Model Of Heterogeneous Medium Research Articles

Evaluation of a practical approach for field scale moisture flow modeling in heterogeneous media at a semiarid site

AbstractA practical approach for modeling field‐scale moisture flow in a highly heterogeneous unsaturated medium is described in this study. The validity of this approach is demonstrated through comparison of the numerical simulations with field observations at a semiarid site located in southcentral Washington State. The methodology is based on upscaling the core scale hydraulic properties and combining power‐law and tensorial connectivity‐tortuosity (PA‐TCT) approaches to derive macroscopic anisotropy parameters for each hydrostratigraphic unit (HSU) identified in the field. Each heterogeneous HSU is approximated by an equivalent homogeneous medium (EHM) model for which PA‐TCT parameters are used in the flow simulations. The available field data on moisture content and matric potential are compared with steady‐state flow simulations based on the mean form of Richards' equation. While the homogenization or averaging of heterogeneities, embedded in the EHM modeling approximation, cannot capture all of the field‐scale variability, the simulated steady‐state moisture and matric potential profiles capture well the central tendency of the field data. This approach is deemed practical for assessing the fate and transport of contaminants in highly heterogeneous unsaturated media at the transport scale of hundreds of meters.

Crustal heterogeneity effects on coseismic deformation: numerical simulation of the 2008 MW 7.9 Wenchuan earthquake

Coseismic deformation of large earthquakes causes significant property damages and fatalities, which requires quantitative research of multiple disciplines such as geodesy, geological investigation, seismic tomography, and seismic dislocation theory. The finite element method accounts for material heterogeneity and geometric complexity, making it suitable for studying the coseismic deformation of large earthquakes. This paper develops a parallel elastic finite element program that utilizes split nodes and high-performance parallel computing technology on the FELAC software platform to study the coseismic deformation of large earthquakes. We verify the accuracy of the parallel elastic finite element program by comparing its results with the analytical solutions from seismic dislocation theory for four ideal earthquake cases. Finally, we established parallel elastic finite element models to study the coseismic deformation of the 2008 Wenchuan earthquake. The simulation results are consistent with the GPS and InSAR data. Coseismic surface deformation results are significantly influenced by medium regional heterogeneity with different layered structures besides the Longmenshan fault. The finite element program lay the foundation for the inversion of the coseismic fault rupture process based on the heterogeneous medium model and complex geometric model.

An in silico testbed for fast and accurate MR labeling of orthopedic implants

One limitation on the ability to monitor health in older adults using magnetic resonance (MR) imaging is the presence of implants, where the prevalence of implantable devices (orthopedic, cardiac, neuromodulation) increases in the population, as does the pervasiveness of conditions requiring MRI studies for diagnosis (musculoskeletal diseases, infections, or cancer). The present study describes a novel multiphysics implant modeling testbed using the following approaches with two examples: (1) an in silico human model based on the widely available Visible Human Project (VHP) cryo-section dataset; (2) a finite element method (FEM) modeling software workbench from Ansys (Electronics Desktop/Mechanical) to model MR radio frequency (RF) coils and the temperature rise modeling in heterogeneous media. The in silico VHP-Female model (250 parts with an additional 40 components specifically characterizing embedded implants and resultant surrounding tissues) corresponds to a 60-year-old female with a body mass index of 36. The testbed includes the FEM-compatible in silico human model, an implant embedding procedure, a generic parameterizable MRI RF birdcage two-port coil model, a workflow for computing heat sources on the implant surface and in adjacent tissues, and a thermal FEM solver directly linked to the MR coil simulator to determine implant heating based on an MR imaging study protocol. The primary target is MR labeling of large orthopedic implants. The testbed has very recently been approved by the US Food and Drug Administration (FDA) as a medical device development tool for 1.5 T orthopedic implant examinations.

