three dimensional volume conserving immersed boundary model for two phase fluid flows abstract we present a volume preserving scheme for two phase immiscible incompressible flows using an immersed boundary method ibm in a three dimensional space the two phase ibm employs a mixture of eulerian and lagrangian variables where the fluid interface is represented by discrete lagrangian markers exerting surface tension forces to the eulerian fluid domain and the markers are advected by the fluid velocity the interactions between the lagrangian markers and the fluid variables are linked by the discretized dirac delta function the present study extends the previous two dimensional research li et al volume preserving immersed boundary methods for two phase fluid flows int j numer meth fluids 69 2012 842 858 to the three dimensional space the key idea of the proposed method is relocating surface points along the normal directions to conserve the total volume we perform a number of numerical experiments to show the efficiency and accuracy of the proposed method

computations of multi fluid flows abstract full numerical simulations of three dimensional flows of two or more immiscible fluids of different densities and viscosities separated by a sharp interface with finite surface tension are discussed the method used is based on a finite difference approximation of the full navier stokes equations and explicit tracking of the interface between the fluids preliminary simulations of the rayleigh taylor instability and the motion of bubbles are shown

an unsplit higher order godunov method for scalar conservation laws in multiple dimensions abstract in this paper we develop an unsplit higher order godunov method for scalar conservation laws in two dimensions the method represents an extension for the special case being considered of methods developed by colella and by van leer in our method we begin with a piecewise bilinear representation of the solution on each grid cell a new piecewise bilinear representation at the next time level is then obtained from a two step procedure in the first step a conservative predictor corrector scheme derived from the integral form of the differential equation and characteristic considerations is used to obtain average values of the solution over grid cells at the new time level next these new average values are then used to construct a limited piecewise bilinear profile for each cell at the new time level the resulting method is shown to satisfy a maximum principle for constant coefficient linear advection computational results are presented comparing the new method to colella s method for linear advection the method is also applied to two model problems from porous media flow miscible and immiscible displacement the new scheme provides accurate resolution of sharp fronts without any significant distortion

a quadratic spline based interface quasi reconstruction algorithm for accurate tracking of two phase flows a new quadratic spline based interface quasi reconstruction algorithm is presented which provides an accurate and continuous representation of the interface in a multiphase domain and facilitates the direct estimation of local interfacial curvature the fluid interface in each of the mixed cells is represented by piecewise parabolic curves and an initial discontinuous plic approximation of the interface is progressively converted into a smooth quadratic spline made of these parabolic curves the conversion is achieved by a sequence of predictor corrector operations enforcing function c 0 and derivative c 1 continuity at the cell boundaries using simple analytical expressions for the continuity requirements the efficacy and accuracy of the current algorithm has been demonstrated using standard test cases involving reconstruction of known static interface shapes and dynamically evolving interfaces in prescribed flow situations these benchmark studies illustrate that the present algorithm performs excellently as compared to the other interface reconstruction methods available in literature quadratic rate of error reduction with respect to grid size has been observed in all the cases with curved interface shapes only in situations where the interface geometry is primarily flat the rate of convergence becomes linear with the mesh size the flow algorithm implemented in the current work is designed to accurately balance the pressure gradients with the surface tension force at any location as a consequence it is able to minimize spurious flow currents arising from imperfect normal stress balance at the interface this has been demonstrated through the standard test problem of an inviscid droplet placed in a quiescent medium finally the direct curvature estimation ability of the current algorithm is illustrated through the coupled multiphase flow problem of a deformable air bubble rising through a column of water

crystal growth and dendritic solidification we present a numerical method which computes the motion of complex solid liquid boundaries in crystal growth the model we solve includes physical effects such as crystalline anisotropy surface tension molecular kinetics and undercooling the method is based on two ideas first the equations of motion are recast as a single history dependent boundary integral equation on the solid liquid boundary a fast algorithm is used to solve the integral equation efficiently second the boundary is moved by solving a hamiltonjacobi type equation on a fixed domain formulated by osher and sethian for a function in which the boundary is a particular level set this equation is solved by finite difference schemes borrowed from the technology of hyperbolic conservation laws the two ideas are combined by constructing a smooth extension of the normal velocity off the moving boundary in a way suggested by the physics of the problem our numerical experiments show the evolution of complex crystalline shapes development of large spikes and corners dendrite formation and side branching and pieces of solid merging and breaking off freely

multidimensional upwind methods for hyperbolic conservation laws abstract we present a class of second order conservative finite difference algorithms for solving numerically time dependent problems for hyperbolic conservation laws in several space variables these methods are upwind and multidimensional in that the numerical fluxes are obtained by solving the characteristic form of the full multidimensional equations at the zone edge and that all fluxes are evaluated and differenced at the same time in particular operator splitting is not used correct behavior at discontinuities is obtained by the use of solutions to the riemann problem and by limiting some of the second order terms numerical results are presented which show that the methods described here yield the same high resolution as the corresponding operator split methods

