Adaptive mixed least squares Galerkin/Petrov finite element method for stationary conduction convection problems

An adaptive mixed least squares Galerkin/Petrov finite element method （FEM） is developed for stationary conduction convection problems. The mixed least squares Galerkin/Petrov FEM is consistent and stable for any combination of discrete velocity and pressure spaces without requiring the Babuska-Brezzi stability condition. Using the general theory of Verfiirth, the posteriori error estimates of the residual type are derived. Finally, numerical tests are presented to illustrate the effectiveness of the method.

A NONLINEAR GALERKIN/PETROV-LEAST SQUARES MIXED ELEMENT METHOD FOR THE STATIONARY NAVIER-STOKES EQUATIONS

A nonlinear Galerkin/Petrov-least squares mixed element (NGPLSME) method for the stationary Navier-Stokes equations is presented and analyzed. The scheme is that Petrov-least squares forms of residuals are added to the nonlinear Galerkin mixed element method so that it is stable for any combination of discrete velocity and pressure spaces without requiring the Babu*lka-Brezzi stability condition. The existence, uniqueness and convergence (at optimal rate) of the NGPLSME solution is proved in the case of sufficient viscosity (or small data).

ON THE BREAKDOWNS OF THE GALERKIN AND LEAST-SQUARES METHODS

The Galerkin and least-squares methods are two classes of the most popular Krylov subspace methOds for solving large linear systems of equations. Unfortunately, both the methods may suffer from serious breakdowns of the same type: In a breakdown situation the Galerkin method is unable to calculate an approximate solution, while the least-squares method, although does not really break down, is unsucessful in reducing the norm of its residual. In this paper we first establish a unified theorem which gives a relationship between breakdowns in the two methods. We further illustrate theoretically and experimentally that if the coefficient matrix of a lienar system is of high defectiveness with the associated eigenvalues less than 1, then the restarted Galerkin and least-squares methods will be in great risks of complete breakdowns. It appears that our findings may help to understand phenomena observed practically and to derive treatments for breakdowns of this type.

Least-Squares及Galerkin谱元方法求解环形区域内的泊松方程

为研究基于Least-Squares变分及Galerkin变分两种形式的谱元方法的求解特性,推导了极坐标系中采用两种变分方法求解环形区域内Poisson方程时对应的弱解形式,采用Chebyshev多项式构造插值基函数进行空间离散,得到两种谱元方法对应的代数方程组,由此分析了系数矩阵结构的特点。数值计算结果显示:Least-Squares谱元方法为实现方程的降阶而引入新的求解变量,使得代数方程组形式更为复杂,但边界条件的处理比Galerkin谱元方法更为简单;两种谱元方法均能求解极坐标系中的Poisson方程且能获得高精度的数值解,二者绝对误差分布基本一致;固定单元内的插值阶数时,增加单元数可减小数值误差,且表现出代数精度的特点,误差降低速度较慢,而固定单元数时,在一定范围内数值误差随插值阶数的增加而减小的速度更快,表现出谱精度的特点;单元内插值阶数较高时,代数方程组系数矩阵的条件数急剧增多,方程组呈现病态,数值误差增大,这一特点限制了单元内插值阶数的取值。研究内容对深入了解两种谱元方法在极坐标系中求解Poisson方程时的特点、进一步采用相关分裂算法求解实际流动问题具有参考价值。

A MESHLESS LOCAL PETROV-GALERKIN METHOD FOR GEOMETRICALLY NONLINEAR PROBLEMS

Nonlinear formulations of the meshless local Petrov-Galerkin （MLPG） method are presented for geometrically nonlinear problems. The method requires no mesh in computation and therefore avoids mesh distortion difficulties in the large deformation analysis. The essential boundary conditions in the present formulation axe imposed by a penalty method. An incremental and iterative solution procedure is used to solve geometrically nonlinear problems. Several examples are presented to demonstrate the effectiveness of the method in geometrically nonlinear problems analysis. Numerical results show that the MLPG method is an effective one and that the values of the unknown variable are quite accurate.

