immersed boundary method for variable viscosity and variable density problems using fast constant coefficient linear solvers ii theory we analyze the stability and convergence of first order accurate and second order accurate timestepping schemes for the navier stokes equations with variable viscosity these schemes are characterized by a mixed implicit explicit treatment of the viscous term in which a numerical parameter lambda determines the degree of splitting between the implicit and explicit contributions the reason for this splitting is that it avoids the need to solve computationally expensive linear systems that may change at each timestep provided the parameter lambda is within a permissible range we prove that the first order accurate and second order accurate schemes are convergent we show further that the efficiency of the second order accurate scheme depends on how lambda is chosen within the permissible range and we discuss choices that work well in practice we use parameters motivated by this analysis to simulate internal gravity waves which arise in stratified fluids with variable density we examine h

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

a brief history of the classification of the finite simple groups first published in bulletin of the american mathematical society in volume 38 issue 3 published by the american mathematical society

active control and drag optimization for flow past a circular cylinder the main objective of this article is to investigate computational methods for the active control and drag optimization of incompressible viscous flow past cylinders using the two dimensional navier stokes equations as the flow model the computational methodology relies on the following ingredients space discretization of the navier stokes equations by finite element approximations time discretization by a second order accurate two step implicit explicit finite difference scheme calculation of the cost function gradient by the adjoint equation approach and minimization of the cost function by a quasi newton method a la bfgs the above methods have been applied to boundary control by rotation of the flow around a circular cylinder and show 30 to 60 drag reduction compared to the fixed cylinder configuration for reynolds numbers in the range from 200 to 1000

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

a parallelized meshfree method with boundary enrichment for large scale cfd a parallel computational implementation of a meshfree method the reproducing kernel particle method rkpm is used for 3 d implicit cfd analysis a novel procedure for implementing the essential boundary condition using the hierarchical enrichment method is presented using this enrichment along the essential boundaries produces results that more closely match experimental and analytical results for a flow past a cylinder problem than does either the finite element method or other meshfree methods that require matrix inversion for the application of essential boundary conditions this technique also allows the efficient parallelization of the algorithm and leads to higher parallel speedups than do other boundary condition implementations many of which involve inherently serial steps this is important because the expense of meshfree computations makes parallelization crucial for large size problems the performance of the parallelization technique and the accuracy of the implicit cfd algorithm are demonstrated in two example problems

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

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

immersed boundary method for variable viscosity and variable density problems using fast constant coefficient linear solvers i numerical method and results we present a general variable viscosity and variable density immersed boundary method that is first order accurate in the variable density case and for problems possessing sufficient regularity second order accurate in the constant density case the viscosity and density are considered material properties and are defined by a dynamically updated tesselation empirical convergence rates are reported for a test problem of a two dimensional viscoelastic shell with spatially varying material properties the reduction to first order accuracy in the variable density case can be avoided by using an iterative scheme although this approach may not be efficient enough for practical use in our time stepping scheme both the inertial and viscous terms are split into two parts a constant coefficient part that is treated implicitly and a variable coefficient part that is treated explicitly this splitting allows the resulting equations to be solved efficiently using fast constant coefficient linear solvers and i

a versatile implicit iterative approach for fully resolved simulation of self propulsion article i nfo we present a computational approach for fully resolved simulation of self propulsion of organisms through a fluid a new implicit iterative algorithm is developed that solves for the swimming velocities of the organism with prescribed deformation kinematics a solution for the surrounding flow field is also obtained this approach uses a constraint based formulation of the problem of self propulsion developed by shirgaonkar et al 1 the approach in this paper is unlike the previous work 1 where a fractional time stepping scheme was used fractional time stepping schemes while efficient for moderate to high reynolds number problems are not suitable for zero or low reynolds number problems where the inertia term in the governing equation is absent or negligible in such cases the implicit iterative algorithm presented here is more appropriate we validate the method by simulating self propulsion of bacterial flagellum jellyfish aurelia aurita and larval zebrafish danio rerio comparison of the computational results with theoretical and experimental results for the test cases is found to be very good