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

a survey of search directions in interior point methods for linear programming a basic characteristic of an interior point algorithm for linear programming is the search direction many papers on interior point algorithms only give an implicit description of the search direction in this report we derive explicit expressions for the search directions used in many well known algorithms comparing these explicit expressions gives a good insight into the similarities and differences between the various algorithms moreover we give a survey of projected gradient and newton directions for all potential and barrier functions this is done both for the affine and projective variants

computing interface motion in compressible gas dynamics a hamilton jacobi level set formulation of the equations of motion for propagating interfaces has been introduced recently by osher and sethian this formulation allows fronts to self intersect develop singularities and change topology the numerical algorithms based on this approach handle topological merging and breaking naturally work in any number of space dimensions and do not require that the moving front be written as a function instead the moving front is embedded as a particular level set in a fixed domain partial differential equation numerical techniques borrowed from hyperbolic conservation laws are then used to accurately capture the complicated surface motion that satisfies the global entropy condition for propagating fronts given by sethian in this paper we analyze the coupling of this level set formulation to a system of conservation laws for compressible gas dynamics we study both conservative and non conservative differencing of the level set function and compare the two approaches to illustrate the capability of the method we study the compressible rayleigh taylor and kelvin helmholtz instabilities for air air and air helium boundaries we perform numerical convergence studies of the method over a range of parameters and analyze the accuracy of this approach applied to these problems

a 3d adaptive mesh refinement algorithm for multimaterial gas dynamics abstract adaptive mesh refinement amr in conjunction with high order upwind finite difference methods has been used effectively on a variety of problems in this paper we discuss an implementation of an amr finite difference method that solves the equations of gas dynamics with two material species in three dimensions an equation for the evolution of volume fractions augments the gas dynamics system the material interface is preserved and tracked from the volume fractions using a piecewise linear reconstruction technique

a dual time method for two dimensional unsteady incompressible flow calculations a method for computing unsteady incompressible viscous flows on moving or deforming meshes is described it uses a well established time marching finite volume flow solver developed for steady compressible flows past rigid bodies time marching methods cannot be applied directly to incompressible flows because the governing equations are not hyperbolic such methods can be extended to steady incompressible flows using an artificial compressibility scheme a time accurate scheme for unsteady incompressible flows is achieved by using an implicit real time discretization and a dual time approach which uses a technique similar to the artificial compressibility scheme results are presented for test cases on both fixed and deforming meshes experimental numerical and theoretical data have been included for comparison where available and reasonable agreement has been achieved 1998 john wiley sons ltd

a level set approach for computing solutions to incompressible two phase flow 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 a new treatment of the level set method allows us to efficiently maintain the level set function as the signed distance from the interface we never have to explicitly reconstruct or find the zero level set consequently we are able to handle arbitrarily complex topologies large density and viscosity ratios and surface tension on relatively coarse grids we use a second order projection method along with a second order upwinded procedure for advecting the momentum and level set equations we consider the motion of air bubbles and water drops we also compute flows with multiple fluids such as air oil and water

a conservative level set method for two phase flow ii in this paper we continue to develop and study the conservative level set method for incompressible two phase flow with surface tension introduced in j comput phys 210 2005 225 246 we formulate a modification of the reinitialization and present a theoretical study of what kind of conservation we can expect of the method a finite element discretization is presented as well as an adaptive mesh control procedure numerical experiments relevant for problems in petroleum engineering and material science are presented for these problems the surface tension is strong and conservation of mass is important problems in both two and three dimensions with uniform as well as non uniform grids are studied from these calculations convergence and conservation is studied good conservation and convergence are observed

