a g space theory and a weakened weak w2 form for a unified formulation of compatible and incompatible methods part ii applications to solid mechanics problems in part i of this paper we have established the g space theory and fundamentals for w 2 formulation part ii focuses on the applications of the g space theory to formulate w 2 models for solid mechanics problems we first define a bilinear form prove some of the important properties and prove that the w 2 formulation will be spatially stable and convergent to exact solutions we then present examples of some of the possible w 2 models including the sfem ns fem es fem ns pim es pim and cs pim we show the major properties of these models 1 they are variationally consistent in a conventional sense if the solution is sought in a proper h space compatible cases 2 they pass the standard patch test when the solution is sought in a proper g space with discontinuous functions incompatible cases 3 the stiffness of the discretized model is reduced compared with the finite element method fem model and possibly to the exact model allowing us to obtain upper bound solutions with respect to both the fem and the exact solutions and 4 the w 2 models are less sensitive to the quality of the mesh and triangular meshes can be used without any accuracy problems these properties and theories have been confirmed numerically via examples solved using a number of w 2 models including compatible and incompatible cases we shall see that the g space theory and the w 2 forms can formulate a variety of stable and convergent numerical methods with the fem as one special case

some remarks on the flux free finite element method for immiscible two fluid flows we give some theoretical considerations on the the flux free finite element method for the generalized stokes interface problem arising from the immiscible two fluid flow problems in the flux free finite element method the flux constraint is posed as another lagrange multiplier to keep the zero flux on the interface as a result the mass of each fluid is expected to be preserved at every time step we first study the effect of discontinuous coefficients viscosity and density on the error of the standard finite element approximations very carefully then the analysis is extended to the flux free finite element method

a gradient augmented level set method for unstructured grids we present and test a gradient augmented level set method for unstructured grids which has been designed to allow for easy integration in parallel flow solvers the means to this end is a hermite formulation where the gradient is stored and advected as an independent variable along with the level set function the method uses narrow band storage in conjunction with a fast marching method which is also designed to take advantage of the gradient augmentation we demonstrate through standard passive advection test cases that the proposed method is able to compute interface motion with performance comparable to popular high performing methods on structured grids

a hybrid ψ v hoc approach for surface tension driven flows in level set framework abstract in this study a higher order compact hoc finite difference scheme is proposed for the simulation of multiphase flows with surface tension effects by coupling the pure streamfunction formulation of navier stokes n s equations with the level set method unlike most of the existing methods for multiphase flows that solve the incompressible navier stokes equations in the velocity pressure form or the stream function vorticity form we recast the governing equations into the biharmonic form of the n s equations which carries stream function as the only variable to be computed the coupling of this formulation with level set ls enables the simulation of two phase flows by the core computation of only the single variable free from discontinuities in the whole physical domain in the process we also propose a novel hybrid hoc algorithm for post processing the pressure on either side of the interface including the jump across it the robustness of this method is established through a series of benchmark problems such as dispersion of capillary waves static bubble and oscillating bubble at equilibrium in a fluid at rest and deforming circular drops in shear flows we compare our numerical results with available benchmark results and excellent match is obtained in all the cases along with accurate representation of overall trends our approach is proved to be very efficient in terms of its easy implementation relatively low computational time and high order of accuracy

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

an eulerian description of fluids containing visco elastic particles equations governing the flow of fluid containing visco hyperelastic particles are developed in an eulerian framework the novel feature introduced here is to write an evolution equation for the strain it is envisioned that this will simplify numerical codes which typically compute the strain on lagrangian meshes moving through eulerian meshes existence results for the flow of linear visco hyperelastic particles in a newtonian fluid are established using a galerkin scheme

a sharp interface method for incompressible two phase flows we present a sharp interface method for computing incompressible immiscible two phase flows it couples the level set and volume of fluid techniques and retains their advantages while overcoming their weaknesses it is stable and robust even for large density and viscosity ratios on the order of 1000 to 1 the numerical method is an extension of the second order method presented by sussman m sussman a second order coupled levelset and volume of fluid method for computing growth and collapse of vapor bubbles journal of computational physics 187 2003 110 136 in which the previous method treated the gas pressure as spatially constant and the present method treats the gas as a second incompressible fluid the new method yields solutions in the zero gas density limit which are comparable in accuracy to the method in which the gas pressure was treated as spatially constant this improvement in accuracy allows one to compute accurate solutions on relatively coarse grids thereby providing a speed up over continuum or ghost fluid methods

a lagrangian nodal integration method for free surface fluid flows abstract we present a lagrangian nodal integration method for the simulation of newtonian and non newtonian free surface fluid flows the proposed nodal lagrangian method uses a finite element mesh to discretize the computational domain and to define the linear shape functions for the unknown nodal variables as in the standard particle finite element method pfem in this approach however the integrals are performed over nodal patches and not over elements and strains stresses are defined at nodes and not at gauss points this allows to limit the drawbacks associated with the remeshing and leads to a more accurate stress computation than in the classical elemental pfem several numerical tests in 2d and in 3d are presented to validate the proposed nodal pfem in all cases the method has shown a very good agreement with analytical solutions and with experimental and numerical results from the literature a thorough comparison between nodal and elemental pfems is also presented focusing on crucial issues such as solution accuracy convergence mass conservation and sensitivity to mesh distortion

regular article a pde based fast local level set method we develop a fast method to localize the level set method of osher and sethian 1988 j comput phys 79 12 and address two important issues that are intrinsic to the level set method a how to extend a quantity that is given only on the interface to a neighborhood of the interface b how to reset the level set function to be a signed distance function to the interface efficiently without appreciably moving the interface this fast local level set method reduces the computational effort by one order of magnitude works in as much generality as the original one and is conceptually simple and easy to implement our approach differs from previous related works in that we extract all the information needed from the level set function or functions in multiphase flow and do not need to find explicitly the location of the interface in the space domain the complexity of our method to do tasks such as extension and distance reinitialization is o n where n is the number of points in space not o n log n as in works by sethian 1996 proc nat acad sci 93 1591 and helmsen and co workers 1996 spie microlithography ix p 253 this complexity estimation is also valid for quite general geometrically based front motion for our localized method

variational formulation for a general dynamic financial equilibrium problem balance law and liability formula abstract an evolutionary model for a multi sector multi instrument financial equilibrium problem with a general utility function different prices for assets and liabilities and including the expenses for the management of the financial institutions is presented in this general case we give the evolutionary financial equilibrium conditions and prove their equivalence with an evolutionary variational inequality from which existence results of the financial equilibrium follow moreover making use of new nonlinear analysis results we develop a lagrange theory which allows one to study the behaviour of the financial equilibrium by means of the lagrange multipliers as a product of this analysis we prove that for the financial model an equilibrium law together with a liability formula must be fulfilled from which the reorganization of the existing financial disequilibrium depends

