handling solid fluid interfaces for viscous flows explicit jump approximation vs ghost cell approaches the ghost cell approaches gca for handling stationary solid boundaries regular or irregular are first investigated theoretically and numerically for the diffusion equation with dirichlet boundary conditions the main conclusion of this part of investigation is that the approximation for the diffusion term has to be second order accurate everywhere in order for the numerical solution to be rigorously second order accurate violating this principle the linear and quadratic gcas have the following shortcomings 1 restrictive constraints on grid size when the viscosity is small 2 susceptibleness to instability of a time explicit formulation for strongly transient flows 3 convergence deterioration to zeroth or first order for solutions with high frequency modes therefore the widely used linear extrapolation for enforcing no slip boundary conditions should be avoided even for regular solid boundaries as a remedy a simple method based on explicit jump approximation eja is proposed eja hinges on the idea that a velocity derivative jump at the boundary reduces to the value of the velocity derivative at the fluid side because the velocity of the stationary boundary is zero although the time marching linear system of eja is not symmetric it is strictly diagonal dominant with positive diagonal entries numerical results show that over a large range of viscosity and grid sizes eja performs much better than gcas in terms of stability and accuracy furthermore the second order convergence of eja does not depend on viscosity and the spectrum of the solution as those of gcas do this paper is written with enough details so that one can reproduce the numerical results

a schur complement formulation for solving free boundary stefan problems of phase change a schur complement formulation that utilizes a linear iterative solver is derived to solve a free boundary stefan problem describing steady state phase change via the isotherm newton approach which employs newton s method to simultaneously and efficiently solve for both interface and field equations this formulation is tested alongside more traditional solution strategies that employ direct or iterative linear solvers on the entire jacobian matrix for a two dimensional sample problem that discretizes the field equations using a galerkin finite element method and employs a deforming grid approach to represent the melt solid interface all methods demonstrate quadratic convergence for sufficiently accurate newton solves but the two approaches utilizing linear iterative solvers show better scaling of computational effort with problem size of these two approaches the schur formulation proves to be more robust converging with significantly smaller krylov subspaces than those required to solve the global system of equations further improvement of performance are made through approximations and preconditioning of the schur complement problem hence the new schur formulation shows promise as an affordable robust and scalable method to solve free boundary stefan problems such models are employed to study a wide array of applications including casting welding glass forming planetary mantle and glacier dynamics thermal energy storage food processing cryosurgery metallurgical solidification and crystal growth

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

solution adaptive cartesian cell approach for viscous and inviscid flows a cartesian cell based approach for adaptively refined solutions of the euler and navier stokes equations in two dimensions is presented grids about geometrically complicated bodies are generated automatically by the recursive subdivision of a single cartesian cell encompassing the entire flow domain where the resulting cells intersect bodies polygonal cut cells are created using modified polygon clipping algorithms the grid is stored in a binary tree data structure that provides a natural means of obtaining cell to cell connectivity and of carrying out solution adaptive mesh refinement the euler and navier stokes equations are solved on the resulting grids using a finite volume formulation the convective terms are up winded a linear reconstruction of the primitive variables is performed providing input states to an approximate riemann solver for computing the fluxes between neighboring cells the results of a study comparing the accuracy and positivity of two classes of cell centered viscous gradient reconstruction procedures is briefly summarized adaptively refined solutions of the navier stokes equations are shown using the more robust of these gradient reconstruction procedures where the results computed by the cartesian approach are compared to theory experiment and other accepted computational results for a series of low and moderate reynolds number flows

a generalized finite difference gfd ale scheme for incompressible flows around moving solid bodies on hybrid meshfree cartesian grids a scheme using the mesh free generalized finite differencing gfd on flows past moving bodies is proposed the aim is to devise a method to simulate flow past an immersed moving body that avoids the intensive remeshing of the computational domain and minimizes data interpolation associated with the established computational fluid methodologies as such procedures are time consuming and are a significant source of error in flow simulation in the present scheme the moving body is embedded and enveloped by a cloud of mesh free nodes which convects with the motion of the body against a background of cartesian nodes the generalized finite difference gfd method with weighted least squares wls approximation is used to discretize the two dimensional viscous incompressible navier stokes equations at the mesh free nodes while standard finite difference approximations are applied elsewhere the convecting motion of the mesh free nodes is treated by the arbitrary lagrangian eulerian ale formulation of the flow equations which are solved by a second order crank nicolson based projection method the proposed numerical scheme was tested on a number of problems including the decaying vortex flow external flows past moving bodies and body driven flows in enclosures