a numerical method for the incompressible navier stokes equations based on an approximate projection in this method we present a fractional step discretization of the time dependent incompressible navier stokes equations the method is based on a projection formulation in which we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields our treatment of the diffusion convection step uses a specialized second order upwind method for differencing the nonlinear convective terms that provides a robust treatment of these terms at a high reynolds number in contrast to conventional projection type discretizations that impose a discrete form of the divergence free constraint we only approximately impose the constraint i e the velocity field we compute is not exactly divergence free the approximate projection is computed using a conventional discretization of the laplacian and the resulting linear system is solved using conventional multigrid methods numerical examples are presented to validate the second order convergence of the method for euler finite reynolds number and stokes flow a second example illustrating the behavior of the algorithm on an unstable shear layer is also presented

lattice boltzmann model for free surface flow and its application to filling process in casting a generalized lattice boltzmann model to simulate free surface is constructed in both two and three dimensions the proposed model satisfies the interfacial boundary conditions accurately a distinctive feature of the model is that the collision processes is carried out only on the points occupied partially or fully by the fluid to maintain a sharp interfacial front the method includes an anti diffusion algorithm the unknown distribution functions at the interfacial region are constructed according to the first order chapman enskog analysis the interfacial boundary conditions are satisfied exactly by the coefficients in the chapman enskog expansion the distribution functions are naturally expressed in the local interfacial coordinates the macroscopic quantities al the interface are extracted from the least square solutions of a locally linearized system obtained from the known distribution functions the proposed method does not require any geometric front construction and is robust for any interfacial topology simulation results of realistic filling process are presented rectangular cavity in two dimensions and hammer box campbell box sheffield box and motorblock in three dimensions to enhance the stability at high reynolds numbers various upwind type schemes are developed free slip and no slip boundary conditions are also discussed

second order accurate volume of fluid algorithms for tracking material interfaces we introduce two new volume of fluid interface reconstruction algorithms and compare the accuracy of these algorithms to four other widely used volume of fluid interface reconstruction algorithms we find that when the interface is smooth e g continuous with two continuous derivatives the new methods are second order accurate and the other algorithms are first order accurate we propose a design criteria for a volume of fluid interface reconstruction algorithm to be second order accurate namely that it reproduce lines in two space dimensions or planes in three space dimensions exactly we also introduce a second order unsplit volume of fluid advection algorithm that is based on a second order finite difference method for scalar conservation laws due to bell dawson and shubin we test this advection algorithm by modeling several different interface shapes propagating in two simple incompressible flows and compare the results with the standard second order operator split advection algorithm although both methods are second order accurate when the interface is smooth we find that the unsplit algorithm exhibits noticeably better resolution in regions where the interface has discontinuous derivatives such as at corners

a balanced force refined level set grid method for two phase flows on unstructured flow solver grids a balanced force refined level set grid method for two phase flows on structured and unstructured flow solver grids is presented to accurately track the phase interface location an auxiliary high resolution equidistant cartesian grid is introduced in conjunction with a dual layer narrow band approach this refined level set grid method allows for parallel efficient grid convergence and error estimation studies of the interface tracking method the navier stokes equations are solved on an unstructured flow solver grid with a novel balanced force algorithm for level set methods based on the recently proposed method by francois et al m m francois s j cummins e d dendy d b kothe j m sicilian m w williams a balanced force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework j comput phys 213 2006 141 173 for volume of fluid methods on structured grids to minimize spurious currents a second order converging curvature evaluation technique for level set methods is presented the results of several different test cases demonstrate the effectiveness of the proposed method showing good mass conservation properties and second order converging spurious current magnitudes

a surface marker algorithm coupled to an area preserving marker redistribution method for three dimensional interface tracking in this work we present a new interface tracking algorithm based on surface markers to reconstruct and advect interfaces in three dimensional space the algorithm is coupled to a marker redistribution method which is area preserving along the interface the interface is described by a set of closed lines which define a coarse lagrangian quadrangular mesh in the fixed computational grid fixed markers are located where the lines cross and are maintained during the whole simulation at each time step the interface is first reconstructed and then all markers are advected by following streamlines in the reconstruction step new markers are determined by computing the intersections of the interface lines with the grid cell faces and by adding area conservation markers between fixed and grid intersection markers intersection and conservation markers defined at the previous time step are discarded the method maintains a smooth geometrical description of the interface for both two dimensional and three dimensional tests with accurate volume conservation even in very challenging situations where the fluid bodies progressively deform and stretch developing localized regions with very high curvature and thin fluid filaments the method compares favorably with front capturing methods such as volume of fluid and level set and other hybrid techniques such as the particle level set method

