robust computational algorithms for dynamic interface tracking in three dimensions front tracking provides sharp resolution of wave fronts through the active tracking of interfaces between distinct materials a major challenge to this method is to handle changes in the interface topology we describe two algorithms implemented in the front tracking code frontier to model dynamic changes in three dimensional interfaces the two methods can be combined to give a hybrid method that is superior to each individual method the success of these algorithms is shown by simulations of rayleigh taylor instability which is an interfacial instability driven by an acceleration directed across a material interface our numerical results are validated by comparing the numerical computation of the velocity of a single rising bubble with an analytic model for the bubble velocity

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

the fixed mesh ale approach for the numerical approximation of flows in moving domains in this paper we propose a method to approximate flow problems in moving domains using always a given grid for the spatial discretization and therefore the formulation to be presented falls within the category of fixed grid methods even though the imposition of boundary conditions is a key ingredient that is very often used to classify the fixed grid method our approach can be applied together with any technique to impose approximately boundary conditions although we also describe the one we actually favor our main concern is to properly account for the advection of information as the domain boundary evolves to achieve this we use an arbitrary lagrangian eulerian framework the distinctive feature being that at each time step results are projected onto a fixed background mesh that is where the problem is actually solved

high order gradients with the shifted boundary method an embedded enriched mixed formulation for elliptic pdes abstract we propose an extension of the embedded boundary method known as shifted boundary method to elliptic diffusion equations in mixed form e g darcy flow heat diffusion problems with rough coefficients etc our aim is to obtain an improved formulation that for linear finite elements is at least second order accurate for both flux and primary variable when either dirichlet or neumann boundary conditions are applied following previous work of nishikawa and mazaheri in the context of residual distribution methods we consider the mixed form of the diffusion equation i e with darcy type operators and introduce an enrichment of the primary variable this enrichment is obtained exploiting the relation between the primary variable and the flux variable which is explicitly available at nodes in the mixed formulation the proposed enrichment mimics a formally quadratic pressure approximation although only nodal unknowns are stored similar to a linear finite element approximation we consider both continuous and discontinuous finite element approximations and present two approaches a non symmetric enrichment which as in the original references only improves the consistency of the overall method and a symmetric enrichment which enables a full error analysis in the classical finite element context combined with the shifted boundary method these two approaches are extended to high order embedded computations and enable the approximation of both primary and flux gradient variables with second order accuracy independently on the type of boundary conditions applied we also show that the the primary variable is third order accurate when pure dirichlet boundary conditions are embedded

fictitious domain approach with hp finite element approximation for incompressible fluid flow we consider the application of fictitious domain approach combined with least squares spectral elements for the numerical solution of fluid dynamic incompressible equations fictitious domain methods allow problems formulated on a complicated shaped domain to be solved on a simpler domain containing least squares spectral element method has been used to develop the discrete model as this scheme combines the generality of finite element methods with the accuracy of spectral methods moreover the least squares methods have theoretical and computational advantages in the algorithmic design and implementation this paper presents the formulation and validation of the fictitious domain least squares spectral element approach for the steady incompressible navier stokes equations the convergence of the approximated solution is verified solving two dimensional benchmark problems demonstrating the predictive capability of the proposed formulation

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

multistability for delayed neural networks via sequential contracting in this paper we explore a variety of new multistability scenarios in the general delayed neural network system geometric structure embedded in equations is exploited and incorporated into the analysis to elucidate the underlying dynamics criteria derived from different geometric configurations lead to disparate numbers of equilibria a new approach named sequential contracting is applied to conclude the global convergence to multiple equilibrium points of the system the formulation accommodates both smooth sigmoidal and piecewise linear activation functions several numerical examples illustrate the present analytic theory

a note on the modified q bernstein polynomials for functions of several variables and their reflections on q volkenborn integration abstract in this paper we consider the modified q bernstein polynomials for functions of several variables on q volkenborn integral and investigate some new interesting properties of these polynomials related to q stirling numbers hermite polynomials and carlitz s type q bernoulli numbers

adaptive mesh refinement and superconvergence for two dimensional interface problems adaptive mesh refinement and the borgers algorithm are combined to generate a body fitted mesh which can resolve the interface with fine geometric details standard linear finite element method bas

