一维非线性对流占优扩散方程特征差分法的两重网格算法

针对一维非线性对流扩散方程，构造了特征差分的两重网络算法，并给出了误差估计和数值算例。此方法是先在粗网格上计算非线性问题，再在细网格上计算线性问题，数值算例表明，在计算精度保持不变的情况下，此算法可以极大提高非线性对流扩散问题的计算效率。     

非定常N-S方程的双重网格数值模拟

本文采用全离散双重网格算法（时间变量采用Eular全隐式格式离散,空间变量采用混合有限元离散）,对非定常Navier-Stokes（N-S）方程进行数值模拟.双重网格算法的基本思想是,首先在粗网格有限元空间X^H上求解一个非线性问题,然后在细网格有限元空间Xh（h＜＜H）上求解一个线性问题.数值实验结果表明：在保持几乎相同精度的前提下,双重网格算法比标准有限元算法节省近一半的计算时间,说明了新算法求解非定常N-S方程的可行性和高效性.     

二维半线性抛物方程的二重网格差分算法

针对二维半线性抛物方程,本文提出了两种二重网格差分算法,并给出了误差估计。该算法能够在粗网格和细网格上线性地求解半线性问题。若重复算法的最后几步可以按粗网格步长任意阶地逼近细网格上的非线性解。     

求解Burger’s方程的两水平有限差分方法(英文)

本文中提出了求解Burger’s方程的两水平方法.新方法只需在粗网格上求解一个网格步长为H的非线性问题,在细网格上求解一个网格步长为h的线性问题.新格式是隐式无条件稳定的,并且能够得到与单水平解相同的收敛阶.由于单水平方法在细网格上求解一个大型非线性问题,所以我们的方法可以节省大量的计算时间.     

对流占优扩散方程的楔形基无网格法

传统的微分方程数值解方法求解对流占优扩散方程时,往往产生数值震荡现象,为了消除数值震荡,本文构建了一种新的数值求解方法――无网格方法进行数值求解。该方法采用配点法并引入一种新的楔形基函数构建了楔形基无网格方法,不需要网格划分,是一种真正的无网格方法,可以避免因为网格划分而影响计算效率。通过对新的楔形基函数的理论分析,证明了本文方法解的存在唯一性。最后,分别通过一维和二维的数值算例,表明该算法计算精度高,可以有效消除对流占优引起的数值震荡,是一种计算对流占优扩散方程数值解的高效方法。     

粘弹性流体的牛顿迭代两重网格方法(英文)

对于非线性粘弹性流体,本文提出了一种有限元两重网格方法.该方法需要首先在粗网格上求解非线性问题,然后在细网格上求解两个线性问题.这两个线性问题具有相同的刚度矩阵,只是右端项不同.我们进一步给出了该方法的误差估计.数值算例验证了理论结论,并且验证了方法的有效性.     

非定常不可压Navier-Stokes方程基于Crank-Nicolson格式的两水平变分多尺度方法

不可压缩粘性流是密度不发生变化的流体运动．它们被用来描述许多重要的物理现象,例如：天气、洋流、绕翼型流动和动脉内的血液流动．Navier-Stokes方程是不可压缩粘性流的基本方程．因此,求解Navier-Stokes方程的数值方法在近几十年得到了广泛的关注．本文主要给出非定常不可压Navier-Stokes方程基于Crank-Nicolson格式的两水平变分多尺度方法．该方法分为两步：第一步,在粗网格上求解稳定的非线性Navier-Stokes系统;第二步,在细网格上求解稳定的线性问题去校正粗网格上的解．通过该方法推导的速度的误差估计关于时间是二阶收敛的．数值实验验证了在粗细网格匹配合理的情形下,本文的方法与直接在细网格上使用单网格的变分多尺度方法相比,可以节约大量的计算时间．     

各向异性扩散问题的一个单元中心型有限体积格式

在辐射流体力学的数值模拟中,扩散算子的高效高精度离散是一个十分重要的问题.本文研究各向异性扩散方程在任意多边形网格上的数值求解问题,我们利用调和平均点和线性精确方法,构造了一个单元中心型有限体积格式.该格式只含有单元中心未知量,满足局部守恒条件,有紧凑的计算模板,在结构四边形网格上退化为一个九点格式.由于调和平均点插值算法是一个具有两点模板的二阶保正算法,因此,采用单元边上的调和平均点为插值节点,使得离散格式十分简洁,容易实施.此外,我们在格式构造中仅采用了二、三维网格的共有拓扑关系,使格式容易向三维问题推广,大部分程序代码可实现二、三维公用.我们采用典型的大变形扭曲网格及典型的扩散算例(包括连续和间断的扩散张量)对所提出的新格式进行了测试,数值算例表明,新格式在许多扭曲的多边形网格上具有二阶精度.     

