B. D. Reddy

D. van Huyssteen, B. D. Reddy

This work studies the approximation of plane problems concerning transversely isotropic elasticity, using a low-order Virtual Element Method (VEM), with a focus on near-incompressibility and near-inextensibility. Additionally, both homogeneous problems, in which the plane of isotropy is fixed; and non-homogeneous problems, in which the fibre direction defining the isotropy plane varies with position, are explored. In the latter case various options are considered for approximating the non-homogeneous fibre directions at element level. Through a range of numerical examples the VEM approximations are shown to be robust and locking-free for several element geometries and for fibre directions that correspond to mild and strong non-homogeneity.

B. J. Grieshaber, F. Rasolofoson, B. D. Reddy

This work studies discontinuous Galerkin (DG) approximations of the boundary value problem for homogeneous transversely isotropic linear elastic bodies. Low-order approximations on triangles are adopted, with the use of three interior penalty DG methods, viz. nonsymmetric, symmetric and incomplete. It is known that these methods are uniformly convergent in the incompressible limit for the isotropic case. This work focuses on behaviour in the inextensible limit for transverse isotropy. An error estimate suggests the possibility of extensional locking, a feature that is confirmed by numerical experiments. 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.

Mebratu F. Wakeni, B. D. Reddy, A. T. McBride

An efficient time-stepping algorithm is proposed based on operator-splitting and the space-time discontinuous Galerkin finite element method for problems in the non-classical theory of thermoelasticity. The non-classical theory incorporates three models; the classical theory based on Fourier's law of heat conduction resulting in a hyperbolic-parabolic coupled system, a non-classical theory of a fully hyperbolic extension, and a combination of the two. The general problem is split into two contractive sub-problems, namely the mechanical phase and the thermal phase. Each sub-problem is discretised using space-time discontinuous Galerkin finite element method resulting each to be stable which then leads to unconditional stability of the global product algorithm. A number of numerical examples are presented to demonstrate the performance and capability of the method.