phase field computations for ternary fluid flows we present a new phase field model for three component immiscible liquid flows with surface tension in the phase field approach the classical sharp interface between the two immiscible fluids is replaced by a transition region across which the properties of fluids change continuously the proposed method incorporates a chemical potential which can eliminate the unphysical phase field profile and a continuous surface tension force formulation from which we can calculate the pressure field directly from the governing equations the capabilities of the method are demonstrated with several examples we compute the ternary phase separation via spinodal decomposition equilibrium phase field profiles pressure field distribution and a three interface contact angle resulting from a spreading liquid lens on an interface the numerical results show excellent agreement with analytical solutions

efficient methods for implicit geometrical representation of complex material microstructures summary we present two methods for initializing distance functions on adaptively refined finite element meshes to represent complex material microstructures from segmented x ray tomographic data implicit microstructure representation combined with the extended fem allows modelers to represent complex material microstructures with consistent mesh quality and accuracy in the first method a level set evolution equation is formulated and solved by the galerkin method on an adaptively refined mesh we show that the convergence and stability of this method is optimal for the order of elements used in the second approach we initialize the distance field by the fast marching method on a uniform grid and then project the solution onto the finite element mesh by least squares we show that this latter approach is superior in speed and accuracy as an example problem both methods are demonstrated in the initialization of distance fields for two inclusion phases within a al 7075 alloy copyright 2014 john wiley sons ltd

a numerical method to study the dynamics of capillary fluid systems we propose a numerical approach to study both the nonlinear dynamics and linear stability of capillary fluid systems in the nonlinear analysis the time dependent fluid region is mapped onto a fixed numerical domain through a coordinate transformation the hydrodynamic equations are spatially discretized with the chebyshev spectral collocation technique while an implicit time advancement is performed using second order backward finite differences the resulting algebraic equations are solved with the iterative newton raphson technique the most novel aspect of the method is the fact that the elements of the jacobian of the discretized system of equations are symbolic functions calculated before running the simulation these functions are evaluated numerically in the newton raphson iterations to find the solution at each time step which reduces considerably the computing time besides this numerical procedure can be easily adapted to solve the eigenvalue problem which determines the linear global modes of the capillary system therefore both the nonlinear dynamics and the linear stability analysis can be conducted with essentially the same algorithm we validate this numerical approach by studying the dynamics of a liquid bridge close to its minimum volume stability limit the results are virtually the same as those obtained with other methods the proposed approach proves to be much more computationally efficient than those other methods finally we show the versatility of the method by calculating the linear global modes of a gravitational jet

analysis of a stokes interface problem we consider a stationary stokes problem with a piecewise constant viscosity coefficient for the variational formulation of this problem we prove a well posedness result in which the constants are uniform with respect to the jump in the viscosity coefficient we apply a standard discretization with a pair of lbb stable finite element spaces the main result of the paper is an infsup result for the discrete problem that is uniform with respect to the jump in the viscosity coefficient from this we derive a robust estimate for the discretization error we prove that the mass matrix with respect to some suitable scalar product yields a robust preconditioner for the schur complement results of numerical experiments are presented that illustrate this robustness property

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 semi implicit level set method for multiphase flows and fluid structure interaction problems in this paper we present a novel semi implicit time discretization of the level set method introduced in 8 for fluid structure interaction problems the idea stems from a linear stability analysis derived on a simplified one dimensional problem the semi implicit scheme relies on a simple filter operating as a pre processing on the level set function it applies to multiphase flows driven by surface tension as well as to fluid structure interaction problems the semi implicit scheme avoids the stability constraints that explicit scheme need to satisfy and reduces significantly the computational cost it is validated through comparisons with the original explicit scheme and refinement studies on two dimensional benchmarks

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 coupled level set and volume of fluid method for computing 3d and axisymmetric incompressible two phase flows we present a coupled level set volume of fluid clsvof method for computing 3d and axisymmetric incompressible two phase flows this method combines some of the advantages of the volume of fluid method with the level set method to obtain a method which is generally superior to either method alone we present direct comparisons between computations made with the clsvof method and computations made with the level set method the volume of fluid method and the boundary integral method we also compare our computations to the exact solution for an oscillating ellipse due to lamb and experimental results obtained for a rising gas bubble in liquid obtained by hnat and buckmaster our computational examples focus on flows in which surface tension forces and changes in topology are dominant features of the flow

a discontinuous galerkin based immersed boundary method a numerical method to approximate partial differential equations on meshes that do not conform to the domain boundaries is introduced the proposed method is conceptually simple and free of user defined parameters starting with a conforming finite element mesh the key ingredient is to switch those elements intersected by the dirichlet boundary to a discontinuous galerkin approximation and impose the dirichlet boundary conditions strongly by virtue of relaxing the continuity constraint at those elements boundary locking is avoided and optimal order convergence is achieved this is shown through numerical experiments in reaction diffusion problems copyright 2008 john wiley sons ltd

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100

a convergence rate theorem for finite difference approximations to delta functions we prove a rate of convergence theorem for approximations to certain integrals over codimension one manifolds in r n the type of manifold involved here is defined by the zero level set of a smooth mapping u r email protected r our approximations are based on the two finite difference methods for discretizing delta functions presented in 16 we included a convergence proof in that paper but only proved rates of convergence in some greatly simplified situations numerical experiments indicated that our two methods were at least first and second order accurate respectively in this note we prove those empirical convergence rates for the two algorithms under fairly general hypotheses

numerical simulation of 3d viscoelastic flows with free surfaces a numerical model is presented for the simulation of viscoelastic flows with complex free surfaces in three space dimensions the mathematical formulation of the model is similar to that of the volume of fluid vof method but the numerical procedures are different a splitting method is used for the time discretization the prediction step consists in solving three advection problems one for the volume fraction of liquid which allows the new liquid domain to be obtained one for the velocity field one for the extra stress the correction step corresponds to solving an oldroyd b fluid flow problem without advection in the new liquid domain two different grids are used for the space discretization the three advection problems are solved on a fixed structured grid made out of small cubic cells using a forward characteristics method the oldroyd b problem without advection is solved using continuous piecewise linear stabilized finite elements on a fixed unstructured mesh of tetrahedrons efficient post processing algorithms enhance the quality of the numerical solution a hierarchical data structure reduces the memory requirements convergence of the numerical method is checked for the pure extensional flow and the filling of a tube numerical results are presented for the stretching of a filament fingering instabilities are obtained when the aspect ratio is large also results pertaining to jet buckling are reported

