extended immersed boundary method using fem and rkpm abstract the proposed extended immersed boundary method eibm has several distinct features in comparison with the immersed boundary ib method in the ib method the interaction between the immersed elastic boundary and the surrounding viscous fluid is replaced with an equivalent body force distribution within the fluid domain the key ingredient of the ib method is the enforcement of the energy input from the equivalent body force into the fluid domain to be the same at all time as that of the elastic forces within the ib in eibm instead of the volumeless immersed elastic boundary we consider the submerged elastic solid which occupies a finite volume within the fluid domain in particular we replace the kinematic and dynamic matching of the fluid solid interface and the effect of the submerged solid with a judiciously chosen collection of equivalent nodal forces calculated in the context of finite element formulations the employment of the finite element representation enables a realistic stress analysis for the submerged solid subject to large deformations finally in eibm we also replace the discretized delta function commonly used in the ib method with the kernel functions in meshless reproducing kernel particle method rkpm with this improvement using the same finite support domain we could improve the discretized delta function from c 1 to c n continuity where n is chosen according to the required smoothness and resolution of such functions and more importantly enables the use of non uniform meshing in the fluid domain these extensions may eventually open doors to the modelling of complex biological fluid involving molecular cellular and flexible vessel flow interactions

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

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

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

conservative load transfer along curved fluid solid interface with non matching meshes we investigate the effect of curvature on the accuracy of schemes used to transfer loads along the interface in coupled fluid solid simulations involving non matching meshes we analyze two types of load transfer schemes for the coupled system a point to element projection schemes and b common refinement schemes the accuracy of these schemes over the curved interface is assessed with the aid of static and transient problems we show that the point to element projection schemes may yield inaccurate load transfer from the source fluid mesh to the target solid mesh leading to a weak instability in the form of spurious oscillations and overshoots in the interface solution the common refinement scheme resolves this problem by providing an accurate transfer of discrete interface conditions across non matching meshes we show theoretically that the accurate transfer preserves the stability of the coupled system while maintaining the energy conservation over a reference interface finally we introduce simple analytical error functions which correlate well with the numerical errors of the load transfer schemes

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

a particle method for history dependent materials abstract a broad class of engineering problems including penetration impact and large rotations of solid bodies causes severe numerical problems for these problems the constitutive equations are history dependent so material points must be followed this is difficult to implement in a eulerian scheme on the other hand purely lagrangian methods typically result in severe mesh distortion and the consequence is ill conditioning of the element stiffness matrix leading to mesh lockup or entanglement remeshing prevents the lockup and tangling but then interpolation must be performed for history dependent variables a process which can introduce errors proposed here is an extension of the particle in cell method in which particles are interpreted to be material points that are followed through the complete loading process a fixed eulerian grid provides the means for determining a spatial gradient because the grid can also be interpreted as an updated lagrangian frame the usual convection term in the acceleration associated with eulerian formulations does not appear with the use of maps between material points and the grid the advantages of both eulerian and lagrangian schemes are utilized so that mesh tangling is avoided while material variables are tracked through the complete deformation history example solutions in two dimensions are given to illustrate the robustness of the proposed convection algorithm and to show that typical elastic behavior can be reproduced also it is shown that impact with no slip is handled without any special algorithm for bodies governed by elasticity and strain hardening plasticity

an immersed interface method for incompressible navier stokes equations the method developed in this paper is motivated by peskin s immersed boundary ib method and allows one to model the motion of flexible membranes or other structures immersed in viscous incompressible fluid using a fluid solver on a fixed cartesian grid the ib method uses a set of discrete delta functions to spread the entire singular force exerted by the immersed boundary to the nearby fluid grid points our method instead incorporates part of this force into jump conditions for the pressure avoiding discrete dipole terms that adversely affect the accuracy near the immersed boundary this has been implemented for the two dimensional incompressible navier stokes equations using a high resolution finite volume method for the advective terms and a projection method to enforce incompressibility in the projection step the correct jump in pressure is imposed in the course of solving the poisson problem this gives sharp resolution of the pressure across the interface and also gives better volume conservation than the traditional ib method comparisons between this method and the ib method are presented for several test problems numerical studies of the convergence and order of accuracy are included

an immersed boundary finite volume method for simulations of flow in complex geometries abstract a new immersed boundary method for simulating flows over or inside complex geometries is developed by introducing a mass source sink as well as a momentum forcing the present method is based on a finite volume approach on a staggered mesh together with a fractional step method both momentum forcing and mass source are applied on the body surface or inside the body to satisfy the no slip boundary condition on the immersed boundary and also to satisfy the continuity for the cell containing the immersed boundary in the immersed boundary method the choice of an accurate interpolation scheme satisfying the no slip condition on the immersed boundary is important because the grid lines generally do not coincide with the immersed boundary therefore a stable second order interpolation scheme for evaluating the momentum forcing on the body surface or inside the body is proposed three different flow problems decaying vortices and flows over a cylinder and a sphere are simulated using the immersed boundary method proposed in this study and the results agree very well with previous numerical and experimental results verifying the accuracy of the present method

