fully implicit methodology for the dynamics of biomembranes and capillary interfaces by combining the level set and newton methods abstract this framework is concerned with the numerical modeling of the dynamics of individual biomembranes and capillary interfaces in a surrounding newtonian fluid a level set approach helps to follow the interface motion our method features the use of high order fully implicit time integration schemes that enable to overcome stability issues related to the explicit discretization of the highly non linear bending force or capillary force at each time step the tangent systems are derived and the resulting nonlinear problems are solved by a newton raphson method based on the signed distance assumption several inexact newton strategies are employed to solve the capillary and vesicle problems and guarantee the second order convergence behavior we address in detail the main features of the proposed method and we report several experiments in the two dimensional case with the aim of illustrating its accuracy and efficiency comparative investigations with respect to the fully explicit scheme depict the stabilizing effect of the new method which allows to use significantly larger time step sizes

computing the dynamics of biomembranes by combining conservative level set and adaptive finite element methods the numerical simulation of the deformation of vesicle membranes under simple shear external fluid flow is considered in this paper a saddle point approach is proposed for the imposition of the fluid incompressibility and the membrane inextensibility constraints through lagrange multipliers defined in the fluid and on the membrane respectively using a level set formulation the problem is approximated by mixed finite elements combined with an automatic adaptive mesh procedure at the vicinity of the membrane boundary numerical experiments show that this combination of the saddle point and adaptive mesh method enhances the robustness of the method the effect of inertia on the stability of the vesicle in a shear flow is also investigated

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

coupling an eulerian fluid calculation to a lagrangian solid calculation with the ghost fluid method we propose a numerical method for modeling multimaterial flows where the domain is decomposed into separate eulerian and lagrangian subdomains that is the equations are written in eulerian form in one subdomain and in lagrangian form in the other subdomain this is of interest for example when considering the effect of underwater explosions on the hull of a ship or the impact of a low speed projectile on a soft explosive target on the one hand high speed fluid flows are traditionally modeled by applying shock capturing schemes to the compressible euler equations to avoid problems associated with tangling of a lagrangian mesh on the other hand solid dynamics calculations are traditionally carried out using lagrangian numerical methods to avoid problems associated with numerical smearing in eulerian calculations we use the ghost fluid method to create accurate discretizations across the eulerian lagrangian interface the numerical method is presented in both one and two spatial dimensions three dimensional extensions to the interface coupling method are straightforward

3d tetrahedral unstructured and anisotropic mesh generation with adaptation to natural and multidomain metric in this paper we present a 3d tetrahedral unstructured and anisotropic mesh generator that is not based on the delaunay frontal or octree method instead it proceeds by local optimizations and uses an anisotropic shape criterion to fit a metric field then we introduce a new 3d metric field that tightens the mesh around interfaces when the calculation domain is divided in several subdomains and a 3d metric field that places enough elements through each subdomain thickness without introducing too many nodes in the other directions finally we show some applications for material forming geometries

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

a sharp interface immersed boundary method for compressible viscous flows an immersed boundary method for computing viscous subsonic compressible flows with complex shaped stationary immersed boundaries is presented the method employs a ghost cell technique for imposing the boundary conditions on the immersed boundaries the current approach leads to a sharp representation of the immersed boundaries a property that is especially useful for flow simulations at high reynolds numbers another unique feature of the method is that it can be applied on cartesian as well as generalized body non conformal curvilinear meshes a mixed second order central difference quick scheme is used which allows a high degree of control over the numerical damping a bilinear interpolation scheme used in conjunction with the ghost cell approach results in second order global as well as local spatial accuracy the solver is parallelized for distributed memory platforms using domain decomposition and message passing interface mpi and salient features of the parallel algorithm are presented the accuracy fidelity and efficiency of the solver are examined by simulating flow past circular cylinders and airfoils and comparing against experimental data and other established results finally we present results from a simulation of wing tip flow at a relatively high reynolds number in order to demonstrate the ability of the solver to model complex non canonical three dimensional flows

the point set method front tracking without connectivity abstract we describe a point set method for extracting the normal curvature and surface area from unordered data points residing on a surface this capability relaxes front tracking s reliance on connectivity between interfacial points and allows front tracking to model topological changes at an interface naturally we use this capability in 2d and 3d front tracking calculations to model coalescence we also describe a simple projection scheme which allows us to suppress parasitic currents in front tracking models

a penalization method to take into account obstacles in incompressible viscous flows from the navier stokes brinkman model a penalization method has been derived by several authors to compute incompressible navier stokes equations around obstacles in this paper convergence theorems and error estimates are derived for two kinds of penalization the first one corresponds to l 2 penalization inducing a darcy equation in the solid body the second one corresponds to a h 1 penalization and induces a brinkman equation in the body numerical tests are performed to confirm the efficiency and accuracy of the method