an immersed boundary method with direct forcing for the simulation of particulate flows we present an improved method for computing incompressible viscous flow around suspended rigid particles using a fixed and uniform computational grid the main idea is to incorporate peskin s regularized delta function approach acta numerica 11 2002 1 into a direct formulation of the fluid solid interaction force in order to allow for a smooth transfer between eulerian and lagrangian representations while at the same time avoiding strong restrictions of the time step this technique was implemented in a finite difference and fractional step context a variety of two and three dimensional simulations are presented ranging from the flow around a single cylinder to the sedimentation of 1000 spherical particles the accuracy and efficiency of the current method are clearly demonstrated

a fixed mesh method for incompressible flow structure systems with finite solid deformations a fixed mesh algorithm is proposed for simulating flow structure interactions such as those occurring in biological systems in which both the fluid and solid are incompressible and the solid deformations are large several of the well known difficulties in simulating such flow structure interactions are avoided by formulating a single set of equations of motion on a fixed eulerian mesh the solid s deformation is tracked to compute elastic stresses by an overlapping lagrangian mesh in this way the flow structure interaction is formulated as a distributed body force and singular surface force acting on an otherwise purely fluid system these forces which depend on the solid elastic stress distribution are computed on the lagrangian mesh by a standard finite element method and then transferred to the fixed eulerian mesh where the joint momentum and continuity equations are solved by a finite difference method the constitutive model for the solid can be quite general for the force transfer standard immersed boundary and immersed interface methods can be used and are demonstrated we have also developed and demonstrated a new projection method that unifies the transfer of the surface and body forces in a way that exactly conserves momentum the interface is still effectively sharp for this approach the spatial convergence of the method is observed to be between first and second order as in most immersed boundary methods for membrane flows the algorithm is demonstrated by the simulations of an advected elastic disk a flexible leaflet in an oscillating flow and a model of a swimming jellyfish

explicit and semi implicit cbs procedures for incompressible viscous flows this paper presents a numerical comparison of two different numerical procedures based on the characteristic based split cbs algorithm for the solution of incompressible flows the first procedure is based on the original semi implicit version of the cbs introduced in the early days of the scheme development the second was proposed more recently in order to have a fully explicit version of the scheme and is based on the introduction of an artificial compressibility type of formulation the interest in fully explicit schemes has increased because of the easy and efficient parallelization the main differences between the two versions of the schemes are highlighted in this work and a numerical comparison is presented on the basis of several benchmark solutions both steady and unsteady flows are considered with the explicit version of the scheme a dual time stepping procedure is needed to recover unsteady solutions copyright 2006 john wiley sons ltd

evolution implementation and application of level set and fast marching methods for advancing fronts abstract a variety of numerical techniques are available for tracking moving interfaces in this review we concentrate on techniques that result from the link between the partial differential equations that describe moving interfaces and numerical schemes designed for approximating the solutions to hyperbolic conservation laws this link gives rise to computational techniques for tracking moving interfaces in two and three space dimensions under complex speed laws we discuss the evolution of these techniques the fundamental numerical approximations involved implementation details and applications in particular we review some work on three aspects of materials sciences semiconductor process simulations seismic processing and optimal structural topology design

two phase viscoelastic jetting a coupled finite difference algorithm on rectangular grids is developed for viscoelastic ink ejection simulations the ink is modeled by the oldroyd b viscoelastic fluid model the coupled algorithm seamlessly incorporates several things 1 a coupled level set projection method for incompressible immiscible two phase fluid flows 2 a higher order godunov type algorithm for the convection terms in the momentum and level set equations 3 a simple first order upwind algorithm for the convection term in the viscoelastic stress equations 4 central difference approximations for viscosity surface tension and upper convected derivative terms and 5 an equivalent circuit model to calculate the inflow pressure or flow rate from dynamic voltage

immersed smoothed finite element method for two dimensional fluid structure interaction problems summary a novel method called immersed smoothed fem using three node triangular element is proposed for two dimensional fluid structure interaction fsi problems with largely deformable nonlinear solids placed within incompressible viscous fluid the fluid flows are solved using the semi implicit characteristic based split method smoothed fems are employed to calculate the transient responses of solids based on explicit time integration the fictitious fluid with two assumptions is introduced to achieve the continuous form of the fsi conditions the discrete formulations to calculate the fsi forces are obtained in terms of the characteristic based split scheme and the algorithm based on a set of fictitious fluid mesh is proposed for evaluating the fsi force exerted on the solid the accuracy stability and convergence properties of immersed smoothed fem are verified by numerical examples investigations on the mesh size ratio indicate that the stability is fairly independent of the wide range of the mesh size ratio no additional volume correction is required to satisfy the incompressible constraints copyright 2012 john wiley sons ltd

an immersed boundary method with formal second order accuracy and reduced numerical viscosity a formally second order accurate immersed boundary method is presented and tested in this paper we apply this new scheme to simulate the flow past a circular cylinder and study the effect of numerical viscosity on the accuracy of the computation by comparing the numerical results with those of a first order method the numerical evidence shows that the new scheme has less numerical viscosity and is therefore a better choice for the simulation of high reynolds number flows with immersed boundaries

numerical simulation of liquid gas phase flow during mold filling abstract a numerical model for simulation of liquid gas phase flow during mold filling is presented the incompressible navier stokes equations are discretized on a fixed cartesian mesh with finite difference method the fractional step scheme is employed for enforcing incompressibility constraint the free surface effects are calculated using the volume of fluid method based on the piecewise linear interface reconstruction and split lagrangian advection of volume fraction field adding limited compressibility to the gas phase led to improvement in convergence rate of poisson equation solver about 2 fold this new concept permits simulation of two phase incompressible free surface flow during mold filling without using outflow boundary condition numerical results are presented for several test cases which demonstrate validity capability and performance of the proposed method in contrast with the traditional fully incompressible approach

a structural optimization method based on the level set method using a new geometry based re initialization scheme structural optimization methods based on the level set method are a new type of structural optimization method where the outlines of target structures can be implicitly represented using the level set function and updated by solving the so called hamilton jacobi equation based on a eulerian coordinate system these new methods can allow topological alterations such as the number of holes during the optimization process whereas the boundaries of the target structure are clearly defined however the re initialization scheme used when updating the level set function is a critical problem when seeking to obtain appropriately updated outlines of target structures in this paper we propose a new structural optimization method based on the level set method using a new geometry based re initialization scheme where both the numerical analysis used when solving the equilibrium equations and the updating process of the level set function are performed using the finite element method the stiffness maximization eigenfrequency maximization and eigenfrequency matching problems are considered as optimization problems several design examples are presented to confirm the usefulness of the proposed method copyright 2010 john wiley sons ltd

