a reconstruction based chebyshev collocation method for the poisson equation an accurate treatment of the gibbs wilbraham phenomenon on irregular interfaces abstract an accurate numerical solution of the laplace equation and poisson equation is desirable in physical problems related to fluid flows and spectral methods are well suited towards this goal however in the presence of an interface the solution is discontinuous owing to the interface jump conditions and thus the numerical approximation is susceptible to the gibbs wilbraham phenomenon leading to a reduced solution accuracy in order to resolve the discontinuity at the interface we propose a reconstruction technique wherein the approximate solution is expressed as the sum of an infinitely differentiable smooth function and a modified heaviside function the smooth function is composed of chebyshev polynomials while the modified heaviside function is expressed in a weak form using the jump conditions across the interface the reconstruction framework allows us to impose the interfacial conditions exactly to motivate further application of the technique we first present the underlying rationale by revisiting the work of wilbraham 1848 13 followed by the numerical implementation we present numerical solution of four equations to illustrate the application of the proposed technique i an ordinary differential equation ode ii the euler bernoulli equation iii the laplace equation and iv the poisson equation the laplace and poisson equation are solved on a square domain embedded with complex interfacial geometries in two dimensions and on a cube with a spherical interface in three dimensions the method achieves error in the maximum norm o 10 14 using approximately 30 modes in each direction and requires a computational time of less than a second for all the problems we also conduct a sensitivity analysis to demonstrate the robustness of the algorithm under sub grid scale perturbations to interface location an alternate derivation for the modified heaviside function is proposed that is distinct from the taylor series expansion used to treat interface conditions in difference methods the pointwise error plots are subsequently presented to demonstrate that the gibbs wilbraham phenomenon is accurately resolved finally we present concluding thoughts on the potential applications of the reconstruction technique

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

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

high order methods for low reynolds number flows around moving obstacles based on universal meshes summary we propose a family of methods for simulating two dimensional incompressible low reynolds number flow around a moving obstacle whose motion is prescribed the methods make use of a universal mesh a fixed background mesh that adapts to the geometry of the immersed obstacle at all times by adjusting a few elements in the neighborhood of the obstacle s boundary the resulting mesh provides a conforming triangulation of the fluid domain over which discretizations of any desired order of accuracy in space and time can be constructed using standard finite element spaces together with off the shelf time integrators we demonstrate the approach by using taylor hood elements to approximate the fluid velocity and pressure to integrate in time we consider implicit runge kutta schemes as well as a fractional step scheme we illustrate the methods and study their convergence numerically via examples that involve flow around obstacles that undergo prescribed deformations copyright 2015 john wiley sons ltd

immersed boundary and overset grid methods assessed for stokes flow due to an oscillating sphere abstract a systematic comparison of three types of immersed boundary methods and an overset grid method is performed for incompressible stokes flow due to an oscillating sphere for each simulation the error between the numerical solution and the analytical solution is reported in terms of the maximum and l2 norms of the errors in the velocity the pressure and the velocity gradient fields the three types of immersed boundary methods are a a diffuse interface immersed boundary method ibm b a sharp interface ibm with standard treatment of the pressure and c a sharp interface ibm with pressure decoupling at interfaces upon grid refinement the velocity gradient and pressure fields generated by methods a and b do not converge to the analytical solution in the maximum norm while the total force on the sphere does converge an inspection of the norms as function of the distance to the surface shows that the lack of convergence mainly occurs in the first layer of grid cells outside the sphere if the pressure poisson equation is not solved inside the sphere method c the velocity gradient and pressure seem to converge in the maximum norm however high resolution is required to achieve a smaller maximum error in the pressure in method c than in methods a and b overall method c is somewhat more accurate than method b and the latter is somewhat more accurate than method a in this assessment the overset grid method clearly outperforms the ibms and is able to produce errors down to one percent in the maximum norm at 32 grid points per sphere diameter within each of the four classes of methods several variants are tested for the overset grid method and in several sharp interface ibms nonincremental projection is found to produce a more accurate pressure than incremental projection does finally variants of method c and the overset grid method are proposed in which the projection method is replaced by an artificial compressibility method while the staggered grid is maintained this simplifies the methods and it strongly reduces the maximum error in the pressure in case c at the expense of an increased maximum error in the velocity divergence

