Simulation of Seawater Intrusion in Coastal Confined Aquifer Using a Point Collocation Method Based Meshfree Model

Seawater intrusion caused by groundwater over-exploitation from coastal aquifers poses a severe problem in many regions. Formulation of proper pumping strategy using a simulation model can assure sustainable supply of fresh water from the coastal aquifers. The focus of the present study is on the development of a numerical model based on Meshfree (MFree) method to study the seawater intrusion problem. For the simulation of seawater intrusion problem, widely used models are based on Finite Difference (FDM) and Finite Element (FEM) Methods, which demand well defined grids/meshes and considerable pre-processing efforts. Here, MFree Point Collocation Method (PCM) based on the Radial Basis Function (RBF) is proposed for the simulation. Diffusive interface approach with density-dependent dispersion and solution of flow and solute transport is adopted. These equations are solved using PCM with appropriate boundary conditions. The developed model has been verified with Henry’s problem, and found to be satisfactory. Further the model has been applied to another established problem and an attempt is made to examine the influence of important system parameters including pumping and recharge on the seawater intrusion. The PCM based MFree model is found computationally efficient as preprocessing is avoided when compared to other numerical methods.

Jacobi Collocation Methods for Solving Generalized Space-Fractional Burgers Equations

The aim of this paper is to obtain the numerical solutions of generalized space-fractional Burgers' equations with initial-boundary conditions by the Jacobi spectral collocation method using the shifted Jacobi-Gauss-Lobatto collocation points. By means of the simplifed Jacobi operational matrix, we produce the diferentiation matrix and transfer the space-fractional Burgers' equation into a system of ordinary diferential equations that can be solved by the fourth-order Runge-Kutta method. The numerical simulations indicate that the Jacobi spectral collocation method is highly accurate and fast convergent for the generalized space-fractional Burgers' equation.

MESHFREE METHOD BASED ON POINT COLLOCATION FOR METAL FORMING SIMULATION

A meshfree method based on reproducing kernel approximation and point collocation is presented for analysis of metal ring compression. The point collocation method is a true meshfree method without the employment of a background mesh. It is shown that, in a point collocation approach, the remesh problem because of the mesh distortion in FEM （finite element method） and the low efficiency in Galerkin-based meshfree method are avoided. The corrected kernel functions are introduced to the stabilization of free-surface boundary conditions. The solution of symmetric ring compression problem is compared with a conventional finite element solution, and reasonable results have been obtained.

Radial point collocation method (RPCM) for solving convection-diffusion problems

In this paper, Radial point collocation method （RPCM）, a kind of meshfree method, is applied to solve convectiondiffusion problem. The main feature of this approach is to use the interpolation schemes in local supported domains based on radial basis functions. As a result, this method is local and hence the system matrix is banded which is very attractive for practical engineering problems. In the numerical examination, RPCM is applied to solve non-linear convection-diffusion 2D Burgers equations. The results obtained by RPCM demonstrate the accuracy and efficiency of the proposed method for solving transient fluid dynamic problems. A fictitious point scheme is adopted to improve the solution accuracy while Neumann boundary conditions exist. The meshfree feature of the nresent method is verv attractive in solving comnutational fluid nroblems.

AHermitian C^(2) Differential Reproducing Kernel Interpolation Meshless Method for the 3D Microstructure-Dependent Static Flexural Analysis of Simply Supported and Functionally Graded Microplates

