Extrapolated Boundary Condition Research Articles

Approximate P3 equation analysis in multi-layer slab media: Steady-state and time-domain based on the diffusion model

This study aimed to establish a steady-state and time-domain solutions for the approximate P3 equation in arbitrary multi-layer slab media for light propagation, based on a diffusion model with an isotropic point source in the first layer and extrapolated boundary conditions. Spatially resolved diffuse reflectance and transmittance were calculated using the steady-state approximation of the P3 equation, whereas temporally resolved diffuse reflectance and transmittance were derived from the time-domain approximation. The validity of the approximate P3 solutions was confirmed by comparing their results with Monte Carlo simulations. In steady-state analysis, the approximate P3 equation demonstrated superior accuracy to the diffusion equation for reflectance, particularly at smaller thicknesses. Accuracy further improved as the absorption coefficient and detection distance increased. For transmittance, the approximate P3 equation closely matched the diffusion equation at low thickness, but divergence occurred with higher absorption. In time-domain analysis, the approximate P3 equation aligned closely with Monte Carlo simulations at peak values, while its numerical values were close approximations of the diffusion equation away from the peak. The potential application of the approximate P3 equation in multi-layer media offers significant advancements for optical non-invasive detection and treatment techniques, enabling the extraction of optical parameters from such media.

Developing an immersed boundary method for compressible flow

AbstractDevelopment of a 3D sharp interface ghost node immersed boundary method (IBM) within a fluid solver based on the compressible Navier–Stokes equations is underway. The objective of the IBM is to accurately apply boundary conditions at fluid–solid interfaces. The Navier–Stokes solver is currently being verified using various test cases, including the classic cylinder in cross flow. As part of this verification process, particular attention was given to investigating the effects of different lateral boundary conditions. The results demonstrate that extrapolation boundary conditions exhibit better agreement with the literature compared to symmetry conditions, in cases with relatively narrow domains. These findings highlight the potential benefits of extrapolation boundary conditions in reducing confinement effects and removing nonphysical waves in external flow problems.

Phase shift migration with absorbing boundary condition for full-matrix imaging heterogenous medium with high quality

Ultrasonic imaging with phased array has been widely employed for nondestructive testing in various fields. Phase shift migration (PSM) is a significant imaging method and shows great advantages for multilayer structures. However, the artifact reflections at the edges of the calculation region emerge in the wavefield reconstruction process, which greatly deteriorate the imaging quality of PSM. To overcome this problem, two absorbing boundary conditions for the wavefield extrapolation of PSM are proposed in this paper. The first absorbing boundary condition renders elements in the boundary regions lower weights in frequency-space domain. The second one follows a nonstationary phase shift scheme and filters the incident waves in the boundary regions in frequency-wavenumber domain. The implementing schemes of the absorbing boundary conditions for both homogenous and heterogenous medium are developed, and the full-matrix imaging method with absorbing boundary conditions is proposed. According to the simulation and experiment, the proposed boundary conditions have strong ability to suppress the background artifacts, and they can improve the signal-to-noise ratio (SNR) and resolution by 2 times when imaging heterogenous medium. Further experiments indicate that the boundary conditions are insensitive to the window functions, and a width of one-sixteenth the length of the scan line is recommended for the absorbing boundary conditions.

A sliding mesh approach to the Lattice Boltzmann Method based on non-equilibrium extrapolation and its application in rotor flow simulation

The recent development of the Lattice Boltzmann Method has been widely applied in various industrial fields. In the application of rotor machinery simulation, the sliding mesh method with grid refinement becomes particularly important for the scenario of large-scale rotating flow computation. A new sliding mesh method based on the extrapolation boundary condition, with only first order interpolation across the interface, is proposed in the current study, which achieved a second order solving precision in two dimensional cases of Taylor-Couette flow and flow over a rotating elliptical cylinder. In the simulation of flow over a rotating propeller, both aerodynamic loads and far-field tonal noise were predicted and matched well with the experimental results.

High order numerical schemes for transport equations on bounded domains

This article is an account of the NABUCO project achieved during the summer camp CEMRACS 2019 devoted to geophysical fluids and gravity flows. The goal is to construct finite difference approximations of the transport equation with nonzero incoming boundary data that achieve the best possible convergence rate in the maximum norm. We construct, implement and analyze the so-called inverse Lax-Wendroff procedure at the incoming boundary. Optimal convergence rates are obtained by combining sharp stability estimates for extrapolation boundary conditions with numerical boundary layer expansions. We illustrate the results with the Lax-Wendroff and O3 schemes.

