奇异边界法:一个新的、简单、无网格、边界配点数值方法

物理力学控制方程的基本解有源点奇异性.因而,传统的观点认为奇异基本解一般不能用做控制方程数值解的基函数;除非源点布置在物理域以外的虚假边界上,与物理边界上的配点分离.后者就是近年来受到广泛关注的基本解方法的基本思路.与这些传统方法不同,文中直接使用基本解做为插值基函数,且源点和配点是同一组物理边界上的离散点.本项工作的一个基本假设是源点奇异时的源点强度因子的存在性.利用待求问题控制方程的已知简单解,提出了一个计算源点强度因子的数值方法,并发现源点强度因子确实存在,且是一个有限值,其大小依赖于边界离散点的分布和边界条件类型.由此,文中提出了一个计算微分方程问题的新数值方法,称为奇异边界方法.该方法数学简单,编程容易,是一个真正的无网格方法.初步数值试验显示该方法精度高,收敛速度快.但有关该方法的数学物理基础还不是十分清楚.     

二阶一致无网格与有限元耦合离散研究

为准确方便地施加本质边界条件,在连续掺混法(Continuous Blending Method,CBM)的框架下,通过增加一个边中节点,发展了采用二阶基底的无网格与二阶有限元的耦合离散方法.Galerkin弱形式的数值积分采用具二阶一致性的3点积分方法(Quadratically Consistent 3-point integration method,QC3).与原本在QC3中采用的Nitsche法相比,所发展的耦合离散方法可像有限元法一样简单高效地施加本质边界条件,不向弱形式中引入额外项,也不依赖于任何人工参数.而且,数值结果还表明,QC3的计算精度也得到进一步提高.     

Meshfree modeling and analysis of physical fields in heterogeneous media

Continuous and discrete variations in material properties lead to substantial difficulties for most mesh-based methods for modeling and analysis of physical fields. The meshfree method described in this paper relies on distance fields to boundaries and to material features in order to represent variations of material properties as well as to satisfy prescribed boundary conditions. The method is theoretically complete in the sense that all distributions of physical properties and all physical fields are represented by generalized Taylor series expansions in terms of powers of distance fields. We explain how such Taylor series can be used to construct solution structures – spaces of functions satisfying the prescribed boundary conditions exactly and containing the necessary degrees of freedom to satisfy additional constraints. Fully implemented numerical examples illustrate the effectiveness of the proposed approach.     

Meshfree weak–strong (MWS) form method and its application to incompressible flow problems

A meshfree weak–strong (MWS) form method has been proposed by the authors' group for linear solid mechanics problems based on a combined weak and strong form of governing equations. This paper formulates the MWS method for the incompressible Navier–Stokes equations that is non‐linear in nature. In this method, the meshfree collocation method based on strong form equations is applied to the interior nodes and the nodes on the essential boundaries; the local Petrov–Galerkin weak form is applied only to the nodes on the natural boundaries of the problem domain. The MWS method is then applied to simulate the steady problem of natural convection in an enclosed domain and the unsteady problem of viscous flow around a circular cylinder using both regular and irregular nodal distributions. The simulation results are validated by comparing with those of other numerical methods as well as experimental data. It is demonstrated that the MWS method has very good efficiency and accuracy for fluid flow problems. It works perfectly well for irregular nodes using only local quadrature cells for nodes on the natural boundary, which can be generated without any difficulty. Copyright © 2004 John Wiley & Sons, Ltd.     

Cavity flow simulation of Carreau–Yasuda non-Newtonian fluids using PIM meshfree method

Lid-driven cavity flow of a purely-viscous non-Newtonian fluid obeying Carreau–Yasuda rheological model is studied numerically using the PIM meshfree method combined with the Characteristic-Based Split-A algorithm. Results are reported for the velocity and pressure profiles at Reynolds numbers as high as 1000 for a non-Newtonian fluid obeying Carreau–Yasuda rheological model. For Newtonian fluids, results obtained from our PIM–CBS-A method show good agreement with benchmark results published in the literature and obtained using finite difference and/or Finite Element Methods. Our numerical results are also consistent with recent published results obtained using another meshfree method called LSM with the advantage that PIM needs less “points” to achieve the same degree of accuracy. Results obtained for the Carreau–Yasuda model reveals the strong effect of the shear-thinning behavior of a fluid on its flow kinematics within the cavity.     

