Error estimation and adaptivity for finite element structural dynamics models under parameter uncertainty

The optimisation of discretisation and stochastic errors under a single criterion is not a simple task. The nature of the errors derived from both phenomena is totally different and so are the measures needed to assess them. Nonetheless, they are related and if either of the errors dominates a problem, any obtained solution is suboptimal. Error estimation research is focused on optimising and bounding the discretisation error only. On the other hand, stochastic research treats error estimation as a black box that ensures enough accuracy to avoid interference with the stochastic process and/or the surrogate of the numerical model, with the only exception of stochastic finite element method. This dissertation presents an adaptive approach to optimise locally the relation between the aforementioned numerical approximations in any stochastic framework. The main novel contribution of this thesis is the development of an algorithm that ensures that all errors are of the same scale after an adaptive process. The numerical problem posed is a structure vibrating steadily under parametric uncertainty, although any partial differential equation could have been selected modifying the refinement strategy. Steady dynamic problems were chosen because they tend to need less intuitive concentration of refinements, the lack of time dependency allows non-conforming meshes and yet, natural frequencies highly influence the solution. The definitions of all measures of error are linked to the relative discretisation error, and are therefore controlled by the algorithm under this single criterion. Another novelty is a new family of residual error estimators based on the Saint- Venant principle rather than on limiting the support of the test function. This new approach allows to unlink the definition of the patch sub-domain from the split of the residual. In addition to the resulting freedom of patch choice, it is proven than the new approach provides enhanced stability to some element centered patch estimators proposed in the past. iii Finally, two minor new contributions are a discrete way to obtain an indicator of refinement for quantities of interest not involving gradients (simpler than the choices already present in literature), and the testing of analogy between error estimators and preconditioners