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

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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.