adaptive unstructured volume remeshing ii application to two and three dimensional level set simulations of multiphase flow in part i adaptive unstructured volume remeshing i the method j comput phys in press we presented an adaptive remeshing algorithm that automatically adjusts the size of the elements of meshes of unstructured triangles 2d and unstructured tetrahedra 3d with time and position in the computational domain in order to efficiently resolve the relevant physical scales here we illustrate the performance of an implementation of the algorithm in finite element level set simulations of deformable droplet and fluid fluid interface interactions breakup and coalescence in multiphase flows the wide range of length scales characterizing the dynamics are accurately resolved as demonstrated by comparison to experiments and to theoretical and sharp interface boundary integral numerical results the computational cost is found to be competitive even with respect to boundary integral methods for the first time using an interface capturing level set method we successfully simulate the inertia driven impact and rebound of a liquid droplet from a liquid interface and find agreement with recent experimental results

uniformization of analytic spaces in this article we give an explicit elementary proof of a local version of resolution of singularities in characteristic zero local means that the centres of blowing up are chosen locally so that a finite number of finite sequences of local blowings up may be required to cover a neighbourhood of a given point hence we use the term uniformization rather than local desingularization although uniformization in our sense is considerably stronger than the original idea of zariski 16 in the last two decades several mathematicians notably abhyankar 1 hironaka and spivakovsky 12 and youssin 14 15 have proposed simpler or more explicit versions of the inductive procedure in hironaka s proof of his great theorem 8 these approaches would seem to lead to some form of uniformization though full details of none of them have yet appeared the idea of seeking an explicit local procedure to determine the centres of blowing up is that a sufficiently good local choice should globalize automatically our method for example gives global resolution of singularities of surfaces in any codimension throughout this paper k denotes either the field of real numbers r or the field of complex numbers c we will work in the category of analytic spaces over k although our techniques apply as well to algebraic spaces over a field

phase field simulations of interfacial dynamics in viscoelastic fluids using finite elements with adaptive meshing this paper describes a novel numerical algorithm for simulating interfacial dynamics of non newtonian fluids the interface between two immiscible fluids is treated as a thin mixing layer across which physical properties vary steeply but continuously the property and evolution of the interfacial layer is governed by a phase field variable that obeys a cahn hilliard type of convection diffusion equation this circumvents the task of directly tracking the interface and produces the correct interfacial tension from the free energy stored in the mixing layer viscoelasticity and other types of constitutive equations can be incorporated easily into the variational phase field framework the greatest challenge of this approach is in resolving the sharp gradients at the interface this is achieved by using an efficient adaptive meshing scheme governed by the phase field variable the finite element scheme easily accommodates complex flow geometry and the adaptive meshing makes it possible to simulate large scale two phase systems of complex fluids in two dimensional and axisymmetric three dimensional implementations the numerical toolkit is applied here to drop deformation in shear and elongational flows rise of drops and retraction of drops and torii some of these solutions serve as validation of the method and illustrate its key features while others explore novel physics of viscoelasticity in the deformation and evolution of interfaces

the geometric minimal models of analytic spaces this paper shows that an analytic space x has a unique maximal model through which every proper surjective morphism from a non singular analytic space to x factors this is called the geometric minimal model of x and characterized by the contraction property of rational curves some other properties such as functoriality the direct product property and the quotient property of the geometric minimal model are also studied here the relation of the geometric minimal model with mori s minimal model is discussed

a grid alignment finite element technique for incompressible multicomponent flows we present a numerical method for solving the multicomponent incompressible navier stokes equations the methods employs a moving grid technique and a projection scheme based on low order crouzeix raviart and p 1 finite elements the computational grid is aligned with the fluid interface at each time step to keep the interpolation order optimal the connectivity of the grid is fixed during the entire time simulation to ensure efficient assembling and parallelization the method is tested on several examples

regular article direct simulations of 2d fluid particle flows in biperiodic domains we propose a method to simulate the motion of 2d rigid particles in a viscous incompressible fluid within the arbitrary lagrangian eulerian framework momentum equations for both the fluid and the particles are discretized and a coupled variational formulation is established by introducing an appropriate finite element approximation a symmetric linear system is obtained this system is solved by an inexact uzawa algorithm the main interest of such simulations lies in the average behaviour of a high number of particles we therefore introduced a biperiodic formulation of the problem which makes it possible to represent many body mixtures at a reasonable computational cost in order to model realistic situations an extra term must be added to the pressure this extra term is shown to be the lagrange multiplier associated with the vertical volume conservation constraint we developed an appropriate unstructured mesh generator so that the biperiodicity of the fields can be treated in a strong sense the question of particle contact is addressed and a simple technique to overcome numerical problems is proposed the influence of periodic lengths is investigated through different simulations the same case is simulated for different sizes of the window and the behaviour of some space averaged quantities is studied

