Model Of Heterogeneous Medium Research Articles

Travelling Waves of Resistance in a Bi-trophic Pest Adaptation Model

We consider a reaction–diffusion system for spatial spread of pest resistance to host plant resistance genes which is based on the Lotka–Volterra predator–prey equations, with logistic growth of the resource level and a diffusion term added to account for spatial spread of the pest. The model is phenotype specific, in which a pest subpopulation's fitness comes down to a balance between its resource assimilation rate and its respiration rate. We derive an expression for the rate of spatial spread of the resistant pest types from an initial point source, and discuss its relevance for adaptive pest resistance management strategies. Using results for an analogous single-species reaction–diffusion model in heterogeneous media, we consider the likely impact of pest-susceptible plant refugia on the speed of the travelling wave of resistant pests, and simultaneously the expected trade-off, in terms of crop yield decrease, when refugia are included. We also explore the possibility that resistance breaking by the pest population is not an inevitable phenomenon, particularly when refugia of the appropriate size are used.

Three-dimensional modelling and optimisation of thermal fields induced in a human body during hyperthermia

In the field of hyperthermia as a cancer therapy, the design of tools to predict as well as to control the thermal fields in biological tissues remains a task of highest importance. In this context, we have developed a three-dimensional numerical model in heterogeneous media in order to compute the rate of volumetric heat generation, produced by means of an electromagnetic irradiation, and the resulting temperature distribution. The electromagnetic power deposition within the tissues is predicted by solving the Maxwell's field equations in the frequency-domain and the thermal process is described by the Pennes' bio-heat transfer equation. All the governing equations have been solved using the boundary element method. Furthermore, our simulation tool has been coupled to an optimisation procedure, whose parameters are the amplitude and phase of each electromagnetic source. The objective is to achieve the highest possible temperature in the tumour without exceeding 42 °C in the surrounding healthy tissues. Simulations based on a three-dimensional heterogeneous model constructed from cross-sectional slices of a person are presented. The studied configurations concern the pelvic region where a tumour is deeply embedded. The efficiency of our optimisation routine in a treatment plan for regional hyperthermia is reported.

Application of the Integral Laguerre Transforms for Forward Seismic Modeling

The paper presents some efficient algorithms based on the application of the integral Laguerre transform for approximation of temporal derivatives. Some specific features of employing this algorithm for the first and the second order equations with respect to time are considered. A few examples of calculation of seismic fields for the layered medium model with drastically contrast elastic parameters and for the 2-D heterogeneous medium model are presented.

APPLICATION OF THE INTEGRAL LAGUERRE TRANSFORMS FOR FORWARD SEISMIC MODELING

The paper presents some efficient algorithms based on the application of the integral Laguerre transform for approximation of temporal derivatives. Some specific features of employing this algorithm for the first and the second order equations with respect to time are considered. A few examples of calculation of seismic fields for the layered medium model with drastically contrast elastic parameters and for the 2-D heterogeneous medium model are presented.

Structural damage accumulation and stable postcritical deformation of composite materials

The macroscopic failure of composite materials is preceded by complex multilevel processes accompanied by accumulation and localization of damaged centers and formation of a failure cluster. Therefore, the study of these mechanisms is one of the basic problems for the mechanics of modern composite materials used in aerospace engineering. The formation of a theory of the stable postcritical deformation of the work-softening media is considered. The pseudo-plastic deformation affected by structural damage of granular composites is investigated within the framework of the considered two-level structurally phenomenological model of heterogeneous media. The stable evolution of the interconnected processes is accompanied by stress redistributions, partial or complete unloading, and strain or damage localization that are one of the main causes of implementation of the postcritical deformation stage. The numerical calculation results of inelastic deformation and failure of the periodic unidirectional fiber-reinforced composites are presented under conditions of the displacement-controlled transverse proportional loading mode. The main mechanisms of the work-softening behavior for the indicated type of materials are described in the macro-homogeneous stress-strain states. Macroscopically, the failure of heterogeneous media as a result of postcritical deformation and the loss of stability of damage accumulation depends on the stiffness of the loading system. When a deformable body is fixed on the closed surface with sufficiently but not infinitely large coefficients of stiffness, it is possible to observe the equilibrium development of the localized volumes of work-softening and damage. The constitutive equations for the work-softening isotropic, transverse isotropic, and orthotropic media are presented. The effect of the loading system on the stability of deformation, damage accumulation, and failure under monotone and nonmonotone triaxial loading was studied. The growth of failure strains with increase in stiffness of the loading system and unequal resistance of heterogeneous body are registered and investigated. A preventive unloading method is offered for the mathematical modeling of the damage accumulation during the testing of the materials on the servo-controlled systems. The displacement-controlled mode is simulated by a series of soft loading and unloading cycles. The detected phenomenon of failure where the unloading leads to stress-strain diagrams with a negative slope of the descending branch was not found either in the displacement or stress-controlled monotone loading mode.

Elaboration of multispectral models for heterogeneous media: Application to the LAI

Most of the models used in remote sensing link a ground parameter to one or more radiometric measurements. These models are defined and validated at local scale considering homogeneous surfaces; however, the application of these models at the satellite scale concerns pixels, which are always heterogeneous. The aim of this work is to propose a method, called spatialized method, which reduces the errors introduced by a change of scale. This method, based on the general results of Raffy (1992) requires the understanding of the models validated on the homogeneous surfaces constituting the pixel and the knowledge of the local spectral domain. The bounds of all possible actual in situ values are given on each pixel. A spatialized model is proposed. An application to the LAI/NDVI model is given with a simulated landscape, in which deciduous trees, conifers, bare soil and water are represented. This new method, valid for any models, surfaces and heterogeneities, appears to be accurate in the large majority of cases.

