a class of elliptic partial differential equations with exponential nonlinearities on demontre l existence d au moins une infinite denombrable de familles a 1 parametre de solutions non triviales continues de holder pour une classe de problemes aux valeurs limites de dirichlet d ordre 2 fortement non lineaires a 2 dimensions et plus

an unconditionally stable fully conservative semi lagrangian method semi lagrangian methods have been around for some time dating back at least to 3 researchers have worked to increase their accuracy and these schemes have gained newfound interest with the recent widespread use of adaptive grids where the cfl based time step restriction of the smallest cell can be overwhelming since these schemes are based on characteristic tracing and interpolation they do not readily lend themselves to a fully conservative implementation however we propose a novel technique that applies a conservative limiter to the typical semi lagrangian interpolation step in order to guarantee that the amount of the conservative quantity does not increase during this advection in addition we propose a new second step that forward advects any of the conserved quantity that was not accounted for in the typical semi lagrangian advection we show that this new scheme can be used to conserve both mass and momentum for incompressible flows for incompressible flows we further explore properly conserving kinetic energy during the advection step but note that the divergence free projection results in a velocity field which is inconsistent with conservation of kinetic energy even for inviscid flows where it should be conserved for compressible flows we rely on a recently proposed splitting technique that eliminates the acoustic cfl time step restriction via an incompressible style pressure solve then our new method can be applied to conservatively advect mass momentum and total energy in order to exactly conserve these quantities and remove the remaining time step restriction based on fluid velocity that the original scheme still had

an existence theorem for weak solutions for a class of elliptic partial differential systems in orlicz spaces abstract we prove the existence of a weak solution of the dirichlet problem for a class of elliptic partial differential systems in separable orlicz sobolev spaces

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

simulation of axisymmetric jets with a finite element navier stokes solver and a multilevel vof approach a multilevel vof approach has been coupled to an accurate finite element navier stokes solver in axisymmetric geometry for the simulation of incompressible liquid jets with high density ratios the representation of the color function over a fine grid has been introduced to reduce the discontinuity of the interface at the cell boundary in the refined grid the automatic breakup and coalescence occur at a spatial scale much smaller than the coarse grid spacing to reduce memory requirements we have implemented on the fine grid a compact storage scheme which memorizes the color function data only in the mixed cells the capillary force is computed by using the laplace beltrami operator and a volumetric approach for the two principal curvatures several simulations of axisymmetric jets have been performed to show the accuracy and robustness of the proposed scheme

existence and uniqueness of weak solutions for a non uniformly parabolic equation in this paper we develop a unifying method to prove the existence and uniqueness of weak solutions for the initial boundary value problem of a non uniformly parabolic equation some well known parabolic equations are the special cases of this equation

on donating regions lagrangian flux through a fixed curve lagrangian flux through a fixed curve segment naturally gives rise to the notion of a donating region a compact point set in which each particle will pass through the curve and contribute to the flux a precise geometric determination of the donating region has not been available until now the author proposes an explicit constructive and analytical definition of the donating region based on two characteristic curves of the flow field viz streaklines and timelines it is also shown that within a time interval of finite size each particle in the proposed donating region has a net effect of going across the fixed curve once the presented analysis might be potentially useful for flow topology studies lagrangian flux calculation and explicit interface tracking

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

interface reconstruction with least squares fit and split advection in three dimensional cartesian geometry in this paper we present and analyze different volume of fluid vof reconstruction and advection algorithms that approximate the interface separating two immiscible fluids in the three dimensional space the paper describes the improvement of the reconstruction when a least square fit algorithm which minimizes a distance functional is applied its performance is tested for several smooth surfaces against other simple reconstruction methods then eulerian lagrangian and mixed split advection schemes are presented and analyzed in particular one advection method is discussed that conserves mass exactly for a divergence free velocity field thus allowing computations to machine precision