an arbitrary lagrangian eulerian method for moving boundary problems and its application to jumping over water we develop an ale arbitrary lagrangian eulerian moving mesh method suitable for solving two dimensional and axisymmetric moving boundary problems including the interaction between a free surface and a solid structure this method employs a body fitted grid system where the gas liquid interface and solid liquid interface are lines of the grid system and complicated dynamic boundary conditions are incorporated naturally and accurately in a finite volume formulation the resulting nonlinear system of mass and momentum conservation is then solved by a fractional step projection method the method is validated on the uniform flow passing a cylinder a two dimensional flow with a solid structure and several problems of bubble dynamics axi symmetrical flows with a free surface for both steady and unsteady flows good agreement with other theoretical numerical and experimental results is obtained a further application is the investigation of a two dimensional mechanical strider a mass spring system interacting with a water surface demonstrating the ability of the method in handling the interaction between a solid structure and a free surface we find that the critical compression required to jump off the water surface varies linearly with spring constant for stiff springs and algebraically with exponent 0 7 for weak springs

modelling merging and fragmentation in multiphase flows with surfer we introduce a new numerical method called surfer for the simulation of two and three dimensional flows with several fluid phases and free interfaces between them we consider incompressible fluids obeying the navier stokes equation with newtonian viscosity in the bulk of each phase capillary forces are taken into account even when interfaces merge or break up fluid interfaces are advanced in time using an exactly volume conserving variant of the volume of fluid algorithm thus allowing for full symmetry between fluid phases the navier stokes equation is solved using staggered finite differences on a mac grid and a split explicit time differencing scheme while incompressibility is enforced using an iterative multigrid poisson solver capillary effects are represented as a stress tensor computed from gradients of the volume fraction function this formulation is completely independent of the topology of interfaces and relatively easy to implement in 3d it also allows exact momentum conservation in the discretized algorithm numerical spurious effects or parasite currents are noticed and compared to similar effects in boltzmann lattice gas methods for immiscible fluids simulations of droplets pairs colliding in 2d and in 3d are shown interface reconnection is performed easily despite the large value of capillary forces during reconnection

evolution implementation and application of level set and fast marching methods for advancing fronts abstract a variety of numerical techniques are available for tracking moving interfaces in this review we concentrate on techniques that result from the link between the partial differential equations that describe moving interfaces and numerical schemes designed for approximating the solutions to hyperbolic conservation laws this link gives rise to computational techniques for tracking moving interfaces in two and three space dimensions under complex speed laws we discuss the evolution of these techniques the fundamental numerical approximations involved implementation details and applications in particular we review some work on three aspects of materials sciences semiconductor process simulations seismic processing and optimal structural topology design

contour surgery a topological reconnection scheme for extended integrations using contour dynamics abstract a numerical algorithm is described which it is believed can accurately model the dynamics of a two dimensional inviscid incompressible fluid with unparalled spatial resolution the fluid is assumed however to be divided into regions of uniform vorticity conservation of vorticity ensuring that this remains true for all time like contour dynamics the algorithm is concerned with following the evolution of the boundaries of vorticity discontinuity contours unlike contour dynamics the algorithm automatically removes vorticity features smaller than a predefined scale for example two contours enclosing the same uniform vorticity merge into one if they are close enough together also the curvature along a contour is not allowed to exceed the inverse of the cutoff scale at present calculations with contour surgery resolve fluid motions extending over four to five orders of magnitude of scales 13 to 20 octaves such high resolution pictures of two dimensional vortex dynamics have been facilitated by and indeed depend critically upon a nonlocal adaptive node adjustment scheme and a variety of tests quantify the accuracy of the technique

efficient implementation of thinc scheme in this paper we improve the multi dimensional thinc tangent of hyperbola for interface capturing scheme f xiao y honma t kono a simple algebraic interface capturing scheme using hyperbolic tangent function int j numer meth fluid 48 2005 1023 the thinc scheme is a vof volume of fluid type method in the original thinc scheme one dimensional thinc scheme was straightforwardly used for multi dimensional cases in this paper we propose the wlic weighed line interface calculation method to extend the thinc scheme to multi dimension in the wlic method the interface is reconstructed by taking an average of line interfaces along x y and z coordinates with the weights calculated from surface normal the wlic method can extend the thinc scheme to multi dimension while maintaining simplicity of implementation and achieve a higher accuracy than the original thinc scheme the wlic method can readily extend the thinc scheme to three dimension

motion of multiple junctions a level set approach 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

