lyapunov based boundary feedback design for parabolic pdes this paper presents a methodology for the construction of simple control lyapunov functionals clfs for boundary controlled parabolic partial differential equations pdes the proposed methodolog

an implicit immersed boundary method for three dimensional fluid membrane interactions we present an implicit immersed boundary method for the incompressible navier stokes equations capable of handling three dimensional membrane fluid flow interactions the goal of our approach is to greatly improve the time step by using the jacobian free newton krylov method jfnk to advance the location of the elastic membrane implicitly the most attractive feature of this jacobian free approach is newton like nonlinear convergence without the cost of forming and storing the true jacobian the generalized minimal residual method gmres which is a widely used krylov subspace iterative method is used to update the search direction required for each newton iteration each gmres iteration only requires the action of the jacobian in the form of matrix vector products and therefore avoids the need of forming and storing the jacobian matrix explicitly once the location of the boundary is obtained the elastic forces acting at the discrete nodes of the membrane are computed using a finite element model we then use the immersed boundary method to calculate the hydrodynamic effects and fluid structure interaction effects such as membrane deformation the present scheme has been validated by several examples including an oscillatory membrane initially placed in a still fluid capsule membranes in shear flows and large deformation of red blood cells subjected to stretching force

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 fast numerical method for solving the three dimensional stokes equations in the presence of suspended particles abstract a new fast numerical method for solving the three dimensional stokes equations in the presence of suspended particles is presented the fluid dynamics equations are solved on a lattice a particle is represented by a set of points each of which moves at the local fluid velocity and is not constrained to lie on the lattice these points are coupled by forces which resist deformation of the particle these forces contribute to the force density in the stokes equations as a result a single set of fluid dynamics equations holds at all points of the domain and there are no internal boundaries particle size shape and deformability may be prescribed computational work increases only linearly with the number of particles so large numbers 500 1000 of particles may be studied efficiently the numerical method involves implicit calculation of the particle forces by minimizing an energy function and solution of a finite difference approximation to the stokes equations using the fourier toeplitz method the numerical method has been implemented to run on all cray computers the implementation exploits the cray s vectorized arithmetic and on machines with insufficient central memory it performs efficient disk 1 o while storing most of the data on disk applications of the method to sedimentation of one two and many particle systems are described trajectories and settling speeds for two particle sedimentation and settling speed for multiparticle sedimentation from initial distributions on a cubic lattice or at random give good quantitative agreement with existing theories

the immersed interface method for simulating prescribed motion of rigid objects in an incompressible viscous flow in the immersed interface method a boundary immersed in a fluid is represented as a singular force in the navier stokes equations this paper presents an explicit approach for computing the singular force to enforce prescribed motion of a rigid boundary in an incompressible viscous flow the tangential component of the singular force is related to the surface vorticity and is calculated from the normal derivative of the velocity the normal component of the singular force is determined from a predictor and a corrector the predictor uses the normal derivative of the vorticity the corrector superposes a homogeneous solution to the pressure poisson equation to achieve the desired normal derivative of the pressure in the current immersed interface method the velocity and the pressure are solved using the mac scheme with the incorporation of jump conditions induced by the singular force and a discontinuous finite body force the body force is applied to obtain the rigid motion of the fluid enclosed by the boundary circular couette flow flow past a cylinder and flow around flappers are simulated to test the accuracy stability and efficiency of the method as well as the effect of the corrector with no stiff springs to model rigid boundaries the method is stable at relatively high reynolds numbers

a comparison of implicit solvers for the immersed boundary equations explicit time discretizations of the immersed boundary method are known to require small timesteps to maintain stability a number of implicit methods have been introduced to alleviate this restriction to allow for a more ecient method but many of these methods still have a stability restriction on the timestep furthermore almost no comparisons have appeared in the literature of the relative computational costs of the implicit methods and the explicit method a recent paper e p newren a l fogelson r d guy and r m kirby unconditionally stable discretizations of the immersed boundary equations journal of computational physics vol 222 702 719 2007 addressed the confusion over stability of immersed boundary discretizations this paper identified the cause of instability in previous immersed boundary discretizations as lack of conservation of energy and introduced a new semi implicit discretization proven to be unconditionally stable i e it has bounded discrete energy the current paper addresses the issue of the eciency of the implicit solvers existing and new methods to solve implicit immersed boundary equations are described systematic comparisons of computational cost are presented for a number of these solution methods for our stable semi implicit immersed boundary discretization and an explicit discretization for two distinct test problems these comparisons show that two of the implicit methods are at least competitive with the explicit method on one test problem and outperform it on the other test problem in which the elastic stiness of the boundary does not dictate the timescale of the fluid motion