an efficient wave based prediction technique for plate bending vibrations the finite element method is a commonly used prediction method for dynamic simulations of plate bending problems a major disadvantage of this method is its practical frequency limitation as its application involves prohibitively large computational efforts at higher frequencies a newly developed wave based prediction technique pushes up the frequency limit through an enhanced computational efficiency this paper discusses the principles of the wave based method for convex plate problems the beneficial convergence rate of the wave based method as compared with the finite element method is verified for various validation examples which include the most commonly encountered boundary conditions

a boundary integral approach to unstable solidification we consider the supercooled stefan problem with a general anisotropic curvature and velocity dependent boundary condition on the moving interface we present numerical methods based on an integral equation formulation and including a new algorithm for moving curves with curvature dependent velocity these methods compute a periodic interface with o t accuracy where t is the time step previous work has been limited to short time spans and achieved slightly less than o t12 accuracy accurate numerical results are seen to agree with the predictions of linear stability theory this agreement has eluded previous authors because their numerical methods suffered from grid effects and their linear stability theory was incorrect we study the long time evolution of an unstable interface our computations exhibit the beginnings of a sidebranching instability when the boundary condition includes anisotropy and tip splitting in the isotropic case

on the convexity of the multiplicative version of karmarkar s potential function karmarkar s potential function is quasi convex but not convex this note investigates the multiplicative version of the potential function and shows that it is not necessarily convex in general but is strictly convex when the corresponding feasible region is bounded this implies that the multiplicative version of the potential function in karmarkar s algorithm is convex since it works on a simplex

cellwise conservative unsplit advection for the volume of fluid method we present a cellwise conservative unsplit ccu advection scheme for the volume of fluid method vof in 2d contrary to other schemes based on explicit calculations of the flux balances the ccu advection adopts a cellwise approach where the pre images of the control volumes are traced backwards through the flow map the donating regions of the fluxes are calculated via the streaklines of the grid intersections represented as polygonal chains whose vertices are determined by backward tracing of particles injected in the flow at different times high order accuracy is obtained from the fourth order runge kutta method where intermediate velocities along pathlines are determined with quadratic temporal and bicubic spatial interpolations the volumes of the donating regions are corrected in order to fulfill the discrete continuity of incompressible flows consequently the calculation produces non overlapping donating regions and pre images with conforming edges to their neighbors resulting in the conservativeness and the boundedness liquid volume fraction inside the interval 0 1 of the ccu advection scheme finally the update of the liquid volume fractions is computed from the intersections of the pre image polygons with the reconstructed interfaces the ccu scheme is tested on several benchmark tests for the vof advection together with the standard piecewise linear interface calculation plic the geometrical errors of the ccu compare favorably with other unsplit vof plic schemes finally potential improvements of the vof method with the use of more precise interface representation techniques and the future extension of the ccu scheme to 3d are discussed cellwise vof update through backward tracing of the cell s pre image accuracy enhanced via bicubic spatial interpolation and polygonal chain streakline the ccu advection scheme is fully conservative and bounded no limitation on the courant number geometrical errors dominated by plic interface representation errors

the nonlinear geometry of linear programming i affine and projective scaling trajectories this series of papers studies a geometric structure underlying karmarkar s projective scaling algorithm for solving linear programming problems a basic feature of the projective scaling algorithm is a vector field depending on the objective function which is defined on the interior of the polytope of feasible solutions of the linear program the geometric structure studied is the set of trajectories obtained by integrating this vector field which we call ptrajectories we also study a related vector field the affine scaling vector field and its associated trajectories called a trajectories the affine scaling vector field is associated to another linear programming algorithm the affine scaling algorithm affine and projective scaling vector fields are each defined for linear programs of a special form called strict standard form and canonical form respectively this paper derives basic properties of p trajectories and a trajectones it reviews the projective and affine scaling algorithms defines the projective and affine scaling vector fields and gives differential equations for p trajectories and a trajectories it shows that projective transformations map p trajectories into p trajectories it presents karmarkar s interpretation of a trajectories as steepest descent paths of the objective function c x with respect to the riemannian geometry ds2 z dx dx x2 restricted to the relative interior of the polytope of feasible solutions p trajectories of a canonical form linear program are radial projections of a trajectories of an associated standard form linear program as a consequence there is a polynomial time linear programming algorithm using the affine scaling vector field of this associated linear program this algorithm is essentially karmarkar s algorithm these trajectories are studied in subsequent papers by two nonlinear changes of variables called legendre transform coordinates and projective legendre transform coordinates respectively it will be shown that p trajectories have an algebraic and a geometric interpretation they are algebraic curves and they are geodesics actually distinguished chords of a geometry isometric to a hilbert geometry on a polytope combinatorially dual to the polytope of feasible solutions the a trajectories of strict standard form linear programs have similar interpretations they are algebraic curves and are geodesics of a geometry isometric to euclidean geometry received by the editors july 28 1986 and in revised form september 28 1987 and march 21 1988 1980 mathematics subject classification 1985 revision primary 90c05 secondary 52a40 34a34 research of the first author partially supported by onr contract n00014 87 k0214 d 1989 american mathematical society 0002 9947 89 1 00 25 per page