reconstructing volume tracking a new algorithm for the volume tracking of interfaces in two dimensions is presented the algorithm is based upon a well defined second order geometric solution of a volume evolution equation the method utilizes local discrete material volume and velocity data to track interfaces of arbitrarily complex topology a linearity preserving piecewise linear interface geometry approximation ensures that solutions generated retain second order spatial accuracy second order temporal accuracy is achieved by virtue of a multidimensional unsplit time integration scheme we detail our geometrically based solution method in which material volume fluxes are computed systematically with a set of simple geometric tasks we then interrogate the method by testing its ability to track interfaces through large controlled topology changes whereby an initially simple interface configuration is subjected to vortical flows numerical results for these strenuous test problems provide evidence for the algorithm s improved solution quality and accuracy

a mesh dependent model for applying dynamic contact angles to vof simulations typical vof algorithms rely on an implicit slip that scales with mesh refinement to allow contact lines to move along no slip boundaries as a result solutions of contact line phenomena vary continuously with mesh spacing this paper presents examples of that variation a mesh dependent dynamic contact angle model is then presented that is based on fundamental hydrodynamics and serves as a more appropriate boundary condition at a moving contact line this new boundary condition eliminates the stress singularity at the contact line the resulting problem is thus well posed and yields solutions that converge with mesh refinement numerical results are presented of a solid plate withdrawing from a fluid pool and of spontaneous droplet spread at small capillary and reynolds numbers

prost a parabolic reconstruction of surface tension for the volume of fluid method volume of fluid vof methods are popular for the direct numerical simulation of time dependent viscous incompressible flow of multiple liquids as in any numerical method however it has its weaknesses namely for flows in which the capillary force is the dominant physical mechanism the lack of convergence with spatial refinement or convergence to a solution that is slightly different from the exact solution has been documented in the literature a well known limiting case for this is the existence of spurious currents for the simulation of a spherical drop with zero initial velocity these currents are present in all previous versions of vof algorithms in this paper we develop an accurate representation of the body force due to surface tension which effectively eliminates spurious currents we call this algorithm prost parabolic reconstruction of surface tension there are several components to this procedure including the new body force algorithm improvements in the projection method for the navier stokes solver and a higher order interface advection scheme the curvature to the interface is calculated from an optimal fit for a quadratic approximation to the interface over groups of cells

applications of q lagrange inversion to basic hypergeometric series a family of g lagrange inversion formulas is given special cases include quadratic and cubic transformations for basic hypergeometric series the g analogs of the so called strange evaluations are also corollaries some new rogers ramanujan identities are given a connection between the work of rogers and andrews and lagrange inversion is stated

numerical simulation of drop impact on a liquid liquid interface with a multiple marker front capturing method the gravity driven motion of a droplet impacting on a liquid liquid interface is studied the full navier stokes equations are solved on a fixed uniform grid using a finite difference front capturing method for the representation of fluid fluid interfaces a coupled level set volume of fluid method m sussman e g puckett a coupled level set and volume of fluid method for computing 3d and axisymmetric incompressible two phase flows j comp phys 162 2000 301 337 is used in which we introduce the novel approach of describing separate interfaces with different marker functions as a consequence we prevent numerical coalescence of the droplet and the liquid liquid interface without excessive local grid refinement to validate our method numerical simulations of the drop impact event are compared with experiments z mohamed kassim e k longmire drop impact on a liquid liquid interface phys fluids 15 2003 3263 3273 furthermore a comparison is made with the numerical results of a esmaeeli g tryggvason direct numerical simulations of bubbly flows part 2 moderate reynolds number arrays j fluid mech 385 1999 325 358 for an array of rising bubbles the investigation shows that the multiple marker approach successfully prevents numerical coalescence of interfaces and adequately captures the effect of surface tension

geometric representations of graded and rational cherednik algebras abstract we provide geometric constructions of modules over the graded cherednik algebra h gr and the rational cherednik algebra h rat attached to a simple algebraic group g together with a pinned automorphism these modules are realized on the cohomology of affine springer fibers of finite type that admit c actions in the rational cherednik algebra case the standard grading on these modules is derived from the perverse filtration on the cohomology of affine springer fibers coming from its global analog hitchin fibers when is trivial we show that our construction gives the irreducible finite dimensional spherical modules l triv of h gr and of h rat we give a formula for the dimension of l triv and give a geometric interpretation of its frobenius algebra structure the rank two cases are studied in further details

canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant this article contains an elementary constructive proof of resolution of singularities in characteristic zero our proof applies in particular to schemes of finite type and to analytic spaces so we recover the great theorems of hironaka we introduce a discrete local invariant rm inv x a whose maximum locus determines a smooth centre of blowing up leading to desingularization to define rm inv x we need only to work with a category of local ringed spaces x x cal o x satisfying certain natural conditions if a in x then rm inv x a depends only on widehat cal o x a more generally rm inv x is defined inductively after any sequence of blowings up whose centres have only normal crossings with respect to the exceptional divisors and lie in the constant loci of rm inv x cdot the paper is self contained and includes detailed examples one of our goals is that the reader understand the desingularization theorem rather than simply know it is true

