a fast algorithm for simulating vesicle flows in three dimensions vesicles are locally inextensible fluid membranes that can sustain bending in this paper we extend the study of veerapaneni et al s k veerapaneni d gueyffier g biros d zorin a numerical method for simulating the dynamics of 3d axisymmetric vesicles suspended in viscous flows journal of computational physics 228 19 2009 7233 7249 to general non axisymmetric vesicle flows in three dimensions although the main components of the algorithm are similar in spirit to the axisymmetric case spectral approximation in space semi implicit time stepping scheme important new elements need to be introduced for a full 3d method in particular spatial quantities are discretized using spherical harmonics and quadrature rules for singular surface integrals need to be adapted to this case an algorithm for surface reparameterization is needed to ensure stability of the time stepping scheme and spectral filtering is introduced to maintain reasonable accuracy while minimizing computational costs to characterize the stability of the scheme and to construct preconditioners for the iterative linear system solvers used in the semi implicit time stepping scheme we perform a spectral analysis of the evolution operator on the unit sphere by introducing these algorithmic components we obtain a time stepping scheme that circumvents the stability constraint on the time step and achieves spectral accuracy in space we present results to analyze the cost and convergence rates of the overall scheme to illustrate the applicability of the new method we consider a few vesicle flow interaction problems a single vesicle in relaxation sedimentation shear flows and many vesicle flows

a boundary integral method for simulating the dynamics of inextensible vesicles suspended in a viscous fluid in 2d we present a new method for the evolution of inextensible vesicles immersed in a stokesian fluid we use a boundary integral formulation for the fluid that results in a set of nonlinear integro differential equations for the vesicle dynamics the motion of the vesicles is determined by balancing the non local hydrodynamic forces with the elastic forces due to bending and tension numerical simulations of such vesicle motions are quite challenging on one hand explicit time stepping schemes suffer from a severe stability constraint due to the stiffness related to high order spatial derivatives and a milder constraint due to a transport like stability condition on the other hand an implicit scheme can be expensive because it requires the solution of a set of nonlinear equations at each time step we present two semi implicit schemes that circumvent the severe stability constraints on the time step and whose computational cost per time step is comparable to that of an explicit scheme we discretize the equations by using a spectral method in space and a multistep third order accurate scheme in time we use the fast multipole method fmm to efficiently compute vesicle vesicle interaction forces in a suspension with a large number of vesicles we report results from numerical experiments that demonstrate the convergence and algorithmic complexity properties of our scheme

simulations of multiphase flows with multiple length scales using moving mesh interface tracking with adaptive meshing in multiphase flows the length scales of thin regions such as thin films between nearly touching drops and thin threads formed during the interface pinch off are usually several orders of magnitude smaller than the size of the drops in this paper a number of extra length criteria for adaptive meshes are developed and implemented in the moving mesh interface tracking method to solve these multiple length scale problems with high fidelity a nominal length scale based on the solutions of laplace s equations with the unit normal vectors of surfaces as the boundary conditions is proposed for the adaptive mesh refinement in the thin regions for almost flat interfaces boundaries which are near to the thin regions the averaged length of the interior edges sharing the two nodes with the boundary edge is introduced for the mesh adaptation the averaged length of the interfacial edges is used for the interior elements near the interfaces but outside of the thin regions for the interior mesh away from the interfaces boundaries different averaged length scales based on the initial mesh are employed for the adaptive mesh refining and coarsening numerous cases are simulated to demonstrate the capability of the proposed schemes in handling multiple length scales which include the relaxation and necking of an elongated droplet droplet droplet head on approaching droplet wall interactions and a droplet pair in a shear flow the smallest length resolved for the thin regions is three orders of magnitude smaller than the largest characteristic length of the problem

integral equation methods for particle simulations in creeping flows abstract integral equation methods for computing the hydrodynamic interactions among solid particles suspended in a creeping flow are presented the particles may have arbitrary shape and they may be suspended in an unbounded or wall bounded fluid the analytic formulation of the integral equation is based on complex variables and the fast multipole based iterative solution procedure requires only o n operations where n is the number of nodes in the discretization of the boundary thus large scale problems with complex geometry can be solve using modest computational resources from the hydrodynamic interactions the particle motions are determined either by computing a sequence of steady state stokes flow problems or by coupling the particles equation of motion thereby including the weak effects of the particles solid inertia examples will include the sedimentation of particles in a quiescent fluid towards or parallel to a plane wall and the motion of neutrally buoyant particles in a shear flow