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

interaction of oscillating filaments a computational study abstract the immersed boundary technique is used to model the interaction of swimming filaments in a viscous incompressible fluid fluid quantities are represented on a grid and the immersed filaments are each represented by a discrete collection of moving points the effects of phase differences and proximity of the filaments on energy dissipation and swimming speeds are studied the computed results are compared with previous asymptotic analysis furthermore we present case studies which exhibit phase un locking phenomena undetectable using previous asymptotic analysis

new cartesian grid methods for interface problems using the finite element formulation new finite element methods based on cartesian triangulations are presented for two dimensional elliptic interface problems involving discontinuities in the coefficients the triangulations in these methods do not need to fit the interfaces the basis functions in these methods are constructed to satisfy the interface jump conditions either exactly or approximately both non conforming and conforming finite element spaces are considered corresponding interpolation functions are proved to be second order accurate in the maximum norm the conforming finite element method has been shown to be convergent with cartesian triangulations these new methods can be used as finite difference methods numerical examples are provided to support the methods and the theoretical analysis

weak imposition of essential boundary conditions in the finite element approximation of elliptic problems with non matching meshes this is the accepted version of the following article codina r and baiges j 2015 weak imposition of essential boundary conditions in the finite element approximation of elliptic problems with non matching meshes int j numer meth engng 104 624 654 doi 10 1002 nme 4815 which has been published in final form at http onlinelibrary wiley com doi 10 1002 nme 4815 abstract

a decoupled augmented iim strategy for incompressible two phase flows with interfaces on irregular domains a decoupled augmented immersed interface method for solving incompressible two phase flows involving both irregular domains and interfaces is presented in order to impose the prescribed velocity at the boundary of the irregular domain singular force as one set of augmented variables is introduced the velocity components at the two fluid interface as another set of augmented variables are introduced to satisfy the continuity condition of the velocity across the interface so that the jump conditions for the velocity and pressure are decoupled across the interface the augmented variables and or the forces along the interface boundary are related to the jumps in both pressure and velocity and the jumps in their derivatives across the interface boundary and applied to the fluid through jump conditions the resulting augmented equation is a couple system of these two sets of augmented variables and the direct application of the gmres is impractical due to larger iterations in this work the novel decoupling of two sets of the augmented variables is proposed and the decoupled augmented equation is then solved by the lu or the gmres method the stokes equations are discretized via the finite difference method with the incorporation of jump contributions on a staggered cartesian grid and solved by the conjugate gradient uzawa type method the numerical results show that second order accuracy for the velocity is confirmed the present method has also been applied to solve for incompressible two phase navier stokes flow with interfaces on irregular domains copyright 2011 john wiley sons ltd

immersed interface methods for stokes flow with elastic boundaries or surface tension a second order accurate interface tracking method for the solution of incompressible stokes flow problems with moving interfaces on a uniform cartesian grid is presented the interface may consist of an elastic boundary immersed in the fluid or an interface between two different fluids the interface is represented by a cubic spline along which the singularly supported elastic or surface tension force can be computed the stokes equations are then discretized using the second order accurate finite difference methods for elliptic equations with singular sources developed in our previous paper siam j numer anal 31 1994 pp 1019 1044 the resulting velocities are interpolated to the interface to determine the motion of the interface an implicit quasi newton method is developed that allows reasonable time steps to be used

