simulating the dynamics of inextensible vesicles by the penalty immersed boundary method in this paper we develop an immersed boundary ib method to simulate the dynamics of inextensible vesicles interacting with an incompressible fluid in order to take into account the inextensibility constraint of the vesicle the penalty immersed boundary pib method is used to virtually decouple the fluid and vesicle dynamics as numerical tests of our current pib method the dynamics of single and multiple inextensible vesicles under shear flows have been extensively explored and compared with the previous literature the method is also validated by a series of convergence study which confirms its consistent first order accuracy on the velocity field the vesicle configuration the vesicle area and the perimeter errors in addition the method is also applied to study a binary component vesicle problem

accurate projection methods for the incompressible navier stokes equations abstract this paper considers the accuracy of projection method approximations to the initial boundary value problem for the incompressible navier stokes equations the issue of how to correctly specify numerical boundary conditions for these methods has been outstanding since the birth of the second order methodology a decade and a half ago it has been observed that while the velocity can be reliably computed to second order accuracy in time and space the pressure is typically only first order accurate in the l norm this paper identifies the source of this problem in the interplay of the global pressure update formula with the numerical boundary conditions and presents an improved projection algorithm which is fully second order accurate as demonstrated by a normal mode analysis and numerical experiments in addition a numerical method based on a gauge variable formulation of the incompressible navier stokes equations which provides another option for obtaining fully second order convergence in both velocity and pressure is discussed the connection between the boundary conditions for projection methods and the gauge method is explained in detail

characteristic cut finite element methods for convection diffusion problems on time dependent surfaces we develop a finite element method for convection diffusion problems on a given time dependent surface for instance modeling the evolution of a surfactant the method is based on a characteristic

three dimensional front tracking we describe a three dimensional front tracking algorithm discuss its numerical implementation and present studies to validate the correctness of this approach based on the results of the two dimensional computations we expect three dimensional front tracking to significantly improve computational efficiencies for problems dominated by discontinuities in some cases for which the interface computations display considerable numerical sensitivity we expect a greatly enhanced capability

higher order finite element methods and pointwise error estimates for elliptic problems on surfaces we define higher order analogues to the piecewise linear surface finite element method studied in g dziuk finite elements for the beltrami operator on arbitrary surfaces in partial differential equations and calculus of variations springer verlag berlin 1988 pp 142 155 and prove error estimates in both pointwise and l 2 based norms using the laplace beltrami problem on an implicitly defined surface gamma as a model pde we define lagrange finite element methods of arbitrary degree on polynomial approximations to gamma which likewise are of arbitrary degree then we prove a priori error estimates in the l 2 h 1 and corresponding pointwise norms that demonstrate the interaction between the pde error that arises from employing a finite dimensional finite element space and the geometric error that results from approximating gamma we also consider parametric finite element approximations that are defined on gamma and thus induce no geometric error computational examples confirm the sharpness of our error estimates

the generalized riemann problem for reactive flows abstract a generalized riemann problem is introduced for the equations of reactive non viscous compressible flow in one space dimension initial data are assumed to be linearly distributed on both sides of a jump discontinuity the resolution of the singularity is studied and the first order variation in time of flow variables is given in exact form

a filtration model with hysteresis abstract we study a filtration model for which a hysteresis effect is accounted in the saturation versus pressure constitutive relation the problem is formulated in a quasi variational inequality we prove the existence of a time global solution

logically rectangular grids and finite volume methods for pdes in circular and spherical domains we describe a class of logically rectangular quadrilateral and hexahedral grids for solving pdes in circular and spherical domains including grid mappings for the circle the surface of the sphere and the three dimensional ball the grids are logically rectangular and the computational domain is a single cartesian grid compared to alternative approaches based on a multiblock data structure or unstructured triangulations this approach simplifies the implementation of numerical methods and the use of adaptive refinement a more general domain with a smooth boundary can be gridded by composing one of the mappings from this paper with another smooth mapping from the circle or sphere to the desired domain although these grids are highly nonorthogonal we show that the high resolution wave propagation algorithm implemented in clawpack can be used effectively to approximate hyperbolic problems on these grids since the ratio between the largest and smallest grids is below 2 for most of our grid mappings explicit finite volume methods such as the wave propagation algorithm do not suffer from the center or pole singularities that arise with polar or latitude longitude grids numerical test calculations illustrate the potential use of these grids for a variety of applications including euler equations shallow water equations and acoustics in a heterogeneous medium pattern formation from a reaction diffusion equation on the sphere is also considered all examples are implemented in the clawpack software package and full source code is available on the web along with matlab routines for the various mappings