a second order projection method for variable density flows this paper describes a second order projection method for variabledensity incompressible flows the method is suitable for both finite amplitude density variations and for fluids that are modeled using a boussinesq approximation it is based on a second order fractional step scheme in which diffusion convection terms are advanced without enforcing the incompressibility condition and the resulting intermediate velocity field is then projected onto the space of discretely divergencefree vector fields the nonlinear convection terms are treated using a godunov type procedure that is second order for smooth flow and remain stable and non oscillatory for nonsmooth flows with low fluid viscosities the method is described for finite amplitude density variation and the simplifications for a boussinesq approximation are sketched numerical results are presented that validate the convergence properties of the method and demonstrate its performance on more realistic problems

the nonlinear geometry of linear programming ii legendre transform coordinates and central trajectories karmarkar s projective scaling algorithm for solving linear programming problems associates to each objective function a vector field defined in the interior of the polytope of feasible solutions of the problem this paper studies the set of trajectories obtained by integrating this vector field called p trajectories as well as a related set of trajectories called a trajectories the a trajectories arise from another linear programming algorithm the affine scaling algorithm the affine and projective scaling vector fields are each defined for linear programs of a special form called standard form and canonical form respectively these trajectories are studied using a nonlinear change of variables called legendre transform coordinates which is a projection of the gradient of a logarithmic barrier function the legendre transform coordinate mapping is given by rational functions and its inverse mapping is algebraic it depends only on the constraints of the linear program and is a one to one mapping for canonical form linear programs when the polytope of feasible solutions is bounded there is a unique point mapping to zero called the center the a trajectories of standard form linear programs are linearized by the legendre transform coordinate mapping when the polytope of feasible solutions is bounded they are the complete set of geodesics of a riemannian geometry isometric to euclidean geometry each a trajectory is part of a real algebraic curve each p trajectory for a canonical form linear program lies in a plane in legendre transform coordinates the p trajectory through 0 in legendre transform coordinates called the central p trajectory is part of a straight line and is contained in the a trajectory through 0 called the central a trajectory each p trajectory is part of a real algebraic curve the central a trajectory is the locus of centers of a family of linear programs obtained by adding an extra equality constraint of the form c x u it is also the set of minima of a parametrized family of logarithmic barrier functions power series expansions are derived for the central a trajectory which is also the central p trajectory these power series have a simple recursive form and are useful in developing higher order analogues of karmarkar s algorithm a trajectories are defined for a general linear program using this definition it is shown that the limit point x 0 of a central a trajectory on the boundary of the feasible solution polytope p is the center of the unique face of p containing x 0 in its relative interior received by the editors october 8 1986 and in revised form june 9 1987 and march 25 1988 1980 mathematics subject classification 1985 revision primary 90c05 secondary 52a40 34a34 the first author was partially supported by onr contract n00014 87 k0214 1989 american mathematical society 0002 9947 89 1 00 25 per page

short note an interface capturing method with a continuous function the thinc method on unstructured triangular and tetrahedral meshes a novel interface capturing method is proposed to compute moving interfaces on unstructured grids with triangular 2d and tetrahedral 3d elements different from the conventional vof volume of fluid method which involves geometric reconstructions of the interface the present method is based on the algebraic reconstruction approach originally developed in the thinc tangent of hyperbola interface capturing scheme by xiao et al 2005 17 a continuous multidimensional hyperbolic tangent function is employed for retrieving the jump like distribution of the indicator function which avoids the explicit geometric representation of the interface and thus substantially reduces the algorithmic complexity in unstructured grids numerical diffusion and smearing are effectively eliminated and the compact thickness of the jump transition layer in the volume fraction is retained throughout the computation even for largely deformed interface the solution quality of the present scheme is comparable to the vof method with plic piecewise linear interface calculation algorithm

