基于流形覆盖的无网格法

针对无网格迦辽金法中形函数构造过于复杂以至计算时间过长的问题,基于流形覆盖思想对节点的自由度重新构造,形函数则使用形式简单的Shepard函数,避免了为得到形函数进行多个矩阵相乘的运算,从而减少了计算时间.利用罚函数法施加边界条件,不仅易于实现,而且精度降低不大.     

一种基于修正SSPH近似的无网格局部Petrov-Galerkin 法

首先采用奇异权函数对对称光滑粒子流体动力学(SSPH)近似进行了修正,使其构造的形函数近似满足d函数性质,方便无网格法中本质边界条件施加;然后应用修正的SSPH 近似法构造试函数,结合以Heaviside 函数为权函数的局部弱形式,提出了一种新的求解弹性静力问题的无网格局部Petrov-Galerkin 法;最后应用新的无网格法计算了一系列数值算例,结果表明：该方法具有良好的精度和收敛性。     

全局弱式无网格方法求解消声器横向模态

应用全局弱式无网格方法求解消声器的横向模态,使用径向基函数点插值法离散本征方程,使用伽辽金加权残数法进行数值积分。分别应用全局弱式无网格方法计算了圆形截面,不规则截面以及含有穿孔截面的本征值和本征向量,计算结果与解析方法和二维有限元方法计算结果吻合较好,并且与二维有限元方法相比,全局弱式无网格方法比较节省计算时间。进而分析了支持域的尺寸以及径向基函数中形状参数对计算精度的影响。     

平面细长梁基于无网格径向基点插值的绝对节点坐标法

为了消除或减弱传统绝对节点坐标法(Absolute Nodal Coordinate Formulation,ANCF)中缩减梁单元的"失真现象",构造了一种适用于描述柔性梁绝对位形的无网格径向基点插值(Radial Point Interpolation Method,RPIM)形函数,提出了柔性梁基于无网格RPIM的ANCF法。传统ANCF梁单元在描述纯弯曲悬臂梁的位形(一段圆弧)时,即便获得精确的单元节点坐标,通过梁单元插值得到的位形与悬臂梁的实际位形存在差异,即失真现象,悬臂梁越弯曲该差异越明显,失真越大。失真导致伪应变的产生,极大地影响数值求解的精度。而RPIM法采用一组场节点离散问题域,通过计算点支持域内的场节点构造形函数,计算点一般位于支持域的中心区域,不同计算点之间的支持域有较多重合的部分,加强了节点之间的联系,能更合理、准确地描述绝对位形,能有效减小失真。研究表明:基于RPIM的ANCF法较传统ANCF法精度更高、计算效率更快、对不等距分布节点的适应性更强,在大变形柔性多体系统动力学领域内具有推广性。     

基于高斯小波函数和线性表达法的开口板自由振动特性研究

研究开口板自振特性的求解方法,将开口部位视为厚度为零的板,引入局域化特性的高斯小波函数作为位移形函数来捕捉厚度突变的情况,提高解的精确度。提出线性表达方法解耦位移形函数与边界条件,基本思路是通过高斯消元法找到约束条件矩阵中线性无关的列向量,将位移形函数中的未知系数转变为线性无关系数列向量的线性表达,从而将有约束问题转变为无约束问题。对四边简支和四边固定的开口板进行分析,结合有限元方法计算结果,讨论解的收敛性和准确性。研究了不同开口尺寸、开口形状对自振频率的影响,得到开口尺寸、开口形状与自振频率关系曲线,对影响的原因进行了解释。最后计算了不同边界约束条件下多开口板的自振频率。     

基于单位分解法的实体壳广义单元模型

基于单位分解的广义有限元法的逼近空间由单位分解函数和局部覆盖函数构成,采用传统有限元形函数作为单位分解函数,其局部覆盖函数的定义不依赖于有限元网格.以十六结点六面体等参单元形函数作为单位分解函数,采用一阶多项式局部覆盖函数建立了十六结点六面体广义单元.在此基础上利用广义有限元法可以灵活构造各向异性逼近空间的特点,根据薄壳的变形特性,对壳体法向挠度和切向位移分别采用一阶和零阶多项式局部覆盖函数,构造了实体薄壳广义单元.计算结果表明：十六结点六面体广义单元和实体薄壳广义单元用于板壳结构分析时具有比相应的常规实体单元更高的收敛性和求解效率,且实体薄壳广义单元比十六结点六面体广义单元具有更高的求解效率.     

