自适应间断有限元方法求解三维欧拉方程

将龙格库塔间断有限元方法(RDDG)与自适应方法相结合,求解三维欧拉方程.区域剖分采用非结构四面体网格,依据数值解的变化采用自适应技术对网格进行局部加密或粗化,减少总体网格数目,提高计算效率.给出四种自适应策略并分析不同自适应策略的优缺点.数值算例表明方法的有效性.     

一种基于预估-修正策略的非定常流网格自适应加密/稀疏方法

给出一种非定常流动数值模拟的网格自适应处理方法.在"求解流动方程-自适应调整网格"的流程中,引入预估-修正步.根据自适应周期内每个时间步上的流场预估解,计算单元上的事后误差估算值.建立考虑解演变的网格自适应指示器,并进行多层次单元加密-稀疏的动态网格自适应处理.在自适应网格上重新计算流场.每个自适应周期中,流动演变区域的网格获得加密;而前一个周期中的特征现象已离开区域的网格被稀疏.应用边界非协调的当地DFD(Domain-Free Discretization)方法求解流动方程.为验证网格自适应处理方法,针对静止圆柱和自推进游鱼的流动进行了数值实验.     

三维非均匀不稳定渗流方程的自适应网格粗化算法

将渗透率自适应网格技术应用于三维非均匀不稳定渗流方程的网格粗化算法中,在渗透率或孔隙度变化异常区域自动采用精细网格,用直接解法求解渗透率或孔隙度变化异常区域的压强分布,在其它区域采用不均匀网格粗化的方法计算,即在流体流速大的区域采用精细网格.用该方法计算了三维非均匀不稳定渗流场的压降解,结果表明三维非均匀不稳定渗流方程的三维非均匀自适应网格粗化算法的解在渗透率或孔隙度异常区的压强分布规律与采用精细网格的解非常逼近,在其它区域压强分布规律与粗化算法的解非常逼近,计算速度比采用精细网格提高100多倍.     

多介质流界面高精度自适应欧拉算法

采用欧拉网格自适应算法捕捉多介质流界面,获得了高精度界面特征,对不同物质引入不同位标函数跟踪界面运动,将位标函数方程与流体动力学方程非耦合求解,在笛卡尔坐标系中运用二阶精度有限体积算法,在保持流场守恒条件下,采用多层网格级对笛卡尔网格嵌套细化,实现了多介质流物质界面的高精度自适应跟踪.方法逻辑简单,大大节省了CPU时间,且能够对局部参数急剧变化的流场(如激波)进行自适应跟踪.     

强爆炸火球辐射流体自适应网格高精度数值模拟

提出一种自适应网格方法,应用于基于Euler方法的强爆炸辐射流体高精度数值求解.通过与Zinn数值结果对比,验证该方法的正确性.研究自适应网格对冲击波和光辐射输出模拟精度的影响,对比不同网格尺度下的计算耗时.在相同的条件下,使用自适应网格与均匀网格加密3倍得到的冲击波超压分布、光辐射输出演化接近,计算效率提升约8.5倍...     

NS方程的各向异性笛卡尔网格法研究

将近年发展起来的用于Euler方程求解的具有局部均匀网格总体非结构特性的笛卡尔网格法推广到NS方程的求解。为了与流场的各向异性相适应、减少网格点数量,提出了一种各向异性网格加密法。另外还研究了分级笛卡尔网格对内点格式稳定性的影响和插值固体边界条件的稳定性。数值结果表明各向异性笛卡尔网格法相对于传统的各向同性网格方法能大量节省网格点数量而且与后者具有同样的精度。     

运动介质中电磁场计算的一种迎风有限元方法

本文提出了运动介质中正弦稳态电磁场问题的一种迎风有限元解法。用伽僚金法求解这类问题,当离散网格的Peclet数大于1时,计算结果会出现伪振荡。为了抑制这种振荡,引入了采用在迎风面与背风面具有不同迎风参数的权函数的迎风有限元法。该方法对一维问题,在均匀网格下能在节点上给出问题的精确解,在一维结果的基础上,提出了相应的二维解法,并用一个二维模型进行了验证。     

完全多重网格法求解光强度传播方程的相位恢复方法

一般多重网格算法求解光强度传播方程是从精细网格层开始计算,精细网格层的初值选择影响算法的收敛速度和求解精度.提出了一种完全多重网格方法求解光强度传播方程的相位恢复方法.基于限制法将最细网格上的方程转化为最粗网格上的方程,求解该方程得到最粗网格上的解;然后对此解用延拓法得到上一层细网格上的解,以得到的解作为此层网格上的初解,利用V循环解此层上的方程,得到此层网格的精确解.依次,直到得到最细网格上的精确解.模拟相位恢复实验结果表明,本文方法具有较快的收敛速度,能够恢复复杂相位分布.     

