Well-posedness and discrete analysis for advection-diffusion-reaction in poroelastic media

Stability analysis for a new model of multi-species convection-diffusion-reaction in poroelastic tissue

In this paper, a generalized particle system (GPS) method, a general method to describe multiple strong form representation based particle methods is described. Gradient, divergence, and Laplacian operators used in various strong form based particle method such as moving particle semi-implicit (MPS) method, smooth particle hydrodynamics (SPH), and peridynamics, can be described by the GPS method with proper selection of parameters. In addition, the application of mixed formulation representation to the GPS method is described. Based on Hu-Washizu principle and Hellinger-Reissner principle, the mixed form refers to the method solving multiple primary variables such as displacement, strain and stress, simultaneously in the FEM method; however for convenience in employing FEM with particle methods, a simple representation in construction only is shown. It is usually applied to finite element method (FEM) to overcome numerical errors including locking issues. While the locking issues do not arise in strong form based particle methods, the mixed form representation in construction only concept applied to GPS method can be the first step for fostering coupling of multi-domain problems, coupling mixed form FEM and mixed form representation GPS method; however it is to be noted that the standard GPS particle method and the mixed for representation construction GPS particle method are equivalent. Two dimensional simple bar and beam problems are presented and the results from mixed form GPS method is comparable to the mixed form FEM results.

We analyse and discretize a mixed formulation for a linearized lubrication fracture model in a poro-elastic medium. The displacement of the medium is expressed in primary variables while the flows in the medium and fracture are written in mixed form, with an additional unknown for the pressure in the fracture. The fracture is treated as a non-planar surface or curve according to the dimension, and the lubrication equation for the flow in the fracture is linearized. The resulting equations are discretized by finite elements adapted to primal variables for the displacement and mixed variables for the flow. Stability and a priori error estimates are derived. A fixed-stress algorithm is proposed for decoupling the computation of the displacement and flow and a numerical experiment is included.

A computational study on dynamic behavior of articular cartilage under cyclic compressive loading and magnetic field.

In the present article, we investigate the biomechanical response of a fiber reinforced solid matrix (soft tissue) saturated with an electrically conducting fluid. A constant magnetic field was exposed to the binary mixture of fluid and deformable porous solid. The governing mechanism of multiphasic deformation was based on the loading imposed at the rigid bony interface. The fluid flow through the cartilage network depends upon the rate of applied compression and strain-dependent permeability of the solid matrix. The components of the mixture were intrinsically incompressible; however, in the derivation of governing dynamics, the visco-elastic behavior of the solid and an interstitial fluid was developed. The continuum mixture theory was employed in modeling solid deformation and local fluid pressure. We incorporated strain-dependent permeability in the governing equations of binary mixture that was found in an early experimental study. The governing non-linear coupled system of partial differential equations was developed for the solid deformation and fluid pressure in the presence of Lorentz forces. In the case of strain-dependent permeability, a numerical solution is computed using the method of lines (MOL), whereas, the exact solution is provided when permeability is kept constant. Graphical results highlight the influence of various physical parameters on both solid displacement and fluid pressure.

<p><span>Fluid injection is one of the main triggers of induced seismicity. Accurate numerical modeling of such processes is crucial for the safety of many affected regions. We propose a high-resolution numerical simulation of the strain localization in elasto-plastic and poro-visco-elasto-plastic media with a particular focus on the fluid pressure distribution. The resolution of our numerical model is 10000 by 10000 grid cells. The simulation is accelerated using graphical processing units (GPUs), thus, the total simulation time is in the order of a few minutes. We implement a pressure-dependent Mohr-Coulomb plastic law and study the influence of fluid pressure on the triggering of shear bands. Mean stress is partitioned between fluid pressure and total pressure. This study is particularly important since the effective stress law (the difference between fluid and total pressures) controls brittle failure. We vary viscosity and permeability as well as initial conditions for fluid pressure to explore the physics of shear bands nucleation. We show that fluid pressure in hydro-mechanically coupled media significantly affects the strain localization pattern compared to only elasto-plastic media. Permeability and viscosity are important parameters that control the fluid pressure distribution in the localized shear zones. This work is a preliminary study to model induced seismicity due to the fluid injection in fluid-saturated rocks described as fully coupled poro-visco-elasto-plastic media. </span></p>