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

a sharp interface immersed boundary method with improved mass conservation and reduced spurious pressure oscillations a method for reducing the spurious pressure oscillations observed when simulating moving boundary flow problems with sharp interface immersed boundary methods ibms is proposed by first identifying the primary cause of these oscillations to be the violation of the geometric conservation law near the immersed boundary we adopt a cut cell based approach to strictly enforce geometric conservation in order to limit the complexity associated with the cut cell method the cut cell based discretization is limited only to the pressure poisson and velocity correction equations in the fractional step method and the small cell problem tackled by introducing a virtual cell merging technique the method is shown to retain all the desirable properties of the original finite difference based ibm while at the same time reducing pressure oscillations for moving boundaries by roughly an order of magnitude

triangular dynamical r matrices and quantization abstract we study some general aspects of triangular dynamical r matrices using poisson geometry we show that a triangular dynamical r matrix r h 2 g always gives rise to a regular poisson manifold using the fedosov method we prove that non degenerate triangular dynamical r matrices i e those such that the corresponding poisson manifolds are symplectic are quantizable and that the quantization is classified by the relative lie algebra cohomology h2 g h

on dynamical r matrices obtained from dirac reduction and their generalizations to affine lie algebras according to etingof and varchenko the classical dynamical yang baxter equation is a guarantee for the consistency of the poisson bracket on certain poisson lie groupoids here it is noticed that dirac reductions of these poisson manifolds give rise to a mapping from dynamical r matrices on a pair to those on another pair where is a chain of lie algebras for which admits a reductive decomposition as several known dynamical r matrices appear naturally in this setting and its application provides new r matrices too in particular we exhibit a family of r matrices for which the dynamical variable lies in the grade zero subalgebra of an extended affine lie algebra obtained from a twisted loop algebra based on an arbitrary finite dimensional self dual lie algebra

a fast computation technique for the direct numerical simulation of rigid particulate flows in this paper we present a computation technique for the direct numerical simulation of freely moving rigid bodies in fluids we solve three dimensional laminar flow problems using a control volume approach the key feature of this approach is that the computational overhead relative to a pure fluid solver to solve for the motion of rigid particle is very small the formulation is convenient for handling irregular geometries we present results for the sedimentation of particles of different shapes convergence tests are presented to assess the order of accuracy of the numerical scheme various test cases are considered and the numerical results are compared with experimental values to validate the code due to the ability to perform fast computations this method has been used for animations and its application to the direct numerical simulation of turbulent particulate flows merits investigation the technique is not restricted to any constitutive model of the suspending fluid hence it may potentially be used in large eddy simulations les or reynolds averaged navier stokes rans type simulations

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

an adaptive formally second order accurate version of the immersed boundary method like many problems in biofluid mechanics cardiac mechanics can be modeled as the dynamic interaction of a viscous incompressible fluid the blood and a visco elastic structure the muscular walls and the valves of the heart the immersed boundary method is a mathematical formulation and numerical approach to such problems that was originally introduced to study blood flow through heart valves and extensions of this work have yielded a three dimensional model of the heart and great vessels in the present work we introduce a new adaptive version of the immersed boundary method this adaptive scheme employs the same hierarchical structured grid approach but a different numerical scheme as the two dimensional adaptive immersed boundary method of roma et al a multilevel self adaptive version of the immersed boundary method ph d thesis courant institute of mathematical sciences new york university 1996 an adaptive version of the immersed boundary method j comput phys 153 2 1999 509 534 and is based on a formally second order accurate i e second order accurate for problems with sufficiently smooth solutions version of the immersed boundary method that we have recently described b e griffith c s peskin on the order of accuracy of the immersed boundary method higher order convergence rates for sufficiently smooth problems j comput phys 208 1 2005 75 105 actual second order convergence rates are obtained for both the uniform and adaptive methods by considering the interaction of a viscous incompressible flow and an anisotropic incompressible viscoelastic shell we also present initial results from the application of this methodology to the three dimensional simulation of blood flow in the heart and great vessels the results obtained by the adaptive method show good qualitative agreement with simulation results obtained by earlier non adaptive versions of the method but the flow in the vicinity of the model heart valves indicates that the new methodology provides enhanced boundary layer resolution differences are also observed in the flow about the mitral valve leaflets