大雷诺数Navier-Stokes方程的两水平亚格子模型稳定化方法

基于两重网格离散方法,提出三种求解大雷诺数定常Navier-Stokes方程的两水平亚格子模型稳定化有限元算法.其基本思想是首先在一粗网格上求解带有亚格子模型稳定项的Navier-Stokes方程,然后在细网格上分别用三种不同的校正格式求解一个亚格子模型稳定化的线性问题,以校正粗网格解.通过适当的稳定化参数和粗细网格尺寸的选取,这些算法能取得最优渐近收敛阶的有限元解.最后,用数值模拟验证三种算法的有效性.     

复杂区域上的一种结构网格生成方法

讨论复杂区域上的一种结构网格生成方法,其主要思想是:以变分形式的Winslow网格生成方法为基础,通过引入网格解扭机制和网格面积均匀化技术,构造出一种新的离散泛函,进而采用一类优化算法求解这一离散泛函的极小化问题,得到所希望的网格.通过分析及大量数值实验表明,这一方法比较健壮,针对二维复杂区域通常能够生成几何品质较优的网格,它在保持Winslow方法优点的同时,克服了它的一些缺点.     

变分同化中水平误差函数的正交小波模拟新方法

背景误差协方差是变分资料同化系统中的一个重要组成部分,能将观测信息从观测点传播到周围的模式格点和垂直层上.为了模拟背景误差协方差中水平误差函数的非均匀性和各向异性,提出了一种用正交小波变换模拟水平误差函数的新方法.试验结果表明：新方法能模拟出水平误差函数中固有的非均匀性和各向异性,极好地表示了背景误差协方差中固有的结构和特征. 关键词： 变分资料同化 背景误差协方差 二维正交小波 水平误差函数     

Adaptive mesh redistribution method for domains with complex boundaries

An adaptive structured mesh redistribution method (ASMRM) that permits smooth transition from non-uniformly distributed boundary points to solution-adaptive interior points and enables the resolution of complex flow in the complex boundary region as well as away from the boundary is proposed. It is a variant of the traditional variational technique. It involves a combination of static and dynamic monitor functions, the former for mesh distribution in the vicinity of a complex boundary and the latter for mesh adaption with the evolving solution elsewhere. Its effectiveness is demonstrated on some example problems, and it is then applied to a chevron nozzle. The proposed method is shown to be capable of generating a mesh with a good balance of orthogonality and smoothness in the entire domain.     

A Hybrid Method for Dynamic Mesh Generation Based on Radial Basis Functions and Delaunay Graph Mapping

Aiming at complex configuration and large deformation, an efficient hybrid method for dynamic mesh generation is presented in this paper, which is based on Radial Basis Functions (RBFs) and Delaunay graph mapping. Based on the computational mesh, a set of very coarse grid named as background grid is generated firstly, and then the computational mesh can be located at the background grid by Delaunay graph mapping technique. After that, the RBFs method is applied to deform the background grid by choosing partial mesh points on the boundary as the control points. Finally, Delaunay graph mapping method is used to relocate the computational mesh by employing area or volume weight coefficients. By applying different dynamic mesh methods to a moving NACA0012 airfoil, it can be found that the RBFs-Delaunay graph mapping hybrid method is as accurate as RBFs and is as efficient as Delaunay graph mapping technique. Numerical results show that the dynamic meshes for all test cases including one two-dimensional (2D) and two three-dimensional (3D) problems with different complexities, can be generated in an accurate and efficient manner by using the present hybrid method.     

抛物方程的多重非均匀网格模型及其应用

提出了抛物方程的多重非均匀网格模型,以准确求解三维空间存在多辐射源的电波传播问题。通过对不同辐射源建立不同的坐标系,并对其仿真空间采用不同的非均匀网格划分,构建了抛物方程的多重非均匀网格模型。在此基础上,实现了三维多辐射源问题的并行计算。实例仿真了空间存在四个辐射源的电波传播特性。结果表明,抛物方程的多重非均匀网格模型能够准确求解多源的空间电磁场分布特性,且在该算例中,并行技术使得抛物方程的计算速度提升了2.41倍,极大地提高了抛物方程对三维多源问题的求解效率。     

嵌套网格技术中的Collar网格和虚拟网格方法

针对嵌套网格技术中的结合部问题,提出了Collar网格和虚拟网格方法.用双曲型微分方程生成的处于结合部的有两个边界分别处于不同固定曲面上的Collar网格,在保证计算网格的生成方便快捷而且网格质量高的前提下,解决了为物体结合部的内外边界点提供有效插值单元的问题.虚拟网格为紧贴物面的面网格,它的作用是将物面边界条件传递给其它网格的边界面,而其本身不作流场计算.计算实践表明,将Collar网格和虚拟网格结合起来应用在嵌套网格技术中能保证几何外形不发生变化,有效地处理各种复杂组合体外形的结合部问题.     