interface tracking and interface capturing techniques for finite element computation of moving boundaries and interfaces we provide an overview of some of the interface tracking and interface capturing techniques we developed for finite element computation of flow problems with moving boundaries and interfaces this category of flow problems includes fluid particle fluid object and fluid structure interactions free surface and two fluid flows and flows with moving mechanical components both classes of techniques are based on stabilized formulations the interface tracking techniques are based on the deforming spatial domain stabilized space time dsd sst formulation where the mesh moves to track the interface the interface capturing techniques developed primarily for free surface and two fluid interface flows are formulated typically over non moving meshes using an advection equation in addition to the flow equations the advection equation governs the evolution of an interface function that marks the location of the interface we also highlight some of the methods we developed to increase the scope and accuracy of these two classes of techniques

novel identities involving genocchi numbers and polynomials arising from applications of umbral calculus abstract the aim of this paper is to deal with applications of umbral calculus on fermionic p adic integral on z p from those applications we derive some new identities on genocchi numbers and polynomials moreover a systemic study of the class of sheffer sequences on the generating function of genocchi polynomials are given

an interface fitted mesh generator and virtual element methods for elliptic interface problems abstract a simple and efficient interface fitted mesh generation algorithm which can produce a semi structured interface fitted mesh in two and three dimensions quickly is developed in this paper elements in such interface fitted meshes are not restricted to simplices but can be polygons or polyhedra especially in 3d the polyhedra instead of tetrahedra can avoid slivers virtual element methods are applied to solve elliptic interface problems with solutions and flux jump conditions algebraic multigrid solvers are used to solve the resulting linear algebraic system numerical results are presented to illustrate the effectiveness of our method

simulation of elastic filaments interacting with a viscous pulsatile flow abstract motivated by the need to understand the motion of ruptured atherosclerosis plaque fragments in a flowing blood we simulated flexible elastic filaments interacting with a viscous pulsatile flow in two dimensions using the immersed boundary method our simulations show that 1 a flexible horizontal filament positioned perpendicularly to the flow is a stable configuration and 2 hydrodynamical forces prevent adhesion of filaments but mutual interaction of vortical fields yields clustering of filaments into groups that may be called aggregation

asymptotic limit in a cell differentiation model with consideration of transcription abstract t cells of the immune system upon maturation differentiate into either th1 or th2 cells that have different functions the decision to which cell type to differentiate depends on the concentrations of transcription factors t bet x 1 and gata 3 x 2 these factors are translated by the mrna whose levels of expression y 1 and y 2 depend respectively on x 1 and x 2 in a nonlinear nonlocal way the population density of t cells t x 1 x 2 y 1 y 2 satisfies a hyperbolic conservation law with coefficients depending nonlinearly and nonlocally on t x 1 x 2 y 1 y 2 while the x i y i satisfy a system of ordinary differential equations we study the long time behavior of and show under some conditions on the parameters of the system of differential equations that the gene expressions in the t cell population aggregate at one two or four points which connect to various cell differentiation scenarios

a hybrid level set front tracking approach for finite element simulations of two phase flows abstract in this paper we give details on the numerical realization of a new finite element method for the simulation of two phase flows which was recently introduced in basting and weismann 2013 the main ingredient is a hybrid representation of the interface between the fluid phases an implicit description of the interface is given by a level set function and an explicit representation is obtained from aligning edges of the computational mesh to the implicitly described interface this step is done by a black box optimization based mesh smoothing approach which does not change the topology of the mesh while guaranteeing optimal mesh quality furthermore we make use of quadratic isoparametric elements to increase the approximation quality of the discrete interface due to the alignment discontinuities of the solution variables pressure can be captured accurately while a variational treatment of the curvature allows for a precise approximation of surface tension we present our time discretization scheme for the coupled navier stokes level set equations and discuss our space discretization based on the so called subspace projection method spm to account for discontinuities across the interface we present two numerical examples for which reference solutions exist we consider the oscillation of a single droplet and provide our results for an established two phase flow benchmark problem