一维非定常对流扩散方程非均匀网格上的高精度紧致差分格式

本文在非均匀网格上给出了求解非定常对流扩散方程的一种高精度紧致差分格式,特别适合边界层和大梯度等问题的求解.从稳态对流扩散方程入手,首先,基于非均匀网格上的泰勒级数展开对空间导数项进行离散,然后对时间项采用二阶向后欧拉差分公式,从而得到一维非定常对流扩散方程在非均匀网格上的三层全隐式紧致差分格式.新格式在时间具有二阶精度,空间具有三到四阶精度,并且是无条件稳定的.最后,通过数值实验验证了本文格式的精确性,以及在处理诸如边界层和大梯度问题上的优势.     

定常对流扩散反应方程非均匀网格上高精度紧致差分格式

本文构造了非均匀网格上求解定常对流扩散反应方程的高精度紧致差分格式.我们首先基于非均匀网格上函数的泰勒级数展开,给出了一阶导数和二阶导数的高阶近似表达式;然后将模型方程变形,借助于对流扩散方程高精度紧致格式构造的方法,结合原模型方程,得到定常对流扩散反应方程的高精度紧致差分格式;最后给出的数值算例验证了本文格式高精度和高分辨率的优点.     

A Newton-Multigrid algrithm for elasto-plastic/viscoplastic problems

This work is concerned with large scaled nonlinear systems of equations resulting from discretization of problems in plasticity and viscoplasticity in the context of the finite element method. The main purpose is to show, how standard linear multigrid methods can be applied for solving the associated linear systems of equations in the frame of the Newton-algorithm. To this end, a so-called Galerkin-approach is used for construction of coarse grid matrices by transformation of fine grid matrices. It will be shown, how this transformation can be performed very efficiently element-by-element wise. Stopping criteria for the inner iteration are based on theories for so-called inexact Newton methods, where the linear systems are only solved approximately, however which preserve the rapid local convergence of Newtons method. In the numerical examples it is demonstrated, how the proposed strategy reduces the CPU-time for large scaled problems compared to solution techniques, where the associated systems of linear equations are solved directly.     

Numerical solution of a parabolic system with coupled nonlinear boundary conditions

A numerical technique has been developed to solve a system that consists of m linear parabolic differential equations with coupled nonlinear boundary conditions. Such a system may represent chemical reactions, chemical lasers and diffusion problems. An implicit finite difference scheme is adopted to discretize the problem, and the resulting system of equations is solved by a novel technique that is a modification of the cyclic odd–even reduction and factorization (CORF) algorithm. At each time level, the system of equations is first reduced to m nonlinear algebraic equations that involve only the m unknown grid points on the nonlinear boundary. Newton's method is used to determine these m unknowns, and the corresponding Jacobian matrix can be computed and updated easily. After convergence is achieved, the remaining unknowns are solved directly. The efficiency of this technique is illustrated by the numerical computations of two examples previously solved by the cubic spline Galerkin method.     

A Composite Grid Method for Moving Conductor Eddy-Current Problem

We present fundamentals and procedures of a composite grid method (CGM) for determining eddy currents in moving conductors. Based on the finite-element method (FEM), CGM uses two separate mesh grids - one coarse and one fine - to calculate in the global region and local region separately. The results of the coarse mesh are interpolated onto the boundary of the fine mesh as its Dirichlet's condition. Then two equations are solved in the fine mesh region in order to obtain the reaction force on the boundary, which is reacted on the coarse mesh to modify its right-hand-side load vector. And the equations in the coarse mesh are re-solved. The iteration continues until the results converge. The advantage of CGM is that it allows two overlapped grids differing greatly in size to be meshed independently. Also, the program is easy to modularize and thus has great flexibility and adaptability. Above all, it ensures good numerical accuracy in each grid set. As an example indicates, CGM is effective in handling 2-D moving conductor eddy-current problems that are tedious to solve by conventional methods such as re-meshing or using a Lagrange multiplier.     