Flow Modeling in Heterogeneous Media in the Context of Geologic Nuclear Waste Repositories

Assessing long-term performance of geologic repositories requires simulation of flow through heterogeneous geologic formations. The effect on flow field of discontinuities such as fracture zones in such media is not only of interest to waste management professionals but also to those involved in petroleum engineering and water resources development. Significant differences in the space and time scales associated with these discontinuities as compared with those associated with bulk geology cause special problems in modeling. The modeling problems are in addition to the very practical problem inherent in proper topological characterization of the discontinuities and also in field measurement of their flow and transport properties. After briefly reviewing various generally used classes of methods for accommodating heterogeneities represented by fractures in numerical models, a different technique of deriving mass balance equations in the presence of fractures is discussed. Compared with full representation of fractures, the proposed technique provides coarser resolution of the flow field, but it is relatively computationally efficient. Two examples of its application are also provided.

Diffusion and reaction among traps: some theoretical and simulation results

Diffusion and reaction in heterogeneous media arise in a host of phenomena in the physical and biological sciences. The determination of the mean survival timeτ (i.e., inverse trapping rate) and relaxation timesTn, n=1,2,3,... (i.e., inverse eigenvalues), associated with diffusion among partially absorbing, static traps with surface rate constantk are problems of long-standing interest. The limitsk=∞ andk=0 correspond to the diffusion-controlled case (i.e., perfect absorbers) and reaction-controlled case (i.e., perfect reflectors), respectively. This paper reviews progress we have made on several basic aspects of this problem: (i) the formulation of rigorous bonding techniques and computational methodologies that enable one to estimate the mean survival time τ and principal relaxation timeT1 (ii) the quantitative characterization of the microstructure of nontrivial continuum (i.e., off-lattice) models of heterogeneous media; and (iii) evaluation of τ and T1 for the same models. We also describe a rigorous link between the mean survival time t and a different effective parameter of the system, namely the fluid permeability tensork associated with Stokes flow through the same porous medium.

Seismic waveform modeling in heterogeneous media by ray perturbation theory

We study the propagation of rays, par axial rays, and Gaussian beams in a medium where slowness differs only slightly from that of a reference medium. Ray theory is developed using a Hamiltonian formalism that is independent of the coordinate system under consideration. Let us consider a ray in the unperturbed medium. The perturbation in slowness produces a change of the trajectory of this ray which may be calculated by means of canonical perturbation theory. We define par axial rays as those rays that propagate in perturbed medium in the vicinity of the perturbed ray. The ray tracing equation for par axial rays may be obtained by a linearization of the canonical ray equations. The linearized equations are then solved by a propagator method. With the help of the propagator we form beams, i. e. families of paraxial rays that depend on a single beam parameter. The results are very general and may be applied to a number of kinematic and dynamic ray tracing problems, like two‐point ray tracing, Gaussian beams, wave front interpolation, etc. The perturbation methods are applied to the study of a few simple problems in which the unperturbed medium is homogeneous. First, we consider a two‐dimensional spherical inclusion with a Gaussian slowness perturbation profile. Second, transmission and reflection problems are examined. We compare results for amplitude and travel time computed by exact and perturbed ray theory. The agreement is excellent and may be improved using an iterative procedure by which we change the reference unperturbed ray whenever the perturbation becomes large. Finally, we apply our technique to a three‐dimensional problem: we calculate the amplitude perturbation and ray deflection produced by the velocity structure under the Mont Dore volcano (central France). Again a comparison shows excellent agreement between exact and perturbed ray theory.

The role of channels in the transport of radon from the soil

Experimental measurements of permeability as a function of scale as well as direct detection of flow indicate macroscopic channels are present at a soil site. A model for heterogeneous media is developed that provides a dimensionless number S, an estimate of the ratio of the exhalation from the channels to the exhalation from the soil matrix, to indicate whether or not channels are important for transport of radon to the atmosphere. The model considers the effects of flow and transverse diffusion in the channels, flow and diffusion in the soil matrix, and coupling between the channels and matrix due to diffusive exchange. Specific predictions depend on soil properties, the geometry of the channels, and the pressure gradient. The model indicates that channels are important for the present soil at times of higher pressure gradients such as can occur with fronts, wind, and atmospheric turbulence. The radon concentration exhausted for steady state upward flow is greater than that exhausted for transient upward flow. On a time‐averaged basis, the contribution of channels to overall exhalation of radon appears moderate. The modeling suggests channeling in soils could be a factor enhancing exhalation of radon to the atmosphere at other locations. The distance of subsurface radon transport is increased by significant channeling, but transverse diffusion into the surrounding matrix is a strong mitigating influence.

RANDOM NETWORKS AND MIXING LAWS

Random networks are investigated as models of heterogeneous media. A general approximate structure is used where the networks are described as a system of embedded networks, and the critical behavior and averaging behavior of such networks are developed. These results are applied to a study of the electrical conductivity of porous media, with special attention to an Archie's law behavior. It appears that the wide range of crack and pore widths in rocks makes the resulting conductivity relatively insensitive to the topology of their interconnections and allows one to make reasonable predictions of rock conductivities, given the distribution of crack and pore widths. It also appears that with low porosity rocks the conductivity is controlled by the microcrack population which only accounts for a fraction of the total porosity. It would seem, therefore, that Archie's law is a feature of some general trend between porosity and crack and pore width distributions rather than a fundamental property of porous media. The law of the geometric mean is an accurate predictor of the physical properties of a mixture of different materials. This mixing law can result from an equal balance of series and parallel arrangements which can be produced by an appropriate distribution of shapes. A brief look is given to problems of anisotropic distributions for the conductivity problem and it is shown how the averaging process greatly dilutes the microscopic anisotropy in producing the macroscopic properties.