Research on the companion solution for a thin plate in the meshless local boundary integral equation method

The meshless local boundary integral equation method is a currently developed numerical method, which combines the advantageous features of Galerkin finite element method ( GFEM ), boundary element method (BEM) and element free Galerkin method (EFGM), and is a truly meshless method possessing wide prospects in engineeringapplications. The companion solution and all the other formulas required in the meshless local boundary integral equation for a thin plate were presented, in order to make this method apply to solve the thin plate problem.     

反平面断裂问题的无单元伽辽金比例边界法

将比例边界法与无单元伽辽金法相结合,建立了反平面断裂分析的无单元伽辽金比例边界法。这是一种边界型无网格法,在环向方向上采用无单元伽辽金法进行离散,因此计算时仅需要边界上的节点信息,不需要边界元所要求的基本解。为了便于施加本质边界条件,通过建立节点值和虚拟节点值之间的关系给出了修正的移动最小二乘形函数。在径向方向上,该方法利用解析的方法求解,因此是一种半解析的数值方法。最后,给出了数值算例,并验证了所提方法后处理简单和计算精度高的特点,适合于求解反平面断裂问题。     

Steady‐state solution of incompressible Navier–Stokes equations using discrete least‐squares meshless method

A fractional step method for the solution of the steady state incompressible Navier–Stokes equations is proposed in this paper in conjunction with a meshless method, named discrete least‐squares meshless (DLSM). The proposed fractional step method is a first‐order accurate scheme, named semi‐incremental fractional step method, which is a general form of the previous first‐order fractional step methods, i.e. non‐incremental and incremental schemes. One of the most important advantages of the proposed scheme is its capability to use large time step sizes for the solution of incompressible Navier–Stokes equations. DLSM method uses moving least‐squares shape functions for function approximation and discrete least‐squares technique for discretization of the governing differential equations and their boundary conditions. As there is no need for a background mesh, the DLSM method can be called a truly meshless method and enjoys symmetric and positive‐definite properties. Several numerical examples are used to demonstrate the ability and the efficiency of the proposed scheme and the discrete least‐squares meshless method. The results are shown to compare favorably with those of the previously published works. Copyright © 2010 John Wiley & Sons, Ltd.     

An efficient discontinuous Galerkin method for aeroacoustic propagation

An efficient discontinuous Galerkin formulation is applied to the solution of the linearized Euler equations and the acoustic perturbation equations for the simulation of aeroacoustic propagation in two‐dimensional and axisymmetric problems, with triangular and quadrilateral elements. To improve computational efficiency, a new strategy of variable interpolation order is proposed in addition to a quadrature‐free approach and parallel implementation. Moreover, an accurate wall boundary condition is formulated on the basis of the solution of the Riemann problem for a reflective wall. Time discretization is based on a low dissipation formulation of a fourth‐order, low storage Runge–Kutta scheme. Along the far‐field boundaries a perfectly matched layer boundary condition is used. For the far‐field computations, the integral formulation of Ffowcs Williams and Hawkings is coupled with the near‐field solver. The efficiency and accuracy of the proposed variable order formulation is assessed for realistic geometries, namely sound propagation around a high‐lift airfoil and the Munt problem. Copyright © 2011 John Wiley & Sons, Ltd.     

A meshless method of a thin plate

A meshless approach to analysis of arbitrary Kirchhoff plates by the local boundary integral equation(LBIE) method is presented. The method combines the advantageous features of, all the three methods: the Galerkin finite element method (GFEM), the boundary element method (BEM) and the element-free Galerkin method (EFGM). It is a truly meshless method, which means that the discretization is independent of geometric subdivision into elements or cells, but is only based on a set of nodes (ordered or scattered) over a domain in question. It involves only boundary integration, however, over a local boundary centered at the node in question; It poses no difficulties in satisfying the essential boundary conditions while leading to banded and sparse system matrices using the moving least square (MLS) approximations. It is shown that high accuracy can be achieved for arbitrary geometries for clamped and simply-supported edge conditions. The method is found to be simple, efficient, and attractive. Project supported by the National Science Foundation of China (No. 19972019).     