modeling a no slip flow boundary with an external force field abstract a novel technique related to peskin s immersed boundary approach is used to introduce solid surfaces into a simulated flow field the navier stokes equations permit the presence of an externally imposed body force that may vary in space and time forces are chosen to lie along a desired surface and to have a magnitude and direction opposing the local flow such that the flow is brought to rest on an element of the surface for unsteady viscous flow the direct calculation of the needed force is facilitated by a feedback scheme in which the velocity is used to iteratively determine the desired value in particular we determine the surface body force from the relation f x 5 t t 0 u x 5 t dt u x 5 t for surface points x 5 velocity u time t and negative constants and examples are presented which include 2d flow around cylinders 3d turbulent channel flow where one boundary is simulated with a force field and turbulent channel flow over a riblet covered surface while the new method may be applied to complex geometries on a non cartesian mesh we have chosen to use a simple cartesian grid all simulations are done with a spectral code in a single computational domain without any mapping of the mesh

variational algorithms and pattern formation in dendritic solidification we present a completely new variational algorithm for computing dendritic solidification this algorithm reproduces the gibbs thomson relation as a balance between bulk and surface energy and is able to operate in the infinite mobility limit with no unphysical time step restriction it may be used with arbitrary non smooth surface energy functions and may include finite kinetic mobility we perform computations with isotropic and anisotropic surface energy from a small irregular initial seed we generate radial tip splitting structures for isotropic energy and parabolic dendrites with side branching for anisotropic energy for anisotropic energy the final structure is determined by the material and environmental properties the initial shape is forgotten for the parabolic dendrite tips we obtain agreement with the ivantsov solutions within a few percent and proper dimensional scaling of lengths and velocities with surface energy

crystal growth and dendritic solidification we present a numerical method which computes the motion of complex solid liquid boundaries in crystal growth the model we solve includes physical effects such as crystalline anisotropy surface tension molecular kinetics and undercooling the method is based on two ideas first the equations of motion are recast as a single history dependent boundary integral equation on the solid liquid boundary a fast algorithm is used to solve the integral equation efficiently second the boundary is moved by solving a hamiltonjacobi type equation on a fixed domain formulated by osher and sethian for a function in which the boundary is a particular level set this equation is solved by finite difference schemes borrowed from the technology of hyperbolic conservation laws the two ideas are combined by constructing a smooth extension of the normal velocity off the moving boundary in a way suggested by the physics of the problem our numerical experiments show the evolution of complex crystalline shapes development of large spikes and corners dendrite formation and side branching and pieces of solid merging and breaking off freely

a correction function method for poisson problems with interface jump conditions abstract in this paper we present a method to treat interface jump conditions for constant coefficients poisson problems that allows the use of standard black box solvers without compromising accuracy the basic idea of the new approach is similar to the ghost fluid method gfm the gfm relies on corrections applied on nodes located across the interface for discretization stencils that straddle the interface if the corrections are solution independent they can be moved to the right hand side rhs of the equations producing a problem with the same linear system as if there were no jumps only with a different rhs however achieving high accuracy is very hard if not impossible with the standard approaches used to compute the gfm correction terms in this paper we generalize the gfm correction terms to a correction function defined on a band around the interface this function is then shown to be characterized as the solution to a pde with appropriate boundary conditions this pde can in principle be solved to any desired order of accuracy as an example we apply this new method to devise a 4th order accurate scheme for the constant coefficients poisson equation with discontinuities in 2d this scheme is based on i the standard 9 point stencil discretization of the poisson equation ii a representation of the correction function in terms of bicubics and iii a solution of the correction function pde by a least squares minimization several applications of the method are presented to illustrate its robustness dealing with a variety of interface geometries its capability to capture sharp discontinuities and its high convergence rate

a second order accurate symmetric discretization of the poisson equation on irregular domains in this paper we consider the variable coefficient poisson equation with dirichlet boundary conditions on an irregular domain and show that one can obtain second order accuracy with a rather simple discretization moreover since our discretization matrix is symmetric it can be inverted rather quickly as opposed to the more complicated nonsymmetric discretization matrices found in other second order accurate discretizations of this problem multidimensional computational results are presented to demonstrate the second order accuracy of this numerical method in addition we use our approach to formulate a second order accurate symmetric implicit time discretization of the heat equation on irregular domains then we briefly consider stefan problems

a finite difference discretization method for heat and mass transfer with robin boundary conditions on irregular domains abstract this paper proposes a finite difference discretization method for simulations of heat and mass transfer with robin boundary conditions on irregular domains the level set method is utilized to implicitly capture the irregular evolving interface and the ghost fluid method to address variable discontinuities on the interface special care has been devoted to providing ghost values that are restricted by the robin boundary conditions specifically it is done in two steps 1 calculate the normal derivative in cells adjacent to the interface by reconstructing a linear polynomial system 2 successively extrapolate the normal derivative and the ghost value in the normal direction using a linear partial differential equation approach this method produces second order accurate solutions for both the poisson and heat equations with robin boundary conditions and first order accurate solutions for the stefan problems the solution gradients are of first order accuracy as expected it is easy to implement in three dimensional configurations and can be straightforwardly generalized into higher order variants the method thus represents a promising tool for practical heat and mass transfer problems involving robin boundary conditions