closed form boundary state feedbacks for a class of 1 d partial integro differential equations in this paper a problem of boundary stabilization of a class of linear parabolic partial integro differential equations p i des in one dimension is considered using the method of backstepping avoiding spatial discretization required in previous efforts the problem is formulated as a design of an integral operator whose kernel is required to satisfy a hyperbolic p i de the kernel p i de is then converted into an equivalent integral equation and by applying the method of successive approximations the equation s well posedness and the kernel s smoothness are established it is shown how to extend this approach to design optimally stabilizing controllers an adaptation mechanism is developed to reduce the conservativeness of the inverse optimal controller and the performance bounds are derived for a broad range of physically motivated special cases feedback laws are constructed explicitly and the closed loop solutions are found in closed form a numerical scheme for the kernel p i de is proposed its numerical effort compares favorably with that associated with operator riccati equations

a boundary condition capturing immersed interface method for 3d rigid objects in a flow to simulate the flow around an object we can replace the object with the fluid enclosed by a singular force we can then simulate the flow on a fixed domain with a fluid fluid interface supporting the singular force in this paper we present a boundary condition capturing approach to determine the singular force for a 3d rigid object we apply a discontinuous body force to enforce the rigid motion of the fluid replacing the object and compute the singular force based on the kinematics of the object due to the singular force and the body force the flow is not smooth across the interface we solve the flow using the immersed interface method our boundary condition capturing immersed interface method is very efficient and stable and its accuracy based on the infinity norm is near second order for the velocity and above first order for the pressure

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

topological phenomena in the real periodic sine gordon theory the set of real finite gap sine gordon solutions corresponding to a fixed spectral curve consists of several connected components a simple explicit description of these components obtained by the authors recently is used to study the consequences of this property in particular this description allows to calculate the topological charge of solutions the averaging of the x derivative of the potential and to show that the averaging of other standard conservation laws is the same for all components

stability analysis for the immersed fiber problem a linear stability analysis is performed on a two dimensional version of the immersed fiber problem formulated by c peskin to model the flow of fluid in the presence of a mesh of moving elastic fibers the purpose of the analysis is to isolate the modes in the solution which are associated with the fiber and thereby determine the effect of the presence of a fiber on the fluid the results are used not only to make conclusions about the stability of the problem but also to suggest guidelines for developing numerical methods for flows with immersed fibers

the dynamics of an elastic membrane using the impulse method a validation study of the impulse method applied to the motion of a closed elastic membrane under tension separating two incompressible inviscid fluids in two dimensions is presented the approach consists of a nonlinear analysis based on a small amplitude perturbation of an exact solution the equations of motion for the fourier coefficients of the solution are developed to two orders beyond the leading order problem the nonlinear terms in the equations depict the coupling of the fourier modes and account for the temporal variation of the tension the last order of the expansion is used to compute frequency corrections to the driving modes solutions for various problems are found and compared with a numerical method based on impulse variables the results show that the numerical periods and amplitudes of the oscillations approach the values predicted by the perturbation analysis as the numerical smoothing parameter is reduced this validates the use of impulse methods for free boundary motion with surface forces

implicit second order immersed boundary methods with boundary mass abstract the immersed boundary method is a computational framework for problems involving the interaction of a fluid and immersed elastic structures immersed boundary computations typically evaluate the elastic forces explicitly in the configuration of the immersed elastic structure in many applications this results in a severe restriction on the time step we present a semi implicit and a fully implicit second order accurate immersed boundary method the methods provide a natural way to handle mass on the immersed elastic structures we demonstrate their performance for a prototypical fluid structure interaction problem the methods are shown to possess superior stability properties that significantly alleviate the typically severe time step restriction of explicit computations

