NOVEL NUMERICAL PROCEDURES FOR LIMIT ANALYSIS OF STRUCTURES: MESH-FREE METHODS AND MATHEMATICAL PROGRAMMING

Current research in the field of limit analysis is focussing on the development of
numerical tools which are sufficiently efficient and robust to be used in engineering
practice. This places demands on the numerical discretisation strategy adopted
as well as on the mathematical programming tools applied, which are the key
ingredients of a typical computational limit analysis procedure. In this research,
the Element-Free Galerkin (EFG) discretisation strategy is used to approximate
the displacement and moment fields in plate and slab problems, and second-order
cone programming (SOCP) is used to solve the resulting discretised formulations.
A numerical procedure using the EFG method and second-order cone programming
for the kinematic limit analysis problem was developed first. The moving
least squares technique was used in combination with a stabilised conforming nodal
integration scheme, both to keep the size of the optimisation problem small and to
provide stable and accurate solutions. The formulation was expressed as a problem
of minimizing a sum of Euclidean norms, which was then transformed into a
form suitable for solution using SOCP.
To improve the accuracy of solutions and to speed-up the computational process,
an efficient h-adaptive EFG scheme was also developed. The naturally conforming
property of meshfree approximations (with no nodal connectivity required) facilitates
the implementation of h-adaptivity. The error in the computed displacement
field was estimated accurately using the Taylor expansion technique. A stabilised
conforming nodal integration scheme was also extended to error estimators, leading
to an efficient and truly meshfree adaptive method.
To obtain an indication of bounds on the solutions obtained, an equilibrium formulation
was also developed. Pure moment fields were approximated using a
moving least squares technique. The collocation method was used to enforce the
strong form of the equilibrium equations and a stabilised conforming nodal integration
scheme was introduced to eliminate numerical instability problems. The von
Mises and Nielsen yield criteria were then enforced by introducing second-order
cone constraints.