Numerical simulation of ground water flow via a new approach to the local radial point interpolation meshless method

A novel approach to local radial point interpolation meshless (LRPIM) method is introduced to investigate the influence of leakage on tidal response in a coastal leaky confined aquifer system, based on a local weighted residual method with the Heaviside step function as the weighting function over a local sub-domain. The present approach is a truly meshless method based only on a number of randomly located nodes. In this approach, neither global background integration mesh nor domain integration is needed. Radial basis functions (RBFs) interpolation is employed in shape function and its derivatives construction for evaluating the local weak form integrals. Due to satisfaction of kronecker delta property in RBF interpolation, no special treatment is needed to impose the essential boundary conditions. In order to obtain the optimum parameters, shape parameters of multiquadrics (MQ)-RBF are tuned and studied. The leakage has a significant impact on the tidal behaviour of the confined aquifer. The numerical results of this research indicate that both tidal amplitude of groundwater head in the aquifer and the distance over which the aquifer can be disturbed by the tide are considerably reduced by leakage. The novelty of the approach is the use of a local Heaviside weight function in the LRPIM which does not need local domain integration and only integrations on the boundary of the local domains are needed. Therefore, in this research a new local Heaviside weight function has been proposed. Numerical results are presented and compared with the results of analytical solution. It is observed that the obtained results agreed very well with the results of analytical solution. The numerical results show that the use of a local Heaviside weight function in the LRPIM is highly accurate, fast and robust. It is also noticed that this novel meshless approach using MQ radial basis is very stable.     

基于局部Petrov-Galerkin离散方案的无网格法

基于局部Petrov-Galerkin离散方案,选用自然邻近插值构造试函数,用Shepard函数作为权函数,提出了一种无网格方法(MNNPG),这种方法充分发挥了局部Petrov-Galerkin法的优势,并且结合了自然邻近插值的特点,方便引入边界条件,由于以Shepard函数的圆形支集作为积分子域,用分片中点插值来完成区域积分,无需额外背景网格,是一种真正的无网格法。本文将该无网格方法用于求解二维弹性力学边值问题,算例结果很好地吻合了精确解,表明该方法具有良好的数值精度和稳定性。     

非均质中厚板的无网格LRPIM动力学分析

用局部加权残值法建立了非均质中厚板的局部径向点插值离散系统方程,采用无网格局部径向点插值法分析了非均质中厚板的自由振动和强迫振动问题。用径向基函数耦合多项式基函数来近似试函数,用四次样条函数做为加权残值法中的权函数。所构造的形函数具有Kronecker delta性质,可以很方便地施加本质边界条件。该方法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。在计算过程中,取积分中的高斯点的材料参数来模拟问题域材料特性的变化。计算结果表明,利用该方法计算非均质中厚板的自由振动和强迫振动问题可以得到具有较高精度的解。     

ELASTIC DYNAMIC ANALYSIS OF MODERATELY THICK PLATE USING MESHLESS LRPIM

A meshless local radial point interpolation method （LRPIM） for solving elastic dy-namic problems of moderately thick plates is presented in this paper. The discretized system equation of the plate is obtained using a locally weighted residual method. It uses a radial basis function （RBF） coupled with a polynomial basis function as a trial function,and uses the quartic spline function as a test function of the weighted residual method. The shape function has the properties of the Kronecker delta function,and no additional treatment is done to impose essen-tial boundary conditions. The Newmark method for solving the dynamic problem is adopted in computation. Effects of sizes of the quadrature sub-domain and influence domain on the dynamic properties are investigated. The numerical results show that the presented method can give quite accurate results for the elastic dynamic problem of the moderately thick plate.     

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。     

一种高效的局部径向基点插值无网格方法

提出了一种弹性动力分析的高效局部径向基点插值无网格方法(MLRPI).该方法采用径向基点插值形函数近似解变量,运用局部Petrov-Galerkin法推导出了相应的离散方程,并根据波动模拟的精度要求,得到某一结点的动力方程.然后采用Newmark常平均加速度法和中心差分法相结合的显式积分格式进行时域积分,得到每个自由度的一种解耦递推格式.最后,对一平面应变问题进行了求解,比较了该文提出的解耦MI.RPI方法、常规MLRPI方法和ANSYS有限元方法的精度和计算时间,结果表明解耦MLRPI方法与常规MLRPI方法的精度相当,但计算效率大大提高.     