NUMERICAL ANALYSIS OF MINDLIN SHELL BY MESHLESS LOCAL PETROV-GALERKIN METHOD

The objectives of this study are to employ the meshless local Petrov-Galerkin method （MLPGM） to solve three-dimensional shell problems. The computational accuracy of MLPGM for shell problems is affected by many factors, including the dimension of compact support domain, the dimension of quadrture domain, the number of integral cells and the number of Gauss points. These factors＇ sensitivity analysis is to adopt the Taguchi experimental design technology and point out the dimension of the quadrature domain with the largest influence on the computational accuracy of the present MLPGM for shells and give out the optimum combination of these factors. A few examples are given to verify the reliability and good convergence of MLPGM for shell problems compared to the theoretical or the finite element results.

求解第二型Fredholm积分方程的迭代快速小波Petrov-Galerkin方法

给出求解带光滑核的第二型Fredholm积分方程的迭代快速小波Petrov-Galerkin方法,并分析该方法的收敛性以及计算复杂度,证明该方法可达到超收敛阶。

LOCAL PETROV-GALERKIN METHOD FOR A THIN PLATE

The meshless local Petrov_Galerkin (MLPG) method for solving the bending problem of the thin plate were presented and discussed. The method used the moving least_squares approximation to interpolate the solution variables, and employed a local symmetric weak form. The present method was a truly meshless one as it did not need a finite element or boundary element mesh, either for purpose of interpolation of the solution, or for the integration of the energy. All integrals could be easily evaluated over regularly shaped domains (in general, spheres in three_dimensional problems) and their boundaries. The essential boundary conditions were enforced by the penalty method. Several numerical examples were presented to illustrate the implementation and performance of the present method. The numerical examples presented show that high accuracy can be achieved for arbitrary grid geometries for clamped and simply_supported edge conditions. No post processing procedure is required to computer the strain and stress, since the original solution from the present method, using the moving least squares approximation, is already smooth enough.

Quasi Ellipsoid Gear Surface Reconstruction Based on Meshless Local Petrov-Galerkin Method and Transmission Characteristic

Special transmission 3D model simulation must be based on surface discretization and reconstruction, but special transmission usually has complicated tooth shape and movement, so present software can＇t provide technical support for special transmission 3D model simulation. Currently, theoretical calculation and experimental method are difficult to exactly solve special transmission contact analysis problem. How to reduce calculation and computer memories consume and meet calculation precision is key to resolve special transmission contact analysis problem. According to 3D model simulation and surface reconstruction of quasi ellipsoid gear is difficulty, this paper employes meshless local Petrov-Galerkin （MLPG） method. In order to reduce calculation and computer memories consume, we disperse tooth mesh into finite points--sparseness points cloud or grid mesh, and then we do interpolation reconstruction in some necessary place of the 3D surface model during analysis. Moving least square method （MLSM） is employed for tooth mesh interpolation reconstruction, there are some advantages to do interpolation by means of MLSM, such as high precision, good flexibility and no require of tooth mesh discretization into units. We input the quasi ellipsoid gear reconstruction model into simulation software, we complete tooth meshing simulation. Simulation transmission ratio during meshing period was obtained, compared with theoretical transmission ratio, the result inosculate preferably. The method using curve reconstruction realizes surface reconstruction, reduce simulation calculation enormously, so special gears simulation can be realized by minitype computer. The method provides a novel solution for special transmission 3D model simulation analysis and contact analysis.

The complex variable meshless local Petrov-Galerkin method of solving two-dimensional potential problems

Based on the complex variable moving least-square（CVMLS） approximation and a local symmetric weak form,the complex variable meshless local Petrov-Galerkin（CVMLPG） method of solving two-dimensional potential problems is presented in this paper.In the present formulation,the trial function of a two-dimensional problem is formed with a one-dimensional basis function.The number of unknown coefficients in the trial function of the CVMLS approximation is less than that in the trial function of the moving least-square（MLS） approximation.The essential boundary conditions are imposed by the penalty method.The main advantage of this approach over the conventional meshless local Petrov-Galerkin（MLPG） method is its computational efficiency.Several numerical examples are presented to illustrate the implementation and performance of the present CVMLPG method.

