Second‐order convex maximum entropy approximants with applications to high‐order PDE

We present a new approach for second‐order maximum entropy (max‐ent) meshfree approximants that produces positive and smooth basis functions of uniform aspect ratio even for nonuniform node sets and prescribes robustly feasible constraints for the entropy maximization program defining the approximants. We examine the performance of the proposed approximation scheme in the numerical solution by a direct Galerkin method of a number of PDEs, including structural vibrations, elliptic second‐order PDEs, and fourth‐order PDEs for Kirchhoff–Love thin shells and for a phase field model describing the mechanics of biomembranes. The examples highlight the ability of the method to deal with nonuniform node distributions and the high accuracy of the solutions. Surprisingly, the first‐order meshfree max‐ent approximants with large supports are competitive when compared with the proposed second‐order approach in all the tested examples, even in the higher order PDEs. Copyright © 2012 John Wiley & Sons, Ltd.     

On the evaluation of the method of finite spheres for the solution of Reissner–Mindlin plate problems using the numerical inf–sup test

In this paper we use the numerical inf–sup test to evaluate both displacement‐based and mixed discretization schemes for the solution of Reissner–Mindlin plate problems using the meshfree method of finite spheres. While an analytical proof of whether a discretization scheme passes the inf–sup condition is most desirable, such a proof is usually out of reach due to the complexity of the meshfree approximation spaces involved. The numerical inf–sup test (Int. J. Numer. Meth. Engng 1997; 40 :3639–3663), developed to test finite element discretization spaces, has therefore been adopted in this paper. Tests have been performed for both regular and irregular nodal configurations. While, like linear finite elements, pure displacement‐based approximation spaces with linear consistency do not pass the inf–sup test and exhibit shear locking, quadratic discretizations, unlike quadratic finite elements, pass the test. Pure displacement‐based and mixed approximation spaces that pass the numerical inf–sup test exhibit optimal or near optimal convergence behaviour. Copyright © 2007 John Wiley & Sons, Ltd.     

Development of a genetic algorithm‐based lookup table approach for efficient numerical integration in the method of finite spheres with application to the solution of thin beam and plate problems

It is observed that for the solution of thin beam and plate problems using the meshfree method of finite spheres, Gaussian and adaptive quadrature schemes are computationally inefficient. In this paper, we develop a novel technique in which the integration points and weights are generated using genetic algorithms and stored in a lookup table using normalized coordinates as part of an offline computational step. During online computations, this lookup table is used much like a table of Gaussian integration points and weights in the finite element computations. This technique offers significant reduction of computational time without sacrificing accuracy. Example problems are solved which demonstrate the effectiveness of the procedure. Copyright © 2006 John Wiley & Sons, Ltd.     

A face‐based smoothed finite element method (FS‐FEM) for 3D linear and geometrically non‐linear solid mechanics problems using 4‐node tetrahedral elements

This paper presents a novel face‐based smoothed finite element method (FS‐FEM) to improve the accuracy of the finite element method (FEM) for three‐dimensional (3D) problems. The FS‐FEM uses 4‐node tetrahedral elements that can be generated automatically for complicated domains. In the FS‐FEM, the system stiffness matrix is computed using strains smoothed over the smoothing domains associated with the faces of the tetrahedral elements. The results demonstrated that the FS‐FEM is significantly more accurate than the FEM using tetrahedral elements for both linear and geometrically non‐linear solid mechanics problems. In addition, a novel domain‐based selective scheme is proposed leading to a combined FS/NS‐FEM model that is immune from volumetric locking and hence works well for nearly incompressible materials. The implementation of the FS‐FEM is straightforward and no penalty parameters or additional degrees of freedom are used. The computational efficiency of the FS‐FEM is found better than that of the FEM. Copyright © 2008 John Wiley & Sons, Ltd.     