a cell based particle method for modeling dynamic interfaces abstract we propose several modifications to the grid based particle method gbpm 21 for moving interface modeling there are several nice features of the proposed algorithm the new method can significantly improve the distribution of sampling particles on the evolving interface unlike the original gbpm where footpoints sampling points tend to cluster to each other the sampling points in the new method tend to be better separated on the interface moreover by replacing the grid based discretization using the cell based discretization we naturally decompose the interface into segments so that we can easily approximate surface integrals as a possible alternative to the local polynomial least square approximation we also study a geometric basis for local reconstruction in the resampling step we will show that such modification can simplify the overall implementations numerical examples in two and three dimensions will show that the algorithm is computationally efficient and accurate

an efficient algorithm for hydrodynamical interaction of many deformable drops an efficient and accurate 3d algorithm for dynamical simulations of many deformable drops with strong hydrodynamical interactions at zero reynolds numbers is developed the drop to medium viscosity ratio and the bond number are arbitrary and the drops are subject to gravity with stationary triply periodic boundary conditions the algorithm at each step is a hybrid of boundary integral and economical multipole techniques with extensive use of rotational transformations and economical truncation of multipole expansions to optimize near field interactions a significant part of the code is the new best paraboloid spline technique for calculating the normal vectors and curvatures on drop surfaces which greatly improves the quality of long time simulations examples show the phenomenon of clustering in a concentrated sedimenting emulsion for 0 25 and 1 which leads to an increase in the average sedimentation velocity with time a high efficiency of the method is demonstrated with two orders of magnitude gains over the standard o n2n2 boundary integral technique for n 102 drops in a periodic cell with n 103 triangular boundary elements per drop so that typical long time dynamical simulations can be performed in a few days or weeks on a standard workstation as compared to the several years which would be required for the same simulations using standard boundary integral techniques the effects of drop triangulation and truncation of multipole expansions on dynamical simulations are assessed

fluid films with curvature elasticity the phenomenology of surfactant fluid film microstructures interspersed in bulk fluids poses significant challenges to continuum theory by using simple models of elastic surfaces chemical physicists have been partially successful in describing the qualitative features of

boundary integral method for the flow of vesicles with viscosity contrast in three dimensions we propose numerical algorithms for the simulation of the dynamics of three dimensional vesicles suspended in viscous stokesian fluid our method is an extension of our previous work s k veerapaneni et al 2011 37 to flows with viscosity contrast this generalization requires a change in the boundary integral formulation of the solution in which a double layer stokes integral is introduced and leads to changes in the fluid dynamics due to the viscosity contrast of the vesicles which can no longer be efficiently resolved with existing algorithms in this paper we describe the algorithms needed to handle flows with viscosity contrast accurately and efficiently we show that a globally semi implicit method does not have any time step stability constraint for flows with single and multiple vesicles with moderate viscosity contrast and the computational cost per simulation unit time is comparable to or less than that of an explicit scheme automatic oversampling adaptation enables us to achieve high accuracy with very low spectral resolution we conduct numerical experiments to investigate the stability accuracy and the computational cost of the algorithms overall our method achieves several orders of magnitude speed up compared to the standard explicit schemes