nonparametric identification of hammerstein systems a discrete time nonlinear hammerstein system is identified and the correlation and frequency domain methods for identification of its linear subsystem are presented the main results concern the estimation of the nonlinear memoryless subsystem no conditions concerning the functional form of the transform characteristic of the subsystem are made and an algorithm for estimation of the characteristic is given the algorithm is simply a nonparametric kernel estimate of the regression function calculated from dependent data it is shown that the algorithm converges to the characteristic of the subsystem regardless of the probability distribution of the input variable pointwise as well as global consistencies are established for lipschitz characteristics the rate of the convergence in probability is o n sup 1 3

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

a fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies application to particulate flow abstract in this article we discuss a methodology that allows the direct numerical simulation of incompressible viscous fluid flow past moving rigid bodies the simulation methods rest essentially on the combination of a lagrange multiplier based fictitious domain methods which allow the fluid flow computations to be done in a fixed flow region b finite element approximations of the navier stokes equations occurring in the global model c time discretizations by operator splitting schemes in order to treat optimally the various operators present in the model the above methodology is particularly well suited to the direct numerical simulation of particulate flow such as the flow of mixtures of rigid solid particles and incompressible viscous fluids possibly non newtonian we conclude this article with the presentation of the results of various numerical experiments including the simulation of store separation for rigid airfoils and of sedimentation and fluidization phenomena in two and three dimensions

spectral methods for the navier stokes equations with one infinite and two periodic directions abstract two numerical methods were designed to solve the time dependent three dimensional incompressible navier stokes equations in boundary layers method a semi infinite domain and mixing layers or wakes method b fully infinite domain their originality lies in the use of rapidly decaying spectral basis functions to approximate the vertical dependence of the solutions combined with one method a or two method b slowly decaying extra functions for each wave vector that exactly represent the irrotational component of the solution at large distances both methods eliminate the pressure term as part of the formulation thus avoiding fractional step time integration they yield rapid convergence and are free of spurious modes in the orr sommerfeld spectra they are also efficient although the operation count is of order n 2 n is the number of modes in the infinite direction these methods have been used for extensive direct numerical simulations of transition and turbulence a new time integration scheme with low storage requirements and good stability properties is also described

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

the superconvergent patch recovery and a posteriori error estimates part 1 the recovery technique this is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation in this paper a general recovery technique is developed for determining the derivatives stresses of the finite element solutions at nodes the implementation of the recovery technique is simple and cost effective the technique has been tested for a group of widely used linear quadratic and cubic elements for both one and two dimensional problems numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent one order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements in particular an o h4 convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time the performance of the proposed technique is compared with the widely used smoothing procedure of global l2 projection and other methods it is found that the derivatives recovered at interelement nodes by using l2 projection are also superconvergent for linear elements but not for quadratic elements numerical experiments on the convergence of the recovered solutions in the energy norm are also presented higher rates of convergence are again observed the results presented in this part of the paper indicate clearly that a new powerful and economical process is now available which should supersede the currently used post processing procedures applied in most codes

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

a dlm fd method for fluid flexible body interactions in this study we extended the distributed lagrange multiplier fictitious domain dlm fd formulation of glowinski et al int j multiphase flow 25 1999 755 for the fluid rigid body interactions to deal with the fluid flexible body interactions by replacing newton s equations of motion for the rigid body with the continuum equations for the general solid material similar to the rigid body case where the dlm is introduced as a pseudo body force to enforce the constraint of rigid body motion of the fictitious fluid in the solid domain the lagrange multiplier in our formulation is to enforce the fictitious fluid to move at the same velocity as the solid for our computational scheme a first order accurate fractional step scheme is employed to decouple the entire system into three sub systems a fluid problem a solid problem and a lagrange multiplier problem the flow problem is solved with the projection method on half staggered grids the solid problem is solved with the lagrangian finite element method and the newton iterative method and the incompressibility of the material is implemented with the penalty function method the proposed method is applied to two typical fluid structure interaction problems the flow driven oscillation of a flexible plate along the flow direction and the self sustained oscillation across the flow direction both results compare favorably with previously reported numerical and experimental results and show that our method is suited to the simulation of the motion of an incompressible non linear elastic material in a fluid