accurate computation of surface stresses and forces with immersed boundary methods many immersed boundary methods solve for surface stresses that impose the velocity boundary conditions on an immersed body these surface stresses may contain spurious oscillations that make them ill suited for representing the physical surface stresses on the body moreover these inaccurate stresses often lead to unphysical oscillations in the history of integrated surface forces such as the coefficient of lift while the errors in the surface stresses and forces do not necessarily affect the convergence of the velocity field it is desirable especially in fluid structure interaction problems to obtain smooth and convergent stress distributions on the surface to this end we show that the equation for the surface stresses is an integral equation of the first kind whose ill posedness is the source of spurious oscillations in the stresses we also demonstrate that for sufficiently smooth delta functions the oscillations may be filtered out to obtain physically accurate surface stresses the filtering is applied as a post processing procedure so that the convergence of the velocity field is unaffected we demonstrate the efficacy of the method by computing stresses and forces that converge to the physical stresses and forces for several test problems

an unfitted discontinuous galerkin method applied to elliptic interface problems in this article an unfitted discontinuous galerkin method is proposed to discretize elliptic interface problems the method is based on the symmetric interior penalty discontinuous galerkin method and can also be interpreted as a generalization of the method given by a hansbo and p hansbo comput methods appl mech engrg 191 2002 pp 5537 5552 we prove the optimal h convergence of the method for arbitrary p in the energy and in the l 2 norm in fact we present an hp error estimate we introduce a variant of the method for which the same hp convergence rates can be proved in the energy norm and which aims at controlling pointwise errors near the interface the analysis includes grids with hanging nodes and the proposed method is symmetric and inherits the attractive locality property of general discontinuous galerkin methods numerical experiments are included

application of a fractional step method to incompressible navier stokes equations a numerical method for computing three dimensional time dependent incompressible flows is presented the method is based on a fractional step or time splitting scheme in conjunction with the approximate factorization technique it is shown that the use of velocity boundary conditions for the intermediate velocity field can lead to inconsistent numerical solutions appropriate boundary conditions for the intermediate velocity field are derived and tested numerical solutions for flows inside a driven cavity and over a backward facing step are presented and compared with experimental data and other numerical results

accuracy improvement for immersed boundary method using lagrangian velocity interpolation abstract the immersed boundary method ibm has been extensively utilized in fluid dynamics simulations with various numerical methods there are two key steps in ibm namely the force distribution and velocity interpolation steps and the kernel function plays important roles in both steps the purpose for the velocity interpolation is to enhance the no slip condition between fluid and boundary surface however with the kernel function designed to satisfy the basic requirements for the force distribution the boundary velocity calculation process in traditional ibm simulations is actually a weighted averaging operation instead of an interpolation this induces large errors in the calculated boundary velocity for general nonlinear velocity distributions in this paper we propose to release the kernel function s role from the boundary velocity calculation and replace it with the classical lagrangian interpolation schemes to obtain the boundary velocity from the local flow field this idea has been tested in several two dimensional periodic systems and both flat and curved surfaces and steady and unsteady flows have been considered our simulation results show that in general a 20 30 accuracy improvement in velocity and flow structure interaction can be achieved with no extra cost in computational complexity and efficiency thanks to the simple mathematical formulation and computational algorithm

systematic derivation of jump conditions for the immersed interface method in three dimensional flow simulation in this paper we systematically derive jump conditions for the immersed interface method siam j numer anal 31 1994 pp 1019 1044 siam j sci comput 18 1997 pp 709 735 to simulate three dimensional incompressible viscous flows subject to moving surfaces the surfaces are represented as singular forces in the navier stokes equations which give rise to discontinuities of flow quantities the principal jump conditions across a closed surface of the velocity the pressure and their normal derivatives have been derived by lai and li appl math lett 14 2001 pp 149 154 in this paper we first extend their derivation to generalized surface parametrization starting from the principal jump conditions we then derive the jump conditions of all first second and third order spatial derivatives of the velocity and the pressure we also derive the jump conditions of first and second order temporal derivatives of the velocity using these jump conditions the immersed interface method is applicable to the simulation of three dimensional incompressible viscous flows subject to moving surfaces where near the surfaces the first and second order spatial derivatives of the velocity and the pressure can be discretized with respectively third and second order accuracy and the first order temporal derivatives of the velocity can be discretized with second order accuracy