the shifted boundary method for embedded domain computations part ii linear advection diffusion and incompressible navier stokes equations abstract we propose a new embedded finite element method for the linear advection diffusion equation and the laminar and turbulent incompressible navier stokes equations the proposed method belongs to the class of surrogate approximate boundary algorithms and is based on the idea of shifting the location where boundary conditions are applied from the true to a surrogate boundary accordingly boundary conditions enforced weakly are appropriately modified to preserve optimal error convergence rates we include the full analysis of stability and convergence of the method in the linear advection diffusion equation and a battery of tests for the case of laminar and turbulent incompressible navier stokes equations we also discuss the conservation properties of the proposed method in all cases

a moving lagrangian interface technique for flow computations over fixed meshes abstract in this paper an enhanced finite element formulation for unsteady incompressible flows with moving interfaces is presented the weak form of the navier stokes equations written using a generalized streamline operator technique is coupled with the movement of the interface between two immiscible fluids defined through an independent moving mesh the position of the interface is updated using a lagrangian formulation in this framework a global mass conservation corrector algorithm and an enhanced element integration technique are proposed to improve accuracy the method is applied to a number of test problems with moving interfaces

a cartesian grid projection method for the incompressible euler equations in complex geometries many problems in fluid dynamics require the representation of complicated internal or external boundaries of the flow here we present a method for calculating time dependent incompressible inviscid flow which combines a projection method with a cartesian grid approach for representing geometry in this approach the body is represented as an interface embedded in a regular cartesian mesh the advection step is based on a cartesian grid algorithm for compressible flow in which the discretization of the body near the flow uses a volume of fluid representation a redistribution procedure is used to eliminate time step restrictions due to small cells where the boundary intersects the mesh the projection step uses an approximate projection based on a cartesian grid method for potential flow the method incorporates knowledge of the body through volume and area fractions along with certain other integrals over the mixed cells convergence results are given for the projection itself and for the time dependent algorithm in two dimensions the method is also demonstrated on flow past a half cylinder with vortex shedding

a fictitious domain method for dirichlet problem and applications abstract in this article we discuss the solution of the dirichlet problem for a class of elliptic operators by a lagrange multiplier fictitious domain method this approach allows the use of regular grids and therefore of fast specialized solvers for problems on complicated geometries the resulting saddle point system can be solved by an uzawa conjugate gradient algorithm in the case of two dimensional problems a quasi optimal preconditioner has been found by fourier analysis and numerical experiments confirm its nice scaling properties the resulting methodology is applied to a nonlinear time dependent problem namely the flow of a viscous plastic medium in a cylindrical pipe showing the potential of this methodology for some classes of nonlinear problems

a note on immersed interface method for three dimensional elliptic equations abstract the immersed interface method proposed by leveque and li 1 is extended to three dimensional elliptic equations of the form x u x x u x f x we study the situation in which there is an irregular interface surface s contained in the solution domain across which and f may be discontinuous or even singular as a result the solution u will usually be nonsmooth or even discontinuous a finite difference approach with a uniform cartesian grid is used in the discretization local truncation error analysis is performed to estimate the accuracy of the numerical solution

the fast solution of poisson s and the biharmonic equations on irregular regions we present fast methods for solving laplace s and the biharmonic equations on irregular regions with smooth boundaries the methods used for solving both equations make use of fast poisson solvers on a rectangular region in which the irregular region is embedded they also both use an integral equation formulation of the problem where the integral equations are fredholm integral equations of the second kind the main idea is to use the integral equation formulation to define a discontinuous extension of the solution to the rest of the rectangular region fast solvers are then used to compute the extended solution aside from solving the equations we have also been able to compute derivatives of the solutions with little loss of accuracy when the data was sufficiently smooth

