线性时变系统二次最优控制问题的保辛近似求解

状态空间的最优控制体系是保守的,其近似算法应当保辛．提出了基于分段常值精细积分方法的保辛摄动近似方法,在同一框架下求解了线性时变LQ最优控制中的计算问题,即变系数矩阵Riccati方程和状态反馈方程．该算法是保辛的,具有很好的数值稳定性和精度．算例验证了算法的有效性．     

基于时域精细积分算法的瞬态传热多宗量反演

基于有限元法和精细积分算法,提出了一种求解瞬态热传导多宗量反演问题的新方法．采用有限元法和精细积分算法分别对空间、时间变量进行离散,可以得到正演问题高精度的半解析数值模型,由此建立了多宗量反演的计算模式,并给出敏度分析的计算公式．对一维和二维的热物性参数、热源项、边界条件等进行了单宗量和多宗量的反演求解,初步考虑了初值和噪音等对反演结果的影响,数值算例验证了该方法的有效性．     

反应、扩散化学动力学方程的有效解法

对扩散、化学反应或瞬态温度场问题，给出了具有4阶精度、自起步的隐式时间积分算法．算例显示，其精度和稳定性都好于四阶Runge-Kutta法，并且保留了原系数矩阵的稀疏存储方式和稀疏矩阵的运算规则，使紧缩存储技术和减少计算时间有效的结合．以旋转填充床内的竞争串联反应为算例，表明该算法是有效的．     

基于超松弛预处理的精细积迭代反演算法

电磁波、声波层析成像技术在地质探测等很多领域有了广泛应用,其中反演算法在层析成像过程中处于核心地位,反演算法的优劣将直接关系层析成像的成败.给出了一种基于超松弛预处理的精细积分迭代反演算法,算法将方程组求解归结为一个常微分方程组初值问题的极限形式,对以此为基础建立的递推算法中指数矩阵利用精细积分法计算,但在此之前利用超松弛法将正演所得病态矩阵预处理,降低条件数以减少测量扰动对反演计算的影响.通过检测板模型恢复测试以及实际资料反演结果比较,预处理精细积分方法在准确性和稳定性上都具有一定的优势,在迭代次数较少时即能得到分辨率较高的反演速度剖面;另外,过程中参数的选择具有一定任意性,同时规则网格尺度可以根据折射初至时数目灵活选取,具有更强的实用性.     

中心焦点判定的形式积分因子方法

本文从积分因子的角度探讨了焦点量的计算方法，给出了形式积分因子的存在性和形式积分因子的系数与焦点量的等价关系，从而给出计算焦点量的简捷算法。     

瞬态动力问题的逐单元矩阵分裂和逐步积分方法

本对瞬态动力问题，结合逐步积分方法提出了一类广义的矩阵分裂和逐单元松弛算法，摆脱了有限元法通常需形成总体刚度矩阵，总体质量矩阵和求解大型稀疏方程组的工作，理论分析和计算实例表明，本的广义矩阵分裂是最优的分裂方案，本的算法物理意义明确，但于编写程序推广应用。     

无约束优化问题的对角稀疏拟牛顿法

对无约束优化问题提出了对角稀疏拟牛顿法,该算法采用了Armijo非精确线性搜索,并在每次迭代中利用对角矩阵近似拟牛顿法中的校正矩阵,使计算搜索方向的存贮量和工作量明显减少,为大型无约束优化问题的求解提供了新的思路．在通常的假设条件下,证明了算法的全局收敛性,线性收敛速度并分析了超线性收敛特征。数值实验表明算法比共轭梯度法有效,适于求解大型无约束优化问题．     

求解奇异摄动边值问题的精细积分法

提出了一种求解一端有边界层的奇异摄动边值问题的精细方法．首先将求解区域均匀离散,由状态参量在相邻节点间的精细积分关系式确定一组代数方程,并将其写成矩阵形式．代入边界条件后,该代数方程组的系数矩阵可化为块三对角形式,针对这一特性,给出了一种高效递推消元方法．由于在离散过程中,精细积分关系式不会引入离散误差,故所提出的方法具有极高的精度．数值算例充分证明了所提出方法的有效性．     

线性互补问题的并行多分裂松弛迭代算法

运用矩阵多重分裂理论,同时考虑并行计算与松弛迭代法,得到一类求解线性互补问题的高效数值算法．当问题的系数矩阵为对角元为正的H-矩阵或对称半正定矩阵时,证明了算法的全局收敛性；该算法与已有算法相比,具有计算量小、计算速度快等特点,因而特别适于求解大规模问题．数值试验的结果说明了算法的有效性．     

精细辛几何算法的误差估计

该文讨论了精细辛几何算法的计算误差,先展开二阶和四阶精细辛几何算法的表达式得到误差同精细剖分数目的关系,然后分析了任意阶精细辛几何算法的误差,得到了一致简洁的结果,总的误差可近似表示为单个精细步长的误差乘以剖分数目,最后讨论了在要求控制精度下剖分数目的选取,该方法克服了算法精度对积分时间步长的依赖性.     