The diffusion equation in multilayered rectangular biological tissue with finite thickness

In biomedical optics, the diffusion model of biological tissue plays an important role. In the biological tissue of any multi-layer rectangular non-matching slab medium, under the rectangular boundary condition, based on the diffusion equation of light propagation in the biological medium, using the extrapolation boundary condition, the exact solution of the diffusion equation Green function of the light in any multi-layer rectangular medium of finite thickness is established and given, and the spatial resolution diffuse reflectivity and transmittance and time resolved diffuse reflectivity and transmittance are calculated in this paper. Different formulas for calculating transmittance are given. We use Monte Carlo to verify our model. The results show that our equation is in good agreement with Monte Carlo simulation. The equation can not only solve the problem of medium which is rectangular laterally, but also solve the problem that the x axis and y axis are infinite in transverse direction, and the medium is semi-infinite in longitudinal direction, and it can also solve the problem that the x, y and z coordinate axes are finite in size. So it's more universal.

Gas transport in self-affine rough microchannels of shale gas reservoir

The matrix of shale gas reservoirs is always characterized by rough surfaces and the roughness of the topography significantly influences the gas flow characteristics. However, most existing models are based on smooth surface. Moreover, the roughness effect leads to a complex velocity distribution and can influence the apparent permeability through its own transport mechanism. In this study, a generalized lattice Boltzmann method with nonequilibrium extrapolation boundary conditions was developed to simulate the rarefied shale gas flow through nanochannels with rough surface. Fractal geometry was introduced to model the rough surface, and the roughness was characterized by two parameters, namely, relative roughness height and fractal dimension. Triple-effect permeability estimation model was adopted to calculate the apparent permeability accounting for multiple roughness characteristics including surface tortuosity, hydraulic tortuosity, and local roughness. Effects of fractal dimension, relative roughness, and tortuosity on velocity distribution were investigated. The effect of pressure and roughness characteristic parameters on the apparent permeability at different Knudsen numbers was also studied. The results showed that the influence of fractal dimension and relative roughness on the permeability was affected by the rarefaction effect. The roughness effect on gas flow behavior became more evident when associated with the enhanced rarefaction effect. Moreover, relative roughness was found to play a significant role in determining the permeability in the transient flow regime.

Time domain transport equations of light in multilayered rectangular biological tissue with semi-infinite medium

In the study of tissue optics, physical models with the boundary of finite size are rarely established. Based on the diffusion equation, under the rectangular boundary conditions, this paper adopts extrapolation boundary conditions to establish a frequency domain model of multilayer medium diffusion equation, according to Fourier transform of frequency domain, turns the frequency domain solution into time domain model. On the basis of the deduced equation, the corresponding calculation process is written and the time resolved diffuse reflectance is calculated. The established model is compared with Monte Carlo simulation of time domain and the traditional medium with semi-infinite thickness, the result shows that our theory is correct. Concurrently, the effects of different boundary conditions on the diffusion equation are compared, it shows that our equations can not only solve the finite boundary problems, when the boundary is infinite, it can replace multilayer medium diffusion model.

Approximate P3 solution for the semi-infinite medium: steady state and time domain.

The steady-state solution of the Green's function obtained by the P3 equation in a semi-infinite medium is presented, the proposed solution is a diffusion-based model. Two time-domain solutions are established: one is the solution under extrapolation boundary condition, which we call the optical parameter method, and the other corresponds to the diffusion equation, which we call the double-diffusion coefficient method. The spatial-resolved reflectance and the time-resolved reflectance are calculated. The Monte Carlo simulation is used to verify the P3 equation. The results show that the P3 steady-state equation and the two time-domain equations are in good agreement with the Monte Carlo simulation. In the steady state, when the distance between the detector and the light source is less than several free paths, the P3 equation is more accurate than the diffusion equation. In other cases, the P3 model and the diffusion model have similar results. However, when the absorption coefficient is large, P3 is more accurate. In the time domain, the optical parameter method is more accurate, and the double-diffusion coefficient method is more consistent with the diffusion equation.

Sensitivity evaluation and selective plane imaging geometry for x-ray-induced luminescence imaging.