an efficient surface integral algorithm applied to unsteady gravity waves a computationally fast method for calculating the unsteady motion of a surface on a two dimensional fluid is described cauchy s integral theorem is used iteratively to solve laplace s equation for successive time derivatives of the surface motion and time stepping is performed using truncated taylor series this allows fairly large time steps to be made for a given accuracy while the required number of spatial points is minimised by using high order differencing formulae this reduces the overall number of required calculations the numerical implementation of the method is found to be accurate and efficient a fairly thorough examination of this implementation is carried out revealing that high accuracies are often achievable using surprisingly few numerical surface points extensive calculations are also performed using modest computing resources some numerical instabilities are identified although these would not usually be significant in practical calculations a model analysis reveals that two of these instabilities can be eliminated by using suitable methods of time stepping should the third steep wave instability become significant it is shown that it can be completely controlled by using high order smoothing techniques at little cost to accuracy using a routine to ensure asymptotic conservation of energy this is confirmed by time stepping a very steep but stable wave over thousands of wave periods 0 1992 academx press inc since all of the interior properties of a body of fluid undergoing inviscid incompressible irrotational motion are fully determined by properties at its boundary it is possible to reduce the calculation of the motion of such a fluid to the evaluation of the motion of its surface alone in a numerical scheme the entire motion can thus be modelled using only a point discretisation of the surface such an approach was first suggested by svendsen in 1971 as a means of calculating the motion of two dimensional gravity waves on water 13 since then the idea has been implemented using a variety of boundary integral or conformal mapping techniques by longuet higgins and cokelet 2 vinje and brevig 3 baker meiron and orszag 4 roberts s fornberg 6 and based on his original formulation by svendsen and co workers 7 although this paper presents yet another numerical method the approach is aimed specifically at producing a computationally fast and efficient numerical scheme in order to be able to study problems of increased complexity in numerically representing complicated surface motions ranging from the breaking of waves to the evolution of instabilities in a train of travelling waves it typically becomes necessary to use a large number of surface points if running times tend to increase too rapidly with the number of points then this can destroy any apparent advantage in being able to treat the whole fluid only in terms of its moving surface the penalty is particularly severe for algorithms that require matrix inversion or factorisation techniques yielding running times per time step that increase like n3 the cube of the number of surface points by using iterative techniques the penalty can be reduced to running times per time step that increase in proportion to n2 while being less severe this still has the effect that increased numbers of surface points become increasingly expensive to calculate the use of a conformal mapping technique 6 can lead to running times per timestep that vary like n log n however this method is severely limited in its ability to resolve steep wave surfaces as are found in breaking waves requiring disproportionately many surface points to achieve a given accuracy it is therefore vital to ensure computational elliciency if it is to be possible accurately to describe wave processes for which large numbers of surface points may be required the most direct way of achieving this is to employ the lowest number of points that can adequately describe a fluid surface in this paper a further improvement is introduced by showing that higher time derivatives of the motion can be calculated directly using exactly the same technique that is used to determine the velocity of the surface at each timestep this yields significantly more information about the motion for relatively little extra computation in turn this allows larger time steps to be performed for a given accuracy so that the overall number of calculations can be reduced the method presented here combines these two ideas into producing a practical numerical algorithm the properties of which are examined in the following pages in appendix a the formulae are derived that make it

a method for computing the tension parameters in convexity preserving spline in tension interpolation this paper is concerned with the problem of convexity preservng orc preserving interpolation by using exponential splines in tension or est s for this purpose the notion of ac preserving interpolant which is usually employed in spline in tension interpolation is refined and the existence ofc preserving est s is established for the so calledc admissible data sets the problem of constructing ac preserving and visually pleasing est is then treated by combining a generalized newton raphson method due to ben israel with a step length technique which serves the need for visual pleasantness the numerical performance of the so formed iterative scheme is discussed for several examples

dynamic simulation of locally inextensible vesicles suspended in an arbitrary two dimensional domain a boundary integral method we consider numerical algorithms for the simulation of hydrodynamics of two dimensional vesicles suspended in a viscous stokesian fluid the motion of vesicles is governed by the interplay between hydrodynamic and elastic forces continuum models of vesicles use a two phase fluid system with interfacial forces that include tension to maintain local surface inextensibility and bending vesicle flows are challenging to simulate on the one hand explicit time stepping schemes suffer from a severe stability constraint due to the stiffness related to high order spatial derivatives in the bending term on the other hand implicit time stepping schemes can be expensive because they require the solution of a set of nonlinear equations at each time step our method is an extension of the work of veerapaneni et al s k veerapaneni d gueyffier d zorin g biros a boundary integral method for simulating the dynamics of inextensible vesicles suspended in a viscous fluid in 2d journal of computational physics 228 7 2009 2334 2353 in which a semi implicit time marching scheme based on a boundary integral formulation of the stokes problem for vesicles in an unbounded medium was proposed in this paper we consider two important generalizations i confined flows within arbitrary shaped stationary moving geometries and ii flows in which the interior to the vesicle and exterior fluids have different viscosity in the rest of the paper we will refer to this as the viscosity contrast these two problems require solving additional integral equations and cause nontrivial modifications to the previous numerical scheme our method does not have severe time step stability constraints and its computational cost per time step is comparable to that of an explicit scheme the discretization is pseudo spectral in space and multistep bdf in time we conduct numerical experiments to investigate the stability accuracy and the computational cost of the algorithm overall our method achieves several orders of magnitude speed up compared to standard explicit schemes as a preliminary validation of our scheme we study the dependence of the inclination angle of a single vesicle in shear flow on the viscosity contrast and the reduced area of the vesicle the lateral migration of vesicles in shear flow the dispersion of two vesicles and the effective viscosity of a dilute suspension of vesicles

