Abstract:   Mullins, in a series of papers, developed a surface dynamics for phase interfaces whose evolution is controlled by mass diffusion within the interface. It is our purpose here to embed Mullin's theory within a general framework based on balance laws for mass and capillary forces in conjunction with a version of the second law, appropriate to a purely mechanical theory, which asserts that the rate at which the free energy increases cannot be greater than the energy inflow plus the power supplied. We develop an appropriate constitutive theory, and deduce general and approximate equations for the evolution of the interface.

Abstract:   Mullins, in a series of papers, developed a surface dynamics for phase interfaces whose evolution is controlled by mass diffusion within the interface. It is our purpose here to embed Mullin's theory within a general framework based on balance laws for mass and capillary forces in conjunction with a version of the second law, appropriate to a purely mechanical theory, which asserts that the rate at which the free energy increases cannot be greater than the energy inflow plus the power supplied. We develop an appropriate constitutive theory, and deduce general and approximate equations for the evolution of the interface.

Abstract:   Mullins, in a series of papers, developed a surface dynamics for phase interfaces whose evolution is controlled by mass diffusion within the interface. It is our purpose here to embed Mullin's theory within a general framework based on balance laws for mass and capillary forces in conjunction with a version of the second law, appropriate to a purely mechanical theory, which asserts that the rate at which the free energy increases cannot be greater than the energy inflow plus the power supplied. We develop an appropriate constitutive theory, and deduce general and approximate equations for the evolution of the interface.

Abstract:   Mullins, in a series of papers, developed a surface dynamics for phase interfaces whose evolution is controlled by mass diffusion within the interface. It is our purpose here to embed Mullin's theory within a general framework based on balance laws for mass and capillary forces in conjunction with a version of the second law, appropriate to a purely mechanical theory, which asserts that the rate at which the free energy increases cannot be greater than the energy inflow plus the power supplied. We develop an appropriate constitutive theory, and deduce general and approximate equations for the evolution of the interface.

Abstract:   Mullins, in a series of papers, developed a surface dynamics for phase interfaces whose evolution is controlled by mass diffusion within the interface. It is our purpose here to embed Mullin's theory within a general framework based on balance laws for mass and capillary forces in conjunction with a version of the second law, appropriate to a purely mechanical theory, which asserts that the rate at which the free energy increases cannot be greater than the energy inflow plus the power supplied. We develop an appropriate constitutive theory, and deduce general and approximate equations for the evolution of the interface.

Abstract:   Mullins, in a series of papers, developed a surface dynamics for phase interfaces whose evolution is controlled by mass diffusion within the interface. It is our purpose here to embed Mullin's theory within a general framework based on balance laws for mass and capillary forces in conjunction with a version of the second law, appropriate to a purely mechanical theory, which asserts that the rate at which the free energy increases cannot be greater than the energy inflow plus the power supplied. We develop an appropriate constitutive theory, and deduce general and approximate equations for the evolution of the interface.

Abstract:   Mullins, in a series of papers, developed a surface dynamics for phase interfaces whose evolution is controlled by mass diffusion within the interface. It is our purpose here to embed Mullin's theory within a general framework based on balance laws for mass and capillary forces in conjunction with a version of the second law, appropriate to a purely mechanical theory, which asserts that the rate at which the free energy increases cannot be greater than the energy inflow plus the power supplied. We develop an appropriate constitutive theory, and deduce general and approximate equations for the evolution of the interface.

Abstract:   Because the stress resulting from compositional inhomogeneities are long range, the local stress, diffusional flux and equilibrium conditions at a point depend on the entire composition distribution in a specimen. For a thin plate with a one-dimensional composition profile, this dependence is simple; the local stress depends on the local composition and on both the average composition and the first moment of the composition profile, neither of which are local. A theory of diffusion and equilibrium in a thin plate is developed, based on a free energy that depends on composition, its gradients and strain, and has a term for chemical effects at the plate boundary. Under certain assumptions, a standard diffusion equation is derived, with all of the non-local stress effects in the boundary conditions. Solutions are altered by these new conditions. Spontaneous bending is often a natural result of diffusion.

Abstract:   Because the stress resulting from compositional inhomogeneities are long range, the local stress, diffusional flux and equilibrium conditions at a point depend on the entire composition distribution in a specimen. For a thin plate with a one-dimensional composition profile, this dependence is simple; the local stress depends on the local composition and on both the average composition and the first moment of the composition profile, neither of which are local. A theory of diffusion and equilibrium in a thin plate is developed, based on a free energy that depends on composition, its gradients and strain, and has a term for chemical effects at the plate boundary. Under certain assumptions, a standard diffusion equation is derived, with all of the non-local stress effects in the boundary conditions. Solutions are altered by these new conditions. Spontaneous bending is often a natural result of diffusion.

Abstract:   We have found that when suprathreshold cathodal stimuli were applied to the epicardium of canine ventricle, impulse propagation originated at a "virtual cathode" with dimensions greater than those of the physical cathode. We report the two- dimensional geometry of the virtual cathode as a function of stimulus strength; the results are compared with the predictions of an anisotropic, bidomain model of cardiac conduction recently developed in our laboratories. Data were collected in six pentobarbital-anesthetized dogs by using a small plaque electrode sewn to the left ventricular epicardium. Arrival times at closely spaced bipolar electrodes oriented radially around a central cathode were obtained as a function of stimulus strength and fiber orientation. The dimensions of the virtual cathode were determined by linear back- extrapolation of arrival times to the time of stimulation. The directional dependence of the conduction velocity was consistent with previous reports: at 1 mA, longitudinal (0- degrees) and transverse (90-degrees) velocities were 0.60 +/- 0.03 and 0.29 +/- 0.02 m/sec, respectively. At 7 mA, the longitudinal velocity was 0.75 +/- 0.05 m/sec, whereas there was no significant change in the transverse velocity. In contrast to conduction velocity, the virtual cathode was smallest in the longitudinal orientation and largest between 45-degrees and 60-degrees. Virtual cathode size was dependent on both orientation and stimulus strength: at 0-degrees, the virtual cathode was small (approximately 1 mm) and relatively constant over the range of 1-7 mA; at oblique orientations (45- degrees-90-degrees), it displayed a roughly logarithmic dependence on stimulus strength, approximately 1 mm at 1 mA and approximately 3 mm at 7 mA. The bidomain, anisotropic model reproduced both the stimulus strength and the fiber-orientation dependence of the virtual cathode geometry when the intracellular and extracellular anisotropies were 10:1 and 4:1, respectively, but not when the two anisotropies were equal. We suggest that the virtual cathode provides a direct measure of the determinants of cardiac activation; its complex geometry appears to reflect the bidomain, anisotropic nature of cardiac muscle.

Abstract:   Simple inequalities are presented for the viscosity solution of a Hamilton-Jacobi equation in N space dimension when neither the initial data nor the Hamiltonian need be convex (or concave). The initial data are uniformly Lipschitz and can be written as the sum of a convex function in a group of variables and a concave function in the remaining variables, therefore including the nonconvex Riemann problem. The inequalities become equalities wherever a "maxmin" equals a "minmax" and thus a representation formula for this problem is then obtained, generalizing the classical Hopf's formulas.

Abstract:   Limit analysis of prismatic torsion bars was the earliest attempt to apply plasticity theory to a continuum. The simplicity of the problem made it feasible to use the two- dimensional Prandtl stress function, defined for the elastic torsion problems, for the plastic stress distributions as well. The gradient of the stress functions for plastic torsion has a constant magnitude, and hence a function of this type assumes the profile of a sand hill. This sand hill analogy of Nadai (1950, The Theory of Flow and Fracture of Solids, McGraw-Hill, U.K.) gave a visual sense of possible non-smoothness of such stress functions and thus discontinuous stress fields. Many stress functions of plastic torsion for relatively simple cross-sections have been constructed graphically. However, collapse modes in terms of warping functions were much less reported. In this paper, we shall establish a duality theorem which relates the correct stress function to the correct warping function, thus providing the means to obtain complete static and kinematic solutions. This dual variational principle leads naturally to a general numerical algorithm which guarantees convergence and accuracy. In this paper, we shall only present three exact solutions to verify the theorem, to demonstrate the possible non-smooth feature of the solutions and to reiterate this effective dual variational approach to limit analysis in general.

Abstract:   We construct a unique weak solution of the nonlinear PDE which asserts each level set evolves in time according to its mean curvature. This weak solution allows us then to define for any compact set GAMMA-0 a unique generalized motion by mean curvature, existing for all time. We investigate the various geometric properties and pathologies of this evolution.

Abstract:   We construct a unique weak solution of the nonlinear PDE which asserts each level set evolves in time according to its mean curvature. This weak solution allows us then to define for any compact set GAMMA-0 a unique generalized motion by mean curvature, existing for all time. We investigate the various geometric properties and pathologies of this evolution.