an interface capturing method with a continuous function the thinc method with multi dimensional reconstruction an interface capturing method with a continuous function is proposed within the framework of the volume of fluid vof method being different from the traditional vof methods that require a geometrical reconstruction and identify the interface by a discontinuous heaviside function the present method makes use of the hyperbolic tangent function known as one of the sigmoid type functions in the tangent of hyperbola interface capturing thinc method f xiao y honma k kono a simple algebraic interface capturing scheme using hyperbolic tangent function int j numer methods fluids 48 2005 1023 1040 to retrieve the interface in an algebraic way from the volume fraction data of multi component materials instead of the 1d reconstruction in the original thinc method a multi dimensional hyperbolic tangent function is employed in the present new approach the present scheme resolves moving interface with geometric faithfulness and compact thickness and has at least the following advantages 1 the geometric reconstruction is not required in constructing piecewise approximate functions 2 besides a piecewise linear interface curved quadratic surface can be easily constructed as well and 3 the continuous multi dimensional hyperbolic tangent function allows the direct calculations of derivatives and normal vectors numerical benchmark tests including transport of moving interface and incompressible interfacial flows are presented to validate the numerical accuracy for interface capturing and to show the capability for practical problems such as a stationary circular droplet a drop oscillation a shear induced drop deformation and a rising bubble

an accurate and efficient method for the incompressible navier stokes equations using the projection method as a preconditioner the projection method is a widely used fractional step algorithm for solving the incompressible navier stokes equations despite numerous improvements to the methodology however imposing physical boundary conditions with projection based fluid solvers remains difficult and obtaining high order accuracy may not be possible for some choices of boundary conditions in this work we present an unsplit linearly implicit discretization of the incompressible navier stokes equations on a staggered grid along with an efficient solution method for the resulting system of linear equations since our scheme is not a fractional step algorithm it is straightforward to specify general physical boundary conditions accurately however this capability comes at the price of having to solve the time dependent incompressible stokes equations at each timestep to solve this linear system efficiently we employ a krylov subspace method preconditioned by the projection method in our implementation the subdomain solvers required by the projection preconditioner employ the conjugate gradient method with geometric multigrid preconditioning the accuracy of the scheme is demonstrated for several problems including forced and unforced analytic test cases and lid driven cavity flows these tests consider a variety of physical boundary conditions with reynolds numbers ranging from 1 to 30000 the effectiveness of the projection preconditioner is compared to an alternative preconditioning strategy based on an approximation to the schur complement for the time dependent incompressible stokes operator the projection method is found to be a more efficient preconditioner in most cases considered in the present work

on error estimates of the projection methods for the navier stokes equations second order schemes we present in this paper a rigorous error analysis of several projection schemes for the approximation of the unsteady incompressible navier stokes equations the error analysis is accomplished by interpreting the respective projection schemes as second order time discretizations of a perturbed system which approximates the navier stokes equations numerical results in agreement with the error analysis are also presented

a numerical method for solving incompressible flow problems with a surface of discontinuity a numerical method for solving problems in which a moving surface of discontinuity separates regions of incompressible flow is presented the method developed is notable in that it does not introduce any artificial smoothing of the change in fluid properties across the surface of discontinuity this results in an increase in accuracy relative to methods which introduce smoothing effects the method was also shown to be fairly versatile problems describing a free surface an immiscible fluid interface and a premixed flame discontinuity were solved there is a limitation however in that the method appears to be most suitable for application to inviscid problems the reason for this limitation and possible approaches toward resolving it are discussed

sharp interface cartesian grid method ii a technique for simulating droplet interactions with surfaces of arbitrary shape a fixed grid sharp interface method is developed to simulate droplet impact and spreading on surfaces of arbitrary shape a finite difference technique is used to discretize the incompressible navier stokes equations on a cartesian grid to compute flow around embedded solid boundaries a previously developed sharp interface method for solid immersed boundaries is used the ghost fluid method gfm is used for fluid fluid interfaces the model accounts for the effects of discontinuities such as density and viscosity jumps and singular sources such as surface tension in both bubble and droplet simulations with a level set representation of the propagating interface large deformations of the boundary can be handled easily the model successfully captures the essential features of interactions between fluid fluid and solid fluid phases during impact and spreading moving contact lines are modeled with contact angle hysteresis and contact line motion on non planar surfaces is computed experimental observations and other simulation results are used to validate the calculations

a matched interface and boundary method for solving multi flow navier stokes equations with applications to geodynamics we have developed a second order numerical method based on the matched interface and boundary mib approach to solve the navier stokes equations with discontinuous viscosity and density on non staggered cartesian grids we have derived for the first time the interface conditions for the intermediate velocity field and the pressure potential function that are introduced in the projection method differentiation of the velocity components on stencils across the interface is aided by the coupled fictitious velocity values whose representations are solved by using the coupled velocity interface conditions these fictitious values and the non staggered grid allow a convenient and accurate approximation of the pressure and potential jump conditions a compact finite difference method was adopted to explicitly compute the pressure derivatives at regular nodes to avoid the pressure velocity decoupling numerical experiments verified the desired accuracy of the numerical method applications to geophysical problems demonstrated that the sharp pressure jumps on the clast newtonian matrix are accurately captured for various shear conditions moderate viscosity contrasts and a wide range of density contrasts we showed that large transfer errors will be introduced to the jumps of the pressure and the potential function in case of a large absolute difference of the viscosity across the interface these errors will cause simulations to become unstable

