关于线弹性问题混合格式稳定化思想的探讨

首先从混合有限元理论出发,探讨线弹性问题混合变分格式所满足的稳定性条件,从而保证解的存在唯一性.使用连续的分片线性函数和分片常数来分别逼近应力和位移,详细分析了混合格式下稳定化的必要性,有助于更加深入地了解稳定化的基本思想.然后,通过在混合格式中引入位移的跳跃惩罚项,展示了一个无闭锁稳定化混合有限元方法,并证明了此方法是稳定的且是线性收敛的.     

几乎不可压缩线性弹性问题的多重网格Uzawa型混合有限元方法

该文针对几乎不可压缩弹性问题,设计了多重网格Uzawa型混合有限元方法,成功克服了"闭锁"现象.通过引入"压力"变量p将弹性问题转化为一个鞍点型系统,对该系统将Uzawa型迭代法和多重网格方法相结合,建立了多重网格和套迭代多重网格Uzawa型混合有限元方法,并给出了该算法的收敛性.数值算例验证了方法的有效性和稳定性.     

新的投影混合稳定化方法在抛物问题中的应用

针对抛物问题提出一种新的投影混合稳定化方法.该方法基于等阶的混合有限元,相比通常的局部投影稳定化方法,增加了新的投影稳定项及压力跳跃项,有效地克服了等阶有限元不满足inf-sup条件而导致的解的不稳定性,也保证了该方法不仅对连续的压力空间适用,且对不连续的压力空间亦适用.本文证明了该方法的稳定性,并给出了误差估计.最后,数值算例验证了该方法的理论分析及有效性.     

弹性力学方程解的整体适定性

本文为一篇综述文章,主要回顾中外数学家在可压缩和不可压缩弹性力学方程平衡态附近经典解的整体适定性方面所取得的关键研究成果.由于这里所涉及的研究思想和方法与研究拟线性波动方程相应问题的思想和方法密切相关,因此也将回顾拟线性波动方程的一些相应问题的理论和研究方法.本文将尽可能简单明了地指出各研究课题的关键困难及克服它们的基本想法,并对其中大部分关键成果给予更为直截了当的证明.本文还将提出几个公开问题并简单讨论其困难所在,以期向更年轻的专家学者抛砖引玉.     

平面弹性力学问题的混合元法的泡函数稳定性及其后验误差估计

该文讨论平面弹性力学问题的混合元法的泡函数稳定性,并导出基于简化的稳定化格式的一种先验误差估计和后验误差估计．这种简化的稳定化格式较通常的格式节省自由度．     

具有弱对称应力的线弹性方程的稳定化格式

利用稳定化方法讨论拉格朗日乘子法得到的具有弱对称应力的线弹性问题. 用线性元和分片常数分别逼近变分问题的应力和位移. 并通过添加稳定项$G_1(\cdot,\cdot)$, $G_2(\cdot,\cdot)$和$G_3(\cdot,\cdot)$ 使相应混合离散变分问题满足弱BB条件. 接着详细研究了变分问题的解与稳定混合有限元解之间的误差估计,最后用两个数值算例验证理论分析的有效性.     

狭吸积环中非轴对称动力学不稳定性的非线性演化

本文在孤子理论的框架下解释非轴对称动力学不稳定性的非线性演化.在长波不可压缩极限下,两维狭吸积环的色散关系与线性KdV方程的相同.我们认为:非线性动力学不稳定性数值模拟中的“行星状”解是KdV方程的孤子解.由于在动力学不稳定性的非线性演化中密度和熵的变化,吸积盘的涡度是不守恒量,这也使角动量在不稳定过程中重新分布.     

G-Brown运动驱动的非线性随机时滞微分方程的稳定化

研究了一类G-Brown运动驱动的非线性随机时滞微分方程的稳定化问题.首先,在一个不稳定的G-Brown运动驱动的非线性随机时滞微分方程的漂移项中设计了时滞反馈控制, 得其相应的控制系统.其次, 利用Lyapunov函数方法给出其相应的控制系统是渐近稳定的充分条件.最后, 通过例子说明了所得的结果.     

夹层结构杆在弹性基础上的弹塑性变形

对位于弹性基底上的、具有可压缩非线性芯子的3层弹塑性杆的弯曲问题进行了研究．研究分析了由2个受力层和1个芯子层组成的3层构件的力学响应．解决了位于弹性基底上的3层杆弯曲的复杂问题．对所给出的弹性解法进行了收敛性检验,以保证该弹性解是可以接受的．计算结果表明,材料的塑性和物理非线性对位于弹性基底上的夹层结构杆的变形影响很大．     

一类可压缩流体驱动问题的有限元方法

多孔介质中渗流驱动问题数值模拟的研究,在采油及许多工程技术领域中有重要意义;一般这类问题对应的数学模型是关于压力、浓度的耦合方程组;不可压缩流体驱动问题有限元、混合元方法在[1,2,8,9]中曾得了成功的研究,文[3,4]研究了一类微可压缩问题,但其理论分析是基于系数函数(浓度的非线性泛函)有不依赖浓度的正的上、下界等     

A Galerkin/least-square finite element formulation for nearly incompressible elasticity/stokes flow

A Galerkin/least-square finite element formulation (GLS) is used to study mixed displacement-pressure formulation of nearly incompressible elasticity. In order to fully incorporate the effect of the residual-based stabilized term to the weak form, the second derivatives of shape functions were also derived and accounted, which can accurately discretize the residual term and improve the GLS method as well as the Petrov–Galerkin method. The numerical studies show that improved stabilized method can effectively remove volumetric locking problem for incompressible elasticity and stabilize the pressure field for stokes flow. When apply GLS to study material nonlinearity, the derivative of tangent modulus at the integration point will be required. Both advantage and disadvantage of using GLS method for nearly incompressible elasticity/stokes flow were demonstrated.     

