Finite element approximation of a strain-limiting elastic model
Finite element approximation of a strain-limiting elastic model
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DOI：
10.1093/imanum/dry065
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发表时间：
2018-05
影响因子：
2.1
通讯作者：
A. Bonito;V. Girault;E. Suli
中科院分区：
文献类型：
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作者：
A. Bonito;V. Girault;E. Suli
我们在 $\mathbb{R}^d$, $d \in \{2,3\}$ 中的有界开域上构造应变限制弹性模型的有限元近似。有限元近似序列显示出对模型唯一弱解的强收敛性。如果模型中的材料参数是 Lipschitz 连续的并且精确解具有额外的规律性，则显示有限元近似序列的收敛速度。如果模型中的材料参数是 Lipschitz 连续的并且精确解具有额外的规律性，则显示有限元近似序列的收敛速度。构建了一种迭代算法来求解由有限元近似产生的非线性代数方程组。迭代算法的一个吸引人的特点是它解耦了模型中非线性的单调和线弹性部分。特别是，我们选择应力张量的分段常数近似（以及位移的连续分段线性近似）使我们能够通过在计算域细分中的每个元素上独立求解具有 $d(d+1)/2$ 未知数的代数系统来计算非线性的单调部分。通过数值实验说明了理论结果。
We construct a finite element approximation of a strain-limiting elastic model on a bounded open domain in $\mathbb{R}^d$, $d \in \{2,3\}$. The sequence of finite element approximations is shown to exhibit strong convergence to the unique weak solution of the model. A rate of convergence for the sequence of finite element approximations is shown provided that the material parameters featuring in the model are Lipschitz continuous and that the exact solution possesses additional regularity. A rate of convergence for the sequence of finite element approximations is shown provided that the material parameters featuring in the model are Lipschitz continuous and that the exact solution possesses additional regularity. An iterative algorithm is constructed for the solution of the system of nonlinear algebraic equations that arises from the finite element approximation. An appealing feature of the iterative algorithm is that it decouples the monotone and linear elastic parts of the nonlinearity in the model. In particular, our choice of piecewise constant approximation for the stress tensor (and continuous piecewise linear approximation for the displacement) allows us to compute the monotone part of the nonlinearity by solving an algebraic system with $d(d+1)/2$ unknowns independently on each element in the subdivision of the computational domain. The theoretical results are illustrated by numerical experiments.