a second order accurate projection method for the incompressible navier stokes equations on non graded adaptive grids we present an unconditionally stable second order accurate projection method for the incompressible navier stokes equations on non graded adaptive cartesian grids we employ quadtree and octree data structures as an efficient means to represent the grid we use the supra convergent poisson solver of c h min f gibou h ceniceros a supra convergent finite difference scheme for the variable coefficient poisson equation on fully adaptive grids cam report 05 29 j comput phys in press a second order accurate semi lagrangian method to update the momentum equation an unconditionally stable backward difference scheme to treat the diffusion term and a new method that guarantees the stability of the projection step on highly non graded grids we sample all the variables at the grid nodes producing a scheme that is straightforward to implement we propose two and three dimensional examples to demonstrate second order accuracy for the velocity field and the divergence free condition in the l1 and l norms

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

some improvements of the fast marching method the fast marching method published by sethian proc natl acad sci usa 93 1996 pp 1591 1595 is an optimally efficient algorithm for solving problems of front evolution where the front speed is monotonic it has been used in a wide variety of applications such as robotic path planning r kimmel and j sethian fast marching methods for computing distance maps and shortest paths tech report 669 cpam university of california berkeley 1996 crack propagation m stolarska et al modelling crack growth by level sets in the extended finite element method comput methods appl mech engrg to appear internat j numer methods engrg 51 2001 pp 943 960 n sukumar d l chopp and b moran extended finite element method and fast marching method for three dimensional fatigue crack propagation j comput phys submitted seismology j sethian and a popovici geophysics 64 1999 pp 516 523 photolithography j sethian fast marching level set methods for three dimensional photolithography development in proceedings of the spie 1996 international symposium on microlithography santa clara ca 1996 and medical imaging r malladi and j sethian proc natl acad sci usa 93 1996 pp 9389 9392 it has also been a valuable tool for the implementation of modern level set methods where it is used to efficiently compute the distance to the front and or an extended velocity function in this paper we improve upon the second order fast marching method of sethian siam rev 41 1999 pp 199 235 by constructing a second order approximation of the interface generated from local data on the mesh the data is interpolated on a single box of the mesh using a bicubic approximation the distance to the front is then calculated by using a variant of newton s method to solve both the level curve equation and the orthogonality condition for the nearest point to a given node the result is a second order approximation of the distance to the interface which can then be used to produce second order accurate initial conditions for the fast marching method and a third order fast marching method

another look at velocity extensions in the level set method the level set method s osher and j a sethian j comput phys 79 1988 pp 12 49 has become a widely used numerical method for moving interfaces e g see the many examples in j sethian level set methods evolving interfaces in geometry fluid mechanics computer vision and material science cambridge university press london 1996 s osher and r fedkiw level set methods and dynamic implicit surfaces springer verlag heidelberg 2002 for many applications the velocity of the interface is known only on the interface while the level set method requires information about the interface speed at least in a neighborhood of grid points near the interface to address this issue velocity extensions are used to map the velocity information on the interface into the rest of the computational domain d adalsteinsson and j sethian j comput phys 118 1995 pp 269 277 this allows the level set method to proceed the velocity extension method presented in d adalsteinsson and j sethian j comput phys 118 1995 pp 269 277 uses the fast marching method j sethian proc natl acad sci 93 1996 pp 1591 1595 j sethian siam rev 41 1999 pp 199 235 and is only a first order approximation for the velocity field near the interface furthermore it can lead to unexpected behavior in some cases this is primarily due to the strictly local solution for the characteristics of the flow near the interface in this paper we look more closely at the characteristics near the interface and then present a modified velocity extension method also based on the fast marching method which handles the characteristcs more accurately in turn this new method will lead to some interesting new possibilities for the fast marching method

an accurate adaptive solver for surface tension driven interfacial flows a method combining an adaptive quad octree spatial discretisation geometrical volume of fluid interface representation balanced force continuum surface force surface tension formulation and height function curvature estimation is presented the extension of these methods to the quad octree discretisation allows adaptive variable resolution along the interface and is described in detail the method is shown to recover exact equilibrium to machine accuracy between surface tension and pressure gradient in the case of a stationary droplet irrespective of viscosity and spatial resolution accurate solutions are obtained for the classical test case of capillary wave oscillations an application to the capillary breakup of a jet of water in air further illustrates the accuracy and efficiency of the method the source code of the implementation is freely available as part of the gerris flow solver