adaptive triangulation of evolving closed or open surfaces by the advancing front method the advancing front method is adapted to describe the evolution of open or closed three dimensional surfaces in terms of an unstructured grid consisting of quadratic triangular elements in science and engineering applications the surface may be identified with a material interface a free boundary or a moving front in the numerical method the geometrical properties of the surface and the coordinates of the triangle vertices are computed either in terms of available analytical expressions or by means of interpolation through an underlying coarse grid the shape and size of the curved triangular elements are determined by the maximum magnitude of the mean or directional local surface curvature two algorithms are implemented the first one for simply connected open surfaces bounded by closed lines and for closed surface with plane symmetry and the second for closed surfaces in the case of open surfaces the discretization front advances from a one dimensional boundary grid that traces the bounding curve the boundary grid is generated either by a one dimensional version of the advancing front method or by requiring a set of criteria based on local line representation with circular arcs in the case of closed surfaces the discretization front emanates from the point of the maximum mean curvature in both cases the algorithm also interpolates to generate the values of surface geometrical or physical variables such as temperature or concentration of a surface active agent the method was tested by following the motion of several passive and active surfaces evolving under the action of specified fields of flow while performing occasional regridding to ensure adequate spatial resolution in one test the large deformation of a viscous drop subjected to an infinite simple shear flow at vanishing reynolds number was computed into the regime where a cigar like shape is established thereby extending previous numerical computations for small and moderate deformations and reproducing experimentally observed shapes overall the adaptive front method emerges as an important tool in numerical studies of free boundaries or moving fronts and should be useful in a broad range of applications

3d numerical simulations of vesicle and inextensible capsule dynamics vesicles are locally inextensible fluid membranes capsules are endowed with in plane shear elasticity mimicking the cytoskeleton of red blood cells rbcs but are extensible while rbcs are inextensible we use boundary integral bi methods based on the green function techniques to model and solve numerically their dynamics we regularize the single layer integral by subtraction of exact identities for the terms involving the normal and the tangential components of the force the stability and precision of bi calculation is enhanced by taking advantage of additional quadrature nodes located in vertices of an auxiliary mesh constructed by a standard refinement procedure from the main mesh we extend the partition of unity technique to boundary integral calculation on triangular meshes the proposed algorithm offers the same treatment of near singular integration regardless whether the source and the target points belong to the same surface or not bending forces are calculated by using expressions derived from differential geometry membrane incompressibility is handled by using two penalization parameters per suspended entity one for deviation of the global area from prescribed value and another for the sum of squares of local strains defined on each vertex extensible or inextensible capsules a model of rbc are studied by storing the position in the reference configuration for each vertex the elastic force is then calculated by direct variation of the elastic energy various nonequilibrium physical examples on vesicles and capsules will be presented and the convergence and precision tests highlighted overall a good convergence is observed with numerical error inversely proportional to the number of vertices used for surface discretization the highest order of convergence allowed by piece wise linear interpolation of the surface

a triangular spectral element method applications to the incompressible navier stokes equations abstract encouraged by the success of spectral elements methods in computational fluid dynamics and p type finite element methods in structural mechanics we wish to extend these ideas to solving high order polynomial approximations on triangular domains as the next generation of spectral element solvers we introduce here a complete formulation using a modal basis which has been implemented in a new code ne t r the new basis has the following properties jacobi polynomials of mixed weights semi orthogonality hierarchical structure generalized tensor warped product variable order and a new apex co ordinate system allowing automated integration with gaussian quadrature we have discussed the formulation using a matrix notation which allows for an easy interpretation of the forward and backward transformations we use this notation to formulate the linear advection and helmholtz equations in an efficient manner and show that we recover the following properties well conditioned matrices asymptotic operation count of o n 3 scaling o n 2 of the spectral radius of the weak convective operator and exponential convergence using polynomials up to 40 having constructed and numerically analysed these equations we are then able to solve the incompressible navier stokes equations using a high order splitting scheme we demonstrate a variety of results showing exponential convergence using both deformed and straight triangular subdomains for both the stokes and navier stokes problems

