Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity
Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity
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DOI：
10.1016/s0045-7825(02)00286-4
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发表时间：
2002-07
影响因子：
7.2
通讯作者：
G. Engel;K. Garikipati;T. Hughes;M. Larson;L. Mazzei;R. Taylor
中科院分区：
文献类型：
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作者：
G. Engel;K. Garikipati;T. Hughes;M. Larson;L. Mazzei;R. Taylor
本文提出了一种求解四阶椭圆型偏微分方程的新的有限元方法，并将其应用于结构力学中的薄弯曲理论问题和应变梯度理论问题。该方法结合了连续Galerkin（CG）方法，间断Galerkin（DG）方法和稳定技术的概念。简要回顾CG方法，DG方法和稳定技术突出了这些方法的优点和缺点，并提出了一种新的方法来解决四阶椭圆问题。提出了一种连续/间断Galerkin（C/DG）方法，该方法采用C_0连续插值函数，且只以主变量形式表示。这种提法相对于更传统的混合方法的优点是，可以避免引入额外的未知数和相关困难。在薄弯曲理论的上下文中，C/DG方法导致一个公式，其中位移是唯一的自由度，并且不需要考虑旋转自由度。C/DG方法的主要特点是通过内边界上的稳定项来弱地执行一阶和高阶导数的连续性。分析了该方法的一致性、稳定性和收敛性。数值实验验证了理论结果，并给出了Bernoulli-Euler梁弯曲、Poisson-Kirchhoff板弯曲和Toupin-Mindlin应变梯度理论在剪切层问题中的应用。
A new finite element method for fourth-order elliptic partial differential equations is presented and applied to thin bending theory problems in structural mechanics and to a strain gradient theory problem. The method combines concepts from the continuous Galerkin (CG) method, the discontinuous Galerkin (DG) method and stabilization techniques. A brief review of the CG method, the DG method and stabilization techniques highlights the advantages and disadvantages of these methods and suggests a new approach for the solution of fourth-order elliptic problems. A continuous/discontinuous Galerkin (C/DG) method is proposed which uses C0-continuous interpolation functions and is formulated in the primary variable only. The advantage of this formulation over a more traditional mixed approach is that the introduction of additional unknowns and related difficulties can be avoided. In the context of thin bending theory, the C/DG method leads to a formulation where displacements are the only degrees of freedom, and no rotational degrees of freedom need to be considered. The main feature of the C/DG method is the weak enforcement of continuity of first and higher-order derivatives through stabilizing terms on interior boundaries. Consistency, stability and convergence of the method are shown analytically. Numerical experiments verify the theoretical results, and applications are presented for Bernoulli–Euler beam bending, Poisson–Kirchhoff plate bending and a shear layer problem using Toupin–Mindlin strain gradient theory.