a convergent finite volume scheme for diffusion on evolving surfaces a finite volume scheme for transport and diffusion problems on evolving hypersurfaces is discussed the underlying motion is assumed to be described by a fixed not necessarily normal velocity field the ingredients of the numerical method are an approximation of the family of surfaces by a family of interpolating simplicial meshes where grid vertices move on motion trajectories a consistent finite volume discretization of the induced transport on the simplices and a proper incorporation of a diffusive flux balance at simplicial faces the semi implicit scheme is derived via a discretization of the underlying conservation law and discrete counterparts of continuous a priori estimates in this geometric setup are proved the continuous solution on the continuous family of evolving surfaces is compared to the finite volume solution on the discrete sequence of simplicial surfaces and convergence of the family of discrete solutions on successively refined meshes is proved under suitable assumptions on the geometry and the discrete meshes furthermore numerical results show remarkable aspects of transport and diffusion phenomena on evolving surfaces and experimentally reflect the established convergence results finally we discuss how to combine the presented scheme with a corresponding finite volume scheme for advective transport on the surface via an operator splitting and present some applications

geometric error of finite volume schemes for conservation laws on evolving surfaces this paper studies finite volume schemes for scalar hyperbolic conservation laws on evolving hypersurfaces of mathbb r 3 r 3 we compare theoretical schemes assuming knowledge of all geometric quantities to practical schemes defined on moving polyhedra approximating the surface for the former schemes error estimates have already been proven but the implementation of such schemes is not feasible for complex geometries the latter schemes in contrast only require easily computable geometric quantities and are thus more useful for actual computations we prove that the difference between approximate solutions defined by the respective families of schemes is of the order of the mesh width in particular the practical scheme converges to the entropy solution with the same rate as the theoretical one numerical experiments show that the proven order of convergence is optimal

cut finite element methods for coupled bulk surface problems we develop a cut finite element method for a second order elliptic coupled bulk surface model problem we prove a priori estimates for the energy and l 2 l2 norms of the error using stabilization terms we show that the resulting algebraic system of equations has a similar condition number as a standard fitted finite element method finally we present a numerical example illustrating the accuracy and the robustness of our approach

a front tracking method for computation of interfacial flows with soluble surfactants a finite difference front tracking method is developed for computations of interfacial flows with soluble surfactants the method is designed to solve the evolution equations of the interfacial and bulk surfactant concentrations together with the incompressible navier stokes equations using a non linear equation of state that relates interfacial surface tension to surfactant concentration at the interface the method is validated for simple test cases and the computational results are found to be in a good agreement with the analytical solutions the method is then applied to study the cleavage of drop by surfactant a problem proposed as a model for cytokinesis h p greenspan on the dynamics of cell cleavage j theor biol 65 1 1977 79 h p greenspan on fluid mechanical simulations of cell division and movement j theor biol 70 1 1978 125 finally the method is used to model the effects of soluble surfactants on the motion of buoyancy driven bubbles in a circular tube and the results are found to be in a good agreement with available experimental data

new exact exponents for two dimensional self avoiding walks the author establishes using standard coulomb gas methods the values of a new set of geometrical exponents that have been previously conjectured these exponents give for any p the number of configurations of p two dimensional self avoiding walks of the same length l which are attached by their ends

renormalization group approach to an abelian sandpile model on planar lattices one important step in the renormalization group rg approach to a lattice sandpile model is the exact enumeration of all possible toppling processes of sandpile dynamics inside a cell for rg transformations here we propose a computer algorithm to carry out such exact enumeration for cells of planar lattices in the rg approach to the bak tang wiesenfeld sandpile model phys rev lett 59 381 1987 and consider both the reduced high rg equations proposed by pietronero vespignani and zapperi pvz phys rev lett 72 1690 1994 and the real height rg equations proposed by ivashkevich phys rev lett 76 3368 1996 using this algorithm we are able to carry out rg transformations more quickly with large cell size e g 3 ifmmode times else texttimes fi 3 cell for the square sq lattice in pvz rg equations which is the largest cell size at the present and find some mistakes in a previous paper phys rev e 51 1711 1995 for sq and plane triangular pt lattices we obtain the only attractive fixed point for each lattice and calculate the avalanche exponent ensuremath tau and the dynamical exponent z our results suggest that the increase of the cell size in the pvz rg transformation does not lead to more accurate results the implication of such result is discussed

