Stony Brook AMS - Downloadable Preprints, 1996

A shock driven material interface between two fluids of different density is unstable. This instability is known as Richtmyer-Meshkov (RM) instability. In this paper, we present a quantitative nonlinear theory of compressible Richtmyer-Meshkov instability in two dimensions. Our nonlinear theory contains no free parameter and provides analytical predictions for the overall growth rate, as well as the growth rates of the bubble and spike, from early to later times for fluids of all density ratios. The theory also includes a general formulation of perturbative nonlinear solutions for incompressible fluids (evaluated explicitly through the fourth order). Our theory shows that the RM unstable system goes through a transition from a compressible and linear one at early times to a nonlinear and incompressible one at later times. Our theoretical predictions are in excellent agreements with the

The material interface between two fluids of different densities is unstable under acceleration by a shock wave. This phenomenon is known as the Richtmyer-Meshkov instability. Theories have failed to provide a quantitatively correct predictions for the growth rates of the unstable interface. Recently the authors have developed a quantitative theory based on the methods of Pade\h'-0.4m'\(aa approximations and of asymptotic matching. In this letter, we extend our theory to the growth rates of the spike and bubble for the systems in two and three dimensions, and for systems with or without phase inversions. Our theoretical predictions are in excellent agreement with the results of full numerical simulations over the full time period from the initial linear (small amplitude) to moderately large amplitude nonlinear regimes.

A material interface between two fluids of different density accelerated by a shock wave is unstable. This instability is known as Richtmyer-Meshkov (RM) instability. Previous theoretical and numerical studies primarily focused on fluids in two dimensions. In this paper, we present the studies of Richtmyer-Meshkov instability in three dimensions in rectangular coordinates. There are three main results: (1) The analysis of linear theory of Richtmyer-Meshkov instability for both reflected shock and reflected rarefaction wave cases. (2) A general formulation of perturbative nonlinear solutions for incompressible RM instability (evaluated explicitly for the impulsive model through the third order). (3) A quantitative nonlinear theory of compressible Richtmyer-Meshkov instability from early to later times. Our nonlinear theory contains no free parameter and provides analytical predictions for the overall growth rate, as well as the growth rates of bubble and spike, of Richtmyer-Meshkov unstable interfaces.

Front Tracking is the combination of two distinct computational traditions. Its purpose is to enable high quality, relatively coarse grid mesh solutions for problems complicated by the presence of fronts, interfaces, and other discontinuities, which may be of considerable geometric complexity. The first of the two algorithmic ideas used in front tracking is upwind differencing, for fluids, materials, and other conservation law based problems arising in physics. The second is computational geometry, from which we draw methods to represent, track, and process the geometric structures defined by the discontinuities. This combination, which is called front tracking, is by now a mature capability in two dimensions, and has been used to achieve definitive solutions to a number of problems which have otherwise resisted solution for decades. Here we report on recent extensions of front tracking to three dimensions and we also present highlights from some recent tw o dimensional front tracking computations.

Front tracking is a computational method in which interfaces or embedded geometrical surfaces are given explicit computational degrees of freedom. It is thus a hybrid method, mixing CFD with computational geometry. We consider here three applications of Front Tracking to the computations of dynamically evolving surfaces. The first two are fluid instabilities, the Rayleigh-Taylor (RT) instability of steady acceleration of a fluid density discontinuity interface, and the related Richtmyer-Meshkov (RM) instability of impulsive acceleration of such interfaces. Our third test problem is surface evolution governed by the Hamilton-Jacobi (HJ) equation, to model the deposition or etching of surfaces in the manufacture of semiconductor chips.

Strong evidence is presented for a renormalization group (RNG) fixed point for the RT mixing problem, including a closed form solution of the RNG equations. Recently developed three dimensional computations for the RT problem are presented here. The RM Front Tracking computations have been validated by comparison to laboratory experiments, agreement with newly derived theories (also presented here), and agreement with known solution symmetries. The HJ surface evolution computations are the first, to the authors' knowledge, for this problem using surface based methodology and with a method to handle surface bifurcations and topology changes.

We present a short introduction to continuum models for the plastic flow of metals. Our emphasis is on the physical principles underlying these models, the nature and validity of approximations involved, and the mathematical structure of the flow equations. Using the framework developed, we derive a simple, but realistic, model describing one-dimensional plastic flow.

Mat. Contemp., vol. 11, 1996, pp. 95-120.

The Hamilton-Jacobi equation describes the dynamics of a hypersurface in R^n. This equation is a nonlinear conservation law and thus has discontinuous solutions. The dependent variable is a surface gradient and the discontinuity is a surface cusp. Here we investigate the intersection of cusp hypersurfaces. The se intersections define (n-1)-dimensional Riemann problems for the Hamilton-Jacobi equation. We propose the class of Hamilton-Jacobi equations as a natural higher-dimensional generalization of scalar equations which allow a satisfactory theory of higher-dimensional Riemann problems. The first main result of this paper is a general framework for the study of higher-dimensional Riemann problems for Hamilton-Jacobi equations. The purpose of the framework is to understand the structure of Hamilton-Jacobi wave interactions in an explicit and constructive manner. Specialized to two-dimensional Riemann problems (i.e., the intersection of cusp curves on surfaces imbedded in R^3), this framework provides explicit solutions to a number of cases of interest. We are specifically interested in models of deposition and etching, important processes for the manufacture of semiconductor chips.

