Stony Brook AMS - Downloadable Preprints, 1999

Both a mixture likelihood method and the EM algorithm are implemented to estimate the time-to-onset-of and the time-to-death-from the tumor of interest in animal carcinogenicity studies. Both methods are implemented using Box's Complex Method for finding the maximum likelihood estimate of parameters for a nonlinear log-likelihood function subject to nonlinear inequality constraints. A comparison of the mixture likelihood method with the EM algorithm suggests that the mixture method may be more efficient for the problem of constrained nonparametric maximum likelihood estimation in carcinogenicity studies. The advantages of using the mixture likelihood method are illustrated with data from benzidine dihydrochloride and caloric restriction studies.

A new statistical approach is developed for estimating the carcinogenic potential of drugs and other chemical substances used by humans. Improved statistical methods are developed for rodent tumorigenicity assays that have interval sacrifices but not cause-of-death data. For such experiments, this paper proposes a nonparametric maximum likelihood estimation method for estimating the distributions of time-to-onset-of and the time-to-death-from the tumor. The log-likelihood function is optimized using a constrained direct-search procedure. Using the maximum likelihood estimators, the number of fatal tumors in an experiment can be imputed. A Monte Carlo Simulation study is conducted for illustration. The accuracy of cause of death attributed by the proposed method tends to increase as the competing risks survival rate increases. By applying the proposed procedure to a real data set, the effect of calorie restriction is investigated. In this study, we found calorie restriction delays the tumor-onset time significantly for pituitary tumors. The present method can result in substantial economic savings by relieving the need for case-by-case assignment of cause of death or context of observation by pathologists. The ultimate goal of the proposed method is to use the imputed number of fatal tumors to modify Peto's IARC test for application to tumorigenicity assays that lack cause-of-death data.

In immiscible three-phase flow, the lead oil bank can split into two, a Buckley-Leverett shock wave followed by a new type of shock wave. Such a nonclassical "transitional" shock wave is common in three-phase flow. Its sensitivity to diffusion implies that capillary pressure must be modeled correctly in order to calculate the flow. In particular, transitional waves arise in WAG flow. They can be calculated by semi-analytic methods, which are helpful in the design of effective WAG recovery strategies.

Radiative heat transfer plays an important part in the Czochralski crystal growth process, but it is difficult to simulate numerically because of the amount of computational time required. This paper describes the parallelization of a code simulating radiative heat transfer in the crystal and in the region above the melt containing gas under the assumption of axial symmetry. This code allows arbitrary grid refinements of the various material interfaces. The code can be used to study the effect of radiative heat transfer near the melt-crystal-gas interface. The results demonstrate how the accuracy of the flux at the interfaces depends on the grid resolution, and parallelization is required to obtain the higher grid resolutions. This paper presents numerous steps which have been taken to validate the code. Timing and the parallel efficiency studies indicate that this code makes efficient use of the computational resources.

New simulation studies, modeling, and theoretical results of the authors and collaborators are presented here. Three-dimensional simulations of the Rayleigh-Taylor instability, carried to late time, give a growth rate h(t) ~ alpha_b*A*g*t^2 with alpha_b ~ 0.08. This value of alpha_b is high but probably consistent with experimental data due to finite compressibility effects. Our modeling has focused on the internal structure of the mixing zone, expressed in terms of the edge velocities. A Chunk Mix model with few adjustable parameters has been proposed, which moreover is mathematically stable, does not require overly smoothed velocities to suppress instabilities, and which avoids thermodynamic complications of atomically mixed or pressure equilibrated equations of state. Our theoretical analysis of the edge velocities has two components: A bubble merger model to obtain a physics based zero parameter determination of alpha_b in agreement with experiment and an analysis based on the center of mass (COM) to predict the ratio alpha_s/alpha_b, again in agreement with experimental data.

The statistical evolution of a planar, randomly perturbed interface subject to Rayleigh-Taylor instability is explored through numerical solution of the Euler equations in two space dimensions. The data set, generated by the front-tracking code {\em FronTier}, is highly resolved and covers a large ensemble of initial perturbations, allowing a more refined analysis of closure issues pertinent to the stochastic modeling of chaotic fluid mixing. We closely approach a two-fold convergence of mean two-phase properties: convergence of the numerical solution under computational mesh refinement, and statistical convergence under increasing ensemble size.

We report results of a numerical study of Richtmyer-Meshkov instability in imploding cylindrical geometry. Numerical solutions are obtained from two different Eulerian hydrodynamics codes, FronTier and RAGE. We consider both single-mode and multi-mode interface perturbations and demonstrate that in each case both codes agree in the mixing zone growth and in the primary features of the interpenetrating bubbles and spikes.