A comparison of finite volume formulations and coupling strategies for two-phase flow in deforming porous media

We have developed a theoretical framework to investigate the deformation of the cartilage's solid phase in response to electrically conducting fluid flow. The purpose is to comprehend compression-induced cartilage stress–relaxation behavior. The biphasic mixture theory forms the model's foundation, which combines the nonlinear strain-dependent permeability found earlier in an experiment. In this investigation, it was assumed that the fluid and solid phases were intrinsically incompressible and nondissipative; however, mathematical modeling also developed for the fluid phase's and solid matrix's viscoelastic behaviors. A set of coupled partial differential equations (PDEs) was developed to characterize the slow rate and fast rate of compression in the presence of Lorentz forces for solid deformation and fluid pressure. The method of lines (MOL) is used to solve the resulting system, and graphs are produced to illustrate the relationship of the magnetic parameter with solid deformation and fluid pressure.

Wave propagation in poroelastic media is a subject that finds applications in many fields of research, from geophysics of the solid Earth to material science. In geophysics, seismic methods are based on the reflection and transmission of waves at interfaces or layers. It is a relevant canonical problem, which has not been solved in explicit form, i.e., the wave response of a single layer, involving three dissimilar media, where the properties of the media are described by Biot's theory. The displacement fields are recast in terms of potentials and the boundary conditions at the two interfaces impose continuity of the solid and fluid displacements, normal and shear stresses, and fluid pressure. The existence of critical angles is discussed. The results are verified by taking proper limits-zero and 100% porosity-by comparison to the canonical solutions corresponding to single-phase solid (elastic) media and fluid media, respectively, and the case where the layer thickness is zero, representing an interface separating two poroelastic half-spaces. As examples, it was calculated the reflection and transmission coefficients for plane wave incident at a highly permeable and compliant fluid-saturated porous layer, and the case where the media are saturated with the same fluid.

A posteriori error estimates by weakly symmetric stress reconstruction for the Biot problem

Parameter‐robust mixed element method for poroelasticity with Darcy‐Forchheimer flow

Convergence of a multi-point flux approximation finite volume scheme for a sharp–diffuse interfaces model for seawater intrusion

We develop non-overlapping domain decomposition methods for the Biot system of poroelasticity in a mixed form. The solid deformation is modeled with a mixed three-field formulation with weak stress symmetry. The fluid flow is modeled with a mixed Darcy formulation. We introduce displacement and pressure Lagrange multipliers on the subdomain interfaces to impose weakly continuity of normal stress and normal velocity, respectively. The global problem is reduced to an interface problem for the Lagrange multipliers, which is solved by a Krylov space iterative method. We study both monolithic and split methods. In the monolithic method, a coupled displacement-pressure interface problem is solved, with each iteration requiring the solution of local Biot problems. We show that the resulting interface operator is positive definite and analyze the convergence of the iteration. We further study drained split and fixed stress Biot splittings, in which case we solve separate interface problems requiring elasticity and Darcy solves. We analyze the stability of the split formulations. Numerical experiments are presented to illustrate the convergence of the domain decomposition methods and compare their accuracy and efficiency.

A discontinuous Galerkin method for a coupled Stokes–Biot problem

In this paper, a new mixed variational form for the Reissner-Mindlin problem is given, which contains two unknowns instead of the classical three ones. A mixed triangle finite element scheme is constructed to get a discrete solution. A new method is put to use for proving the uniqueness of the solutions in both continuous and discrete mixed variational formulations. The convergence and error estimations are obtained with the help of different norms. Numerical experiments are given to verify the validity of the theoretical analysis.