We also define elementary waves as Riemann solutions which possess a common group velocity. Our second main result, for elementary waves, is a complete characterization in terms of algebraic constraints on the data. When satisfied, these constraints allow a consistently defined closed form expression for the solution. We also give a computable characterization for the admissibility of an elementary wave which is inductive in the codimension of the wave, and which generalizes the classical Oleinik condition for scalar conservation laws in one dimension.

In this paper, we study the shock driver Richtmeyer-Meskov (RM) instability in three dimensions using a high resolution numerical method, the TVD scheme with artificial compression. We study the RM instability in three physical and geometrical configurations: the instability driven by a single transit shock, the plane re-shock by a reflecting wall and the sphereical re-shock problem. We report the eistence of a threshold between a stable and an unstable interface after the second shock transition.

Front Tracking is a methodology which uses computational geometry to provide numerical solutions of enhanced quality for problems of computational physics. It is particularly applicable to the computation of solutions with important jump discontinuities, or fronts.

Fronts are represented as lower dimensional data structures, which in the spirit of non-manifold geometries used in CAD systems, support surfaces, curves (at surface intersections) and nodes (at common intersection points of three surfaces). These are expressed in the front tracking algorithm as a simplicial complex: data structures representing non intersecting smooth objects of given dimension, and boundary and coboundary operators to link them to lower and higher dimensional smooth objects. In the solution to problems of computational physics, these fronts evolve and can change their topology, forcing the need for fast retriangulation, intersection detection, and intersection removal routines.

The methodology has been applied to a variety of problems in computational physics, including fluid dynamics, elasto-plastic deformations, deposition and etching for manufacture of semiconductors, and flow in porous media. Characteristically, this method gives solutions of improved accuracy and convergence properties.

The main results presented here are the existence of both two- and three-dimensional implementations of these ideas. In other words, we show that the computational complexities inherent in this approach can be overcome, and the computational advantages realized. The two-dimensional version (moving curves in ${\Bbb R}^2$) is by now highly mature, and can handle very complex geometries, with bifurcation of topology where the front remains physical at all times. In this case, bifurcation issues for a number of physical problems have been completely resolved. Our algorithm for bifurcation extends to three dimensions, where it has been partially implemented.

This article proposes tree-structured logistic regression modeling for over-dispersed binomial data. Recursive partitioning is performed using a combination of statistical tests and residual analysis. The splitting criterion in cross-validation is based on the deviance function. A nested grid algorithm to estimate the bootstrap parameters is developed. The regression tree procedure provides a new approach to explore the relationship between the binomial response and explanatory variables in detail. The proposed procedure is applied to model the relationship between the incidence of malformation, and dose and fetal weight using data from a developmental experiment conducted at the National Center for Toxicological Research. A conditional Gaussian chain model is used to account for the effect of fetal weight by dose.

Resin Transfer Molding (RTM) has drawn interest in recent years as an attractive technique for the manufacture of advanced fiber reinforced composite materials. A major issue in this new manufacturing process is the reduction of voids during the resin fill process so that products with high quality are manufactured. Process modeling is particularly useful in understanding, designing, and optimizing the process conditions. The purpose of this paper is to illustrate the important application of mathematical and numerical modeling to this industrial problem. First, an overview of the RTM process, its manufacturing problems,and related b ackground issues is given. A survey of various RTM models developed in recent years by researchers in this field are then presented. Finally, as an application, a novel two phase flow model, developed recently by the authors, is proposed to study the formation and migration of the macro voids,a major manufacturing problem. The unique feature of this model is the identification of local pressure as a major mobilization factor of these macro voids. It is demonstrated that the model is in good agreement with experimental results.

Materials processing systems, such as the Czochralski crystal growth system, are often characterized by the presence of a number of distinct materials and phases with significantly different thermophysical and transport properties. They may also contain irregular boundaries, moving interfaces and free surfaces. The understanding of the complex transport phenomena in these systems is of vital importance for the design and fabrication of equipment and the optimization and control of the manufacturing process. High performance, high resolution numerical simulation can prove to be an effective tool for the understanding of these transport mechanisms. Massively parallel computers promise to deliver the extensive computer resources required by these simulations. This paper presents a parallel implementation of a high resolution numerical scheme which has been developed to simulate the Czochralski crystal growth processes. The scheme employs adaptive grid generation and curvilinear finite volume discretization to solve the transport equations in a domain with complex geometries. Selected results are presented to demonstrate the feasibility and potential of introducing parallel computations into crystal growth process modeling and simulation.

The central content and purpose of this paper is to explain the scientific necessity for stochastic models, and certain key mathematical and theoretical issues which need to be addressed in this area in order to achieve the scientific goals proposed here. These two ideas occupy the first two sections of this paper. In the next two sections of this paper, we illustrate these ideas by examining them from the point of view of fluid mixing. Here the discussion becomes more specific, and is based on properties of solutions of the Euler and Navier-Stokes equations, Darcy's law, the Buckley-Leverett equations and related equations. Because the scientific issues which we examine concern deep interactions between the nonlinear and stochastic aspects of the behavior of solutions of partial differential equations, neither of which is presently in a definitive or final stage, we appeal to a variety of methods for scientific understanding: theoretical analysis, numerical simulations, and analysis of experimental and field data.