Adaptive Time Multirate Iterative Algorithms For Linear And Nonlinear Poroelasticity Models

The poroelasticity model can be divided into linear poroelasticity model and nonlinear poroelasticity model according to the linear and nonlinear stress-strain relationships.Poroelasticity models have applications in carbon dioxide sequestration,biomechanics,materials science,environmental engineering,reservoir engineering and other fields.The linear poroelasticity model can be used in biomedicine.For example,finite element model of poroelasticity based on the poroelasticity model can be used to predict the changes in the biomechanical behavior of lumbar disc caused by intervertebral disc degeneration.The nonlinear poroelasticity model can be used to simulate some biological tissues,such as the soft tissue of orbit and lung organs.The study of poroelasticity models will help with the development of human diagnosis,treatment of pulmonary diseases and human organs.In this paper,the adaptive time multirate iterative algorithm is proposed for linear and nonlinear poroelasticity models,which is a new solver for linear and nonlinear poroelasticity models.For the linear poroelasticity models,firstly,using the multi-physical field reconstruction to reconstruct the linear poroelasticity models into a generalized Stokes problem of displacement vector field and a diffusion problem of other pseudo pressure fields.Secondly,due to the requirement of convergence,different m should be selected in different grids.And m is the time step of the variable that changes slowly over time is the multiple of the time step of the variable that changes fast over time.Therefore,proposing the adaptive time multirate iterative algorithm to solve the reconstructed problem,that is,using the multi-physical finite element method to discretize the spatial variables and using the adaptive time multirate iterative algorithm to discretize the time variable.Then,constructing a new error indicator to estimate the posterior error,introducing the dual problem to prove that the numerical solution error of this method is bounded.Finally,using numerical examples to verify the performance of the algorithm.For the nonlinear poroelasticity models,firstly,using the multi-physical field reconstruction to reconstruct the nonlinear poroelasticity models into a generalized Stokes problem of displacement vector field and a diffusion problem of other pseudo pressure fields,giving the linearization algorithm of the nonlinear term in combination with the Newton iterative method.Secondly,due to the requirement of convergence,different m should be selected in different grids.Therefore,proposing the adaptive time multirate iterative algorithm to solve the reconstructed problem,that is,using the multi-physical finite element method to discretize the spatial variables and using the adaptive time multirate iterative algorithm to discretize the time variable.Then,constructing a new error indicator to estimate the posterior error,introducing the auxiliary problem to prove that the numerical solution error of this method is bounded.Finally,using numerical examples to verify the performance of the algorithm.