finite volume treatment of dispersion relation preserving and optimized prefactored compact schemes for wave propagation in developing suitable numerical techniques for computational aero acoustics the dispersion relation preserving drp scheme by tam and co workers and the optimized prefactored compact opc scheme by ashcroft and zhang have shown desirable properties of reducing both dissipative and dispersive errors these schemes originally based on the finite difference attempt to optimize the coefficients for better resolution of short waves with respect to the computational grid while maintaining pre determined formal orders of accuracy in the present study finite volume formulations of both schemes are presented to better handle the nonlinearity and complex geometry encountered in many engineering applications linear and nonlinear wave equations with and without viscous dissipation have been adopted as the test problems highlighting the principal characteristics of the schemes and utilizing linear and nonlinear wave equations with different wavelengths as the test cases the performance of these approaches is documented for the linear wave equation there is no major difference between the drp and opc schemes for the nonlinear wave equations the finite volume version of both drp and opc schemes offers substantially better solutions in regions of high gradient or discontinuity

a hybrid extended finite element level set method for modeling phase transformations a hybrid numerical method for modelling the evolution of sharp phase interfaces on fixed grids is presented we focus attention on two dimensional solidification problems where the temperature field evolves according to classical heat conduction in two subdomains separated by a moving freezing front the enrichment strategies of the extended finite element method x fem are employed to represent the jump in the temperature gradient that governs the velocity of the phase boundary a new approach with the x fem is suggested for this class of problems whereby the partition of unity is constructed with c1 polynomials and enriched with a c0 function this approach leads to jumps in temperature gradient occurring only at the phase boundary and is shown to significantly improve estimates for the front velocity temporal derivatives of the temperature field in the vicinity of the phase front are obtained with a projection that employs discontinuous enrichment in conjunction with a finer finite difference grid the level set method is used to represent the evolution of the phase interface an iterative procedure is adopted to satisfy the constraints on the temperature field on the phase boundary the robustness and utility of the method is demonstrated with several benchmark problems of phase transformation copyright 2002 john wiley sons ltd

maximal positive boundary value problems as limits of singular perturbation problems we study three types of singular perturbations of a symmetric positive system of partial differential equations on a domain 2 c r in all cases the limiting behavior is given by the solution of a maximal positive boundary value problem in the sense of friedrichs the perturbation is either a second order elliptic term or a term large on the complement of q the first corresponds to a sort of viscosity and the second to physical systems with vastly different properties in i2 and outside u the results show that in the limit of zero viscosity or infinitely large difference the behavior is described by a maximal positive boundary value problem in 2 the boundary condition is determined in a simple way from the system and the singular terms 1 introduction in this paper we study boundary value roblems for singularly perturbed systems of partial differential equations the unperturbed system is of first order and positive symmetric in the sense of friedrichs we study three sorts of singular perturbations 1 the addition of e times a positive second order elliptic system in a domain fi 2 3 the addition of xp or p d dt with a 1 and p strictly positive in the exterior of n and zero in s2 in the first case we study the solutions he to the dirichlet problem in s2 bardos brezis and brezis 1 have shown that in this case ue u weakly in l2 fi where u is uniquely determined as the solution of a maximal positive boundary value problem for the unperturbed opera tor here one has the phenomenon of loss of boundary conditions in the limit and the existence of boundary layers for uc near 3i2 we prove several more refined estimates on ue in particular a uniform bound in l 2 fi for any rj 0 these estimates yield a proof of the strong convergence of uc to u in all hs u for s this singular perturbation problem is well known and much studied in contrast to the problems of the second sort mentioned above in the latter problems we study the cauchy problems

a conservative immersed interface method for large eddy simulation of incompressible flows we propose a conservative second order accurate immersed interface method for representing incompressible fluid flows over complex three dimensional solid obstacles on a staggered cartesian grid the method is based on a finite volume discretization of the incompressible navier stokes equations which is modified locally in cells that are cut by the interface in such a way that accuracy and conservativity are maintained a level set technique is used for description and tracking of the interface geometry so that an extension of the method to moving boundaries and flexible walls is straightforward numerical stability is ensured for small cells by a conservative mixing procedure discrete conservation and sharp representation of the fluid solid interface render the method particularly suitable for large eddy simulations of high reynolds number flows accuracy second order grid convergence and robustness of the method is demonstrated for several test cases inclined channel flow at re 20 flow over a square cylinder at re 100 flow over a circular cylinder at re 40 re 100 and re 3900 as well as turbulent channel flow with periodic constrictions at re 10 595