This work develops a Hermitian C^(2) differential reproducing kernel interpolation meshless(DRKIM)method within the consistent couple stress theory(CCST)framework to study the three-dimensional(3D)microstructuredependent static flexural behavior of a functionally graded(FG)microplate subjected to mechanical loads and placed under full simple supports.In the formulation,we select the transverse stress and displacement components and their first-and second-order derivatives as primary variables.Then,we set up the differential reproducing conditions(DRCs)to obtain the shape functions of the Hermitian C^(2) differential reproducing kernel(DRK)interpolant’s derivatives without using direct differentiation.The interpolant’s shape function is combined with a primitive function that possesses Kronecker delta properties and an enrichment function that constituents DRCs.As a result,the primary variables and their first-and second-order derivatives satisfy the nodal interpolation properties.Subsequently,incorporating ourHermitianC^(2)DRKinterpolant intothe strong formof the3DCCST,we develop a DRKIM method to analyze the FG microplate’s 3D microstructure-dependent static flexural behavior.The Hermitian C^(2) DRKIM method is confirmed to be accurate and fast in its convergence rate by comparing the solutions it produces with the relevant 3D solutions available in the literature.Finally,the impact of essential factors on the transverse stresses,in-plane stresses,displacements,and couple stresses that are induced in the loaded microplate is examined.These factors include the length-to-thickness ratio,the material length-scale parameter,and the inhomogeneity index,which appear to be significant.

ASYMPTOTIC ERROR EXPANSIONS OF QUADRATIC SPLINE COLLOCATION SOLUTIONS FOR TWO-POINT BOUNDARY VALUE PROBLEMS

In this paper, we consider the following problem:The quadratic spline collocation, with uniform mesh and the mid-knot points are taken as the collocation points for this problem is considered. With some assumptions, we have proved that the solution of the quadratic spline collocation for the nonlinear problem can be written as a series expansions in integer powers of the mesh-size parameter. This gives us a construction method for using Richardson’s extrapolation. When we have a set of approximate solution with different mesh-size parameter a solution with high accuracy can he obtained by Richardson’s extrapolation.

求解测地线边界值问题的均匀B样条配点法

曲面上的测地线在图像处理、机器人学、数控加工等领域有着广泛的应用.针对参数曲面,提出了采用二阶、三阶均匀B样条配点法求解测地线边界值问题.首先建立了测地线边界值问题的离散控制方程,在给定初始逼近后,该方程可采用近似牛顿迭代法进行求解;然后推导了B样条配点法的计算精度;最后在Matlab2016b软件环境下,结合球面、圆环面以及样条曲面对所提出的方法进行了验证.实验考察了不同方法的计算精度、时间成本以及计算效率,结果表明三阶B样条配点法获得相同计算精度所需的时间通常更少,即该方法在计算效率方面具有优势.当计算测地线长度误差为0.01%时,三阶B样条配点法相较于Chen的近似测地线法节省时间成本5.5%~26.9%.对于二阶B样条配点法,其计算效率与Kasap的中心差分法相当.所提出的方法计算结果为连续曲线,无需额外的插值函数,一个潜在的应用场景为自动铺带轨迹规划.

耦合拉格朗日-欧拉方法及其在海洋工程中的应用

Combining the strengths of Lagrangian and Eulerian descriptions,the coupled Lagrangian–Eulerian methods play an increasingly important role in various subjects.This work reviews their development and application in ocean engineering.Initially,we briefly outline the advantages and disadvantages of the Lagrangian and Eulerian descriptions and the main characteristics of the coupled Lagrangian–Eulerian approach.Then,following the developmental trajectory of these methods,the fundamental formulations and the frameworks of various approaches,including the arbitrary Lagrangian–Eulerian finite element method,the particle-in-cell method,the material point method,and the recently developed Lagrangian–Eulerian stabilized collocation method,are detailedly reviewed.In addition,the article reviews the research progress of these methods with applications in ocean hydrodynamics,focusing on free surface flows,numerical wave generation,wave overturning and breaking,interactions between waves and coastal structures,fluid–rigid body interactions,fluid–elastic body interactions,multiphase flow problems and visualization of ocean flows,etc.Furthermore,the latest research advancements in the numerical stability,accuracy,efficiency,and consistency of the coupled Lagrangian–Eulerian particle methods are reviewed;these advancements enable efficient and highly accurate simulation of complicated multiphysics problems in ocean and coastal engineering.By building on these works,the current challenges and future directions of the hybrid Lagrangian–Eulerian particle methods are summarized.