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

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

the approximation properties of generalized bernstein polynomials of two variables the positive linear operators of bernstein type of two variables are considered and the theorems about approximation and about the order of approximation of the sequence of these operators are established

a regularization method for the numerical solution of periodic stokes flow we introduce a regularization method that gives a smooth formulation for the fundamental solution to stokes flow driven by an infinite triply periodic array of point forces with this formulation the velocity at any spatial location may be calculated including at and very near the point forces these locations typically lead to numerical difficulties due to the singularity within the stokeslet when using other methods for computational efficiency we build upon previous methods in which the periodic stokeslet is split into two rapidly decaying sums one in physical space and one in reciprocal or fourier space we present two validation studies of our method first we compute the drag coefficient for periodic arrays of spheres with a variety of concentrations of sphere packings and second we prescribe a force density onto a periodic array of spheres compute the resulting nearby velocity field and compare these velocities to those computed using an immersed boundary method formulation the drag coefficients computed with our method are within 0 63 of previously published values the velocity field comparison shows a relative error of about 0 18 in the l 2 norm we then apply our numerical method to a periodic arrangement of sinusoidal swimmers by systematically varying their spacing in three directions we are able to explore how their spacing affects their collective swimming speed

level set methods an overview and some recent results the level set method was devised by s osher and j a sethian 1988 j comput phys 79 12 49 as a simple and versatile method for computing and analyzing the motion of an interface in two or three dimensions bounds a possibly multiply connected region the goal is to compute and analyze the subsequent motion of under a velocity field v this velocity can depend on position time the geometry of the interface and the external physics the interface is captured for later time as the zero level set of a smooth at least lipschitz continuous function x t i e t x x t 0 is positive inside negative outside and is zero on t topological merging and breaking are well defined and easily performed in this review article we discuss recent variants and extensions including the motion of curves in three dimensions the dynamic surface extension method fast methods for steady state problems diffusion generated motion and the variational level set approach we also give a user s guide to the level set dictionary and technology and couple the method to a wide variety of problems involving external physics such as compressible and incompressible possibly reacting flow stefan problems kinetic crystal growth epitaxial growth of thin films vortex dominated flows and extensions to multiphase motion we conclude with a discussion of applications to computer vision and image processing

a fictitious domain method for external incompressible viscous flow modeled by navier stokes equations abstract in this article we discuss a fictitious domain method for the numerical solutions of three dimensional elliptic problems with dirichlet boundary conditions and also of the navier stokes equations modeling incompressible viscous flow the methodology for the navier stokes equations described here takes a systematic advantage of time discretization by operator splitting in order to treat separately advection imbedding and incompressibility due to the decoupling fast elliptic solvers can be used to treat the incompressibility condition even if the original problem is taking place on a nonregular geometry the resulting methodology is applied to two dimensional unsteady external incompressible viscous flow problems and three dimensional stokes problems

the immersed interface method for the navier stokes equations with singular forces abstract peskin s immersed boundary method has been widely used for simulating many fluid mechanics and biology problems one of the essential components of the method is the usage of certain discrete delta functions to deal with singular forces along one or several interfaces in the fluid domain however the immersed boundary method is known to be first order accurate and usually smears out the solutions in this paper we propose an immersed interface method for the incompressible navier stokes equations with singular forces along one or several interfaces in the solution domain the new method is based on a second order projection method with modifications only at grid points near or on the interface from the derivation of the new method we expect fully second order accuracy for the velocity and nearly second order accuracy for the pressure in the maximum norm including those grid points near or on the interface this has been confirmed in our numerical experiments furthermore the computed solutions are sharp across the interface nontrivial numerical results are provided and compared with the immersed boundary method meanwhile a new version of the immersed boundary method using the level set representation of the interface is also proposed in this paper

the numerical approximation of a delta function with application to level set methods it is shown that a discrete delta function can be constructed using a technique developed by anita mayo the fast solution of poisson s and the biharmonic equations on irregular regions siam j sci comput 21 1984 285 299 for the numerical solution of elliptic equations with discontinuous source terms this delta function is concentrated on the zero level set of a continuous function in two space dimensions this corresponds to a line and a surface in three space dimensions delta functions that are first and second order accurate are formulated in both two and three dimensions in terms of a level set function the numerical implementation of these delta functions achieves the expected order of accuracy