a standard test set for numerical approximations to the shallow water equations in spherical geometry a suite of seven test cases is proposed for the evaluation of numerical methods intended for the solution of the shallow water equations in spherical geometry the shallow water equations exhibit the major difficulties associated with the horizontal dynamical aspects of atmospheric modeling on the spherical earth these cases are designed for use in the evaluation of numerical methods proposed for climate modeling and to identify the potential trade offs which must always be made in numerical modeling before a proposed scheme is applied to a full baroclinic atmospheric model it must perform well on these problems in comparison with other currently accepted numerical methods the cases are presented in order of complexity they consist of advection across the poles steady state geostrophically balanced flow of both global and local scales forced nonlinear advection of an isolated low zonal flow impinging on an isolated mountain rossby haurwitz waves and observed atmospheric states one of the cases is also identified as a computer performance algorithm efficiency benchmark for assessing the performance of algorithms adapted to massively parallel computers

a critical analysis of rayleigh taylor growth rates abstract recent simulations of rayleigh taylor instability growth rates display considerable spread we provide evidence that differences in numerical dissipation effects mass diffusion and viscosity due to algorithmic differences and differences in simulation duration are the dominant factors that produce such different results within the simulation size and durations explored here we provide evidence that the principal discrepancies are due to numerical dispersion through comparison of simulations using different algorithms we present new 3d front tracking simulations that show tentative agreement with the range of reported experimental values we begin an exploration of new physical length scales that may characterize a transition to a new rayleigh taylor mixing regime

universal finite size scaling functions with exact nonuniversal metric factors using exact partition functions and finite size corrections for the ising model on finite square plane triangular and honeycomb lattices and extending a method j phys 19 l1215 1986 to subtract leading singular terms from the free energy we obtain universal finite size scaling functions for the specific heat internal energy and free energy of the ising model on these lattices with exact nonuniversal metric factors

an immersed boundary method for interfacial flows with insoluble surfactant in this paper an immersed boundary method is proposed for the simulation of two dimensional fluid interfaces with insoluble surfactant the governing equations are written in a usual immersed boundary formulation where a mixture of eulerian flow and lagrangian interfacial variables are used and the linkage between these two set of variables is provided by the dirac delta function the immersed boundary force comes from the surface tension which is affected by the distribution of surfactant along the interface by tracking the interface in a lagrangian manner a simplified surfactant transport equation is derived the numerical method involves solving the navier stokes equations on a staggered grid by a semi implicit pressure increment projection method where the immersed interfacial forces are calculated at the beginning of each time step once the velocity value and interfacial configurations are obtained surfactant concentration is updated using the transport equation in this paper a new symmetric discretization for the surfactant concentration equation is proposed that ensures the surfactant mass conservation numerically the effect of surfactant on drop deformation in a shear flow is investigated in detail

semi implicit surface tension formulation with a lagrangian surface mesh on an eulerian simulation grid we present a method for applying semi implicit forces on a lagrangian mesh to an eulerian discretization of the navier stokes equations in a way that produces a sparse symmetric positive definite system the resulting method has semi implicit and fully coupled viscosity pressure and lagrangian forces we apply our new framework for forces on a lagrangian mesh to the case of a surface tension force which when treated explicitly leads to a tight time step restriction by applying surface tension as a semi implicit lagrangian force the resulting method benefits from improved stability and the ability to take larger time steps the resulting discretization is also able to maintain parasitic currents at low levels

arbitrary lagrangian eulerian finite element method for computation of two phase flows with soluble surfactants a finite element scheme based on a coupled arbitrary lagrangian eulerian and lagrangian approach is developed for the computation of interface flows with soluble surfactants the numerical scheme is designed to solve the time dependent navier stokes equations and an evolution equation for the surfactant concentration in the bulk phase and simultaneously an evolution equation for the surfactant concentration on the interface second order isoparametric finite elements on moving meshes and second order isoparametric surface finite elements are used to solve these equations the interface resolved moving meshes allow the accurate incorporation of surface forces marangoni forces and jumps in the material parameters the lower dimensional finite element meshes for solving the surface evolution equation are part of the interface resolved moving meshes the numerical scheme is validated for problems with known analytical solutions a number of computations to study the influence of the surfactants in 3d axisymmetric rising bubbles have been performed the proposed scheme shows excellent conservation of fluid mass and of the total mass of the surfactant

self organized critical state of sandpile automaton models we study a general bak tang wiesenfeld type automaton model of self organized criticality in which the toppling conditions depend on local height but not on its gradient we characterize the critical state and determine its entropy for an arbitrary finite lattice in any dimension the two point correlation function is shown to satisfy a linear equation the spectrum of relaxation times describing the approach to the critical state is also determined exactly