a geometrical area preserving volume of fluid advection method a new class of algorithms that preserve mass exactly for incompressible flows on a cartesian mesh are presented they amount to piecewise linear area preserving mappings of tessellations of the plane they are equivalent to volume of fluid vof advection methods which are decomposed into an eulerian implicit scheme in one direction followed by a lagrangian explicit step in the other one it is demonstrated that mass conservation is exact for incompressible flows and that there are no undershoots or overshoots of the volume fraction which thus always remains constrained between 0 and 1

a level set formulation of eulerian interface capturing methods for incompressible fluid flows a level set formulation is derived for incompressible immiscible navier stokes equations separated by a free surface the interface is identified as the zero level set of a smooth function eulerian finite difference methods based on this level set formulation are proposed these methods are robust and efficient and are capable of computing interface singularities such as merging and reconnection numerical experiments are presented to demonstrate the effectiveness of the methods

a fast pressure correction method for incompressible two fluid flows we have developed a new pressure correction method for simulating incompressible two fluid flows with large density and viscosity ratios the method s main advantage is that the variable coefficient poisson equation that arises in solving the incompressible navier stokes equations for two fluid flows is reduced to a constant coefficient equation which can be solved with an fft based fast poisson solver this reduction is achieved by splitting the variable density pressure gradient term in the governing equations the validity of this splitting is demonstrated from our numerical tests and it is explained from a physical viewpoint in this paper the new pressure correction method is coupled with a mass conserving volume of fluid method to capture the motion of the interface between the two fluids but in general it could be coupled with other interface advection methods such as level set phase field or front tracking first we verified the new pressure correction method using the capillary wave test case up to density and viscosity ratios of 10 000 then we validated the method by simulating the motion of a falling water droplet in air and comparing the droplet terminal velocity with an experimental value next the method is shown to be second order accurate in space and time independent of the vof method and it conserves mass momentum and kinetic energy in the inviscid limit also we show that for solving the two fluid navier stokes equations the method is 10 40 times faster than the standard pressure correction method which uses multigrid to solve the variable coefficient poisson equation finally we show that the method is capable of performing fully resolved direct numerical simulation dns of droplet laden isotropic turbulence with thousands of droplets using a computational mesh of 1024 3 points

a high order projection method for tracking fluid interfaces in variable density incompressible flows we present a numerical method for computing solutions of the incompressible euler or navier stokes equations when a principal feature of the flow is the presence of an interface between two fluids with different fluid properties the method is based on a second order projection method for variable density flows using an approximate projection formulation the boundary between the fluids is tracked with a second order volume of fluid interface tracking algorithm we present results for viscious rayleigh taylor problems at early time with equal and unequal viscosities to demonstrate the convergence of the algorithm we also present computational results for the rayleigh taylor instability in air helium and for bubbles and drops in an air water system without surface tension to demonstrate the behavior of the algorithm on problems with large density and viscosity contrasts

ripple a new model for incompressible flows with free surfaces a new free surface flow model ripple is summarised ripple obtains finite difference solutions for incompressible flow problems having significant surface tension forces at free surfaces of arbitrarily complex topology the key innovation is the continuum surface force model which represents surface tension as a localised volume force other features include a higher order momentum advection model a volume of fluid free surface treatment and an efficient two step projection solution method ripple s unique capabilities are illustrated with two example problems low gravity jet induced tank flow and the collision and coalescence of two cylindrical rods

projection methods for two velocity two pressure models for flows of heterogeneous mixtures in this paper a numerical analysis of two velocity two pressure models for flows of solid particles and fluids is presented first a formal exploitation of the weak formulation of such models asserts that they are amenable to integration via projection methods the challenging issues in the algorithm development for these models are then documented and suitable numerical methodologies for their remedy are devised subsequently an algorithm for the integration of the models of interest is proposed this is a two phase projection method on collocated grids that utilizes a fractional step time marching scheme it is further endowed with an interface detection and treatment methodology to properly account for the stiffness induced by the presence of moving and deforming material interfaces the efficiency and robustness of the proposed numerical method are assessed in a series of numerical experiments that are delineated in the last part of this paper