locally corrected semi lagrangian methods for stokes flow with moving elastic interfaces we present a new method for computing two dimensional stokes flow with moving interfaces that respond elastically to stretching the interface is moved by semi lagrangian contouring a distance function is introduced on a tree of cells near the interface transported by a semi lagrangian time step and then used to contour the new interface the velocity field in a periodic box is calculated as a potential integral resulting from interfacial and body forces using a technique based on ewald summation with analytically derived local corrections the interfacial stretching is found from a surprisingly natural formula a test problem with an exact solution is constructed and used to verify the speed accuracy and robustness of the approach

a computational model of the cochlea using the immersed boundary method this thesis describes a two dimensional computational model of the cochlea inner ear solved using a modified form of peskin s immersed boundary method this method is an extension of peskin s method that was originally applied to solving a model of the heart 1972 this method solves the time dependent incompressible navier stokes equations in the presence of immersed boundaries the fluid equations are specified on a fixed eulerian grid while the immersed boundaries are specified on a moving lagrangian grid the immersed boundaries are used to represent the structures of the cochlea the rigid bone the actively moving stapes and the flexible membranes the immersed boundaries exert forces locally on the fluid these local forces are seen by the fluid as external forces that are added to the other forces pressure and viscous acting on the fluid the computational method divides into three separate parts first the forces exerted by the points of the immersed boundary are calculated and extended to the fluid grid second the fluid equations are solved and the velocities are restricted to the immersed boundary points finally the immersed boundary points are moved to their new locations at the new local velocities the immersed boundary forces are either specified such that the boundary moves at a given velocity zero or non zero or they are specified such that the forces obey some given force law for example hooke s law these forces can be calculated either explicitly or implicitly or a combination of both for the fluid the navier stokes equations are solved on a doubly periodic rectangular grid in a second order accurate manner using a projection method developed by bell colella and glaz 1988 the extension and restriction of the boundary forces and fluid velocities to and from the fluid grid are also done in a second order accurate manner the cochlea is modelled as two fluid chambers separated by a flexible partition whose stiffness varies exponentially along its length the stapes is represented by a moving piston and the bony outer walls are allowed to be either straight or tapered a travelling wave propagates along the flexible partition under the influence of the moving piston and the dependence of this wave on cochlea geometry and partition stiffness is studied also included are studies of bone conduction and transient signal analysis finally comparisons of the model results are made to experimental data and asymptotic results

analysis of a one dimensional model for the immersed boundary method numerical methods are studied for the one dimensional heat equation with a singular forcing term u t u xx c t delta x alpha t the delta function delta x is replaced by a discrete approximation d h x and the resulting equation is solved by a crank nicolson method on a uniform grid the accuracy of this method is analyzed for various choices of d h the case where c t is specified and also the case where c is determined implicitly by a constraint on the solution at the point a are studied these problems serve as a model for the immersed boundary method of peskin for incompressible flow problems in irregular regions some insight is gained into the accuracy that can be achieved and the importance of choosing appropriate discrete delta functions

an interface fitted subspace projection method for finite element simulations of particulate flows abstract a novel finite element method for the direct numerical simulation of particles in a newtonian carrier fluid is presented the proposed method is based on a fictitious or one domain formulation and a subspace projection method to account for the rigid body motion of the particles underlying equations are posed in an ale arbitrary lagrangian eulerian formulation allowing for moving computational meshes the mesh is adapted to the particles by a novel mesh smoothing approach guaranteeing both mesh optimality and a sharp representation of the particles boundaries we show that by using quadratic taylor hood finite elements for the discretization of the navier stokes equations and isoparametric elements to represent the geometry second order convergence with respect to the energy norm can be achieved as opposed to a sharp h 1 2 result which holds if no mesh adaptation is performed we present numerical examples in 2d confirming the theoretical convergence results furthermore we validate the approach by simulating the sedimentation process of a single particle and show the potential of the method by simulating a lid driven cavity flow with 100 particles

