Intrinsically locking-free formulations for problems in structural mechanics

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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.