GALERKIN-PETROV METHODS OF TOEPLITZ OPERATORS ON DIRICHLET SPACE

The convergence of several Galerkin-Petrov methods, including polynomial collocation and analytic element collocation methods of Toeplitz operators on Dirichlet space, is established. In particular, it is shown that such methods converge if the basis and test function own certain circular symmetry.

A meshless local Petrov–Galerkin method for solving the neutron diffusion equation

The goal of this study is to solve the neutron diffusion equation by using a meshless method and evaluate its performance compared to traditional methods. This paper proposes a novel method based on coupling the meshless local Petrov–Galerkin approach and the moving least squares approximation. This computational procedure consists of two main steps. The first involved applying the moving least squares approximation to construct the shape function based on the problem domain. Then, the obtained shape function was used in the meshless local Petrov–Galerkin method to solve the neutron diffusion equation.Because the meshless method is based on eliminating the mesh-based topologies, the problem domain was represented by a set of arbitrarily distributed nodes. There is no need to use meshes or elements for field variable interpolation. The process of node generation is simply and fully automated, which can save time. As this method is a local weak form, it does not require any background integration cells and all integrations are performed locally over small quadrature domains. To evaluate the proposed method,several problems were considered. The results were compared with those obtained from the analytical solution and a Galerkin finite element method. In addition, the proposed method was used to solve neutronic calculations in thesmall modular reactor. The results were compared with those of the citation code and reference values. The accuracy and precision of the proposed method were acceptable. Additionally, adding the number of nodes and selecting an appropriate weight function improved the performance of the meshless local Petrov–Galerkin method. Therefore, the proposed method represents an accurate and alternative method for calculating core neutronic parameters.

变系数Volterra型积分微分方程的2种Legendre谱Galerkin数值积分方法

为了进一步提高求解Volterra型积分微分的数值精度,针对一种变系数Volterra型积分微分方程,提出了2种Legendre谱Galerkin数值积分法。采用Galerkin Legendre数值积分对Volterra型积分微分方程的积分项进行预处理,对其构造Legendre tau格式,同时用Chebyshev-Gauss-Lobatto配置点对变系数和积分项部分进行计算,并通过对方程的定义区间进行分解,提出了一种多区间Legendre谱Galerkin数值积分法。该方法的格式对于奇数阶模型具有对称结构。此外,通过引入Volterra型积分微分方程的最小二乘函数,构造了Legendre谱Galerkin最小二乘数值积分法。该方法对应的代数方程系数矩阵是对称正定的。数值算例验证了这2种Legendre谱Galerkin数值积分方法的高阶精度和有效性。

A LEAST SQUARE BASED WEAK GALERKIN FINITE ELEMENT METHOD FOR SECOND ORDER ELLIPTIC EQUATIONS IN NON-DIVERGENCE FORM

This article is devoted to establishing a least square based weak Galerkin method for second order elliptic equations in non-divergence form using a discrete weak Hessian operator.Naturally,the resulting linear system is symmetric and positive definite,and thus the algorithm is easy to implement and analyze.Convergence analysis in the H2 equivalent norm is established on an arbitrary shape regular polygonal mesh.A superconvergence result is proved when the coefficient matrix is constant or piecewise constant.Numerical examples are performed which not only verify the theoretical results but also reveal some unexpected superconvergence phenomena.

A complex variable meshless local Petrov-Galerkin method for transient heat conduction problems

On the basis of the complex variable moving least-square （CVMLS） approximation, a complex variable meshless local Petrov-Galerkin （CVMLPG） method is presented for transient heat conduction problems. The method is developed based on the CVMLS approximation for constructing shape functions at scattered points, and the Heaviside step function is used as a test function in each sub-domain to avoid the need for a domain integral in symmetric weak form. In the construction of the well-performed shape function, the trial function of a two-dimensional （2D） problem is formed with a one-dimensional （1D） basis function, thus improving computational efficiency. The numerical results are compared with the exact solutions of the problems and the finite element method （FEM）. This comparison illustrates the accuracy as well as the capability of the CVMLPG method.