smooth manifolds for semilinear wave equations on r2 on the existence of almost periodic breathers abstract in this paper we prove several nonexistence and existence results for certain real solutions to semilinear wave equations in one space dimension these solutions represent time almost periodic free vibrations which decay toward a constant equilibrium solution as the spatial variable goes to infinity our nonexistence results are of two kinds we first prove that the requirements of almost periodicity in time and of spatial decay at infinity force such solutions to be trivial whenever their spectrum does not interact with the nonlinearity in a certain sense we also prove non existence results without requiring any spectral conditions but this is at the expense of having to impose more stringent conditions on the nonlinearities such as certain convexity properties in the vicinity of the equilibrium solution finally we prove an existence result for time almost periodic free vibrations whose profiles decay exponentially rapidly toward a constant equilibrium solution to accomplish this we convert the wave equation into a dynamical system in which the original spatial variable plays the role of time we then embed that dynamical system into an appropriate banach space of time almost periodic functions so that the one parameter family of stable and unstable manifolds which we construct carry the solutions that we seek the major difficulty to overcome in our construction is a small divisor problem it is related to the fact that the spectrum of the infinitesimal generator for the linearized flow is a pure point spectrum without gaps we discuss several examples and we also stress some important qualitative differences which distinguish the almost periodic case from the purely periodic one

sharp interface immersed boundary level set method for wave body interactions a sharp interface cartesian grid method for the large eddy simulation of two phase turbulent flows interacting with moving bodies is presented the overall approach uses a sharp interface immersed boundary formulation and a level set ghost fluid method for solid fluid and fluid fluid interface treatments respectively a four step fractional step method is used for velocity pressure coupling and a lagrangian dynamic smagorinsky subgrid scale model is adopted for large eddy simulations a simple contact angle boundary condition treatment that conforms to the immersed boundary formulation is developed a variety of test cases of different scales ranging from bubble dynamics water entry and exit landslide generated waves to ship hydrodynamics are performed for validation extensions for high reynolds number ship flows using wall layer models are also considered

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

flair fluz line segment model for advection and interface reconstruction abstract a computational technique for solving fluid problems with free surfaces and interfaces is presented the conventional cell volume fraction approach is employed for tracking the interfaces however for surface advection and its reconstruction a new and more accurate flair flux line segment model for advection and interface reconstruction algorithm is developed the surface is approximated by a set of line segments fitted at the boundary of every two neighboring computational cells a criterion is developed for identifying the line segment orientation by inspecting the cell volume fractions the new cell volume fraction field is obtained by integrating the advected area underneath the interface line segment as an example this technique is applied to the capillary driven viscous flow of an initially elliptic two dimensional fluid zone the problem is posed mathematically as a solution of the navier stokes equations with moving free surface boundary conditions the damping motion of the fluid zone is observed through transport of the free surface which is related to the instantaneous internal velocity field under the influence of surface tension and viscous forces

isogeometric analysis of free surface flow abstract this paper presents the first application of isogeometric analysis a new computational technology built on higher order and higher continuity basis functions employed in computer aided design and computer graphics to the computation of free surface phenomena described using the level set approach the method is based on the variational framework that is suitable for discretization by standard finite elements as well as the basis functions employed in isogeometric analysis the underlying numerical formulation globally conserves mass and preserves a sharp air water interface for the entire length of the simulation the numerical tests indicate that the proposed methodology gives an accurate description of the free surface behavior in both quasi steady and dynamic regimes furthermore very good per degree of freedom accuracy is obtained when higher order and higher continuity isogeometric discretizations are employed in free surface computations

toward efficient and accurate interface capturing on arbitrary hybrid unstructured grids the thinc method with quadratic surface representation and gaussian quadrature abstract a novel interface capturing scheme is proposed to compute moving interface on arbitrary hybrid unstructured grids different from conventional volume of fluid vof schemes that require complicated geometric manipulations for interface reconstructions the present method can be viewed as a hybrid geometric algebraic type vof approach i e the interface is implictly retrieved from an algebraic function that effectively makes use of the geometrical information such as normal direction and curvature of the interface unlike previous versions of the thinc tangent of hyperbola interface capturing method the interface is represented by a quadratic surface for grid cells of arbitrary shapes in this scheme the gaussian quadrature is used to estimate the integration of the cell wise multi dimensional hyperbolic tangent reconstruction function which is then used to retrieve the interface from the volume fraction value of the target cell the gaussian quadrature is also used to compute the numerical fluxes from the reconstruction function numerical accuracy for reconstruction and flux computation can be effectively improved by increasing quadrature points the whole solution procedure follows the finite volume method for advection transport and is thus simple and easy to use for cells of arbitrary shapes as verified in the benchmark tests in this paper the presented scheme so called thinc qq thinc method with quadratic surface representation and gaussian quadrature scheme shows significantly improved geometrical fidelity of interface representation particularly for curved surface despite algorithmic simplicity the solution quality of thinc qq is comparable to other existing vof methods with plic geometrical interface reconstructions and thus an accurate and efficient vof scheme of great practical significance for unstructured grids