a stable fluid structure interaction solver for low density rigid bodies using the immersed boundary projection method dispersion of low density rigid particles with complex geometries is ubiquitous in both natural and industrial environments we show that while explicit methods for coupling the incompressible navier stokes equations and newton s equations of motion are often sufficient to solve for the motion of cylindrical particles with low density ratios for more complex particles such as a body with a protrusion they become unstable we present an implicit formulation of the coupling between rigid body dynamics and fluid dynamics within the framework of the immersed boundary projection method similarly to previous work on this method the resulting matrix equation in the present approach is solved using a block lu decomposition each step of the block lu decomposition is modified to incorporate the rigid body dynamics we show that our method achieves second order accuracy in space and first order in time third order for practical settings only with a small additional computational cost to the original method our implicit coupling yields stable solution for density ratios as low as 10 4 we also consider the influence of fictitious fluid located inside the rigid bodies on the accuracy and stability of our method

a fast immersed boundary method for external incompressible viscous flows using lattice green s functions a new parallel computationally efficient immersed boundary method for solving three dimensional viscous incompressible flows on unbounded domains is presented immersed surfaces with prescribed motions are generated using the interpolation and regularization operators obtained from the discrete delta function approach of the original peskin s immersed boundary method unlike peskin s method boundary forces are regarded as lagrange multipliers that are used to satisfy the no slip condition the incompressible navier stokes equations are discretized on an unbounded staggered cartesian grid and are solved in a finite number of operations using lattice green s function techniques these techniques are used to automatically enforce the natural free space boundary conditions and to implement a novel block wise adaptive grid that significantly reduces the run time cost of solutions by limiting operations to grid cells in the immediate vicinity and near wake region of the immersed surface these techniques also enable the construction of practical discrete viscous integrating factors that are used in combination with specialized half explicit runge kutta schemes to accurately and efficiently solve the differential algebraic equations describing the discrete momentum equation incompressibility constraint and no slip constraint linear systems of equations resulting from the time integration scheme are efficiently solved using an approximation free nested projection technique the algebraic properties of the discrete operators are used to reduce projection steps to simple discrete elliptic problems e g discrete poisson problems that are compatible with recent parallel fast multipole methods for difference equations numerical experiments on low aspect ratio flat plates and spheres at reynolds numbers up to 3700 are used to verify the accuracy and physical fidelity of the formulation

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

a moving control volume approach to computing hydrodynamic forces and torques on immersed bodies abstract we present a moving control volume cv approach to computing hydrodynamic forces and torques on complex geometries the method requires surface and volumetric integrals over a simple and regular cartesian box that moves with an arbitrary velocity to enclose the body at all times the moving box is aligned with cartesian grid faces which makes the integral evaluation straightforward in an immersed boundary ib framework discontinuous and noisy derivatives of velocity and pressure at the fluid structure interface are avoided and far field smooth velocity and pressure information is used we re visit the approach to compute hydrodynamic forces and torques through force torque balance equations in a lagrangian frame that some of us took in a prior work bhalla et al 2013 13 we prove the equivalence of the two approaches for ib methods thanks to the use of peskin s delta functions both approaches are able to suppress spurious force oscillations and are in excellent agreement as expected theoretically test cases ranging from stokes to high reynolds number regimes are considered we discuss regridding issues for the moving cv method in an adaptive mesh refinement amr context the proposed moving cv method is not limited to a specific ib method and can also be used for example with embedded boundary methods

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

a flexible inner outer preconditioned gmres algorithm a variant of the gmres algorithm is presented that allows changes in the preconditioning at every step there are many possible applications of the new algorithm some of which are briefly discussed in particular a result of the flexibility of the new variant is that any iterative method can be used as a preconditioner for example the standard gmres algorithm itself can be used as a preconditioner as can cgnr or cgne the conjugate gradient method applied to the normal equations however the more appealing utilization of the method is in conjunction with relaxation techniques possibly multilevel techniques the possibility of changing preconditioners may be exploited to develop efficient iterative methods and to enhance robustness a few numerical experiments are reported to illustrate this fact

quantization of symplectic dynamical r matrices and the quantum composition formula in this paper we quantize symplectic dynamical r matrices over a possibly nonabelian base the proof is based on the fact that the existence of a star product with a nice property called strong invariance is sufficient for the existence of a quantization we also classify such quantizations and prove a quantum analogue of the classical composition formula for coboundary dynamical r matrices

