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

固体力学中非线性和非光滑问题的混合离散化

• 批准号: 255721882
• 负责人: Professorin Dr.-Ing. Stefanie Reese
• 金额: --
• 依托单位: Arbeitsgruppe 3: Wissenschaftliches Rechnen
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2014
• 资助国家: 德国
• 起止时间: 2013-12-31 至 2021-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/255721882?language=en
• 关键词: Hybrid; discretizations; solid; mechanics; non

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.

现代有限元方法目前在新材料，创新产品和生产过程的建设，设计和开发中起着重要作用。尽管过去成功进行了研究，但仍然存在许多开放问题，例如人造僵硬的效果，数值不稳定性和不想要的网格失真敏感性。在该项目中，特别重点是几何和材料非线性，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用性相关性。不连续的Galerkin（DG）方法可以看作是连续方法的概括，因此为改进上述字段中数值计算提供了其他功能和选项。这是有代价的，因为DG方法比同一网格上的连续离散化需要更多的自由度和记忆消耗。为了改善这个问题，我们研究了混合不连续的盖尔金方法，允许通过静态凝结大大降低全球自由度。 除了团队内的跨学科研究工作外，我们还确定了优先计划中三个不同团队的联合科学目标。我们的目的是探索固体力学中混合不连续的galerkin近似值的潜力和局限性，并识别，开发和分析相关方法，从而可以在收敛性，鲁棒性和稳定性方面改善性能，而不必增加数值的努力，而不必增加效力的结果，以提高效果的结果，以提高效率的研究，以提高效果的研究，以提高效率的研究。应开发新的对称混合DG方法，以模拟损伤和可塑性以及多规模问题的通用材料模型。 DG概念为适应性开辟了全新的可能性，应根据适当的错误估计来利用。在界面领域，将开发和研究新的有前途的离散化计划，用于接触，分层和非匹配网格，从混合DG概念中得出。只要有利，就将利用有关连续方法的现有知识，以比较和传输相关的技术从连续有限元和iSOOGEOMETRIC方法到DG近似值，反之亦然。例如，与DG方法相反，DG方法允许元素之间的最大不连续性，同几何方法基于元素之间的最大平滑度。在混合贴片方法中，等几何方法的效率将提高。在补丁中，使用最大光滑度，而在不连续的方法之间则使用网格灵活性。