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

non oscillatory continuous fem for transport and shallow water flows abstract the paper presents a sign preserving continuous finite element model fem for coupled transport equations and in particular for the depth integrated shallow water equations the model incorporates an enhanced flux corrected transport method fct with a high order solution given by a generalization of the characteristic based split finite element method cbs and a predictor positive algorithm founded in a finite element upwind methodology numerical experiments explore the efficacy of the procedure with emphasis in the simulation of flows with evolutionary boundaries determined by null values of any flow property and in severe conditions of propagation such as dam breaking waves

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

a class of orlicz sobolev spaces with applications to variational problems involving nonlinear hill s equations necessary and sufficient conditions are proved for a b 2 young function g with independent variable t to be convex resp concave in t2 in terms of inequalities between the second derivative of g and the first derivative of its legendre transform g with independent variable s it is then proven that a young function g is convex resp concave in t2 if and only if g is concave resp convex in s2 these results along with another set of inequalities for functions g convex resp concave in t2 allow the proof of the uniform convexity and thereby of the reflexivity with respect to luxemburg s norm f g inf k 0 d g f k 1 of the orlicz space lg over an open domain rn with lebesgue measure d when applied to g t t pp and g s s p p with p 1 p 1 1 the preceding results lead to the shortest proof to date of two clarkson s inequalities and of the reflexivity of lp spaces for 1 p finally some of these results are used to solve by direct methods variational problems associated with the existence question of periodic orbits for a class of nonlinear hill s equations these variational problems are formulated on suitable orlicz sobolev spaces wmlg and thereby allow for nonlinear terms which may grow faster than any power of the variable

an l lp mesh adaptive method for computing unsteady bi fluid flows this paper discusses the contribution of mesh adaptation to high order convergence of unsteady multi fluid flow simulations on complex geometries the mesh adaptation relies on a metric based method controlling the lp norm of the interpolation error and on a mesh generation algorithm based on an anisotropic delaunay kernel the mesh adaptive time advancing is achieved thanks to a transient fixed point algorithm to predict the solution evolution coupled with a metric intersection in the time procedure in the time direction we enforce the equidistribution of the error i e the error minimization in l norm this adaptive approach is applied to an incompressible navier stokes model combined with a level set formulation discretized on triangular and tetrahedral meshes applications to interface flows under gravity are performed to evaluate the performance of this method for this class of discontinuous flows copyright 2010 john wiley sons ltd

oddls a new unstructured mesh finite element method for the analysis of free surface flow problems this paper introduces a new stabilized finite element method based on the finite calculus comput methods appl mech eng 1998 151 233 267 and arbitrary lagrangian eulerian techniques comput methods appl mech eng 1998 155 235 249 for the solution to free surface problems the main innovation of this method is the application of an overlapping domain decomposition concept in the statement of the problem the aim is to increase the accuracy in the capture of the free surface as well as in the resolution of the governing equations in the interface between the two fluids free surface capturing is based on the solution to a level set equation the navier stokes equations are solved using an iterative monolithic predictor corrector algorithm encyclopedia of computational mechanics wiley new york 2004 where the correction step is based on imposing the divergence free condition in the velocity field by means of the solution to a scalar equation for the pressure examples of application of the oddls formulation for overlapping domain decomposition level set to the analysis of different free surface flow problems are presented copyright 2008 john wiley sons ltd

a conservative level set method suitable for variable order approximations and unstructured meshes this paper presents a formulation for free surface computations capable of handling complex phenomena such as wave breaking without excessive mass loss or smearing of the interface the formulation is suitable for discretizations using finite elements of any topology and order or other approaches such as isogeometric and finite volume methods furthermore the approach builds on standard level set tools and can therefore be used to augment existing implementations of level set methods with discrete conservation properties implementations of the method are tested on several difficult two and three dimensional problems including two incompressible air water flow problems with available experimental results linear and quadratic approximations on unstructured tetrahedral and trilinear approximations on hexahedral meshes were tested global conservation and agreement with experiments as well as computations by other researchers are obtained