velocity correction projection methods for incompressible flows we introduce and study a new class of projection methods namely the velocity correction methods in standard form and in rotational form for solving the unsteady incompressible navier stokes equations we show that the rotational form provides improved error estimates in terms of the h1 norm for the velocity and of the l2 norm for the pressure we also show that the class of fractional step methods introduced in s a orsag m israeli and m deville j sci comput 1 1986 pp 75 111 and k e karniadakis m israeli and s a orsag j comput phys 97 1991 pp 414 443 can be interpreted as the rotational form of our velocity correction methods thus to the best of our knowledge our results provide the first rigorous proof of stability and convergence of the methods in those papers we also emphasize that contrary to those of the above groups our formulations are set in the standard l2 setting and consequently they can be easily implemented by means of any variational approximation techniques in particular the finite element methods

numerical modeling of the acoustic wave propagation across a homogenized rigid microstructure in the time domain homogenization of a thin micro structure yields effective jump conditions that incorporate the geometrical features of the scatterers these jump conditions apply across a thin but nonzero thickness interface whose interior is disregarded this paper aims i to propose a numerical method able to handle the jump conditions in order to simulate the homogenized problem in the time domain ii to inspect the validity of the homogenized problem when compared to the real one for this purpose we adapt an immersed interface method originally developed for standard jump conditions across a zero thickness interface doing so allows us to handle arbitrary shaped interfaces on a cartesian grid with the same efficiency and accuracy of the numerical scheme than those obtained in an homogeneous medium numerical experiments are performed to test the properties of the numerical method and to inspect the validity of the homogenization problem

a balanced force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework a new balanced force algorithm is presented for modeling interfacial flow with surface tension the algorithm is characterized by a pressure correction method with the interfaces represented by volume fractions within this flow algorithm we devise a continuous e g continuum surface tension model and a sharp e g a ghost fluid method interface representation of the surface tension induced interfacial pressure jump condition the sharp interface representation is achieved by temporarily reconstructing distance functions from volume fractions we demonstrate that a flow algorithm designed to legislate force balance retains an exact balance between surface tension forces and the resulting pressure gradients this balance holds for both continuous and sharp representations of interfacial surface tension the algorithm design eliminates one of the elusive impediments to more accurate models of surface tension driven flow the remaining of which is accurate curvature estimation to validate our formulation we present results for an equilibrium static drop in two and three dimensions having an arbitrary density jump across the interface we find that the sharp surface tension method yields an abrupt pressure jump across the interface whereas the continuous surface tension method results in a smoother transition both methods however yield spurious velocities of the same order the origin of which is due solely to errors in curvature dynamic results are also presented to illustrate the versatility of the method

removing the stiffness from interfacial flows with surface tension a new formulation and new methods are presented for computing the motion of fluid interfaces with surface tension in two dimensional irrotational and incompressible fluids through the laplace young condition at the interface surface tension introduces high order terms both nonlinear and nonlocal into the dynamics this leads to severe stability constraints for explicit time integration methods and makes the application of implicit methods difficult this new formulation has all the nice properties for time integration methods that are associated with having a linear constant coefficient highest order term that is using this formulation we give implicit time integration methods that have no high order time step stability constraint associated with surface tension and are explicit in fourier space the approach is based on a boundary integral formulation and applies more generally even to problems beyond the fluid mechanical context here they are applied to computing with high resolution the motion of interfaces in hele shaw flows and the motion of free surfaces in inviscid flows governed by the euler equations one hele shaw computation shows the behavior of an expanding gas bubble over long time as the interface undergoes successive tip splittings and finger competition a second computation shows the formation of a very ramified interface through the interaction of surface tension with an unstable density stratification in euler flows the computation of a vortex sheet shows its roll up through the kelvin helmholtz instability this motion culminates in the late time self intersection of the interface creating trapped bubbles of fluid this is we believe a type of singularity formation previously unobserved for such flows in 2d finally computations of falling plumes in an unstably stratified boussinesq fluid show a very similar behavior

