arising in flow through porous media are given. © 1995 John Wiley & Sons, Inc.     

Numerical simulation of the unsteady aerodynamic interaction of two plane airfoil cascades

A method for the calculation of unsteady aerodynamic interaction of two plane airfoil cascades that are in relative motion in a subsonic flow of ideal gas is developed. This interaction provides a two-dimensional approximation of the flow in a stage of an axial turbomachine. The method is based on the reduction of the problem to the calculation of the unsteady flow in a single interblade passage of each of the cascades. The calculation uses generalized space-time periodicity relations corresponding to the unsteady process of interest. The calculation is based on the direct numerical integration of the non-stationary gas dynamics equations with the use of the finite difference Godunov-Kolgan-Rodionov scheme of the second approximation order with respect to time and space. The calculation procedure includes the determination of the acoustic fields that are generated by the stage in the incident flow and in the flow behind it. The results of the calculations that illustrate the accuracy of the numerical solution and the capabilities of the method are presented.     

Error measurement and FEM benchmark for phase field modeling

Solid-state phase-transformation often deals with coupled field variables like mechanical stress or electric potential on different scales compared to the size of phase transition. Further, highly nonlinear equations rule the phase parameter. Thus, required mesh density to approximate phase field variables is often higher than for coupled fields. Finite element techniques applying h- or p-adaptivity can help to increase the efficiency of such simulations. Therefore, it is important to define a benchmark test to verify numerical approaches. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)