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

xgfm recovering convergence of fluxes in the ghost fluid method abstract we introduce a simple but effective correction to the ghost fluid method gfm introduced by liu et al 2000 30 for capturing sharp interface conditions when solving poisson equations with discontinuities our method is shown to alleviate the gfm lack of convergence for the flux across the interfaces pointwise convergence of the gradient is recovered by ensuring consistent jump conditions with an iterative process compared to the gfm the discretization stencil and the symmetric positive definiteness of the resulting linear system are not modified only the right hand side of the linear system is affected we first show that the original gfm is consistent with the second order accurate symmetric scheme of gibou et al 2002 17 for dirichlet conditions in separate regions of the computational domain where the parameters and the solution are smooth but with matching conditions at the interface that are not fully consistent in general our proposed method recovers consistency and is shown to produce convergent gradients through several numerical examples in two and three spatial dimensions

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

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

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

projection method i convergence and numerical boundary layers this is the first of a series of papers on the subject of projection methods for viscous incompressible flow calculations the purpose of these papers is to provide a thorough understanding of the

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

analysis of finite difference schemes for unsteady navier stokes equations in vorticity formulation in this paper we provide stability and convergence analysis for a class of finite difference schemes for unsteady incompressible navier stokes equations in vorticity stream function formulation the no slip boundary condition for the velocity is converted into local vorticity boundary conditions thom s formula wilkes formula or other local formulas in the earlier literature can be used in the second order method while high order formulas such as briley s formula can be used in the fourth order compact difference scheme proposed by e and liu the stability analysis of these long stencil formulas cannot be directly derived from straightforward manipulations since more than one interior point is involved in the formula the main idea of the stability analysis is to control local terms by global quantities via discrete elliptic regularity for stream function we choose to analyze the second order scheme with wilkes formula in detail in this case we can avoid the complicated technique necessitated by the strang type high order expansions as a consequence our analysis results in almost optimal regularity assumption for the exact solution the above methodology is very general we also give a detailed analysis for the fourth order scheme using a 1 d stokes model

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

the immersed interface multigrid methods for interface problems new multigrid methods are developed for the maximum principle preserving immersed interface method applied to second order linear elliptic and parabolic pdes that involve interfaces and discontinuities for elliptic interface problems the multigrid solver developed in this paper works while some other multigrid solvers do not for linear parabolic equations we have developed the second order maximum principle preserving finite difference scheme in this paper we use the crank nicolson scheme to deal with the diffusion part and an explicit scheme for the first order derivatives numerical examples are also presented

smoothing properties of implicit finite difference methods for a diffusion equation in maximum norm we prove a regularity property of finite difference schemes for the heat or diffusion equation u t delta u in maximum norm with large time steps for a class of time discretizations including l stable single step methods and the second order backward difference formula with the usual second order laplacian we show that solutions of the scheme gain first spatial differences boundedly and also second differences except for logarithmic factors with respect to nonhomogeneous terms a weaker property is shown for the crank nicolson method as a consequence we show that the numerical solution of a convection diffusion equation with an interface can allow o h truncation error near the interface and still have a solution with uniform o h 2 accuracy and first differences of uniform accuracy almost o h 2

a coupling interface method for elliptic interface problems we propose a coupling interface method cim under cartesian grid for solving elliptic complex interface problems in arbitrary dimensions where the coefficients the source terms and the solutions may be discontinuous or singular across the interfaces it consists of a first order version cim1 and a second order version cim2 in one dimension the cim1 is derived from a linear approximation on both sides of the interface the method is extended to high dimensions through a dimension by dimension approach to connect information from each dimension a coupled equation for the first order derivatives is derived through the jump conditions in each coordinate direction the resulting stencil uses the standard 5 grid points in two dimensions and 7 grid points in three dimensions similarly the cim2 is derived from a quadratic approximation in each dimension in high dimensions a coupled equation for the principal second order derivatives u x k x k is derived through the jump conditions in each coordinate direction the cross derivatives are approximated by one side interpolation this approach reduces the number of grid points needed for one side interpolation the resulting stencil involves 8 grid points in two dimensions and 12 14 grid points in three dimensions a numerical study for the condition number of the resulting linear system of the cim2 in one dimension has been performed it is shown that the condition number has the same behavior as that of the discrete laplacian independent of the relative location of the interface in a grid cell further we also give a proof of the solvability of the coupling equations provided the curvature k of the interface satisfies kh

