改进EFG法用于旋转梁的刚柔耦合动力学研究

采用全域插值广义移动最小二乘(IGMLS)的改进无网格伽辽金(EFG)法,对旋转中心刚体-柔性梁系统的刚柔耦合动力学特性进行研究。利用考虑了柔性梁横向变形引起的非线性耦合项的一次近似耦合模型,根据Hamilton原理和EFG离散方法得到刚柔耦合系统的无网格动力学离散方程。在大范围运动未知的情况下,采用数值方法对刚柔耦合系统进行动力响应的仿真计算,并对EFG法的主要影响因素进行了讨论分析。通过与有限元法的数值计算结果对比,验证EFG法用于刚柔耦合系统动力学研究的有效性及可行性,并为无网格法用于更复杂的柔性多体动力学的研究提供了理论依据。     

正交各向异性结构的三维无网格法稳态传热模型及应用

利用无网格迦辽金(Element-free Galerkin,EFG)法建立了正交各向异性结构三维稳态传热分析的计算模型,并推导了其三维EFG法传热离散控制方程。基于该模型编写程序对正交各向异性材料热喷嘴和压力容器的稳态传热算例进行了分析,发现在相同节点分布下三维EFG法温度场比有限元结果更接近参考解,三维EFG传热模型的计算精度比有限元法高,从而验证了该模型的正确性和优越性。同时,对比了各向同性与各向异性结构的温度场分布规律和温度幅值,研究了三维热导率因子及三个材料方向角对传热性能的影响,并给出了这些参数的合理取值范围。结果表明:热导率因子和材料方向角对温度场影响很大,增大热导率因子和材料方向角可使最高温度下降且温度梯度变小;导热方向会随材料方向角发生旋转,三维热导率因子决定主导热方向。对于正交各向异性材料热喷嘴和压力容器,为取得较好的传热效果,建议三维热导率因子在8∶1∶16～16∶1∶32范围内取值,三个材料方向角在45～60°范围内取相同值。在三维复合材料传热结构设计中,合理选取热导率因子和材料方向角可增强结构传热性能、减小温度梯度。     

正交各向异性材料结构热变形和热应力分析的无网格法计算模型及应用正交各向异性材料结构热变形和热应力分析的无网格法计算模型及应用

利用无网格伽辽金法（Element-free Galerkin,EFG）建立了正交各向异性材料结构热变形和热应力分析的计算模型,并推导了正交各向异性材料结构热弹性问题的EFG法离散控制方程。选择复合材料冷却栅管算例验证了计算模型和程序的正确性,利用该计算模型分析了具有不同材料方向角及热导率因子、热膨胀系数因子和主次泊松比因子的汽轮机叶轮,得到了其热变形总位移和Mises应力,讨论了材料方向角和上述正交各向异性材料因子对其热变形总位移和Mises应力的影响规律,给出了这些参数的合理取值范围,并选取一组参数与各向同性材料结构进行了热变形和热应力对比分析。结果表明,基于EFG法的热变形总位移和Mises应力的计算精度比有限元法高,材料方向角同时影响热变形总位移和Mises应力的大小和方向,而正交各向异性材料因子只影响热变形总位移和Mises应力的大小,不影响其方向。在复合材料结构设计过程中,合理选取材料方向角和正交各向异性材料因子可有效减小结构热变形和热应力。      

非线性Burgers方程求解的自适应Euler-Lagrange无单元Galerkin方法

建立了求解非线性Burgers方程的自适应Euler-Lagrange无单元Galerkin(adaptiveEuler-Lagrange element-free Galerkin,AELEFG)方法.该方法将Euler形式的非线性Burgers方程转化成Lagrange形式的纯扩散方程,使用节点自适应无单元Galerkin(element-free Galerkin,EFG)方法求解该扩散方程,并沿特征路径反向追踪对对流项进行处理.数值结果表明,运用AELEFG方法求解非线性Burgers方程具有较高的精度及稳定性.     

修正的内部基扩充无网格法求解多裂纹应力强度因子

修正的内部基扩充无网格Galerkin法求解了多裂纹应力强度因子。采用特征距离对内部基扩充无网格法进行修正,应用变分原理推导了系统离散方程,给出相互作用能量积分计算混合型模式下的应力强度因子的公式。求解3个平面应力条件下的多裂纹问题,并与其他数值方法的计算结果进行比较。数值算例表明:修正的内部基扩充无网格Galerkin法可以方便、有效地求解多裂纹问题,在不增加附加节点和自由度的情况下便可以得到较高精度的计算结果。     

基于多边形支持域的无单元Galerkin方法研究

本文针对传统无单元Galerkin方法不能直接施加本质边界条件的缺点,提出了基于多边形支持域的无单元Galerkin方法.该方法将计算点的支持域由矩形或圆形扩展为多边形,使得移动最小二乘形函数满足Kronecker函数性质,进而使无单元Galerkin方法可以直接施加本质边界条件.此外,该方法将积分背景网格与多边形支持域关联,可以避免重复的节点搜索,提高了无单元Galerkin方法的计算效率.数值结果表明,基于多边形支持域的无单元Galerkin方法不但具有较高的计算效率,且与稳定化方案耦合,可以成功克服对流占优引起的数值不稳定问题.     