a three dimensional hybrid meshfree cartesian scheme for fluid body interaction a numerical method based on a hybrid meshfree cartesian grid is developed for solving three dimensional fluid solid interaction fsi problems involving solid bodies undergoing large motion the body is discretized and enveloped by a cloud of meshfree nodes the motion of the body is tracked by convecting the meshfree nodes against a background of cartesian grid points spatial discretization of second order accuracy is accomplished by the combination of a generalized finite difference gfd method and conventional finite difference fd method which are applied to the meshfree and cartesian nodes respectively error minimization in gfd is carried out by singular value decomposition the discretized equations are integrated in time via a second order fractional step projection method a time implicit iterative procedure is employed to compute the new evolving position of the immersed bodies together with the dynamically coupled solution of the flow field the present method is applied on problems of free falling spheres and tori in quiescent flow and freely rotating spheres in simple shear flow good agreement with published results shows the ability of the present hybrid meshfree cartesian grid scheme to achieve good accuracy an application of the method to the self induced propulsion of a deforming fish like swimmer further demonstrates the capability and potential of the present approach for solving complex fsi problems in 3d copyright 2011 john wiley sons ltd

signalizer functors proper 2 generated cores and nonconnected groups as has been well understood since its inception the signalizer functor method works most smoothly in finite groups satisfying some connectedness condition in particular in groups with a connected sylow 2 subgroup the original approach to nonconnectivity disposed of the problem once and for all by completely classifying all groups of sectional 2 rank at most 4 and then determining those with a nonconnected sylow 2 subgroup as a corollary by a theorem of macwilliams 141 every section of a nonconnected 2group can be generated by at most four elements even though this approach was very effective it was extremely elaborate for the actual proof of the classification of such groups runs to 464 pages l 11 and in addition depends upon a number of other classification theorems concerning groups with specified types of sylow 2 subgroups moreover although the sectional 2 rank 4 theorem has been invoked at many other points in the classification of finite simple groups the analysis of nonconnected groups has remained its primary application hence in attempting to simplify the existing classification of the finite simple groups one of the first natural questions to consider is whether there exists a shorter more direct way of treating nonconnectivity indeed harada has recently achieved just such a direct approach by a short elementary fusion argument 121 his result can be stated as follows here a k group is any finite group whose composition factors are among the known simple groups

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

on the order of accuracy of the immersed boundary method higher order convergence rates for sufficiently smooth problems the immersed boundary method is both a mathematical formulation and a numerical scheme for problems involving the interaction of a viscous incompressible fluid and a visco elastic structure in m c lai simulations of the flow past an array of circular cylinders as a test of the immersed boundary method ph d thesis courant institute of mathematical sciences new york university 1998 m c lai c s peskin an immersed boundary method with formal second order accuracy and reduced numerical viscosity j comput phys 160 2000 705 719 lai and peskin introduced a formally second order accurate immersed boundary method but the convergence properties of their algorithm have only been examined computationally for problems with nonsmooth solutions consequently in practice only first order convergence rates have been observed in the present work we describe a new formally second order accurate immersed boundary method and demonstrate its performance for a prototypical fluid structure interaction problem involving an immersed viscoelastic shell of finite thickness studied over a broad range of reynolds numbers we consider two sets of material properties for the viscoelastic structure including a case where the material properties of the coupled system are discontinuous at the fluid structure interface for both sets of material properties the true solutions appear to possess sufficient smoothness for the method to converge at a second order rate for fully resolved computations

an immersed interface method for simulating the interaction of a fluid with moving boundaries in the immersed interface method boundaries are represented as singular force in the navier stokes equations which enters a numerical scheme as jump conditions recently we systematically derived all the necessary spatial and temporal jump conditions for simulating incompressible viscous flows subject to moving boundaries in 3d with second order spatial and temporal accuracy near the boundaries sheng xu z jane wang systematic derivation of jump conditions for the immersed interface method in three dimensional flow simulation siam j sci comput 2006 in press in this paper we implement the immersed interface method to incorporate these jump conditions in a 2d numerical scheme we study the accuracy efficiency and robustness of our method by simulating taylor couette flow flow induced by a relaxing balloon flow past single and multiple cylinders and flow around a flapping wing our results show that 1 our code has second order accuracy in the infinity norm for both the velocity and the pressure 2 the addition of an object introduces relatively insignificant computational cost 3 the method is equally effective in computing flow subject to boundaries with prescribed force or boundaries with prescribed motion

