线弹性断裂力学问题的扩展自然单元法

为了更加有效地求解线弹性断裂问题,提出了扩展自然单元法。该方法基于单位分解的思想,在自然单元法的位移模式中加入扩展项表征不连续位移场和裂纹尖端奇异场。通过水平集方法确定裂纹面和裂纹尖端区域,并基于虚位移原理推导了平衡方程的离散线性方程。由于自然单元法的形函数满足Kronecker delta函数性质,本质边界条件易于施加。混合模式裂纹的应力强度因子由相互作用能量积分方法计算。数值算例结果表明扩展自然单元法可以方便地求解线弹性断裂力学问题。     

模拟三维裂纹问题的自适应多尺度扩展有限元法

为了在大型结构分析中考虑小裂纹或以小的代价提高裂纹附近求解精度,该文建立了分析三维裂纹问题的自适应多尺度扩展有限元法。基于恢复法评估三维扩展有限元后验误差,大于给定误差值的单元进行细化。所有尺度单元采用八结点六面体单元,采用六面体任意结点单元连接不同尺度单元。采用互作用积分法计算三维应力强度因子。三维Ⅰ型裂纹和Ⅰ-Ⅱ复合型裂纹算例分析表明了该方法的正确性和有效性。     

裂纹扩展问题的改进XFEM算法

利用扩展有限元法计算裂尖附近应力、位移场,进而得到裂尖应力强度因子和开裂角;水平集法描述、追踪裂纹,并由单元结点水平集值判别单元类型;将二者结合起来分析处理裂纹扩展问题。针对水平集判别倾斜裂纹单元类型的不足,分析问题的原因,并给出解决方案。最后,通过典型算例分析,表明将扩展有限元法与水平集法结合分析裂纹扩展问题时具有不需网格重构,裂纹与网格相互独立的特点;同时验证了笔者提出解决方案的准确性和可行性。     

HT有限元在Ⅰ、Ⅱ与Ⅲ型复合弹性断裂问题中的应用

探讨了HT有限元应用于Ⅰ、Ⅱ和Ⅲ型复合裂纹的弹性断裂问题。分析了Ⅲ型弹性断裂问题的HT有限元方法及高阶奇异性应力强度因子KΙΙΙ,同时,对Ⅰ和Ⅱ型断裂问题的HT有限元原理及断裂强度因子KΙ和KΙΙ的计算也进行了阐述。特别地,在计算三个强度因子时,引入了一种新的方法——附加试函数法,它主要用于满足裂尖特殊的边界条件,提高了三个奇异应力强度因子的精确性与可靠性。最后,根据HT有限元计算结果,讨论了奇异应力强度因子无量纲化系数K/Kc随裂纹单元特殊T函数项数、细划单元数、单元高斯点数及裂尖不同附加试函数的变化规律;获得了应力强度因子精确度和可靠度,并与其它有限元结果进行了比较,阐述了此方法的优越性。     

求解双材料界面裂纹应力强度因子的扩展有限元法

基于双材料界面裂纹尖端的基本解,构造扩展有限元法(e Xtended Finite Element Methods,XFEM)裂尖单元结点的改进函数。有限元网格剖分不遵从材料界面,考虑3种类型的结点改进函数:弱不连续改进函数、Heaviside改进函数和裂尖改进函数,建立XFEM的位移模式,给出计算双材料界面裂纹应力强度因子(Stress Intensity Factors,SIFs)的相互作用积分方法。数值结果表明:XFEM无需遵从材料界面剖分网格,该文的方法能够准确评价双材料界面裂纹尖端的SIFs。     

垂直界面裂纹应力强度因子的加料有限元分析

为求解裂尖位于界面上的垂直双材料界面裂纹应力强度因子,发展了一种加料有限元方法。该方法应用Williams本征函数展开和线性变换方法求解裂尖渐进位移场,将该位移场加入常规单元位移模式中,得到加料垂直界面裂纹单元和过渡单元的位移模式,给出加料有限元方程。建立了典型垂直界面裂纹平面问题的加料有限元模型,求解加料有限元方程直接得到应力强度因子,与文献结果对比表明该方法具有较高的精度,可方便地推广应用于垂直界面裂纹的计算分析。     

发展基于CB壳单元的扩展有限元模拟三维任意扩展裂纹

该文提出了一种新的基于连续体壳单元的扩展有限元格式,以用于对曲面上任意形状裂纹的扩展问题进行模拟。扩充形函数的构造和应力强度因子的计算都是基于三维实体单元进行,因此可以模拟复杂的三维断裂情况,壳体厚度的变化也可以得到考虑。三维应力强度因子的计算公式被引入到这种方法中。为模拟裂纹扩展,三维最大能量释放率准则被用作裂纹扩展准则。计算结果显示了曲面上的裂纹扩展路径可以与网格无关,并且由于在裂纹尖端的单元设置了具有奇异性的形函数,裂尖应力场被精确捕捉,从而证明了这种方法的优越性。     