X-ray-induced luminescence (XIL) is a hybrid x-ray/optical imaging modality that employs nanophosphors that luminescence in response to x-ray irradiation. X-ray-activated phosphorescent nanoparticles have potential applications in radiation therapy as theranostics, nanodosimeters, or radiosensitizers. Extracting clinically relevant information from the luminescent signal requires the development of a robust imaging model that can determine nanophosphor distributions at depth in an optically scattering environment from surface radiance measurements. The applications of XIL in radiotherapy will be limited by the dose-dependent sensitivity at depth in tissue. We propose a novel geometry called selective plane XIL (SPXIL), and apply it to experimental measurements in optical gel phantoms and sensitivity simulations. An imaging model is presented based on the selective plane geometry which can determine the detected diffuse optical signal for a given x-ray dose and nanophosphor distribution at depth in a semi-infinite, optically homogenous material. The surface radiance in the model is calculated using an analytical solution to the extrapolated boundary condition. Y2 O3 :Eu3+ nanoparticles are synthesized and inserted into various optical phantom in order to measure the luminescent output per unit dose for a given concentration of nanophosphors and calibrate an imaging model for XIL sensitivity simulations. SPXIL imaging with a dual-source optical gel phantom is performed, and an iterative Richardson-Lucy deconvolution using a shifted Poisson noise model is applied to the measurements in order to reconstruct the nanophosphor distribution. Nanophosphor characterizations showed a peak emission at 611nm, a linear luminescent response to tube current and nanoparticle concentration, and a quadratic luminescent response to tube voltage. The luminescent efficiency calculation accomplished with calibrated bioluminescence mouse phantoms determines 1.06 photons were emitted per keV of x-ray radiation absorbed per g/mL of nanophosphor concentration. Sensitivity simulations determined that XIL could detect a concentration of 1mg/mL of nanophosphors with a dose of 1cGy at a depth ranging from 2 to 4cm, depending on the optical parameters of the homogeneous diffuse optical environment. The deconvolution applied to the SPXIL measurements could resolve two sources 1cm apart up to a depth of 1.75cm in the diffuse phantom. We present a novel imaging geometry for XIL in a homogenous, diffuse optical environment. Basic characterization of Y2 O3 :Eu3+ nanophosphors are presented along with XIL/SPXIL measurements in optical gel phantoms. The diffuse optical imaging model is validated using these measurements and then calibrated in order to execute initial sensitivity simulations for the dose-depth limitations of XIL imaging. The SPXIL imaging model is used to perform a deconvolution on a dual-source phantom, which successfully reconstructs the nanophosphor distributions.

Modified Outlet Boundary Condition Schemes for Large Density Ratio Lattice Boltzmann Models

Outlet boundary conditions (OBCs) and their numerical descriptions are critical to computational fluid dynamics (CFD) since they have significant influence on the numerical accuracy and stability. They present significant challenges to the two-phase lattice Boltzmann (LB) method, especially in the limit of large density ratio. In this study, three commonly used OBCs: convection boundary condition (CBC), Neumann boundary condition (NBC), and extrapolation boundary condition (EBC), are investigated and improved on basis of two LB models for large density ratios (single and double distribution function models). The existing numerical schemes for the OBCs are not directly applicable to the LB models because of the deviation of the momentum balance at the outlet boundary. The deviation becomes substantial at a large density ratio. Thus, in this work, modified OBC schemes are proposed to make the OBCs suitable for the two-phase LB models by adding an independent equation to obtain the outlet velocity. Numerical tests on droplet flowing in a channel are performed to evaluate the performance of the modified OBC schemes. Results indicate that the modified OBC schemes may be extended to tackle large density ratio situations. The modified NBC and EBC schemes are only suitable for the LB model with single distribution function. Three modified CBC schemes exhibit optimum performance for both single and double distribution function LB models which can be implemented for large density ratios.

A preliminary verification of the floating reference measurement method for non-invasive blood glucose sensing

In the non-invasive sensing of blood glucose by near-infrared diffuse reflectance spectroscopy, the spectrum is highly susceptible to the unstable and complicated background variations from the human body and the environment. In in vitro analyses, background variations are usually corrected by the spectrum of a standard reference sample that has similar optical properties to the analyte of interest. However, it is hard to find a standard sample for the in vivo measurement. Therefore, the floating reference measurement method is proposed to enable relative measurements in vivo, where the spectra under some special source-detector distance, defined as the floating reference position, are insensitive to the changes in glucose concentration due to the absorption effect and scattering effect. Because the diffuse reflectance signals at the floating reference positions only reflect the information on background variations during the measurement, they can be used as the internal reference. In this paper, the theoretical basis of the floating reference positions in a semi-infinite turbid medium was discussed based on the steady-state diffusion equation and its analytical solutions in a semi-infinite turbid medium (under the extrapolated boundary conditions). Then, Monte-Carlo (MC) simulations and in vitro experiments based on a custom-built continuous-moving spatially resolving double-fiber NIR measurement system, configured with two types of light source, a super luminescent diode (SLD) and a super-continuum laser, were carried out to verify the existence of the floating reference position in 5%, 10% and 20% Intralipid solutions. The results showed that the simulation values of the floating reference positions are close to the theoretical results, with a maximum deviation of approximately 0.3mm in 1100–1320nm. Great differences can be observed in 1340–1400nm because the optical properties of Intralipid in this region don not satisfy the conditions of the steady-state diffusion equation. For the in vitro experiments, floating reference positions exist in 1220nm and 1320nm under two types of light source, and the results are quite close. However, the reference positions obtained from experiments are further from the light source compared with those obtained in the MC simulation. For the turbid media and the wavelengths investigated, the difference is up to 1mm. This study is important for the design of optical fibers to be applied in the floating reference measurement.