amplitude ratios for critical systems in the c 2 universality class we study the finite size corrections of the critical dense polymer cdp and the dimer models on n rectangular lattice we find that the finite size corrections in the cdp and dimer models depend in a crucial way on the parity of n and a change of the parity of n is equivalent to the change of boundary conditions we present a set of universal amplitude ratios for amplitudes in finite size correction terms of critical systems in the universality class with central charge c 2 the results are in perfect agreement with a perturbated conformal field theory under the assumption that all analytical corrections coming from the operators which belongs to the tower of the identity our results inspire many interesting problems for further studies

an error estimate for finite volume methods for multidimensional conservation laws in this paper an l ll error estimate for a class of finite volume methods for the approximation of scalar multidimensional conservation laws is obtained these methods can be formally high order accurate and are defined on general triangulations the error is proven to be of order ft 4 where h represents the size of the mesh via an extension of kuznetsov approximation theory for which no estimate of the total variation and of the modulus of con tinuity in time are needed the result is new even for the finite volume method constructed from monotone numerical flux functions

interface tracking for axisymmetric flows a front tracking method for inviscid gas dynamics is presented the key constructions and algorithms used in the code are described and the interrelations between shock capturing interface dynamics computational geometry grid construction and parallelism are discussed for the code as a whole validation is carried out by comparing the single mode bubble velocity for rayleigh taylor instability with theoretical models and experimental results the calculations are validated by mesh refinement studies and by the comparison of the asymptotic limit of the minimum radius r rm min rightarrow infty to a pure planar computation in two dimensions

exact multileg correlation functions for the dense phase of branching polymers in two dimensions we consider branching polymers on the planar square lattice with open boundary conditions and exactly calculate correlation functions of k polymer chains that connect two lattice sites with a large distance r apart for odd number of polymer chains k we find that besides the standard power law factor the leading term also has a logarithmic multiplier

a hybrid numerical method for interfacial fluid flow with soluble surfactant we address a significant difficulty in the numerical computation of fluid interfaces with soluble surfactant that occurs in the physically representative limit of large bulk peclet number pe at the high values of pe in typical fluid surfactant systems there is a transition layer near the interface in which the surfactant concentration varies rapidly and large gradients at the interface must be resolved accurately to evaluate the exchange of surfactant between the interface and bulk flow we use the slenderness of the layer to develop a fast and accurate hybrid numerical method that incorporates a separate singular perturbation analysis of the dynamics in the transition layer into a full numerical solution of the interfacial free boundary problem the accuracy and efficiency of the method is assessed by comparison with a more traditional numerical approach that uses finite differences on a curvilinear coordinate system exterior to the bubble without the separate transition layer reduction the traditional method implemented here features a novel fast calculation of fluid velocity off the interface

liquid vapor transformations with surfactants phase field model and isogeometric analysis surfactants are compounds that find energetically favorable to be located at the boundaries between fluids they are able to modify the properties of those interfaces for example reducing surface tension here we propose a new model for liquid vapor flows with surfactants which captures the dynamics of the surfactant and accounts for phase transformations in the fluid the aforementioned model is derived from a free energy functional by using a coleman noll approach the proposed theory emanates from the isothermal navier stokes korteweg equations which describe single component two phase flow and naturally allow for phase transformations we believe that our model has significant potential to study the influence of surfactants in vaporization and condensation processes from a numerical point of view the proposed model poses significant challenges to existing discretization methods including stiffness in space and time internal and boundary layers as well as higher order partial differential operators to overcome these challenges we propose algorithms based on isogeometric analysis which permit an accurate and efficient discretization finally we illustrate the viability of the theoretical framework and the effectiveness of our algorithms by solving several numerical problems in two and three dimensions

the nitsche xfem dg space time method and its implementation in three space dimensions in a recent paper c lehrenfeld and a reusken siam j numer anal 51 2013 pp 958 983 a new finite element discretization method for a class of two phase mass transport problems is presented and analyzed the transport problem describes mass transport in a domain with an evolving interface across the evolving interface a jump condition has to be satisfied the discretization in that paper is a space time approach which combines a discontinuous galerkin dg technique in time with an extended finite element method xfem using the nitsche method the jump condition is enforced in a weak sense while the emphasis in that paper was on the analysis and one dimensional numerical experiments the main contribution of this paper is the discussion of implementation aspects for the spatially three dimensional case as the space time interface is typically given only implicitly as the zero level of a level set function we construct a piecewise planar approximation of the space time interface this disc