超高速碰撞问题的三维物质点法

简要介绍了物质点法（material point method）的离散原理,通过引入Johnson-Cook材料模型和Mie-Gruneisen状态方程,将其用于超高速碰撞问题的分析中,并编制了相应三维物质点法程序MPM3D。该方法避免了拉格朗日格式因网格畸变产生的数值困难,也克服了欧拉格式材料界面跟踪问题以及因非线性对流扩散项而引起的数值困难。利用该程序对Taylor杆高速碰撞问题和空间碎片防护超高速碰撞问题进行了数值模拟,所得数值结果与实验结果基本吻合,验证了程序的正确性,说明了物质点法在分析超高速碰撞问题时相对于有限元法的优势。     

MPS与GPU结合数值模拟LNG液舱晃荡

液舱晃荡是一种在外部激励作用下部分装载的液舱内液体的波动现象,它会对液舱结构强度和运输船舶稳性产生危害.移动粒子半隐式法(moving particle semi-implicit,MPS)是一种典型的无网格粒子类方法,可以有效地模拟剧烈的液舱晃荡问题.但MPS方法存在计算效率低的缺点,难以模拟大规模三维问题,而GPU并行加速技术已广泛应用于科学计算领域.因此,本文将MPS方法与GPU并行加速技术相结合,采用CUDA程序语言编写,自主开发了MPSGPU-SJTU求解器,对三维液化天然气(liquefiednatural gas, LNG)型液舱晃荡进行了数值模拟.通过三种不同粒子间距的数值模拟,验证了求解器的收敛性,其中最大计算粒子数达到了200多万.与其他研究结果相比,MPSGPU-SJTU求解器能够准确地预测壁面砰击压力,并且捕捉晃荡过程中自由面的大幅度变形和强非线性破碎现象.相比CPU求解器的计算时间,GPU并行加速技术可以大幅度地减小计算时长,提高MPS方法的计算效率.本文将LNG型液舱与方型液舱的晃荡进行对比,结果表明在高充液率下LNG型液舱可以有效地减小晃荡幅值和壁面砰击压力.但在中低充液率下,LNG型液舱则会加剧晃荡,自由面呈现明显的三维特征.本文还进一步研究了水和LNG两种不同介质的液舱晃荡现象,数值模拟结果表明二者的流场基本相似,砰击压力则正比于液体密度.      

基于无网格稳定化方案求解非稳态强对流问题的自适应节点加密技术

无网格Taylor最小二乘(MFLS)稳定化方案可有效地消除无网格Galerkin方法求解对流占优问题时产生的数值伪振荡,但当对流作用很强或纯对流时,它的求解效果不尽人意.因此,本文基于MFLS稳定化方案给出了一种自适应节点加密技术.该技术将无网格方法中背景积分单元作为自适应节点加密时物理量梯度指标的控制单元,并计算该控制单元上的物理量梯度指标;然后将其与给定的物理量梯度指标限进行比较,标识出大梯度区域从而进行自适应节点加密.数值实验表明,当求解对流作用很强的问题或纯对流问题时,这种基于MFLS稳定化方案的自适应节点加密技术不仅能有效地标示出数值振荡区域,而且可以彻底地消除数值伪振荡.     

Numerical simulation of the flow in straight blade agitator with the MPS method

The three‐dimensional flow in a straight blade agitator with free surface on top is simulated using a grid‐free method named moving particle semi‐implicit method. The agitator has six rotor blades matched with six stationary guide blades. The mechanism and phenomena of the flow are investigated in the area between two adjacent stationary guide blades. Eddies near each tip of the rotational blades are predicted, and they move with the motion of the rotor blades but in opposite rotation direction of the rotor. The rotation axis of the eddies is traced and annular vortices, which are made by the eddies, are studied. The pressure pulsation in the rotation process is also predicted with this particle method. Copyright © 2010 John Wiley & Sons, Ltd.     

A Truly Boundary-Only Meshfree Method Applied to Kirchhoff Plate Bending Problems

The boundary particle method (BPM) is a truly boundary-only collocation scheme, whose basis function is the high-order nonsingular general solution or singular fundamental solution, based on the recursive composite multiple reciprocity method (RC-MRM). The RC-MRM employs the high-order composite differential operator to solve a much wider variety of inhomogeneous problems with boundary-only collocation nodes while significantly reducing computational cost via a recursive algorithm. In this study, we simulate the Kirchhoff plate bending problems by the BPM based on the RC-MRM. Numerical results show that this approach produces accurate solutions of plates subjected to various loadings with boundary-only discretization.