Implicit formulation with the boundary element method for nonlinear radiation of water waves

An accurate and efficient numerical method is presented for the two-dimensional nonlinear radiation problem of water waves. The wave motion that occurs on water due to an oscillating body is described under the assumption of ideal fluid flow. The governing Laplace equation is effectively solved by utilizing the GMRES (Generalized Minimal RESidual) algorithm for the boundary element method (BEM) with quadratic approximation. The intersection or corner singularity in the mixed Dirichlet–Neumann problem is resolved by introducing discontinuous elements. The fully implicit trapezoidal rule is used to update solutions at new time-steps, by considering stability and accuracy. Traveling waves generated by the oscillating body are absorbed downstream by the damping zone technique. To avoid the numerical instability caused by the local gathering of grid points, the re-gridding technique is employed, so that all the grids on the free surface may be re-distributed with an equal distance between them. The nonlinear radiation force is evaluated by means of the acceleration potential. For a mixed Dirichlet–Neumann problem in a computational domain with a wavy top boundary, the present BEM yields numerical solutions for the quadratic rate of convergence with respect to the number of boundary elements. It is also demonstrated that the present time-marching and radiation condition work successfully for nonlinear radiation problems of water waves. The results obtained from this study concur reasonably well with other numerical computations.     

Trajectory Optimization for a Multi-Stage Launch Vehicle Using Time Finite Element and Direct Collocation Methods

An indirect time finite element method is applied to solve the trajectory optimization problem for a multi-stage launch vehicle, and the numerical results are compared with the numerical solutions obtained by using a direct collocation and nonlinear program-ming method. A nonlinear programming problem is solved by the sequential quadratic programming algorithm with an augmented Lagrangian merit function, and the converged Lagrange multiplier is used for estimating the costate variables of the optimal trajectory. As a numerical example, a multi-stage launch vehicle trajectory optimization problem with a control variable constraint is solved, and the results are compared.     

An adaptive level set method based on two‐level uniform meshes and its application to dislocation dynamics

In this paper, we present an adaptive level set method for motion of high codimensional objects (e.g., curves in three dimensions). This method uses only two (or a few fixed) levels of meshes. A uniform coarse mesh is defined over the whole computational domain. Any coarse mesh cell that contains the moving object is further divided into a uniform fine mesh. The coarse‐to‐fine ratios in the mesh refinement can be adjusted to achieve optimal efficiency. Refinement and coarsening (removing the fine mesh within a coarse grid cell) are performed dynamically during the evolution. In this adaptive method, the computation is localized mostly near the moving objects; thus, the computational cost is significantly reduced compared with the uniform mesh over the whole domain with the same resolution. In this method, the level set equations can be solved on these uniform meshes of different levels directly using standard high‐order numerical methods. This method is examined by numerical examples of moving curves and applications to dislocation dynamics simulations. This two‐level adaptive method also provides a basis for using locally varying time stepping to further reduce the computational cost. Copyright © 2013 John Wiley & Sons, Ltd.     

A method of solving nonstationary three-dimensional problems of hydroelasticity with allowance for fluid failure

A numerical–analytical method of solving a class of nonlinear problems of dynamic hydroelasticity is proposed. The essence of the method consists in the expansion of all unknowns in trigonometric Fourier series in terms of the angular coordinate with subsequent application of the finite-difference method over a two-dimensional grid. It has been generally used to solve linear problems. Good accuracy of the developed algorithm is demonstrated on the basis of numerical experiments and trial calculations, and several specific examples have been solved. A number of effects associated with different behaviours of an ideally elastic and rupturable fluid, as well as a nonlinear dependence of the wave fields on the load amplitude are observed.     

Time‐parallel implicit integrators for the near‐real‐time prediction of linear structural dynamic responses

The time‐parallel framework for constructing parallel implicit time‐integration algorithms (PITA) is revisited in the specific context of linear structural dynamics and near‐real‐time computing. The concepts of decomposing the time‐domain in time‐slices whose boundaries define a coarse time‐grid, generating iteratively seed values of the solution on this coarse time‐grid, and using them to time‐advance the solution in each time‐slice with embarrassingly parallel time‐integrations are maintained. However, the Newton‐based corrections of the seed values, which so far have been computed in PITA and related approaches on the coarse time‐grid, are eliminated to avoid artificial resonance and numerical instability. Instead, the jumps of the solution on the coarse time‐grid are addressed by a projector which makes their propagation on the fine time‐grid computationally feasible while avoiding artificial resonance and numerical instability. The new PITA framework is demonstrated for a complex structural dynamics problem from the aircraft industry. Its potential for near‐real‐time computing is also highlighted with the solution of a relatively small‐scale problem on a Linux cluster system. Copyright © 2006 John Wiley & Sons, Ltd.