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 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

symmetry preserving discretization of turbulent flow we propose to perform turbulent flow simulations in such manner that the difference operators do have the same symmetry properties as the underlying differential operators i e the convective operator is represented by a skew symmetric coefficient matrix and the diffusive operator is approximated by a symmetric positive definite matrix mimicking crucial properties of differential operators forms in itself a motivation for discretizing them in a certain manner we give it a concrete form by noting that a symmetry preserving discretization of the navier stokes equations is stable on any grid and conserves the total mass momentum and kinetic energy for the latter the physical dissipation is to be turned off of coarse being stable on any grid the choice of the grid may be based on the required accuracy solely and the main question becomes how accurate is a symmetry preserving discretization its accuracy is tested for a turbulent flow in a channel by comparing the results to those of physical experiments and previous numerical studies the comparison is carried out for a reynolds number of 5600 which is based on the channel width and the mean bulk velocity based on the channel half width and wall shear velocity the reynolds number becomes 180 the comparison shows that with a fourth order symmetry preserving method a 64 64 32 grid suffices to perform an accurate numerical simulation

geometric integration over irregular domains with application to level set methods we present a geometric approach for calculating integrals over irregular domains described by a level set function this procedure can be used to evaluate integrals over a lower dimensional interface and may be used to evaluate the contribution of singular source terms this approach produces results that are second order accurate and robust to the perturbation of the interface location on the grid moreover since we use a cell wise approach this procedure can be easily extended to quadtree and octree grids we demonstrate the second order accuracy and the robustness of the method in two and three spatial dimensions

a cartesian grid embedded boundary method for the heat equation on irregular domains we present an algorithm for solving the heat equation on irregular time dependent domains it is based on the cartesian grid embedded boundary algorithm of johansen and colella j comput phys 147 2 60 85 for discretizing poisson s equation combined with a second order accurate discretization of the time derivative this leads to a method that is second order accurate in space and time for the case where the boundary is moving we convert the moving boundary problem to a sequence of fixed boundary problems combined with an extrapolation procedure to initialize values that are uncovered as the boundary moves we find that in the moving boundary case the use of crank nicolson time discretization is unstable requiring us to use the l sub 0 stable implicit runge kutta method of twizell gumel and arigu

quadrature rules for triangular and tetrahedral elements with generalized functions quadrature rules are developed for exactly integrating products of polynomials and generalized functions over triangular and tetrahedral domains these quadrature rules greatly simplify the implementation of finite element methods that involve integrals over volumes and interfaces that are not coincident with the element boundaries specifically the integrands considered here consist of a quadratic polynomial multiplied by a heaviside or dirac delta function operating on a linear polynomial this form allows for exact integration of expressions obtained from linear finite elements over domains and interfaces defined by a linear level set function exact quadrature rules are derived that involve fixed quadrature point locations with weights that depend continuously on the nodal level set values compared with methods involving explicit integration over subdomains the quadrature rules developed here accommodate degenerate interface geometries without any need for special consideration and provide analytical jacobian information describing the dependence of the integrals on the nodal level set values the accuracy of the method is demonstrated for a simple conduction problem with the neumann and robin type boundary conditions copyright 2007 john wiley sons ltd

the immersed boundary method a projection approach a new formulation of the immersed boundary method with a structure algebraically identical to the traditional fractional step method is presented for incompressible flow over bodies with prescribed surface motion like previous methods a boundary force is applied at the immersed surface to satisfy the no slip constraint this extra constraint can be added to the incompressible navier stokes equations by introducing regularization and interpolation operators the current method gives prominence to the role of the boundary force acting as a lagrange multiplier to satisfy the no slip condition this role is analogous to the effect of pressure on the momentum equation to satisfy the divergence free constraint the current immersed boundary method removes slip and non divergence free components of the velocity field through a projection the boundary force is determined implicitly without any constitutive relations allowing the present formulation to use larger cfl numbers compared to some past methods symmetry and positive definiteness of the system are preserved such that the conjugate gradient method can be used to solve for the flow field examples show that the current formulation achieves second order temporal accuracy and better than first order spatial accuracy in l 2 norms for one and two dimensional test problems results from two dimensional simulations of flows over stationary and moving cylinders are in good agreement with those from previous experimental and numerical studies

a series solution of the exact equation for thick orthotropic plates by considering three dimensional elasticity without any initial assumptions the authors obtain the state equations for an orthotropic body a series solution for a simply supported rectangular thick plate with arbitrary ratio between thickness and width under any given load is presented every fundamental equation of three dimensional elasticity can be exactly satisfied and all the nine elastic constants can also be taken into account by the present method numerical results are obtained and compared with those of reissner s and ambartsumyan s theories and some references