Abstract:   Seismic traveltimes can be computed efficiently on a regular grid by an upwind finite-difference method. The method solves a conservation law that describes changes in the gradient components of the traveltime field. The traveltime field itself is easily obtained from the solution of the conservation law by numerical integration. The conservation law derives from the eikonal equation, and its solution depicts the first- arrival-time field. The upwind finite-difference scheme can be implemented in fully vectorized form, in contrast to a similar scheme proposed recently by Vidale. The resulting traveltime field is useful both in Kirchhoff migration and modeling and in seismic tomography. Many reliable methods exist for the numerical solution of conservation laws, which appear in fluid mechanics as statements of the conservation of mass, momentum, etc. A first-order upwind finite-difference scheme proves accurate enough for seismic applications. Upwind schemes are stable because they mimic the behavior of fluid flow by using only information taken from upstream in the fluid. Other common difference schemes are unstable, or overly dissipative, at shocks (discontinuities in flow variables), which are time gradient discontinuities in our approach to solving the eikonal equation.

Abstract:   Seismic traveltimes can be computed efficiently on a regular grid by an upwind finite-difference method. The method solves a conservation law that describes changes in the gradient components of the traveltime field. The traveltime field itself is easily obtained from the solution of the conservation law by numerical integration. The conservation law derives from the eikonal equation, and its solution depicts the first- arrival-time field. The upwind finite-difference scheme can be implemented in fully vectorized form, in contrast to a similar scheme proposed recently by Vidale. The resulting traveltime field is useful both in Kirchhoff migration and modeling and in seismic tomography. Many reliable methods exist for the numerical solution of conservation laws, which appear in fluid mechanics as statements of the conservation of mass, momentum, etc. A first-order upwind finite-difference scheme proves accurate enough for seismic applications. Upwind schemes are stable because they mimic the behavior of fluid flow by using only information taken from upstream in the fluid. Other common difference schemes are unstable, or overly dissipative, at shocks (discontinuities in flow variables), which are time gradient discontinuities in our approach to solving the eikonal equation.

Abstract:   We derive an "eikonal-curvature" equation to describe the propagation of action potential wavefronts in myocardium. This equation is used to study the effects of fiber orientation on propagation in the myocardial wall. There are significant computational advantages to the use of an eikonal-curvature equation over a full ionic model of action potential spread. With this model, it is shown that the experimentally observed misalignment of spreading action potential "ellipses" from fiber orientation in level myocardial surfaces is adequately explained by the rotation of fiber orientation through the myocardial wall. Additionally, it is shown that apparently high propagation velocities on the epicardial and endocardial surfaces are the result of propagation into the midwall region and acceleration along midwall fibers before reemergence at an outer surface at a time preceding what could be accomplished with propagation along the surface alone.

Abstract:   Hamilton-Jacobi (H-J) equations are frequently encountered in applications, e.g., in control theory and differential games. H-J equations are closely related to hyperbolic conservation laws-in one space dimension the former is simply the integrated version of the latter. Similarity also exists for the multidimensional case, and this is helpful in the design of difference approximations. In this paper high-order essentially nonoscillatory (ENO) schemes for H-J equations are investigated, which yield uniform high-order accuracy in smooth regions and sharply resolve discontinuities in the derivatives. The ENO scheme construction procedure is adapted from that for hyperbolic conservation laws. The schemes are numerically tested on a variety of one-dimensional and two-dimensional problems, including a problem related to control optimization, and high-order accuracy in smooth regions, good resolution of discontinuities in the derivatives, and convergence to viscosity solutions are observed.

Abstract:   We study shock reflection for the two 2D Burgers equation. This model equation is an asymptotic limit of the Euler equations, and retains many of the features of the full equations. A von Neumann type analysis shows that the 2D Burgers equation has detachment, sonic, and Crocco points in complete analogy with gas dynamics. Numerical solutions support the detachment/sonic criterion for transition from regular to Mach reflection. There is also strong numerical evidence that the reflected shock in the 2D Burgers Mach reflection forms a smooth wave near the Mach stem, as proposed by Colella and Henderson in their study of the Euler equations.

Abstract:   This paper is concerned with the asymptotic behavior, as epsilon arrow pointing down and to the right 0, of the solution (u(epsilon), upsilon(epsilon)) Of the second initial-boundary value problem of the reaction-diffusion system: [GRAPHICS] where gamma > 0 is a constant. When upsilon is-an-element-of (- 2 square-root 3/9, 2 square-root 3/9), f is bistable in the sense that the ordinary differential equation u(t) = f(u, upsilon) has two stable solutions u = h-(upsilon) and u = h+(upsilon) and one unstable solution u = h0(upsilon), where h- (upsilon) , h0(upsilon) , and h+(upsilon) are the three solutions of the algebraic equation f(u, upsilon) = 0 . We show that, when the initial data of upsilon is in the interval (-2 square-root 3/9, 2 square-root 3/9) , the solution (u(epsilon), upsilon(epsilon)) of the system tends to a limit (u, upsilon) which is a solution of a free boundary problem, as long as the free boundary problem has a unique classical solution. The function u is a ''phase'' function in the sense that it coincides with h+(upsilon) in one region OMEGA+ and with h- (upsilon) in another region OMEGA- . The common boundary (free boundary or interface) of the two regions OMEGA- and OMEGA+ moves with a normal velocity equal to V(upsilon), where V(.) is a function that can be calculated. The local (in time) existence of a unique classical solution to the free boundary problem is also established. Further we show that if initially u(., 0) - h0(upsilon(.,0)) takes both positive and negative values, then an interface will develop in a short time O(epsilon\ln epsilon\) near the hypersurface where u(x, 0) - h0(upsilon(x, 0)) = 0.

Abstract:   Because the stress resulting from compositional inhomogeneities are long range, the local stress, diffusional flux and equilibrium conditions at a point depend on the entire composition distribution in a specimen. For a thin plate with a one-dimensional composition profile, this dependence is simple; the local stress depends on the local composition and on both the average composition and the first moment of the composition profile, neither of which are local. A theory of diffusion and equilibrium in a thin plate is developed, based on a free energy that depends on composition, its gradients and strain, and has a term for chemical effects at the plate boundary. Under certain assumptions, a standard diffusion equation is derived, with all of the non-local stress effects in the boundary conditions. Solutions are altered by these new conditions. Spontaneous bending is often a natural result of diffusion.

Abstract:   Because the stress resulting from compositional inhomogeneities are long range, the local stress, diffusional flux and equilibrium conditions at a point depend on the entire composition distribution in a specimen. For a thin plate with a one-dimensional composition profile, this dependence is simple; the local stress depends on the local composition and on both the average composition and the first moment of the composition profile, neither of which are local. A theory of diffusion and equilibrium in a thin plate is developed, based on a free energy that depends on composition, its gradients and strain, and has a term for chemical effects at the plate boundary. Under certain assumptions, a standard diffusion equation is derived, with all of the non-local stress effects in the boundary conditions. Solutions are altered by these new conditions. Spontaneous bending is often a natural result of diffusion.

Abstract:   Because the stress resulting from compositional inhomogeneities are long range, the local stress, diffusional flux and equilibrium conditions at a point depend on the entire composition distribution in a specimen. For a thin plate with a one-dimensional composition profile, this dependence is simple; the local stress depends on the local composition and on both the average composition and the first moment of the composition profile, neither of which are local. A theory of diffusion and equilibrium in a thin plate is developed, based on a free energy that depends on composition, its gradients and strain, and has a term for chemical effects at the plate boundary. Under certain assumptions, a standard diffusion equation is derived, with all of the non-local stress effects in the boundary conditions. Solutions are altered by these new conditions. Spontaneous bending is often a natural result of diffusion.

Abstract:   Because the stress resulting from compositional inhomogeneities are long range, the local stress, diffusional flux and equilibrium conditions at a point depend on the entire composition distribution in a specimen. For a thin plate with a one-dimensional composition profile, this dependence is simple; the local stress depends on the local composition and on both the average composition and the first moment of the composition profile, neither of which are local. A theory of diffusion and equilibrium in a thin plate is developed, based on a free energy that depends on composition, its gradients and strain, and has a term for chemical effects at the plate boundary. Under certain assumptions, a standard diffusion equation is derived, with all of the non-local stress effects in the boundary conditions. Solutions are altered by these new conditions. Spontaneous bending is often a natural result of diffusion.

Abstract:   This paper discusses a generalized Stefan problem which allows for supercooling and superheating and for capillarity in the interface between phases. Simple solutions are obtained indicating the chief differences between this problem and the classical Stefan problem. A weak formulation of the general problem is given.

Abstract:   A stable algorithm is proposed for image restoration based on the "mean curvature motion" equation. Existence and uniqueness of the "viscosity" solution of the equation are proved, a L(infinity) stable algorithm is given, experimental results are shown, and the subjacent vision model is compared with those introduced recently by several vision researchers. The algorithm presented appears to be the sharpest possible among the multiscale image smoothing methods preserving uniqueness and stability.

