Robust and Efficient Finite Element Discretizations for Higher-Order Gradient Formulations

高阶梯度公式的稳健且高效的有限元离散化

• 批准号: 392564687
• 负责人: Professor Dr.-Ing. Daniel Balzani
• 金额: --
• 依托单位: Mathematisches Institut
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2017
• 资助国家: 德国
• 起止时间: 2016-12-31 至 2023-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/392564687?language=en
• 关键词: Robust; Efficient; Finite; Element; Discretizations

The simulation of complex mechanical engineering problems, including e.g., a complex material behavior or geometrical singularities, requires robust numerical discretization methods. More and more, nonlocal approaches including higher-order derivatives are considered in order to e.g., capture a length-scale dependent material response or to cure mesh-dependent solutions in case of evolving microscopic material degradation. These modeling approaches pose new challenges with view to their algorithmic treatment since classical methods are not directly applicable. This is due to the higher-order derivatives taken into account which induce a switch from a second-order to a fourth-order partial differential equation (PDE), which leads to complicated ansatz functions in the approximation. Mixed methods that allow for standard ansatz functions are desirable. A naive approach that splits the problem in two second order problems fails in that it approximates the wrong solution. This was observed in the context of the Kirchhoff plate problem for the Ciarlet-Raviart method and is known as Sapondjan paradox. Therefore, new mixed formulations and discretizations are constructed and analyzed in this project for the problem of gradient elasticity and gradient damage, which circumvent this effect and lead to robust and reliable approximations of the solution. The gradient of the displacement will be the only independent variable and it will be discretized with standard Lagrange ansatz functions. This enables the integration of the new discretizations in existing software packages and leads to an efficient approximation of the solution. The key idea is to characterize derivatives as functions that are rotation-free. Besides the introduction of new formulations and suitable discretizations, the error analysis, the implementation, the computations of benchmark problems and the comparison of the new discretizations with existing ones are in the focus of this proposal. A further part is devoted to the a posteriori analysis of these problems and defines efficient and reliable error estimators. Singularities of the standard elasticity problem usually appear if the underlying domain is not convex. The solution of the gradient elasticity problem then usually also has a singularity in the sense that the solution does not lie in the Sobolev space H³, and therefore, discretizations of the problem show a suboptimal convergence behavior. The error estimators that should be defined in the project eventually lead to a mesh-adaptive algorithm, which is indispensable to exploit the computational power in this situation.

复杂的机械工程问题（包括复杂的材料行为或数值奇点）的模拟需要鲁棒的离散化方法，越来越多地考虑包括高阶导数在内的非局部方法，以便捕获长度尺度相关的材料响应。或者在不断变化的微观材料退化的情况下解决依赖于网格的解决方案，这些建模方法对其算法处理提出了新的挑战，因为经典方法不能直接适用，这是由于考虑了导致的高阶导数。从二阶偏微分方程 (PDE) 切换到四阶偏微分方程 (PDE)，这会导致近似中出现复杂的 ansatz 函数，因此需要一种能够在两秒内分解问题的简单方法。阶问题失败的原因是它逼近了错误的解。这是在 Ciarlet-Raviart 方法的基尔霍夫板问题的背景下观察到的，被称为 Sapondjan 悖论。因此，新的混合公式和在该项目中，针对梯度弹性和梯度损伤问题构建并分析了离散化，从而规避了这种效应，并得出了稳健且可靠的解近似值。位移的梯度将是唯一的自变量，并且将通过以下方式进行离散化。标准拉格朗日 ansatz 函数。这使得新的离散化能够集成到现有软件包中，并导致解决方案的有效近似。关键思想是将导数描述为无旋转的函数。除了引入新的公式和合适的离散化之外，误差分析、实现、基准问题的计算以及新离散化与现有离散化的比较也是本提案的重点。如果基础域不是凸的，则标准弹性问题的奇异性通常会出现，并且梯度弹性问题的解通常也具有奇异性。不位于 Sobolev 空间 H3 中，因此，问题的离散化表现出次优的收敛行为。应在项目中定义的误差估计器最终会产生网格自适应算法，这对于利用计算能力是必不可少的。这种情况。