immersed finite element method abstract in this paper the immersed finite element method ifem is proposed for the solution of complex fluid and deformable structure interaction problems encountered in many physical models in ifem a lagrangian solid mesh moves on top of a background eulerian fluid mesh which spans over the entire computational domain hence the mesh generation is greatly simplified moreover both fluid and solid domains are modeled with the finite element methods and the continuity between the fluid and solid sub domains are enforced via the interpolation of the velocities and the distribution of the forces with the reproducing kernel particle method rkpm delta function in comparison with the immersed boundary ib method the higher ordered rkpm delta function enables the fluid domain to have nonuniform spatial meshes with arbitrary geometries and boundary conditions the use of such kernel functions may eventually open doors to multi scale and multi resolution modelings of complex fluid structure interaction problems rigid and deformable spheres dropping in channels are simulated to demonstrate the unique capabilities of the proposed method the results compare well with the experimental data to the authors knowledge these are the first solutions that deal with particulate flows with very flexible solids

a stabilized finite element method using a discontinuous level set approach for the computation of bubble dynamics a novel numerical method for solving three dimensional two phase flow problems is presented this method combines a quadrature free discontinuous galerkin method for the level set equation with a pressure stabilized finite element method for the navier stokes equations the main challenge in the computation of such flows is the accurate evaluation of surface tension forces this involves the computation of the curvature of the fluid interface in the context of the discontinuous galerkin method we show that the use of a curvature computed by means of a direct derivation of the level set function leads to inaccurate and oscillatory results a more robust second order least squares computation of the curvature that filters out the high frequencies and produces converged results is presented this whole numerical technology allows to simulate a wide range of flow regimes with large density ratios to accurately capture the shape of the deforming interface of the bubble and to maintain good mass conservation

an edge based smoothed finite element method es fem for static free and forced vibration analyses of solids abstract this paper presents an edge based smoothed finite element method es fem to significantly improve the accuracy of the finite element method fem without much changing to the standard fem settings the es fem can use different shape of elements but prefers triangular elements that can be much easily generated automatically for complicated domains in the es fem the system stiffness matrix is computed using strains smoothed over the smoothing domains associated with the edges of the triangles intensive numerical results demonstrated that the es fem possesses the following excellent properties 1 the es fem model possesses a close to exact stiffness it is much softer than the overly stiff fem and much stiffer than the overly soft ns fem model 2 the results are often found superconvergence and ultra accurate much more accurate than the linear triangular elements of fem and even more accurate than those of the fem using quadrilateral elements with the same sets of nodes 3 there are no spurious non zeros energy modes found and hence the method is also temporally stable and works well for vibration analysis and 4 the implementation of the method is straightforward and no penalty parameter is used and the computational efficiency is better than the fem using the same sets of nodes in addition a novel domain based selective scheme is proposed leading to a combined es ns fem model that is immune from volumetric locking and hence works very well for nearly incompressible materials these properties of the es fem are confirmed using examples of static free and forced vibration analyses of solids

an immersed discontinuous finite element method for the stokes problem with a moving interface abstract we present a discontinuous immersed finite element ife method for incompressible interfacial flows that are governed by the stokes equations the method is based on a cartesian mesh with elements cut by the moving interface on this fixed unfitted mesh we employ an immersed q 1 q 0 finite element space constructed according to the location of the interface and pertinent interface jump conditions as such the smearing of solution across the interface is greatly reduced the interface represented by a sequence of marker points is advected on the fixed background mesh by the local fluid velocity the mesh is locally refined near the interface to further improve accuracy compared with the phase field method on adaptive meshes our method can achieve the same level of accuracy with much less degrees of freedoms we present some numerical examples to validate and demonstrate the capability of the proposed method

parallel re initialization of level set functions on distributed unstructured tetrahedral grids level set functions are employed to track interfaces in various application areas including simulation of two phase flows and image segmentation often a re initializing algorithm is incorporated to transform a numerically instable level set function to a signed distance function in this note we present a parallel algorithm for re initializing level set functions on unstructured three dimensional tetrahedral grids the main idea behind this new domain decomposition approach is to combine a parallel brute force re initializing algorithm with an efficient way to compute distances between the interface and grid points time complexity and error analysis of the algorithm are investigated detailed numerical experiments demonstrate the accuracy and scalability on up to 128 processes

analysis of an xfem discretization for stokes interface problems we consider a stationary stokes interface problem in the discretization the interface is not aligned with the triangulation for the discretization we use the p 1 extended finite element space p 1 xfem for the pressure and the standard conforming p 2 finite element space for the velocity since this pair is not necessarily lbb stable a consistent stabilization term known from the literature is added for the discrete bilinear form an inf sup stability result is derived which is uniform with respect to h mesh size parameter the viscosity quotient mu 1 mu 2 and the position of the interface in the triangulation based on this discretization error bounds are derived an optimal preconditioner for the stiffness matrix corresponding to this pair p 1 xfe for pressure and p 2 fe for velocity is presented the preconditioner has block diagonal form with a multigrid preconditioner for the velocity block and a new schur complement preconditioner optimality of this block preconditioner

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

free surface flows under impacting droplets a numerical method which fulfils the free surface boundary conditions and extrapolates the fluid velocity into empty grid cells outside the fluid region on a fixed cartesian grid system is presented the complex three dimensional vortex structures formed via surface vortex interaction and induction between vortices have been computed using the proposed technique implemented within a level set method for both vertical and oblique droplet impacts in incompressible fluids the present results have been validated through numerical tests which confirm zero tangential shear at the free surface and comparisons with experimental observations of cavity and vortex ring formation underneath the impact location in some cases transitions from a concentric vortex ring to a fully three dimensional vortex structure has been confirmed whilst the primary vortex ring is initiated at the highly curved contact surface between the droplet and receiving surface azimuthal instabilities are manifested in the shear layer around the cavity crater developing after the vertical impact resulting in axial counter rotating vorticity between the cavity and descending vortex ring underlying mechanisms which induce local deformation of the free surface creating a so called scar due to the sub surface vortices at the oblique impacts are also discussed