Abstract:   Recent theoretical advances in the mathematical treatment of geometric interface motion make more tractable the theory of a wide variety of materials science problems where the interface velocity is not controlled by long-range-diffusion. Among the interface motion problems that can be modelled as geometric are certain types of phase changes, crystal growth, domain growth, grain growth. ion beam and chemical etching, and coherency stress driven interface migration. We provide an introduction to nine mathematical methods for solving such problems, give the limits of applicability of the methods, and discuss the relations among them theoretically and their uses in computation. Comparisons of some of them are made by displaying how the same physical problems are treated in the various applicable methods.

Abstract:   The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence may now be proved by very efficient and striking arguments. The range of important applications of these results is enormous. This article is a self-contained exposition of the basic theory of viscosity solutions.

Abstract:   A fully nonlinear evolution equation governing the propagation of a premixed flame through a large-scale spatially periodic shear flow is derived, and steady-state solutions are obtained numerically. The gas density is assumed to be constant across the flame, but the local normal burning speed is allowed to vary with the local strain and curvature along the flame front in order to investigate the influence of the length scale of the external flow on the average propagation speed of the wrinkled flame. At fixed values of the amplitude of the flow- field variations an increase in the length scale (relative to the flame thickness) is found to result in an increase in the average flame propagation speed, in accordance with the predictions of earlier theoretical investigations and with experimental observations for the regime of large-scale turbulence. The propagation speed of the wrinkled flame is calculated to exhibit the experimentally observed bending effect, the tendency of the rate of change of the burning velocity to decrease with increasing turbulence intensity at low fixed turbulence Reynolds numbers. It is shown also how the average flame speed depends on the ratio of the transverse to longitudinal length scale associated with the periodic flow.

Abstract:   A fully nonlinear evolution equation governing the propagation of a premixed flame through a large-scale spatially periodic shear flow is derived, and steady-state solutions are obtained numerically. The gas density is assumed to be constant across the flame, but the local normal burning speed is allowed to vary with the local strain and curvature along the flame front in order to investigate the influence of the length scale of the external flow on the average propagation speed of the wrinkled flame. At fixed values of the amplitude of the flow- field variations an increase in the length scale (relative to the flame thickness) is found to result in an increase in the average flame propagation speed, in accordance with the predictions of earlier theoretical investigations and with experimental observations for the regime of large-scale turbulence. The propagation speed of the wrinkled flame is calculated to exhibit the experimentally observed bending effect, the tendency of the rate of change of the burning velocity to decrease with increasing turbulence intensity at low fixed turbulence Reynolds numbers. It is shown also how the average flame speed depends on the ratio of the transverse to longitudinal length scale associated with the periodic flow.

Abstract:   This paper reports the progress made in multiquadrics (MQ) as a spatial approximation scheme for systems of governing equations of engineering and physics by minimizing the spatial truncation errors without excessive refinement. Although MQ is defined over the general n-dimensional real space, this paper is limited to two spatial dimensions defined over a general non- convex irregular region containing inhomogeneously scattered nodes. We have developed a strictly conservative interpolation scheme over such irregular regions from which the partial derivative estimates are obtained. In addition, we developed a non-iterative scheme to be used with domain decomposition to ensure derivative continuity over contiguous regions. Jump discontinuities for shock and material interfaces are likewise treated by appropriate modification of the algorithm. We have compared the relative errors of the derivative estimates defined over an irregular region consisting of inhomogeneously scattered nodes obtained by the MQ and Voronoi mesh schemes. The MQ relative errors of the derivative estimates are three orders of magnitude better than those obtained from the Voronoi mesh method. (In our previous papers, we have shown that MQ is superior in its derivative estimates over regular gridded regions.) We have also used MQ to estimate derivatives within a very narrow "shock" region with similar excellent results. While comparing spatial approximation schemes for PDE's, we found the MQ results to be superior in accuracy and were calculated by far fewer operations than standard finite difference schemes. Other authors have likewise used MQ successfully to solve integral equations.

Abstract:   For realistic interfacial energies, the equations of anisotropic motion-by-curvature exhibit backward-parabolic behavior over portions of their domain, thereby inducing phenomena such as the formation of facets and wrinkles. In this paper, a physically consistent regularized equation that may be used to analyze such phenomena is derived.

Abstract:   We describe all multiscale causal, local, stable and shape preserving filterings. This classification contains the classical "morphological" operators, and some new ones.

Abstract:   Stationary premixed flames in dual-source flow are considered. The significant features of the dual-source system are that the sources are of finite strength, and that a stagnation point is located between the sources. A new mathematical model for front propagation and advection is introduced that tracks the front along streamlines. The equations for the stationary fronts of the dual-source system are solved numerically. The assumption of constant-density potential flow is made to simplify the problem and to illustrate the effects of the geometry alone. It is shown that for sufficiently slow burning velocity (or equivalently, small source separation), three stationary states exist for closed, free flames, but one of them is unstable. In addition, several types of burner-attached flames are observed. Quasi-stationary evolution of a closed, free flame exhibits a change of topology and hysteresis. Nonclosed flames are predicted if local extinction due to flow strain is allowed.

Abstract:   Stationary premixed flames in dual-source flow are considered. The significant features of the dual-source system are that the sources are of finite strength, and that a stagnation point is located between the sources. A new mathematical model for front propagation and advection is introduced that tracks the front along streamlines. The equations for the stationary fronts of the dual-source system are solved numerically. The assumption of constant-density potential flow is made to simplify the problem and to illustrate the effects of the geometry alone. It is shown that for sufficiently slow burning velocity (or equivalently, small source separation), three stationary states exist for closed, free flames, but one of them is unstable. In addition, several types of burner-attached flames are observed. Quasi-stationary evolution of a closed, free flame exhibits a change of topology and hysteresis. Nonclosed flames are predicted if local extinction due to flow strain is allowed.

Abstract:   We study the limiting behavior of solutions to appropriately rescaled versions of the Allen-Cahn equation, a simplified model for dynamic phase transitions. We rigorously establish the existence in the limit of a phase-antiphase interface evolving according to mean curvature motion. This assertion is valid for all positive time, the motion interpreted in the generalized sense of Evans-Spruck and Chen-Giga-Goto after the onset of geometric singularities.

Abstract:   This paper describes a method for tracking contact discontinuities and material interfaces that arise in the solution of hyperbolic systems of conservation laws. Numerical results arc presented to show that the fronts are resolved to within a mesh interval and smooth portions of the solution are computed to within the accuracy of the underlying numerical scheme.

Abstract:   A numerical study was conducted to understand how a vortex, when convected at moderate speeds across a premixed flame, can induce velocities that pull the flame along with the vortex, causing flame elongation and unsteady flame stretch. If the vortex-induced velocity that opposes flame motion is sufficiently large, the flame cannot propagate over the vortex and thus temporarily remains attached to the moving vortex. A flame attachment criterion is discussed; when the criterion is met the vortex forms cusps and pockets in the flame structure similar to those observed experimentally. The net result of increasing the vortex convection velocity is to reduce the residence time of the vortex in the flame, which reduces the degree of flame wrinkling. Vortex pairs that exert an extensive strain on the flame were found to have significantly longer residence times of interaction than vortices that exert compressive strain; this difference in residence time helps to explain why extensive strain on a flame is more probable in turbulent flames than compressive strain. The calculated images of the laminar flame shape show encouraging agreement with experiment, which is another indication that flame-interface simulations are a promising way to represent very wrinkled turbulent premixed flames in a numerically efficient manner.

Abstract:   The level-set flow of Evans-Spruck and Chen-Giga-Goto is generalized to a Riemannian manifold, using recent techniques of Crandall-Ishii for viscosity solutions. Generally speaking, the motion is not unique for noncompact closed sets, but the definition can be modified to make the motion unique. We give examples to show: (1) a smooth set can develop an interior that originates from infinity (2) in the case of a Grayson neckpinch, the evolving function u(x,t) need not remain C2.

Abstract:   We consider a macroscopic model of the excitation process in the anisotropic myocardium involving the transmembrane, extracellular, and extracardiac potentials upsilon, u(e), and u0. The model is described by a reaction-diffusion (R-D) system, and the component upsilon exhibits a front-like behavior reflecting the features of the excitation process. In numerical simulations, the presence of a moving excitation layer imposes severe constraints on the time and space steps to achieve stability and accuracy; consequently, application of the model is very costly in terms of computer time. An approximate model has been derived from the R-D system by means of a singular perturbation technique, and it is described by an eikonal equation, nonlinear and elliptic, in the activation time psi(x). Larger space steps are possible with this equation. From psi(x), we can derive, for a given instant t, the transmembrane potential upsilon and subsequently, by solving an elliptic problem, we can compute the corresponding extracellular and extracardiac potentials u(e) and u0. The results of the R-D and the eikonal models applied to a portion of the ventricular wall are in excellent agreement; moreover, the eikonal model requires only a small fraction of the computer time needed by the R-D system. Therefore, for large- scale simulations of the excitation process, only the eikonal model has been used, and we investigate its ability to cope with complex situations such as front-front collisions and related potential patterns.