verification and validation for turbulent mixing abstract verification and validation constitute a relatively new requirement for numerical analysis it is one which results from the growing practical use of computation as a basis for engineering decisions while a number of procedures have been proposed for these tasks their application to the more difficult cases remains a research issue here we explain methods developed by ourselves and collaborators for verification and validation of simulations for chaotic multiscale flows specifically for turbulent mixing simulations

a hybrid immersed boundary and immersed interface method for electrohydrodynamic simulations in this paper we develop a hybrid immersed boundary ib and immersed interface method iim to simulate the dynamics of a drop under an electric field in navier stokes flows within the leaky dielectric framework with piecewise constant electric properties in each fluid the electric stress can be treated as an interfacial force on the drop interface thus both the electric and capillary forces can be formulated in a unified immersed boundary framework the electric potential satisfies a laplace equation which is solved numerically by an augmented immersed interface method which incorporates the jump conditions naturally along the normal direction the incompressible navier stokes equations for the fluids are solved using a projection method on a staggered mac grid and the potential is solved at the cell center the interface is tracked in a lagrangian manner with mesh control by adding an artificial tangential velocity to transport the lagrangian markers to ensure that the spacing between markers is uniform throughout the computations a series of numerical tests for the present scheme have been conducted to illustrate the accuracy and applicability of the method we first compute the potential and its gradient electric field to perform the accuracy check for the present augmented iim we then check the convergence of the interfacial electric force and the fluid variables we further run a series of simulations with different permittivity and conductivity ratios and compare with the results obtained by the small deformation theory and other numerical results in literature in addition we also study the electric effect for a drop under shear flow

a front tracking algorithm for limited mass diffusion the advantage of front tracking over capturing methods for the numerical simulation of discontinuity surfaces in fluid flow is to eliminate the numerical diffusion of mass across a fluid interface for accurate modeling of an interface between miscible fluids however physical mass diffusion may be significant and must be added to the calculation the main purpose of this paper is to introduce a new front tracking algorithm in which the physical diffusion of mass across a tracked interface is included the accuracy and convergence properties of this algorithm are discussed the new algorithm is a subgrid algorithm in the sense that the asymptotic fine grid behavior is identical to conventional untracked capturing methods while the coarse grid behavior is improved it is thus most suitable for modeling small values of physical mass diffusion for which adequate numerical resolution is computationally demanding the mixing rates for the 3d rayleigh taylor instability of miscible fluids based on this algorithm agree with experimental values

a phase field model with temperature dependent constraint this paper is concerned with a phase field model with temperature dependent constraint for the order parameter in the kinetic equation of the order parameter we take account of diffusion as well as undercooled superheated state which is modelled by means of a hysteresis input output relation the input is the temperature field and the output is the order parameter in this paper we shall show that there exists a solution globally in time in a physically reasonable sense when the coefficient of interfacial energy is sufficiently small

a computational method for the coupled solution of reaction diffusion equations on evolving domains and manifolds in this paper we devise a moving mesh finite element method for the approximate solution of coupled bulk surface reaction diffusion equations on an evolving two dimensional domain fundamental to the success of the method is the robust generation of bulk and surface meshes for this purpose we use a novel moving mesh partial differential equation mmpde approach the developed method is applied to model problems with known analytical solutions these experiments indicate second order spatial and temporal accuracy coupled bulk surface problems occur frequently in many areas in particular in the modelling of eukaryotic cell migration and chemotaxis we apply the method to a model of the two way interaction of a migrating cell in a chemotactic field where the bulk region corresponds to the extracellular region and the surface to the cell membrane

multiscale models for fluid mixing recent work of the authors and colleagues on the turbulent mixing of compressible fluids is developed and extended with an emphasis on the multiscale aspects of this work specifically we study an interplay between micro and macro aspects of mixing

a hybrid variational front tracking level set mesh generator for problems exhibiting large deformations and topological changes we present a method for generating 2 d unstructured triangular meshes that undergo large deformations and topological changes in an automatic way we employ a method for detecting when topological changes are imminent via distance functions and shape skeletons when a change occurs we use a level set method to guide the change of topology of the domain mesh this is followed by an optimization procedure using a variational formulation of active contours that seeks to improve boundary mesh conformity to the zero level contour of the level set function our method is advantageous for arbitrary lagrangian eulerian ale type methods and directly allows for using a variational formulation of the physics being modeled and simulated including the ability to account for important geometric information in the model such as for surface tension driven flow furthermore the meshing procedure is not required at every time step and the level set update is only needed during a topological change hence our method does not significantly affect computational cost