convergence of numerical approximations of the incompressible navier stokes equations with variable density and viscosity we consider numerical approximations of incompressible newtonian fluids having variable possibly discontinuous density and viscosity since solutions of the equations with variable density and viscosity may not be unique numerical schemes may not converge if the solution is unique then approximate solutions computed using the discontinuous galerkin method to approximate the convection of the density and stable finite element approximations of the momentum equation converge to the solution if the solution is not unique a subsequence of these approximate solutions will converge to a solution

high fidelity anisotropic adaptive variational multiscale method for multiphase flows with surface tension we propose in this work an adaptive variational multiscale method for two fluid flows with surface tension a level set function is used to provide a precise position of the interfaces the implementation of the surface tension in the context of the continuum surface force is proposed to circumvent the capillary time step restriction the obtained system is then solved using a new derived variational multiscale stabilized finite element method designed to handle the abrupt changes at the interface and large density and viscosity ratios combined with an a posteriori error estimator we show that anisotropic mesh adaptation provides highly stretched elements at the interfaces and thus yield an accurate modeling framework for two phase incompressible isothermal flows stable and accurate results are obtained for all two and three dimensional numerical examples to the best of our knowledge these are the first simulation results for representative time dependent three dimensional two fluid flow problems using an implicit treatment of the surface tension and a dynamic unstructured anisotropic mesh adaptation

a boundary condition capturing method for incompressible flame discontinuities abstract in this paper we propose a new numerical method for treating two phase incompressible flow where one phase is being converted into the other e g the vaporization of liquid water we consider this numerical method in the context of treating discontinuously thin flame fronts for incompressible flow this method was designed as an extension of the ghost fluid method 1999 j comput phys 152 457 and relies heavily on the boundary condition capturing technology developed in liu et al 2000 j comput phys 154 15 for the variable coefficient poisson equation and in kang et al in press j comput phys for multiphase incompressible flow our new numerical method admits a sharp interface representation similar to the method proposed in helenbrook et al 1999 j comput phys 148 366 since the interface boundary conditions are handled in a simple and straightforward fashion the code is very robust e g no special treatment is required to treat the merging of flame fronts the method is presented in three spatial dimensions with numerical examples in one two and three spatial dimensions

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

adaptive unstructured volume remeshing ii application to two and three dimensional level set simulations of multiphase flow in part i adaptive unstructured volume remeshing i the method j comput phys in press we presented an adaptive remeshing algorithm that automatically adjusts the size of the elements of meshes of unstructured triangles 2d and unstructured tetrahedra 3d with time and position in the computational domain in order to efficiently resolve the relevant physical scales here we illustrate the performance of an implementation of the algorithm in finite element level set simulations of deformable droplet and fluid fluid interface interactions breakup and coalescence in multiphase flows the wide range of length scales characterizing the dynamics are accurately resolved as demonstrated by comparison to experiments and to theoretical and sharp interface boundary integral numerical results the computational cost is found to be competitive even with respect to boundary integral methods for the first time using an interface capturing level set method we successfully simulate the inertia driven impact and rebound of a liquid droplet from a liquid interface and find agreement with recent experimental results

three dimensional volume conserving immersed boundary model for two phase fluid flows abstract we present a volume preserving scheme for two phase immiscible incompressible flows using an immersed boundary method ibm in a three dimensional space the two phase ibm employs a mixture of eulerian and lagrangian variables where the fluid interface is represented by discrete lagrangian markers exerting surface tension forces to the eulerian fluid domain and the markers are advected by the fluid velocity the interactions between the lagrangian markers and the fluid variables are linked by the discretized dirac delta function the present study extends the previous two dimensional research li et al volume preserving immersed boundary methods for two phase fluid flows int j numer meth fluids 69 2012 842 858 to the three dimensional space the key idea of the proposed method is relocating surface points along the normal directions to conserve the total volume we perform a number of numerical experiments to show the efficiency and accuracy of the proposed method

uniform preconditioners for a parameter dependent saddle point problem with application to generalized stokes interface equations we consider an abstract parameter dependent saddle point problem and present a general framework for analyzing robust schur complement preconditioners the abstract analysis is applied to a generalized stokes problem which yields robustness of the cahouet chabard preconditioner motivated by models for two phase incompressible flows we consider a generalized stokes interface problem application of the general theory results in a new schur complement preconditioner for this class of problems the robustness of this preconditioner with respect to several parameters is treated results of numerical experiments are given that illustrate robustness properties of the preconditioner

a g space theory and a weakened weak w2 form for a unified formulation of compatible and incompatible methods part i theory this paper introduces a g space theory and a weakened weak form w2 using the generalized gradient smoothing technique for a unified formulation of a wide class of compatible and incompatible methods the w2 formulation works for both finite element method settings and mesh free settings and w2 models can have special properties including softened behavior upper bounds and ultra accuracy part i of this paper focuses on the theory and fundamentals for w2 formulations a normed g space is first defined to include both continuous and discontinuous functions allowing the use of much more types of methods techniques to create shape functions for numerical models important properties and a set of useful inequalities for g spaces are then proven in the theory and analyzed in detail these properties ensure that a numerical method developed based on the w2 formulation will be spatially stable and convergent to the exact solutions as long as the physical problem is well posed the theory is applicable to any problems to which the standard weak formulation is applicable and can offer numerical solutions with special properties including close to exact stiffness upper bounds and ultra accuracy copyright 2009 john wiley sons ltd

new existence theorems for quasi variational inequalities and applications to financial models abstract in this paper we present a financial economy in the case when the financial volumes depend on time and on the expected solution in order to take into account the influence of the expected equilibrium distribution for assets and liabilities on the investments on all financial instruments we derive the quasi variational formulation which characterizes the equilibrium of the dynamical financial model the main result of the paper is a general existence theorem for quasi variational inequalities under general assumptions which is also applied to the financial model we also recall some concepts on the infinite dimensional duality and study some numerical examples

a contoured continuum surface force model for particle methods a surface tension model is essential to simulate multiphase flows with deformed interfaces this study develops a contoured continuum surface force ccsf model for particle methods a color function that varies sharply across the interface to mark different fluid phases is smoothed in the transition region where the local contour curvature can be regarded as the interface curvature the local contour passing through each reference particle in the transition region is extracted from the local profile of the smoothed color function the local contour curvature is calculated based on the taylor series expansion of the smoothed color function whose derivatives are calculated accurately according to the definition of the smoothed color function two schemes are proposed to specify the smooth radius fixed scheme where 2 r e r e particle interaction radius is assigned to all particles in the transition region and varied scheme where r e and 2 r e are assigned to the central and edged particles in the transition region respectively numerical examples including curvature calculation for static circle and ellipse interfaces deformation of square droplet to a circle 2d and 3d droplet deformation in shear flow and droplet coalescence are simulated to verify the ccsf model and compare its performance with those of other methods the ccsf model with the fixed scheme is proven to produce the most accurate curvature and lowest parasitic currents among the tested methods