differential equation based constrained reinitialization for level set methods a partial differential equation based reinitialization method is presented in the framework of a localized level set method two formulations of the new reinitialization scheme are derived these formulations are modifications of the partial differential equation introduced by sussman et al m sussman p smereka s osher a level set approach for computing solutions to incompressible two phase flow j comput phys 114 1994 146 159 and in particular improvements of the second order accurate modification proposed by russo and smereka g russo p smereka a remark on computing distance functions j comput phys 163 2000 51 67 the first formulation uses the least squares method to explicitly minimize the displacement of the zero level set within the reinitialization the overdetermined problem which is solved in the first formulation of the new reinitialization scheme is reduced to a determined problem in another formulation such that the location of the interface is locally preserved within the reinitialization the second formulation is derived by systematically minimizing the number of constraints imposed on the reinitialization scheme for both systems the resulting algorithms are formulated in a three dimensional frame of reference and are remarkably simple and efficient the new formulations are second order accurate at the interface when the reinitialization equation is solved with a first order upwind scheme and do not diminish the accuracy of high order discretizations of the level set equation the computational work required for all components of the localized level set method scales with o n detailed analyses of numerical solutions obtained with different discretization schemes evidence the enhanced accuracy and the stability of the proposed method which can be used for localized and global level set methods

derivation and validation of a novel implicit second order accurate immersed boundary method a novel implicit second order accurate immersed boundary method ibm for simulating the flow around arbitrary stationary bodies is developed implemented and validated in this paper the ibm is used to efficiently take into account the existence of bodies within the fluid domain the flow domain consist of simple cartesian cells whereas the body can be arbitrary at the triangulated interface of the body and the fluid the immersed boundary the coefficients obtained from discretizing the navier stokes equations are closed with a second order accurate interpolation arising from the immersed boundary condition employed at the interface two different conditions are developed in this paper and it is shown that for the mirroring method the resulting coefficients lead to a well posed and diagonally dominant system which can be efficiently solved with a preconditioned krylov sub space solver the immersed boundary condition generates a fictitious reversed velocity field inside the immersed boundary which is excluded from the continuity equation to account for the presence of the ib in the pressure correction equation resulting in no mass flux over the ib the force acting on the object from the fluid is determined by integrating the pressure and the viscous forces over the object the method is validated by simulating the flow around a sphere for a range of re numbers it is shown that the drag is very well in agreement with experimental data accuracy and convergence studies are employed proving the second order accuracy of the method and showing the superiority in convergence rate compared to other ibm finally the drag force of a cluster of non spherical particles is employed to show the generality and potential of the method

motion of multiple junctions a level set approach abstract a coupled level set method for the motion of multiple junctions is proposed the new method extends the hamilton jacobi level set formulation of osher and sethian it retains the feature of tracking fronts by following level sets and allows the specification of arbitrary velocities on each front the diffusion equation is shown to generate curvature dependent motion and this is used to develop an algorithm to move multiple junctions with curvature dependent speed systems of reaction diffusion equations are shown to possess inherent properties which prohibit efficient numerical solutions when applied to curvature dependent motion

an immersed boundary method for simulating vesicle dynamics in three dimensions we extend our previous immersed boundary ib method for 3d axisymmetric inextensible vesicle in navier stokes flows hu et al 2014 17 to general three dimensions despite a similar spirit in numerical algorithms to the axisymmetric case the fully 3d numerical implementation is much more complicated and is far from straightforward a vesicle membrane surface is known to be incompressible and exhibits bending resistance as in 3d axisymmetric case instead of keeping the vesicle locally incompressible we adopt a modified elastic tension energy to make the vesicle surface patch nearly incompressible so that solving the unknown tension lagrange multiplier for the incompressible constraint can be avoided nevertheless the new elastic force derived from the modified tension energy has exactly the same mathematical form as the original one except the different definitions of tension the vesicle surface is discretized on a triangular mesh where the elastic tension and bending force are calculated on each vertex lagrangian marker in the ib method of the triangulation a series of numerical tests on the present scheme are conducted to illustrate the robustness and applicability of the method we perform the convergence study for the immersed boundary forces and the fluid velocity field we then study the vesicle dynamics in various flows such as quiescent simple shear and gravitational flows our numerical results show good agreements with those obtained in previous theoretical experimental and numerical studies

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

high order ale schemes for incompressible capillary flows the spatial discretization of problems with moving boundaries is considered incorporating the temporal evolution of not just the mechanical variables but also of the nodal positions of the moving mesh the outcome is a system of differential algebraic equations dae of index 2 or in the case of inertialess flow just 1 from the dae formulation it its possible to define strategies to build schemes of arbitrary accuracy we introduce here several schemes of order 2 and 3 that avoid the solution of a nonlinear system involving simultaneously the mechanical variables and the geometrical ones this class of schemes has been the one adopted by the majority of practitioners of computational fluid dynamics up to now the proposed schemes indeed achieve the design accuracy and further show stability restrictions that are not significantly more severe than those of popular first order schemes the numerical experimentation is performed on capillary problems discretized by both div stable p 2 p 1 p 1 p 1 and equal order p 1 p 1 stabilized finite elements and incorporating surface tension and triple contact line effects

