Numerical Approximation of the Solution of an Obstacle Problem Modelling the Displacement of Elliptic Membrane Shells via the Penalty Method
Numerical Approximation of the Solution of an Obstacle Problem Modelling the Displacement of Elliptic Membrane Shells via the Penalty Method
复制标题
通过惩罚法模拟椭圆膜壳位移的障碍问题求解的数值逼近
DOI：
10.1007/s00245-024-10112-x
复制
发表时间：
2024
影响因子：
1.8
通讯作者：
Piersanti, Paolo
中科院分区：
文献类型：
--
作者：
Meixner, Aaron;Piersanti, Paolo
在本文中，我们建立了一个数值格式的收敛性的基础上，有限元法，为时间无关的问题建模的线性弹性椭圆形膜壳的变形受到保持限制在半空间。而不是近似原来的变分不等式管理这个障碍问题，我们近似的惩罚版本的问题正在考虑中。适当的耦合之间的惩罚参数和网格大小，然后将导致我们建立的离散惩罚问题的解决方案的原始变分不等式的解决方案的收敛性。我们还建立了收敛的Brezis-Sibony计划所考虑的问题。由于这种迭代方法，我们可以近似的离散惩罚问题的解决方案，而不必求助于非线性优化工具。最后，我们提出了数值模拟验证我们的新的理论结果。
In this paper we establish the convergence of a numerical scheme based, on the Finite Element Method, for a time-independent problem modelling the deformation of a linearly elastic elliptic membrane shell subjected to remaining confined in a half space. Instead of approximating the original variational inequalities governing this obstacle problem, we approximate the penalized version of the problem under consideration. A suitable coupling between the penalty parameter and the mesh size will then lead us to establish the convergence of the solution of the discrete penalized problem to the solution of the original variational inequalities. We also establish the convergence of the Brezis–Sibony scheme for the problem under consideration. Thanks to this iterative method, we can approximate the solution of the discrete penalized problem without having to resort to nonlinear optimization tools. Finally, we present numerical simulations validating our new theoretical results.