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

模拟复杂的机械工程问题，包括，复杂的材料行为或几何奇异性，需要强大的数值离散化方法。越来越多地考虑包括高阶导数的非局部方法，以便例如，捕获依赖于长度尺度的材料响应，或者在不断发展的微观材料退化的情况下固化依赖于网格的解决方案。这些建模方法提出了新的挑战，鉴于他们的算法处理，因为经典的方法是不直接适用的。这是由于考虑到高阶导数，其引起从二阶到四阶偏微分方程（PDE）的切换，这导致在近似中的复杂的反函数。考虑到标准的反求函数的混合方法是可取的。将问题分解为两个二阶问题的天真方法失败了，因为它近似了错误的解决方案。这是在Ciarlet-Raviart方法的Kirchhoff板问题的背景下观察到的，被称为Sapondjan悖论。因此，新的混合配方和离散化构造和分析在这个项目中的梯度弹性和梯度损伤的问题，它规避了这种影响，并导致强大的和可靠的近似的解决方案。位移的梯度将是唯一的独立变量，并且将用标准的拉格朗日插值函数对其进行离散化。这使得新的离散化集成在现有的软件包，并导致一个有效的近似的解决方案。其关键思想是将导数刻画为无旋转的函数。除了引入新的公式和合适的离散化，误差分析，实施，基准问题的计算和新的离散化与现有的比较是本建议的重点。另一部分是专门的后验分析这些问题，并定义有效和可靠的误差估计。标准弹性力学问题的奇异性通常出现在基础区域不是凸的情况下。梯度弹性问题的解通常也有一个奇异性，即解不在索伯列夫空间H 3中，因此，问题的离散化表现出次优收敛行为。误差估计，应在项目中定义最终导致网格自适应算法，这是必不可少的，以利用在这种情况下的计算能力。

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.