a cartesian based embedded geometry technique with adaptive high order finite differences for compressible flow around complex geometries an immersed boundary methodology to solve the compressible navier stokes equations around complex geometries in cartesian fluid dynamics solvers is described the objective of the new approach is to enable smooth reconstruction of pressure and viscous stresses around the embedded objects without spurious numerical artifacts a standard level set represents the boundary of the object and defines a fictitious domain into which the flow fields are smoothly extended boundary conditions on the surface are enforced by an approach inspired by analytic continuation each fluid field is extended independently constrained only by the boundary condition associated with that field unlike most existing methods no jump conditions or explicit derivation of them from the boundary conditions are required in this approach numerical stiffness that arises when the fluid solid interface is close to grid points of the mesh is addressed by preconditioning in addition the embedded geometry technique is coupled with a stable high order adaptive discretization that is enabled around the object boundary to enhance resolution the stencils used to transition the order of accuracy of the discretization are derived using the summation by parts technique that ensures stability applications to shock reflections shock ramp interactions and supersonic and low mach number flows over two and three dimensional geometries are presented

a second order projection method for the incompressible navier stokes equations in this paper we describe a second order projection method for the time dependent incom pressible navier stokes equations as in the original projection method developed by chorin we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields by introducing more coupling between the diffusion onvection step and the projection step we obtain a temporal discretiza tion that is second order accurate our treatment of the diffusion convection step uses a specialized higher order godunov method for differencing the nonlinear convective terms that provides a robust treatment of these terms at high reynolds number the godunov procedure is second order accurate for smooth flow and remains stable for discontinuous initial data even in the zero viscosity limit we approximate the projection directly using a galerkin procedure that uses a local basis for discretely divergence free vector fields numerical results are presented validating the convergence properties of the method we also apply the method to doubly periodic shear layers to assess the performance of the method on more difficult

development of an immersed boundary phase field lattice boltzmann method for neumann boundary condition to study contact line dynamics the implementation of neumann boundary condition in the framework of immersed boundary method ibm is presented in this paper to simulate contact line dynamics using a phase field lattice boltzmann method immersed boundary method 10 is known as an efficient algorithm for modelling fluid solid interaction abundance of prominent works have been devoted to refine ibm 1 11 12 however they are mainly restricted to problems with dirichlet boundary condition research that implements the neumann boundary condition in ibm is very limited to the best of our knowledge this deficiency significantly limits the application of ibm in computational fluid dynamics cfd since physical phenomena associated with neumann boundary conditions are extremely diverse the difficulty is attributed to the fact that implementation of neumann boundary condition is much more complex than that of dirichlet boundary condition in the present work we initiate the first endeavour to implement neumann boundary condition in ibm with assistance of its physical interpretation rather than simple mathematical manipulation concretely speaking rooted from physical conservation law the neumann boundary condition is considered as contribution of flux from the boundary to its relevant physical parameter in a control volume moreover the link between the flux and its corresponding flow field variable is directly manipulated through the immersed boundary concept in this way the neumann boundary conditions can be implemented in ibm the developed method is applied together with phase field lattice boltzmann method to study contact line dynamics the phase field method 27 39 which becomes increasingly popular in multiphase flow simulation can efficiently capture complex interface topology and naturally resolve the contact line singularity meanwhile the lattice boltzmann method is known as an alternative to model fluid dynamics and holds good prospect to simulate multiphase flows with complex geometry 38 in this context the developed immersed boundary phase field lbm is verified in detail for both steady and unsteady contact line problems tests show that the proposed method can correctly reproduce both equilibrium results and dynamic process

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

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

maximum principle preserving schemes for interface problems with discontinuous coefficients new finite difference methods using cartesian grids are developed for elliptic interface problems with variable discontinuous coefficients singular sources and nonsmooth or even discontinuous solutions the new finite difference schemes are constructed to satisfy the sign property of the discrete maximum principle using quadratic optimization techniques the methods are shown to converge under certain conditions using comparison functions the coefficient matrix of the resulting linear system of equations is an m matrix and is coupled with a multigrid solver numerical examples are also provided to show the efficiency of the proposed methods