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

adaptive finite element method for a phase field bending elasticity model of vesicle membrane deformations in this paper a three dimensional adaptive finite element method is developed for a variational phase field bending elasticity model of vesicle membrane deformations using a mixed finite element formulation residual type a posteriori error estimates are derived for the associated nonlinear system of equations and they are used to introduce the mesh refinement and coarsening the resulting mesh adaptivity significantly improves the efficiency of the phase field simulation of vesicle membranes and enhances its capability in handling complex shape and topological changes the effectiveness of the adaptive method is further demonstrated through numerical examples

a vortex level set method for the two way coupling of an incompressible fluid with colliding rigid bodies we present a vortex method for the simulation of the interaction of an incompressible flow with rigid bodies the method is based on a penalization technique where the system is considered as a single flow subject to the navier stokes equation with a penalization term that enforces continuity at the solid fluid interface and rigid motion inside the solid level set functions are used to capture interfaces compute rigid motions inside the solid bodies and model collisions between bodies a vortex in cell algorithm is built on this method numerical comparisons with existing 3d methods on problems of sedimentation and collision of spheres are provided to illustrate the capabilities of the method

an immersed interface method for solving incompressible viscous flows with piecewise constant viscosity across a moving elastic membrane this paper presents an implementation of the second order accurate immersed interface method to simulate the motion of the flexible elastic membrane immersed in two viscous incompressible fluids with different viscosities which further develops the work reported in tan et al z j tan d v le k m lim b c khoo an immersed interface method for the incompressible navier stokes equations with discontinuous viscosity across the interface submitted for publication focussing mainly on the fixed interface problems in this work we introduce the velocity components at the membrane as two augmented unknown interface variables to decouple the originally coupled jump conditions for the velocity and pressure three forms of augmented equation are derived to determine the augmented variables to satisfy the continuous condition of the velocity the velocity at the membrane which determine the motion of the membrane is then solved by the gmres iterative method the forces calculated from the configuration of the flexible elastic membrane and the augmented variables are interpolated using cubic splines and applied to the fluid through the jump conditions the position of the flexible elastic membrane is updated implicitly using a quasi newton method bfgs within each time step the navier stokes equations are solved on a staggered cartesian grid using a second order accurate projection method with the incorporation of spatial and temporal jump conditions in addition we also show that the inclusion of the temporal jump contributions has non negligible effect on the simulation results when the grids are crossed by the membrane using the above method we assess the effect of different viscosities on the flow solution and membrane motion

an immersed boundary method for simulating the dynamics of three dimensional axisymmetric vesicles in navier stokes flows in this paper we develop a simple immersed boundary method to simulate the dynamics of three dimensional axisymmetric inextensible vesicles in navier stokes flows instead of introducing a lagrange s multiplier to enforce the vesicle inextensibility constraint we modify the model by adopting a spring like tension to make the vesicle boundary nearly inextensible so that solving for the unknown tension can be avoided we also derive a new elastic force from the modified vesicle energy and obtain exactly the same form as the originally unmodified one in order to represent the vesicle boundary we use fourier spectral approximation so we can compute the geometrical quantities on the interface more accurately a series of numerical tests on the present scheme have been conducted to illustrate the applicability and reliability of the method we first perform the accuracy check of the geometrical quantities of the interface and the convergence check for different stiffness numbers as well as fluid variables then we study the vesicle dynamics in quiescent flow and in gravity finally the shapes of vesicles in poiseuille flow are investigated in detail to study the effects of the reduced volume the confinement and the mean flow velocity the numerical results are shown to be in good agreement with those obtained in literature

an adaptive level set approach for incompressible two phase flows we present a numerical method using the level set approach for solving incompressible two phase flow with surface tension in the level set approach the free surface is represented as the zero level set of a smooth function this has the effect of replacing the advection of density which has steep gradients at the free surface with the advection of the level set function which is smooth in addition the free surface can merge or break up with no special treatment we maintain the level set function as the signed distance from the free surface in order to accurately compute flows with high density ratios and stiff surface tension effects in this work we couple the level set scheme to an adaptive projection method for the incompressible navier stokes equations in order to achieve higher resolution of the free surface with a minimum of additional expense we present two dimensional axisymmetric and fully three dimensional results of air bubble and water drop computations

a phase field model for the mixture of two incompressible fluids and its approximation by a fourier spectral method abstract a phase field model for the mixture of two incompressible fluids is presented in this paper the model is based on an energetic variational formulation it consists of a navier stokes system linear momentum equation coupled with a cahn hilliard equation phase field equation through an extra stress term and the transport term the extra stress represents the phase induced capillary effect for the mixture due to the surface tension a fourier spectral method for the numerical approximation of this system is proposed and analyzed numerical results illustrating the robustness and versatility of the model are presented