a method for analysis of stability of flows in ribbed annuli a spectrally accurate algorithm for the linear stability analysis of pressure gradient driven flows in corrugated annuli is presented the algorithm is suitable to analyze three dimensional disturbances in the form of spiral traveling waves as well as in the form of streamwise vortices a separate algorithm for the analysis of axisymmetric disturbances is presented the discretization method relies on fourier expansions in the streamwise and circumferential directions and on chebyshev expansions in the radial direction the immersed boundary conditions ibc method is used to enforce the physical boundary conditions at the corrugated walls numerous tests confirm the spectral accuracy of the results the performance of the algorithm is consistent with the theoretical predictions

estimation of boundary parameters in general heterodirectional linear hyperbolic systems abstract in this paper we extend recent results on state and boundary parameter estimation in coupled systems of linear partial differential equations pdes of the hyperbolic type consisting of n rightward and one leftward convecting equations to the general case which involves an arbitrary number of pdes convecting in both directions two adaptive observers are derived based on swapping design where one introduces a set of filters that can be used to express the system states as linear static combinations of the filter states and the unknown parameters standard parameter identification laws can then be applied to estimate the unknown parameters one observer which requires sensing at both boundaries generates estimates of the boundary parameters and system states while the second observer estimates the parameters from sensing limited to the boundary anti collocated with the uncertain parameters proof of boundedness of the adaptive laws is offered and sufficient conditions ensuring exponential convergence are derived the theory is verified in simulations

compensation of wave actuator dynamics for nonlinear systems the problem of stabilization of pde ode cascades has been solved in the linear case for several pde classes whereas in the nonlinear case the problem has been solved only for the transport delay pde namely for compensation of an arbitrary delay at the input of a nonlinear plant motivated by a specific engineering application in off shore drilling we solve the problem of stabilization of the cascade of a wave pde with a general nonlinear ode due to the presence of nonlinearities of arbitrary growth and the time reversibility of the wave pde and due to the possibility of using arguments based on lyapunov functionals or explicit solutions several stability analysis approaches are possible we present stability results in the h 2 times h 1 and c 1 times c 0 norms for general nonlinear odes as well as in the h 1 times l 2 norm for linear odes we specialize our general design for wave pde ode cascades to the case of a wave pde whose uncontrolled end does not drive an ode but is instead governed by a nonlinear robin boundary condition a nonlinear spring as in the friction law in drilling this is the first global stabilization result for wave equations that incorporate non collocated destabilizing nonlinearities of superlinear growth we present two numerical examples one with a nonlinear ode and one with a nonlinear spring at the uncontrolled boundary of the wave pde

an efficient semi implicit immersed boundary method for the navier stokes equations the immersed boundary method is one of the most useful computational methods in studying fluid structure interaction on the other hand the immersed boundary method is also known to require small time steps to maintain stability when solved with an explicit method many implicit or approximately implicit methods have been proposed in the literature to remove this severe time step stability constraint but none of them give satisfactory performance in this paper we propose an efficient semi implicit scheme to remove this stiffness from the immersed boundary method for the navier stokes equations the construction of our semi implicit scheme consists of two steps first we obtain a semi implicit discretization which is proved to be unconditionally stable this unconditionally stable semi implicit scheme is still quite expensive to implement in practice next we apply the small scale decomposition to the unconditionally stable semi implicit scheme to construct our efficient semi implicit scheme unlike other implicit or semi implicit schemes proposed in the literature our semi implicit scheme can be solved explicitly in the spectral space thus the computational cost of our semi implicit schemes is comparable to that of an explicit scheme our extensive numerical experiments show that our semi implicit scheme has much better stability property than an explicit scheme this offers a substantial computational saving in using the immersed boundary 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

a three dimensional computational method for blood flow in the heart 1 immersed elastic fibers in a viscous incompressible fluid abstract this paper describes the numerical solution of the 3 dimensional equations of motion of a viscous incompressible fluid that contains an immersed system of elastic fibers implementation details such as vectorization and the efficient use of external memory are discussed the method is applied to the damped vibrations of a fiber wound toroidal tube and empirical evidence of convergence is presented