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

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

comments on an improved free surface capturing method based on cartesian cut cell mesh for water entry and exit problems in the paper entitled an improved free surface capturing method based on cartesian cut cell mesh for water entry and exit problems wang wang 2009 1 proc r soc a 465 1843 1868 doi 10 1098 rspa 2008 0410 2 the present authors earlier work qian et al 2006 3 proc r soc a 462 21 42 doi 10 1098 rspa 2005 1528 4 has been specifically applied to the study of water entry and exit of solid objects an extended boundary condition retaining the term owing to acceleration of moving boundaries in the momentum equation has been implemented for calculating the pressure gradient at solid surfaces and based on their numerical experiments it was concluded by wang and wang that without this term the calculation will substantially under predict the impact forces and may even break down therefore a more complex procedure based on the exact solution of a riemann problem for moving boundaries was implemented in this short comment by applying the authors free surface capturing code to the same flow problem of water entry of a wedge it can however be demonstrated that the results from implementing the new pressure boundary condition are nearly identical to that of employing the original boundary condition without the acceleration term indicating that its effects on the simulation results are minimal a further examination of the implementation details on the pressure boundary condition also supports this conclusion 1 ref 5 2 lookup doi 10 1098 rspa 2008 0410 3 ref 4 4 lookup doi 10 1098 rspa 2005 1528

one fluid formulation for fluid structure interaction with free surface abstract a simple and efficient computational framework is presented for the simulation of fluid structure interaction problems involving rigid body and multiphase flows in the context of hydrodynamics unlike existing publications this method does not solve the general motion of rigid bodies in the lagrangian form of newton s law derived from the distributed lagrange multiplier treatment of the rigid body a new set of governing equations is presented on the fully eulerian one fluid formulation to solve the problem numerically the complex problem is separated into three parts balance of the momentum and mass dynamic problem evolving of the heaviside function by the external velocity geometric problem and rigid motion projection kinematic problem the conservation of mass and momentum is guaranteed by the multiphase fluid solver the water air and structure coupling is accomplished by the smeared interface a new way of initialisation and convection of the rigid heaviside function is designed for an arbitrary shape to deal with rigid velocity vector a linear least square method is proposed the excellent agreement between the numerical experiment and the reference data from experiments demonstrate the validity and applicability of the new methodology

a new volume of fluid advection algorithm a new vof advection algorithm is presented termed the stream scheme the algorithm uses a linear piecewise method for free surface reconstruction coupled to a unique fully multidimensional method of cell boundary flux integration the performance of the new algorithm is compared against other vof advection algorithms using a variety of standard advection tests in general the performance of the stream scheme appears superior to existing multidimensional and dimensionally split advection algorithms

a positivity preserving and conservative variational scheme for phase field modeling of two phase flows abstract we present a positivity preserving variational scheme for the phase field modeling of incompressible two phase flows with high density ratio the variational finite element technique relies on the allen cahn phase field equation for capturing the phase interface on a fixed eulerian mesh with mass conservative and energy stable discretization the mass conservation is achieved by enforcing a lagrange multiplier which has both temporal and spatial dependence on the underlying solution of the phase field equation to make the scheme energy stable in a variational sense we discretize the spatial part of the lagrange multiplier in the phase field equation by the mid point approximation the proposed variational technique is designed to reduce the spurious and unphysical oscillations in the solution while maintaining the second order accuracy of both spatial and temporal discretizations we integrate the allen cahn phase field equation with the incompressible navier stokes equations for modeling a broad range of two phase flow and fluid fluid interface problems the coupling of the implicit discretizations corresponding to the phase field and the incompressible flow equations is achieved via nonlinear partitioned iterative procedure comparison of results between the standard linear stabilized finite element method and the present variational formulation shows a remarkable reduction of oscillations in the solution while retaining the boundedness of the phase indicator field we perform a standalone test to verify the accuracy and stability of the allen cahn two phase solver we examine the convergence and accuracy properties of the coupled phase field solver through the standard benchmarks of the laplace young law and a sloshing tank problem two and three dimensional dam break problems are simulated to assess the capability of the phase field solver for complex air water interfaces involving topological changes on unstructured meshes finally we demonstrate the phase field solver for a practical offshore engineering application of wave structure interaction