Multi-component elastic reverse time migration based on the P- and S-wave separated velocity–stress equations

The vector P- and S-seismograms in an elastic common-shot gather generated with a P-wave source in a two-dimensional model using a finite difference solution of the P- and S-wave separated velocity–stress equations can be imaged by two independent ERTMs based on the same equations. The inputs as boundary conditions for reverse-time extrapolation are the pure vector P- and S-waves, respectively. The vector P-wave image components can be obtained by the normalized correlation operation between the vector P-source wavefields and receiver wavefields, which are obtained by extrapolating the vector P-seismograms in reverse-time using the same equations as the forward modeling. The vector S-wave image components can be obtained by a similar method. Compared with the conventional ERTM, this method can minimize the artifacts caused by the crosstalk between different wave modes and can preserve the phase and amplitude attributes of migration images very well. Furthermore, the polarity-reversal of the vector S-wave data can be corrected automatically during the imaging process, so destructive interferences between data from adjacent sources do not exist. Numerical examples with synthetic data have demonstrated the feasibility and effectiveness of this method for complex structures.

The Judgment of Garbage: End-of-Pipe Treatment and Waste Reduction

Many scholars have argued that systems for treating waste impede organizations from preventing waste in the first place. They theorize that end-of-pipe (EOP) treatment diminishes the incentive to avoid creating waste in the production process and obscures the information necessary to devise prevention techniques. This prediction has been widely accepted, influencing both policy and practice, despite both a lack of supporting empirical evidence and the existence of a counterprediction. In this paper, we use data describing U.S. manufacturing establishments from 1991 to 2005 to test the connection between EOP treatment and waste reduction. Our findings show that EOP treatment is associated with an initial jump in reported waste, followed by ongoing reduction. We analyze these results by exploring mechanisms that may drive this relationship. For practitioners, our paper provides critical guidance about strategies for reducing waste. For scholars of environmental management, our paper provides new insight on when facilities accomplish “source reduction” of process waste. For broader management theories of operations and organizational design, our analysis provides new insight on boundary conditions for extrapolation from existing theories. Finally, our paper provides new guidance for the formulation of effective regulatory policy. This paper was accepted by Christian Terwiesch, operations management.

Evaluation of outflow boundary conditions for two-phase lattice Boltzmann equation

Outflow boundary condition (OBC) is a critical issue in computational fluid dynamics. As a type of numerical method for fluid flows, the lattice Boltzmann equation (LBE) method has gained much success in a variety of complex flows, and certain OBCs have been suggested for the LBE in simulating simple single-phase flows. However, very few discussions on the OBCs have been made for the two-phase LBE method. In this work, three types of OBCs that are widely used in the LBE for single-phase flows, i.e., the Neumann boundary condition, the convective boundary condition, and the extrapolation boundary condition, are extended to a two-phase LBE method and their performances are investigated. The comprehensive results of several two-phase flows show that these boundary conditions behave quite differently in the simulations of two-phase flows. Specifically, it is found that the Neumann boundary condition and the extrapolation boundary condition give rather poor predictions, while the type of convective boundary conditions work well, although the choice of the convection velocity has some slight influences on the results. We also apply these OBC schemes to some other two-phase models, and similar observations are found.

Photon diffusion in a homogeneous medium bounded externally or internally by an infinitely long circular cylindrical applicator V Steady-state fluorescence

As Part V in our series, this paper examines steady-state fluorescence photon diffusion in a homogenous medium that contains a homogenous distribution of fluorophores, and is enclosed by a "concave" circular cylindrical applicator or is enclosing a "convex" circular cylindrical applicator, both geometries being infinite in the longitudinal dimension. The aim is to predict by analytics and examine with the finite-element method the changing characteristics of the fluorescence-wavelength photon-fluence rate and the ratio (sometimes called the Born ratio) of it versus the excitation-wavelength photon-fluence rate, with respect to the source-detector distance. The analysis is performed for a source and a detector located on the medium-applicator interface and aligned either azimuthally or longitudinally in both concave and convex geometries. When compared to its steady-state counterparts on a semi-infinite medium-applicator interface with the same line-of-sight source-detector distance, the fluorescence-wavelength photon-fluence rate reduces faster along the longitudinal direction and slower along the azimuthal direction in the concave geometry, and conversely in the convex geometry. However, the Born ratio increases slower in both azimuthal and longitudinal directions in the concave geometry and faster in both directions in the convex geometry, respectively, when compared to that in the semi-infinite geometry.