curvilinear immersed boundary method for simulating fluid structure interaction with complex 3d rigid bodies the sharp interface curvib approach of ge and sotiropoulos l ge f sotiropoulos a numerical method for solving the 3d unsteady incompressible navier stokes equations in curvilinear domains with complex immersed boundaries journal of computational physics 225 2007 1782 1809 is extended to simulate fluid structure interaction fsi problems involving complex 3d rigid bodies undergoing large structural displacements the fsi solver adopts the partitioned fsi solution approach and both loose and strong coupling strategies are implemented the interfaces between immersed bodies and the fluid are discretized with a lagrangian grid and tracked with an explicit front tracking approach an efficient ray tracing algorithm is developed to quickly identify the relationship between the background grid and the moving bodies numerical experiments are carried out for two fsi problems vortex induced vibration of elastically mounted cylinders and flow through a bileaflet mechanical heart valve at physiologic conditions for both cases the computed results are in excellent agreement with benchmark simulations and experimental measurements the numerical experiments suggest that both the properties of the structure mass geometry and the local flow conditions can play an important role in determining the stability of the fsi algorithm under certain conditions the fsi algorithm is unconditionally unstable even when strong coupling fsi is employed for such cases however combining the strong coupling iteration with under relaxation in conjunction with the aitken s acceleration technique is shown to effectively resolve the stability problems a theoretical analysis is presented to explain the findings of the numerical experiments it is shown that the ratio of the added mass to the mass of the structure as well as the sign of the local time rate of change of the force or moment imparted on the structure by the fluid determine the stability and convergence of the fsi algorithm the stabilizing role of under relaxation is also clarified and the upper bound of the under relaxation coefficient required for stability is derived

a simple level set method for solving stefan problems a simple level set method for solving stefan problems is presented this method can be applied to problems involving dendritic solidification our method consists of an implicit finite difference scheme for solving the heat equation and a level set approach for capturing the front between solid and liquid phases of a pure substance our method is accurate with respect to some exact solutions of the stefan problem results indicate that this method can handle topology changes and complicated interfacial shapes and that it can numerically simulate many of the physical features of dendritic solidification

svd gfd scheme to simulate complex moving body problems in 3d space abstract the present paper presents a hybrid meshfree and cartesian grid method for simulating moving body incompressible viscous flow problems in 3d space the method combines the merits of cost efficient and accurate conventional finite difference approximations on cartesian grids with the geometric freedom of generalized finite difference gfd approximations on meshfree grids error minimization in gfd is carried out by singular value decomposition svd the arbitrary lagrangian eulerian ale form of the navier stokes equations on convecting nodes is integrated by a fractional step projection method the present hybrid grid method employs a relatively simple mode of nodal administration nevertheless it has the geometrical flexibility of unstructured mesh based finite volume and finite element methods boundary conditions are precisely implemented on boundary nodes without interpolation the present scheme is validated by a moving patch consistency test as well as against published results for 3d moving body problems finally the method is applied on low reynolds number flapping wing applications where large boundary motions are involved the present study demonstrates the potential of the present hybrid meshfree and cartesian grid scheme for solving complex moving body problems in 3d

sharp interface cartesian grid method iii solidification of pure materials and binary solutions a numerical technique is presented for computing dendritic growth of crystals from pure melts and binary solutions the governing equations are solved on a fixed cartesian mesh and the immersed phase boundary is treated as a sharp solid fluid interface the interface is tracked using a level set field a finite difference scheme is presented that incorporates the immersed phase boundary with only a small change to a standard cartesian grid poisson solver the scheme is simple to implement in three dimensions the results from our calculations show excellent agreement with two dimensional microscopic solvability theory for pure material solidification it is shown that the method predicts dendrite tip characteristics in excellent agreement with the theory the sharp interface treatment allows discontinuous material property variation at the solid liquid interface this facilitates sharp interface simulations of dendritic solidification of binary solutions

short note a volume of fluid approach for crystal growth simulation a new approach to simulating the dendritic growth of pure metals based on a recent volume of fluid vof method with plic piecewise linear interface calculation reconstruction of the interface is presented the energy equation is solved using a diffuse interface method which avoids the need to apply the thermal boundary conditions directly at the solid front the thermal gradients at both sides of the interface which are needed to obtain the front velocity are calculated with the aid of a distance function to the reconstructed interface the advection equation of a discretized solid fraction function is solved using the unsplit vof advection method proposed by lopez et al j comput phys 195 2004 718 742 extended to three dimensions by hernandez et al int j numer methods fluids 58 2008 897 921 and the interface curvature is computed using an improved height function technique which provides second order accuracy the proposed methodology is assessed by comparing the numerical results with analytical solutions and with results obtained by different authors for the formation of complex dendritic structures in two and three dimensions