基于交叉节点对无网格Galerkin法的改进算法研究

针对无网格Galerkin法刚度矩阵的稀疏存储实现难、节点与积分点的全局搜索效率低等问题,该文基于交叉节点对及其循环组装整体刚度矩阵的思想,利用CSR格式存储刚度矩阵,通过局部搜索方法来搜寻节点与积分点,提出了一种采用三角形网格进行积分计算的无网格Galerkin法。通过数值算例对比了不同节点规模的刚度矩阵存储消耗,以及节点与积分点的搜索效率。结果表明所提出算法在满足计算精度的前提下,能有效地节省存储空间和提高节点与积分点的搜索效率,并对复杂形状的几何模型具有良好的适应性。     

无网格法模拟复合型疲劳裂纹的扩展

本文提出了用无网格Galerkin法模拟构件在复合变形作用下疲劳裂纹扩展路径并预估其疲劳寿命的方法。该法能够自然模拟疲劳裂纹的扩展,不需要网格重构,避免了裂纹扩展过程中的精度受损。应用无网格数值结果计算了J积分和应力强度因子IK和IIK;按照最大周向应力理论获得了裂纹扩展偏斜角。基于最小应变能密度因子理论,确定了裂纹扩展量aD,并能获得疲劳载荷的循环周数ND。文末对数值模拟结果和实验拟合结果进行了对照。     

金属体积成形的无网格Galerkin模拟

为避免金属体积成形有限元法模拟中网格畸变造成网格重划和模拟精度降低,采用无网格法模拟金属体积成形.利用无网格法近似位移场,建立金属体积成形的无网格法连续性控制方程,采用罚函数法施加本征边界条件和体积不变条件,基于Markov变分原理推导了金属体积成形的无网格Galerkin求解列式.用数值计算法求解该列式,实现金属体积成形的无网格模拟.数值结果表明,无网格法能有效处理金属体积成形中出现的大变形,避免了网格畸变和重划,具有较高的模拟精度.     

线性弹性柔性壳非协调有限元计算模型

本文基于Ciarlet-Lods-Miara定义的柔性壳模型提出一种Galerkin非协调有限元离散格式.首先,对积分区域进行Delaunay三角剖分,并在三角网格上对位移前两个分量用一次Lagrange多项式逼近,对第三个分量(即法向位移)用非协调Morley元逼近.其次,讨论了构造的Galerkin非协调有限元离散格式解的存在性、唯一性和先验误差估计.最后对特殊边界条件下的锥壳采用该方法进行数值实验,计算出不同网格下锥壳的位移,并通过分析数值实验结果证明有限元离散格式的收敛性和有效性.     

无单元法研究现状及展望

无单元法是众多无网格方法中较有代表性的一种,形式简单、明确,计算精度高。因其具有仅需离散的结点信息、解答具有高次连续性、能较好地反映应力高梯度分布并便于跟踪裂纹的扩展过程等优点,无单元法自问世以来获得了广泛的重视,已成为计算力学领域的一个研究热点。文中着重分析了无单元法研究中的热点问题及解决方法,介绍了该方法目前的一些应用范围,并指出其可能的发展方向。     

Topology optimization using bi-directional evolutionary structural optimization based on the element-free Galerkin method

In this article, the bi-directional evolutionary structural optimization (BESO) method based on the element-free Galerkin (EFG) method is presented for topology optimization of continuum structures. The mathematical formulation of the topology optimization is developed considering the nodal strain energy as the design variable and the minimization of compliance as the objective function. The EFG method is used to derive the shape functions using the moving least squares approximation. The essential boundary conditions are enforced by the method of Lagrange multipliers. Several topology optimization problems are presented to show the effectiveness of the proposed method. Many issues related to topology optimization of continuum structures, such as chequerboard patterns and mesh dependency, are studied in the examples.     

Heat transfer analysis of composite slabs using meshless element Free Galerkin method

This paper deals with three dimensional heat transfer analysis of composite slabs using meshless element free Galerkin method. The element free Galerkin method (EFG) method utilizes moving least square (MLS) approximants to approximate the unknown function of temperature Tx). These approximants are constructed by using a weight function, a basis function and a set of coefficients that depends on position. Penalty and Lagrange multiplier techniques have been used to enforce the essential boundary conditions. MATLAB codes have been developed to obtain the EFG results. Two new basis functions namely trigonometric and polynomial have been proposed. A comparison has been made among the results obtained using existing (linear) and proposed (trigonometric and polynomial) basis functions for three dimensional heat transfer in composite slabs. The effect of penalty parameter on EFG results has also been discussed. The results obtained by EFG method are compared with those obtained by finite element method     

小波基无单元法及其工程应用

论述数值计算中新的小波基无单元方法,即用小波基函数取代传统无单元方法中的幂级数基之后,使无单元法具有了小波变换的局域化和多分辨率等优良特性,并能有效地克服有限单元法的网格敏感性和单元之间应力不连续现象,从而不但拓展和丰富了无单元法的理论内容,也为其工程应用开辟了新的途径。     

