Hybrid discretizations in solid mechanics for non-linear and non-smooth problems

现代有限元方法目前在新材料、创新产品和生产工艺的构建、设计和开发中发挥着重要作用。尽管过去的研究取得了成功，但仍然存在许多有待解决的问题，如人工加劲效应、数值不稳定性和不理想的网格畸变灵敏度。在这个项目中，特别关注几何和材料非线性、几乎不可压缩、各向异性和广义材料以及接触和界面模型，因为这些领域具有重要的理论和实践意义。不连续伽辽金（DG）方法可以看作是连续方法的推广，从而为上述领域的数值计算的改进提供了额外的特征和选择。这是有代价的，因为DG方法比在同一网格上的连续离散化需要更多的自由度和内存消耗。为了改善这个问题，我们研究了混合不连续伽辽金方法，允许通过静态冷凝显著降低全局自由度。我们的跨学科小组代表三个研究领域：应用力学，数值分析和科学计算。除了团队内部的跨学科研究工作之外，我们还与优先项目中的三个不同团队确定了共同的科学目标。我们的目标是探索混合不连续伽辽金近似在固体力学中的潜力和局限性，并识别、开发和分析相关方法，这些方法可以在不增加数值努力的情况下提高收敛性、鲁棒性和稳定性。混合方法的第一次基准测试显示出有希望的结果，这要求在优先方案富有成效的科学环境中进行更多的调查。针对广义材料的损伤和塑性模型以及多尺度问题，需要发展新的对称混合DG方法。DG概念为适应性开辟了全新的可能性，这种可能性应基于适当的误差估计加以利用。在界面领域，新的有前途的离散方案接触，分层和非匹配网格，由混合DG概念衍生，将开发和研究。现有的连续方法知识将被利用，只要这是有利的，比较和转移相关技术从连续有限元和等几何方法到DG近似，反之亦然。例如，与允许元素之间最大不连续的DG方法相反，等几何方法基于元素之间的最大平滑度。等几何方法的效率将在混合补丁方法中得到提高。在patch内使用最大平滑度，而在两者之间使用不连续方法提供网格灵活性。

Modern finite element methods currently play an important role in the construction, design and development of new materials, innovative products and production processes. Despite successful research in the past, there are still many open problems, e.g., artificial stiffening effects, numerical instabilities and undesired mesh distortion sensitivity. Within this project, a special focus is on geometrical and material non-linearities, nearly incompressible, anisotropic and generalized materials as well as contact and interface models, since these fields are of great theoretical and practical relevance. Discontinuous Galerkin (DG) methods may be seen as generalizations of continuous methods, thus offering additional features and options for the improvement of numerical computations in the aforementioned fields. This comes at a cost, as DG methods require far more degrees of freedom and memory consumption than continuous discretizations on the same mesh. To improve this issue, we investigate hybrid discontinuous Galerkin methods allowing for a significant reduction of global degrees of freedom via static condensation.Our interdisciplinary group represents three research fields: Applied Mechanics, Numerical Analysis and Scientific Computing. Beyond the interdisciplinary research work within the team, we identified joint scientific goals with three different teams within the priority programme. Our aim is to explore the potential and the limitations of hybrid discontinuous Galerkin approximations in solid mechanics and to identify, develop and analyze related methods, which allow for an improvement of the performance in terms of convergence, robustness and stability without increasing the numerical effort.A first benchmarking of the hybrid methods showed promising results, which call for more investigations in the fruitful scientific environment of the priority programme. New symmetric hybrid DG methods shall be developed for the simulation of generalized material models in damage and plasticity as well as multi-scale problems. The DG concept opens up entirely new possibilities for adaptivity which shall be exploited based on proper error estimates. Within the field of interfaces, new promising discretization schemes for contact, delamination and non-matching meshes, being derived from the hybrid DG concept, will be developed and investigated. Existing knowledge about continuous methods will be exploited, whenever this is advantageous, to compare and transfer related technologies from continuous finite element and isogeometric methods to DG approximations and vice versa. For example, in contrast to DG methods, which allow for maximal discontinuity between the elements, isogeometric methods are based on maximal smoothness between the elements. Efficiency of isogeometric methods will be improved in a hybrid patch-wise approach. Within the patches maximal smoothness is used, while in between a discontinuous approach provides mesh flexibility.