the ls stag method a new immersed boundary level set method for the computation of incompressible viscous flows in complex moving geometries with good conservation properties this paper concerns the development of a new cartesian grid immersed boundary ib method for the computation of incompressible viscous flows in two dimensional irregular geometries in ib methods the computational grid is not aligned with the irregular boundary and of upmost importance for accuracy and stability is the discretization in cells which are cut by the boundary the so called cut cells in this paper we present a new ib method called the ls stag method which is based on the mac method for staggered cartesian grids and where the irregular boundary is sharply represented by its level set function this implicit representation of the immersed boundary enables us to calculate efficiently the geometry parameters of the cut cells we have achieved a novel discretization of the fluxes in the cut cells by enforcing the strict conservation of total mass momentum and kinetic energy at the discrete level our discretization in the cut cells is consistent with the mac discretization used in cartesian fluid cells and has the ability to preserve the five point cartesian structure of the stencil resulting in a highly computationally efficient method the accuracy and robustness of our method is assessed on canonical flows at low to moderate reynolds number taylor couette flow flows past a circular cylinder including the case where the cylinder has forced oscillatory rotations finally we will extend the ls stag method to the handling of moving immersed boundaries and present some results for the transversely oscillating cylinder flow in a free stream

cut element based immersed boundary method for moving geometries in compressible liquid flows with cavitation the conservative immersed interface method for representing complex immersed solid boundaries or phase interfaces on cartesian grids is improved and extended to allow for the simulation of weakly compressible fluid flows through moving geometries we demonstrate that an approximation of moving interfaces by a level set field results in unphysical oscillations in the vicinity of sharp corners when dealing with weakly compressible fluids such as water by introducing an exact reconstruction of the cut cell properties directly based on a surface triangulation of the immersed boundary we are able to recover the correct flow evolution free of numerical artifacts the new method is based on cut elements it provides sub cell resolution of the geometry and handles flows through narrow closing or opening gaps in a straightforward manner we validate our method with canonical flows around oscillating cylinders we demonstrate that the method allows for an accurate prediction of flows around moving obstacles in weakly compressible liquid flows with cavitation effects in particular we show that the cavitating flow through a closing fuel injector control valve which is an example for a complex application with interaction of stationary and moving parts can be predicted by the method

computation of solid liquid phase fronts in the sharp interface limit on fixed grids a finite difference formulation is applied to track solid liquid boundaries on a fixed underlying grid the interface is not of finite thickness but is treated as a discontinuity and is explicitly tracked the imposition of boundary conditions exactly on a sharp interface that passes through the cartesian grid is performed using simple stencil readjustments in the vicinity of the interface attention is paid to formulating difference schemes that are globally second order accurate in x and t error analysis and grid refinement studies are performed for test problems involving the diffusion and convection diffusion equations and for stable solidification problems issues concerned with stability and change of phase of grid points in the evolution of solid liquid phase fronts are also addressed it is demonstrated that the field calculation is second order accurate while the position of the phase front is calculated to first order accuracy furthermore the accuracy estimates hold for the cases where there is a property jump across the interface unstable solidification phenomena are simulated and an attempt is made to compare results with previously published work the results indicate the need to begin an effort to benchmark computations of instability phenomena

numerical approximation of a thermally driven interface using finite elements a two dimensional finite element model for dendritic solidification has been developed that is based on the direct solution of the energy equation over a fixed mesh the model tracks the position of the sharp solid liquid interface using a set of marker points placed on the interface the simulations require calculation of the temperature gradients on both sides of the interface in the direction normal to it at the interface the heat flux is discontinuous due to the release of latent heat during the solidification melting process two ways to calculate the temperature gradients at the interface evaluating their interpolants at gauss points were proposed using known one and two dimensional solutions to stable solidification problems the stefan problem it was shown that the method converges with second order accuracy when applied to the unstable solidification of a crystal into an undercooled liquid it was found that the numerical solution is extremely sensitive to the mesh size and the type of approximation used to calculate the temperature gradients at the interface i e different approximations and different meshes can yield different solutions the cause of these difficulties is examined the effect of different types of interpolation on the simulations is investigated and the necessary criteria to ensure converged solutions are established copyright 2003 john wiley sons ltd