an immersed boundary method using unstructured anisotropic mesh adaptation combined with level sets and penalization techniques the interest on embedded boundary methods increases in computational fluid dynamics cfd because they simplify the mesh generation problem in the case of the navier stokes equations the same simplifications occur for the simulation of multi physics flows the coupling of fluid solid interactions in situation of large motions or deformations to give a few examples nevertheless an accurate treatment of the wall boundary conditions remains an issue of the method in this work the wall boundary conditions are easily taken into account through a penalization technique and the accuracy of the method is recovered using mesh adaptation thanks to the potential of unstructured meshes several classical examples are used to demonstrate that claim

on two presentations of the affine weyl groups of classical types abstract the main result of the paper is to get the transition formulae between the alcove form and the permutation form of w wa where wa is an affine weyl group of classical type on the other hand we get a new characterization for the alcove form of an affine weyl group element which has a much simpler form compared with that in j y shi j london math soc 35 1987 42 55 as applications we give an affirmative answer to a conjecture of h eriksson and k eriksson preprint 1998 concerning the characterization of the inverse table of w wa we also describe the number s w in terms of the permutation form of w wa

a semi lagrangian high order method for navier stokes equations we present a semi lagrangian method for advection diffusion and incompressible navier stokes equations the focus is on constructing stable schemes of second order temporal accuracy as this is a crucial element for the successful application of semi lagrangian methods to turbulence simulations we implement the method in the context of unstructured spectral hp element discretization which allows for efficient search interpolation procedures as well as for illumination of the nonmonotonic behavior of the temporal advection error of the form o tk xp 1 t we present numerical results that validate this error estimate for the advection diffusion equation and we document that such estimate is also valid for the navier stokes equations at moderate or high reynolds number two and three dimensional laminar and transitional flow simulations suggest that semi lagrangian schemes are more efficient than their eulerian counterparts for high order discretizations on nonuniform grids

simulating the deformation of vesicle membranes under elastic bending energy in three dimensions in this paper we study the three dimensional deformation of a vesicle membrane under the elastic bending energy with prescribed bulk volume and surface area both static and dynamic deformations are considered a newly developed energetic variational formulation is employed to give an effective eulerian description efficient time and spatial discretizations are considered and implemented numerical experiments illustrate some fascinating phenomena that are of interests in real applications

the constrained reinitialization equation for level set methods based on the constrained reinitialization scheme d hartmann m meinke w schroder differential equation based constrained reinitialization for level set methods j comput phys 227 2008 6821 6845 a new constrained reinitialization equation incorporating a forcing term is introduced two formulations for high order constrained reinitialization hcr are presented combining the simplicity and generality of the original reinitialization equation m sussman p smereka s osher a level set approach for computing solutions to incompressible two phase flow j comput phys 114 1994 146 159 in terms of high order standard discretization and the accuracy of the constrained reinitialization scheme in terms of interface displacement the novel hcr schemes represent simple extensions of standard implementations of the original reinitialization equation the results evidence the significantly increased accuracy and robustness of the novel schemes

an adaptive version of the immersed boundary method a computational setting for the immersed boundary method employing an adaptive mesh refinement is presented enhanced accuracy for the method is attained locally by covering an immersed boundary vicinity with a sequence of nested progressively finer rectangular grid patches which dynamically follow the immersed boundary motion the set of equations describing the interaction between a non stationary viscous incompressible fluid and an immersed elastic boundary is solved by coupling a projection method specially designed for locally refined meshes to an implicit formulation of the immersed boundary method the main contributions of this work concern the formulation and the implementation of a multilevel self adaptive version of the immersed boundary method on locally refined meshes this approach is tested for a particular two dimensional model problem for which no significant difference is found between the solutions obtained on a mesh refined locally around the immersed boundary and on the associated uniform mesh built with the resolution of the finest level

metric tensors for anisotropic mesh generation it has been amply demonstrated that significant improvements in accuracy and efficiency can be gained when a properly chosen anisotropic mesh is used in the numerical solution for a large class of problems which exhibit anisotropic solution features in practice an anisotropic mesh is commonly generated as a quasi uniform mesh in the metric determined by a tensor specifying the shape size orientation of elements thus it is crucial to choose an appropriate metric tensor for anisotropic mesh generation and adaptation in this paper we develop a general formula for the metric tensor for use in any spatial dimension the formulation is based on error estimates for polynomial preserving interpolation on simiplicial elements numerical results in two dimensions are presented to demonstrate the ability of the metric tensor to produce anisotropic meshes with correct mesh concentration and good overall quality the procedure developed in this paper for defining the metric tensor can also be applied to other types of error estimates