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

coupling of two absorbing boundary conditions for 2d time domain simulations of free surface gravity waves the numerical simulation of nonlinear gravity waves propagating at the surface of a perfect fluid is now usually solved by totally nonlinear time domain numerical models in two dimensions and this approach is being extended to three dimensions the original initial boundary value problem is posed in an unbounded region extending horizontally up to infinity to model the sea its numerical solution requires truncating the domain at a finite distance unfortunately no exact nonreflecting boundary condition on the truncating surface exists in this time domain formulation the proposed strategy is based on the coupling of two previously known methods in order to benefit from their different and complementary bandwidth the numerical beach very efficient in the high frequency range and a piston like neumann condition asymptotically ideal for low frequencies the coupling method gives excellent results in the whole range of frequencies of interest and is as easy to implement in nonlinear as in linear versions one of its major advantages is that it does not require any spectral knowledge of the incident waves

a newton krylov solver for remapping based volume of fluid methods a new conservative remapping method is presented here for generating multi dimensional fluxes used for the time integration of hyperbolic equations through single stage unsplit advection a solution to the lagrangian mesh location is generated that converges the volumes of all remapped flux definitions to the volumes of equivalent one dimensional flux definitions at their corresponding cell faces the method is most feasible within localized subdomains of an eulerian solution domain the main utility of the method is the generation of fluxes in the vicinity of discontinuities and sharp gradients that does not introduce physically spuriously extrema while locally conserving mass in generating solutions to lagrangian mesh location by solving a nonlinear equation system the new remap requires the anchoring of the lagrangian mesh removal of degrees of freedom to make the problem definite the method is based on characteristic tracing along trajectories assumed to be linear this paper investigates the convergence and sensitivity of the lagrangian mesh solver to mesh and timestep size in assessing its merit for application in transient cfd codes for single stage unsplit advection volume of fluid methods the new remap suppresses the introduction of undershoots overshoots and wisps in combination with a conservative remapping phase the new method for fixing the location of the lagrangian mesh eliminates the need for heuristic redistribution or clipping procedures for spurious extrema removal

symmetry preserving discretization of turbulent flow we propose to perform turbulent flow simulations in such manner that the difference operators do have the same symmetry properties as the underlying differential operators i e the convective operator is represented by a skew symmetric coefficient matrix and the diffusive operator is approximated by a symmetric positive definite matrix mimicking crucial properties of differential operators forms in itself a motivation for discretizing them in a certain manner we give it a concrete form by noting that a symmetry preserving discretization of the navier stokes equations is stable on any grid and conserves the total mass momentum and kinetic energy for the latter the physical dissipation is to be turned off of coarse being stable on any grid the choice of the grid may be based on the required accuracy solely and the main question becomes how accurate is a symmetry preserving discretization its accuracy is tested for a turbulent flow in a channel by comparing the results to those of physical experiments and previous numerical studies the comparison is carried out for a reynolds number of 5600 which is based on the channel width and the mean bulk velocity based on the channel half width and wall shear velocity the reynolds number becomes 180 the comparison shows that with a fourth order symmetry preserving method a 64 64 32 grid suffices to perform an accurate numerical simulation

a characterization of minimal orlicz sobolev norms in the affine class abstract the orlicz sobolev norm is invariant only under rigid motions but is not invariant under volume preserving affine transformations on r n in this paper we will consider the orlicz sobolev norm in the affine class and characterize its minimum