基于Kirchhoff板中裂纹尖端辛本征解的有限元应力恢复方法

对于含穿透裂纹的板结构,裂纹尖端应力场及应力强度因子的计算精度对评估板的安全性具有非常重要的影响。基于含裂纹Kirchhoff板弯曲问题中裂纹尖端场的辛本征解析解,该文提出了一个提高裂纹尖端应力场计算精度的有限元应力恢复方法。首先利用常规有限元程序对含裂纹板弯曲问题进行分析,得到裂纹尖端附近的单元节点位移;然后根据节点位移确定辛本征解中的待定系数,得到裂纹尖端附近应力场的显式表达式。数值结果表明,该方法给出的应力分析精度得到较大提高,并具有良好的数值稳定性。     

裂纹板贴补应力分析

本文采用含裂纹无限大板特殊基本解和合力边界条件,用体积力法对含裂纹金属薄板的胶贴补强问题进行应力分析。使用一满足胶贴层位移连续条件的剪切单元,把问题转化为对裂纹板和贴片的分析。由于使用的特殊基本解精确满足裂纹面自由力边界条件,避免了对裂纹尖端附近的奇异场进行离散处理,因而可以比较精确地求出裂纹尖端附近的应力分布,同时由于单位集中力引起的裂纹尖端应力强度因子可以解析得到,因而可以较准确地反映出用应力强度因子的降低来表征的贴补效果。作为贴补计算的例子,文中计算了受拉力和剪力作用时,含中心裂纹的金属裂纹板在贴补前后裂纹尖端应力强度因子的降低,给出了贴片的厚度、弹性模量和尺寸及肢贴层厚度等对贴补效果的影响。     

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。     

Natural element approximations involving bubbles for treating mechanical models in incompressible media

In this paper, a new approach is proposed to address issues associated with incompressibility in the context of the meshfree natural element method (NEM). The NEM possesses attractive features such as interpolant shape functions or auto‐adaptive domain of influence, which alleviates some of the most common difficulties in meshless methods. Nevertheless, the shape functions can only reproduce linear polynomials, and in contrast to moving least squares methods, it is not easy to define interpolations with arbitrary approximation consistency. In order to treat mechanical models involving incompressible media in the framework of mixed formulations, the associated functional approximations must satisfy the well‐known inf–sup, or LBB condition. In the proposed approach, additional degrees of freedom are associated with some topological entities of the underlying Delaunay tessellation, i.e. edges, triangles and tetrahedrons. The associated shape functions are computed from the product of the NEM shape functions related to the original nodes. Different combinations can be used to construct new families of NEM approximations. As these new approximations functions are not related to any node, as they vanish at the nodes, from now on we refer these shape functions as bubbles. The shape functions can be corrected enforcing different reproducing conditions, when they are used as weights in the moving least square (MLS) framework. In this manner, the effects of the obtained higher approximation consistency can be evaluated. In this work, we restrict our attention to the 2D case, and the following constructions will be considered: (a) bubble functions associated with the Delaunay triangles, called b1‐NEM and (b) bubble functions associated with the Delaunay edges, called b2‐NEM. We prove that all these approximation schemes allow direct enforcement of essential boundary conditions. The bubble‐NEM schemes are then used to approximate the displacements in the linear elasticity mixed formulation, the pressure being approximated by the standard NEM. The numerical LBB test is passed for all the bubble‐NEM approximations, and pressure oscillations are removed in the incompressible limit. Copyright © 2005 John Wiley & Sons, Ltd.     

Combination of Newmark method and natural element method for elastodynamics

The natural element method (NEM) is a meshless method. The trial and test functions of the NEM are constructed using natural neighbor interpolations which are based on the Voronoi tessellation of a set of nodes. The NEM interpolation is linear between adjacent nodes on the boundary of the convex hull, which makes imposition of essential boundary conditions easy to implement. We investigate the performance of the NEM combined with the Newmark method for problems of elastodynamics in this article. Applications are considered for a cantilever beam with different initial load conditions. The NEM numerical results are compared with the finite element method. NEM shows promise for these applications.     

Calculation of the derivative of interpolation shape function for three-dimensional natural element method

The natural element method (NEM) is a special meshless method. Its shape functions are constructed using natural neighbor node interpolations based on the concepts of Voronoi tessellation. The NEM interpolation is linear between adjacent nodes on the boundary of the convex hull, which facilitates imposition of essential boundary conditions. However, for a three-dimensional problem, the computation of shape function derivative of NEM is still very complicated even with the non-Sibson interpolation function, which makes the NEM an unpopular numerical method. In this paper, we adopt the direct mathematical derivative technique, and after some rigorous deduction, finally obtain the shape function derivative expression of three-dimensional NEM. Compared with the Lasserre algorithm, this algorithm is more intuitionistic and can be conveniently programmed. The NEM numerical results for cantilever beams verify the correctness of the shape function derivative expression of NEM derived in this paper.     

