A p-TYPE SUPERCONVERGENT RECOVERY METHOD FOR FE ELASTIC STABILITY ANALYSIS OF EULER BEAMS1)

本文将有限元p型超收敛算法应用于欧拉梁弹性稳定分析。该法基于有限元解答中失稳载荷和失稳模态结点位移的超收敛特性,建立了单元上失稳模态近似满足的线性常微分方程边值问题,在每个单元上,对该边值问题采用一个高次元进行求解,获得失稳模态的超收敛解,再将失稳模态的超收敛解代入瑞利商的解析表达式,最终获得失稳载荷的超收敛解。该法思路简明,通过少量计算即可显著提高失稳载荷和失稳模态的精度与收敛阶。数值算例表明,该法高效、可靠,值得进一步研究和推广到各类杆系结构。     

Recursive super-convergence computation for multi-dimensional problems via one-dimensional element energy projection technique

This paper presents a strategy for computation of super-convergent solutions of multi-dimensional problems in the finite element method (FEM) by recursive application of the one-dimensional (1D) element energy projection (EEP) technique. The main idea is to conceptually treat multi-dimensional problems as generalized 1D problems, based on which the concepts of generalized 1D FEM and its consequent EEP formulae have been developed in a unified manner. Equipped with these concepts, multi-dimensional problems can be recursively discretized in one dimension at each step, until a fully discretized standard finite element (FE) model is reached. This conceptual dimension-by-dimension (D-by-D) discretization procedure is entirely equivalent to a full FE discretization. As a reverse D-by-D recovery procedure, by using the unified EEP formulae together with proper extraction of the generalized nodal solutions, super-convergent displacements and first derivatives for two-dimensional (2D) and three-dimensional (3D) problems can be obtained over the domain. Numerical examples of 3D Poisson’s equation and elasticity problem are given to verify the feasibility and effectiveness of the proposed strategy.     

Higher‐order surface FEM for incompressible Navier‐Stokes flows on manifolds

Stationary and instationary Stokes and Navier‐Stokes flows are considered on two‐dimensional manifolds, ie, on curved surfaces in three dimensions. The higher‐order surface FEM is used for the approximation of the geometry, velocities, pressure, and Lagrange multiplier to enforce tangential velocities. Individual element orders are employed for these various fields. Streamline‐upwind stabilization is employed for flows at high Reynolds numbers. Applications are presented, which extend classical benchmark test cases from flat domains to general manifolds. Highly accurate solutions are obtained, and higher‐order convergence rates are confirmed.     

Finite element simulations on the mechanical properties of MHS materials

Finite element simulations are carried out to examine the mechanical behavior of the metallic hollow sphere (MHS) material during their large plastic deformation and to estimate the energy absorbing capacity of these materials under uniaxial compression. A simplified model is proposed from experimental observations to describe the connection between the neighboring spheres, which greatly improves the computation efficiency. The effects of the governing physical and geometrical parameters are evaluated; whilst a special attention is paid to the plateau stress, which is directly related to the energy absorbing capacity. Finally, the empirical functions of the relative material density are proposed for the elastic modulus, yield strength and plateau stress for FCC packing arrangement of hollow spheres, showing a good agreement with the experimental results obtained in our previous study. The project supported by the Hong Kong Research Grant Council (RGC) (HKUST 6079/00E) and the National Natural Science Foundation of China (10532020).     

具有单元分裂功能的间断有限元方法

介绍一种能够模拟材料开裂过程的有限元方法,该方法引入单元分裂和界面分离技术,结合具体的破坏准则,模拟材料变形中的破坏,同时还可以方便地处理材料中任意分布的界面结构．以三点弯曲实验为例,通过数值模拟结果和实验数据的比较,验证了该方法的适用性．     

Superconvergence analysis of an energy stable scheme with three step backward differential formula-finite element method for nonlinear reaction–diffusion equation

A three step backward differential formula scheme is proposed for nonlinear reaction–diffusion equation and superconvergence results are studied with Galerkin finite element method unconditionally. Energy stability is testified for the constructed scheme with an artificial term. Splitting technique is utilized to get rid of the ratio between the time step size and the subdivision parameter . Temporal error estimate in H²-norm is derived, which leads to the boundedness of the solutions of the time-discrete equations. Unconditional spatial error estimate in L²-norm is deduced which help bound the numerical solutions in L^∞-norm. Superconvergent property of in H¹-norm with order is obtained by taking difference between two time levels of the error equations unconditionally. The global superconvergent property is deduced through the above results. Two numerical examples show the validity of the theoretical analysis.     

Superconvergence Analysis of a BDF-Galerkin FEM for the Nonlinear Klein-Gordon-Schrödinger Equations with Damping Mechanism

The focus of this paper is on a linearized backward differential formula (BDF) scheme with Galerkin FEM for the nonlinear Klein-Gordon-Schrödinger equations (KGSEs) with damping mechanism. Optimal error estimates and superconvergence results are proved without any time-step restriction condition for the proposed scheme. The proof consists of three ingredients. First, a temporal-spatial error splitting argument is employed to bound the numerical solution in certain strong norms. Second, optimal error estimates are derived through a novel splitting technique to deal with the time derivative and some sharp estimates to cope with the nonlinear terms. Third, by virtue of the relationship between the Ritz projection and the interpolation, as well as a so-called "lifting'' technique, the superconvergence behavior of order $O(h^2+\tau^2)$ in $H^1$-norm for the original variables are deduced. Finally, a numerical experiment is conducted to confirm our theoretical analysis. Here, $h$ is the spatial subdivision parameter, and $\tau$ is the time step.