Structural Reliability Assessment by a Modified Spectral Stochastic Meshless Local Petrov-Galerkin Method

This study presents a new tool for solving stochastic boundary-value problems. This tool is created by modify the previous spectral stochastic meshless local Petrov-Galerkin method using the MLPG5 scheme. This modified spectral stochastic meshless local Petrov-Galerkin method is selectively applied to predict the structural failure probability with the uncertainty in the spatial variability of mechanical properties. Except for the MLPG5 scheme, deriving the proposed spectral stochastic meshless local Petrov-Galerkin formulation adopts generalized polynomial chaos expansions of random mechanical properties. Predicting the structural failure probability is based on the first-order reliability method. Further comparing the spectral stochastic finite element-based and meshless local Petrov-Galerkin-based predicted structural failure probabilities indicates that the proposed spectral stochastic meshless local Petrov-Galerkin method predicts the more accurate structural failure probability than the spectral stochastic finite element method does. In addition, generating spectral stochastic meshless local Petrov-Galerkin results are considerably time-saving than generating Monte-Carlo simulation results does. In conclusion, the spectral stochastic meshless local Petrov-Galerkin method serves as a time-saving tool for solving stochastic boundary-value problems sufficiently accurately.

基于Voronoi结构的无网格局部Petrov-Galerkin方法

基于自然邻结点近似位移函数提出了一种用于求解弹性力学平面问题的无网格局部Petrov-Galerkin方法.这种方法在结构的求解域Ω内任意布置离散的结点,并且利用需求结点的自然邻结点和Voronoi结构来构造整体求解的近似位移函数.对于构造好的近似位移函数,在局部的Delaunay三角形子域上采用局部Petrov-Galerkin方法建立整体求解的平衡控制方程,这样平衡方程的积分可在背景三角形积分网格的形心上解析计算得到,而采用标准Galerkin方法的自然单元法需要三个数值积分点.该方法能够准确地施加边界条件,得到的系统矩阵是带状稀疏矩阵,对软件用户来说,它还是一种完全的、真正的无网格方法.所得计算结果表明,该方法的计算精度与有限元法四边形单元相当,但计算和形成系统平衡方程的时间比有限元法四边形单元提高了将近一倍,是一种理想的数值求解方法.

定常的Navier-Stokes方程的非线性Galerkin/Petrov最小二乘混合元法

给出定常的Navier_Stokes方程的一种非线性Galerkin/Petrov最小二乘混合元法 ,该方法是将余量形式的Petrov最小二乘方法与非线性Galerkin混合元结合起来 ,使得速度和压力的混合元空间无需满足离散的Babu ka_Brezzi稳定性条件 ,从而使得它们的有限元空间可以任意选择· 并证明该方法的解的存在唯一性和收敛性·

弹性地基板分析的局部Petrov-Galerkin方法

利用弹性地基板控制微分方程的等效积分对称弱形式和对解变量(挠度)采用移动最小二乘近似函数进行插值,研究了无网格局部Petrov-Galerkin(MLPG)方法在弹性地基板弯曲问题中的应用。它不需要任何形式的网格划分,所有的积分都在规则形状的子域及其边界上进行,并用罚因子法施加本质边界条件。数值算例表明,MLPG方法不但能够求解弹性静力学问题,而且在求解弹性地基板问题时仍具有收敛快,精度高的特点。

用无网格局部Petrov-Galerkin方法分析Winkler弹性地基板

利用Winkler弹性地基板控制微分方程的等效积分对称弱形式,同时对变量(挠度)采用移动最小二乘近似函数进行插值,研究了无网格局部Petrov Galerkin方法在弹性地基板弯曲问题中的应用.它不需要任何形式的网格划分,所有的积分都在规则形状的子域及其边界上进行,并用罚因子法施加本质边界条件.数值算例说明,无网格局部Petrov Galerkin法不但能够求解弹性静力学问题,而且在求解弹性地基板问题时仍具有收敛快、稳定性好和精度高的特点.