Artificial boundary method for the exterior Stokes flow in three dimensions

In this paper, the artificial boundary method is considered for the numerical simulation of the exterior Stokes flow in three dimensions. First, an exact relation between the normal stress and the velocity field is obtained on a spherical artificial boundary. With the relation specified on the artificial boundary, the original problem is reduced to a new one only defined on a finite domain. After that, an variational problem equivalent to the reduced problem is derived. By truncating the series term in the formulation, a sequence of approximate variational problems are obtained, which can then be solved with a suitable finite‐element scheme. Finally, a numerical example is presented to show the performance of the method. Copyright © 2003 John Wiley & Sons, Ltd.     

Time‐accurate stabilized finite‐element model for weakly nonlinear and weakly dispersive water waves

Introduction of a time‐accurate stabilized finite‐element approximation for the numerical investigation of weakly nonlinear and weakly dispersive water waves is presented in this paper. To make the time approximation match the order of accuracy of the spatial representation of the linear triangular elements by the Galerkin finite‐element method, the fourth‐order time integration of implicit multistage Padé method is used for the development of the numerical scheme. The streamline‐upwind Petrov–Galerkin (SUPG) method with crosswind diffusion is employed to stabilize the scheme and suppress the spurious oscillations, usually common in the numerical computation of convection‐dominated flow problems. The performance of numerical stabilization and accuracy is addressed. Treatments of various boundary conditions, including the open boundary conditions, the perfect reflecting boundary conditions along boundaries with irregular geometry, are also described. Numerical results showing the comparisons with analytical solutions, experimental measurements, and other published numerical results are presented and discussed. Copyright © 2007 John Wiley & Sons, Ltd.     

Singular boundary method for 2D dynamic poroelastic problems

This paper extends a strong-form meshless boundary collocation method, named the singular boundary method (SBM), for the solution of dynamic poroelastic problems in the frequency domain, which is governed by Biot equations in the form of mixed displacement–pressure formulation. The solutions to problems are represented by using the fundamental solutions of the governing equations in the SBM formulations. To isolate the singularities of the fundamental solutions, the SBM uses the concept of the origin intensity factors to allow the source points to be placed on the physical boundary coinciding with collocation points, which avoids the auxiliary boundary issue of the method of fundamental solutions (MFS). Combining with the origin intensity factors of Laplace and plane strain elastostatic problems, this study derives the SBM formulations for poroelastic problems. Five examples for 2D poroelastic problems are examined to demonstrate the efficiency and accuracy of the present method. In particular, we test the SBM to the multiply connected domain problem, the multilayer problem and the poroelastic problem with corner stress singularities, which are all under varied ranges of frequencies.     

Kinetic meshless method for compressible flows

We present a grid‐free or meshless approximation called the kinetic meshless method (KMM), for the numerical solution of hyperbolic conservation laws that can be obtained by taking moments of a Boltzmann‐type transport equation. The meshless formulation requires the domain discretization to have very little topological information; a distribution of points in the domain together with local connectivity information is sufficient. For each node, the connectivity consists of a set of nearby nodes which are used to evaluate the spatial derivatives appearing in the conservation law. The derivatives are obtained using a modified form of the least‐squares approximation. The method is applied to the Euler equations for inviscid flow and results are presented for some 2‐D problems. The ability of the new scheme to accurately compute inviscid flows is clearly demonstrated, including good shock capturing ability. Comparisons with other grid‐free methods are made showing some advantages of the current approach. Copyright © 2007 John Wiley & Sons, Ltd.     

无单元Galerkin法和边界元耦合法

无网格方法与有限元法或边界元法耦合是无网格方法处理边界条件的方法之一,在无网格方法中研究无网格方法与有限元法或边界元法耦合的研究显得非常重要.本文在无单元Galerkin法和边界元法的基础上,基于无单元Galerkin法子域和边界元法子域的界面上位移连续和面力平衡条件,提出了一种新的无单元Galerkin法和边界元法的直接耦合方法,对弹性力学问题详细推导了在整个求解域上的耦合公式.与以往的耦合法相比,这种方法简单直观,不需要增加新的耦合区域,也不需要建立新的逼近函数来保证界面位移的连续性.算例结果表明,该方法具有较好的计算精度.     