用无网格局部径向点插值法分析非均质中厚板的弯曲问题

用无网格局部径向点插值法分析了非均质中厚板的弯曲问题.利用虚位移原理推导了中厚板的离散系统方程.采用径向基函数耦合多项式基函数来近似试函数,用四次样条函数作为加权残值公式中的权函数.所构造成的形函数具有Kronecker delta性质,可以很方便地施加本质边界条件.此方法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,是一种真正的无网格方法.在计算过程中,取积分中的高斯点的材料参数来模拟问题域材料特性的变化.算例结果表明这种无网格方法具有效率高、精度高和易于实现等优点.     

大变形问题分析的局部Petrov-Galerkin法

在微机电系统(MEMS)的建模和模拟研究中,大变形或大移动要充分予以考虑.用有限元法分析这类问题,由于难以避免的网格畸变,使模拟效率精度降低甚至失效,无网格方法(Meshless Method)则能在分析这类问题时显示出明显的优势,无网格局部Petrov-Galerkin(MLPG)法被誉为是一种有发展前景的真正无网格法.本文进一步发展了MLPG法,通过对任意的离散分布节点采用局部径向基函数构造插值形函数和Heaviside权函数,分析方程采用局部加权弱形式离散,建立了变量仅依赖于初始构型的完全Lagrange分析格式,最后用Newton-Raphson法迭代求解.文中分析了悬臂梁典型算例和微机电开关非线性大变形问题,通过与有限元结果的比较,表明本文提出的大变形问题无网格局部Petrov-Galerkin法具有稳定性好及收敛性快等优点.     

无网格局部Petrov-Galerkin方法在弹塑性断裂力学问题中的应用

采用无网格局部Petroy-Galerkin方法来分析弹塑性断裂力学问题.这种无网格方法采用移动最小二乘法(MLS)来构造近似试函数和采用Heaviside函数作为加权残值法中的权函数,由于近似函数不满足KroneckerDelta条件,因此采用直接插值法来施加本质边界条件.如果不考虑体力,所形成的整体刚度矩阵只包含局部边界积分,而不包含局部域积分和奇异积分.采用增量Newton-Raphson迭代法来求解弹塑性增量形式的局部Petrov-Galerkin方程.数值算例结果表明,该文方法对于弹塑性断裂力学问题的求解是可行的和有效的,并且所得到的结果具有较好的精度.     

用杂交边界点法分析简支薄板的热弯曲问题

该文利用杂交边界点法对简支薄板的热弹性弯曲进行了分析计算.采用薄板的热弹性理论,通过薄板的修正变分原理建立了各向同性薄板的边界局部积分方程,域内变量使用基本解插值,而边界上的变量则用移动最小二乘法近似.计算时仅需边界上离散点的信息,无论变量近似还是数值积分都不需要网格,因此该方法是一种纯边界类型无网格方法.数值算例表明,杂交边界点法在分析薄板的热弯曲问题时具有效率高、精度高和收敛性好等优点.     

弹性力学问题的局部边界积分方程方法

提出了弹性力学平面问题的局部边界积分方程方法。这种方法是一种无网格方法,它采用移动最小二乘近似试函数,且只包含中心在所考虑节点的局部边界上的边界积分。它易于施加本质边界条件。所得系统矩阵是一个带状稀疏矩阵。它组合了伽辽金有限元法、整体边界元法和无单元伽辽金法的优点。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。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,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。     

A MESH-FREE ALGORITHM FOR DYNAMIC IMPACT ANALYSIS OF HYPERELASTICITY

A mesh-free method based on local Petrov-Galerkin formulation is presented to solve dynamic impact problems of hyperelastic material.In the present method,a simple Heaviside test function is chosen for simplifying domain integrals.Trial function is constructed by using a radial basis function(RBF)coupled with a polynomial basis function,in which the shape function possesses the kronecker delta function property.So,additional treatment is not required for imposing essential boundary conditions.Governing equations of impact problems are established and solved node by node by using an explicit time integration algorithm in a local domain,which is very similar to that of the collocation method except that numerical integration can be implemented over local domain in the present method.Numerical results for several examples show that the present method performs well in dealing with the dynamic impact problem of hyperelastic material.     

Numerical analysis of Biot's consolidation process by radial point interpolation method

An algorithm is proposed to solve Biot's consolidation problem using meshless method called a radial point interpolation method (radial PIM). The radial PIM is advantageous over the meshless methods based on moving least-square (MLS) method in implementation of essential boundary condition and over the original PIM with polynomial basis in avoiding singularity when shape functions are constructed. Two variables in Biot's consolidation theory, displacement and excess pore water pressure, are spatially approximated by the same shape functions through the radial PIM technique. Fully implicit integration scheme is proposed in time domain to avoid spurious ripple effect. Some examples with structured and unstructured nodes are studied and compared with closed-form solution or finite element method solutions.     

