an averaging model for chaotic system with periodic time varying parameter in this paper we provide a mathematical justification to explain the dynamics of chaotic system with periodic time varying parameter which have been illustrated by some of us in a previous paper 1 based on an equivalent averaging model it is proved that such a parametric time varying system follows the same trajectory of its averaging model provided that the parameter is varied periodically with a sufficiently high frequency some other observations related with this class of chaotic systems are also remarked in this paper

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

simulating the dynamics of inextensible vesicles by the penalty immersed boundary method in this paper we develop an immersed boundary ib method to simulate the dynamics of inextensible vesicles interacting with an incompressible fluid in order to take into account the inextensibility constraint of the vesicle the penalty immersed boundary pib method is used to virtually decouple the fluid and vesicle dynamics as numerical tests of our current pib method the dynamics of single and multiple inextensible vesicles under shear flows have been extensively explored and compared with the previous literature the method is also validated by a series of convergence study which confirms its consistent first order accuracy on the velocity field the vesicle configuration the vesicle area and the perimeter errors in addition the method is also applied to study a binary component vesicle problem

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

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 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

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

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

numerical simulation of a cylinder in uniform flow in this study a virtual boundary technique is applied to the numerical simulation of stationary and moving cylinders in uniform flow this approach readily allows the imposition of a no slip boundary within the flow field by a feedback forcing term added to the momentum equations in the present work this technique is used with a high order finite difference method effectively eliminating spurious oscillations caused by the feedback forcing when used with spectrally discretized flow solvers very good agreement is found between the present calculations and previous computational and experimental results for steady and time dependent flow at low reynolds numbers

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

on positive recurrence of constrained diffusion processes let g c r k be a convex polyhedral cone with vertex at the origin given as the intersection of half spaces g i i 1 n where n i and d i denote the inward normal and direction of constraint associated with g i respectively stability properties of a class of diffusion processes constrained to take values in g are studied under the assumption that the skorokhod problem defined by the data n i d i i 1 n is well posed and the skorokhod map is lipschitz continuous explicit conditions on the drift coefficient b of the diffusion process are given under which the constrained process is positive recurrent and has a unique invariant measure define c i d i i 0 i 1 n then the key condition for stability is that there exists 0 and a bounded subset a of g such that for all x g a b x c and dist b x c where c denotes the boundary of c

simulation of flexible filaments in a uniform flow by the immersed boundary method an improved version of the immersed boundary ib method is developed for simulating flexible filaments in a uniform flow the proposed ib method is based on an efficient navier stokes solver adopting the fractional step method and a staggered cartesian grid system the fluid motion defined on an eulerian grid and the filament motion defined on a lagrangian grid are independently solved and their interaction force is explicitly calculated using a feedback law a direct numerical method is developed to calculate the filament motion under the constraint of inextensibility when applied to the case of a swinging filament analogous to a rope pendulum the proposed method gave results very similar to those of the analytical solution derived using the perturbation method for a flexible filament flapping in a uniform flow the mechanism by which small vortex processions are produced was investigated the bistable property of the system was observed by altering the filament length and the effects of the boundary condition at the fixed end simply supported or clamped were studied for two side by side filaments in a uniform flow both in phase flapping and out of phase flapping were reproduced in the present simulations a repulsive force was included in the formulation to handle collisions between the free ends of side by side filaments undergoing out of phase flapping

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

a switching scheme for synthesizing attractors of dissipative chaotic systems in this paper a periodic parameter switching scheme is proposed for synthesizing a large class of hyperbolic attractors of continuous time and autonomous dissipative chaotic systems depending linearly on a single real bifurcation parameter it is illustrated by numerical simulations that a wide range of hyperbolic attractors can be obtained by this new scheme the scheme can also be considered as an effective way for control and anticontrol of chaos