基于流形覆盖思想的无网格方法的研究

本语言基于流形思想,利用有限覆盖,单位分解等概念,引入建立在覆盖上的覆盖函数和具有紧支撑特性的单位分解函数,建立场逼近的近似表达,由弱形式的Galerkin变分得到数值分析模型,结合边界条件用于边值问题的求解,由此建立了一类新的无网格数值方法,论文采用这种方法分析了平面弹性问题,分析了体积闭锁现象,h、p型收敛性等,提出了一种选择覆盖大小的方案,且对狭长城采用了椭圆覆盖形式,取得了比较好的效果。     

A Galerkin spectral method based on helical‐wave decomposition for the incompressible Navier–Stokes equations

This paper presents a global Galerkin spectral method for solving the incompressible Navier–Stokes equations in three‐dimensional bounded domains. The method is based on helical‐wave decomposition (HWD), which uses the vector eigenfunctions of the curl operator as orthogonal basis functions. We shall first review the general theory of HWD in an arbitrary simply connected domain, along with some new developments. We then employ the HWD to construct a Galerkin spectral method. The current method innovates the existing HWD‐based spectral method by (a) adding a series of auxiliary fields to the HWD of the velocity field to fulfill the no‐slip boundary condition and to settle the convergence problem of the HWD of the curl fields, and (b) providing a pseudo‐spectral method that utilizes a fast spherical harmonic transform algorithm and Gaussian quadrature to calculate the nonlinear term in the Navier–Stokes equations. The auxiliary fields are uniquely determined by solving the Stokes and Stokes‐like equations under adequate boundary conditions. The implementation of the method under the spherical geometry is presented in detail. Several numerical examples are provided to validate the proposed method. The method can be easily extended to other domains once the helical‐wave bases, which depend only on the geometry of the domains, are available. Copyright © 2015 John Wiley & Sons, Ltd.     

A 3D shell-like approach using element-free Galerkin method for analysis of thin and thick plate structures

A new efficient meshless method based on the element-free Galerkin method is proposed to analyze the static deformation of thin and thick plate structures in this paper. Using the new 3D shell-like kinematics in analogy to the solid-shell concept of the finite element method, discretization is carried out by the nodes located on the upper and lower surfaces of the structures. The approximation of all unknown field variables is carried out by using the moving least squares (MLS) approximation scheme in the in-plane directions, while the linear interpolation is applied through the thickness direction. Thus, different boundary conditions are defined only using displacements and penalty method is used to enforce the essential boundary conditions. The constrained Galerkin weak form, which incorporates only displacement degrees of freedom (d.o.f.s), is derived. A modified 3D constitutive relationship is adopted in order to avoid or eliminate some self-locking effects. The numeric efficiency of the proposed meshless formulation is illustrated by the numeric examples.     

加权最小二乘无网格法

在最小二乘法和移动最小二乘近似的基础上提出了加权最小二乘无网格法．该方法除节点外又引入了一些辅助点，控制方程在所有节点和辅助点处的残差用最小二乘法予以消除，边界条件用罚函数法引入．另外对移动最小二乘近似进行了改进，并给出了最小二乘法中泛函的简化格式，因而提高了计算效率．与配点法相比，新方法精度高，稳定性好，并且系数矩阵是对称正定矩阵．与Galerkin法相比，该方法不需要进行高斯积分，因而计算量小．算例表明该方法具有效率高、精度高和稳定性好等优点，并且易于实现．     

基于核重构思想的配点型无网格方法的研究--一维问题

无网格方法按其离散原理可分为Galerkin型、配点型等。其中Galerkin型无网格方法的实施需要背景网格，不属于真正的无网格法；配点型无网格方法的实施不需要背景网格，是真正的无网格法。本文首先介绍了重构核点法的基本原理，然后基于核重构思想，与配点法相结合，以一维问题为例，研究了配点型无网格方法，对该方法构造过程中的近似函数及其导数的计算、修正函数的计算及方法的实现等问题进行了探讨。并结合若干典型算例，检验了其计算精度与收敛姓。     

Galerkin domain decomposition procedures for parabolic equations on rectangular domain

Galerkin domain decomposition procedures for parabolic equations with three cases of boundary conditions on rectangular domain are discussed. These procedures are non‐iterative and non‐overlapping ones. They rely on implicit Galerkin method in the sub‐domains and integral mean method on the inter‐domain boundaries to present explicit flux calculation. Thus, the parallelism can be achieved by the use of these procedures. Two kinds of approximating schemes are presented. Because of the explicit nature of the flux calculation, a less severe time‐step constraint is derived to preserve stability. To bound L²‐norm error estimates, new elliptic projections are established and analyzed. Numerical experiments are provided to confirm theoretical results. Copyright © 2009 John Wiley & Sons, Ltd.     