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

a numerical method for suspension flow abstract peskin s j comput phys 25 220 1977 immersed boundary technique is modified to give a new numerical method for studying a fluid with suspended elastic particles as before the presence of the suspended particles is transmitted to the fluid through a force density term in the fluid equations as a result one set of equations holds in the entire computational domain eliminating the need to apply boundary conditions on the surface of suspended objects the new method computes the force density by discretizing the stress strain constitutive equations for an elastic solid on a grid using data provided by clusters of lagrangian points this approach clearly specifies the material properties of the suspended objects a simple data structure for the lagrangian points makes it easy to model suspended solids with arbitrary shape and size the method is validated by comparing numerical results for elastic vibrations and particle settling in viscous fluids with theory and analysis the capability of the method to do a wide range of problems is illustrated by qualitative results for lubrication and cavity flow problems

travelling wave solutions for the 2 1 dimensional nizhnik novikov veselov equation abstract using an auxiliary ordinary differential equation method we have obtained many exact travelling wave solutions for the 2 1 dimensional nizhnik novikov veselov equation which is considered as a model for an incompressible fluid the key of this method is to construct appropriate travelling wave functions in terms of solving a system of nonlinear algebraic equations the method is also extended to the case of other nonlinear partial differential equations with solitary wave solutions triangular periodic wave solutions and elliptic function solutions et al

a numerical study of electro migration voiding by evolving level set functions on a fixed cartesian grid a numerical method for studying migration of voids driven by surface diffusion and electric current in a metal conducting line is developed the mathematical model involves moving boundaries governed by a fourth order nonlinear partial differential equation which contains a nonlocal term corresponding to the electrical field and a nonlinear term corresponding to the curvature numerical challenges include efficient computation of the electrical field with sufficient accuracy to afford fourth order differentiation along the void boundary and to capture singularities arising in topological changes we use the modified immersed interface method with a fixed cartesian grid to solve for the electrical field and the fast local level set method to update the position of moving voids numerical examples are performed to demonstrate the physical mechanisms by which voids interact under electromigration

an immersed interface method for a 1d poroelasticity problem with discontinuous coefficients we introduce an immersed interface method iim based on a staggered grid for the 1d biot s poroelasticity model when the coefficients have discontinuities along material interfaces the iim uses a standard finite difference method away from the interfaces and modifies the numerical schemes near or on the interfaces to treat the irregularities using the method of undetermined coefficients a second order accuracy in the infinity norm has been derived for both pressure and displacement regardless of the position of the interfaces we present some numerical results to confirm the theoretical error estimates

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

the 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

a fictitious domain distributed lagrange multiplier based fluid structure interaction scheme with hierarchical b spline grids abstract we present a numerical scheme for fluid structure interaction based on hierarchical b spline grids and fictitious domain distributed lagrange multipliers the incompressible navier stokes equations are solved over a cartesian grid discretised with b splines the fluid grid near the immersed solids is refined locally using hierarchical b splines the immersed solid is modelled as geometrically exact beam discretised with standard linear lagrange shape functions the kinematic constraint at the fluid solid interface is enforced with distributed lagrange multipliers the unconditionally stable and second order accurate generalised method is used for integration in time for both the fluid and solid domains a fully implicit and fully coupled solution scheme is developed by using the newton raphson method to solve the non linear system of equations obtained with galerkin weak formulation first the spatial and temporal convergence of the proposed scheme is assessed by studying steady and unsteady flow past a fixed cylinder then the scheme is applied to several benchmark problems to demonstrate the efficiency and robustness of the proposed scheme the results obtained with the present scheme are compared with the reference values

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

fourth order accurate evaluation of integrals in potential theory on exterior 3d regions we present two methods for the rapid high order accurate evaluation of integrals in potential theory on general unbounded 3d regions our methods allow for direct calculation of derivatives of the integrals as well one of the methods uses a fourth order compact stencil and the other uses a nonstandard variant of richardson extrapolation both methods involve calculation of discontinuities in high order derivatives of the integrals across the boundary of the integration region the extrapolation method in addition involves correction for the discontinuities in truncation error the number of operations required for the methods is essentially equal to twice the number of operations needed to solve poisson s equation on a regular grid both methods avoid problems associated with using quadrature methods to evaluate integrals with singular kernels numerical results are presented for experiments on a variety of geometries in free space

intertwiner realization of a simple non standard r matrix using jimbo s method we realize the non standard r matrix r12 1 2 x y as an intertwiner between non generic irreducible representations of the quantum affine algebra uq sl2