偶应力理论的径向点插值无网格法研究

偶应力理论引入了位移二阶导数和相应的高阶边界条件,在有限元实施过程中,需要实现位移函数C1连续,这对传统有限元方法来说是一个苛刻的要求。无网格法能够实现位移函数的高阶导数连续,该文基于应变梯度偶应力理论下的虚功原理,导出了其无网格实施方法。无网格法的形函数一般不具有插值特性,本质边界条件施加困难,为克服这一困难,采用了带有多项式基的径向点插值法,构造的形状函数具有插值特性,可直接施加本质边界条件。数值结果表明,该方法数值结果稳定、精度高。     

基于一阶及二阶DIC方法的常应变剪切带的测量误差分析

制作了水平和倾斜的常应变剪切带,使用数字图像相关方法(DIC)对剪切带内、外的位移场和应变场进行了测量,分析了形函数、子区尺寸及测点间距对测量结果的影响。结果表明:在剪切带边界附近,一阶或二阶DIC方法的位移误差限均较大,这是因为这些区域的子区覆盖了一部分剪切带;子区尺寸越大,或测点距离剪切带边界越近,位移误差限越大;一般对于剪切带边界附近测点,二阶DIC方法比一阶DIC方法的位移误差限小;随着测点间距的增加,一阶或二阶DIC方法的剪切带的法向剪应变坐标曲线由陡峭变得平缓,剪切带宽度逐渐增加。     

波在各向异性介质中传播规律的无网格法数值模拟

采用基于紧支距离基函数近似的配点型无网格方法对波在各向异性层状介质中的传播规律进行了数值模拟,得到了应力波的传播历程,并与冲击载荷作用下的有限元计算结果值进行了比较。该方法所得到的结果与有限元计算的结果吻合较好。说明该方法可以有效地模拟波在各向异性材料中的传播过程。     

流固耦合问题的网格更新与信息传递新方法

研究流固耦合问题中的网格技术。针对流体域的网格移动,提出基于映射结构化网格的插值更新的新方法,采用映射插值函数计算流体网格节点位移并与初始网格坐标值叠加,以获取流体新的节点坐标。对二维正方形、梭形及三维立方体流场网格更新开展数值计算。计算表明,该方法可保持原网格的拓扑关系,且更新速度快,更新质量好。使用约束反力分配法和投影插值法分别传递流体域到结构域、结构域到流体域的信息,运用基于该方法的自编程序对典型形体的结构流固耦合界面进行信息传递计算模拟。通过对比传递前后结构与流体的作用,验证了基于该方法的数值传递效果理想。     

Free vibration analysis of thin plates using Hermite reproducing kernel Galerkin meshfree method with sub-domain stabilized conforming integration

A Hermite reproducing kernel (HRK) Galerkin meshfree formulation is presented for free vibration analysis of thin plates. In the HRK approximation the plate deflection is approximated by the deflection as well as slope nodal variables. The nth order reproducing conditions are imposed simultaneously on both the deflectional and rotational degrees of freedom. The resulting meshfree shape function turns out to have a much smaller necessary support size than its standard reproducing kernel counterpart. Obviously this reduction of minimum support size will accelerate the computation of meshfree shape function. To meet the bending exactness in the static sense and to remain the spatial stability the domain integration for stiffness as well as mass matrix is consistently carried out by using the sub-domain stabilized conforming integration (SSCI). Subsequently the proposed formulation is applied to study the free vibration of various benchmark thin plate problems. Numerical results uniformly reveal that the present method produces favorable solutions compared to those given by the high order Gauss integration (GI)-based Galerkin meshfree formulation. Moreover the effect of sub-domain refinement for the domain integration is also investigated.     