the development of a free surface capturing approach for multidimensional free surface flows in closed containers a new surface capturing method is developed for numerically simulating viscous free surface flows in partially filled containers the method is based on the idea that the flow of two immiscible fluids within a closed container is governed by the equations of motion for an incompressible viscous nonhomogeneous variable density fluid by computing the flow fields in both the liquid and gas regions in a consistent manner the free surface can be captured as a discontinuity in the density field thereby eliminating the need for special free surface tracking procedures the numerical algorithm is developed using a conservative implicit finite volume discretization of the equations of motion the algorithm incorporates the artificial compressibility method in conjunction with a dual time stepping strategy to maintain a divergence free velocity field a slope limited high order muscl scheme is adopted for approximating the inviscid flux terms while the viscous fluxes are centrally differenced the capabilities of the surface capturing method are demonstrated by calculating solutions to several two and three dimensional problems

integrability of vector and multivector fields associated with interior point methods for linear programming in the feasible region of a linear programming problem a number of desirably good directions have been defined in connexion with various interior point methods each of them determines a contravariant vector field in the region whose only stable critical point is the optimum point some interior point methods incorporate a two or higher dimensional search which naturally leads us to the introduction of the corresponding contravariant multivector field we investigate the integrability of those multivector fields i e whether a contravariantp vector field isxp forming is enveloped by a family ofxq s q p or envelops a family ofxq s q p in j a schouten s terminology wherexq is aq dimensional manifold immediate consequences of known facts are 1 the directions hitherto proposed arex1 forming with the optimum point of the linear programming problem as the stable accumulation point and 2 there is anx2 forming contravariant bivector field for which the center path is the critical submanifold most of the meaningfulp vector fields withp 3 are notxp forming in general though they envelop that bivector field this observation will add another circumstantial evidence that the bivector field has a kind of invariant significance in the geometry of interior point methods for linear programming for a kind of appendix it is noted that if we have several objectives i e in the case of multiobjective linear programs extension to higher dimensions is easily obtained

development of a dispersively accurate conservative level set scheme for capturing interface in two phase flows a two step interface capturing scheme implemented within the framework of conservative level set method is developed in this study to simulate the gas water two phase fluid flow in addition to solving the pure advection equation which is used to advect the level set function for tracking interface both nonlinear and stabilized features are taken into account for the level set function so that a sharply varying interface can be stably predicted to preserve the conservative property the mapping given by x x u t is performed between two coordinates x and x for the advection diffusion equation in a flow field with velocity u to capture the interface the flux term capable of compressing the level set contours is also adopted in the construction of linear inviscid burgers equation which is indispensable in the level set method to resolve the physically sharp interface without incurring contact discontinuity oscillations a damping term which is nonlinear in terms of the level set function and can render an adequate artificial diffusion to stabilize the contact surface is added into the reinitialization step in the modified level set method for accurately predicting the level set function the advection scheme for solving the linear inviscid burgers equation in the advection step of the modified level set method is developed to accommodate the true dispersion relation the solution computed from the resulting two dimensional dispersion relation preserving advection scheme can minimize the phase error less artificial damping is needed to damp the oscillations in the vicinity of contact surface and the interface can be less numerically smeared for the sake of programming simplicity the incompressible two phase flow will be discretized in non staggered grids without incurring checkerboard oscillations by the developed explicit compact scheme for the approximation of pressure gradient terms for the verification of the proposed two dimensional dispersion relation preserving scheme and the non staggered incompressible flow solver three benchmark problems have been chosen in this study the proposed conservative level set method for capturing the interface in incompressible fluid flows is also verified by solving the dam break bubble rising in water droplet falling in water and rayleigh taylor instability problems good agreements with the referenced solutions are demonstrated for all the four investigated problems

on a comparative study of an accurate spatial discretization method for one dimensional continuous systems abstract this work provides an in depth investigation on advantages of a recently developed new global spatial discretization method over the assumed modes method and a clear description of the procedure and validity of the new method and its feasibility for arbitrary boundary conditions a general formulation of the new spatial discretization method is given for second and fourth order continuous systems whose displacements are divided into internal terms and boundary induced terms and two examples that consider the longitudinal vibration of a rod and the transverse vibration of a tensioned euler bernoulli beam are used to demonstrate the new spatial discretization method in the two examples natural frequencies mode shapes harmonic steady state responses and transient responses of the systems are calculated using the new spatial discretization method and the assumed modes method and results are compared with those from exact analyses convergence of the new spatial discretization method is investigated using different sets of trial functions for internal and boundary induced terms while the new spatial discretization method has additional degrees of freedom at boundaries of a continuous system compared with other global spatial discretization methods it has the following advantages 1 compared with the assumed modes method the new method gives better results in calculating eigensolutions and dynamic responses of the system and allows more terms to be retained in a trial function expansion due to the slowly growing condition number of the mass matrix of the system and 2 compared with the exact eigenfunction expansion method the new method can use sinusoidal functions as trial functions for the internal term rather than complicated eigenfunctions of the system in the expansion solution