Global space-time Trefftz DG schemes for the time-dependent linear wave equation

In this paper we are concerned with Trefftz discretizations of the time-dependent linear wave equation in anisotropic media in arbitrary space dimensional domains $\Omega \subset \mathbb{R}^d~ (d\in \mathbb{N})$. We propose two variants of the Trefftz DG method, define novel plane wave basis functions based on rigorous choices of scaling transformations and coordinate transformations, and prove that the corresponding approximate solutions possess optimal-order error estimates with respect to the meshwidth $h$ and the condition number of the coefficient matrices, respectively. Besides, we propose the global Trefftz DG method combined with local DG methods to solve the time-dependent linear nonhomogeneous wave equation in anisotropic media. In particular, the error analysis holds for the (nonhomogeneous) Dirichlet, Neumann, and mixed boundary conditions from the original PDEs. Furthermore, a strategy to discretize the model in heterogeneous media is proposed. The numerical results verify the validity of the theoretical results, and show that the resulting approximate solutions possess high accuracy.

Three-dimensional Trefftz computational grains for the micromechanical modeling of heterogeneous media with coated spherical inclusions

Three-dimensional Trefftz computational grains for the micromechanical modeling of heterogeneous media with coated spherical inclusions

Computation of ray-Born seismograms using isochrons

Seismic modeling in heterogeneous media is accomplished by using either approximate or fully numerical methods. A popular approximate method is ray-Born modeling, which requires the computation of 3D integrals. We have developed an integration technique for accurate and, under certain circumstances, efficient evaluation of the ray-Born integrals in the time domain. The 3D integrals are split into several 2D integrals, each of which gives the wavefield at a certain time, so that the waveform at each time step is computed independently of all other times. We compute seismograms for 3D heterogeneous acoustic media using this technique and compare these seismograms with seismograms computed using two other modeling methods: frequency-domain ray-Born modeling and finite-difference modeling of the acoustic wave equation. Our method can also be applied to elastic ray-Born modeling. Velocity models with smooth scatterers and the SEG/EAGE overthrust model are used for comparison. The ray-Born seismograms computed using the time- and frequency-domain ray-Born modeling methods are identical, as expected. The comparison between the ray-Born modeling and the finite-difference-modeling method indicates that the waveforms are similar for both types of velocity models. We evaluate the discrepancies in terms of multiple scattering and multipathing.

Kramers-Kronig Relations and the Properties of Conductivity and Permittivity in Heterogeneous Media

The macroscopic electric permittivity of a given medium may depend on frequency, but this frequency dependence cannot be arbitrary, its real and imaginary parts are related by the well-known Kramers-Kronig relations. Here, we show that an analogous paradigm applies to the macroscopic electric conductivity. If the causality principle is taken into account, there exist Kramers-Kronig relations for conductivity, which are mathematically equivalent to the Hilbert transform. These relations impose strong constraints that models of heterogeneous media should satisfy to have a physically plausible frequency dependence of the conductivity and permittivity. We illustrate these relations and constraints by a few examples of known physical media. These extended relations constitute important constraints to test the consistency of past and future experimental measurements of the electric properties of heterogeneous media.

An NAD Scheme with Wavenumber Error Optimized for 2D Scalar Wave Equation

This article presents a wavenumber error optimized nearly analytic discrete differentiator (WONAD) using Taylor polynomial constraints and wavenumber domain optimizing in the function space expanded by the displacement and its gradient. We also apply the differentiator in a 2D scalar‐wave equation for seismic‐wave forward modeling in heterogeneous media. Subsequent analysis shows the WONAD generates both low numerical dispersion and numerical error. The maximum phase velocity error of the WONAD is as low as 3.12%, even in an extreme case with only two sampling points per minimum wavelength when the Courant number is 0.5. In our numerical experiments, the maximum relative error of the WONAD in a simple harmonic wavefield is less than 1.12% after a 300 s simulation. On the same grids, both the numerical dispersion and the numerical error of the WONAD are lower than what have been found in cases using traditional high‐order methods, such as the 24th‐order Lax–Wendroff correction (LWC) method and so on. To simulate a wavefield without visible numerical dispersion, the computational efficiency of the WONAD is 341%, 651%, and 316%, compared with that of the fourth‐order staggered grid method and the fourth‐order and 24th‐order LWC methods, while the phase misfit of the WONAD is still lower than the other three methods. The WONAD, showing a promising prospective in large‐scale long‐time seismic modeling, performs well in computational efficiency, simulation accuracy, and numerical stability.

Latent Waves in Heterogeneous Media

The dynamics of latent waves, i.e., their reflection from a solid side and interaction with each other, has been investigated numerically within the framework of the heterogeneous-medium model with a gasdynamic core.

