Researches On Numerical Methods For Poroelasticity

Biot's poroelastic theory has been used in a diverse range of scientific and engineering applications because of its ubiquity and unique properties,for exam-ple,environmental engineering and biomechanical engineering which are closely related to our lives.Therefore,the poroelasticity has attracted more and more scholars' attention.However,in practical applications,due to the geometric diversity of materials and the heterogeneity of material properties,solving the porous elastic model becomes more complicated.So,numerical simulation has very important practical value and significance.In this thesis,we mainly focus on the the finite element methods of poroelasticity.First,in the first chapter,we introduce the research background of the poroe-lasticity in detail,discuss the locking problem in numerical simulation and the ex-isting literatures,and then put forward our solution to the corresponding problem.Considering the complex coupling of the porous elastic model and the amount of calculation in the simulation process,we give a coupled hybrid mixed finite element method for solving the poroelasticity in Chapter 2.We present a hybrid mixed element method for the pressure and Darcy velocity of the fluid phase and a continuous Galerkin finite element method for the displacement of the sol-id phase in the poroelasticity equation.By replacing discreting the velocity in H(div,?),we use completely discontinuous piecewise polynomial functions and ensure the continuity of the normal fluxes over element interfaces by adding ap-propriate constraints.A particular advantage of this method is that it can be implemented element-wise,i.e.,it allows for static condensation,local degrees of freedom can be eliminated on the element level,yielding global systems for the degree of freedom on the mesh skeleton only.The method is consistent and lo-cally mass conservative.By introducing the energy norm,we obtain the stability of this system.The optimal error estimates are derived for both semidiscrete and fully discrete schemes.Finally,numerical results illustrate the accuracy of the method.In Chapter 3,we consider a stabilized hybrid mixed finite element method for Biot's model.The hybrid P1-RTO-PO discretization of the displacement-pressure-Darcy's velocity system of Biot's model presented in Chapter 2 is not uniformly stable with respect to the physical parameters,resulting in some issues in nu-merical simulations.To alleviate such problems,we stabilize the hybrid scheme with edge(face)bubble functions and show the well-posedness with respect to physical and discretization parameters,which provide optimal error estimates of the stabilized method.We introduce a perturbation of the bilinear form of the displacement which allows for the elimination of the bubble functions.Togeth-er with eliminating Darcy's velocity by hybridization,we obtain an eliminated system whose size is the same as the classical P1-RT0-P0 discretization.Based on the well-posedness of the eliminated system,we design block preconditioners that are parameter-robust.Numerical experiments are presented to confirm the theoretical results of the stabilized scheme as well as the block preconditioners.Modeling poroelasticity in heterogeneous media,or the interaction of Biot's system with other fluid flow or mechanical systems,needs interface conditions across the interface,some of which involve the stress tensor,so the mixed formula-tion with the stress tensor as one of its primary variables allows a very simple way to enforce the interface conditions.In Chapter 4,we consider total stress tensor-displacement formulation of the solid phase based on the Hellinger-Reissner vari-ational principle,which is suitable for the analysis of nearly incompressible elastic materials.Then we have a symmetric discrete formulation.However,the linear space pair of total stress tensor and displacement usually violate the inf-sup con-dition.In order to give an effective discrete system which can avoid pressure locking,we develop a Brezzi-Pitkaranta stabilization method for the finite ele-ment space pair.For the time discrete,we use the backward Euler method.We prove the stability of this system and derive the optimal error estimates for the fully discrete system.Finally,we present some numerical experiments in the end to confirm the theoretical analysis.The stabilized method presented in Chapter 4 can overcome the locking,how-ever,it will make the calculation cost increase to a certain degree.In order to give a similar stabilization method,we consider total stress tensor-displacement for-mulation of the solid phase based on the Hellinger-Reissner variational principle and apply the lowest equal-order finite element triple P1-Pl-P1 to approximate three variables.The displacement projection method based on two local Gauss integrals is introduced to overcome the deficiency of inf-sup condition.Similarly,by introducing the energy norm,we obtain the stability of this scheme.For the time discrete,we use the backward Euler method,and the optimal error esti-mates for the fully discrete systems are derived.Finally,numerical experiments are given to verify the rationality of our theoretical analysis and the effectiveness in eliminating locking phenomena.