interactions of breaking waves with a current over cut cells by design of the external and internal wave current generators the objective of this paper is to extend our efficient navier stokes solver t li p troch j de rouck wave overtopping over a sea dike j comput phys 198 2004 686 726 for modelling of interactions between breaking waves and a current over a cut cell grid based on a dynamic subgrid scale sgs model this solver is constructed by a novel vof finite volume approach coupled with surface tension when studying waves following a positive current our external generator creates the combined inflow motions of waves and a current which is viewed as one type of wavy inflow conditions for cases of waves against strong currents our internal generator describes the opposing current by incorporating the source function to the continuity and momentum equations as a net driving force acting on the fluid elements lying within the finite thickness source region the outgoing waves downstream are dissipated with a breaking type wave absorber placed in the tank extremity five test cases recommended are of distinctly different applications of interest characterized by overtopping of following waves over sloping and vertical structures under the grid refinement effects the results in 2d and 3d are in close agreement with the experimental data available in terms of the surface wave additionally the performance of convergence in computations is also investigated including full discussion for waves on beaches between 2d and 3d by visualization of the motions that describe the physics of turbulence it has been shown that our solver can capture most of the significant features in wave current interactions varying with three different current speeds positive zero negative

a volume of fluid based simulation method for wave impact problems in this paper some aspects of water impact and green water loading are considered by numerically investigating a dambreak problem and water entry problems the numerical method is based on the navier stokes equations that describe the flow of an incompressible viscous fluid the equations are discretised on a fixed cartesian grid using the finite volume method even though very small cut cells can appear when moving an object through the fixed grid the method is stable the free surface is displaced using the volume of fluid method together with a local height function resulting in a strictly mass conserving method the choice of boundary conditions at the free surface appears to be crucial for the accuracy and robustness of the method for validation results of a dambreak simulation are shown that can be compared with measurements a box has been placed in the flow as a model for a container on the deck of an offshore floater on which forces are calculated the water entry problem has been investigated by dropping wedges with different dead rise angles a cylinder and a cone into calm water with a prescribed velocity the resulting free surface dynamics with the sideways jets has been compared with photographs of experiments also a comparison of slamming coefficients with theory and experimental results has been made finally a drop test with a free falling wedge has been simulated

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

mass momentum and energy conserving mamec discretizations on general grids for the compressible euler and shallow water equations the paper explains a method by which discretizations of the continuity and momentum equations can be designed such that they can be combined with an equation of state into a discrete energy equation the resulting mamec discretizations conserve mass momentum as well as energy although no explicit conservation law for the total energy is present essential ingredients are i discrete convection that leaves the discrete energy invariant and ii discrete consistency between the thermodynamic terms of particular relevance is the way in which finite volume fluxes are related to nodal values the method is an extension of existing methods based on skew symmetry of discrete operators because it allows arbitrary equations of state and a larger class of grids than earlier methods the method is first illustrated with a one dimensional example on a highly stretched staggered grid in which the mamec method calculates qualitatively correct results and a non skew symmetric finite volume method becomes unstable a further example is a two dimensional shallow water calculation on a rectilinear grid as well as on an unstructured grid the conservation of mass momentum and energy is checked and losses are found negligible up to machine accuracy

an efficient interface preserving level set redistancing algorithm and its application to interfacial incompressible fluid flow in sussman smereka and osher j comp phys 94 1994 pp 146 159 a numerical scheme was presented for computing incompressible air water flows using the level set method crucial to the above method was a new iteration method for maintaining the level set function as the signed distance from the zero level set in this paper we implement a constraint along with higher order difference schemes in order to make the iteration method more accurate and efficient accuracy is measured in terms of the new computed signed distance function and the original level set function having the same zero level set we apply our redistancing scheme to incompressible flows with noticeably better resolved results at reduced cost we validate our results with experiment and theory we show that our distance level set scheme with the added constraint competes well with available interface tracking schemes for basic advection of an interface we perform basic accuracy checks and more stringent tests involving complicated interfacial structures as with all level set schemes our method is easy to implement

a second order boundary fitted projection method for free surface flow computations this paper describes a new approach to the high fidelity simulation of axisymmetric free surface flows a boundary fitted grid is coupled with a new projection method for the solution of the navier stokes equations with second order accuracy in space and time two variants of this new method are developed by adapting two existing algorithms suitable for prescribed velocity boundary conditions to the case of prescribed normal and tangential stresses at the free boundary a normal mode analysis for a fixed boundary problem confirms the second order accuracy of the algorithms the approach is validated by comparison with a rayleigh plesset solution for an oscillating spherical bubble with an analysis of shape oscillations and with existing results for the buoyant rise of a deforming bubble for reynolds numbers up to 200 and weber numbers up to 12 in addition to the simulation of axisymmetric free surface flows of intrinsic interest the present approach is suitable for the validation of genuinely three dimensional calculations