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

a penalty projection algorithm for a monolithic fluid structure interaction solver in this paper we propose a new iterative penalty projection algorithm for a monolithic fluid structure interaction solver projection methods that split the computation of the velocity from the pressure are very popular in fluid dynamics since the boundary errors generated by the projection method are localized mainly near the boundary layers where the incorrect pressure boundary conditions are imposed however when solid regions are taken into account the pressure projected field cannot satisfy fully coupled boundary constraints imposed on external solid surfaces such as stress free conditions and due to the rigidity of the medium the boundary errors propagate deeply in the interior in order to reduce the projection errors we propose a one step penalty projection method in the fluid domain and an iterative penalty projection method in the solid region this technique decouples the pressure velocity degrees of freedom and as a consequence the computational cost in order to verify the accuracy and robustness of the proposed method we compare the results between this splitting velocity pressure algorithm and the coupled one these numerical results show stability and robustness of the proposed algorithm with a strong reduction of the computational effort a new penalty projection algorithm for a monolithic fsi solver is proposed projection method error is localized where pressure boundary conditions are imposed a one step penalty together with an iterative projection algorithm is used this technique reduces the problem degrees of freedom and the computational cost numerical results show stability and reduction of cpu effort for large problems

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

modeling and simulation of fish like swimming modeling and simulation of two dimensional flows past deformable bodies are considered the incompressible navier stokes equations are discretized in space onto a fixed cartesian mesh and the displacement of deformable objects through the fluid is taken into account using a penalization method the interface between the solid and the fluid is tracked using a level set description so that it is possible to simulate several bodies freely evolving in the fluid as an illustration of the methods fish like locomotion is analyzed in terms of propulsion efficiency underwater maneuvering and school swimming are also explored

a new mathematical formulation and fast algorithm for fully resolved simulation of self propulsion we present a computational algorithm for fully resolved numerical simulation frs of rigid and deforming bodies moving in fluids given the deformation of the body in its own reference frame the method solves for the swimming velocity of the body together with the surrounding flow field and the hydrodynamic forces on the body we provide the mathematical foundation of the algorithm based on distributed lagrange multipliers and show that it naturally connects with vortex methods through a vorticity source at the interface we demonstrate applications to rigid and flexible bodies membranes and bodies with a propelling membrane attached to them in contrast to some existing methods the swimming velocity of the body is not prescribed but is computed along with the forces without requiring a body fitted grid the algorithm is designed to be fast efficient and easy to implement in existing fluid dynamics codes for practical solid fluid problems in engineering and biology

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

immersed boundary method for flow around an arbitrarily moving body for the simulation of flow around an arbitrarily moving body an immersed boundary method is developed in a non inertial reference frame that is fixed to the body the navier stokes equation is formulated in a conservative form such that the force terms due to the rotation and the translational and rotational accelerations are included in the nonlinear term in order to satisfy the no slip condition on the body surface momentum forcing and mass source sink are applied on the body surface or inside the body the numerical method is based on a finite volume approach on a staggered mesh together with a fractional step method the present numerical method is applied to both the forced motion and fluid structure interaction problems in the latter we solve fully coupled navier stokes and dynamic equations for the moving body without introducing any iteration four different flow problems are tested and the results are in excellent agreements with previous numerical and experimental ones

immersed boundary methods for general finite difference and finite volume navier stokes solvers we present an immersed boundary algorithm for incompressible flows with complex boundaries suitable for cartesian or curvilinear grid system the key stages of any immersed boundary technique are the interpolation of a velocity field given on a mesh onto a general boundary a line in 2d a surface in 3d and the spreading of a force field from the immersed boundary to the neighboring mesh points to enforce the desired boundary conditions on the immersed boundary points we propose a technique that uses the reproducing kernel particle method w k liu s jun y f zhang reproducing kernel particle methods int j numer methods fluids 20 8 1995 1081 1106 for the interpolation and spreading unlike other methods presented in the literature the one proposed here has the property that the integrals of the force field and of its moment on the grid are conserved independent of the grid topology uniform or non uniform cartesian or curvilinear the technique is easy to implement and is able to maintain the order of the original underlying spatial discretization applications to two and three dimensional flows in cartesian and non cartesian grid system with uniform and non uniform meshes are presented

