Intrinsically locking-free formulations for problems in structural mechanics

在结构力学问题的数值解的背景下，术语锁定是指在预渐近范围内的次优收敛率现象，其中该预渐近范围的大小取决于参数，例如壳的长细。锁紧的结果是位移太小，以及振荡，寄生应力在数值解。这个问题在有限元法（FEM）的早期就已经为人所知，但它也存在于其他离散化方法中，如无网格法、配置法和有限元的等几何版本。事实上，锁定的起源并不是有限元所特有的，而是潜在的物理问题和用来表述它的微分方程的固有性质。尽管这一领域的出版物数量庞大，但仍存在许多悬而未决的问题。特别是，许多消除锁定的方法，如减少集成和混合方法，可用于许多应用和离散化方法，但不能直接转移到其他离散化方案中。在文献中，解除锁的方法大多是在FEM的背景下发现的。对于新的离散化方法，如基于t样条的等几何分析，必须发展新的方法。所提议的研究项目的目的是开发一种方法，这种方法不去除离散化水平上的锁定，而是在潜在问题的数学公式中先验地避免锁定。因此，选择特定的ansatz空间或正交规则变得过时了。如果这是成功的，锁定的起源是避免的，问题本质上是无锁定的，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，采用了两种不同的策略：第一，控制方程的分层再参数化，第二，所谓的混合位移方法，其中某些物理量的近似空间是在特殊设计的微分算子的帮助下由替代变量构建的，以保证无锁定的结果。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，许多悬而未决的问题必须得到回答和解决。这涉及到某些辅助条件的执行、近似空间的连续性要求、避免膜锁定在壳中的分层重新参数化、将先前的工作扩展到体积锁定以及非结构化离散化和扭曲网格的验证。

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.