A domain decomposition preconditioner for steady groundwater flow in porous media

In this paper an algorithm is presented based on the additive Schwarz method for steady groundwater flow in a porous medium. The subproblems in the algorithm correspond to the problem on a coarse grid and some overlapping subdomains. It will be shown that the rate of convergence is independent of the mesh parameters and discontinuities of the coefficients, and depends on the overlap ratio.     

Evolution of microstructure in unstable porous media flow: A relaxational approach

We study the flow of two immiscible fluids of different density and mobility in a porous medium. If the heavier phase lies above the lighter one, the interface is observed to be unstable. The two phases start to mix on a mesoscopic scale and the mixing zone grows in time—an example of evolution of microstructure. A simple set of assumptions on the physics of this two‐phase flow in a porous medium leads to a mathematically ill‐posed problem—when used to establish a continuum free boundary problem. We propose and motivate a relaxation of this “nonconvex” constraint of a phase distribution with a sharp interface on a macroscopic scale. We prove that this approach leads to a mathematically well‐posed problem that predicts shape and evolution of the mixing profile as a function of the density difference and mobility quotient. © 1999 John Wiley & Sons, Inc.     

A porous media approach for plantar tissue during gait

Recently a new computational model, based on the thermodynamically constrained averaging theory, has been proposed to predict tumor initiation and proliferation and afterwards to study plantar tissue mechanics. The foot tissue is modeled as an elastic porous medium, in large strain regime and completely filled by a fluid phase. By considering the interstitial fluid, it is possible to mimic the viscoelastic behavior of the plantar tissue observed experimentally. Being the global response of the bi-phase system viscoelastic, it is shown that the duration of stance as well as of each of gait cycle has an influence on tissue stress field. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Localisation in granular media: Particle approach,homogenisation and continuum modelling

The individual motion of grains in granular material has a strong influence on the macroscopic material behaviour, which is in particular the case for the phenomena of strain localisation in shear zones and justifies the need for techniques that incorporate a micro-macro transition. In this contribution, granular media are investigated in three steps. Firstly, a microscopic particle-based modelling is set up, where individual grains are considered as rigid uncrushable particles while their motion is obtained through Newton's equations of state. The inter-particle contact forces are thereby determined via constitutive contact-force formulations, which have to account for the envisaged material behaviour. The second step is the homogenisation of the obtained particle's displacements and contact forces through a particle-centre-based strategy towards continuum quantities. Therefore, Representative Elementary Volumes (REV) are introduced on the mesoscale and the specific construction of the REV boundary leads to the understanding of granular media as a micropolar continuum. Finally, in order to verify the homogenisation strategy, a continuum based micropolar model is applied to model localisation phenomena and a comparative study of the results is carried out in a qualitative way. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A continuum theory of superconductivity

Abstract. A continuum theory of superconductivity is formulated for a mixture consisting of three species: a superelectron fluid, a conducting fluid, and a conducting elastic solid. Each one of the three species is subject to their own electro-magnetic (E-M) fields and motions. Irreversible thermodynamics are used to obtain constitutive equations. Field equations, boundary and initial conditions are given. A special case is obtained, suitable for mathematical analysis and applications. The Pippard theory of superconductivity is shown to be a special case of the present theory.     

A quantum mechanical approach to a fuzzy theory

Recently, we proposed a general measurement theory for classical and quantum systems (i.e., “objective fuzzy measurement theory”). In this paper, we propose “subjective fuzzy measurement theory”, which is characterized as the statistical method of the objective fuzzy measurement theory. Our proposal of course has a lot of advantages. For example, we can directly see “membership functions” (= “fuzzy sets”) in this theory. Therefore, we can propose the objective and the subjective methods of membership functions. As one of the consequences, we assert the objective (i.e., individualistic) aspect of Zadeh's theory. Also, as a quantum application, we clarify Heisenberg's uncertainty relation.     

On the representations of solutions in the theory of fluid-saturated porous media

In the present paper the linear theory of the liquid-saturatedporous medium consisting of a microscopically incompressiblesolid skeleton containing microscopically incompressible liquidis considered. First, the representation of Galerkin type solutionof equations of motion is obtained. Then, the representationtheorem of Galerkin type of system of the equations of steadyoscillations is presented. Finally, the general solution ofthe system of homogeneous equations of the steady oscillationsin terms of one harmonic function and four metaharmonic functionsis established.     

A continuum theory of dense suspensions

A continuum theory is introduced for viscous fluids carrying dense suspensions (such as blood) or emulsions of arbitrary shape and inertia. Suspended particles possess microinertia that make the mixture an anisotropic fluid whose viscosity changes with motion and orientation of suspensions. The microinertia balance law coupled with the equations of motion of an anisotropic fluid govern the ultimate outcome. By means of the second law of thermodynamics, constitutive equations are obtained in terms of the frame-independent tensors. In a special case, a theory of bar-like suspensions is obtained. The field equations, boundary and initial conditions are given for both the arbitrarily-shaped suspensions and the bar-like suspensions. The theory is demonstrated with the solution of the channel flow problem. The mean viscosity of the fluid with suspensions is determined. The motions of suspensions down flow are demonstrated.     