an explicit finite difference scheme for simulation of moving particles we present an explicit finite difference scheme for direct simulation of the motion of solid particles in a fluid the method is based on a second order maccormack finite difference solver for the flow and newton s equations for the particles the fluid is modeled with fully compressible mass and momentum balances the technique is intended to be used at moderate particle reynolds number several examples are shown including a single stationary circular particle in a uniform flow between two moving walls a particle dropped in a stationary fluid at particle reynolds number of 20 the drafting kissing and tumbling of two particles and 100 particles falling in a closed box

on strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method we consider a problem stemming from recent models of phase transitions in stimulus responsive hydrogels wherein a sharp interface separates swelled and collapsed phases extended finite element methods that approximate the local solution with an enriched basis such that the mesh need not explicitly fit the interface geometry are emphasized attention is focused on the weak enforcement of the normal configurational force balance and various options for evaluating the jump in the normal component of the solute flux at the interface we show that as the reciprocal interfacial mobility vanishes it plays the role of a penalty parameter enforcing a pure dirichlet constraint eventually triggering oscillations in the interfacial velocity we also examine alternative formulations employing a lagrange multiplier to enforce this constraint it is shown that the most convenient choice of basis for the lagrange multiplier results in oscillations in the multiplier field and a decrease in accuracy and rate of convergence in local error norms suggesting a lack of stability in the discrete formulation under such conditions neither the direct evaluation of the gradient of the approximation at the phase interface nor the interpretation of the lagrange multiplier field provide a robust means to obtain the jump in the normal component of solute flux fortunately the adaptation and use of local domain integral methodologies considerably improves the flux evaluations several example problems are presented to compare and contrast the various techniques and methods

a singular value decomposition svd based generalized finite difference gfd method for close interaction moving boundary flow problems in this paper we present a study on a singular value decomposition svd based generalized finite difference gfd method and a nodal selection scheme for moving body boundary flow problems formulated on a hybrid cartesian cum meshfree grid system the present study shows that the svd based method is more robust and accurate than the conventional least squares based gfd scheme a nodal selection scheme is also introduced to overcome the problem of numerical instability associated with the clustering of computational nodes such nodal clustering occurs dynamically when moving bodies or boundaries approach within close proximity of each other resulting in the overlap of their meshfree grids the nodal scheme is applied to close interaction flow problems as exemplified by the squeezing action of a circular cylinder through a very narrow slot and the close proximity bypass interaction of two oscillating circular cylinders

a differentially interpolated direct forcing immersed boundary method for predicting incompressible navier stokes equations in time varying complex geometries a dispersion relation preserving dual compact scheme developed in cartesian grids is applied together with the immersed boundary method to solve the flow equations in irregular and time varying domains the artificial momentum forcing term applied at certain points in cells containing fluid and solid allows an imposition of velocity condition to account for the motion of solid body we develop in this study a differential based interpolation scheme which can be easily extended to three dimensional simulation the results simulated from the proposed immersed boundary method agree well with other numerical and experimental results for the chosen benchmark problems the accuracy and fidelity of the ib flow solver developed to predict flows with irregular boundaries are therefore demonstrated

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

efficient symmetric discretization for the poisson heat and stefan type problems with robin boundary conditions we present a novel and efficient method for solving the poisson equation the heat equation and stefan type problems with robin boundary conditions over potentially moving arbitrarily shaped domains the method utilizes a level set framework thus it has all of the benefits of a sharp implicitly represented interface such as the ease of handling complex topological changes this method is straightforward to implement and leads to a linear system that is symmetric and positive definite which can be inverted efficiently with standard iterative methods this approach is second order accurate for both the poisson and heat equations and first order accurate for the stefan problem we demonstrate the accuracy in the l 1 and l norms

multigrid approach to overset grid communication the nonaligned multigrid method is introduced as an alternative to boundary point interpolation for communication in overset grid systems the nonaligned multigrid method treats the individual grids as levels in a multigrid solution procedure this technique was able to converge 69 faster than standard boundary point interpolation for mach 2 flow over a 5 deg ramp and 28 faster for a more complex reflected shock case for these test cases the largest increase in convergence rate came from applying the nonaligned multigrid method to the communication between overset grids further coarsening of the global grid was next in effectiveness and coarsening of local grids gave little improvement

an embedded boundary formulation for large eddy simulation of turbulent flows interacting with moving boundaries a non boundary conforming formulation for simulating complex turbulent flows with dynamically moving boundaries on fixed cartesian grids is proposed the underlying finite difference solver for the filtered incompressible navier stokes equations is based on a second order fractional step method on a staggered grid to satisfy the boundary conditions on an arbitrary immersed interface the velocity field at the grid points near the interface is reconstructed using momentum forcing without smearing the sharp interface the concept of field extension is also introduced to treat the points emerging from a moving solid body to the fluid laminar flow cases and large eddy simulations les are presented to demonstrate the formal accuracy and range of applicability of the method in particular simulations of laminar flow induced by the harmonic in line oscillation of a circular cylinder in quiescent fluid and from a transversely oscillating cylinder in a free stream are presented and compared to reference simulations and experiments les of turbulent flow over a traveling wavy wall and transitional flow through a bileaflet prosthetic heart valve are also shown all results are in very good agreement with reference results in the literature

