Space-time finite elements for a simplified thermo-metallurgical problem relevant to casting

The non-linear thermo-metallurgical problem, relevant for the cooling of a molten metal including the macro-segregation that occurs during the cooling process, is studied here. Due to the strong non-linearities involved in phase transformations, it is necessary to use a fine resolution in space-time in a finite element approximation in order to meet accuracy requirements. We derive space-time FE-methods based on the discontinuous and continuous Galerkin method in time for the energy equation. This formulation integrates the stored energy exactly for a given heat flux. When macro-segregation is incorporated into the model, the problem can be formulated in such a way that the phase-transition drives a flow of species. In addition, diffusion is possible throughout the domain. The model can be further rewritten using a potential approach. By this approach for modelling macro-segregation, we are able to obtain discretizations that guarantee that the balance equations are satisfied, and it is possible to solve the phase-transition problem either as a field problem or as a local problem (defined by a local evolution rule).     

基于Cosserat理论的头发动态模拟*

针对基于弹性杆理论的超螺旋模型中对动力学方程的求解影响头发模拟实时性的问题,采用Cosserat弹性杆理论对头发建模,引入角速度变量,并加入头发运动平衡的固有约束,得到改进的拉格朗日动力学运动方程;然后,将头发单体离散化,用角速度和四元数作为状态变量简化动力学方程,用半显式的欧拉方程加速方程的执行,大大降低了系统的运行时间,提高了模拟的速度,在满足实时性要求的同时提高了头发模拟的真实度。     

有限体积法求解立体角离散误差

为解决采用有限体积法求解辐射传输方程时引起离散误差的问题,从有限体积法立体角内热流密度的求解过程出发,运用泰勒级数推导了近似误差的表达式,并分析影响离散误差的各种因素,包括离散射线数,辐射强度分布,假散射及网格比.根据分析结果,采用辐射强度的连续分布模型,模拟了各种因素对误差的影响.结果表明,辐射强度的分布函数对于有限体积法的离散误差有一定程度影响.当增加网格数及减小网格尺寸比时,可以有效地减小离散误差.假散射与离散误差间呈较强的非线性耦合关系.     

多量级应急数据无监督离散化方法研究

在对连续的多量级应急数据进行离散化时,采用传统无监督离散化方法难以找出量级变化点完成离散。提出一种针对多量级应急数据的无监督离散化方法。根据量级变化的差异性将离散数据由大到小排序,结合拟合函数和二阶导数计算得到准确的量级变化点作为数据截断点,将截断所得较大数据构成的离散类移出待离散数据集,不断重复上述操作直到待离散数据满足设定的离散系数阈值,最终完成全部数据的离散。实验结果表明,该方法实现了地震相关多量级应急数据的均匀离散,其离散系数较等频离散化、层次聚类离散化等传统方法更低,可有效离散化隐藏多量级差异的应急数据。     

Computing electromagnetic eigenmodes with continuous Galerkin approximations

Costabel and Dauge proposed a variational setting to solve numerically the time-harmonic Maxwell equations in 3D polyhedral geometries, with a continuous approximation of the electromagnetic field. In order to remove spurious eigenmodes, three computational strategies are then possible. The original method, which requires a parameterization of the variational formulation. The second method, which is based on an a posteriori filtering of the computed eigenmodes. And the third method, which uses a mixed variational setting so that all spurious modes are removed a priori. In this paper, we discuss the relative merits of the approaches, which are illustrated by a series of 3D numerical examples.     

A Numerical Study of Diagonally Split Runge–Kutta Methods for PDEs with Discontinuities

Diagonally split Runge–Kutta (DSRK) time discretization methods are a class of implicit time-stepping schemes which offer both high-order convergence and a form of nonlinear stability known as unconditional contractivity. This combination is not possible within the classes of Runge–Kutta or linear multistep methods and therefore appears promising for the strong stability preserving (SSP) time-stepping community which is generally concerned with computing oscillation-free numerical solutions of PDEs. Using a variety of numerical test problems, we show that although second- and third-order unconditionally contractive DSRK methods do preserve the strong stability property for all time step-sizes, they suffer from order reduction at large step-sizes. Indeed, for time-steps larger than those typically chosen for explicit methods, these DSRK methods behave like first-order implicit methods. This is unfortunate, because it is precisely to allow a large time-step that we choose to use implicit methods. These results suggest that unconditionally contractive DSRK methods are limited in usefulness as they are unable to compete with either the first-order backward Euler method for large step-sizes or with Crank–Nicolson or high-order explicit SSP Runge–Kutta methods for smaller step-sizes. We also present stage order conditions for DSRK methods and show that the observed order reduction is associated with the necessarily low stage order of the unconditionally contractive DSRK methods. The work of C.B. Macdonald was partially supported by an NSERC Canada PGS-D scholarship, a grant from NSERC Canada, and a scholarship from the Pacific Institute for the Mathematical Sciences (PIMS). The work of S. Gottlieb was supported by AFOSR grant number FA9550-06-1-0255. The work of S.J. Ruuth was partially supported by a grant from NSERC Canada.     

基于Cramer＇SV的连续属性离散化算法

在类一属性相关离散化方法的基础上,提出一种基于Cramer’s V的连续属性离散化算法CVM,该方法利用统计学中的Cramer’s V来量化类一属性相关度,以保证离散后的类一属性相关度最大。与CADD和CAIM算法的实验比较以及对离散化后的数据进行C4．5分类测试,表明CVM算法性能良好,其离散化的数据明显地提高了分类器的预测精度。     

一种改进的启发式离散化算法及应用

Nguyen S.H提出的布尔逻辑和粗糙集理论相结合的离散化算法是粗糙集理论中的离散化算法在思想上的重大突破。通过定义分界点来区分Nguyen S.H离散化算法中定义的断点对决策系统的分辨关系是否有贡献,并仅取分界点集作为初始断点集,使得初始断点数目较大幅度地降低,提出了一种改进的启发式离散化算法并应用于一个实际的决策系统的连续属性离散化。应用实例表明改进算法较大程度地减小了算法空间复杂性和时间复杂性,具有正确性和实用性。     

Implicit discrete-time fast terminal sliding mode control with disturbance compensation

In this paper, an implicit discrete-time fast terminal sliding mode (DFTSM) control with disturbance compensation is designed and analyzed for uncertain high-order systems. First, a recursive discrete sliding surface is constructed based on implicit Euler technique, which can completely eliminate discretization chattering so as to significantly reduce the boundary layer of sliding mode motion. With the help of a high-order disturbance compensator, the accuracy limitation of implicit DFTSM control systems is overcome by increasing the order of sliding mode. Then the finite-time convergence of implicit DFTSM is proved for the first time, and the influence relationship of control parameters on the convergence speed and control accuracy of the algorithm is established. Finally, two numerical examples are provided to demonstrate the effectiveness and superiority of the proposed design approach.     

带输入延时的连续时间系统的离散化

含多项式插值的Runge-Kutta方法应用于对带输入延时的连续时间系统的离散化中. 与传统的离散化方法相比, 本文提出的方法是有效且精度高阶的. 此方法的精度与Runge-Kutta法及插值多项式的精度紧密相关. 本文讨论了离散化方法的近似精度阶及最大可达的精度阶. 除此之外, 也分析了方法的输入状态稳定性. 为保证相应离散系统的稳定性, 可通过考察RK法的绝对稳定域来选择采样时间. 特别当RK法是A-稳定时, 可以不受稳定性的约束选择采样时间. 最后提供了一个数值例子来证明方法的优越性.