A continuum theory of dense suspensions

A continuum theory is introduced for viscous fluids carrying dense suspensions (such as blood) or emulsions of arbitrary shape and inertia. Suspended particles possess microinertia that make the mixture an anisotropic fluid whose viscosity changes with motion and orientation of suspensions. The microinertia balance law coupled with the equations of motion of an anisotropic fluid govern the ultimate outcome. By means of the second law of thermodynamics, constitutive equations are obtained in terms of the frame-independent tensors. In a special case, a theory of bar-like suspensions is obtained. The field equations, boundary and initial conditions are given for both the arbitrarily-shaped suspensions and the bar-like suspensions. The theory is demonstrated with the solution of the channel flow problem. The mean viscosity of the fluid with suspensions is determined. The motions of suspensions down flow are demonstrated.     

On the mathematical transport theory in microporous media: The billiard approach

This paper is an expository of the main dynamical properties of billiards, which depend on the shape of the walls of the container, and the recent developments like the introduction of an external field, which mimic the coupling with a thermostat.The class of dynamical system dealt with in this paper exhibits characteristics of hybrid systems as it links discrete and continuous, deterministic and stochastic dynamics.The contents are focused on applications. Specifically, transport dynamics in highly-confined regions has been of interest in the last few decades because of industrial and medical applications. Aspects of confined transport remain elusive, considering that in microporous membranes, whose size pores is about that of the molecules, the transport is sometimes ballistic, and sometimes diffusive. The classical kinetic and macroscopic approach can not be directly applied because collisions of particle fluid with walls prevail. The microscopic mathematical billiard theory can be applied as a mathematical tool since the interstices between obstacles can be considered as the pores of the membranes.     

Application of homogenization theory related to Stokes flow in porous media

We consider applications, illustration and concrete numerical treatments of some homogenization results on Stokes flow in porous media. In particular, we compute the global permeability tensor corresponding to an unidirectional array of circular fibers for several volume-fractions. A 3-dimensional problem is also considered.     

Some aspects in practical estimation of groundwater contaminant concentrations

Spatial distribution of groundwater contaminant concentration has special characteristics such as approximate symmetric profile, for example, in the transversal direction to groundwater flow direction, a certain ratio in directional propagation distances, etc. To obtain a geophysically appropriate semivariogram which is a key factor in estimation of groundwater contaminant concentration at desired locations, these special characteristics should be considered. Specifically, the concepts of symmetry and ratio are considered in this paper. By applying these two concepts, significant improvement of semivariograms, estimation variances, and final estimation results compared with the ones by conventional approaches which usually do not account for symmetry and ratio are shown using field experimental data.     

From discrete to continuum: A Young measure approach

The passage from atomistic to continuum models is usually done via G\Gamma-convergence with respect to the weak topology of some Sobolev space; the obtained continuum energy, in a one dimensional model, is then convex. These kind of results are not optimal for problems related to materials which may undergo to phase transitions. We present here a new simple way for dealing with these problems. Our method consists in rewriting the discrete energy in terms of particular measures and taking the G\Gamma-limit with respect to the weak * convergence of measures. The continuum energy arising from a linear chain of discrete mass points interacting with only the nearest neighbours turns out to be written in terms of Young measures. While, if the discrete mass points interact not only with the nearest neighbours but also with the second nearest neighbours we obtain a continuum problem in which appears a ``multiple Young measure" representing multiple levels of interaction. In this way we obtain a novel continuum problem which is able to capture the ``microstructure" at two different levels.     

An approach to constructing models of multiphase elastic porous media

A novel approach to describing the behaviour of multiphase elastic porous media is proposed. The average values of the physical quantities needed to describe the motions of porous media are formulated using an integral relation. The validity of this relation is taken as the fundamental hypothesis. The integral definition of the average values enables integral relations to be devised for the average values from the integral laws of conservation of mass, momentum and energy and the increase in entropy. Along with the average values, the integral relations contain new variables that can be identified with generalized thermodynamic forces, which can be used to take into account the phase interaction in a porous medium. The integral relations are used to derive differential equations for the rate of entropy change and Gibbs relations for a porous medium as a basis for obtaining the constitutive relations. Relationships between the thermomechanical parameters of the model are established from the Gibbs relations under additional assumptions. The equation for the rate of entropy change can be used to establish relations between the generalized thermodynamic forces and fluxes. A complete system of differential equations in the defining parameters, which describes the motion of multiphase elastic porous media, is finally obtained.     

Modeling the effective permeability of microcracked materials using continuum and lattice micromechanics

Using the framework of the mean-field homogenization method, we propose a continuum and a lattice version of the cascade micromechanics model for the estimation of the effective permeability of microcracked materials. Estimates for the critical microcrack density below which the REV becomes effectively impermeable are derived for both the continuum and the lattice idealizations. A validation example that compares model predictions with direct numerical simulations for the effective permeability as a function of the microcrack volume fraction is provided. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)