Abstract:   We describe all multiscale movie filtering which are causal, local, shape preserving and galilean invariant.

Abstract:   A constrained optimization type of numerical algorithm for removing noise from images is presented. The total variation of the image is minimized subject to constraints involving the statistics of the noise. The constraints are imposed using Lagrange multipliers. The solution is obtained using the gradient-projection method. This amounts to solving a time dependent partial differential equation on a manifold determined by the constraints. As t --> infinity the solution converges to a steady state which is the denoised image. The numerical algorithm is simple and relatively fast. The results appear to be state-of-the-art for very noisy images. The method is noninvasive, yielding sharp edges in the image. The technique could be interpreted as a first step of moving each level set of the image normal to itself with velocity equal to the curvature of the level set divided by the magnitude of the gradient of the image, and a second step which projects the image back onto the constraint set.

Abstract:   In this paper we shall describe a numerical method for the solution of curve flow problems in which the normal velocity of the curve depends locally on the position, normal and curvature of the curve. The method involves approximating the curve by a number of line elements (segments) which are only allowed to move in a direction normal to the element. Hence the normal of each line element remains constant throughout the evolution. In regions of high curvature elements naturally tend to accumulate. The method easily deals with the formation of cusps as found in flame propagation problems and is computationally comparable to a naive marker particle method. As a test of the method we present a number of numerical experiments related to mean curvature flow and flows associated with flame propagation and bushfires.

Abstract:   This paper deals with the mathematical theory of invariant curve evolution. We present a high-level procedure for the formulation of geometric heat flows which are invariant with respect to a given Lie group. This approach is based on the classical theory of differential invariants. The affine group is then analyzed in detail. Indeed, we give a rather complete description of the properties of the affine geometric heat equation. We moreover extend the results of [38] from the convex to the nonconvex case. The paper concludes with a summary of recent applications of curve evolution theory to image analysis.

Abstract:   In this work we investigate, by means of numerical simulations, the performance of two mathematical models describing the spread of excitation in a three-dimensional block representing anisotropic cardiac tissue. The first model is characterized by a reaction-diffusion system in the transmembrane and extracellular potentials v and u. The second model is derived from the first by means of a perturbation technique. It is characterized by an eikonal equation, nonlinear and elliptic in the activation time psi(x). The level surfaces psi(x) = t represent the wave-front positions. The numerical procedures based on the two models were applied to test functions and to excitation processes elicited by local stimulations in a relatively small block. The results are in excellent agreement, and for the same problem the computation time required by the eikonal equation is a small fraction of that needed for the reaction-diffusion system. Thus we have strong evidence that the eikonal equation provides a reliable and numerically efficient model of the excitation process. Moreover, numerical simulations have been performed to validate an approximate model for the extracellular potential based on knowledge of the excitation sequence. The features of the extracellular potential distribution affected by the anisotropic conductivity of the medium were investigated.

Abstract:   We consider an evolution which starts as a flow of smooth surfaces in nonparametric form propagating in space with normal speed equal to the mean curvature of the current surface The boundaries of the surfaces are assumed to remain fixed. G. Huisken has shown that if the boundary of the domain over which this flow is considered satisfies the ''mean curvature'' condition of H. Jenkins and J. Serrin (that is, the boundary of the domain is convex ''in the mean'') then the corresponding initial boundary value problem with Dirichlet boundary data the smooth initial data admits a smooth SolUtion for all time. In this paper we consider the case of arbitrary domains with smooth boundaries not necessarily satisfying the condition of Jenkins-Serrin. In this case, even if the flow starts with smooth initial data and homogeneous Dirichlet boundary data, singularities may develop in finite time at the boundary of the domain and the solution will not satisfy the boundary condition. We prove, however. existence of solutions that are smooth inside the domain for all time and become smooth up to the boundary after elapsing of a sufficiently long period of time. From that moment on such solutions assume the boundary values in the classical sense. We also give sufficient conditions that guarantee the existence of classical solutions for all time t greater-than-or-equal-to 0. In addition. we establish estimates of the rate at which solutions tend to zero as t --> infinity.

Abstract:   We consider the problem of coexistence of two competing species mediated by the presence of a predator. We employ a reaction- diffusion model equation with Lotka-Volterra interaction, and speculate that the possibility of coexistence is enhanced by differences in the diffusion rates of the prey and their predator. In the limit where the diffusion rate of the prey tends to zero, a new equation is derived and the dynamics of spatial segregation is discussed by means of the interfacial dynamics approach. Also, we show that spatial segregation permits periodic and chaotic dynamics for certain parameter ranges.

Abstract:   The connection between the weak theories for a class of geometric equations and the asymptotics of appropriately rescaled reaction-diffusion equations is rigorously established. Two different scalings are studied. In the first, the limiting geometric equation is a first-order equation; in the second, it is a generalization of the mean curvature equation. Intrinsic definitions for the geometric equations are obtained, and uniqueness under a geometric condition on the initial surface is proved. In particular, in the case of the mean curvature equation, this condition is satisfied by surfaces that are strictly starshaped, that have positive mean curvature, or that satisfy a condition that interpolates between the positive mean curvature and the starshape conditions.

Abstract:   The connection between the weak theories for a class of geometric equations and the asymptotics of appropriately rescaled reaction-diffusion equations is rigorously established. Two different scalings are studied. In the first, the limiting geometric equation is a first-order equation; in the second, it is a generalization of the mean curvature equation. Intrinsic definitions for the geometric equations are obtained, and uniqueness under a geometric condition on the initial surface is proved. In particular, in the case of the mean curvature equation, this condition is satisfied by surfaces that are strictly starshaped, that have positive mean curvature, or that satisfy a condition that interpolates between the positive mean curvature and the starshape conditions.

Abstract:   An algorithm for shape offsetting is presented that is based on level-set propagation. This algorithm avoids the topological problems encountered in traditional offsetting algorithms, and it deals with curvature singularities by including an 'entropy condition' in its numerical implementation.

Abstract:   A simple phase field model for one component melt growth is presented. which includes anisotropy in a certain form. The formation of various dendritic patterns can be shown by a series of numerical simulations of this model. Qualitative relations between the shapes of crystals and some physical parameters are discussed. Also it is shown that noises give a crucial influence on the side branch structure of dendrites in some situations.

Abstract:   We consider applications of Hamilton-Jacobi equations for which the initial data is only assumed to be in L(infinity). Such problems arise for example when one attempts to describe several characteristic singularities of the compressible Euler equations such as contact and acoustic surfaces, propagating from the same discontinuous initial front. These surfaces represent the level sets of solutions to a Hamilton-Jacobi equation which belongs to a special class. For such Hamilton- Jacobi equations we prove the existence and regularity of solutions for any positive time and convergence to initial data along rays of geometrical optics at any point where the gradient of the initial data exists. Finally, we present numerical algorithms for efficiently capturing singular fronts with complicated topologies such as corners and cusps. The approach of using Hamilton-Jacobi equations for capturing fronts has been used in [14] for fronts propagating with curvature-dependent speed.

Abstract:   The propagation of a two-dimensional wave front in an excitable medium is dependent on the curvature of the front; current theories of excitable reaction-diffusion models predict that, when reaction is much faster than diffusion, the normal wave speed (N) is approximately related to the curvature of the wave front (kappa), the plane wave speed (c), and the diffusion coefficient of the propagator variable (D), by the ''eikonal'' equation, N = c - Dkappa. We show that a simple model for intracellular calcium (Ca2+) wave propagation does not obey the eikonal equation, and postulate an alternative curvature equation that is dependent on the parameter values used in the model. This new curvature relation is confirmed by numerical simulations. We raise the possibility that different models for Ca2+ wave propagation will have qualitatively different spiral wave patterns, providing a new way of distinguishing between proposed models. The theory developed here also necessitates a reconsideration of methods previously used to measure the intracellular diffusion coefficient of Ca2+.

Abstract:   The propagation of a two-dimensional wave front in an excitable medium is dependent on the curvature of the front; current theories of excitable reaction-diffusion models predict that, when reaction is much faster than diffusion, the normal wave speed (N) is approximately related to the curvature of the wave front (kappa), the plane wave speed (c), and the diffusion coefficient of the propagator variable (D), by the ''eikonal'' equation, N = c - Dkappa. We show that a simple model for intracellular calcium (Ca2+) wave propagation does not obey the eikonal equation, and postulate an alternative curvature equation that is dependent on the parameter values used in the model. This new curvature relation is confirmed by numerical simulations. We raise the possibility that different models for Ca2+ wave propagation will have qualitatively different spiral wave patterns, providing a new way of distinguishing between proposed models. The theory developed here also necessitates a reconsideration of methods previously used to measure the intracellular diffusion coefficient of Ca2+.