A Lagrangian reproducing kernel particle method for metal forming analysis

A Meshless approach based on a Reproducing Kernel Particle Method is developed for metal forming analysis. In this approach, the displacement shape functions are constructed using the reproducing kernel approximation that satisfies consistency conditions. The variational equation of materials with loading-path dependent behavior and contact conditions is formulated with reference to the current configuration. A Lagrangian kernel function, and its corresponding reproducing kernel shape function, are constructed using material coordinates for the Lagrangian discretization of the variational equation. The spatial derivatives of the Lagrangian reproducing kernel shape functions involved in the stress computation of path-dependent materials are performed by an inverse mapping that requires the inversion of the deformation gradient. A collocation formulation is used in the discretization of the boundary integral of the contact constraint equations formulated by a penalty method. By the use of a transformation method, the contact constraints are imposed directly on the contact nodes, and consequently the contact forces and their associated stiffness matrices are formulated at the nodal coordinate. Numerical examples are given to verify the accuracy of the proposed meshless method for metal forming analysis.     

Synchronized reproducing kernel interpolant via multiple wavelet expansion

In this paper, a new partition of unity – the synchronized reproducing kernel (SRK) interpolant – is derived. It is a class of meshless shape functions that exhibit synchronized convergence phenomenon: the convergence rate of the interpolation error of the higher order derivatives of the shape function can be tuned to be that of the shape function itself. This newly designed synchronized reproducing kernel interpolant is constructed as an series expansion of a scaling function kernel and the associated wavelet functions. These wavelet functions are constructed in a reproducing procedure, simultaneously with the scaling function kernel, by directly enforcing certain orders of vanishing moment conditions. To the authors knowledge, this unique interpolant is the first of its kind to be constructed, and to be used in numerical computations, both in concept and in practice. The new interpolants are in fact a group of special hierarchial meshless bases, and similar counterparts may exist in spline interpolation method, other meshless methods, Galerkin-wavelet method, as well as the finite element method. A detailed account of the subject is presented, and the mathematical principle behind the construction procedure is further elaborated. Another important discovery of this study is that the 1st order wavelet together with the scaling function kernel can be used as a weighting function in Petrov-Galerkin procedures to provide a stable numerical computation in some pathological problems. Benchmark problems in advection-diffusion problems, and Stokes flow problem are solved by using the synchronized reproducing kernel interpolant as the weighting function. Reasonably good results have been obtained. This may open the door for designing well behaved Galerkin procedures for numerical computations in various constrained media.     

Filters, reproducing kernel, and adaptive meshfree method

 Reproducing kernel, with its intrinsic feature of moving averaging, can be utilized as a low-pass filter with scale decomposition capability. The discrete convolution of two nth order reproducing kernels with arbitrary support size in each kernel results in a filtered reproducing kernel function that has the same reproducing order. This property is utilized to separate the numerical solution into an unfiltered lower order portion and a filtered higher order portion. As such, the corresponding high-pass filter of this reproducing kernel filter can be used to identify the locations of high gradient, and consequently serves as an operator for error indication in meshfree analysis. In conjunction with the naturally conforming property of the reproducing kernel approximation, a meshfree adaptivity method is also proposed. Received: 31 July 2002 / Accepted: 3 March 2003 The support of this work by the NSF/DARPA OPAAL Program under the grant DMS 98-74015 to UCLA is greatly acknowledged.     

Quasi-convex reproducing kernel meshfree method

A quasi-convex reproducing kernel approximation is presented for Galerkin meshfree analysis. In the proposed meshfree scheme, the monomial reproducing conditions are relaxed to maximizing the positivity of the meshfree shape functions and the resulting shape functions are referred as the quasi-convex reproducing kernel shape functions. These quasi-convex meshfree shape functions are still established within the framework of the classical reproducing or consistency conditions, namely the shape functions have similar form as that of the conventional reproducing kernel shape functions. Thus this approach can be conveniently implemented in the standard reproducing kernel meshfree formulation without an overmuch increase of computational effort. Meanwhile, the present formulation enables a straightforward construction of arbitrary higher order shape functions. It is shown that the proposed method yields nearly positive shape functions in the interior problem domain, while in the boundary region the negative effect of the shape functions are also reduced compared with the original meshfree shape functions. Subsequently a Galerkin meshfree analysis is carried out by employing the proposed quasi-convex reproducing kernel shape functions. Numerical results reveal that the proposed method has more favorable accuracy than the conventional reproducing kernel meshfree method, especially for structural vibration analysis.     