an arbitrary lagrangian eulerian method for three phase flows with triple junction points we present a moving mesh method suitable for solving two dimensional and axisymmetric three liquid flows with triple junction points this method employs a body fitted unstructured mesh where the interfaces between liquids are lines of the mesh system and the triple junction points if exist are mesh nodes to enhance the accuracy and the efficiency of the method the mesh is constantly adapted to the evolution of the interfaces by refining and coarsening the mesh locally dynamic boundary conditions on interfaces in particular the triple points are therefore incorporated naturally and accurately in a finite element formulation in order to allow pressure discontinuity across interfaces double values of pressure are necessary for interface nodes and triple values of pressure on triple junction points the resulting non linear system of mass and momentum conservation is then solved by an uzawa method with the zero resultant condition on triple points reinforced at each time step the method is used to investigate the rising of a liquid drop with an attached bubble in a lighter liquid

on the cohomology of fano varieties and the springer correspondence abstract in this paper we compute the cohomology of the fano varieties of k planes in the smooth complete intersection of two quadrics in p 2 g 1 using springer theory for symmetric spaces

a continuum method for modeling surface tension abstract a new method for modeling surface tension effects on fluid motion has been developed interfaces between fluids of different properties or colors are represented as transition regions of finite thickness across which the color variable varies continuously at each point in the transition region a force density is defined which is proportional to the curvature of the surface of constant color at that point it is normalized so that the conventional description of surface tension on an interface is recovered when the ratio of local transition region thickness to local radius of curvature approaches zero the continuum method eliminates the need for interface reconstruction simplifies the calculation of surface tension enables accurate modeling of two and three dimensional fluid flows driven by surface forces and does not impose any modeling restrictions on the number complexity or dynamic evolution of fluid interfaces having surface tension computational results for two dimensional flows are given to illustrate the properties of the method

a front tracking method for viscous incompressible multi fluid flows abstract a method to simulate unsteady multi fluid flows in which a sharp interface or a front separates incompressible fluids of different density and viscosity is described the flow field is discretized by a conservative finite difference approximation on a stationary grid and the interface is explicitly represented by a separate unstructured grid that moves through the stationary grid since the interface deforms continuously it is necessary to restructure its grid as the calculations proceed in addition to keeping the density and viscosity stratification sharp the tracked interface provides a natural way to include surface tension effects both two and three dimensional full numerical simulations of bubble motion are presented

numerical equivalence defined on chow groups of noetherian local rings in the present paper we define a notion of numerical equivalence on chow groups or grothendieck groups of noetherian local rings which is an analogue of that on smooth projective varieties under a mild condition it is proved that the chow group modulo numerical equivalence is a finite dimensional vector space as in the case of smooth projective varieties numerical equivalence on local rings is deeply related to that on smooth projective varieties for example if grothendieck s standard conjectures are true then a vanishing of chow group of local rings modulo numerical equivalence can be proven using the theory of numerical equivalence the notion of numerically roberts rings is defined it is proved that a cohen macaulay local ring of positive characteristic is a numerically roberts ring if and only if the hilbert kunz multiplicity of a maximal primary ideal of finite projective dimension is always equal to its colength numerically roberts rings satisfy the vanishing property of intersection multiplicities we shall prove another special case of the vanishing of intersection multiplicities using a vanishing of localized chern characters

numerical schemes for the hamilton jacobi and level set equations on triangulated domains borrowing from techniques developed for conservation law equations numerical schemes which discretize the hamilton jacobi h j level set and eikonal equations on triangulated domains are presented the first scheme is a provably monotone discretization for the h j equations unfortunately the basic scheme lacks lipschitz continuity of the numerical hamiltonian by employing a virtual edge flipping technique local lipschitz continuity of the numerical flux is restored on acute triangulations next schemes are introduced and developed based on the weaker concept of positive coefficient approximations for homogeneous hamiltonians these schemes possess a discrete maximum principle on arbitrary triangulations and exhibit local lipschitz continuity of the numerical hamiltonian finally a class of petrov galerkin approximations is considered these schemes are stabilized via a least squares bilinear form the petrov galerkin schemes do not possess a discrete maximum principle but generalize to high order accuracy discretization of the level set equation also requires the numerical approximation of a mean curvature term a simple mass lumped galerkin approximation is presented in section 6 and analyzed using maximum principle analysis the use of unstructured meshes permits several forms of mesh adaptation which have been incorporated into numerical examples these numerical examples include discretizations of convex and nonconvex forms of the h j equation the eikonal equation and the level set equation