an adaptive method to capture weldlines during the injection mold filling weldlines are generally unavoidable during the process of injection mold filling with even moderate complexity when two or more melts meet which greatly influence the quality of the products in this paper we propose a new method to numerically simulate the mold filling process and to capture the weldline adaptively based on the level set ghost method the cases where the weldlines are caused by two melt fronts in the injection mold filling process are analyzed and our special interest is devoted to confirming the position and the shape of the weldline depending on the new adaptive technique based on the level set function the numerical results show that the proposed method is feasible to detect the influence of the inset size the inset shape and the inset center place on the weldline

an existence theorem for weak solutions for a class of elliptic partial differential systems in general orlicz sobolev spaces abstract i prove the existence of a weak solution for the dirichlet problem of a class of elliptic partial differential systems a i x x u x d u x b i x u x d u x 0 in general orlicz sobolev spaces w 0 1 l m r n where i 1 n 1 n u r n is a vector valued function and the summation convention is used throughout with i j running from 1 to n and running from 1 to n

an improved free surface capturing method based on cartesian cut cell mesh for water entry and exit problems the free surface capturing method based on cartesian cut cell mesh is extended to the water entry and exit fields with body fluid interaction the governing equations are the incompressible euler equations for a variable density fluid system with a free surface which is treated as a contact discontinuity the solver is based on the artificial compressibility method with a dual time stepping technique for time advancing and the finite volume method with a high resolution godunov type upwind scheme on spatial discretization for solving the numerical problem caused by the extension the acceleration term of body is introduced for the new pressure condition of moving solid boundary and the exact riemann solution is used to calculate the flux of the solid boundary in addition a new solution of gradients based on the least squares idea is presented for simply calculating the gradients finally test cases show that the present method is available and can be successfully applied in various water entry and exit phenomena

fourth order interface tracking in two dimensions via an improved polygonal area mapping method we present an improved pam ipam method as the first fourth order interface tracking method whose convergence rates are independent of c 1 derivative discontinuities of the interface as an improved version of the polygonal area mapping pam method q zhang and p l f liu j comput phys 227 2008 pp 4063 4088 the accuracy of the ipam method is achieved via i augmenting the abstract data structure of pam to faithfully represent multiple components of material regions within a single cell ii removing restrictive assumptions of pam iii adjusting the volume of represented cell material regions via polygon ear removal and iv maintaining a relation h l r hh alpha between the eulerian grid size h and the lagrangian length scale that measures the distance between adjacent interface markers unlike volume of fluid methods and level set methods merging and separation of the tracked material is not automatically handled by the ipam method instead flexible subgrid resolutions

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 modified sor method for the poisson equation in unsteady free surface flow calculations convergence difficulties that sometimes occur if the successive overrelaxation sor method is applied to the poisson equation on a region with irregular free boundaries are analyzed it is shown that these difficulties are related to the treatment of the free boundaries and caused by the appearance of complex eigenvalues in the system of discrete equations when standard centered differences are used after a modification of this system of equations such that the complex eigenvalues become small a modified sor method is presented where two relaxation factors are used alternately the method leads to fast convergence without requiring specific information about the complex eigenvalues