A quasi‐linear reproducing kernel particle method

Reproducing kernel particle method (RKPM) has been applied to many large deformation problems. RKPM relies on polynomial reproducing conditions to yield desired accuracy and convergence properties but requires appropriate kernel support coverage of neighboring nodes to ensure kernel stability. This kernel stability condition is difficult to achieve for problems with large particle motion such as the fragment‐impact processes that commonly exist in extreme events. A new reproducing kernel formulation with ‘quasi‐linear’ reproducing conditions is introduced. In this approach, the first‐order polynomial reproducing conditions are approximately enforced to yield a nonsingular moment matrix. With proper error control of the first‐order completeness, nearly second‐order convergence rate in L2 norm can be achieved while maintaining kernel stability. The effectiveness of this quasi‐linear RKPM formulation is demonstrated by modeling several extremely large deformation and fragment‐impact problems. Copyright © 2016 John Wiley & Sons, Ltd.     

A meshfree unification: reproducing kernel peridynamics

This paper is the first investigation establishing the link between the meshfree state-based peridynamics method and other meshfree methods, in particular with the moving least squares reproducing kernel particle method (RKPM). It is concluded that the discretization of state-based peridynamics leads directly to an approximation of the derivatives that can be obtained from RKPM. However, state-based peridynamics obtains the same result at a significantly lower computational cost which motivates its use in large-scale computations. In light of the findings of this study, an update to the method is proposed such that the limitations regarding application of boundary conditions and the use of non-uniform grids are corrected by using the reproducing kernel approximation.     

A Hermite DRK interpolation-based collocation method for the analyses of Bernoulli–Euler beams and Kirchhoff–Love plates

A Hermite differential reproducing kernel (DRK) interpolation-based collocation method is developed for solving fourth-order differential equations where the field variable and its first-order derivatives are regarded as the primary variables. The novelty of this method is that we construct a set of differential reproducing conditions to determine the shape functions of derivatives of the Hermite DRK interpolation, without directly differentiating it. In addition, the shape function of this interpolation at each sampling node is separated into a primitive function possessing Kronecker delta properties and an enrichment function constituting reproducing conditions, so that the nodal interpolation properties are satisfied for the field variable and its first-order derivatives. A weighted least-squares collocation method based on this interpolation is developed for the static analyses of classical beams and plates with fully simple and clamped supports, in which its accuracy and convergence rate are examined, and some guidance for using this method is suggested.     

A meshfree weak-strong (MWS) form method for the unsteady magnetohydrodynamic (MHD) flow in pipe with arbitrary wall conductivity

In this paper a meshfree weak-strong (MWS) form method is considered to solve the coupled equations in velocity and magnetic field for the unsteady magnetohydrodynamic flow throFor this modified estimaFor this modified estimaFor this modified estimaugh a pipe of rectangular and circular sections having arbitrary conducting walls. Computations have been performed for various Hartman numbers and wall conductivity at different time levels. The MWS method is based on applying a meshfree collocation method in strong form for interior nodes and nodes on the essential boundaries and a meshless local Petrov–Galerkin method in weak form for nodes on the natural boundary of the domain. In this paper, we employ the moving least square reproducing kernel particle approximation to construct the shape functions. The numerical results for sample problems compare very well with steady state solution and other numerical methods.     

Treatment of discontinuity in the reproducing kernel element method

A discontinuous reproducing kernel element approximation is proposed in the case where weak discontinuity exists over an interface in the physical domain. The proposed method can effectively take care of the discontinuity of the derivative by truncating the window function and global partition polynomials. This new approximation keeps the advantage of both finite element methods and meshfree methods as in the reproducing kernel element method. The approximation has the interpolation property if the support of the window function is contained in the union of the elements associated with the corresponding node; therefore, the continuity of the primitive variables at nodes on the interface is ensured. Furthermore, it is smooth on each subregion (or each material) separated by the interface. The major advantage of the method is its simplicity in implementation and it is computationally efficient compared to other methods treating discontinuity. The convergence of the numerical solution is validated through calculations of some material discontinuity problems. Copyright © 2005 John Wiley & Sons, Ltd.