a lattice boltzmann dynamic immersed boundary scheme for the transport of deformable inertial capsules in low re flows abstract in this work a dynamic immersed boundary method combined with a bgk lattice boltzmann technique is developed and critically discussed the fluid evolution is obtained on a three dimensional lattice with 19 reticular velocities d3q19 computational molecule while the immersed body surface is modeled as a collection of lagrangian points responding to an elastic potential and a bending resistance a moving least squares reconstruction is used to accurately interpolate flow quantities and the forcing field needed to enforce the boundary condition on immersed bodies the proposed model is widely validated against well known benchmark data for rigid and deformable objects rigid transport is validated by computing the settling of a sphere under gravity for five different conditions then the tumbling of inertial particles with different shape is considered recovering the jefferey orbit for a prolate spheroid moreover the revolution period for an oblate spheroid and for a disk like particle is obtained as a function of the reynolds number the existence of a critical reynolds number is demonstrated for both cases above which revolution is inhibited the transport of deformable objects is also considered the steady deformation of a membrane under shear for three different values of the mechanical stiffness is assessed then the tumbling of a weakly deformable spheroid under shear is systematically analyzed as a function strain stiffness bending resistance and membrane mass

parabolic pdes with hysteresis and quasivariational inequalities w0 being prescribed as an initial output of w its input output relation is illustrated in fig 1 1 see 1 part i iii 2 for the precise de nition in the gure fa and fd are continuous and nondecreasing functions on r such that fa fd on r we refer for some results on this sort of model to 1 4 as is well known 1 part i iii 2 1 2 is equivalent to the following variational inequality wt x t iu x t w x t 3 0 x t qt 1 3

finite element discretization error analysis of a general interfacial stress functional the finite element discretization of a variable interfacial stress tensor sigma gamma x is considered variable interfacial tension is included as a special case interfacial stress for example occurs in two phase flow problems in the weak formulation the interfacial stress gives rise to a functional which is supported on the interface gamma a finite element discretization of this functional is presented and analyzed the discretization admits almost independent meshes for the approximation of the interface and the approximation of the flow variables the main result is an mathcal o h k 1 2 error bound for the interfacial stress functional in a natural norm if the discrete interface is an mathcal o h k 1 approximation of gamma the bound is shown to be sharp in numerical experiments

eulerian finite element methods for parabolic equations on moving surfaces three new eulerian finite element methods for parabolic pdes on a moving surface gamma t are presented and compared in numerical experiments these are space time galerkin methods which are derived from a weak formulation in space and time the trial and test spaces contain the traces on the space time manifold of an outer prismatic finite element space the numerical experiments show that two of the methods converge with second order with respect to both the time step size and the spatial mesh width

the immersed interface method for elliptic equations with discontinuous coefficients and singular sources the authors develop finite difference methods for elliptic equations of the form nabla cdot beta x nabla u x kappa x u x f x in a region omega in one or two space dimension

connectivity free front tracking method for multiphase flows with free surfaces in this study a connectivity free front tracking method is developed to simulate multiphase flows with free surfaces this method is based on the point set method which does not require any connectivities between interfacial points to represent the interface the main advantage of the connectivity free approach is the easiness in re constructing the interface when large topology change occurs it requires an indicator field to be constructed first based on the existing interface and the surface curvature and normal are then computed using the indicator field here we adopt the reproducing kernel particle method rkpm interpolation function that provides the ability to deal with free surface flows and the flexibility of using non uniform meshes when local fine resolution is needed a points regeneration scheme is developed to construct smooth interfaces and to automatically handle topology changes the mass conservation is verified by performing a single vortex advection test several 2 d and 3 d numerical tests including an oscillating droplet dam breaking two droplet impacting and multi bubble merging are presented to show the accuracy and the robustness of the method

a diffuse interface method for two phase flows with soluble surfactants a method is presented to solve two phase problems involving soluble surfactants the incompressible navier stokes equations are solved along with equations for the bulk and interfacial surfactant concentrations a nonlinear equation of state is used to relate the surface tension to the interfacial surfactant concentration the method is based on the use of a diffuse interface which allows a simple implementation using standard finite difference or finite element techniques here finite difference methods on a block structured adaptive grid are used and the resulting equations are solved using a nonlinear multigrid method results are presented for a drop in shear flow in both 2d and 3d and the effect of solubility is discussed

