Convergent finite element approximations for problems of near-incompressible and near-inextensible transversely isotropic linear elasticity - Rasolofoson, Faraniaina (Department of Mathematics and Applied Mathematics, 2019)
This work comprises a detailed theoretical and computational study of the boundary value problem for transversely isotropic linear elastic bodies. The main objective is the development and implementation of low-order finite element methods that are uniformly convergent in the incompressible and inextensible limits. The first step in the investigation is a study of the constitutive relation for transversely isotropic elasticity, and establishment of conditions on the five material parameters under which the relation is pointwise stable. This forms the basis for a study of well-posedness of the weak displacement-based formulation. Conforming finite element approximations are studied. The error estimate indicates the possibility of extensional locking; on the other hand, anisotropy, measured as the ratio of Young’s moduli in the fibre and transverse directions, plays a role in minimizing or even eliminating volumetric locking behaviour. Extensional locking is circumvented with the use of selective under-integration, in the context of low-order quadrilateral elements. Its equivalence with mixed and perturbed Lagrangian methods are shown. A series of numerical results illustrates the various features of the formulations considered. In a second approach, interior penalty or discontinuous Galerkin (DG) formulations of the problem are considered. Low-order approximations on triangles are adopted, with the use of three interior penalty discontinuous Galerkin methods, viz. nonsymmetric, symmetric and incomplete. It is known that these methods are uniformly convergent in the incompressible limit for the case of isotropy. This property carries over to the transversely isotropic case for moderate anisotropy. An error estimate suggests the possibility of extensional locking, and under-integration of the extensional edge terms is proposed as a remedy. This modification is shown to lead to an error estimate that is consistent with locking-free behaviour. Numerical tests confirm the uniformly convergent behaviour, at an optimal rate, of the under-integrated scheme.