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造、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，但仍然存在许多悬而未决的问题，如人工硬化效应、数值不稳定性和不希望看到的网格扭曲敏感性等。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，因为这些领域具有很大的理论和实践意义。不连续Galerkin(DG)方法可以看作是连续方法的推广，从而为改进上述领域的数值计算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进行连续离散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Galerkin方法，允许通过静态凝聚显著减少全局自由度。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外，我们还在优先方案内与三个不同的团队确定了联合科学目标。我们的目标是探索混合间断Galerkin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，以便在不增加数值工作量的情况下提高收敛、稳健性和稳定性。混合方法的第一次基准测试显示了良好的结果，这需要在优先计划富有成效的科学环境中进行更多的研究。应发展新的对称混合DG方法来模拟损伤和塑性的广义材料模型以及多尺度问题。DG概念为适应性开辟了全新的可能性，应根据适当的误差估计加以利用。在界面领域，将发展和研究基于混合DG概念的接触、分层和非匹配网格的新的离散化方法。现有的关于连续方法的知识将被利用，只要这是有利的，比较和转移相关技术从连续有限元和等距方法到DG近似，反之亦然。例如，与允许元素之间最大不连续性的DG方法不同，等距方法基于元素之间的最大光滑度。在混合面片方法中，等距方法的效率将得到提高。在面片内部使用最大的光滑度，而在面片之间使用不连续的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.