Abstract
In this Ph.D. thesis we develop equilibrated flux a posteriori error estimates for poro-mechanical and poro-plasticity problems.Based on these estimations we propose adaptive algorithms for the numerical solution of problems in soil mechanics.The first chapter deals with linear poro-elasticity problems.Using equilibrated H({rm div})-conforming flux reconstructions of the Darcy velocity and the mechanical stress tensor, we obtain a guaranteed upper bound on the error.We apply this estimate in an adaptive algorithm balancing the space and time discretisation error components in simulations in two space dimensions.The main contribution of this chapter is the symmetric reconstruction of the stress tensor.In the second chapter we propose another reconstruction technique for the stress tensor, while considering nonlinear elasticity problems.By imposing the symmetry of the tensor only weakly, we reduce computation time and simplify the implementation.We prove that the estimate obtained using this stress reconstuction is locally and globally efficient for a wide range of hyperelasticity problems.We add a linearization error estimator, enabling us to introduce adaptive stopping criteria for the linearization solver.The third chapter adresses the industrial application of the obtained results.We apply an adaptive algorithm to three-dimensional poro-mechanical problems involving elasto-plastic mechanical behavior laws.