an adaptive residual based splitting approach for the penalized navier stokes equations the interest on immersed boundary methods ibm is increasing in computational fluid dynamics as they simplify the mesh generation problem in this work we consider an approach based on the addition of a penalty term to the navier stokes equations to account for the wall boundary conditions to discretize the resulting equations we use a residual distribution approach previously developed by some of the authors to adapt the method to the ibm considered we developed a new formulation of residual distribution based on a strang splitting method in time coupling an implicit asymptotic integration procedure of the penalization ode with a simplified explicit residual distribution for the navier stokes equations the first method provides an operator which is exact up to orders 2 with the penalty parameter assuming values of the order of 10 10 a modification of the solution gradient reconstruction necessary for the evaluation of the viscous fluxes is also introduced in the paper this guarantees that correct physical values of the viscous stresses are recovered in vicinity of the solid we show formally and numerically that the approach proposed is second order accurate for smooth solutions we evaluate its potential for ibm by coupling the resulting method with unstructured mesh adaptation on wall boundaries several steady and time dependent tests are used to show the promising features of the method proposed

an immersed interface method for viscous incompressible flows involving rigid and flexible boundaries we present an immersed interface method for the incompressible navier stokes equations capable of handling both rigid and flexible boundaries the immersed boundaries are represented by a number of lagrangian control points in order to ensure that the no slip condition on the rigid boundary is satisfied singular forces are applied on the fluid the forces are related to the jumps in pressure and the jumps in the derivatives of both pressure and velocity and are interpolated using cubic splines the strength of the singular forces at the rigid boundary is determined by solving a small system of equations at each timestep for flexible boundaries the forces that the boundary exerts on the fluid are computed from the constitutive relation of the flexible boundary and are applied to the fluid through the jump conditions the position of the flexible boundary is updated implicitly using a quasi newton method bfgs within each timestep the navier stokes equations are discretized on a staggered cartesian grid by a second order accurate projection method for pressure and velocity and the overall scheme is second order accurate

a gradient augmented level set method with an optimally local coherent advection scheme the level set approach represents surfaces implicitly and advects them by evolving a level set function which is numerically defined on an eulerian grid here we present an approach that augments the level set function values by gradient information and evolves both quantities in a fully coupled fashion this maintains the coherence between function values and derivatives while exploiting the extra information carried by the derivatives the method is of comparable quality to weno schemes but with optimally local stencils performing updates in time by using information from only a single adjacent grid cell in addition structures smaller than the grid size can be located and tracked and the extra derivative information can be employed to obtain simple and accurate approximations to the curvature we analyze the accuracy and the stability of the new scheme and perform benchmark tests

fast marching methods fast marching methods are numerical schemes for computing solutions to the nonlinear eikonal equation and related static hamilton jacobi equations based on entropy satisfying upwind schemes and fast sorting techniques they yield consistent accurate and highly efficient algorithms they are optimal in the sense that the computational complexity of the algorithms is o n log n where n is the total number of points in the domain the schemes are of use in a variety of applications including problems in shape offsetting computing distances from complex curves and surfaces shape from shading photolithographic development computing first arrivals in seismic travel times construction of shortest geodesics on surfaces optimal path planning around obstacles and visibility and reflection calculations in this paper we review the development of these techniques including the theoretical and numerical underpinnings provide details of the computational schemes including higher order versions and demonstrate the techniques in a collection of different areas

computational parametric willmore flow we propose a new algorithm for the computation of willmore flow this is the l 2 gradient flow for the willmore functional which is the classical bending energy of a surface willmore flow is described by a highly nonlinear system of pdes of fourth order for the parametrization of the surface the spatially discrete numerical scheme is stable and consistent the discretization relies on an adequate calculation of the first variation of the willmore functional together with a derivation of the second variation of the area functional which is well adapted to discretization techniques with finite elements the algorithm uses finite elements on surfaces we give numerical examples and tests for piecewise linear finite elements a convergence proof for the full algorithm remains an open question

letter to the editor third order modification of newton s method in this paper we present a new modification of newton s method for solving non linear equations analysis of convergence shows that the new method is cubically convergent numerical examples show that the new method can compete with the classical newton s method

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

a fictitious domain approach for a mixed finite element method solving the two phase stokes problem with surface tension forces abstract in this article we study a mixed finite element formulation for solving the stokes problem with general surface forces that induce a jump of the normal trace of the stress tensor on an interface that splits the domain into two subdomains equality of velocities is assumed at the interface the interface conditions are taken into account with multipliers a suitable lagrangian functional is introduced for deriving a weak formulation of the problem a specificity of this work is the consideration of the interface with a fictitious domain approach the latter is inspired by the xfem approach in the sense that cut off functions are used but it is simpler to implement since no enrichment by singular functions is provided in that context getting convergence for the dual variables defined on the interface is non trivial for that purpose an augmented lagrangian technique stabilizes the convergence of the multipliers which is important because their value would determine the dynamics of the interface in an unsteady framework theoretical analysis is provided where we show that a discrete inf sup condition independent of the mesh size is satisfied for the stabilized formulation this guarantees optimal convergence rates that we observe with numerical tests the capacity of the method is demonstrated with robustness tests and with an unsteady model tested for deformations of the interface that correspond to ellipsoidal shapes in dimension 2