On solving large strain hyperelastic problems with the natural element method

In this paper, an extension of the natural element method (NEM) is presented to solve finite deformation problems. Since NEM is a meshless method, its implementation does not require an explicit connectivity definition. Consequently, it is quite adequate to simulate large strain problems with important mesh distortions, reducing the need for remeshing and projection of results (extremely important in three‐dimensional problems). NEM has important advantages over other meshless methods, such as the interpolant character of its shape functions and the ability of exactly reproducing essential boundary conditions along convex boundaries. The α‐NEM extension generalizes this behaviour to non‐convex boundaries. A total Lagrangian formulation has been employed to solve different problems with large strains, considering hyperelastic behaviour. Several examples are presented in two and three dimensions, comparing the results with the ones of the finite element method. NEM performs better showing its important capabilities in this kind of applications. Copyright © 2004 John Wiley & Sons, Ltd.     

A natural neighbour-based moving least-squares approach for the element-free Galerkin method

The element-free Galerkin method (EFG) and the natural element method (NEM) are two well known and widely used meshless methods. Whereas the EFG method can represent moving boundaries like cracks only by modifying the weighting functions the NEM requires an adaptation of the nodal set-up. But on the other hand the NEM is computationally more efficient than EFG. In this paper a new concept for the automatic adjustment of nodal influence domains in the EFG method is presented in order to obtain an efficiency similar to the NEM. This concept is based on the definition of natural neighbours for each meshless node which can be determined from a Voronoi diagram of the nodal set-up. In this approach adapted nodal influence domains are obtained by interpolating the distances to the natural neighbours depending on the direction. In the paper we show that this concept leads, especially for problems with grading node density, to a reduced number of influencing nodes at the interpolation points and consequently a significant reduction of the numerical effort. Copyright © 2006 John Wiley & Sons, Ltd.     

Dispersive properties of the natural element method

 The Natural Element Method (NEM) is a mesh-free numerical method for the solution of partial differential equations. In the natural element method, natural neighbor coordinates, which are based on the Voronoi tesselation of a set of nodes, are used to construct the interpolant. The performance of NEM in two-dimensional linear elastodynamics is investigated. A standard Galerkin formulation is used to obtain the weak form and a central-difference time integration scheme is chosen for time history analyses. Two different applications are considered: vibration of a cantilever beam and dispersion analysis of the wave equations. The NEM results are compared to finite element and analytical solutions. Excellent dispersive properties of NEM are observed and good agreement with analytical solutions is obtained.     

自然元与无限元耦合方法在岩土工程粘弹性分析中的应用

自然单元法是一种新的偏微分方程数值解法,由于其位移插值函数采用无网格的方式构造且形函数满足插值性质,从而克服传统有限元方法对单元网格信息的依赖,大大简化数值计算的前处理过程,同时又能像有限元那样准确施加边界条件,在岩土工程中具有广阔的应用前景;介绍了自然元与无限元的基本原理,针对在处理岩土工程无限域或半无限域问题时需要人为确定边界条件而带来计算误差的问题,引入无限元模拟无穷远处边界条件,与自然元相结合形成耦合分析方法;并根据粘弹性理论,采用Laplace插值,编制了基于自然元与无限元耦合方法的二维粘弹性分析程序,通过算例验证了算法的正确性,结果也表明相对于纯自然单元法,耦合方法能够显著提高分析结果的精度,在此基础上拓展了自然单元法在岩土工程中的应用范围.     

响应面建模方法的比较分析

 工程应用中选用拟合精度和拟合效率较高的响应面建模方法是其应用的关键。首先描述主要响应面建模方法的逼近函数形式、参数估计方法及其基本特性；然后引入4个有代表性的函数，并分别利用上述响应面建模方法对这4个函数进行响应面拟合；最后从拟合精度和拟合效率两方面比较分析上述响应面建模方法。结果表明，基于径向基函数的响应面建模方法最适于实际问题的响应面建模。     

Reproducing kernel enhanced local radial basis collocation method

Standard radial basis functions (RBFs) offer exponential convergence, however, the method is suffered from the large condition numbers due to their ‘nonlocal’ approximation. The nonlocality of RBFs also limits their applications to small‐scale problems. The reproducing kernel functions, on the other hand, provide polynomial reproducibility in a ‘local’ approximation, and the corresponding discrete systems exhibit relatively small condition numbers. Nonetheless, reproducing kernel functions produce only algebraic convergence. This work intends to combine the advantages of RBFs and reproducing kernel functions to yield a local approximation that is better conditioned than that of the RBFs, while at the same time offers a higher rate of convergence than that of reproducing kernel functions. Further, the locality in the proposed approximation allows its application to large‐scale problems. Error analysis of the proposed method is also provided. Numerical examples are given to demonstrate the improved conditioning and accuracy of the proposed method. Copyright © 2007 John Wiley & Sons, Ltd.