三维空间Stokes问题的两个各向异性非协调混合有限元逼近

构造了两个可用于求解三维Stokes问题的各向异性非协调混合有限元格式,在不需要通常的辅助空间的情况下给出相应的收敛性分析和最优的误差估计。这两种单元具有构造简单,整体自由度少的特点,是目前较为理想的单元。该方法对进一步设计相关的自适应算法有潜在的应用价值。     

自适应天线结构位移最优控制

为保证大型天线结构的形状精度,传统的方法是使结构具有足够的刚度。当采用自适应结构控制天线形变规律的时候,可以大大降低结构的重量并使结构具有更高的形状精度。针对嵌入压电作动器的自适应抛物面天线结构,建立了以天线反射面最佳吻合抛物面的精度和作动器能耗为综合目标的多目标优化控制模型,模型以结构强度和作动器性态作为约束条件,并用线性加权和法转化为单目标二次规划问题。算例表明,可以用较少的作动器,实现大型天线结构的高精度控制。     

求解Stokes方程的一种自适应变分多尺度方法

针对Stokes方程,本文给出一种新的变分多尺度方法.此方法通过构造特殊投影并采用局部化技巧优化了经典变分多尺度方法.同时,本文对此方法进行了后验误差估计,并给出可靠的后验误差估计子及相应的自适应算法.此自适应变分多尺度方法计算简单且易于程序实现.L型区域问题和方腔流问题的数值结果验证了方法的有效性和稳定性.     

一种改进的水平集方法在Navier-Stokes问题形状优化中的应用

本文将经典的形状灵敏度分析方法与一种改进的水平集方法相结合,给出了Navier-Stokes问题形状优化的一种新方法。该算法是在固定的Euler网格上进行计算且在优化过程中不需要对水平集函数进行重新初始化,从而可以有效地节省计算时间。数值算例说明该算法是稳定、高效的。     

复杂二维区域有限元网格自适应生成方法

在结合推进波前法和法向偏移法各自的优势的基础上,提出了新的网格自动自适应生成方法。采用预先检查节点距离的方式,显著地提高了网格生成的效率和可靠性。利用中间过程点的网格尺寸信息改进了生成节点的迭代公式,使得节点的生成更为迅速,并有效地提高了单元质量。密度控制方式独立于网格生成算法,控制灵活,便于采用纯数学手段来控制网格的疏密分布,能有效地实现自适应。     

二维抛物型奇异摄动问题的移动网格方法

奇异摄动问题在力学、声学、光学、工程等领域有广泛的应用。研究了一类含源项二维抛物型奇异摄动问题,通过坐标变换和有限体积方法,构造了该问题在空间移动网格上的数值格式,给出了网格移动时的网格迭代方法和解的更新方法,提出了局部加密的自适应移动网格算法。数值实验的结果表明,与均匀网格上求解的结果相比,自适应移动网格方法能更好地体现解在局部区域的特性,具有更理想的求解精度。     

Stokes问题的一个新的三角形Hermite型二阶格式

本文构造了一个新的协调三角形Hermite型单元,其形函数空间仅为完全二次多项式.针对定常Stokes问题,我们由此构造了一个新的二阶格式.同时,给出了其最优误差估计.     

Stokes问题的一个新的三角形Hermite型二阶格式

本文构造了一个新的协调三角形Hermite型单元,其形函数空间仅为完全二次多项式。针对定常Stokes问题,我们由此构造了一个新的二阶格式。同时,给出了其最优误差估计。     

有限元网格修正的自适应分析及其应用

本文在对有限元变量连续条件分析的基础上,将应力误差范数用于计算结果的误差估计,使非结构化网格生成系统与有限元计算有机地结合起来,并将网格单元修正的自适应分析应用于二维应力集中问题的研究,从而实现了有限元最佳化离散,提高了有限元数值求解的可靠性和近似程度。     

无网格时域自适应算法在振动问题中的应用

为了提高无网格方法计算振动问题时的计算效率和计算精度,开发了无网格时域自适应算法。该方法在时域内采用时域自适应算法,将每个变量在每个时间步内离散成关于时间的展开形式,通过控制展开阶数保证自适应算法的精度,可以更准确的描述变量随时间的变化;同时将时空耦合的初值问题转化为一系列的空间边值问题,并采用无网格方法递推求解。该方法可以提高计算效率并弥补时间步长较多或较大时造成的精度损失,并通过数值算例验证了其准确性。     

耦合对流传热的Stokes流体形状优化设计

本文研究了耦合对流传热的Stokes流体中的形状优化问题.利用不可压缩的定常Stokes方程耦合对流传热的模型来描述流体的特性,运用形状导数方法分析依赖于区域的状态方程解的极小化问题.通过引入共轭状态方程,计算出目标函数的微分形式,并构造求解该形状优化问题的梯度型算法.数值实验的结果验证了所用方法的有效性和可行性.     