partially implicit motion of a sharp interface in navier stokes flow we develop a numerical method for the coupled motion of navier stokes flow with an elastic interface of zero thickness which exerts tension and bending forces on the fluid the interface motion is made partially implicit by approximating a backward euler step in the high wavenumbers as in the small scale decomposition method of hou lowengrub and shelley this modified step is combined with the method of beale and layton j t beale a t layton a velocity decomposition approach for moving interfaces in viscous fluids j comput phys 228 2009 3358 67 the fluid velocity is found by computing the stokes velocity and a more regular remainder the resulting scheme is second order in space and first order in time it can be made second order in time by extrapolation the discontinuities in the pressure and velocity gradient are preserved the partially implicit method allows much larger time steps than an explicit method with negligible added effort the formulas in the fourier transform for the implicit approximation in high wavenumbers are similar to those derived in hou and shi t y hou z shi an efficient semi implicit immersed boundary method for the navier stokes equations j comput phys 227 2008 9138 69 in a different context

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

a finite element technique for multifluid incompressible flow using eulerian grids the paper presents a finite element method for 3d incompressible fluid flows with capillary free boundaries it uses a fixed eulerian grid of 10 nodes p2 p1 tetrahedra and tracks the free boundary using a six nodes p2 triangular surface grid in order to improve the mass conservation properties of the method a local enrichment of the finite element basis in the elements intersected by the free boundaries is employed in addition to the surface tracking it also advects a smooth indicator function for an easy identification of the fluid properties in the different parts of the domain the advective part of the navier stokes equations is split and integrated with a characteristic method the remaining generalized stokes problem is resolved by means of an inexact outer inner uzawa iteration with a properly chosen preconditioner the performance of this technique is evaluated on several problems involving droplets in viscous liquids

smoothed profile method for particulate flows error analysis and simulations abstract we re formulate and analyze a new method for particulate flows the so called smoothed profile method spm first proposed in y nakayama r yamamoto simulation method to resolve hydrodynamic interactions in colloidal dispersions phys rev e 71 2005 036707 which uses a fixed computational mesh without conformation to the geometry of the particles the method represents the particles by certain smoothed profiles to construct a body force term added into the navier stokes equations spm imposes accurately and efficiently the proper conditions at the particle fluid interface in particular while the original method employs a fully explicit time integration scheme we develop a high order semi implicit splitting scheme which we implement in the context of spectral hp element discretization first we show that the modeling error of spm has a non monotonic dependence on the time step size t it is a function of t where is the kinematic viscosity of fluid and is the interface thickness of the smoothed profile subsequently we present several steady and unsteady simulations including flow past 3d complex shaped particles and compare against direct numerical simulations and the force coupling method fcm

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

least squares spectral method for the two dimensional stokes interface problems abstract this paper deals with the pseudo spectral least squares method to approximate the solution of the stokes interface problem that is the stokes equations with discontinuous viscosity and singular source term the stokes interface equations are adjusted in each sub domain along with the interface conditions then the equations are reformulated as a first order system by introducing stress tensor as a new independent variable the continuous and discrete norm least squares functionals using legendre and chebyshev weights for the first order system are defined and shown to be equivalent to appropriate product norms the spectral convergence of the proposed method is presented numerical examples are also given to demonstrate the method and analysis

a simplified reproducing kernel method for 1 d elliptic type interface problems abstract in this paper an efficient numerical method is proposed for the 1d elliptic type interface problems we first construct a broken reproducing kernel space and then apply the simplified reproducing kernel method to the interface problems the convergence of the proposed method is discussed finally some representative numerical examples are presented to illustrate the accuracy and robustness of the proposed method

asymptotic phases in a cell differentiation model t cells of the immune system upon maturation differentiate into either th1 or th2 cells that have different functions the decision to which cell type to differentiate depends on the concentrations of transcription factors t bet x1 and gata 3 x2 the population density of the t cells t x1 x2 satisfies a conservation law t x1 f1 x2 f2 g where fi depends on t x1 x2 and in a nonlinear nonlocal way on it is proved that as t t x1 x2 converges to a linear combination of 1 2 or 4 dirac measures numerical simulations and their biological implications are discussed