The Lanczos-Chebyshev Pseudospectral Method for Solution of Differential Equations

In this paper, we propose to replace the Chebyshev series used in pseudospectral methods with the equivalent Chebyshev economized power series that can be evaluated more rapidly. We keep the rest of the implementation the same as the spectral method so that there is no new mathematical principle involved. We show by numerical examples that the new approach works well and there is indeed no significant loss of solution accuracy. The advantages of using power series also include simplicity in its formulation and implementation such that it could be used for complex systems. We investigate the important issue of collocation point selection. Our numerical results indicate that there is a clear accuracy advantage of using collocation points corresponding to roots of the Chebyshev polynomial.

Mode Ⅲ fracture analysis by Trefftz boundary element method

This paper presents a hybrid Trefftz （HT） boundary element method （BEM） by using two indirect techniques for mode III fracture problems. Two Trefftz complete functions of Laplace equation for normal elements and a special purpose Trefftz function for crack elements are proposed in deriving the Galerkin and the collocation techniques of HT BEM. Then two auxiliary functions are introduced to improve the accuracy of the displacement field near the crack tips, and stress intensity factor （SIF） is evaluated by local crack elements as well. Furthermore, numerical examples are given, including comparisons of the present results with the analytical solution and the other numerical methods, to demonstrate the efficiency for different boundary conditions and to illustrate the convergence influenced by several parameters. It shows that HT BEM by usingthe Galerkin and the collocation techniques is effective for mode III fracture problems.

New Perspective to Isogeometric Analysis:Solving Isogeometric Analysis Problem by Fitting Load Function

Isogeometric analysis(IGA)is introduced to establish the direct link between computer-aided design and analysis.It is commonly implemented by Galerkin formulations(isogeometric Galerkin,IGA-G)through the use of nonuniform rational B-splines(NURBS)basis functions for geometric design and analysis.Another promising approach,isogeometric collocation(IGA-C),working directly with the strong form of the partial differential equation(PDE)over the physical domain defined by NURBS geometry,calculates the derivatives of the numerical solution at the chosen collocation points.In a typical IGA,the knot vector of the NURBS numerical solution is only determined by the physical domain.A new perspective on the IGAmethod is proposed in this study to improve the accuracy and convergence of the solution.Solving the PDE with IGA can be regarded as fitting the load function defined on the NURBS geometry(right-hand side)with derivatives of the NURBS numerical solution(left-hand side).Moreover,the design of the knot vector has a close relationship to theNURBS functions to be fitted in the area of data fitting in geometric design.Therefore,the detected feature points of the load function are integrated into the initial knot vector of the physical domainto construct thenewknot vector of thenumerical solution.Then,they are connected seamlessly with the IGA-C framework for its great potential combining the accuracy and smoothness merits with the computational efficiency,which we call isogeometric collocation by fitting load function(IGACL).In numerical experiments,we implement our method to solve 1D,2D,and 3D PDEs and demonstrate the improvement in accuracy by comparing it with the standard IGA-C method.We also verify the superiority in the accuracy of our knot selection scheme when employed in the IGA-G method,which we call isogeometric Galerkin by fitting load function(IGA-GL).

An Adaptive Collocation Method with Weighted Extended PHT-Splines

This paper presents an adaptive collocation method with weighted extended PHT-splines.The authors modify the classification rules for basis functions based on the relation between the basis vertices and the computational domain. The Gaussian points are chosen to be collocation points since PHT-splines are C1 continuous. The authors also provide relocation techniques to resolve the mismatch problem between the number of basis functions and the number of interpolation conditions. Compared to the traditional Greville collocation method, the new approach has improved accuracy with fewer oscillations. Several numerical examples are also provided to test our the proposed approach.