a fast iterative algorithm for elliptic interface problems a fast second order accurate iterative method is proposed for the elliptic equation grad cdot beta x y grad u f x y in a rectangular region omega in two space dimensions we assume that there is an irregular interface across which the coefficient beta the solution u and its derivatives and or the source term f may have jumps we are especially interested in the cases where the coefficients beta are piecewise constant and the jump in beta is large the interface may or may not align with an underlying cartesian grid the idea in our approach is to precondition the differential equation before applying the immersed interface method proposed by leveque and li siam j numer anal 4 1994 pp 1019 1044 in order to take advantage of fast poisson solvers on a rectangular region an intermediate unknown function the jump in the normal derivative across the interface is introduced our discretization is equivalent to using a second order difference scheme for a corresponding poisson equation in the region and a second order discretization for a neumann like interface condition thus second order accuracy is guaranteed a gmres iteration is employed to solve the schur complement system derived from the discretization a new weighted least squares method is also proposed to approximate interface quantities from a grid function numerical experiments are provided and analyzed the number of iterations in solving the schur complement system appears to be independent of both the jump in the coefficient and the mesh size

the blob projection method for immersed boundary problems a new finite difference numerical method for modeling the interaction between flexible elastic membranes and an incompressible fluid in a two dimensional domain is presented the method differs from existing methods in the way the forces exerted by the membranes on the fluid are modeled these are described by a collection of regularized point forces and the velocity field they induce is computed directly on a regular cartesian grid via a smoothed dipole potential comparisons between this method and the immersed boundary method of peskin and mcqueen are presented the results show that the method proposed here preserves volumes better and has a higher order of convergence

hypam a hybrid continuum particle model for incompressible free surface flows three major issues associated with numerical simulations of complex free surface flows viz interface tracking fragmentation and large physical jumps are addressed by a new hybrid continuum particle model hypam the new model consists of three parts 1 the polygonal area mapping method q zhang p l f liu a new interface tracking method the polygonal area mapping method j comput phys 227 8 2008 4063 4088 2 a new algorithm that decomposes the interested water phase into a continuum zone a buffer zone and a particle zone based on material topology and graph theory 3 a passive response assumption in which the air phase is assumed to respond passively to the continuum part of the water phase the incompressible inviscid euler equations and the equations describing the free fall of rigid bodies are used as the governing equations for the continuum buffer zone and the particle zone respectively and separately a number of examples including water droplet impact solitary wave propagation and dam break problems are simulated for the illustration and validation of hypam it is shown that hypam is more accurate and versatile than a continuum based volume of fluid model one major contribution of this work is the single phase decomposition algorithm useful for many other hybrid formulations neglecting surface tension viscosity and particle interactions hypam is currently limited to mildly fragmented free surface flows with high reynolds and weber numbers

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

numerical approximation of internal discontinuity interface problems this work focuses on the finite element discretization of boundary value problems whose solution features either a discontinuity or a discontinuous conormal derivative across an interface inside the computational domain the interface is characterized via a level set function the discontinuities are accounted for by using suitable extension operators whose numerical implementation requires a very low computational effort after carrying out the error analysis numerical results to validate our approach are presented in one two and three dimensions

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

a boundary condition capturing method for poisson s equation on irregular domains interfaces have a variety of boundary conditions or jump conditions that need to be enforced the ghost fluid method gfm was developed to capture the boundary conditions at a contact discontinuity in the inviscid euler equations and has been extended to treat more general discontinuities such as shocks detonations and deflagrations and compressible viscous flows in this paper a similar boundary condition capturing approach is used to develop a new numerical method for the variable coefficient poisson equation in the presence of interfaces where both the variable coefficients and the solution itself may be discontinuous this new method is robust and easy to implement even in three spatial dimensions furthermore the coefficient matrix of the associated linear system is the standard symmetric matrix for the variable coefficient poisson equation in the absence of interfaces allowing for straightforward application of standard black box solvers

continuity constrained least squares interpolation for sfo suppression in immersed boundary methods a new immersed boundary interpolation for discrete forcing methods is presented the method decreases spurious oscillations in the pressure field and consequently in the body force calculations which are a common issue in several immersed boundary methods the method applies a continuity constraint in the least squares interpolation and guarantees that adjacent interpolation polynomials are continuous between each other this approach strictly enforces continuity in the flow reconstruction domain reducing time discontinuities caused by the boundary conditions applied at the immersed boundary several moving body problems test cases are simulated including a three dimensional one to demonstrate the new method s capability of computing stable pressure fields even for coarse grids and small cfl numbers which are known to increase the pressure oscillations