efficient numerical methods for multiple surfactant coated bubbles in a two dimensional stokes flow we present efficient and highly accurate numerical methods to compute the deformation of surfactant coated two dimensional bubbles in a slow viscous flow surfactant acts to locally alter the surface tension and thereby change the nature of the interface motion in this paper we restrict our attention to the case of a dilute insoluble surfactant the convection diffusion equation for the surfactant concentration on the interface is coupled with the stokes equations in the fluid domain through a boundary condition based on the laplace young condition the stokes equations are first recast as an integral equation and then solved using a fast multipole accelerated iterative procedure the computational cost per time step is only o n log n operations with n being the number of discretization points on the interface the bubble interfaces are described by a spectral mesh and is advected according to the fluid velocity in such a manner so as to preserve equal arc length spacing of marker points this equal arc length framework has the dual advantage of dynamically maintaining the spatial mesh and allowing efficient implicit treatment of the stiffest terms in the dynamics several phenomenologically different examples are presented

universality in critical exponents for toppling waves of the btw sandpile model on two dimensional lattices abstract universality and scaling for systems driven to criticality by a tuning parameter has been well studied however there are very few corresponding studies for the models of self organized criticality e g the bak tang and wiesenfeld btw sandpile model it is well known that every avalanche of the btw sandpile model may be represented as a sequence of waves and the asymptotic probability distributions of all waves and last waves have critical exponents 1 and 11 8 respectively by an inversion symmetry hu ivashkevich lin and priezzhev showed that in the btw sandpile model the probability distribution of dissipating waves of topplings that touch the boundary of the system shows a power law relationship with critical exponent 5 8 and the probability distribution of those dissipating waves that are also last in an avalanche has an exponent of 1 phys rev lett 85 2000 4048 such predictions have been confirmed by extensive numerical simulations of the btw sandpile model on square lattices very recently we used monte carlo simulations to find that the waves of the btw model on square honeycomb triangular and random lattices have the same set of critical exponents

nitsche xfem with streamline diffusion stabilization for a two phase mass transport problem we consider an unsteady convection diffusion equation which models the transport of a dissolved species in two phase incompressible flow problems the so called henry interface condition leads to a jump condition for the concentration at the interface between the two phases in a hansbo and p hansbo comput methods appl mech engrg 191 2002 pp 5537 5552 for the purely elliptic stationary case an extended finite element method xfem is combined with a nitsche type method and optimal error bounds are derived these results were extended to the unsteady case in a reusken and t nguyen j fourier anal appl 15 2009 pp 663 683 in the latter paper convection terms are also considered but assumed to be small in many two phase flow applications however convection is the dominant transport mechanism hence there is a need for a stable numerical method for the case of a convection dominated transport equation in this paper we address this topic and study the streamline diffusion sta

exact partition functions of the ising model on m n planar lattices with periodic aperiodic boundary conditions the grassmann path integral approach is used to calculate exact partition functions of the ising model on m n square sq plane triangular pt and honeycomb hc lattices with periodic periodic pp periodic antiperiodic pa antiperiodic periodic ap and antiperiodic antiperiodic aa boundary conditions the partition functions are used to calculate and plot the specific heat c kb as a function of temperature kbt j we find that for the n n sq lattice c kb for pa and ap boundary conditions are different from those for aa boundary conditions but for the n n pt and hc lattices c kb for ap pa and aa boundary conditions have the same values our exact partition functions might also be useful for understanding the effects of lattice structures and boundary conditions on critical finite size corrections of the ising model

mudpack multigrid portable fortran software for the efficient solution of linear elliptic partial differential equations mudpack is a package of portable fortran subprograms which use multigrid iteration for solving real or complex elliptic partial differential equations the solution regions are rectangles in two dimensions and boxes in three dimensions any combination of periodic dirichlet and mixed derivative boundary conditions is allowed the equations are automatically discretized using second order finite differencing the package will vectorize on cray computers examples are given which demonstrate ease of use efficiency and applicability to a wide range of problems

on a system of nonlinear pde s for phase transitions with vector order parameter and convective effect the paper deals with a system of nonlinear pde s which describes a phase field model with convection and temperature dependent constraint to the vector order parameter existence of solutions for the system under consideration is proved by the method of yosida approximation and fixed point arguments

geometrical factor and thermal properties of the potts model based on the connection between the q state potts model qpm and a q state bond correlated percolation model qbcpm we define a non percolating geometrical factor g and percolating geometrical factor gp which depend only on q and geometrical properties of the system the thermal properties of the qpm could be derived simply from g and g the idea is confirmed by an exact calculation for one dimensional qpm and may be extended to other ising like spin models our formulation also provides a geometrical meaning for the finite size scaling and broadening at first order phase transitions of the spin model