An exponential for the solutions model of collocation method of the HIV infection CD4＋T cells

In this paper, an exponential method is presented for the approximate solutions of the HIV infection model of CD4＋T. The method is based on exponential polynomi- als and collocation points. This model problem corresponds to a system of nonlinear ordinary differential equations. Matrix relations are constructed for the exponential functions. By aid of these matrix relations and the collocation points, the proposed technique transforms the model problem into a system of nonlinear algebraic equations. By solving the system of the algebraic equations, the unknown coefficients are com- puted and thus the approximate solutions are obtained. The applications of the method for the considered problem are given and the comparisons are made with the other methods.

基于核重构思想的最小二乘配点型无网格方法

介绍重构核点法的基本原理和近似函数的构造方法,并基于核重构思想,应用配点法和最小二乘原理,离散微分方程,建立求解的代数方程,提出了一种基于核重构思想的最小二乘配点型无网格方法.与一般配点法相比,该方法的系数矩阵是有对称正定的,计算精度高,稳定性好.该方法的实施不需要背景网格,不需要进行高斯积分,与Galerkin法相比,具有计算量小、边界条件处理简单的特点,是一种真正的无网格法.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.文中结合若干典型算例,检验了该方法的有效性.

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。

基于点插值的配点型无网格法解Helmholtz问题

基于点插值法的思想,用三角函数作为基函数在局部支持域内构造具有Kroneckerδ函数性、单位分解性、高阶连续性、再生性和紧支性的形函数。用配点法离散微分方程,得到了具有稀疏带状性的系数矩阵,用GMERS方法求解代数方程组,分别研究了Helmholtz问题的边界层问题和波传播问题。通过数值算例可以发现,给出的数值结果非常接近于精确解,且随着节点的增加,其精确度越来越高,具有良好的收敛性。

极坐标系下泊松方程的拟谱方法

极坐标系下的泊松方程,由于坐标原点的奇异性,给谱方法的实施带来了很大的困难。本文提出了一种新的拟谱方法,在径向上求解区域为[-1,1],采用标准的Gauss-Lobatto配置点;而在角方向上配置点均匀分布在[0,π]内。通过对配置点及空间导数矩阵的处理,成功解决了坐标奇异问题。同时也避免了配置点在原点附近的集中,极大改善了矩阵条件数,减小了舍入误差,从而提高了解的精度。数值实验表明,该方法具有很高的精度。

基于径向基函数逼近的结构动力响应计算方法

提出将"时间间隔"替换"空间距离"作为径向基函数的自变量,利用径向基函数逼近的思想,结合加权余量配点法,用于结构动力响应的数值分析.并且针对结构动力学的特点,发展了位移、速度、加速度联合插值的径向基函数表达式,提出了精密计算的概念和标准.根据实际算例表明,该方法对比传统的Newmark法、Wilson-θ法、Runge-Kutta(龙格-库塔)法,在求解强刚性动力学方程,结构瞬态段的动力响应方面具有明显的优势,其计算精度与精细时程积分法相当.该方法与计算效率相关的动力特征矩阵,以及问题的自由度无关,因此针对大规模问题具有很好的适用性,是构建结构动力响应计算方法的新途径.

夹层扁薄球壳的非线性稳定

以幂函数为试函数，用配点法计算受轴对称分布荷载的夹层扁薄球壳的非线性稳定．夹层壳采用Ｒｅｉｓｎｅｒ模型［１］．当球壳退化为圆板，在均布荷载作用下，本法同幂级数法［２］取得了相同的结果．在相同的精度要求下，本法的幂函数所取项数仅为幂级数法的１／１３。

任意变厚度的旋转扁薄壳非线性稳定的幂函数解法

以幂函数为试函数,两次使用配点法成功地分离了耦合的大挠度方程,从而导出变厚度旋转扁壳非线性稳定的计算式。支座可以是弹性的。本文给出了均布或多项式分布荷载作用下,线性或多项式型变厚度的圆锥壳、球壳、余弦壳或四次多项式型旋转壳的上、下临界荷载。均布荷载作用下指数型变厚度球壳的上临界荷载同其他方法的结果作了比较。用配点法编写的程序具有收敛范围大、精度高、通用性强和计算时间少的优点。