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

a conservative adaptive projection method for the variable density incompressible navier stokes equations in this paper we present a method for solving the equations governing time dependent variable density incompressible flow in two or three dimensions on an adaptive hierarchy of grids the method is based on a projection formulation in which we first solve advection diffusion equations to predict intermediate velocities and then project these velocities onto a space of approximately divergence free vector fields our treatment of the first step uses a specialized second order upwind method for differencing the nonlinear convection terms that provides a robust treatment of these terms suitable for inviscid and high reynolds number flow density and other scalars are advected in such a way as to maintain conservation if appropriate and free stream preservation our approach to adaptive refinement uses a nested hierarchy of logically rectangular girds with simultaneous refinement of the girds in both space and time the integration algorithm on the grid hierarchy is a recursive procedure in which coarse grids are advanced in time fine grids are advanced multiple steps to reach the same time as the coarse grids and the data at different levels are then synchronized the single grid algorithm is described briefly but the emphasis here is on the time stepping procedure for the adaptive hierarchy numerical examples are presented to demonstrate the algorithms s accuracy and convergence properties and illustrate the behavior of the method an additional example demonstrates the performance of the method on a more realistic problem namely a three dimensional variable density shear layer

a coupled level set and volume of fluid method for computing 3d and axisymmetric incompressible two phase flows we present a coupled level set volume of fluid clsvof method for computing 3d and axisymmetric incompressible two phase flows this method combines some of the advantages of the volume of fluid method with the level set method to obtain a method which is generally superior to either method alone we present direct comparisons between computations made with the clsvof method and computations made with the level set method the volume of fluid method and the boundary integral method we also compare our computations to the exact solution for an oscillating ellipse due to lamb and experimental results obtained for a rising gas bubble in liquid obtained by hnat and buckmaster our computational examples focus on flows in which surface tension forces and changes in topology are dominant features of the flow

a novel roe solver for incompressible two phase flow problems abstract in this paper a novel roe type riemann solver is developed for solving incompressible two phase flow problems the two phase flow is modeled by coupling the artificial compressibility based navier stokes equations and an interface advection equation in a conservative level set framework the proposed roe solver does not assume a locally constant density field while evaluating the flux jacobian as a result the solver is capable of capturing the exact jump conditions across the contact and shear waves which in turn helps in enhancing the accuracy of the interface motion moreover mass conservation characteristics are significantly improved by adopting conservative level set framework together with finite volume discretization of the system of equations a dual time stepping method with explicit pseudo time marching scheme is used for modeling the unsteady terms the efficacy of the proposed solver is demonstrated using various standard incompressible two phase flow problems involving low amplitude sloshing broken dam rayleigh taylor instability and droplet splash

polynomial affine algorithms for linear programming the method of steepest descent with scaling affine scaling applied to the potential functionq logc x i 1n logxi solves the linear programming problem in polynomial time forq n ifq n then the algorithm terminates in no more than o n2l iterations if q n sqrt n withq o n then it takes no more than o nl iterations a modified algorithm using rank 1 updates for matrix inversions achieves respectively o n4l and o n3 5l arithmetic computions

a class of polynomial variable metric algorithms for linear optimization in the paper the behaviour of interior point algorithms is analyzed by using a variable metric method approach a class of polynomial variable metric algorithms is given achieving o n l iterations for solving a canonical form linear optimization problem with respect to a wide class of riemannian metrics wheren is the number of dimensions and a fixed value it is shown that the vector fields of several interior point algorithms for linear optimization is the negative riemannian gradient vector field of a linear a potential or a logarithmic barrier function for suitable riemannian metrics

a hybrid particle level set method for improved interface capturing in this paper we propose a new numerical method for improving the mass conservation properties of the level set method when the interface is passively advected in a flow field our method uses lagrangian marker particles to rebuild the level set in regions which are underresolved this is often the case for flows undergoing stretching and tearing the overall method maintains a smooth geometrical description of the interface and the implementation simplicity characteristic of the level set method our method compares favorably with volume of fluid methods in the conservation of mass and purely lagrangian schemes for interface resolution the method is presented in three spatial dimensions