immersed finite element method for rigid body motions in the incompressible navier stokes flow abstract the immersed finite element method ifem takes places as a method developed for the purpose of solving effectively fluid structure interaction problems including multi physics ones in the original ifem the reproducing kernel particle method rkpm playing a role of discrete dirac delta function is employed to distribute the interacting force on the structure to the surrounding fluid and calculate the velocity on the structure induced from the background fluid in this paper fluid structure interaction fsi problems in 2d between the incompressible navier stokes flow and rigid structure are considered and we make use of the transformed finite element basis functions to replace discrete dirac delta functions such as rkpm this replacement makes the numerical support of fsi force distributed to the fluid smaller than that in case where the discrete dirac delta function is used in the finite element formulation reducing the support size of distributed fsi force affects the accuracy of numerical solutions near the structure the comparison of our numerical solution for particulate flows shows this fact well we calculate particulate flows of rigid circular and rod type disks and compare them with the preceding results of reliability to show a good agreement to them moreover the interaction motion for rod type of rigid structure in the fluid is traced

a taylor galerkin method for convective transport problems a method is described to derive finite element schemes for the scalar convection equation in one or more space dimensions to produce accurate temporal differencing the method employs forward time taylor series expansions including time derivatives of second and third order which are evaluated from the governing partial differential equation this yields a generalized time discretized equation which is successively discretized in space by means of the standard bubnov galerkin finite element method the technique is illustrated first in one space dimension with linear elements and euler leap frog and crank nicolson time stepping several interesting relations with standard galerkin and recently developed petrov galerkin methods emerge and the new taylor galerkin schemes are found to exhibit particularly high phase accuracy with minimal numerical damping the method is successively extended to deal with variable coefficient problems and multi dimensional situations

direct numerical simulations of droplet emulsions in sliding bi periodic frames using the level set method we present a direct numerical simulation technique for droplet emulsions of newtonian newtonian system in simple shear flow the lees edwards type boundary condition has been incorporated with the sliding bi periodic frame of hwang et al w r hwang m a hulsen h e h meijer direct simulation of particle suspensions in sliding bi periodic frames j comput phys 194 2004 742 for the continuous flow field problem and the level set method with the continuous surface stress css formulation has been used for accurate description of the sharp interfaces based on the standard velocity pressure formulation of the finite element method we use the mortar element method for the implementation of the sliding periodicity and employ the discontinuous galerkin dg method for the stabilization of the interface advection equation we present numerical results on the morphological development for a single two and multiple drops in sliding bi periodic frames for the demonstration of the feasibility of the present method in investigation of the relationship between the morphology and the bulk material responses such as the shear viscosity and the first normal stress difference

gas bubble with a moving contact line rising in an inclined channel at finite reynolds number abstract the dynamics of a gas bubble with a moving contact line rising in an inclined channel is investigated the solution of this free boundary problem is determined numerically by using a level set method coupled with a finite difference solution of the navier stokes equations results are presented as a function of reynolds number re bond number and the angle of inclination steadily rising bubbles are found for small values of both the reynolds and bond number with increasing reynolds number the steady solution first bifurcates to a time periodic oscillation and then the numerical results imply period doubling large values of reynolds or bond number cause the bubble to either detach from the wall or to rupture

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

evolutionary variational inequalities applied to financial equilibrium problems in an environment of risk and uncertainty abstract in this paper we present an evolutionary model for multi sector multi instrument financial equilibrium problems in particular we consider the case in which the variance covariance matrices or more generally the utility function associated with risk perception the financial volumes held by the sectors the optimal portfolio composition and the instrument prices are time dependent

an extended finite element method applied to levitated droplet problems we consider a standard model for incompressible two phase flows in which a localized force at the interface describes the effect of surface tension if a level set method is applied then the approximation of the interface is in general not aligned with the triangulation this causes severe difficulties w r t the discretization and often results in large spurious velocities in this paper we reconsider a modified extended finite element method xfem which in previous papers has been investigated for relatively simple two phase flow model problems and apply it to a physically realistic levitated droplet problem the results show that due to the extension of the standard fe space one obtains much better results in particular for large interface tension coefficients furthermore a certain cut off technique results in better efficiency without sacrificing accuracy copyright 2010 john wiley sons ltd

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

combined immersed boundary finite difference methods for three dimensional complex flow simulations a second order accurate highly efficient method is developed for simulating unsteady three dimensional incompressible flows in complex geometries this is achieved by using boundary body forces that allow the imposition of the boundary conditions on a given surface not coinciding with the computational grid the governing equations therefore can be discretized and solved on a regular mesh thus retaining the advantages and the efficiency of the standard solution procedures two different forcings are tested showing that while the quality of the results is essentially the same in both cases the efficiency of the calculation strongly depends on the particular expression a major issue is the interpolation of the forcing over the grid that determines the accuracy of the scheme this ranges from zeroth order for the most commonly used interpolations up to second order for an ad hoc velocity interpolation the present scheme has been used to simulate several flows whose results have been validated by experiments and other results available in the literature finally in the last example we show the flow inside an ic piston cylinder assembly at high reynolds number to our knowledge this is the first example in which the immersed boundary technique is applied to a full three dimensional complex flow with moving boundaries and with a reynolds number high enough to require a subgrid scale turbulence model

convergence of a non stiff boundary integral method for interfacial flows with surface tension boundary integral methods to simulate interfacial flows are very sensitive to numerical instabilities in addition surface tension introduces nonlinear terms with high order spatial derivatives into the interface dynamics this makes the spatial discretization even more difficult and at the same time imposes a severe time step constraint for stable explicit time integration methods a proof of the convergence of a reformulated boundary integral method for two density fluid interfaces with surface tension is presented the method is based on a scheme introduced by hou lowengrub and shelley j comp phys 114 1994 pp 312 338 to remove the high order stability constraint or stiffness some numerical filtering is applied carefully at certain places in the discretization to guarantee stability the key of the proof is to identify the most singular terms of the method and to sho through energy estimates that these terms balance one another the analysis is at a time continuous space discrete level but a fully discrete case for a simple hele shaw interface is also studied the time discrete analysis shows that the high order stiffness is removed and also provides an estimate of how the cfl constraint depends on the curvature and regularity of the solution the robustness of the method is illustrated with several numerical examples a numerical simulation of an unstably stratified two density interfacial flow shows the roll up of the interface the computations proceed up to a time where the interface is about to pinch off and trapped bubbles of fluid are formed the method remains stable even in the full nonlinear regime of motion another application of the method shows the process of drop formation in a falling single fluid