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

a stable partitioned fsi algorithm for rigid bodies and incompressible flow part i model problem analysis a stable partitioned algorithm is developed for fluid structure interaction fsi problems involving viscous incompressible flow and rigid bodies this added mass partitioned amp algorithm remains stable without sub iterations for light and even zero mass rigid bodies when added mass and viscous added damping effects are large the scheme is based on a generalized robin interface condition for the fluid pressure that includes terms involving the linear acceleration and angular acceleration of the rigid body added mass effects are handled in the robin condition by inclusion of a boundary integral term that depends on the pressure added damping effects due to the viscous shear forces on the body are treated by inclusion of added damping tensors that are derived through a linearization of the integrals defining the force and torque added damping effects may be important at low reynolds number or for example in the case of a rotating cylinder or rotating sphere when the rotational moments of inertia are small in this first part of a two part series the properties of the amp scheme are motivated and evaluated through the development and analysis of some model problems the analysis shows when and why the traditional partitioned scheme becomes unstable due to either added mass or added damping effects the analysis also identifies the proper form of the added damping which depends on the discrete time step and the grid spacing normal to the rigid body the results of the analysis are confirmed with numerical simulations that also demonstrate a second order accurate implementation of the amp scheme

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

a robust incompressible navier stokes solver for high density ratio multiphase flows abstract this paper presents a robust adaptive numerical scheme for simulating high density ratio and high shear multiphase flows on locally refined staggered cartesian grids that adapt to the evolving interfaces and track regions of high vorticity the algorithm combines the interface capturing level set method with a variable coefficient incompressible navier stokes solver that is demonstrated to stably resolve material contrast ratios of up to six orders of magnitude the discretization approach ensures second order pointwise accuracy for both velocity and pressure with several physical boundary treatments including velocity and traction boundary conditions the paper includes several test cases that demonstrate the order of accuracy and algorithmic scalability of the flow solver to ensure the stability of the numerical scheme in the presence of high density and viscosity ratios we employ a consistent treatment of mass and momentum transport in the conservative form of discrete equations this consistency is achieved by solving an additional mass balance equation which we approximate via a strong stability preserving runga kutta time integrator and by employing the same mass flux obtained from the mass equation in the discrete momentum equation the scheme uses higher order total variation diminishing tvd and convection boundedness criterion cbc satisfying limiter to avoid numerical fluctuations in the transported density field the high order bounded convective transport is done on a dimension by dimension basis which makes the scheme simple to implement we also demonstrate through several test cases that the lack of consistent mass and momentum transport in non conservative formulations which are commonly used in practice or the use of non cbc satisfying limiters can yield very large numerical error and very poor accuracy for convection dominant high density ratio flows our numerical scheme also uses well balanced surface tension and gravity force discretizations in the hydrostatic limit we show that the well balanced formulation mitigates spurious flow currents and achieves discrete force balance between the pressure gradient and surface tension or gravity

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

on the hyper elastic formulation of the immersed boundary method abstract the immersed boundary ib method is both a mathematical formulation and a numerical method for fluid structure interaction problems in which immersed incompressible visco elastic bodies or boundaries interact with an incompressible fluid previous formulations of the ib method were not able to treat appropriately immersed materials of finite nonzero thickness modeled by general hyper elastic constitutive laws because of the lack of appropriate transmission conditions between the immersed body and the surrounding fluid in the case of a nonzero jump in normal stress at the solid fluid interface such a jump does not arise when the solid is comprised of fibers that run parallel to the interface but typically does arise in other cases e g when the solid contains elastic fibers that terminate at the solid fluid interface we present a derivation of the ib method that takes into account in an appropriate way the missing term the derivation presented in this paper starts from a separation of the stress that appears in the principle of virtual work the stress is divided into its fluid like and solid like components and each of these two terms is treated in its natural framework i e the eulerian framework for the fluid like stress and the lagrangian framework for the solid like stress we describe how the ib method can be used in conjunction with standard formulations of continuum mechanics models for immersed incompressible elastic materials and present some illustrative numerical experiments

