二维等谱问题研究的计算数学框架

等谱问题是数学、物理诸学科关注的一个热点问题,本文总结并诠释了二维等谱问题的内在计算数学性质与规律:利用镜像反演讨论等谱对的几何结构(不等距而谱相等);把一般文献中假定的特殊三角形扩展到一般的三角形或者矩形;研究特征函数的正交结构,把特定的Laplace等谱问题扩展到一般零边值的二阶线性椭圆算子等谱问题.指出合理的粗网格对于研究等谱问题及其计算的重要性:两个连续问题等谱成立的充分必要条件是存在自然粗网格使其离散问题谱相等.文中给出的数值例子与特征值近似逼近验证了相应的结论,所用的方法原则上可用于研究三维乃至高维的PDE等谱问题.     

An Evaluation of HPF and MPI Approaches and Performance in Unstructured Finite Element Simulations

The High Performance Fortran (HPF) language and the Message Passing Interface (MPI) are two widely used methods to achieve parallelism on today's clusters and multiprocessor supercomputers. HPF is a distinct language providing extensions to Fortran 90/95 to express parallel execution paths and regions. MPI is a library of communication calls that can be inserted into modern high-level languages (C and Fortran). This paper discusses the use of the two approaches in a parallel finite element application for liquid composite manufacturing process modeling. The unstructured nature of the code provides an excellent opportunity to test both the computation and communication effectiveness of the two approaches. We discuss performance results based on implementations conducted on a modern massively parallel computing platform with a highly tuned processor interconnection network.     

Numerical integration using sparse grids

We present new and review existing algorithms for the numerical integration of multivariate functions defined over d-dimensional cubes using several variants of the sparse grid method first introduced by Smolyak [49]. In this approach, multivariate quadrature formulas are constructed using combinations of tensor products of suitable one-dimensional formulas. The computing cost is almost independent of the dimension of the problem if the function under consideration has bounded mixed derivatives. We suggest the usage of extended Gauss (Patterson) quadrature formulas as the one‐dimensional basis of the construction and show their superiority in comparison to previously used sparse grid approaches based on the trapezoidal, Clenshaw–Curtis and Gauss rules in several numerical experiments and applications. For the computation of path integrals further improvements can be obtained by combining generalized Smolyak quadrature with the Brownian bridge construction. This revised version was published online in August 2006 with corrections to the Cover Date.     

FeCrAl丝网负载FAU沸石膜的制备及其催化性能

采用水热法制备FeCrAl丝网负载FAU沸石膜,考察了合成液老化、载体预涂晶种以及晶种液中添加聚乙烯醇(PVA)等因素的影响,评价了FeCrAl丝网负载FAU沸石膜对模型吸热燃料正辛烷裂解的催化活性。结果表明,载体表面预涂Y分子筛晶种后有助于增加载体表面分子筛负载量,采用老化后的合成液,效果更显著,并且更易在金属丝上生长出一层连续致密的FAU沸石膜。在相同水热合成条件下,载体表面用含PVA的晶种液预涂晶种后,可增加分子筛负载量。正辛烷在FeCrAl丝网负载FAU沸石膜上的初始裂解转化率(7.2%～13.2%,500 ℃)高于其在空白载体上的裂解转化率(4.4%),FAU沸石膜的初始催化活性不仅与载体表面分子筛负载量有关,也与分子筛膜的形貌有关。     

重叠型非匹配网格抛物型方程的有限元法及收敛性分析

基于单位分解技术,本文介绍了重叠型非匹配网格抛物型方程初边值问题的有限元方法,分别给出了半离散有限元、全离散有限元格式和收敛性分析的结果,并给出了数值例子.     

一类二元有理插值函数的存在性及算法

文[3]构造了对于矩形网格上基于二元Newton插值公式的一类二元有理插值函数,并给出了其存在性的充分条件.本文进一步证明了这类二元有理插值函数存在性的必要条件,特别地,当m=n时,给出了具有三角形结构的系数矩阵的判别方法,该方法计算简便且具有承袭性,文章最后给出的实例说明了方法的有效性.     

The piecewise polynomial collocation method for nonlinear weakly singular Volterra equations

Second-kind Volterra integral equations with weakly singular kernels typically have solutions which are nonsmooth near the initial point of the interval of integration. Using an adaptation of the analysis originally developed for nonlinear weakly singular Fredholm integral equations, we present a complete discussion of the optimal (global and local) order of convergence of piecewise polynomial collocation methods on graded grids for nonlinear Volterra integral equations with algebraic or logarithmic singularities in their kernels.

     

三维非结构网格Euler方程的LU-SGS算法及其改进

将LU-SGS隐式时间推进格式运用到非结构网格Euler方程的求解中,并对传统LU-SGS格式进行改进,结合网格重排序,发展了一套效率更高的三维Euler方程求解器.以M6机翼及超临界LANN机翼的跨音速无粘流场为算例,将改进LU-SGS格式与四步龙格-库塔显式格式及传统LU-SGS格式进行了比较.计算结果表明:所有格式的计算结果与实验结果都符合很好;传统LU-SGS格式计算效率为显式格式的3倍多,而改进LU-SGS格式计算效率为显式格式的7倍多.     

基于非结构网格的不可压流动耦合求解器

采用非结构化网格有限容积法求解了不可压N-S方程组,对流项采用GAMMA格式,扩散项采用二阶中心差分格式建立离散方程,用SOAR算法处理压力与速度的耦合关系,得到了一种求解不可压N-S方程的非结构网格耦合求解器。通过方腔顶盖驱动流、后台阶绕流以及方腔自然对流等几个典型的算例,考察了求解器的计算精度及收敛特性,并与SIMPLE算法进行了比较,结果表明该求解器是有效可行的。     

二维自适应结构网格的变分生成方法

提出-一种二维结构网格的自适应生成方法,即采用变分方程控制网格的疏密程度以及光滑性、正交性和正规性,并且所有的控制方程具有-致的量纲.几个数值算例表明,给出的网格自适应生成方法能够较好地收敛到方程的唯-解,同时产生性质良好的二维自适应结构网格.