on the combined effects of surface tension force calculation and interface advection on spurious currents within volume of fluid and level set frameworks this paper deals with the comparison of eulerian methods to take into account the capillary contribution in the vicinity of a fluid fluid interface eulerian methods are well known to produce additional vorticity close to the interface that leads to non physical spurious currents numerical equilibrium between pressure gradient and capillary force for the static bubble test case within a vof framework has been reached in 35 with the height function technique 14 35 however once the bubble is translated in a uniform flow spurious currents are maintained by slight errors induced by translation schemes in this work two main points are investigated the ability of volume of fluid and level set methods to accurately calculate the curvature and the magnitude of spurious currents due to errors in the calculation of the curvature after advection in both translating and rotating flows the spurious currents source term is expressed from the vorticity equation and used to discuss and compare the methods simulations of gas liquid taylor flow at low capillary number show that the flow structure and the bubble velocity can be significantly affected by spurious currents

a stabilized conforming nodal integration for galerkin mesh free methods domain integration by gauss quadrature in the galerkin mesh free methods adds considerable complexity to solution procedures direct nodal integration on the other hand leads to a numerical instability due to under integration and vanishing derivatives of shape functions at the nodes a strain smoothing stabilization for nodal integration is proposed to eliminate spatial instability in nodal integration for convergence an integration constraint ic is introduced as a necessary condition for a linear exactness in the mesh free galerkin approximation the gradient matrix of strain smoothing is shown to satisfy ic using a divergence theorem no numerical control parameter is involved in the proposed strain smoothing stabilization the numerical results show that the accuracy and convergent rates in the mesh free method with a direct nodal integration are improved considerably by the proposed stabilized conforming nodal integration method it is also demonstrated that the gauss integration method fails to meet ic in mesh free discretization for this reason the proposed method provides even better accuracy than gauss integration for galerkin mesh free method as presented in several numerical examples copyright 2001 john wiley sons ltd

a three field stabilized finite element method for fluid structure interaction elastic solid and rigid body limit summary we propose a full eulerian framework for solving fluid structure interaction fsi problems based on a unified formulation in which the fsis are modelled by introducing an extra stress in the momentum equation the obtained three field velocity pressure and stress system is solved using a stabilized finite element method the key feature of this unified formulation is the ability to describe different kind of interactions between the fluid and the structure which can be either elastic or a perfect rigid body without the need of treating this last case via penalization the level set method combined with a dynamic anisotropic mesh adaptation is used to track the fluid solid interface copyright 2015 john wiley sons ltd

simulation of surface tension in 2d and 3d with smoothed particle hydrodynamics method the methods for simulating surface tension with smoothed particle hydrodynamics sph method in two dimensions and three dimensions are developed in 2d surface tension model the sph particle on the boundary in 2d is detected dynamically according to the algorithm developed by dilts g a dilts moving least squares particle hydrodynamics ii conservation and boundaries international journal for numerical methods in engineering 48 2000 1503 1524 the boundary curve in 2d is reconstructed locally with lagrangian interpolation polynomial in 3d surface tension model the sph particle on the boundary in 3d is detected dynamically according to the algorithm developed by haque and dilts a haque g a dilts three dimensional boundary detection for particle methods journal of computational physics 226 2007 1710 1730 the boundary surface in 3d is reconstructed locally with moving least squares mls method by transforming the coordinate system it is guaranteed that the interface function is one valued in the local coordinate system the normal vector and curvature of the boundary surface are calculated according to the reconstructed boundary surface and then surface tension force can be calculated surface tension force acts only on the boundary particle density correction is applied to the boundary particle in order to remove the boundary inconsistency the surface tension models in 2d and 3d have been applied to benchmark tests for surface tension the ability of the current method applying to the simulation of surface tension in 2d and 3d is proved

retrieving topological information for phase field models the phase field approach has become a popular tool in modeling interface motion microstructure evolution and more recently the shape transformation of vesicle membranes under elastic bending energy while it is advantageous to employ phase field models in numerical simu lations to automatically handle topological changes to the microstructures or the configurations of vesicle membranes detecting topological events may also become important for many applications such as those in the simulation of blood cells motivated by such considerations a new quantity is formulated to retrieve some topological information based on the phase field formulation and to capture the occurrence of topological events it can also be used as a control method to avoid unphys ical changes of topology due to the numerical methods should it become necessary for particular practical applications through numerical experiments we demonstrate the effectiveness and the robustness of the new quantity in detecting the topology of fluid bubbles and vesicle membranes

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

theoretical and numerical analysis on a thermo elastic system with discontinuities a second order accurate numerical scheme is proposed for a thermo elastic system which models a bar made of two distinct materials the physical parameters involved may be discontinuous across the joint of the two materials where there might be also singular heat and or force sources the solution components the temperature and the displacement may change rapidly across the joint by transforming the system into a different one time marching schemes can be used for the new system which is well posed the immersed interface method is employed to deal with the discontinuities of the coefficients and the singular sources the proposed numerical method can fit both explicit and implicit formulation for the implicit version a stable and fast prediction correction scheme is also developed convergence analysis shows that our method is second order accurate at all grid points in spite of the discontinuities across the interface numerical experiments are performed to support the theoretical analysis in this paper

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

approximation of time dependent multi component viscoelastic fluid flow abstract in this article we analyse a fully discrete approximation to the time dependent viscoelasticity equations allowing for multicomponent fluid flow the oldroyd b constitutive equation is used to model the viscoelastic stress for the discretization time derivatives are replaced by backward difference quotients and the non linear terms are linearized by lagging appropriate factors the modeling equations for the individual fluids are combined into a single system of equations using a continuum surface model the numerical approximation is stabilized by using an supg approximation for the constitutive equation under a small data assumption on the true solution existence of the approximate solution is proven a priori error estimates for the approximation in terms of the mesh parameter h the time discretization parameter t and the supg coefficient are also derived numerical simulations of viscoelastic fluid flow involving two immiscible fluids are also presented