Dispersion effects modeling while forecasting physical-geological parameters of heterogeneous media

A problem of forecasting the parameters of physical-chemical model for heterogeneous media based on available standard data selection considered as a field of corresponding param­eters dispersion is being considered. Dispersion of parameters is considered as an observed component of medium heterogeneity effects manifestation, which is simulated by a function of concentration based on diffusion equations. Diffusion operator is being designed based on fundamental solutions of diffusion equations, which made possible to obtain prognostic field of dispersion for sought-for parameter. A problem of reconstruction of heterogeneities density concentration in distribution of expected parameter as a solution of integral equation of the first kind is formulated. On the basis of Mamdani rule of logical conclusion chain rules of calculation for resultant dispersion field for prediction of heterogeneities of final parameters have been designed. An example of modeling the dispersion field for presence of oil by one of conditional oil deposits is presented, which demonstrates validity of new notions introduced.

Finite Difference Method for Solving Acoustic Wave Equation Using Locally Adjustable Time-steps

Explicit finite difference method has been widely used for seismic modeling in heterogeneous media with strong discontinuities in physical properties. In such cases, due to stability considerations, the time step size is primarily determined by the medium with higher wave speed propagation, resulting that the higher the speed, the lower the time step needs to be to ensure stability throughout the whole domain. Therefore, the use of different temporal discretizations can greatly reduce the computational cost involved when solving this kind of problem. In this paper we propose an algorithm for the local temporal discretization setting named Region Triangular Transition (RTT), which allows the local temporal discretizations to be related by any integer value that enables these discretizations to operate at the stability limit of the finite difference approximations used.

WE-E-108-05: Evaluation of the XRAD 225Cx MC Source Model in Heterogeneous Mediums

Purpose: This study was performed to benchmark a Monte Carlo source model of the XRAD 225Cx irradiator under heterogeneous conditions. Methods: The BEAM/EGS was used to model 225 kV photon beams from a small animal irradiator (Precision XRAD225Cx). Benchmarking the model in heterogeneous media was performed in a heterogeneous phantom (4×4×4 cm3) composed of solid water and cortical bone. The 10 mm field size virtual source from BEAMnrc was used to compute the 3D dose distribution in DOSXYZnrc. Benchmarking the simulation against measurements was performed using EBT2 film. Heterogeneous conditions were further validated using the phase space file for the 20 mm field size. Gamma analysis was performed to further validate the beam model for the heterogeneous configuration phantom. Results: The MC simulation indicates that there is a 2.4 times increased dose deposition in cortical bone, in agreement with published data. This increased dose deposition, however, was not observed in film measurements and it is investigated further. Gamma analysis was performed for 20 mm field size applicator and the Result of the analysis for a confined region. The single beam with 20 mm diameter circular field irradiation demonstrates a good agreement within the region of interest, with 94% of the calculated data meeting the 5%/0.5 mm criterion. Conclusion: Our kV Monte Carlo source model demonstrates excellent agreement with measurement in heterogeneous conditions.

OC-0438: Experimental validation of monte carlo pencil beam scanning model in heterogeneous media for proton therapy