conservative and bounded volume of fluid advection on unstructured grids abstract this paper presents a novel eulerian lagrangian piecewise linear interface calculation plic volume of fluid vof advection method which is three dimensional unsplit and discretely conservative and bounded the approach is developed with reference to a collocated node based finite volume two phase flow solver that utilizes the median dual mesh constructed from non convex polyhedra the proposed advection algorithm satisfies conservation and boundedness of the liquid volume fraction irrespective of the underlying flux polyhedron geometry which differs from contemporary unsplit vof schemes that prescribe topologically complicated flux polyhedron geometries in efforts to satisfy conservation instead of prescribing complicated flux polyhedron geometries which are prone to topological failures our vof advection scheme the non intersecting flux polyhedron advection nifpa method builds the flux polyhedron iteratively such that its intersection with neighboring flux polyhedra and any other unavailable volume is empty and its total volume matches the calculated flux volume during each iteration a candidate nominal flux polyhedron is extruded using an iteration dependent scalar the candidate is subsequently intersected with the volume guaranteed available to it at the time of the flux calculation to generate the candidate flux polyhedron the difference in the volume of the candidate flux polyhedron and the actual flux volume is used to calculate extrusion during the next iteration the choice in nominal flux polyhedron impacts the cost and accuracy of the scheme however it does not impact the methods underlying conservation and boundedness as such various robust nominal flux polyhedron are proposed and tested using canonical periodic kinematic test cases zalesak s disk and two and three dimensional deformation the tests are conducted on the median duals of a quadrilateral and triangular primal mesh in two dimensions and on the median duals of a hexahedral wedge and tetrahedral primal mesh in three dimensions comparisons are made with the adaptation of a conventional unsplit vof advection scheme to our collocated node based flow solver depending on the choice in the nominal flux polyhedron the nifpa scheme presented accuracies ranging from zeroth to second order and calculation times that differed by orders of magnitude for the nominal flux polyhedra which demonstrate second order accuracy on all tests and meshes the nifpa method s cost was comparable to the traditional topologically complex second order accurate vof advection scheme

a plic vof method suited for adaptive moving grids this paper presents a new variant of the volume of fluid vof color function c advection algorithm based on the piecewise linear interface construction plic method suitable for use on general moving grids from several existing methods for reconstructing the linear interface we adopted the least squares volume of fluid interface reconstruction algorithm lvira which can be easily implemented on general grids the distinguishing step in the advection algorithm that takes into account the grid movement is the construction of the donating region containing the fluid passing through corresponding cell faces in a single time step the donating regions are constructed utilizing fluid velocity in cell corners relative to grid corner velocities the method is conservative as it complies with the space conservation law scl and requires a proper definition of the grid velocities and fluxes due to the grid movement the accuracy of the presented advection algorithm is assessed with standard test cases it is comparable with other plic based algorithms on fixed grids while the applicability on adaptive moving grids enables a considerable reduction in the number of grid cells

one parameter families of sturm liouville eigenfunctions and of periodic solutions for a class of nonlinear hill s equations i introduction in cyclic accelerators and in intersecting storage rings the orbits of the particles are periodically perturbed by strongly nonlinear electromagnetic forces whenever the two counterrotating beams undergo head on collisions at the interaction regions around the machine such perturbations in turn generate nonlinear transverse oscillations about the ideal orbits which can generally be described by nonlinear hill s equations of the form z 8 nz 8 f z d i 1 see

a geometrical predictor corrector advection scheme and its application to the volume fraction function we present a multidimensional eulerian advection method for interfacial and incompressible flows in two dimensional cartesian geometry in the scheme we advect the grid nodes backwards along the streamlines to compute the pre images of the grid lines these pre images are approximated by continuous piecewise linear lines the enforcement of the discrete version of the incompressibility constraint is a very important issue to determine correctly the flux polygons and to reduce considerably the integration discretization and interpolation numerical errors the proposed method compares favorably with other previous multidimensional advection methods as long as the initial interface line is well reconstructed conversely we show that when the interface is very fragmented the total numerical error is completely dominated by the reconstruction error and in these conditions it is very difficult to assess which advection scheme is the most reliable one

an improved plic vof method for tracking thin fluid structures in incompressible two phase flows a drawback common to plic schemes is the difficulty of reconstructing the interface when filaments of a thickness smaller than the cell size are present in this work we present an improvement of the reconstruction method proposed by lopez et al a volume of fluid method based on multidimensional advection and spline interface reconstruction j comput phys 195 2004 718 for two dimensional flows which allows tracking fluid structures thinner than the cell size by allowing the interface to be represented in each cell by two non contiguous linear segments the method is based on using markers that are placed every time step at the mid points of the reconstructed cell interface segments the markers are used to improve the orientation and location of the interface segments although the method essentially remains of a vof type numerical tests show that a substantial improvement in accuracy is achieved over previous volume of fluid methods and that the proposed method compares well even with a recent hybrid markers and volume of fluid method