numerical simulation of drop impact on a liquid liquid interface with a multiple marker front capturing method the gravity driven motion of a droplet impacting on a liquid liquid interface is studied the full navier stokes equations are solved on a fixed uniform grid using a finite difference front capturing method for the representation of fluid fluid interfaces a coupled level set volume of fluid method m sussman e g puckett a coupled level set and volume of fluid method for computing 3d and axisymmetric incompressible two phase flows j comp phys 162 2000 301 337 is used in which we introduce the novel approach of describing separate interfaces with different marker functions as a consequence we prevent numerical coalescence of the droplet and the liquid liquid interface without excessive local grid refinement to validate our method numerical simulations of the drop impact event are compared with experiments z mohamed kassim e k longmire drop impact on a liquid liquid interface phys fluids 15 2003 3263 3273 furthermore a comparison is made with the numerical results of a esmaeeli g tryggvason direct numerical simulations of bubbly flows part 2 moderate reynolds number arrays j fluid mech 385 1999 325 358 for an array of rising bubbles the investigation shows that the multiple marker approach successfully prevents numerical coalescence of interfaces and adequately captures the effect of surface tension

an adaptive mesh redistribution method for the incompressible mixture flows using phase field model a phase field model which describes the motion of mixtures of two incompressible fluids is presented by liu and shen c liu j shen a phase field model for the mixture of two incompressible fluids and its approximation by a fourier spectral method phys d 179 2003 211 228 the model is based on an energetic variational formulation in this work we develop an efficient adaptive mesh method for solving a phase field model for the mixture flow of two incompressible fluids it is a coupled nonlinear system of navier stokes equations and allen cahn phase equation phase field equation through an extra stress term and the transport term the numerical strategy is based on the approach proposed by li et al r li t tang p w zhang moving mesh methods in multiple dimensions based on harmonic maps j comput phys 170 2001 562 588 to separate the mesh moving and pde evolution in the pde evolution part the phase field equation is numerically solved by a conservative scheme with a lagrange multiplier and the coupled incompressible navier stokes equations are solved by the incremental pressure correction projection scheme based on the semi staggered grid method in the mesh moving part the mesh points are iteratively redistributed by solving the euler lagrange equations with a parameter free monitor function in each iteration the pressure and the phase are updated on the resulting new grid by a conservative interpolation formula while the velocity is re mapped in a non conservative approach a simple method for preserving divergence free is obtained by projecting the velocity onto the divergence free space after generating the new mesh at the last iterative step numerical experiments are presented to demonstrate the effectiveness of the proposed method for solving the incompressible mixture flows

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

an active contour model for image segmentation based on elastic interaction the task of image segmentation is to partition an image into non overlapping regions based on intensity or textural information the active contour methods provide an effective way for segmentation in which the boundaries of the objects are detected by evolving curves in this paper we propose a new edge based active contour method which uses a long range and orientation dependent interaction between image boundaries and the moving curves while maintaining the edge fidelity as a result this method has a large capture range and is able to detect sharp features of the images the velocity field for the moving curves generated by this elastic interaction is calculated using the fast fourier transform fft method level set representation is used for the moving curves so that the topological changes during the evolution are handled automatically this new method is derived based on the elastic interaction between line defects in solids dislocations although it is derived originally for two dimensional segmentation we also extend it to three dimensions the features of the new method are examined by experiments on both synthetic images and medical images of blood vessels comparisons are made with the existing active contour methods

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

a high order projection method for tracking fluid interfaces in variable density incompressible flows we present a numerical method for computing solutions of the incompressible euler or navier stokes equations when a principal feature of the flow is the presence of an interface between two fluids with different fluid properties the method is based on a second order projection method for variable density flows using an approximate projection formulation the boundary between the fluids is tracked with a second order volume of fluid interface tracking algorithm we present results for viscious rayleigh taylor problems at early time with equal and unequal viscosities to demonstrate the convergence of the algorithm we also present computational results for the rayleigh taylor instability in air helium and for bubbles and drops in an air water system without surface tension to demonstrate the behavior of the algorithm on problems with large density and viscosity contrasts

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

a level set method for computing solutions to viscoelastic two phase flow abstract a finite element code based on the level set method is developed for simulating the motion of viscoelastic two phase flow problems this method is a generalization of the finite difference approach described in 1 4 for computing solutions to two phase problems of inviscid and viscous fluids the marchuk yanenko operator splitting technique is used to decouple the difficulties associated with the nonlinear convection term the incompressibility constraint the viscoelastic term and the interface motion problem the nonlinear convection problem is solved using a least squares conjugate gradient algorithm and the stokes like problem is solved using a conjugate gradient algorithm the constitutive equation is solved using a scheme that guarantees the positive definiteness of the configuration tensor while the convection term in the constitutive equation is discretized using a third order upwinding scheme the code is verified by performing a convergence study to show that the results are independent of the mesh and time step sizes using our code we have studied the deformation of drops in simple shear and pressure driven flows and of bubbles in gravity driven flows over a wide range of dimensionless capillary ca and deborah numbers de for a newtonian bubble rising in a quiescent viscoelastic liquid we find that there are limiting values of the parameters de and ca above which the bubble assumes a characteristic shape with a cusp like trailing edge the front of the bubble however remains round as the local viscoelastic and viscous stresses act to round the bubble in a pressure driven flow the drop is stretched so that its front which is closer to the channel center remains round and the trailing edge which is closer to the channel wall becomes sharp these numerical results are in agreement with the experimental observations

numerical simulation of free surface incompressible liquid flows surrounded by compressible gas a numerical model for the three dimensional simulation of liquid gas flows with free surfaces is presented the incompressible navier stokes equations are assumed to hold in the liquid domain in the gas domain the velocity is disregarded the pressure is supposed to be constant in each connected component of the gas domain and follows the ideal gas law the gas pressure is imposed as a normal force on the liquid gas interface an implicit splitting scheme is used to decouple the physical phenomena given the gas pressure on the interface the method described in j comput phys 155 1999 439 int j numer meth fluids 42 7 2003 697 is used to track the liquid domain and to compute the velocity and pressure fields in the liquid then the connected components of the gas domain are found using an original numbering algorithm finally the gas pressure is updated from the ideal gas law in each connected component of gas the implementation is validated in the frame of mould filling numerical results in two and three space dimensions show that the effect of pressure in the bubbles of gas trapped by the liquid cannot be neglected

a topology optimization method based on the level set method incorporating a fictitious interface energy this paper proposes a new topology optimization method which can adjust the geometrical complexity of optimal configurations using the level set method and incorporating a fictitious interface energy derived from the phase field method first a topology optimization problem is formulated based on the level set method and the method of regularizing the optimization problem by introducing fictitious interface energy is explained next the reaction diffusion equation that updates the level set function is derived and an optimization algorithm is then constructed which uses the finite element method to solve the equilibrium equations and the reaction diffusion equation when updating the level set function finally several optimum design examples are shown to confirm the validity and utility of the proposed topology optimization method