Stationary motion of a viscous drop taking into account its deformation

One investigates the stationary axisymmetric motion of a drop of a viscous incompressible fluid in the flow of a viscous incompressible fluid in the domain of moderate motion velocities (the Reynolds' number ～1–100), various surface tensions and relations between the viscosities inside and outside the drop. One gives a numerical algorithm for the computations. One gives some examples of flows for some values of the defining parameters.     

病态代数方程求解的一种改进精细积分法

通过在病态代数方程精细积分法的基础上增加一个迭代改善算法,建立了病态代数方程求解的改进精细积分法．该方法进一步提高了病态代数方程精细积分法的精度和效率,具有良好的应用前景．算例证明了该方法在病态代数方程求解中的有效性．     

一种自适应的四阶Newton-Cotes求积方法

本文给出了一种基于四阶Newton-Cotes公式的自适应求积算法，该算法能根据给定的容许误差，由计算机自动选取积分步长，克服了由于被积函数的性态不好而导致积分较复杂的缺陷．     

一种显式子步应力点积分算法及其在SMA数值模拟中的应用

形状记忆合金（shape memory alloys,简称SMA）具有复杂的热力本构关系,为了模拟SMA及其组合结构复杂的受力和变形行为,在数值模拟中需要采用可靠且高效的应力点积分算法．隐式应力点回映算法已经成功应用于形状记忆合金的数值模拟,但在复杂加载条件下,荷载增量较大时有可能导致整体非线性迭代求解不收敛．推广了局部误差控制的显式子步积分算法,首次将其应用于形状记忆合金及其组合结构这类热力相变问题的应力点积分,并通过数值算例对所提算法和隐式应力点回映算法进行了比较．数值结果表明:对于大规模数值模拟和计算,整体子步步数决定着总体计算时间;所提出的修正Euler自动子步方案可以有效减少整体子步步数,在保证相同计算精度的前提下能够大幅提高有限元计算效率,因而更适合大规模形状记忆合金智能结构的数值模拟.     

A hybrid approach for the integration of a rational function

A hybrid integration algorithm obtaining an indefinite integral of a rational function (say q/r, q and r are polynomials) with floating-point but real coefficients is proposed. The algorithm consists of four steps and is based on combinations of symbolic and numeric computations (hybrid computation). The first step is a hybrid preprocessing stage. An integrand is decomposed into rational and logarithmic parts by using an approximate Horowitz' method which allows floating-point coefficients. Here, we replace the Euclidean GCD algorithm with an approximate-GCD algorithm which was proposed by Sasaki and Noda recently. It is easy to integrate the rational part. The logarithmic part is integrated numerically in the second step. Zeros of a denominator of it are computed by the numerical Durand-Kerner method which computes all zeros of a polynomial equation simultaneously. The integrand is then decomposed into partial fractions in the third step. Coefficients of partial fractions are determined by residue theory. Finally, in the fourth step, partial fractions are transformed into the resulting indefinite integral by using well-known rules of integrals. The hybrid algorithm proposed here gives both indefinite integrals and accurate values of definite integrals. Numerical errors in the hybrid algorithm depend only on errors in the second step. The algorithm evaluates some problems where numerical methods are inefficient or incapable, or a pure symbolic method is theoretically insufficient.     

Three-dimensional simulation of high compressible flows using a multi-time-step integration technique with subcycles

An algorithm to simulate three-dimensional high compressible flows using the finite element method and a multi-time-step integration technique with subcycles is presented in this work. An explicit two-step Taylor–Galerkin scheme is adopted to integrate in time the continuum equations. When explicit schemes are used, the time-steps must satisfy the CFL stability conditions. If the smallest critical time-step is adopted uniformly for the whole domain, the integration scheme may consume a large amount of CPU time. Multi-time-step integration techniques are very suitable in these cases because elements and nodes are separated into several groups and a different time-step is assigned to each group. In this way, each group of elements is integrated with a time interval which is much closer to the critical time-steps of the elements in the group. This results in great computational savings, mainly when element sizes and properties are very different, leading to significant differences in the local critical time-step values. Multi-time-steps integration techniques are also very useful in transient problems, taking into account that at the end of each subcycle, values of the unknowns at the same time level are obtained. The multi-time-step algorithm is applied to analyze the supersonic flow (Mach=8.5) past a sphere immersed in a non-viscous flow, and the results and computational performance are compared with those obtained when a uniform time-step is used over the whole domain.     

Computation of viscous or nonviscous conservation law by domain decomposition based on asymptotic analysis

An accurate and efficient numerical method has been developed for a nonlinear diffusion convection-dominated problem. The scheme combines asymptotic methods with usual solution techniques for hyperbolic problems. After having localized shock or corner layers and rescaling, first terms of the inner expansion are computed. Using the same concepts gives a method to compute a very accurate solution of the nonlinear conservation law. Because our numerical scheme is based on a uniform approximation throughout the domain, the shock is localized very accurately and there is practically no smearing out. Numerical computations are presented. Another novel feature is the ability to break down the problem according to subdomains of different local behavior, based on asymptotic analysis, which may make it feasible to do computations with different processors.