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

an immersed boundary method for fluid flexible structure interaction an immersed boundary method for simulation of fluid flexible structure interaction has been proposed in the method an efficient navier stokes solver adopting the fractional step method and a staggered cartesian grid system is used to solve the incompressible fluid motion in an eulerian domain on the other hand a moving lagrangian grid is used to discretize the structure domain and the equation of structure motion is derived by the energy method and is solved by an iterative method the fluid structure interaction is formulated by an additional momentum forcing which is obtained on the lagrangian grid using the structure motion equation and is spread to the nearby eulerian girds by applying a smoothed approximation of the dirac delta function in this way the massive boundary is handled easily without any special treatment while for the neutrally buoyant case our formulation becomes equivalent with previous ones in the simulations we mainly use small additional boundary mass toward biofluid applications three numerical examples are simulated a lid driven cavity with a deformable volume at the bottom a swimming jellyfish and a deformable circular ring moving through a channel with contraction the results are compared with those of previous studies and the theoretical solutions the temporal and spatial convergence rates of the present method are specified the tendency of the numerical solution as the additional boundary mass approaches zero is investigated

prediction of wall pressure fluctuation in turbulent flows with an immersed boundary method the objective of this paper is to assess the accuracy and efficiency of the immersed boundary ib method to predict the wall pressure fluctuations in turbulent flows where the flow dynamics in the near wall region is fundamental to correctly predict the overall flow the present approach achieves sufficient accuracy at the immersed boundary and overcomes deficiencies in previous ib methods by introducing additional constraints a compatibility for the interpolated velocity boundary condition related to mass conservation and the formal decoupling of the pressure on this surfaces the immersed boundary approximated domain method ib adm developed in the present study satisfies these conditions with an inexpensive computational overhead the ib adm correctly predicts the near wall velocity pressure and scalar fields in several example problems including flows around a very thin solid object for which incorrect results were obtained with previous ib methods in order to have sufficient near wall mesh resolution for les and dns computations the present approach uses local mesh refinement the present method has been also successfully applied to computation of the wall pressure space time correlation in dns of turbulent channel flow on grids not aligned with the boundaries when applied to a turbulent flow around an airfoil the computed flow statistics the mean rms flow field and power spectra of the wall pressure are in good agreement with experiment

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

parameter switching in a generalized duffing system finding the stable attractors this paper presents a simple periodic parameter switching method which can find any stable limit cycle that can be numerically approximated in a generalized duffing system in this method the initial value problem of the system is numerically integrated and the control parameter is switched periodically within a chosen set of parameter values the resulted attractor matches with the attractor obtained by using the average of the switched values the accurate match is verified by phase plots and hausdorff distance measure in extensive simulations

a pre conditioned implicit direct forcing based immersed boundary method for incompressible viscous flows a novel immersed boundary ib method based on an implicit direct forcing idf scheme is developed for incompressible viscous flows the key idea for the present idf method is to use a block lu decomposition technique in momentum equations with taylor series expansion to construct the implicit ib forcing in a recurrence form which imposes more accurate no slip boundary conditions on the ib surface to accelerate the ib forcing convergence during the iterative procedure a pre conditioner matrix is introduced in the recurrence formulation of the ib forcing a jacobi type parameter is determined in the pre conditioner matrix by minimizing the frobenius norm of the matrix function representing the difference between the ib forcing solution matrix and the pre conditioner matrix in addition the pre conditioning parameter is restricted due to the numerical stability in the recurrence formulation consequently the present pre conditioned idf pidf enables accurate calculation of the ib forcing within a few iterations we perform numerical simulations of two dimensional flows around a circular cylinder and three dimensional flows around a sphere for low and moderate reynolds numbers the result shows that pidf yields a better imposition of no slip boundary conditions on the ib surfaces for low reynolds number with a fairly larger time step than ib methods with different direct forcing schemes due to the implicit treatment of the diffusion term for determining the ib forcing finally we demonstrate the robustness of the present pidf scheme by numerical simulations of flow around a circular array of cylinders flows around a falling sphere and two sedimenting spheres in gravity

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