the implicit closest point method for the numerical solution of partial differential equations on surfaces many applications in the natural and applied sciences require the solutions of partial differential equations pdes on surfaces or more general manifolds the closest point method is a simple and accurate embedding method for numerically approximating pdes on rather general smooth surfaces however the original formulation is designed to use explicit time stepping this may lead to a strict time step restriction for some important pdes such as those involving the laplace beltrami operator or higher order derivative operators to achieve improved stability and efficiency we introduce a new implicit closest point method for surface pdes the method allows for large stable time steps while retaining the principal benefits of the original method in particular it maintains the order of accuracy of the discretization of the underlying embedding pde it works on sharply defined bands without degrading the accuracy of the method and it applies to general smooth surfaces it also is very simple and may be applied to a rather general class of surface pdes convergence studies for the in surface heat equation and a fourth order biharmonic problem are given to illustrate the accuracy of the method we demonstrate the flexibility and generality of the method by treating flows involving diffusion reaction diffusion and fourth order spatial derivatives on a variety of interesting surfaces including surfaces of mixed codimension

bond percolation in frustrated systems conwe study occurrence and properties of percolation of occupied bonds in systems with random interactions and hence frustration we develop a general argument somewhat like peierls argument by which we show that in z d d 2 percolation occurs for all possible interactions provided they are bounded away from zero if the parameter p e 0 1 regulating the density of occupied bonds is high enough if the interactions are i i d random variables then we determine bounds on the values of p for which percolation occurs for all almost all but not all almost none but some or none of the interactions motivations of this work come from the rigorous analysis of phase transitions in frustrated statistical mechanics systems

level set jet schemes for stiff advection equations the semijet method abstract many interfacial phenomena in physical and biological systems are dominated by high order geometric quantities such as curvature here a semi implicit method is combined with a level set jet scheme to handle stiff nonlinear advection problems the new method offers an improvement over the semi implicit gradient augmented level set method previously introduced by requiring only one smoothing step when updating the level set jet function while still preserving the underlying methods higher accuracy sample results demonstrate that accuracy is not sacrificed while strict time step restrictions can be avoided

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

godunov type method on non structured meshes for three dimensional moving boundary problems abstract this paper presents a numerical method for the computation of compressible flows in domains whose boundaries move in a well defined predictable manner the method uses the space time formulation by godunov while the discretization is conducted on non structured tetrahedral meshes using roe s approximate riemann solver an implicit time stepping and a muscl type interpolation the computation of the geometrical parameters required to take into account the movement of the boundaries is described examples including the calculation of the flow in the cylinder of an internal combustion engine illustrates the possibilities of the method

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 hybrid cartesian immersed boundary method for simulating flows with 3d geometrically complex moving bodies a numerical method is developed for solving the 3d unsteady incompressible navier stokes equations in cartesian domains containing immersed boundaries of arbitrary geometrical complexity moving with prescribed kinematics the governing equations are discretized on a hybrid staggered non staggered grid layout using second order accurate finite difference formulas the discrete equations are integrated in time via a second order accurate dual time stepping artificial compressibility iteration scheme unstructured triangular meshes are employed to discretize complex immersed boundaries the nodes of the surface mesh constitute a set of lagrangian control points used to track the motion of the flexible body at every instant in time the influence of the body on the flow is accounted for by applying boundary conditions at cartesian grid nodes located in the exterior but in the immediate vicinity of the body by reconstructing the solution along the local normal to the body surface grid convergence tests are carried out for the flow induced by an oscillating sphere in a cubic cavity which show that the method is second order accurate the method is validated by applying it to calculate flow in a cartesian domain containing a rigid sphere rotating at constant angular velocity as well as flow induced by a flapping wing the ability of the method to simulate flows in domains with arbitrarily complex moving bodies is demonstrated by applying to simulate flow past an undulating fish like body and flow past an anatomically realistic planktonic copepod performing an escape like maneuver euver

a remark on gauge symmetries in ising spin glasses this paper studies a particular line in the parameter space of the fk random interaction random cluster model for spin glasses following katsura k and mazza m we show that after averaging over the random couplings the occupied fk bonds have exactly a bernoulli distribution comparison with explicit calculations on trees confirms the marginal role of fk percolation in determining phase transitions

a continuum method for modeling surface tension abstract a new method for modeling surface tension effects on fluid motion has been developed interfaces between fluids of different properties or colors are represented as transition regions of finite thickness across which the color variable varies continuously at each point in the transition region a force density is defined which is proportional to the curvature of the surface of constant color at that point it is normalized so that the conventional description of surface tension on an interface is recovered when the ratio of local transition region thickness to local radius of curvature approaches zero the continuum method eliminates the need for interface reconstruction simplifies the calculation of surface tension enables accurate modeling of two and three dimensional fluid flows driven by surface forces and does not impose any modeling restrictions on the number complexity or dynamic evolution of fluid interfaces having surface tension computational results for two dimensional flows are given to illustrate the properties of the method