Compact local integrated-RBF approximations for second-order elliptic differential problems

This paper presents a new compact approximation method for the discretisation of second-order elliptic equations in one and two dimensions. The problem domain, which can be rectangular or non-rectangular, is represented by a Cartesian grid. On stencils, which are three nodal points for one-dimensional problems and nine nodal points for two-dimensional problems, the approximations for the field variable and its derivatives are constructed using integrated radial basis functions (IRBFs). Several pieces of information about the governing differential equation on the stencil are incorporated into the IRBF approximations by means of the constants of integration. Numerical examples indicate that the proposed technique yields a very high rate of convergence with grid refinement.     

The density profile for the Klein-Kramers equation near an absorbing wall

We derive asymptotic series for the expansion coefficients of a function in terms of the Pagani functions, which occur in the boundary layer solutions of the Klein-Kramers equation. The results enable us to determine the density profile in the stationary solution of this equation near an absorbing wall from the numerically determined velocity distribution at the wall, with an accuracy of about 2%. We also obtain information about the analytic behavior of the density profile: this profile increases near the wall with the square root of the distance to the wall. Finally, the asymptotic analysis leads to an understanding of the slow convergence of variational approximations to the solution of the absorbing-wall problem and of the exponents that occur when one studies the variational approximations to various quantities of interest as functions of the number of terms in the variational ansatz. This is used to obtain a better variational estimate for the density at the wall.     

Multi-objective optimization of ring stiffened cylindrical shells using a genetic algorithm

In this paper, the genetic algorithm (GA) method is used for the multi-objective optimization of ring stiffened cylindrical shells. The objective functions seek the maximum fundamental frequency and minimum structural weight of the shell subjected to four constraints including the fundamental frequency, the structural weight, the axial buckling load, and the radial buckling load. The optimization process contains six design variables including the shell thickness, the number of stiffeners, the width and height of stiffeners, the stiffeners eccentricity distribution order, and the stiffeners spacing distribution order. The real coding scheme is used for representing the solution string, while the generation number-based adaptive penalty function is applied for penalizing infeasible solutions. In analytical solution, the Ritz method is applied and the stiffeners are treated as discrete elements. Some examples of simply supported cylindrical shells with nonuniform eccentricity distribution and nonuniform rings spacing distribution are provided to demonstrate the optimality of the solution obtained by the GA technique. The effects of objective weighting coefficients and bounding values of the design variables on the optimum solution are studied for various cases. The results show that the optimal solution can vary with the weighting coefficients significantly. It is also found that extreme reduction and augmentation in turn in the structural weight and fundamental frequency can be simultaneously achieved by selecting suitable stiffeners’ geometrical parameters and distributions. Furthermore, the bounding values of the design variables have great effects on the optimum results.     

Variational model for one-dimensional quantum magnets

A new variational technique for investigation of the ground state and correlation functions in 1D quantum magnets is proposed. A spin Hamiltonian is reduced to a fermionic representation by the Jordan–Wigner transformation. The ground state is described by a new non-local trial wave function, and the total energy is calculated in an analytic form as a function of two variational parameters. This approach is demonstrated with an example of the XXZ-chain of spin-1/2 under a staggered magnetic field. Generalizations and applications of the variational technique for low-dimensional magnetic systems are discussed.     

Approximate solutions of the Liouville equation. III. Variational principles and projection operators

Aspects of stationary variational principles for the Laplace-transformed Liouville equation are discussed. Projection techniques are used to derive new stationary principles applicable to the space orthogonal to the space spanned by functions occurring in the conservation laws. As a result, any trial function automatically leads to results satisfying the conservation laws. The procedure is also applied to the parity-even and parity-odd distributions which obey equations governed by the square of the Liouville operator. The technique is extended to eliminate the one-body additive contribution to the solution exactly. Finally, the ideas of the moment method, which leads to the continued-fraction representation of autocorrelation functions, are applied to variational principles. We find continued-fraction variational principles such that a zero trial function yields the usual representation. However, a trial function representing noninteracting particles contains the results of the moment method and in addition yields the exact analytic behavior for free particles.Work supported by a grant from the National Science Foundation.