Purpose/Objective To validate experimentally a GATE/GEANT4-based(G4) Monte Carlo (MC) model in heterogeneous media for dedicated pencil beam scanning in proton therapy. Comparisons between measurements and MC simulations using G4 and PENELOPE-proton are presented. A comparison against analytical modeling from commercial TPS is also investigated. This work evaluates the impact of heterogeneities on range prediction, beam shape and depth dose changes. Materials and Methods The MC model for pencil beam based on G4 has been validated in water and PMMA phantoms (Grevillot et al Phys. Med. Biol.(2011)) reproducing pristine Bragg peaks for a series of individual energies (from 100 to 226.7 MeV) with 0.7 mm range and 0.2 mm spot size accuracy. The same optical model was implemented in PENELOPE-proton. In order to validate the beam model in heterogeneous media, phantoms made of stacked slabs with different densities and known compositions were used. Two experimental test cases including solid water (SW), lung (LN-300) and bone (SB3) tissue-equivalent material were investigated. Depth-dose distributions for a monoenergetic single spot and 10x10cm² composite fields were measured using Gafchromic EBT3 films and the ionization chamber (IC) PPC05 in all configurations. To measure accurately the Bragg peak position, a stack of films of 2x2cm² was inserted in the last centimeter of the proton range. Results Figure 1 shows results for one heterogeneous configuration. All doses-to-medium were converted to dose to water using stopping power ratios. Dose distributions were arbitrarily normalized in the middle of the second SW region. Bragg peak positions are reproduced by MC simulations within 1mm in both configurations (Table 1). IC measurements, G4 (binary-cascade) and PENELOPE-proton simulations are within 2%/2 mm. Point-to-point mean difference of 1.2% is observed between G4 (precompound) and measurement in the first 15 cm of the phantom and increased to 7.2% after bone insert until 286.5mm depth. In lung and bone slabs, EBT3 films and G4 binary-cascade are inagreement within 0.77% while a mean point-to-point difference up to 1.26% is observed with G4 precompound model. The uncertainty (1σ) on EBT3 films was evaluated to be at 2.75% which included readout process and dose calibration against IC (TRS-398). A statistical uncertainty of 0.1% was achieved for MC simulations. Conclusions The Bragg peak position is predicted with 1mm precision for all MC simulations, even though ionization potential values for phantom slabs were calculated using classical additive rules. G4/GATE beam model reproduce depth-dose behavior of proton transport regarding both IC and EBT3 measurement in heterogeneities.

Hyperbolic model of a multivelocity heterogeneous medium

The model of a multivelocity multicomponent medium, which is based on the conservation laws and whose equations belong to a hyperbolic type, has been presented. A number of problems on disintegration of an ar- bitrary discontinuity in a dust-laden gas have been calculated with the Courant-Isaacson-Rees numerical method. modeling. Introduction. Earlier models of heterogeneous media were not hyperbolic, which gave rise to nonphysical ef- fects of various kind (associated, e.g., with the presence of waves propagating with infinitely large velocities) when these models were used. Furthermore, the Cauchy problem did not necessarily turn out to be correct for these models describing flow of heterogeneous media, which made it difficult to use numerical methods (1). Therefore, subsequently the main effort went into the development of hyperbolic models of heterogeneous media. The first hyperbolic model of a two-phase medium was proposed in (2), where several pressures (according to the number of fractions) were used to attain hyperbolicity. Since the number of unknown quantities expressing conser- vation laws in such a system increased, additional equations were needed to close it. It is noteworthy that these addi- tional equations have no rigorous physical substantiation. Currently available models of heterogeneous media (see, e.g., (3-7)) are also based on the concept of several pressures. In the present work, we propose a different approach to construction of a heterogeneous-medium model, which is based solely on conservation laws and whose essence is reduced to postulating the existence of a certain state of a multicomponent mixture (which will be called the mixture as a whole); this state is characterized by the averaged values of velocity, density, etc. Applying the laws of conservation of mass, momentum, and energy to the mixture as a whole introduced in this manner, we obtain equations coincident in form with gasdynamic equations, which must be satisfied by the averaged variables. To these relations, there are added equations expressing conservation laws for in- dividual components of the mixture. Below, it will be shown that the heterogeneous-medium model constructed in this manner is hyperbolic. It is noteworthy that introduction of the state of the mixture as a whole becomes topical where we must take account of, e.g., the viscous or heat-conducting properties of a medium (remaining within the framework of the hyperbolic equations), since the terms responsible for the indicated effects are included precisely in the equa- tions for the averaged motion, i.e., for the mixture as a whole. Such an approach was previously used in the model of a one-velocity heterogeneous medium (8-10). Multivelocity-Medium Model. To simplify the presentation, phase and chemical transformations in the mix- ture will be disregarded, as will be mass forces. The system of equations of the n-component mixture with first m compressible fractions, which describes flow of a multivelocity heterogeneous medium, involves the equations of the laws of conservation of mass, momentum, and energy for the mixture as a whole: ∂ρ ∂t + ∂ρu ∂x = 0 ,

Elastic waves in suspensions

A model of heterogeneous medium taking into account the friction between the particles and liquid, as well as the relaxation of the small-size particles to the equilibrium on the stress, has been proposed to describe the propagation of the elastic waves in a suspension. A system of wave equations describing the propagation of a plane longitudinal wave has been formulated for the components of the medium. Analytical expressions for the sound velocity in a suspension has been obtained in the approximation in which the particles are completely carried away by liquid in the limiting cases in which the particles are in equilibrium under stress with the liquid or equilibrium is absent. The dependence of the sound velocity in the medium on the volumetric portion and the size of the inclusions has been studied. The obtained results agree with the experimental data and obtained analytical expressions for the sound velocity. The dynamics of the components of the medium at the propagation of the plane longitudinal monochromatic wave has been studied.