Abstract:   The propagation of a two-dimensional wave front in an excitable medium is dependent on the curvature of the front; current theories of excitable reaction-diffusion models predict that, when reaction is much faster than diffusion, the normal wave speed (N) is approximately related to the curvature of the wave front (kappa), the plane wave speed (c), and the diffusion coefficient of the propagator variable (D), by the ''eikonal'' equation, N = c - Dkappa. We show that a simple model for intracellular calcium (Ca2+) wave propagation does not obey the eikonal equation, and postulate an alternative curvature equation that is dependent on the parameter values used in the model. This new curvature relation is confirmed by numerical simulations. We raise the possibility that different models for Ca2+ wave propagation will have qualitatively different spiral wave patterns, providing a new way of distinguishing between proposed models. The theory developed here also necessitates a reconsideration of methods previously used to measure the intracellular diffusion coefficient of Ca2+.

Abstract:   Computation of three-dimensional (3-D) Rayleigh-Taylor instability in compressible fluids is performed on a MIMD computer. A second-order TVD scheme is applied with a fully parallelized algorithm to the 3-D Euler equations. The computational program is implemented for a 3-D study of bubble evolution in the Rayleigh-Taylor instability with varying bubble aspect ratio and for large-scale simulation of a 3-D random fluid interface. The numerical solution is compared with the experimental results by Taylor.

Abstract:   A level set formulation for the solution of the Hamilton-Jacobi equation F(x, y, u, u(x), u(y)) = 0 is Presented, where u is prescribed on a set of closed bounded noncharacteristic curves. A time dependent Hamilton-Jacobi equation is derived such that the zero level set at various time t of this solution is precisely the set of points (x, y) for which u(x, y) = t. This gives a fast and simple numerical method for generating the viscosity solution to F = 0. The level set capturing idea was first introduced by Osher and Sethian [J. Comput. Phys., 79 (1988), pp. 12-49], and the observation that this is useful for an important computer vision problem of this type was then made by Kimmel and Bruckstein in [Technion (Israel) Computer Science Report, CIS #9209, 1992] following Bruckstein [Comput. Vision Graphics Image Process, 44 (1988), pp. 139-154]. Finally, it is noted that an extension to many space dimensions is immediate.

Abstract:   A level set formulation for the solution of the Hamilton-Jacobi equation F(x, y, u, u(x), u(y)) = 0 is Presented, where u is prescribed on a set of closed bounded noncharacteristic curves. A time dependent Hamilton-Jacobi equation is derived such that the zero level set at various time t of this solution is precisely the set of points (x, y) for which u(x, y) = t. This gives a fast and simple numerical method for generating the viscosity solution to F = 0. The level set capturing idea was first introduced by Osher and Sethian [J. Comput. Phys., 79 (1988), pp. 12-49], and the observation that this is useful for an important computer vision problem of this type was then made by Kimmel and Bruckstein in [Technion (Israel) Computer Science Report, CIS #9209, 1992] following Bruckstein [Comput. Vision Graphics Image Process, 44 (1988), pp. 139-154]. Finally, it is noted that an extension to many space dimensions is immediate.

Abstract:   A new affine invariant scale-space for planar curves is presented in this work. The scale-space is obtained from the solution of a novel nonlinear curve evolution equation which admits affine invariant solutions. This flow was proved to be the affine analogue of the well known Euclidean shortening flow. The evolution also satisfies properties such as causality, which makes it useful in defining a scale-space. Using an efficient numerical algorithm for curve evolution, this continuous affine flow is implemented, and examples are presented. The affine-invariant progressive smoothing property of die evolution equation is demonstrated as well.

Abstract:   Image-processing transforms must satisfy a list of formal requirements. We discuss these requirements and classify them into three categories: ''architectural requirements'' like locality, recursivity and causality in the scale space, ''stability requirements'' like the comparison principle and ''morphological requirements'', which correspond to shape- preserving properties (rotation invariance, scale invariance, etc.). A complete classification is given of all image multiscale transforms satisfying these requirements. This classification yields a characterization of all classical models and includes new ones, which all are partial differential equations. The new models we introduce have more invariance properties than all the previously known models and in particular have a projection invariance essential for shape recognition. Numerical experiments are presented and compared. The same method is applied to the multiscale analysis of movies. By introducing a property of Galilean invariance, we find a single multiscale morphological model for movie analysis.

Abstract:   A formalism for a flamelet's evolution of its spatial distribution is derived from a field equation which is slightly more general than Williams' field equation. Unlike Williams' field equation, the field equation used here, though non- linear, has the property that an arbitrary linear combination of interface solutions (Heavyside type of functions) is also a solution. We therefore can describe the location of the flamelet with two interfaces rather than one, both moving relative to the flow in the same direction. The volume between these two interfaces is on average conserved; this makes it possible to define a probability density for the spatial distribution of the flamelet, and thereby derive equations describing the evolution of the spatial distribution of folds and wrinkles of the flame front. Three main conclusions are reached in this paper using this formalism, through the exact analytical study of a flamelet in an arbitrary 1-d velocity field, and through the numerical study of a flamelet in a simulated 2-d turbulent velocity field. (1) The rate of advancement u(M) of the average location of the flame front can be smaller than the turbulent flame speed u(T) at short times, and sometimes even smaller than the laminar flame speed u(L) (at short times). It is shown, in the case of an arbitrary 1-d velocity field, that u(M) = u(T) only after cusps have formed on the flamelet, and u(M) < u(L) < u(T) before. (2) If the turbulence is too weak or too strong compared with the laminar flame speed, the dispersion of the flame is, at short times, increased by the turbulence and reduced by the laminar flame speed. (3) The dispersion of the flame is skewed towards the direction of the flame's propagation at all times, even before cusp formation.

Abstract:   Bence, Merriman and Osher [BMO] have proposed a new numerical algorithm for computing mean curvature flow, in terms of solutions of the usual heat equation, continually reinitialized after short time steps. This paper employs nonlinear semigroup theory to reconcile their algorithm with the ''level-set'' approach to mean curvature flow of Osher-Sethian [OS], Evans- Spruck [ES], and Chen-Giga-Goto [CGG].

Abstract:   A new algorithm that determines the evolution of a surface eroding under reactive-ion etching is presented. The surface motion is governed by both the Hamilton-Jacobi equation and the entropy condition for a given etch rate. The trajectories of ''shocks'' and ''rarefaction waves'' are then directly tracked, and thus this method may be regarded as a generalization of the method of characteristics. This allows slope discontinuities to be accurately calculated without artificial diffusion. The algorithm is compared with ''geometric'' surface evolution methods, such as the line-segment method.

Abstract:   We propose a new model for active contours based on a geometric partial differential equation. Our model is intrinsec, stable (satisfies the maximum principle) and permits a rigorous mathematical analysis. It enables us to extract smooth shapes (we cannot retrieve angles) and it can be adapted to find several contours simultaneously. Moreover, as a consequence of the stability, we can design robust algorithms which can be engineed with no parameters in applications. Numerical experiments are presented.

Abstract:   A new approach to digital implementation of continuous-scale mathematical morphology is presented. The approach is based on discretization of evolution equations associated with continuous multiscale morphological operations. Those equations, and their corresponding numerical implementation, can be derived either directly from mathematical morphology definitions or from curve evolution theory. The advantages of the proposed approach over the classical discrete morphology are demonstrated.

Abstract:   Numerical modeling of propagating fronts in non-uniform two- dimensional flow fields is performed in order to simulate the effect of such flows on premixed flame fronts. In particular, the influence of the flow disturbance intensity (u') on the mean front propagation rate (S-T) is examined. A second-order numerical technique is employed that combines the level set (G- equation) formulation to describe the self-propagation of the front and a multidimensional upwind technique to describe the convection of the front by the flow field. In this way the effect of the non-dimensional disturbance intensity (u'/S-L) on the non-dimensional propagation rate (S-T/S-L) at values of u'/S-L >> 1 is computed. The dependence of the laminar propagation speed (S-L) on the flame stretch (including both the front curvature and the velocity strain effects) is incorporated in this formulation. We focus on front propagation in simulated Taylor-Couette flows in the ''Taylor vortex'' regime and the results are found to compare favorably with recent experiments on the propagation of isothermal chemical fronts in this flow. The formation of ''islands'' of reactants is observed and its relation to front propagation rates is discussed.

Abstract:   The time necessary to cover a path on a grey-scale image is the sum of the grey-level values along the path. The geodesic time between two points in a grey-scale image is defined as the smallest amount of time allowing to link these points. The geodesic time allows the definition of generalized geodesic distances, erosions, dilations, and skeletons by influence zones. An application to minimal path extraction on grey-scale images is presented.

Abstract:   A mathematical model is developed for melting of a multilayered medium while a heat source traverses one boundary. The finite- element method uses moving meshes, front-tracking using spines, an automatic time-step algorithm, and an efficient solution of the linearized equations. A novel solution method allows the fixed-mesh code to work unchanged but allows a moving mesh in other problems. The finite-element method is applied when the heater mesh moves with respect to the multilayered medium mesh. The same technique allows parallel processing for finite- element codes. The model is applied to several test problems and then to the title problem.