A two-phase flow model for an acceleration-driven compressible fluid mixing layer is applied to an initially planar/cylindrical/spherical fluid configuration. A conservative form of the one-dimensional compressible equations is derived under the assumption that the volume fraction is continuous. With a hyperbolic equation for the effective interface velocity, the model supports traveling discontinuities in the fluid concentration gradient. The primary examples of this wave type are the moving boundaries of a finite mixing layer, which determine the instability growth rate. Closures for interfacial terms, previously derived for planar incompressible mixing, are re-interpreted and shown to be applicable to other one-dimensional mixing problems of interest. The equations of motion for an incompressible mixing layer in planar, cylindrical, or spherical geometry are solved exactly, up to a history integral of a function of the edge trajectories, and without assuming incompressible flow outside the layer. Full solutions are obtained by numerically integrating a coupled system of ordinary differential equations for the volume fraction characteristics. Results for self-similar Rayleigh-Taylor mixing in planar geometry are compared to the work of others. This comparison suggests that the shape of the fluid concentration profile is primarily a consequence of mass conservation, parameterized by the expansion ratio of the mixing zone edges. Thus, predicting this quantity is not a stringent test of the modeling assumptions.

An attractive approach for simulation of solid dynamics is to combine an Eulerian finite difference method with material interface tracking. The fixed Eulerian computational mesh is not subject to mesh distortion, and the tracking eliminates spurious numerical diffusion at interfaces and the need for mixed-material computational cells. We have developed such an approach within the framework of the front tracking method, as implemented in the FronTier code. Our two-dimensional solid dynamics code is based on a fully conservative formulation of the governing equations for large-strain deformation, a hyperelastic equation of state that allows for large volumetric change, and a rate-dependent plasticity model for high strain rates; it features conservative finite differencing, a Riemann solver that accounts for the nonlinearity of longitudinal waves, and an implicit method for integrating the plastic source term. This paper presents an overview of the FronTier Solid code and some preliminary applications to high-velocity impact and shock-accelerated interface problems.

In this paper we formulate a multiphase model with nonequilibrated temperatures but with equal velocities and pressures for each species. Turbulent mixing is driven by diffusion in these equations. The closure equations are defined in part by reference to a more exact chunk mix model developed by the authors and coworkers which has separate pressures, temperatures, and velocities for each species. There are two main results in this paper. The first is to identify a thermodynamic constraint, in the form of a process dependence, for pressure equilibrated models. The second is to determine the diffusion coefficients needed for the closure of the equilibrated pressure multiphase flow equations, in the incompressible case. The diffusion coefficients depend on entrainment times, derived from the chunk mix model. These entrainment times are determined here first via general formulas and then explicitly for Rayleigh-Taylor and Richtmyer-Meshkov large time asymptotic flows. We also determine volume fractions for these flows, using the chunk mix model.

We report on direct measurement of flow-relevant geometrical properties of the void space in a suite of 4 samples of Fontainebleau sandstone ranging from 7.5% to 22% porosity. We present measured distributions of coordination number, channel length, throat size and pore volume, and of correlations between throat-size/pore-volume and nearest neighbor pore-volume/pore volume determined for these samples. We also present quantitative characterization of the distributions measured. The measurements are obtained from computer analysis of three dimensional, synchrotron X-ray computed microtomographic images. The effects of finite sample volume are investigated. The accuracy of the numerical algorithms employed is investigated using a simulated image of hexagonal closed packed spheres.

Recent simulations of Rayleigh-Taylor instability growth rates display considerable spread. We propose that differences in numerical dissipation effects (mass diffusion and viscosity) due to algorithmic differences and differences in simulation duration are the dominant factors that produce such different results. Within the simulation size and durations explored here, we have explained principal discrepancies as due to numerical dispersion through comparison of simulations using different algorithms. Furthermore, we have tentatively identified viscosity as having the larger role of these two dissipative effects over the time range examined here. We present new 3D front tracking simulations that show agreement with the range of reported experimental values. We begin an exploration of new physical length scales, that may characterize a transition to a new Rayleigh-Taylor mixing regime.

The statistical evolution of a planar, randomly perturbed fluid interface subject to Rayleigh-Taylor instability is explored through numerical simulation in two space dimensions. The data set, generated by the front-tracking code FronTier, is highly resolved and covers a large ensemble of initial perturbations. We closely approach a two-fold convergence of the mean two-phase flow: convergence of the numerical solution under computational mesh refinement, and statistical convergence under increasing ensemble size. Quantities that appear in the two-phase averaged Euler equations are computed directly and analyzed for numerical and statistical convergence. Bulk averages show a high degree of convergence, whereas interfacial averages are convergent only in the outer portions of the mixing zone, which are comprised of coherent arrays of bubbles and mushrooming jets. Comparison with the familiar bubble/spike penetration law $h=\alpha Agt^2$ is complicated by the lack of scale invariance, inability to carry the simulations to late time, the increasing Mach numbers of the bubble/spike tips, and sensitivity to the method of data analysis. Finally, we use the simulation data to analyze some constitutive properties of turbulent mixing layers.