A high‐order scheme for the incompressible Navier–Stokes equations with open boundary condition

We present a numerical scheme to solve the incompressible Navier–Stokes equations with open boundary condition. After replacing the incompressibility constraint by the pressure Poisson equation, the key is how to give an appropriate boundary condition for the pressure Poisson equation. We propose a new boundary condition for the pressure on the open boundary. Some numerical experiments are presented to verify the accuracy and stability of scheme. Copyright © 2013 John Wiley & Sons, Ltd.     

Parallelization of a vorticity formulation for the analysis of incompressible viscous fluid flows

A parallel computer implementation of a vorticity formulation for the analysis of incompressible viscous fluid flow problems is presented. The vorticity formulation involves a three‐step process, two kinematic steps followed by a kinetic step. The first kinematic step determines vortex sheet strengths along the boundary of the domain from a Galerkin implementation of the generalized Helmholtz decomposition. The vortex sheet strengths are related to the vorticity flux boundary conditions. The second kinematic step determines the interior velocity field from the regular form of the generalized Helmholtz decomposition. The third kinetic step solves the vorticity equation using a Galerkin finite element method with boundary conditions determined in the first step and velocities determined in the second step. The accuracy of the numerical algorithm is demonstrated through the driven‐cavity problem and the 2‐D cylinder in a free‐stream problem, which represent both internal and external flows. Each of the three steps requires a unique parallelization effort, which are evaluated in terms of parallel efficiency. Copyright © 2002 John Wiley & Sons, Ltd.     

弹性力学问题的局部Petrov—Galerkin方法

提出了弹性力学平面问题的局部Petrov-Galerkin方法,这是一种真正的无网格方法。这种方法采和移动最小二乘近似函数作为试函数,并且采用移动最小二乘近似函数的权函数作为加权残值法加权函数;同时这种方法只包含中心在所考虑点处的规则局部区域上以及局部边界上的积分,所得系统矩阵是一个带状稀疏矩阵,该方法可以容易推广到求解非线性问题以及非均匀介质的力学问题。还计算了两个弹性力学平面问题的例子,给出了位移和能量的索波列夫模及其相对误差。所得计算结果证明：该方法是一种具有收敛快、精度高、简便有效的通用方法;在工程中具有广阔的应用前景。     

Boundary discretization for high‐order discontinuous Galerkin computations of tidal flows around shallow water islands

In this paper some preliminary results concerning the application of the high‐order discontinuous Galerkin (DG) method for the resolution of realistic problems of tidal flows around shallow water islands are presented. In particular, tidal flows are computed around the Rattray island located in the Great Barrier Reef. This island is a standard benchmark problem well documented in the literature providing useful in situ measurements for validation of the model. Realistic elements of the simulation are a tidal flow forcing, a variable bathymetry and a non‐trivial coastline. The computation of tidal flows in shallow water around an island is very similar to the simulation of the Euler equations around bluff bodies in quasi‐steady flows. The main difference lies in the high irregularity of islands' shapes and in the fact that, in the framework of large‐scale ocean models, the number of elements to represent an island is drastically limited compared with classical engineering computations. We observe that the high‐order DG method applied to shallow water flows around bluff bodies with poor linear boundary representations produces oscillations and spurious eddies. Surprisingly those eddies may have the right size and intensity but may be generated by numerical diffusion and are not always mathematically relevant. Although not interested in solving accurately the boundary layers of an island, we show that a high‐order boundary representation is mandatory to avoid non‐physical eddies and spurious oscillations. It is then possible to parametrize accurately the subgrid‐scale processes to introduce the correct amount of diffusion in the model. The DG results around the Rattray island are eventually compared with current measurements and reveal good agreement. Copyright © 2008 John Wiley & Sons, Ltd.