| With the development of medicine and biomechanics,mathematical modeling and numerical simulation of various physical phenomena in biophysiology have attracted widespread attention.The physical phenomena that control biophysiology are often complex structures with multi-scale characteristics,involving multiple physical quantities such as displacement,velocity,pressure,and stress.This poses significant challenges to related modeling and numerical simulations.The theory of porous media mathematically describes the relevant physical quantities on a scale window between macro and micro scales through homogenization methods,and formularizes each physical quantity,providing a very ideal model framework for this article.This paper critically reviews the basic principles of porous media.theory,and pays special attention to the hypothesis of the continuous two-phase model,which is often used to describe the flow phenomenon of fluid in biological tissues.The control equation of the model consists of Darcy’s law,the law of conservation of mass.the law of conservation of momentum,and the constitutive equation.Research on biomechanics has shown that the fluid velocity and pressure gradient in Darcy’s law are not simple linear rela.tionships,but complex nonlinear relationships.Biological tissues undergo deformation under pressure,and the fluid in them often exhibits viscous behavior.Based on these characteristics,this paper proposes a nonlincar porous viscoclastic model for the first time.This model simultaneously considers the interaction between the viscosity of fluids in porous media and the nonlinear Darcy’s law.In terms of numerical simulation.considering the complexity of the model,this paper investigates two lowest order numerical methods,namely the P1-RT0-P0 mixed finite element method corresponding to the three field problem(displacement velocity pressure)and the P1-P1 mixed finite element method corresponding to the two field problem(displacement pressure).The numerical performance of the two methods in terms of error convergence speed and whether locking phenomenon occurs is discussed,The feasibility of using traditional mixed finite element methods to numerically solve nonlinear porous viscoelastic models has been obtained. |