Iterative schemes for mixed finite element methods with applications to elasticity and compressible flow problems

Summary Iterative schemes for mixed finite element methods are proposed and analyzed in two abstract formulations. The first one has applications to elliptic equations and incompressible fluid flow problems, while the second has applications to linear elasticity and compressible Stokes problems. These schemes are constructed through iteratively penalizing the mixed finite element scheme, of which iterated penalty method and augmented Lagrangian method are special cases. Convergence theorems are demonstrated in abstract formulations in Hilbert spaces, and applications to individual physical problems are considered as examples. Theoretical analysis and computational experiments both show that the proposed schemes have very fast convergence; a few iterations are normally enough to reduce the iterative error to a prescribed precision. Numerical examples with continuous and discontinuous coefficients are presented.     

Finite element approximation of the elasticity spectral problem on curved domains

We analyze the finite element approximation of the spectral problem for the linear elasticity equation with mixed boundary conditions on a curved non-convex domain. In the framework of the abstract spectral approximation theory, we obtain optimal order error estimates for the approximation of eigenvalues and eigenvectors. Two kinds of problems are considered: the discrete domain does not coincide with the real one and mixed boundary conditions are imposed. Some numerical results are presented.     

Semilinear Mixed Problems on Hilbert Complexes and Their Numerical Approximation

Arnold, Falk, and Winther recently showed (Bull. Am. Math. Soc. 47:281–354, 2010) that linear, mixed variational problems, and their numerical approximation by mixed finite element methods, can be studied using the powerful, abstract language of Hilbert complexes. In another recent article (arXiv:), we extended the Arnold–Falk–Winther framework by analyzing variational crimes (à la Strang) on Hilbert complexes. In particular, this gave a treatment of finite element exterior calculus on manifolds, generalizing techniques from surface finite element methods and recovering earlier a priori estimates for the Laplace–Beltrami operator on 2- and 3-surfaces, due to Dziuk (Lecture Notes in Math., vol. 1357:142–155, 1988) and later Demlow (SIAM J. Numer. Anal. 47:805–827, 2009), as special cases. In the present article, we extend the Hilbert complex framework in a second distinct direction: to the study of semilinear mixed problems. We do this, first, by introducing an operator-theoretic reformulation of the linear mixed problem, so that the semilinear problem can be expressed as an abstract Hammerstein equation. This allows us to obtain, for semilinear problems, a priori solution estimates and error estimates that reduce to the Arnold–Falk–Winther results in the linear case. We also consider the impact of variational crimes, extending the results of our previous article to these semilinear problems. As an immediate application, this new framework allows for mixed finite element methods to be applied to semilinear problems on surfaces.     

Consistency,stability, a priori and a posteriori errors for Petrov-Galerkin methods applied to nonlinear problems

Summary. In an abstract framework we present a formalism which specifies the notions of consistency and stability of Petrov-Galerkin methods used to approximate nonlinear problems which are, in many practical situations, strongly nonlinear elliptic problems. This formalism gives rise to a priori and a posteriori error estimates which can be used for the refinement of the mesh in adaptive finite element methods applied to elliptic nonlinear problems. This theory is illustrated with the example: in a two dimensional domain with Dirichlet boundary conditions. Received June 10, 1992 / Revised version received February 28, 1994     

Stabilized two-grid discretizations of locking free for the elasticity eigenvalue problem

In this paper, we propose two stabilized two-grid finite element discretizations for nearly incompressible elasticity eigenvalue problem and give the error estimates of eigenvalues and eigenfunctions for the schemes. Numerical experiments are provided to validate our theoretical analysis and exhibit that our schemes are locking free and highly efficient.     

Towards a unified analysis of mixed methods for elasticity with weakly symmetric stress

We propose a framework for unified analysis of mixed methods for elasticity with weakly symmetric stress. Based on a commuting diagram in the weakly symmetric elasticity complex and extending a previous stability result, stable mixed methods are obtained by combining Stokes stable and elasticity stable finite elements. We show that the framework can be used to analyze most existing mixed methods for the linear elasticity problem with elementary techniques. We also show that some new stable mixed finite elements are obtained.     

Numerical resolvent methods for constrained problems in mechanics

Resolvent methods are presented for generating systematically iterative numerical algorithms for constrained problems in mechanics. The abstract framework corresponds to a general mixed finite element subdifferential model, with dual and primal evolution versions, which is shown to apply to problems of fluid dynamics, transport phenomena and solid mechanics, among others. In this manner, Uzawa’s type methods and penalization-duality schemes, as well as macro-hybrid formulations, are generalized to non necessarily potential nonlinear mechanical problems.     

一类基于变量分离的稳定化混合有限元方法

Based on a seperated model for saddle-point problems, we develop a new sta-bilized mixed finite element method. Such a model consists of two subproblems with respect to the primal and the dual variables, respectively. We show that the new method is coercive and that optimal error bounds hold. As an application,the nearly incompressible elastic problem is analyzed with our method.