isogeometric fem bem simulations of drop capsule and vesicle dynamics in stokes flow we develop an algorithm for the three dimensional simulation of the dynamics of soft objects drops capsules vesicles under creeping flow conditions loop elements are used to describe the shape of the soft objects this surface representation is used both for membrane solver based on finite element method fem and for the fluid solver based on the boundary element method bem this isogeometric analysis of the low reynolds fluidstructure interaction problem is then coupled to high order explicit time stepping or second order implicit time stepping algorithm for vesicles simulation a preconditioner is designed for the resolution of the surface velocity incompressibility constraint which is treated by the use of a local lagrange multiplier a mesh quality preserving algorithm is introduced to improve the control mesh quality over long simulation times we test the proposed algorithm on capsule and vesicle dynamics in various flows and study its convergence properties showing a second order convergence o n2 with mesh number of elements

a grid based particle method for moving interface problems we propose a novel algorithm for modeling interface motions the interface is represented and is tracked using quasi uniform meshless particles these particles are sampled according to an underlying grid such that each particle is associated to a grid point which is in the neighborhood of the interface the underlying grid provides an eulerian reference and local sampling rate for particles on the interface it also renders neighborhood information among the meshless particles for local reconstruction of the interface the resulting algorithm which is based on lagrangian tracking using meshless particles with eulerian reference grid can naturally handle control topological changes moreover adaptive sampling of the interface can be achieved easily through local grid refinement with simple quad oct tree data structure extensive numerical examples are presented to demonstrate the capability of our new algorithm

interfacial dynamics in stokes flow via a three dimensional fully implicit interfacial spectral boundary element algorithm since the pioneering work of youngren and acrivos g k youngren a acrivos on the shape of a gas bubble in a viscous extensional flow j fluid mech 76 1976 433 442 30 years ago interfacial dynamics in stokes flow has been implemented through explicit time integration of boundary integral schemes which require that the time step is sufficiently small to ensure numerical stability to avoid this difficulty we have developed an efficient fully implicit time integration algorithm based on a mathematically rigorous combination of implicit formulas with a jacobian free three dimensional newton method the resulting algorithm preserves the stability of the employed implicit formula and thus it has strong stability properties e g it is not affected by the courant condition or by physical stiffness such as that associated with the critical conditions of interfacial deformation in our work the numerical solution of our implicit algorithm is achieved through our spectral boundary element method our numerical results for free suspended droplets are in excellent agreement with experimental findings analytical predictions and earlier computational results at both subcritical and supercritical conditions and establish the properties of our fully implicit spectral boundary element algorithm

spectral collocation on triangular elements we consider the poisson problem on a segment of the unit disc and on triangles on the segment we transform the poisson problem by means of polar coordinates in these new coordinates we have a problem in a rectangle which can easily be mapped onto the square here standard chebyshev collocation techniques can be applied then the segment is mapped onto a triangle where the same spectral scheme may be used by numerical tests we observed the expected high spectral accuracy due to the corner singularity a singular behaviour of the solution can be expected here we improved the accuracy by auxiliary mapping techniques further it is shown that finite difference preconditioning can be successfully applied in order to construct an efficient iterative solver finally a domain decomposition technique is applied to the patching of a rectangular and a triangular element

nonlinear observer design in the siegel domain we extend the method of kazantzis and kravaris systems control lett 34 1998 pp 241 247 for the design of an observer to a larger class of nonlinear systems the extended method is applicable to any real analytic observable nonlinear system it is based on the solution of a first order singular nonlinear pde this solution yields a change of state coordinates which linearizes the error dynamics under very general conditions the existence and uniqueness of the solution is proved lyapunov s auxiliary theorem and siegel s theorem are obtained as corollaries the technique is constructive and yields a method for constructing approximate solutions

a fast algorithm for spherical grid rotations and its application to singular quadrature we present a fast and accurate algorithm for evaluating singular integral operators on smooth surfaces that are globally parametrized by spherical coordinates problems of this type arise for example in simulating stokes flows with particulate suspensions and in multiparticle scattering calculations for smooth surfaces spherical harmonic expansions are commonly used for geometry representation and the evaluation of the singular integrals is carried out with a spectrally accurate quadrature rule on a set of rotated spherical grids we propose a new algorithm that interpolates function values on the rotated spherical grids via hybrid nonuniform ffts the algorithm has a small complexity constant and the cost of applying the quadrature rule is nearly optimal mathcal o p 4 log p for a spherical harmonic expansion of degree p