The rate of expansion of a planar, incompressible two-fluid mixing layer subject to a time-dependent acceleration $g(t)$ has been accurately measured by Dimonte and Schneider for several different $g(t)$ and a wide range of density ratio. Their measurements can be replicated by solving a phenomenological force law for the interpenetrating bubbles and spikes that comprise these mixing layers. The coefficients in this model are calibrated with respect to the asymptotic growth rates for impulsive and constant accelerations, and are so far successful in predicting bubble and spike growth for other acceleration histories. We use this information to analyze the inertial and buoyant effects that are implied by the experimental data and the known physics of the coherent structures comprising the mixing layer.

We study theoretically a potential motion of fluid with a free boundary and with no external forces. We derive and integrate explicitly the equations describing the flow evolution from the initial to a highly non-linear stage in 2D and 3D cases. We study the influence of initial conditions in a wide range of parameters, and we show that the structure of bubbles and jets is determined by the distribution of the initial velocity. The local dynamics of highly symmetric 3D flows has a universal form when expressed in dimensionless units. At a fixed length scale, the 3D solutions depend strongly on the flow symmetry. Bubbles and jets conserve a near-circular contour, while a 2D well-developed structure would tend to break into a 3D structure. We show that the dependence on initial conditions can result in non-uniqueness of late-time asymptotes of the regular bubble motion.

We study theoretically steady flows generated by the Rayleigh-Taylor instability. Three-dimensional motion of bubbles and jets is assumed to be periodic in the plane and the flow is invariant with respect to the symmetry group of rectangle p2mm. For an incompressible inviscid fluid there is a family of steady solutions, and local stability analysis for the solutions is performed analytically over the whole range of the family parameters. We show that low-symmetric steady flows are unstable. 3D bubbles in RTI tend to conserve a near-circular contour and cannot be transformed into 2D bubbles continuously. We discuss the dynamics of the bubbles from the initial perturbation to the turbulent regime.

The 3DMA code is designed to provide statistical analysis of the geometrical distribution of the phases in a two or three dimensional image of a bi-phase material. The code has been applied to the analysis of void and grain phases of rock, cellulose fiber networks, fish movement, and the structure of neurons. For literature studies based upon the 3DMA code see references \cite{LLCJS96,LinVen99,LVDW99}. This guide provides general installation and execution instructions for the 3DMA code, version Dec. 1999. In addition to general algorithms employed in all four applications, algorithms to support specialized computation for each of these applications are also contained in the Dec. 99 version. This manual describes only the general algorithms and those designed for applications analogous to that for rock microstructure.

Numerical simulations for Rayleigh-Taylor (RT) and Richtmyer-Meshkov (RM) instabilities for three dimensional axisymmetric fluids have been successfully conducted by a front tracking method. In the first simulation, we consider single mode RT instability for an axially symmetric perturbed interface in a circular tube. We compare the computed bubble velocity with theory and laboratory experiment and find excellent agreement. The second simulation studies the RM instability for an axisymmetric sphere driven by an imploding shock. We investigate the growth rate of perturbation, the reshock process and the phase inversion phenomenon.

In this paper, we establish a Front Tracking method to solve Rayleigh-Taylor (RT) and Richtmyer-Meshkov (RM) Instabilities in axi-symmetric cylindrical and spherical geometries. Validation is carried out by comparing the computed single mode bubble velocity with various theoretical models and experimental results. We also validate our results by comparing three different front propagation algorithms, mesh refinement and the comparison of the asymptotic limit, as the minimum radius r_min tends to infinity, to a pure planar computation in two dimensions. In the cylindrical RT simulations, we study the influence of the geometry on the bubble velocities. We achieve convergence of bubble velocities as the minimum radius r_min tends to zero. We observe an interesting monotonic dependence of the bubble velocity on r_min. For the RM simulations, we perform a detailed study of the growth rate of fingers at an unstable shell driven by an imploding spherical shock. A qualitative understanding of this system has been achieved. We observe that the rotational or translational symmetry of initial axi-symmetric perturbations will not be preserved in our curved geometry simulations. These symmetries of the planar Euler equations are not symmetries of the axi-symmetric equations due to the symmetry breaking presence of the radial source term. This symmetry breaking is thus a distinctive feature of axi-symmetric flows, not present in the case of rectangular geometry.