relations between the coulomb gas picture and conformal invariance of two dimensional critical models partition functions of critical 2d models on a torus can be derived from their microscopic formulation and their free field representation in the continuum limit this is worked out explicitly for theo n andq state potts model forn orq integer we recover results obtained from conformal invariance but our procedure also extends to nonintegral values in the latter case the expansion on characters of the virasoro algebra involves real coefficients of either sign the operator content of both models is discussed in detail

a finite element method for surface pdes matrix properties we consider a recently introduced new finite element approach for the discretization of elliptic partial differential equations on surfaces the main idea of this method is to use finite element spaces that are induced by triangulations of an outer domain to discretize the partial differential equation on the surface the method is particularly suitable for problems in which there is a coupling with a problem in an outer domain that contains the surface for example two phase flow problems it has been proved that the method has optimal order of convergence both in the h 1 and in the l 2 norm in this paper we address linear algebra aspects of this new finite element method in particular the conditioning of the mass and stiffness matrix is investigated for the two dimensional case we present an analysis which proves that the effective spectral condition number of the diagonally scaled mass matrix and the diagonally scaled stiffness matrix behaves like h 3 ln h and h 2 ln h respectively where h is the mesh size of the outer triangulation

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

scalar conservation laws on constant and time dependent riemannian manifolds abstract in this paper we establish well posedness for scalar conservation laws on closed manifolds m endowed with a constant or a time dependent riemannian metric for initial values in l m in particular we show the existence and uniqueness of entropy solutions as well as the l 1 contraction property and a comparison principle for these solutions throughout the paper the flux function is allowed to depend on time and to have non vanishing divergence furthermore we derive estimates of the total variation of the solution for initial values in bv m and we give in the case of a time independent metric a simple geometric characterisation of flux functions that give rise to total variation diminishing estimates

a front tracking method for the computations of multiphase flow direct numerical simulations of multiphase flows using a front tracking method are presented the method is based on writing one set of governing equations for the whole computational domain and treating the different phases as one fluid with variable material properties interfacial terms are accounted for by adding the appropriate sources as functions at the boundary separating the phases the unsteady navier stokes equations are solved by a conventional finite volume method on a fixed structured grid and the interface or front is tracked explicitly by connected marker points interfacial source terms such as surface tension are computed on the front and transferred to the fixed grid advection of fluid properties such as density is done by following the motion of the front the method has been implemented for fully three dimensional flows as well as for two dimensional and axisymmetric ones first the method is described for the flow of two or more isothermal phases the representation of the moving interface and its dynamic restructuring as well as the transfer of information between the moving front and the fixed grid are discussed applications and extensions of the method to homogeneous bubbly flows atomization flows with variable surface tension solidification and boiling are then presented

improved volume conservation in the computation of flows with immersed elastic boundaries this paper introduces a recipe for the construction of a finite difference divergence operator the coefficients of which are completely determined once an interpolation scheme has been chosen substitution of this divergence operator for the previously used divergence based on central differences makes a dramatic improvement in the overall volume conservation that is observed in an immersed boundary computation this improvement is particularly important for computations in which an elastic boundary separates chambers containing fluid at substantially different pressures a situation which is prominent in cardiac fluid dynamics during the contraction of the ventricles

a finite element method for elliptic equations on surfaces in this paper a new finite element approach for the discretization of elliptic partial differential equations on surfaces is treated the main idea is to use finite element spaces that are induced by triangulations of an outer domain to discretize the partial differential equation on the surface the method is particularly suitable for problems in which there is a coupling with a flow problem in an outer domain that contains the surface we give an analysis that shows that the method has optimal order of convergence both in the h 1 and in the l 2 norm results of numerical experiments are included that confirm this optimality

a cut finite element method for coupled bulk surface problems on time dependent domains in this contribution we present a new computational method for coupled bulk surface problems on time dependent domains the method is based on a space time formulation using discontinuous piecewise linear elements in time and continuous piecewise linear elements in space on a fixed background mesh the domain is represented using a piecewise linear level set function on the background mesh and a cut finite element method is used to discretize the bulk and surface problems in the cut finite element method the bilinear forms associated with the weak formulation of the problem are directly evaluated on the bulk domain and the surface defined by the level set essentially using the restrictions of the piecewise linear functions to the computational domain in addition a stabilization term is added to stabilize convection as well as the resulting algebraic system that is solved in each time step we show in numerical examples that the resulting method is accurate and stable and results in well conditioned algebraic systems independent of the position of the interface relative to the background mesh