弹性静力问题的无网格弱-强形式结合法

基于改进的移动最小二乘(MLS)二阶导数近似,建立了一种求解弹性静力问题的无网格弱-强形式结合法(MLS-MWS)。该方法采用节点离散求解域,通过MLS构造形函数,将求解域划分为边界域和内部域,并分别使用控制方程的局部弱形式和强形式来建立离散系统方程。对强形式中涉及的近似函数二阶导数计算,提出了一种将其转化为求两次一阶导数的方法,与传统方法相比,该方法计算简单、精度高。MLS-MWS法结合了弱、强形式无网格法的优点,Neumann边界条件容易满足,并且只需在边界区域进行积分。文中应用该方法分析了两个弹性力学平面问题,分析结果表明本文方法具有良好的精度和收敛性。     

Local Petrov-Galerkin method for a thin plate

The meshless local Petrov-Galerkin (MLPG) method for solving the bendingproblem 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 symmetricweak form. The present method was a truly meshless one as it did not need a finite elementor boundary element mesh, either for purpose of interpolation of the solution, or for theintegration of the energy. All integrals could be easily evaluated over regularly shapeddomains ( in general, spheres in three-dimensional problems ) and their boundaries. Theessential boundary conditions were enforced by the penalty method. Several numericalexamples were presented to illustrate the implementation and performance of the presentmethod. The numerical examples presented show that high accuracy can be achieved forarbitrary grid geometries for clamped and simply-supported edge conditions. No postprocessing procedure is required to computer the strain and stress, since the originalsolution from the present method, using the moving least squares approximation, is already smooth enough.     

Extension of domain‐free discretization method to simulate compressible flows over fixed and moving bodies

This paper is the first endeavour to present the local domain‐free discretization (DFD) method for the solution of compressible Navier–Stokes/Euler equations in conservative form. The discretization strategy of DFD is that for any complex geometry, there is no need to introduce coordinate transformation and the discrete form of governing equations at an interior point may involve some points outside the solution domain. The functional values at the exterior dependent points are updated at each time step to impose the wall boundary condition by the approximate form of solution near the boundary. Some points inside the solution domain are constructed for the approximate form of solution, and the flow variables at constructed points are evaluated by the linear interpolation on triangles. The numerical schemes used in DFD are the finite element Galerkin method for spatial discretization and the dual‐time scheme for temporal discretization. Some numerical results of compressible flows over fixed and moving bodies are presented to validate the local DFD method. Copyright © 2006 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).     

壳结构的无网格局部Petrov-Galerkin方法

无网格近似函数具有高度光滑性,能够很好的逼近曲壳表面及其位移场。无网格局部Petrov-Galerkin方法不论插值还是离散都不需要单元,是一种真正的无网格方法。本文基于无网格局部Petrov-Galerkin方法的基本原理,采用移动最小二乘插值,利用控制微分方程弱形式,建立了Mindlin壳结构的无网格局部Petrov-Galerkin分析方法,用屋顶壳、受夹圆柱壳、几何非线性圆柱壳作为计算实例分析了求解精度、收敛性和稳定性,并与精确解和有限元计算结果进行了对比,表明该方法计算精度高及收敛性好。     

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.     

Development of circular function‐based gas‐kinetic scheme (CGKS) on moving grids for unsteady flows through oscillating cascades

In this paper, the circular function‐based gas‐kinetic scheme (CGKS), which was originally developed for simulation of flows on stationary grids, is extended to solve moving boundary problems on moving grids. Particularly, the unsteady flows through oscillating cascades are our major interests. The main idea of the CGKS is to discretize the macroscopic equations by the finite volume method while the fluxes at the cell interface are evaluated by locally reconstructing the solution of the continuous Boltzmann Bhatnagar–Gross–Krook equation. The present solver is based on the fact that the modified Boltzmann equation, which is expressed in a moving frame of reference, can recover the corresponding macroscopic equations with Chapman–Enskog expansion analysis. Different from the original Maxwellian function‐based gas‐kinetic scheme, in improving the computational efficiency, a simple circular function is used to describe the equilibrium state of distribution function. Considering that the concerned cascade oscillating problems belong to cases that the motion of surface boundary is known a priori, the dynamic mesh method is suitable and is adopted in the present work. In achieving the mesh deformation with high quality and efficiency, a hybrid dynamic mesh method named radial basic functions‐transfinite interpolation is presented and applied for cascade geometries. For validation, several numerical test cases involving a wide range are investigated. Numerical results show that the developed CGKS on moving grids is well applied for cascade oscillating flows. And for some cases where nonlinear effects are strong, the solution accuracy could be effectively improved by using the present method. Copyright © 2017 John Wiley & Sons, Ltd.