Intrinsically locking-free formulations for problems in structural mechanics

针对结构力学问题的本质无锁公式

• 批准号: 452589815
• 负责人: Professor Dr.-Ing. Manfred Bischoff
• 金额: --
• 依托单位: Institut für Baustatik und Baudynamik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/452589815?language=en
• 关键词: Intrinsically; locking; free; formulations; problems

In the context of numerical solutions of problems in structural mechanics, the term locking means the phenomenon of a sub-optimal rate of convergence in the preasymptotic range, where the magnitude of this preasymptotic range depends on a parameter, e.g. the slenderness of a shell. Locking results in displacements being too small as well as oscillating, parasitic stress in the numerical solution.The problem is known since the early days of the finite element method (FEM) but it also occurs for other discretization methods, like mesh-free methods, collocation methods and the isogeometric version of FEM. In fact, the origin of locking is not specific to FEM but it is an intrinsic property of the underlying physical problem and the differential equations by which it is formulated. In spite of the enourmous amount of publications in this area, still a lot of open questions persist. In particular, there is the problem that the many methods to remove locking, like reduced integration and mixed methods, which are available for numerous applications and discretization methods, are not directly transferrable to other discretization schemes. In the literature, methods to remove locking are found mostly in the context of FEM. For new discretization methods, e.g. isogeometric analysis on the basis of T-splines, new methods have to be developed.The aim of the proposed research project is the development of methods that do not remove locking on the level of discretization but avoid it a priori in the mathematical formulation of the underlying problem. Thus, choice of specific ansatz spaces or quadrataure rules become obsolete. If this is successful, the origin of locking is avoided, the problem is intrinsically locking-free and application of any arbitrary discretization scheme leads to locking-free numerical results.In terms of methods, two different strategies are pursued: first, hierarchic reparametrization of the governing equations and, second, the so-called mixed displacement method, in which the approximation spaces for certain physical quantities are constructed from surrogate variables with the help of specially designed differential operators in a way that guarantees locking-free results. The fundamental validity of this approach is already confirmed. Yet, numerous open questions have to be answered and problems solved before these methods are generally applicable. This concerns enforcement of certain subsidiary conditions, continuity requirements for the approximation spaces, a hierarchic reparametrization to avoid membrane locking in shells, extension of previous work to volumetric locking and validation for unstructured discretizations and distorted meshes.

在结构力学问题的数值解中，术语锁定是指在前渐近范围内的次优收敛率现象，其中前渐近范围的大小取决于一个参数，例如壳体的细长度。锁定会导致位移过小以及数值解中的振荡和寄生应力。这个问题在有限元法（FEM）的早期就已经知道了，但它也会出现在其他离散方法中，如无网格法，配点法和等几何版本的FEM。事实上，锁定的起源并不是FEM所特有的，而是基本物理问题和微分方程的内在属性。尽管这方面的出版物数量很多，但仍然存在许多悬而未决的问题。特别地，存在的问题是，许多方法来消除锁定，如减少集成和混合方法，这是可用于许多应用程序和离散化方法，不能直接转移到其他离散化方案。在文献中，发现解除锁定的方法大多是在有限元法的背景下。对于新的离散化方法，例如基于T样条的等几何分析，必须开发新的方法，拟议的研究项目的目的是开发不消除离散化水平上的锁定，但在基本问题的数学公式中先验地避免锁定的方法。因此，选择特定的空间或正交规则变得过时。如果这是成功的，锁定的起源就被避免了，问题本质上是无锁定的，并且任何任意离散化方案的应用都会导致无锁定的数值结果。在方法方面，采用两种不同的策略：第一，控制方程的分层重新参数化，第二，所谓的混合位移法，其中某些物理量的近似空间是在特别设计的微分算子的帮助下从代理变量以保证无锁定结果的方式构造的。这种方法的基本有效性已经得到证实。然而，在这些方法普遍适用之前，必须回答许多悬而未决的问题并解决问题。这涉及强制执行的某些附属条件，连续性要求的近似空间，层次重新参数化，以避免膜锁定壳，扩展以前的工作，体积锁定和验证非结构化离散化和扭曲的网格。