Simulation of generation of ultradisperse particles upon irradiation of metals by a high-power electron beam

A model of formation of ultradisperse particles in the plasma torch emerging during evaporation of a metal target by a high-power electron beam is described. A model of heterogeneous media is proposed for describing the plasma torch dynamics taking into account heat conduction, heat transfer and friction between components, relaxation of components to equilibrium, condensation, and evaporation and coagulation of drops as a result of their collisions. Numerical simulation of the generation of ultradisperse particles in the plasma torch formed during irradiation of a metal target by a powerful electron beam is performed. The size distribution of ultradisperse particles is obtained for various regimes of irradiation and cooling.

Seismic scalar wave equation with variable coefficients modeling by a new convolutional differentiator

Studying seismic wavefields in the Earth's interior requires an accurate calculation of wave propagation using accurate and efficient numerical techniques. In this paper, we present an alternative method for accurately and efficiently modeling seismic wavefields using a convolutional generalized orthogonal polynomial differentiator. Our approach uses optimization and truncation to form a localized operator. This preserves the fine structure of the wavefield in complex media and avoids non-causal interaction when parameter discontinuities are present in the medium. We demonstrate this approach for scalar wavefield modeling in heterogeneous media and conclude that the method could be readily extended to elastic wavefield calculations. Our numerical results indicate that this method can suppress numerical dispersion and allow for the study of wavefields in heterogeneous structures. The results hold promise not only for future seismic studies, but also for any field that requires high-precision numerical solution of partial differential equation with variable coefficients.

Hybrid integral transforms analysis of the bioheat equation with variable properties

Pennes’ equation is the most frequently employed model to describe heat transfer processes within living tissues, with numerous applications in clinical diagnostics and thermal treatments. A number of analytical solutions were provided in the literature that represent the temperature distribution across tissue structures, but considering simplifying assumptions such as uniform and linear thermophysical properties and blood perfusion rates. The present work thus advances such analysis path by considering a heterogeneous medium formulation that allows for spatially variable parameters across the tissue thickness. Besides, the eventual variation of blood perfusion rates with temperature is also accounted for in the proposed model. The Generalized Integral Transform Technique (GITT) is employed to yield a hybrid numerical–analytical solution of the bioheat model in heterogeneous media, which reduces to the exact solution obtained via the Classical Integral Transform Method for a linear formulation with uniform coefficients. The open source UNIT code (“UNified Integral Transforms”) is utilized to obtain numerical results for a set of typical values of the governing parameters, in order to illustrate the convergence behavior of the proposed eigenfunction expansions and inspect the importance of accounting for spatially variable properties in predicting the thermal response of living tissues to external stimulus.

The Riemann problem for one-velocity model of multicomponent mixture

The exact and approximate solutions of the Riemann problem are given for a one-velocity model of heterogeneous medium which takes into account the internal forces of interfraction interaction. The shock adiabat for mixture, coordinated with the equations of one-velocity model, was used in constructing solutions with shock waves.

Investigation of the stability of a plane-channel suspension flow with account for finite particle volume fraction

Within the framework of the two-fluid approach, a variant of a heterogeneous-medium model which takes into account a finite volume fraction of the inclusions and a small but finite phase velocity slip is proposed. The interphase momentum exchange is described by the Stokes force with the Brinkman correction for the finite particle volume fraction. The suspension viscosity depends on the particle volume fraction in accordance with the Einstein formula. Within the framework of the model constructed, a formulation of the problem of linear stability of plane-parallel two-phase flows is proposed. As an example, the stability of a channel suspension flow is considered. The system of equations for small disturbances with the boundary conditions is reduced to an eigenvalue problem for a fourth-order ordinary differential equation. Using the orthogonalization method, the dependence of the critical Reynolds number on the governing nondimensional parameters of the problem is studied numerically. It is shown that taking a finite volume fraction of the inclusions into account significantly affects the laminar-turbulent transition limit.