Explicit form and efficient computation of MLS shape functions and their derivatives

This work presents a general and efficient way of computing both diffuse and full derivatives of shape functions for meshless methods based on moving least‐squares approximation (MLS) and interpolation. It is an extension of the recently introduced consistency approach based on Lagrange multipliers which provides a general framework for constrained MLS along with robust algorithms for the computation of shape functions and their diffuse derivatives. The particularity of the proposed algorithms is that they do not involve matrix inversion or linear system solving. The previous approach is limited to diffuse derivatives of the shape functions and not their full derivatives which are usually much more expensive to obtain. In the present paper we propose to efficiently compute the full derivatives by a new algorithm based on the formal differentiation of the previous one. In this way, we obtain a unified low‐cost consistent methodology for evaluating the shape functions and both their diffuse and full derivatives. In the second part of the paper we introduce explicit forms of MLS shape functions in 1D, 2D and 3D for an arbitrary number of nodes. These forms are especially useful for comparing finite element and MLS approximations. Finally we present a general architecture of an MLS program. Copyright © 2000 John Wiley & Sons, Ltd.     

基于一阶剪切变形理论和移动最小二乘近似的加肋板屈曲临界荷载求解

针对加肋板屈曲临界荷载的求解,提出了一种基于一阶剪切变形理论和移动最小二乘近似的无网格方法。该方法将加肋板的肋条和平板分开考虑,肋条用梁模型来模拟,按照一阶剪切变形理论和移动最小二乘近似给出平板和肋条的无网格近似位移场,再利用板和肋条交界上的位移协调条件推导出将肋条的节点参数转换成板节点参数的公式,最后通过转换公式,将板和肋条的势能叠加,由最小势能原理得到描述整个加肋板线性屈曲行为的控制方程。该文方法相对有限元的优势在于加肋板肋条不必沿网格线布置,即使肋条位置改变也不需要网格重构。文末通过几个算例比较了该文方法解和采用实体单元的ANSYS 有限元解,两者较为接近,证明了该文方法的准确性。     

超大变形分析无网格法并行计算

鉴于无网格法相较于其他技术计算任务更重,以及求解问题本身的高度复杂和大强度计算,并行计算对此显得尤具吸引力。重构核质点法(RKPM)是一种对固体和结构进行大应变弹塑性分析的常用无网格法,考虑到它能精确建模超大变形问题而无网格扭曲现象,以及能方便地对需要精细化的区域进行简单改变质点定义即可求解,在此仅集中讨论该方法。并行程序包括网格分区预分析和并行计算,后者包括各处理器上分区间的显式信息传递。文中用基于图像的Metis程序来进行网格分区,由于其重定义技术能应用于不同几何部件的共享区域,该程序常见于基于网格的分析中。并行模拟和MPI信息传递在SGI Onyx3900超级计算机上完成。文中对不同分区的并行计算效果和性能进行了比较分析,并给出了无网格法与有限元法的对比结果。     

An adaptive‐hybrid meshfree approximation method

It is now commonly agreed that the global radial basis functions (GRBF) method is an attractive approach for approximating smooth functions. This superiority does not come free; one must find ways to circumvent the associated problem of ill‐conditioning and the high computational cost for solving dense matrix systems. We previously proposed different variants of adaptive methods for selecting proper trial subspaces so that the instability caused by inappropriately shaped parameters were minimized. In contrast, the compactly supported radial basis functions (CSRBF) are more relaxing on the smoothness requirements. By settling with the algebraic order of convergence only, the CSRBF method, provided the support radii are properly chosen, can approximate functions with less smoothness. The reality is that end users must know the functions to be approximated a priori to decide which method to be used; this is not practical if one is solving a time‐evolving partial differential equation. The solution could be smooth at the beginning but the formation of shocks may come later. In this paper, we propose a hybrid algorithm making use of both GRBF and CSRBF with other developed techniques for meshfree approximation with minimal fine tuning. The first contribution here is an adaptive node refinement scheme. Second, we apply the GRBFs (with adaptive subspace selection) on the adaptively generated data sites, and lastly, the CSRBF (with adaptive support selection) that can be used as a blackbox algorithm for robust approximations to a wider class of functions and for solving PDEs. Copyright © 2011 John Wiley & Sons, Ltd.