Abstract:   A mathematical model is developed for melting of a multilayered medium while a heat source traverses one boundary. The finite- element method uses moving meshes, front-tracking using spines, an automatic time-step algorithm, and an efficient solution of the linearized equations. A novel solution method allows the fixed-mesh code to work unchanged but allows a moving mesh in other problems. The finite-element method is applied when the heater mesh moves with respect to the multilayered medium mesh. The same technique allows parallel processing for finite- element codes. The model is applied to several test problems and then to the title problem.

Abstract:   The authors define a new class of filters for noise elimination and edge enhancement by using shock filters and anisotropic diffusion. Some nonlinear partial differential equations used as models for these filters are studied. The authors develop recursive and unconditional stable schemes which drastically reduce the computational effort of the algorithms. A new fast recursive approach to linear Gaussian filters is also shown by using the heat equation.

Abstract:   The dynamics of inviscid multicomponent fluids may be modelled by the Euler equations, augmented by one (or more) additional species equation(s). Attempts to compute solutions for extended Euler models in conservation form, show strong oscillations and other computational inaccuracies near material interfaces. These are due to erroneous pressure fluctuations generated by the conservative wave model. This problem does not occur in single component computations and arises only in the presence of several species. A nonconservative (primitive) Euler formulation is proposed, which results in complete elimination of the oscillations. The numerical algorithm uses small viscous perturbations to remove leading order conservation errors and is conservative to the order of numerical approximation. Numerical experiments show clean monotonic solution profiles, with acceptably small conservation error for shocks of weak to moderate strengths. (C) 1994 Academic Press, Inc.

Abstract:   A coupled level set method for the motion of multiple junctions is proposed. The new method extends the ''Hamilton-Jacobi'' level set formulation of Osher and Sethian. It retains the feature of tracking fronts by following level sets and allows the specification of arbitrary velocities on each front, The diffusion equation is shown to generate curvature dependent motion and this is used to develop an algorithm to move multiple junctions with curvature-dependent speed. Systems of reaction-diffusion equations are shown to possess inherent properties which prohibit efficient numerical solutions when applied to curvature-dependent motion. (C) 1994 Academic Press, Inc.

Abstract:   The premixed flame situated in a spatially periodic flow field is examined using the passive propagation model with the local flame speed affected by stretch and nonequidiffusion. Numerical solution shows that the average flame speed increases with either increasing fluctuation amplitude or increasing wavelength of the imposed flow field, and that the flame surface can locally extinguish for sufficiently large fluctuation amplitude of the imposed flow. Perturbation solutions in the weakly wrinkled flame and the thin flame limits are presented. The formation of comers on the flame surface in the thin flame limit is illustrated, and the structure of the comer is further found to resemble that of the Bunsen flame. The premixed flame situated in a two-dimensional periodic flow field is also analyzed in the Huygens limit, leading to the observation that flame surface discontinuities exist in the form of cones.

Abstract:   An accurate finite difference scheme for the flow by curvature in R2 is presented, and its convergence theorem is stated. The numerical scheme has a correction term which is effective in locating points uniformly and the effect prevents the computation from breaking down.

Abstract:   The nonlinear interfacial instability of a liquid jet in a coflowing compressible airstream is studied numerically. A high-resolution scheme which has second-order accuracy in space and time is coupled with a Lagrangian marker particle algorithm to visualize the large-scale motion of the interfaces in compressible flow. A numerical algorithm based on an approximate equation of state of a compressible liquid is developed to allow this two-fluid system to be governed by the nonlinear unsteady Euler equations in conservative form. The initial growth of small disturbances given by the simulations agrees well with linear theory. The process of jet disruption in compressible flow is demonstrated to consist of the formation of liquid spikes, interweaving of the gas and liquid and stretching and detachment of the liquid main center core.

Abstract:   The nonlinear interfacial instability of a liquid jet in a coflowing compressible airstream is studied numerically. A high-resolution scheme which has second-order accuracy in space and time is coupled with a Lagrangian marker particle algorithm to visualize the large-scale motion of the interfaces in compressible flow. A numerical algorithm based on an approximate equation of state of a compressible liquid is developed to allow this two-fluid system to be governed by the nonlinear unsteady Euler equations in conservative form. The initial growth of small disturbances given by the simulations agrees well with linear theory. The process of jet disruption in compressible flow is demonstrated to consist of the formation of liquid spikes, interweaving of the gas and liquid and stretching and detachment of the liquid main center core.

Abstract:   A level set approach for computing solutions to incompressible two-phase flow is presented. The interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function. We use a second-order projection method which implements a second-order upwinded procedure for differencing the convection terms. A new treatment of the level set method allows us to include large density and viscosity ratios as well as surface tension. We consider the motion of air bubbles in water and falling water drops in air. (C) 1994 Academic Press, Inc.

Abstract:   A certain class of surface motions, including those of a relativistic membrane minimizing the three-dimensional volume swept out in Minkowski space, is shown to be equivalent to three-dimensional steady-state irrotational inviscid isentropic gas dynamics. The SU(infinity) Nahm equations turn out to correspond to motions where the time t at which the surface moves through the point r is a harmonic function of the three space coordinates. The solution also implies the linearisation of a non-trivial-looking scalar field theory.

Abstract:   I. We study Brakke's motion of varifolds by mean curvature in the special case that the initial surface is an integral cycle, giving a new existence proof by mean of elliptic regularization. Under a uniqueness hypothesis, we obtain a weakly continuous family of currents solving Brakke's motion. II. These currents remain within the corresponding level-set motion by mean curvature, as defined by Evans-Spruck and Chen- Giga-Goto. Now let T0 be the reduced boundary of a bounded set of finite perimeter in R(n). If the level-set motion of the support of T0 does not develop positive Lebesgue measure, then there corresponds a unique integral n-current T, partial derivative = T0, whose time-slices form a unit density Brakke motion. Using Brakke's Regularity Theorem, spt T is smooth H(n)-almost everywhere. In consequence, almost every level-set of the level-set flow is smooth H(n)-almost everywhere in space-time.

Abstract:   The authors treat some applications of Hamilton-Jacobi equations to the study of a flame front propagation model and the rendez-vous problem. The solution of both problems requires the determination of the level sets of the viscosity solution for the corresponding equation. In the flame front propagation model described here, it is assumed that the evolution is driven by a vector field satisfying a transversality condition at time t = 0. The evolution in the normal direction with variable velocity c(x) greater than or equal to 0 is considered as a special case. This approach is constructive, permitting the numerical solution of such problems.

Abstract:   We study the limiting behavior (the macroscopic limit) of an appropriately scaled spin system with Glauber-Kawasaki dynamics. We rigorously establish the existence in the limit of an interface evolving according to motion by mean curvature. This limit is valid for all positive times, past possible geometric singularities of the motion, which is interpreted in the viscosity sense.

Abstract:   We study the limiting behavior (the macroscopic limit) of an appropriately scaled spin system with Glauber-Kawasaki dynamics. We rigorously establish the existence in the limit of an interface evolving according to motion by mean curvature. This limit is valid for all positive times, past possible geometric singularities of the motion, which is interpreted in the viscosity sense.

Abstract:   We describe a numerical technique to generate logically rectangular body-fitted interior and exterior grids. The technique is based on solving a Hamilton-Jacobi-type equation for a propagating level set function, using techniques borrowed from hyperbolic conservation laws. Coordinate grid lines are kept smooth through curvature terms which regularize the equation of motion, and upwind difference schemes which satisfy the correct entropy conditions of front propagation. The resulting algorithm can be used to generate two- and three- dimensional interior and exterior grids around reasonably complex bodies which may contain sharp corners and significant variations in curvature. The technique may also be easily extended to problems of boundary-fitted moving grids. (C) 1994 Academic Press, Inc.

Abstract:   We analyze the problem of stationary self-propagating fronts in potential flow. The issues of local existence and uniqueness for solutions of the ODE describing stationary fronts, multiplicity of solutions and linearized stability of a stationary front as a solution of the (hyperbolic) evolution equation are addressed. The results are illustrated in the case of the dual-source system, which is a simple model of a combustion system in which local extinction may arise. Model extensions for combustion applications are presented.

Abstract:   In this paper, we study generalized ''viscosity'' solutions of the mean curvature evolution which were introduced by Chen, Giga, and Goto and by Evans and Spruck. We devote much of our attention to solutions whose initial value is a compact, smooth, rotationally symmetric hypersurface given by rotating a graph around an axis. Our main result is the regularity of the solution except at isolated points in spacetime and estimates on the number of such points.

Abstract:   Evolutions of closed planar polygons are studied in this work. In the first part of the paper, the general theory of linear polygon evolutions is presented, and two specific problems are analyzed. The first one is a polygonal analog of a novel affine-invariant differential curve evolution, for which the convergence of planar curves to ellipses was proved. In the polygon case, convergence to polygonal approximation of ellipses, polygonal ellipses, is proven. The second one is related to cyclic pursuit problems, and convergence, either to polygonal ellipses or to polygonal circles, is proven. In the second part, two possible polygonal analogues of the well-known Euclidean curve shortening flow are presented. The models follow from geometric considerations. Experimental results show that an arbitrary initial polygon converges to either regular or irregular polygonal approximations of circles when evolving according to the proposed Euclidean flows.