direct numerical simulation of particulate flows with an overset grid method abstract we evaluate an efficient overset grid method for two dimensional and three dimensional particulate flows for small numbers of particles at finite reynolds number the rigid particles are discretised using moving overset grids overlaid on a cartesian background grid this allows for strongly enforced boundary conditions and local grid refinement at particle surfaces thereby accurately capturing the viscous boundary layer at modest computational cost the incompressible navier stokes equations are solved with a fractional step scheme which is second order accurate in space and time while the fluid solid coupling is achieved with a partitioned approach including multiple sub iterations to increase stability for light rigid bodies through a series of benchmark studies we demonstrate the accuracy and efficiency of this approach compared to other boundary conformal and static grid methods in the literature in particular we find that fully resolving boundary layers at particle surfaces is crucial to obtain accurate solutions to many common test cases with our approach we are able to compute accurate solutions using as little as one third the number of grid points as uniform grid computations in the literature a detailed convergence study shows a 13 fold decrease in cpu time over a uniform grid test case whilst maintaining comparable solution accuracy

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 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 boundary method with divergence free velocity interpolation and force spreading abstract the immersed boundary ib method is a mathematical framework for constructing robust numerical methods to study fluid structure interaction in problems involving an elastic structure immersed in a viscous fluid the ib formulation uses an eulerian representation of the fluid and a lagrangian representation of the structure the lagrangian and eulerian frames are coupled by integral transforms with delta function kernels the discretized ib equations use approximations to these transforms with regularized delta function kernels to interpolate the fluid velocity to the structure and to spread structural forces to the fluid it is well known that the conventional ib method can suffer from poor volume conservation since the interpolated lagrangian velocity field is not generally divergence free and so this can cause spurious volume changes in practice the lack of volume conservation is especially pronounced for cases where there are large pressure differences across thin structural boundaries the aim of this paper is to greatly reduce the volume error of the ib method by introducing velocity interpolation and force spreading schemes with the properties that the interpolated velocity field in which the structure moves is at least c 1 and satisfies a continuous divergence free condition and that the force spreading operator is the adjoint of the velocity interpolation operator we confirm through numerical experiments in two and three spatial dimensions that this new ib method is able to achieve substantial improvement in volume conservation compared to other existing ib methods at the expense of a modest increase in the computational cost further the new method provides smoother lagrangian forces tractions than traditional ib methods the method presented here is restricted to periodic computational domains its generalization to non periodic domains is important future work

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

quantization of classical dynamical r matrices with nonabelian base we construct some classes of dynamical r matrices over a nonabelian base and quantize some of them by constructing dynamical pseudo twists in the sense of xu this way we obtain quantizations of r matrices obtained in earlier work of the second author with schiffmann and varchenko a part of our construction may be viewed as a generalization of the donin mudrov nonabelian fusion construction we apply these results to the construction of equivariant star products on poisson homogeneous spaces which include some homogeneous spaces introduced by de concini

an efficient conservative cut cell method for rigid bodies interacting with viscous compressible flows a cartesian cut cell method for viscous flows interacting with freely moving boundaries is presented the method enables a sharp resolution of the embedded boundaries and strictly conserves mass momentum and energy a new explicit runge kutta scheme pc rk is introduced by which the overall computational time is reduced by a factor of up to 2 5 the new scheme is a predictor corrector type reformulation of a popular class of runge kutta methods which substantially reduces the computational effort for tracking the moving boundaries and subsequently reinitializing the solver impairing neither stability nor accuracy the structural motion is computed by an implicit scheme with good stability properties due to a strong coupling strategy and the conservative discretization of the flow solver at the material interfaces a new formulation for the treatment of small cut cells is proposed with high accuracy and robustness for arbitrary geometries based on a weighted taylor series approach solved via singular value decomposition the efficiency and the accuracy of the new method are demonstrated for several three dimensional cases of laminar and turbulent particulate flow it is shown that the new method remains fully conservative even for large displacements of the boundaries leading to a fast convergence of the fluid solid coupling while spurious force oscillations inherent to this class of methods are effectively suppressed the results substantiate the good stability and accuracy properties of the scheme even on relatively coarse meshes an efficient cartesian cut cell method for freely moving solid boundaries is presented the new predictor corrector runge kutta scheme pc rk enables simulation speedups of up to 2 5 mass momentum and energy are strictly conserved at the sharply resolved solid fluid interfaces the method exhibits very good stability and accuracy properties even for large displacements of the boundaries spurious force oscillations inherent to this class of methods are suppressed