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

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

numerical simulation of dendritic solidification with convection three dimensional flow a numerical method for the simulation of the effect of melt flow on the three dimensional growth of a dendrite is described the method is an extension of the technique for two dimensional flow described in al rawahi and tryggvason j comput phys 180 2002 471 and is based on the explicit tracking of connected marker points that describe the liquidsolid interface an explicit projection method is used to solve the energy and the navier stokes equations on a regular stationary grid and the solidified region is represented by setting the velocities in the solid phase to zero the latent heat released during solidification is calculated using the normal temperature gradient near the interface the method is validated by a comparison with an exact solution for a stefan problem and a grid refinement study the simulations show that the speed of a dendrite arm growing into the flow is increased due to an increase in the temperature gradient on the upstream side and the formation of side branches is promoted as in two dimensions the effect of the flow on the growth of dendrite arms growing in the downstream direction is smaller than in two dimensions due to a smaller wake

front tracking finite element method for dendritic solidification 765 abstract a two dimensional finite element method capable of tracking sharp interfaces is developed the method is based on a fixed mesh of bilinear isoparametric elements and is extremely simple and easy to use the interfaces are tracked with a set of marker points that define their position at all times several different approaches to finding the interface position and velocity are discussed and their effect on the convergence rate is examined it is shown through numerical examples that the approximation to the interface position converges with second order accuracy in two dimensional problems as opposed to previously developed finite difference algorithms that are only first order accurate the approximation to the interface velocity is shown to be of first order a number of examples are examined including several that provide a careful comparison with previously published results

cartesian grid method for gas kinetic scheme on irregular geometries a cartesian grid method combined with a simplified gas kinetic scheme is presented for subsonic and supersonic viscous flow simulation on complex geometries under the cartesian mesh the boundaries are represented by a set of direction oriented boundary points and the computational grid points are classified into four different categories the fluid point the solid point the drop point and the interpolation point a constrained weighted least square method is employed to evaluate the physical quantities at the interpolation points different boundary conditions including isothermal boundary adiabatic boundary and euler slip boundary are presented by different interpolation strategies we adopt a simplified gas kinetic scheme as the flux solver for both subsonic and supersonic flow computations the methodology of constructing a simplified kinetic flux function can be extended to other flow systems a few numerical examples are used to validate the cartesian grid method and the simplified flux solver the reconstruction scheme for recovering the boundary conditions of compressible viscous and heat conducting flow with a cartesian mesh can provide a smooth distribution of physical quantities at solid boundary and present an accurate solution for the flow study with complex geometry

sharp interface cartesian grid method i an easily implemented technique for 3d moving boundary computations a cartesian grid method is developed for the simulation of incompressible flows around stationary and moving three dimensional immersed boundaries the embedded boundaries are represented using level sets and treated in a sharp manner without the use of source terms to represent boundary effects the narrow band distance function field in the level set boundary representation facilitates implementation of the finite difference flow solver the resulting algorithm is implemented in a straightforward manner in three dimensions and retains global second order accuracy the accuracy of the finite difference scheme is established and shown to be comparable to finite volume schemes that are considerably more difficult to implement moving boundaries are handled naturally the pressure solver is accelerated using an algebraic multigrid technique adapted to be effective in the presence of moving embedded boundaries benchmarking of the method is performed against available numerical as well as experimental results

regular article an accurate cartesian grid method for viscous incompressible flows with complex immersed boundaries a cartesian grid method has been developed for simulating two dimensional unsteady viscous incompressible flows with complex immersed boundaries a finite volume method based on a second order accurate central difference scheme is used in conjunction with a two step fractional step procedure the key aspects that need to be considered in developing such a solver are imposition of boundary conditions on the immersed boundaries and accurate discretization of the governing equation in cells that are cut by these boundaries a new interpolation procedure is presented which allows systematic development of a spatial discretization scheme that preserves the second order spatial accuracy of the underlying solver the presence of immersed boundaries alters the conditioning of the linear operators and this can slow down the iterative solution of these equations the convergence is accelerated by using a preconditioned conjugate gradient method where the preconditioner takes advantage of the structured nature of the underlying mesh the accuracy and fidelity of the solver is validated by simulating a number of canonical flows and the ability of the solver to simulate flows with very complicated immersed boundaries is demonstrated

