Derivation, simulation and validation of poroelastic models in dental biomechanics

Poroelasticity and mechanics of growth are playing an increasingly relevant role in biomechanics. This work is a self- contained and holistic presentation of the modeling and simulation of non-linear poroelasticity with and without growth inhomogeneities. Balance laws of poroelasticity are derived in Cartesian coordinates. These allow to write the governing equations in a form that is general but also readily implementable. Closure relations are formally derived from the study of dissipation. We propose an approximation scheme for the poroelasticity problem based on an implicit Euler method for the time discretization and a finite element method for the spatial discretization. The non-linear system is solved by means of Newton's method. Time integration of the growth tensor is discussed for the specific case in which the rate of inelastic deformations is prescribed. We discuss the stability of the mixed finite element discretization of the arising saddle-point problem. We show that a linear finite element approximation of both the unknowns, that is not LBB compliant for the elasticity problem, is nevertheless stable when applied to the linearized poroelasticity problem. This choice enables a fast assembling phase. The discretization of the poroelastic system may present unphysical oscillations if the spatial and temporal step-sizes are not properly chosen. We study the source of these wiggles by comparing the pressure Schur complement to a reaction- diffusion problem. From our analysis, we define a novel Péclet number for the poroelastic system and we show how it depends on the shear and bulk moduli of the solid phase. This number allows to introduce a stability condition that ensures that the solution is free of unphysical oscillations. If this condition on the Péclet number is not met, we introduce a fluid pressure Laplacian stabilization in order to remove the wiggles. This stabilization technique depends on a numerical parameter, whose optimal value is given by the derived Péclet number. Finally, we propose a coupled elastic-poroelastic model for the simulation of a tooth-periodontal ligament system. Because of the high resolution required by this system, we develop an efficient multigrid Newton's method for the non-linear poroelasticity system. The stability condition has again a significant influence on the performances of this solver. If the condition on the Péclet number is not satisfied on all levels of the multigrid algorithm, poor convergence rates or even divergence of the solver can be observed. The stabilization of the coarse grid operators with the optimal fluid pressure Laplacian method is a simple and efficient method to improve the convergence rate of the multigrid solver applied to this saddle-point system. We validate our coupled model against experimental measurements realized by the group of Prof. Bourauel at the University of Bonn