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

a consistent direct discretization scheme on cartesian grids for convective and conjugate heat transfer a new discretization scheme on cartesian grids namely a consistent direct discretization scheme is proposed for solving incompressible flows with convective and conjugate heat transfer around a solid object the navier stokes and the pressure poisson equations are discretized directly even in the immediate vicinity of a solid boundary with the aid of the consistency between the face velocity and the pressure gradient from verifications in fundamental flow problems the present method is found to significantly improve the accuracy of the velocity and the wall shear stress it is also confirmed that the numerical results are less sensitive to the courant number owing to the consistency between the velocity and pressure fields the concept of the consistent direct discretization scheme is also explored for the thermal field the energy equations for the fluid and solid phases are discretized directly while satisfying the thermal relations that should be valid at their interface it takes different forms depending on the thermal boundary conditions dirichlet isothermal and neumann adiabatic iso heat flux boundary conditions for convective heat transfer and a fluid solid thermal interaction for conjugate heat transfer the validity of these discretizations is assessed by comparing the simulated results with analytical solutions for the respective thermal boundary conditions and it is confirmed that the present schemes also show high accuracy for the thermal field a significant improvement for the conjugate heat transfer problems is that the second order spatial accuracy and numerical stability are maintained even under severe conditions of near practical physical properties for the fluid and solid phases

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

a 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

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

a well conditioned augmented system for solving navier stokes equations in irregular domains an augmented method based on a cartesian grid is proposed for the incompressible navier stokes equations in irregular domains the irregular domain is embedded into a rectangular one so that a fast poisson solver can be utilized in the projection method unlike several methods suggested in the literature that set the force strengths as unknowns which often results in an ill conditioned linear system we set the jump in the normal derivative of the velocity as the augmented variable the new approach improves the condition number of the system for the augmented variable significantly using the immersed interface method we are able to achieve second order accuracy for the velocity numerical results and comparisons to benchmark tests are given to validate the new method a lid driven cavity flow with multiple obstacles and different geometries are also presented

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

stability characteristics of the virtual boundary method in three dimensional applications the refined stability analysis of the virtual boundary method proposed by goldstein et al 1994 and modified by saiki and biringen 1996 is carried out for applications to three dimensional turbulent flows in complex geometry the precise stability boundaries in the forcing parameter space for various time advancing schemes are provided under the assumption that the virtual boundary points are densely distributed from these and the relevant investigation of frequency of the forced system the optimum gains of the feedback forcing are suggested stability regimes of the runge kutta schemes of various order are much wider than those of the adams bashforth schemes specially the third order runge kutta scheme allows the use of an order one cfl number in the integration of the feedback forcing rendering the method applicable to turbulent flows with complex boundaries the three dimensional turbulent flow caused by a surface mounted box was simulated using a spectral method for evaluation confirming the stability limit proposed by theoretical estimate the method was then applied to simulations of the flow around an impulsively starting cylinder and of the rough wall turbulent boundary layer flow

a unified mathematical framework and an adaptive numerical method for fluid structure interaction with rigid deforming and elastic bodies many problems of interest in biological fluid mechanics involve interactions between fluids and solids that require the coupled solution of momentum equations for both the fluid and the solid in this work we develop a mathematical framework and an adaptive numerical method for such fluid structure interaction fsi problems in which the structure may be rigid deforming or elastic we employ an immersed boundary ib formulation of the problem that permits us to avoid body conforming discretizations and to use fast cartesian grid solvers rigidity and deformational kinematic constraints are imposed using a formulation based on distributed lagrange multipliers and a conventional ib method is used to describe the elasticity of the immersed body we use cartesian grid adaptive mesh refinement amr to discretize the equations of motion and thereby obtain a solution methodology that efficiently captures thin boundary layers at fluid solid interfaces as well as flow structures shed from such interfaces this adaptive methodology is validated for several benchmark problems in two and three spatial dimensions in addition we use this scheme to simulate free swimming including the maneuvering of a two dimensional model eel and a three dimensional model of the weakly electric black ghost knifefish

an immersed interface method for stokes flows with fixed moving interfaces and rigid boundaries we present an immersed interface method for solving the incompressible steady stokes equations involving fixed moving interfaces and rigid boundaries irregular domains the fixed moving interfaces and rigid boundaries are represented by a number of lagrangian control points in order to enforce the prescribed velocity at the rigid boundaries singular forces are applied on the fluid at these boundaries the strength of singular forces at the rigid boundary is determined by solving a small system of equations for the deformable interfaces the forces that the interface exerts on the fluid are calculated from the configuration position of the deformed interface the jumps in the pressure and the jumps in the derivatives of both pressure and velocity are related to the forces at the fixed moving interfaces and rigid boundaries these forces are interpolated using cubic splines and applied to the fluid through the jump conditions the positions of the deformable interfaces are updated implicitly using a quasi newton method bfgs within each time step in the proposed method the stokes equations are discretized via the finite difference method on a staggered cartesian grid with the incorporation of jump contributions and solved by the conjugate gradient uzawa type method numerical results demonstrate the accuracy and ability of the proposed method to simulate incompressible stokes flows with fixed moving interfaces on irregular domains

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

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

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