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

a fast solver for the stokes equations with distributed forces in complex geometries we present a new method for the solution of the stokes equations the main features of our method are 1 it can be applied to arbitrary geometries in a black box fashion 2 it is second order accurate and 3 it has optimal algorithmic complexity our approach to which we refer as the embedded boundary integral method ebi is based on anita mayo s work for the poisson s equation the fast solution of poisson s and the biharmonic equations on irregular regions siam journal on numerical analysis 21 1984 285 299 we embed the domain in a rectangular domain for which fast solvers are available and we impose the boundary conditions as interface jump conditions on the velocities and tractions we use an indirect boundary integral formulation for the homogeneous stokes equations to compute the jumps the resulting equations are discretized by nystrom s method the rectangular domain problem is discretized by finite elements for a velocity pressure formulation with equal order interpolation bilinear elements q1 q1 stabilization is used to circumvent the inf sup condition for the pressure space for the integral equations fast matrix vector multiplications are achieved via an n logn algorithm based on a block representation of the discrete integral operator combined with kernel independent singular value decomposition to sparsify low rank blocks the regular grid solver is a krylov method conjugate residuals combined with an optimal two level schwartz preconditioner for the integral equation we use gmres we have tested our algorithm on several numerical examples and we have observed optimal convergence rates

regular article a spectral element method for particulate stokes flow a numerical procedure is presented for solving the equations of stokes flow past a fixed bed of rigid particles and the equations describing the motion of a suspension of rigid particles upon which a specified force and torque is exerted for general flow configurations and arbitrary particle shapes the problem is formulated in terms of an integral equation of the first kind for the distribution of the boundary traction incorporating a green function that observes the periodicity of the flow and the geometry of the boundaries of the flow accompanied by appropriate boundary conditions and integral constraints the integral equation is solved for two dimensional flow by a spectral element orthogonal collocation method two important components of the numerical method are a preconditioning of the linear system that arises from the discretization of integral equation followed by reduction to remove the eigenfunctions corresponding to the null eigenvalue over each particle surface and b a physically motivated iterative solution of the master linear system based on particle clustering it is found that for the purpose of computing the force and torque exerted on fixed particles and the velocity of translation and angular velocity of rotation of freely suspended particles the orthogonal collocation method has significant advantages over the trapezoidal discretization the iterative solution of the integral equation converges even for closely spaced particles where each particle is treated as a cluster

elastic surface substrate interactions a theory for three dimensional finite deformations of elastic solids with conforming elastic films attached to their bounding surfaces is described the gurtin murdoch theory incorporating elastic resistance of the film to strain is generalized to account for the effects of intrinsic flexural resistance this modification yields a model that can be used to describe equilibrium deformations in the presence of compressive surface stress fields an associated variational theory is given and material symmetry considerations are discussed the theory is illustrated by examples

a numerical method for simulating the dynamics of 3d axisymmetric vesicles suspended in viscous flows we extend shravan k veerapaneni denis gueyffier denis zorin george biros a boundary integral method for simulating the dynamics of inextensible vesicles suspended in a viscous fluid in 2d journal of computational physics 228 7 2009 2334 2353 to the case of three dimensional axisymmetric vesicles of spherical or toroidal topology immersed in viscous flows although the main components of the algorithm are similar in spirit to the 2d case spectral approximation in space semi implicit time stepping scheme the main differences are that the bending and viscous force require new analysis the linearization for the semi implicit schemes must be rederived a fully implicit scheme must be used for the toroidal topology to eliminate a cfl type restriction and a novel numerical scheme for the evaluation of the 3d stokes single layer potential on an axisymmetric surface is necessary to speed up the calculations by introducing these novel components we obtain a time scheme that experimentally is unconditionally stable has low cost per time step and is third order accurate in time we present numerical results to analyze the cost and convergence rates of the scheme to verify the solver we compare it to a constrained variational approach to compute equilibrium shapes that does not involve interactions with a viscous fluid to illustrate the applicability of method we consider a few vesicle flow interaction problems the sedimentation of a vesicle interactions of one and three vesicles with a background poiseuille flow

a fast algorithm for simulating multiphase flows through periodic geometries of arbitrary shape this paper presents a new boundary integral equation bie method for simulating particulate and multiphase flows through periodic channels of arbitrary smooth shape in two dimensions the authors consider a particular system multiple vesicles suspended in a periodic channel of arbitrary shape to describe the numerical method and test its performance rather than relying on the periodic green s function as classical bie methods do the method combines the free space green s function with a small auxiliary basis and imposes periodicity as an extra linear condition as a result we can exploit existing free space solver libraries quadratures and fast algorithms and handle a large number of vesicles in a geometrically complex channel spectral accuracy in space is achieved using the periodic trapezoid rule and product quadratures while a first order semi implicit scheme evolves particles by treating the vesicle channel interactions explicitly new constraint correction formulas are introduced that pre