Analytical solutions for diffuse fluorescence spectroscopy/imaging in biological tissues Part I: zero and extrapolated boundary conditions

The mathematical model for diffuse fluorescence spectroscopy/imaging is represented by coupled partial differential equations (PDEs), which describe the excitation and emission light propagation in soft biological tissues. The generic closed-form solutions for these coupled PDEs are derived in this work for the case of regular geometries using the Green's function approach using both zero and extrapolated boundary conditions. The specific solutions along with the typical data types, such as integrated intensity and the mean time of flight, for various regular geometries were also derived for both time- and frequency-domain cases.

Analytical solutions for diffuse fluorescence spectroscopy/imaging in biological tissues Part II: comparison and validation

The analytical solutions for the coupled diffusion equations that are encountered in diffuse fluorescence spectroscopy/imaging for regular geometries were compared with the well-established numerical models, which are based on the finite element method. Comparison among the analytical solutions obtained using zero boundary conditions and extrapolated boundary conditions (EBCs) was also performed. The results reveal that the analytical solutions are in close agreement with the numerical solutions, and solutions obtained using EBCs are more accurate in obtaining the mean time of flight data compared to their counterpart. The analytical solutions were also shown to be capable of providing bulk optical properties through a numerical experiment using a realistic breast model.

SYNOPTIC MAPPING OF CHROMOSPHERIC MAGNETIC FLUX

We used daily full-disk Ca II 854.2?nm magnetograms from the Synoptic Optical Long Term Investigations of the Sun (SOLIS) facility to study the chromospheric magnetic field from 2006 April through 2009 November. We determined and corrected previously unidentified zero offsets in the SOLIS magnetograms. By tracking the disk passages of stable unipolar regions, the measured net flux densities were found to systematically decrease from the disk center to the limb by a factor of about two. This decrease was modeled using a thin flux tube model with a difference in signal formation height between the center and limb sides. Comparison of photospheric and chromospheric observations shows that their differences are largely due to horizontal spreading of magnetic flux with increasing height. The north polar magnetic field decreased nearly linearly with time during our study period while the south polar field was nearly constant. We used the annual change in the viewing angle of the polar regions to estimate the radial and meridional components of the polar fields and found that the south polar fields were tilted away from the pole. Synoptic maps of the chromospheric radial flux density distribution were used as boundary conditions for extrapolation of the field from the chromosphere into the corona. A comparison of modeled and observed coronal hole boundaries and coronal streamer positions showed better agreement when using the chromospheric rather than the photospheric synoptic maps.

EVOLUTION OF RELATIVE MAGNETIC HELICITY AND CURRENT HELICITY IN NOAA ACTIVE REGION 11158

Both magnetic and current helicities are crucial ingredients for describing the complexity of active-region magnetic structure. In this Letter, we present the temporal evolution of these helicities contained in NOAA active region 11158 during five days from 2011 February 12 to 16. The photospheric vector magnetograms of the Helioseismic and Magnetic Imager on board the Solar Dynamic Observatory were used as the boundary conditions for the coronal field extrapolation under the assumption of nonlinear force-free field, from which we calculated both relative magnetic helicity and current helicity. We construct a time-altitude diagram in which altitude distribution of the magnitude of current helicity density is displayed as a function of time. This diagram clearly shows a pattern of upwardly propagating current helicity density over two days prior to the X2.2 flare on February 15 with an average propagation speed of {approx}36 m s{sup -1}. The propagation is synchronous with the emergence of magnetic flux into the photosphere, and indicative of a gradual energy buildup for the X2.2 flare. The time profile of the relative magnetic helicity shows a monotonically increasing trend most of the time, but a pattern of increasing and decreasing magnetic helicity above the monotonic variation appears prior tomore » each of two major flares, M6.6 and X2.2, respectively. The physics underlying this bump pattern is not fully understood. However, the fact that this pattern is apparent in the magnetic helicity evolution but not in the magnetic flux evolution makes it a useful indicator in forecasting major flares.« less