Abstract:   In this paper, area preserving multi-scale representations of planar curves are described. This allows smoothing without shrinkage at the same time preserving all the scale-space properties. The representations are obtained deforming the curve via geometric heat flows while simultaneously magnifying the plane by a homethety which keeps the enclosed area constant. When the Euclidean geometric heat now is used, the resulting representation is Euclidean invariant, and similarly it is affine invariant when the affine one is used. The flows are geometrically intrinsic to the curve, and exactly satisfy all the basic requirements of scale-space representations. In the case of the Euclidean heat flow, it is completely local as well. The same approach is used to define length preserving geometric flows. A similarity (scale) invariant geometric heat flow is studied as well in this work.

Abstract:   Shape modeling is an important constituent of computer vision as well as computer graphics research. Shape models aid the tasks of object representation and recognition. This paper presents a new approach to shape modeling which retains some of the attractive features of existing methods and overcomes some of their limitations. Our techniques can be applied to model arbitrarily complex shapes, which include shapes with significant protrusions, and to situations where no a priori assumption about the object's topology is made. A single instance of our model, when presented with an image having more than one object of interest, has the ability to split freely to represent each object. This method is based on the ideas developed by Osher and Sethian to model propagating solid/liquid interfaces with curvature dependent speeds. The interface (front) is a closed, nonintersecting, hypersurface flowing along its gradient field with constant speed or a speed that depends on the curvature, It is moved by solving a ''Hamilton-Jacobi'' type equation written for a function in which the interface is a particular level set. A speed term synthesized from the image is used to stop the interface in the vicinity-of object boundaries. The resulting equation of motion is solved by employing entropy-satisfying upwind finite difference schemes. We present a variety of ways of computing evolving front, including narrow bands, reinitializations, and different stopping criteria. The efficacy of the scheme is demonstrated with numerical experiments on some synthesized images and some low contrast medical images.

Abstract:   This paper considers the questions of convergence of: (i) MUSCL type (i.e. second-order, TVD) finite-difference approximations towards the entropic weak solution of scalar, one-dimensional conservation laws with strictly convex flux and (ii) higher- order schemes (filtered to ''preserve'' an upper-bound on some weak second-order finite differences) towards the viscosity solution of scalar, multi-dimensional Hamilton-Jacobi equations with convex Hamiltonians.

Abstract:   Level-set approach to the motion of surfaces is presented. Applications to geometrical models in condensed matter physics are given. The finger solution and its generalizations, which were reported very recently, are derived in a simple way.

Abstract:   In this paper, a curve evolution approach for the computation of geodesic curves on 3D surfaces is presented. The algorithm is based on deforming, via the curve shortening flow, an arbitrary initial curve ending at two given surface points. The 3D curve shortening flow is first transformed into an equivalent 2D one. This 2D flow is implemented, using an efficient numerical algorithm for curve evolution with fixed end points.

Abstract:   We prove the convergence of an approximation scheme recently proposed by Bence, Merriman, and Osher for computing motions of hypersurfaces by mean curvature. Our proof is based on viscosity solutions methods.

Abstract:   We present some difference approximation schemes which converge to the entropy solution of a scalar conservation law having a convex flux. The numerical methods described here take their origin from approximation schemes for Hamilton-Jacobi-Bellman equations related to optimal control problems and exhibit several interesting features: the convergence result still holds for quite arbitrary time steps, the main assumption for convergence can be interpreted as a discrete analogue of Oleinik's entropy condition, numerical diffusion around the shocks is very limited. Some tests are included in order to compare the performances of these methods with other classical methods (Godunov, TVD).

Abstract:   We continue Our investigation of the ''level-set'' technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature. The principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by K. Brakke: we prove that almost every level set of the solution to the mean curvature evolution PDE is in fact a unit-density varifold moving according to its mean curvature. In particular, a.e. level set is endowed with a kind of ''geometric structure.'' The proof utilizes compensated compactness methods to pass to limits in various geometric expressions.

Abstract:   We discuss three different affine invariant evolution processes for smoothing planar curves. The first one is derived from a geometric heat-type flow, both the initial and the smoothed curves being differentiable. The second smoothing process is obtained from a discretization of this affine heat equation. In this case, the curves are represented by planar polygons. The third process is based on B-spline approximations. For this process, the initial curve is a planar polygon, and the smoothed curves are differentiable and even analytic. We show that, in the limit, all three affine invariant smoothing processes collapse any initial curve into an elliptic point.

Abstract:   A method is introduced to decrease the computational labor of the standard level set method for propagating interfaces. The fast approach uses only points close to the curve at every time step. We describe this new algorithm and compare its efficiency and accuracy with the standard level set approach. (c) 1995 Academic Press, Inc.

Abstract:   We present a nerv algorithm for determining minimal length paths between two regions on a three dimensional surface, The numerical implementation is based on finding equal geodesic distance contours from a given area, These contours are calculated as zero sets of a bivariate function designed to evolve so as to track the equal distance curves on the given surface, The algorithm produces all paths of minimal length between the source and destination areas on the surface given as height values on a rectangular grid.

Abstract:   A new algorithm for recovering depth to a Lambertian C-1 smooth object given its gray-level image under uniform illumination from the viewing direction is presented. To recover depth, an almost arbitrarily initialized surface is numerically propagated on a rectangular grid, so that a level set of this surface tracks the height contours of the depth function. The image shading controls the propagation of the surface. When the light direction is tilted with respect to the viewing direction the problem is solved by tracking the projection of equal- height contours defined with respect to the light source direction. This projection approach provides a solution that overcomes ambiguity problems encountered in previous work, while the level set approach of implementing the contour propagation overcomes numerical problems and some of the topology problems of the evolving contours. (C) 1995 Academic Press, Inc.

Abstract:   A new algorithm for recovering depth to a Lambertian C-1 smooth object given its gray-level image under uniform illumination from the viewing direction is presented. To recover depth, an almost arbitrarily initialized surface is numerically propagated on a rectangular grid, so that a level set of this surface tracks the height contours of the depth function. The image shading controls the propagation of the surface. When the light direction is tilted with respect to the viewing direction the problem is solved by tracking the projection of equal- height contours defined with respect to the light source direction. This projection approach provides a solution that overcomes ambiguity problems encountered in previous work, while the level set approach of implementing the contour propagation overcomes numerical problems and some of the topology problems of the evolving contours. (C) 1995 Academic Press, Inc.

Abstract:   We present a controlled image smoothing and enhancement method based on a curvature flow interpretation of the geometric heat equation. Compared to existing techniques, the model has several distinct advantages. (i) It contains just one enhancement parameter. (ii) The scheme naturally inherits a stopping criterion from the image; continued application of the scheme produces no further change. (iii) The method is one of the fastest possible schemes based on a curvature-controlled approach.

Abstract:   We study a sharp-interface model for phase transitions which incorporates the interaction of tile phase boundaries with the walls of a container Omega. In this model, the interfaces move by their mean curvature and are normal to partial derivative Omega. We first establish local-in-time existence and uniqueness of smooth solutions for the mean curvature equation with a normal contact angle condition. We then discuss global solutions by interpreting the equation and the boundary condition in a weak (viscosity) sense. Finally, we investigate the relation of the aforementioned model with a transition- layer model. We prove that if Omega is convex, the transition- layer solutions converge to the sharp-interface solutions as the thickness of the layer tends to zero. We conclude with a discussion of the difficulties that arise in establishing this result in nonconvex domains.

Abstract:   We apply a level set formulation to the problem of surface advancement in a two-dimensional topography simulation of deposition, etching, and lithography processes in integrated circuit fabrication. The level set formulation is based on solving a Hamilton-Jacobi type equation for a propagating level set function, using techniques borrowed from hyperbolic conservation laws. Topological changes, corner a nd cusp development, a nd accurate determination of geometric properties such as curvature and normal direction are naturally obtained in this setting. The equations of motion of a unified model, including the effects of isotropic and unidirectional deposition and etching, visibility, surface diffusion, reflection, and material dependent etch/deposition rates are presented and adapted to a level set formulation. The development of this model and algorithm naturally extends to three dimensions in a straightforward manner and is described in part II of this paper (in press). (C) 1995 Academic Press, Inc.