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

accurate and efficient boundary integral methods for electrified liquid bridge problems we derive and implement boundary integral methods for axisymmetric liquid bridge problems in the presence of an axial electric field the liquid bridge is bounded by solid parallel electrodes placed perpendicular to the axis of symmetry and held at a constant potential difference the fluid is assumed to be nonconducting and has permittivity different from that of the passive surrounding medium the problem reduces to the solution of two harmonic problems for the fluid and voltage potential inside the bridge and another harmonic problem for the voltage potential outside the bridge the shape of the moving interface is determined by the imposition of stress as well as kinematic and electric field boundary conditions the former condition accounting for discontinuous electric stresses across the interface we propose fast and highly accurate boundary integral methods based on fast summations of appropriate series representations of axisymmetric green s functions in bounded geometries we implement our method to calculate equilibrium shapes for electrified liquid bridges in the absence and presence of gravity such calculations appear in the literature using finite element methods and our boundary integral approach is a fast and accurate alternative

regularized single and double layer integrals in 3d stokes flow abstract we present a numerical method for computing the single layer stokeslet and double layer stresslet integrals in stokes flow the method applies to smooth closed surfaces in three dimensions and achieves high accuracy both on and near the surface the singular stokeslet and stresslet kernels are regularized and for the nearly singular case corrections are added to reduce the regularization error these corrections are derived analytically for both the stokeslet and the stresslet using local asymptotic analysis for the case of evaluating the integrals on the surface as needed when solving integral equations we design high order regularizations for both kernels that do not require corrections this approach is direct in that it does not require grid refinement or special quadrature near the singularity and therefore does not increase the computational complexity of the overall algorithm numerical tests demonstrate the uniform convergence rates for several surfaces in both the singular and near singular cases as well as the importance of corrections when two surfaces are close to each other

a new high order immersed interface method for solving elliptic equations with imbedded interface of discontinuity this paper presents a new high order immersed interface method for elliptic equations with imbedded interface of discontinuity compared with the original second order immersed interface method of r j leveque z li the immersed interface method for elliptic equations with discontinuous coefficients and singular sources siam j numer anal 31 1994 1001 25 the new method achieves arbitrarily high order accuracy for derivatives at an irregular grid point by imposing only two physical jump conditions together with a wider set of grid stencils the new interface difference formulas are expressed in a general explicit form so that they can be applied to different multi dimensional problems without any modification the new interface algorithms of up to o h 4 accuracy have been derived and tested on several one and two dimensional elliptic equations with imbedded interface compared to the standard second order immersed interface method the test results show that the new fourth order immersed interface method leads to a significant improvement in accuracy of the numerical solutions the proposed method has potential advantages in the application to two phase flow because of its high order accuracy and simplicity in applications

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

high order quadratures for the evaluation of interfacial velocities in axi symmetric stokes flows we propose new high order accurate methods to compute the evolution of axi symmetric interfacial stokes flow the velocity at a point on the interface is given by an integral over the surface quadrature rules to evaluate these integrals are developed using asymptotic expansions of the integrands both locally about the point of evaluation and about the poles where the interface crosses the axis of symmetry the local expansions yield methods that converge to the chosen order pointwise for fixed evaluation point the pole expansions yield corrections that remove maximal errors of low order introduced by singular behaviour of the integrands as the evaluation point approaches the poles an interesting example of roundoff error amplification due to cancellation is also addressed the result is a uniformly accurate fifth order method second order pointwise fifth order and uniform fifth order methods are applied to compute three sample flows each of which presents a different computational difficulty an initially bar belled drop that pinches in finite time a drop in a strain flow that approaches a steady state and a continuously extending drop in each case the fifth order methods significantly improve the ability to resolve the flow the examples furthermore give insight into the effect of the corrections needed for uniformity we determine conditions under which the pointwise method is sufficient to obtain resolved results and others under which the corrections significantly improve the results