on the reinitialization procedure in a narrow band locally refined level set method for interfacial flows a local level set algorithm for simulating interfacial flows described by the two dimensional incompressible navier stokes equations is presented the governing equations are solved using a finite difference discretization on a cartesian grid and a second order approximate projection method the level set transport and reinitialization equations are solved in a narrow band around the interface using an adaptive refined grid which is reconstructed every time step and refined using a simple uniform cell splitting operation within the band instabilities at the border of the narrow band are avoided by smoothing the level set function in the outer part of the band the influence of different pde based reinitialization strategies on the accuracy of the results is investigated the ability of the proposed method to accurately compute interfacial flows is discussed using different tests namely the advection of a circle of fluid in two different time reversed vortex flows the advection of zalesak s rotating disk the propagation of small amplitude gravity and capillary waves at the interface between two superposed viscous fluids in deep water and a classical test of rayleigh taylor instability with and without surface tension effects the interface location error and area loss for some of the results obtained are compared with those of a recent particle level set method copyright 2005 john wiley sons ltd

a level set projection model of lipid vesicles in general flows a new numerical method to model the dynamic behavior of lipid vesicles under general flows is presented a gradient augmented level set method is used to model the membrane motion to enforce the volume and surface incompressibility constraints a four step projection method is developed to integrate the full navier stokes equations this scheme is implemented on an adaptive non graded cartesian grid convergence results are presented along with sample two dimensional results of vesicles under various flow conditions

electrohydrodynamics of three dimensional vesicles a numerical approach a three dimensional numerical model of vesicle electrohydrodynamics in the presence of dc electric fields is presented the vesicle membrane is modeled as a thin capacitive interface through the use of a semi implicit gradient augmented level set jet scheme the enclosed volume and surface area are conserved both locally and globally by a new navier stokes projection method the electric field calculations explicitly take into account the capacitive interface by an implicit immersed interface method formulation which calculates the electric potential field and the transmembrane potential simultaneously the results match well with previously published experimental analytic and two dimensional computational works

a stable numerical method for the dynamics of fluidic membranes we develop a finite element scheme to approximate the dynamics of two and three dimensional fluidic membranes in navier stokes flow local inextensibility of the membrane is ensured by solving a tangential navier stokes equation taking surface viscosity effects of boussinesq scriven type into account in our approach the bulk and surface degrees of freedom are discretized independently which leads to an unfitted finite element approximation of the underlying free boundary problem bending elastic forces resulting from an elastic membrane energy are discretized using an approximation introduced by dziuk numer math 111 55 80 2008 the obtained numerical scheme can be shown to be stable and to have good mesh properties finally the evolution of membrane shapes is studied numerically in different flow situations in two and three space dimensions the numerical results demonstrate the robustness of the method and it is observed that the conservation properties are fulfilled to a high precision

on optimal triangular meshes for minimizing the gradient error construction of optimal triangular meshes for controlling the errors in gradient estimation for piecewise linear interpolation of data functions in the plane is discussed using an appropriate linear coordinate transformation rigorously optimal meshes for controlling the error in quadratic data functions are constructed it is shown that the transformation can be generated as a curvilinear coordinate transformation for anyc data function with nonsingular hessian matrix using this transformation a construction of nearly optimal meshes for general data functions is described and the error equilibration properties of these meshes discussed in particular it is shown that equilibration of errors is not a sufficient condition for optimality a comparison of meshes generated under several different criteria is made and their equilibrating properties illustrated

a modification of newton method with third order convergence in this paper we present a new modification of newton method for solving non linear equations analysis of convergence shows that the new method is cubically convergent per iteration the new method requires two evaluations of the function and one evaluation of its first derivative thus the new method is preferable if the computational costs of the first derivative are equal or more than those of the function itself its practical utility is demonstrated by numerical examples

new anisotropic a priori error estimates we prove a priori anisotropic estimates for the l 2 and h 1 interpolation error on linear finite elements the full information about the mapping from a reference element is employed to separate the contribution to the elemental error coming from different directions this new h 1 error estimate does not require the maximal angle condition the analysis has been carried out for the 2d case but may be extended to three dimensions numerical experiments have been carried out to test our theoretical 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

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

simulation of two fluid flows using a finite element level set method application to bubbles and vesicle dynamics a new framework for two fluid flows using a finite element level set method is presented and verified through the simulation of the rising of a bubble in a viscous fluid this model is then enriched to deal with vesicles which mimic red blood cells mechanical behavior by introducing a lagrange multiplier to constrain the inextensibility of the membrane moreover high order polynomial approximation is used to increase the accuracy of the simulations a validation of this model is finally presented on known behaviors of vesicles under flow such as tank treading and tumbling motions