Abstract:   We undertake to develop a general theory of two-dimensional shape by elucidating several principles which any such theory should meet. The principles are organized around two basic intuitions: first, if a boundary were changed only slightly, then, in general, its shape would change only slightly. This leads us to propose an operational theory of shape based on incremental contour deformations. The second intuition is that not all contours are shapes, but rather only those that can enclose ''physical'' material. A theory of contour deformation is derived from these principles, based on abstract conservation principles and Hamilton-Jacobi theory. These principles are based on the work of Sethian (1985a, c), the Osher-Sethian (1988), level set formulation the classical shock theory of Lax (1971; 1973), as well as curve evolution theory for a curve evolving as a function of the curvature and the relation to geometric smoothing of Gage-Hamilton-Grayson (1986; 1989). The result is a characterization of the computational elements of shape: deformations, parts, bends, and seeds, which show where to place the components of a shape. The theory unifies many of the diverse aspects of shapes, and leads to a space of shapes (the reaction/diffusion space), which places shapes within a neighborhood of ''similar'' ones. Such similarity relationships underlie descriptions suitable for recognition.

Abstract:   This paper presents two-dimensional direct numerical simulations of a passive flame surface passing through homogeneous isotropic turbulence. The flame was represented by a field variable, G(x, t), whose isocontours constitute flame surfaces. One well known complication in analyzing premixed combustion in a homogeneous environment is decoupling the effect of the decaying turbulent velocity field from the dynamics of the flame surface. To overcome this, the velocity field was made stationary by introducing a random forcing term into the Navier Stokes equations. Forcing was done over two different ranges of wavenumbers (k(f) = 10-14, and k(f) = 80- 84) thus creating turbulence with different length scales and inertial range power laws. By comparing the response of the flame to the two types of turbulence it was possible to determine the effect the spectral distribution energy has on the surface topology and mean rate of propagation. Indeed, the flames were found to be remarkably sensitive to the spectral distribution of the turbulent energy, and not just its magnitude. Furthermore, a k(-5/3) inertial range was shown to produce a flame surface that was preferentially wrinkled at intermediate to small scales for purely geometric reasons. By defining a surface area spectrum it was possible to rationalize this result by recognizing that flame surface area is closely related to the dissipation spectrum of the scalar field. Collectively the results suggest that knowledge of the energy spectrum al a minimum is required to predict a turbulent flame speed under general circumstances.

Abstract:   The formation of drops resulting from the breakup of an axisymmetric Newtonian liquid jet injected vertically into another immiscible Newtonian liquid at various Reynolds numbers is investigated here. The full transient from startup to breakup into drops was simulated numerically by solving the time-dependent axisymmetric equations of motion and continuity using a combination of the volume-of-fluid (VOF) and continuous-surface-force (CSF) methods. The numerical simulation results compare well with previous experimental data and are significantly more accurate than previous simplified analyses based on drop formation before and after jetting over a wide range of conditions. (C) 1995 American Institute of Physics.

Abstract:   We present a new implementation of an algorithm aimed at recovering a 3D shape from its 2D gray-level picture. In order to reconstruct the shape of the object, an almost arbitrarily initialized 3D function is propagated on a rectangular grid, so that a level set of this function tracks the height contours of the shape. The method imports techniques from differential geometry, fluid dynamics, and numerical analysis and provides an accurate shape from shading algorithm. The method solves some topological problems and gracefully handles cases of non- smooth surfaces that give rise to shocks in the propagating contours. Real and synthetic images of 3D profiles were submitted to the algorithm and the reconstructed surfaces are presented, demonstrating the effectiveness of the proposed method.

Abstract:   The level-set approach of Osher & Sethian to tracking interfaces is successfully adapted to the simulation of a premixed turbulent open V-flame including the effects of exothermicity and baroclinicity. In accord with experimental observations this algorithm, along with a flame anchoring scheme, predicts flame cusping for a case in which a strong vortex pair interacts with the flame front. The computed velocity and scalar statistics obtained for the turbulent V- flame compare reasonably well with experimental results by Cheng & Shepherd, and demonstrate the importance of flame- generated vorticity in the determination of flame dynamics and product velocity characteristics.

Abstract:   The medial axis transform (MAT) of a shape, better known as its skeleton, is frequently used in shape analysis and related areas. In this paper a new approach for determining the skeleton of an object is presented. The boundary is segmented at points of maximal positive curvature and a distance map from each of the segments is calculated. The skeleton is then located by applying simple rules to the zero sets of distance map differences. A framework is proposed for numerical approximation of distance maps that is consistent with the continuous case and hence does nor suffer from digitization bias due to metrication errors of the implementation on the grid. Subpixel accuracy in distance map calculation is obtained by using gray-level information along the boundary of the shape in the numerical scheme. The accuracy of the resulting efficient skeletonization algorithm is demonstrated by several examples. (C) 1995 Academic Press, Inc.

Abstract:   A new approach for the reconstruction of a smooth three- dimensional object from its two-dimensional gray-level image is presented. An algorithm based on topological properties of simple smooth surfaces is provided to solve the problem of global reconstruction. Classifying singular points in the shading image as maxima, minima, and two kinds of saddle points serves as the key to the solution of the problem. The global reconstruction procedure, being deterministic and using topological properties of the surface, performs better than other approaches proposed so far that are based on classification of singular points according to the behavior of characteristics in their neighborhood. The proposed algorithm is simple and easy to implement and lends itself to a parallel implementation. (C) 1995 Academic Press, Inc.

Abstract:   We show how optimization of the Nordstrom and Mumford-Shah functionals can be used to develop a type of curve-evolution that is able to preserve salient features of closed curves while simultaneously suppressing noise and irrelevant details. The idea is to characterize a curve by means of its angle- function and apply the appropriate dynamics to this representation. Upon convergence, the resulting form of the contour is reconstructed from the representation.

Abstract:   This paper introduces a discrete scheme for mean curvature motion using a morphological image processing approach. An axiomatic approach of image processing and the mean curvature partial differential equation (PDE) are briefly presented, then the properties of the proposed scheme are studied. In particular, consistency and convergence are proved. The applications of mean curvature motion in image denoising and form evolution are developed and experiences are presented.

Abstract:   We apply a level set formulation to the problem of surface advancement in three-dimensional topography simulation of deposition, etching, and lithography processes in integrated circuit fabrication. The level set formulation is based on solving a Hamilton-Jacobi-type equation for a propagating level set function, using techniques borrowed from hyperbolic conservation laws. Topological changes, corner and cusp development, and accurate determination of geometric properties such as curvature and normal direction are naturally obtained in this setting. The equations of motion of a unified model, including the effects of isotropic and unidirectional deposition and etching, visibility, surface diffusion, reflection, and material dependent etch/deposition rates are presented and adapted to a level set formulation. In Part I of this paper, the basic equations and algorithms for two- dimensional simulations were developed. In this paper, the extension to three dimensions is presented. We show a large collection of simulations, including three-dimensional etching and deposition into cavities under the effects of visibility, directional and source flux functions, evolution of lithographic profiles, discontinuous etch rates through multiple materials, and non-convex sputter yield flux functions. In Part III of this paper, effects of reflection and re-emission and surface diffusion Will be presented. (C) 1995 Academic Press, Inc.

Abstract:   In this paper, we study generalized ''viscosity'' solutions of the mean curvature evolution which were introduced by Chen, Giga, and Goto and by Evans and Spruck. We devote much of our attention to solutions whose initial value is a compact, smooth, rotationally symmetric hypersurface given by rotating a graph around an axis. Our main result is the regularity of the solution except at isolated points in spacetime and estimates on the number of such points.

Abstract:   A new geometric approach to the affine geometry of curves in the plane and to affine-invariant curve shortening is presented. We describe methods of approximating the affine curvature with discrete finite difference approximations, based on a general theory of approximating differential invariants of Lie group actions by joint invariants. Applications to computer vision are indicated. (C) 1996 Academic Press, Inc.

Abstract:   The kinematic effects of a space-time forced velocity held upon a thin premixed flame, stabilized above a circular cross- section burner, are studied in order to point out the non- linearities due to a sufficiently high velocity perturbation level whose RMS amplitudes remain nonetheless inferior to the normal burning velocity. The present calculation proposes to seek a solution using the characteristics method, without any linearized calculation, to express these effects. A front evolution equation is interpreted as the differentiated form of a conservation equation of the radial distance between two points of the front. These modelling results are used to interpret experiments of a vibrating flame subjected to a space-time sinusoidal velocity held. In this last case, the limit of cusps formation is represented as a similarity law expressing the nondimensional perturbation amplitude versus a Strouhal number of the aero-acoustic reactive medium.

Abstract:   We present a fully conservative, high resolution approach to front tracking for nonlinear systems of conservation laws in two space dimensions. An underlying uniform Cartesian grid is used, with some cells cut by the front into two subcells. The front is moved by solving a Riemann problem normal to each segment of the front and using the motion of the strongest wave to give an approximate location of the front at the end of the time step. A high resolution finite volume method is then applied on the resulting slightly irregular grid to update all cell values. A ''large time step'' wave propagation algorithm is used that remains stable in the small cut cells with a time step that is chosen with respect to the uniform grid cells. Numerical results on a radially symmetric problem show that pointwise convergence with order between 1 and 2 is obtained in both the cell values and location of the front. Other computations are also presented. (C) 1996 Academic Press, Inc.