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

a rescaling scheme with application to the long time simulation of viscous fingering in a hele shaw cell in this paper we present a time and space rescaling scheme for the computation of moving interface problems the idea is to map time space such that the interfaces can evolve exponentially fast in the new time scale while the area volume enclosed by the interface remains unchanged the rescaling scheme significantly reduces the computation time especially for slow growth and enables one to accurately simulate the very long time dynamics of moving interfaces we then implement this scheme in a hele shaw problem examine the dynamics for a number of different injection fluxes and present the largest and most pronounced viscous fingering simulations to date

a moving mesh interface tracking method for 3d incompressible two phase flows an interface tracking method using an unstructured moving mesh has been developed for simulating three dimensional incompressible and immiscible two phase flows the interface mesh is moved in a lagrangian fashion a local mesh adaptation method is used to capture the changing interface curvature maintain good mesh quality and deal with large deformation the interface is of zero thickness so the jump and continuity conditions across the interface are implemented directly without any smoothing of the properties of the two fluids this is theoretically beneficial compared to other methods that allow the fluids properties to continuously vary in an interface region the curvature for interfacial tension calculation is geometrically computed by a least squares parabola fitting method a mesh separation scheme for interfacial flows is employed to handle topological transition the numerical technique is tested and validated by several cases which include a two layer couette flow an oscillating droplet surrounded by gas and a droplet in shear flow the results obtained from numerical simulations are found to be in excellent agreement with analytical solutions and experimental results a collapsing ligament in a quiescent gas flow and a modulated jet pinching are simulated to show the robustness of the method

interfacial dynamics for stokes flow theoretical and computational aspects of the method of interfacial dynamics for flow at vanishing reynolds number are considered the mathematical formulation relies on the boundary integral representation that expresses the flow in terms of distributions of stokes flow singularities over the interfaces the densities of the distributions are identified with the jump in the hydrodynamic traction due to interfacial in plane and transverse tensions the interfacial velocity or the strength of a hydrodynamic potential the numerical procedure involves describing the interfaces in terms of interfacial marker points that reproduce the evolving shapes of the interfaces by global or local interpolation solving integral equations of the second kind for the interfacial velocity or for the density of a hydrodynamic potential and computing the motion of the marker points while simultaneously updating interfacial fields relevant to the dynamics including the concentration of a surfactant and the position of interfacial point particles at an equilibrium configuration interfaces exhibiting isotropic tension elastic tensions viscous and incompressible behavior are considered the mathematical modeling of the tensions and bending moments developing over interfaces with a membrane like constitution is discussed in the context of the theory of thin shells to facilitate the numerical implementation the coupling of the interfacial mechanics to the hydrodynamics by means of interface force and torque balances is formulated in global cartesian coordinates recent progress in the implementation of boundary element methods is reviewed and areas for further research are identified

numerical simulation of 3d bubbles rising in viscous liquids using a front tracking method the rise of bubbles in viscous liquids is not only a very common process in many industrial applications but also an important fundamental problem in fluid physics an improved numerical algorithm based on the front tracking method originally proposed by tryggvason and his co workers has been validated against experiments over a wide range of intermediate reynolds and bond numbers using an axisymmetric model j hua j lou numerical simulation of bubble rising in viscous liquid j comput phys 22 2007 769 795 in the current paper this numerical algorithm is further extended to simulate 3d bubbles rising in viscous liquids with high reynolds and bond numbers and with large density and viscosity ratios representative of the common air water two phase flow system to facilitate the 3d front tracking simulation mesh adaptation is implemented for both the front mesh on the bubble surface and the background mesh on the latter mesh the governing navier stokes equations for incompressible newtonian flow are solved in a moving reference frame attached to the rising bubble specifically the equations are solved using a finite volume scheme based on the semi implicit method for pressure linked equations simple algorithm and it appears to be robust even for high reynolds numbers and high density and viscosity ratios the 3d bubble surface is tracked explicitly using an adaptive unstructured triangular mesh the numerical model is integrated with the software package paramesh a block based adaptive mesh refinement amr tool developed for parallel computing paramesh allows background mesh adaptation as well as the solution of the governing equations in parallel on a supercomputer further peskin distribution function is applied to interpolate the variable values between the front and the background meshes detailed sensitivity analysis about the numerical modeling algorithm has been performed the current model has also been applied to simulate a number of cases of 3d gas bubbles rising in viscous liquids e g air bubbles rising in water simulation results are compared with experimental observations both in aspect of terminal bubble shapes and terminal bubble velocities in addition we applied this model to simulate the interaction between two bubbles rising in a liquid which illustrated the model s capability in predicting the interaction dynamics of rising bubbles