H(curl)空间中椭圆最优控制问题的自适应有限元算法

针对H(curl)空间椭圆型最优控制问题提出了一种自适应有限元方法。首先将H(curl)空间Maxwell方程的最优控制模型转化为偏微分方程组,给出了偏微分方程组的解得正则性。其次利用自适应有限元方法求解此偏微分方程组,同时讨论了方法的后验误差及收敛性。最后通过数值算例给出了该方法的数值结果,验证了有限元方法的有效性和可靠性。这一方法可以应用于更复杂的最优控制问题的求解。     

具分布观测的年龄相关的种群扩散系统生育率控制的非线性问题

本文讨论了具分布观测的年龄相关的种群扩散系统最优生育率控制的非线性问题,证明了系统解的存在唯一性和最优生育率控制的存在性,得到了控制为最优的一阶必要条件及其由偏微分方程组和变分不等式组成的最优性组,并进而建立了最优生育率控制的反馈表达式。这些结果可为种群扩散系统中的最优控制问题的实际研究提供理论基础。     

A two-phase computational scheme for solving bang-bang control problems

This paper focuses on numerical methods for solving time-optimal control problems using discrete-valued controls. A numerical Two-Phase Scheme, which combines admissible optimal control problem formulation with enhanced branch-and-bound algorithms, is introduced to efficiently solve bang-bang control problems in the field of engineering. In Phase I, the discrete restrictions are relaxed, and the resulting continuous problem is solved by an existing optimal control solver. The information on switching times obtained in Phase I is then used in Phase II wherein the discrete-valued control problem is solved using the proposed algorithm. Two numerical examples, including a third-order system and the F-8 fighter aircraft control problem, are presented to demonstrate the use of this proposed scheme. Comparing to STC and CPET methods proposed in the literature, the proposed scheme provides a novel method to find a different switching structure with a better minimum time for the F-8 fighter jet control problem.     

Optimal search of distributed arrival time control

Distributed arrival time control (DATC) is a heuristic feedback control algorithm for scheduling which has been developed for a real-time distributed scheduling of heterarchical systems. It has been renowned not only for its fast solution searching algorithm but also for its flexibility to changing environment. However, the optimality of this heuristic method has not been analytically explained until recently because it has been designed to discover a near optimal solution instead of the true optimum. In this paper, we provide a novel optimal search method for the DATC scheduling problem by introducing a scalar cost function over the vector space of time and show the existence and location of true optima for the DATC scheduling problem through geometric approach. This geometrical interpretation enables us to find the true optimal by direct projection without iterations like previous DATC approaches. Based on the true optimum found, we evaluate the optimality of DATC algorithms by examining their dependency on initial conditions and explain their intrinsic causality mechanism for the discrepancy with true optimum. The implication of this study is on the new viewpoint over the vector space of DATC, which not only solves the optimality issue of DATC but also provides a new direction of direct search approach like projection method for the true optimum.     

Reanalysis‐based optimal design of trusses

This paper presents a structural reanalysis method and its applications in optimal design of trusses. This reanalysis technique is derived primarily on the basis of a reduced basis formulation, and it has several advantages over previous reduced basis methods. In particular, the reduced system is uncoupled by using a Gram–Schmidt orthonormalization procedure and an error measure is introduced to adaptively monitor whether a good approximate solution is achieved. The latter aspect makes this reanalysis method suitable for use in optimal design problems because the changes in design variables usually vary during a design process. Discussions are presented on the implementation of this reanalysis method using both mathematical programming and optimality criteria‐based optimization schemes. Finally, several example problems of optimal truss design are used to validate the proposed reanalysis‐based design procedure. The presented numerical results indicate that the new reanalysis technique affects very slightly the accuracy of the optimal solutions and it does speed up the design process when the system analysed is large. Copyright © 2000 John Wiley & Sons, Ltd.     

Case Studies in Optimization: Catenary Problem

This is the second paper in a series presenting case studies in modern large-scale constrained optimization 9 In this paper, we consider the shape of a hanging chain, which, in equilibrium, minimizes the potential energy of the chain. In addition to the tutorial aspects of this paper, we also emphasize the importance of certain modeling issues such as convex vs. nonconvex formulations of given problem. We will present several models of the problem and demonstrate differences in the number of iterations and solution time.     

Optimal inspection of a finite population

This article presents a method for determining an optimal quality control (QC) inspection frequency for a manufacturing process where a specified number of items will be produced and where, if the process goes out of control and generates a defective item, it stays out of control. In addition, there is a QC inspection procedure than can detect a defective item. The frequency of inspection is based on minimizing the total cost. The total cost includes the cost of QC inspections plus the cost of manufacturing defective items. In this application the cost of manufacturing defective items is, after identifying a defective item, the cost of checking previously manufactured items until it is determined when the process went out of control, i.e., until a good item is found.