ripple a new model for incompressible flows with free surfaces a new free surface flow model ripple is summarised ripple obtains finite difference solutions for incompressible flow problems having significant surface tension forces at free surfaces of arbitrarily complex topology the key innovation is the continuum surface force model which represents surface tension as a localised volume force other features include a higher order momentum advection model a volume of fluid free surface treatment and an efficient two step projection solution method ripple s unique capabilities are illustrated with two example problems low gravity jet induced tank flow and the collision and coalescence of two cylindrical rods

multiphase dynamics in arbitrary geometries on fixed cartesian grids in this work a mixed eulerian lagrangian algorithm calledelafint eulerian lagrangian algorithm for interface tracking is developed further and applied to compute flows with solid fluid and fluid fluid interfaces the method is capable of handling fluid flows in the presence of both irregularly shaped solid boundaries and moving boundaries on a fixed cartesian grid the field equations are solved on the underlying fixed grid using a collocated variable pressure based formulation the moving boundary is tracked explicitly by the lagrangian translation of marker particles the moving boundary passes through the grid and the immersed boundary technique is used to handle its interaction with the underlying grid the internal solid boundaries are dealt with by using a cut cell technique particular attention is directed toward conservation and consistency in the vicinity of both phase boundaries the complex geometry feature has been tested for a variety of flow problems the performance of the immersed boundary representation is demonstrated in the simulation of newtonian liquid drops the combination of the two features is then employed in the simulation of motion of drops through constricted tubes the capabilities developed here can be useful for solving flow problems involving moving and stationary complex boundaries

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 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

a complete optimality condition in the inverse problem of optimal control a complete optimality condition in the standard inverse problem of optimal control for the multi input case is established the optimality of a feedback control law is characterized completely in terms of a new geometric condition as well as the well known return difference condition the new condition not only provides a better insight into the well known sensitivity reduction property of optimal control for the multi input case but also indicates an essential difference between the solutions of the single and multi input inverse problem

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

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

the embedded finite difference method for the poisson equation in a domain with an irregular boundary and dirichlet boundary conditions the poisson equation subject to dirichlet boundary conditions on an irregular domain can be treated by embedding the region in a rectangular domain and solving using finite differences over the rectangle the crucial issue is the discretization of the boundaries of the irregular domain in the past both linear and quadratic boundary treatments have been used and error bounds have been derived in both cases showing that the linear case gives uniform second order accuracy whereas the quadratic case gives third order accuracy at the boundaries and second order accuracy internally thus it has been recommended that the linear boundary treatment be used as it is simpler gives rise to a symmetric matrix formulation and has uniform accuracy the present work shows that this argument is inadequate because the coefficients of the error terms also play an important role we demonstrate this in the 1 d case by determining explicit expressions for the error for both the linear and quadratic boundary treatments it is shown that for the linear case the coefficient of error is in general large enough to dominate the calculation and that therefore it is necessary to use a quadratic boundary treatment in order to obtain errors comparable with those obtained for a regular domain we go on to show that the 1 d expressions for error can be used to approximate the boundary error for 2 d problems and that for the linear treatment the boundary error again dominates

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

computation of three dimensional dendrites with finite elements starting from an initial seed crystal inside an undercooled liquid the solid phase begins to grow rapidly and develops unstable growth patterns some growth directions are preferred because of anisotropic parameters in the physical model this results in the development of dendrites the physical model includes the heat equation for both the liquid and solid phases the gibbs thomson law couples velocity and curvature of the interface and the temperature we describe a numerical method that enables us to compute dendritic growth of crystals in two and three space dimensions the method consists of two coupled finite element algorithms the first one solves the heat equation the other operates on a discretization of the free boundary and computes the evolution of this moving interface the two methods work with totally independent grids by using timedependent locally refined and coarsened adaptive meshes in both methods we are able to reach a spatial resolution necessary to compute dendritic growth in two and three space dimensions

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

a fixed grid sharp interface method for bubble dynamics and phase change abstract a numerical method has been developed for direct simulation of bubble dynamics with large liquid to vapor density ratio and phase change the numerical techniques are based on a fixed grid finite volume method capable of treating the interface as a sharp discontinuity the unsteady axisymmetric navier stokes equations and energy equation in both liquid and vapor phases are computed the mass momentum and energy conditions are explicitly matched at the phase boundary to determine the interface shape and movement the cubic b spline is used in conjunction with a fairing algorithm to yield smooth and accurate information of curvatures nondimensional parameters including reynolds weber and jakob numbers are varied to offer insight into the physical and numerical characteristics of the bubble dynamics based on the present sharp interface approach bubble dynamics for density ratio of 1600 or higher with and without phase change can be successfully computed