Analyzing three-dimensional potential problems with the improved element-free Galerkin method

The potential problem is one of the most important partial differential equations in engineering mathematics. A potential problem is a function that satisfies a given partial differential equation and particular boundary conditions. It is independent of time and involves only space coordinates, as in Poisson’s equation or the Laplace equation with Dirichlet, Neumann, or mixed conditions. When potential problems are very complex, both in their field variable variation and boundary conditions, they usually cannot be solved by analytical solutions. The element-free Galerkin (EFG) method is a promising method for solving partial differential equations on which the trial and test functions employed in the discretization process result from moving least-squares (MLS) interpolants. In this paper, by employing improved moving least-squares (IMLS) approximation, we derive the formulas for an improved element-free Galerkin (IEFG) method for three-dimensional potential problems. Because there are fewer coefficients in the IMLS approximation than in the MLS approximation, and in the IEFG method, fewer nodes are selected in the entire domain than in the conventional EFG method, the IEFG method should result in a higher computing speed.     

Volumetric locking in the element free Galerkin method

A new formulation of the Element Free Galerkin (EFG) method is developed for the modelling of incompressible materials. Beginning with a mixed variational principle, a selective reduced integration procedure is developed by implementing nodal quadrature. Numerical examples are provided which compare the performance of the proposed technique to the standard EFG approximation. These studies illustrate the capability of the new formulation to eliminate volumetric locking. For the standard method, however, the degree of volumetric locking is shown to be a function of the local support sizes (domains of influence) of the EFG shape functions. Copyright © 1999 John Wiley & Sons, Ltd.     

Topology optimization of continuum structures with displacement constraints based on meshless method

In this paper, the element free Galerkin method (EFG) is applied to carry out the topology optimization of continuum structures with displacement constraints. In the EFG method, the matrices in the discretized system equations are assembled based on the quadrature points. In the sense, the relative density at Gauss quadrature point is employed as design variable. Considering the minimization of weight as an objective function, the mathematical formulation of the topology optimization subjected to displacement constraints is developed using the solid isotropic microstructures with penalization interpolation scheme. Moreover, the approximate explicit function expression between topological variables and displacement constraints are derived. Sensitivity of the objective function is derived based on the adjoint method. Three numerical examples are used to demonstrate the feasibility and effectiveness of the proposed method.     

The interpolating element-free Galerkin (IEFG) method for two-dimensional potential problems

In this paper, the moving least-squares (MLS) approximation and the interpolating moving least-squares (IMLS) method proposed by Lancaster are discussed first. A new method for deriving the MLS approximation is presented, and the IMLS method is improved. Compared with the IMLS method proposed by Lancaster, the shape function of the improved IMLS method in this paper is simpler so that the new method has higher computing efficiency. Then combining the shape function of the improved IMLS method with Galerkin weak form of the potential problem, the interpolating element-free Galerkin (IEFG) method for the two- dimensional potential problem is presented, and the corresponding formulae are obtained. Compared with the conventional element-free Galerkin (EFG) method, the boundary conditions can be applied directly in the IEFG method, which makes the computing efficiency higher. For the purposes of demonstration, some selected numerical examples are solved using the IEFG method.     

Improved element-free Galerkin method for two-dimensional potential problems

Potential difficulties arise in connection with various physical and engineering problems in which the functions satisfy a given partial differential equation and particular boundary conditions. These problems are independent of time and involve only space coordinates, as in Poisson's equation or the Laplace equation with Dirichlet, Neuman, or mixed conditions. When the problems are too complex, they usually cannot be solved with analytical solutions. The element-free Galerkin (EFG) method is a meshless method for solving partial differential equations on which the trial and test functions employed in the discretization process result from moving least-squares (MLS) interpolants. In this paper, by using the weighted orthogonal basis function to construct the MLS interpolants, we derive the formulae of an improved EFG (IEFG) method for two-dimensional potential problems. There are fewer coefficients in the improved MLS (IMLS) approximation than in the MLS approximation, and in the IEFG method fewer nodes are selected in the entire domain than in the conventional EFG method. Hence, the IEFG method should result in a higher computing speed.     

Meshless approach to shape optimization of linear thermoelastic solids

This paper presents a formulation for shape optimization in thermoelasticity using a meshless method, namely the element‐free Galerkin method. Two examples are treated in detail and comparisons with previously published finite element analysis results demonstrate the excellent opportunities the EFG offers for solving these types of problems. Smoother stresses, no remeshing, and better accuracy than finite element solutions, permit answers to shape optimization problems in thermoelasticity that are practically unattainable with the classical FEM without remeshing. For the thermal fin example, the EFG finds finger shapes that are missed by the FEM analysis, and the objective value is greatly improved compared to the FEM solution. A study of the influence of the number of design parameters is performed and it is observed that the EFG can give better results with a smaller number of design parameters than is possible with traditional methods. Copyright © 2001 John Wiley & Sons, Ltd.