modeling elastic shells immersed in fluid we describe a numerical method to simulate an elastic shell immersed in a viscous incompressible fluid the method is developed as an extension of the immersed boundary method using shell equations based on the kirchhoff love and the planar stress hypotheses a detailed derivation of the shell equations used in the numerical method is presented this derivation as well as the numerical method uses techniques of differential geometry our main motivation for developing this method is its use in constructing a comprehensive threedimensional computational model of the cochlea the inner ear the central object of study within the cochlea is the basilar membrane which is immersed in fluid and whose elastic properties rather resemble those of a shell we apply the method to a specific example which is a prototype of a piece of the basilar membrane and study the convergence of the method in this case some typical features of cochlear mechanics are already captured in this simple model in particular numerical experiments have shown a traveling wave propagating from the base to the apex of the model shell in response to external excitation in the fluid c 2004 wiley periodicals inc

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

an immersed 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

removing the stiffness of elastic force from the immersed boundary method for the 2d stokes equations the immersed boundary method has evolved into one of the most useful computational methods in studying fluid structure interaction on the other hand the immersed boundary method is also known to suffer from a severe timestep stability restriction when using an explicit time discretization in this paper we propose several efficient semi implicit schemes to remove this stiffness from the immersed boundary method for the two dimensional stokes flow first we obtain a novel unconditionally stable semi implicit discretization for the immersed boundary problem using this unconditionally stable discretization as a building block we derive several efficient semi implicit schemes for the immersed boundary problem by applying the small scale decomposition to this unconditionally stable discretization our stability analysis and extensive numerical experiments show that our semi implicit schemes offer much better stability property than the explicit scheme unlike other implicit or semi implicit schemes proposed in the literature our semi implicit schemes can be solved explicitly in the spectral space thus the computational cost of our semi implicit schemes is comparable to that of an explicit scheme but with a much better stability property

a 3d immersed interface method for fluid solid interaction abstract in immersed interface methods solids in a fluid are represented by singular forces in the navier stokes equations and flow jump conditions induced by the singular forces directly enter into numerical schemes this paper focuses on the implementation of an immersed interface method for simulating fluid solid interaction in 3d the method employs the mac scheme for the spatial discretization the rk4 scheme for the time integration and an fft based poisson solver for the pressure poisson equation a fluid solid interface is tracked by lagrangian markers intersections of the interface with mac grid lines identify finite difference stencils on which jump contributions to finite difference schemes are needed to find the intersections and to interpolate jump conditions from the lagrangian markers to the intersections parametric triangulation of the interface is used the velocity of the lagrangian markers is interpolated directly from surrounding mac grid nodes with interpolation schemes accounting for jump conditions numerical examples demonstrate that 1 the method has near second order accuracy in the infinity norm for velocity and the accuracy for pressure is between first and second order 2 the method conserves the volume enclosed by a no penetration boundary and 3 the method can efficiently handle multiple moving solids with ease

two dimensional simulations of valveless pumping using the immersed boundary method flow driven by pumping without valves is examined motivated by biomedical applications cardiopulmonary resuscitation cpr and the human fetus before the development of the heart valves the direction of flow inside a loop of tubing which consists of almost rigid and flexible parts is investigated when the boundary of one end of the flexible segment is forced periodically in time despite the absence of valves net flow around the loop may appear in these simulations the magnitude and even the direction of this flow depend on the driving frequency of the periodic forcing

control of 1d parabolic pdes with volterra nonlinearities part ii analysis for a class of stabilizing boundary controllers for nonlinear 1d parabolic pdes introduced in a companion paper we derive bounds for the gain kernels of our nonlinear volterra controllers prove the convergence of the series in the feedback laws and establish the stability properties of the closed loop system we show that the state transformation is at least locally invertible and include an explicit construction for computing the inverse of the transformation using the inverse we show l 2 and h 1 exponential stability and explicitly construct the exponentially decaying closed loop solutions we then illustrate the theoretical results on an analytically tractable example

control of 1 d parabolic pdes with volterra nonlinearities part i design boundary control of nonlinear parabolic pdes is an open problem with applications that include fluids thermal chemically reacting and plasma systems in this paper we present stabilizing control designs for a broad class of nonlinear parabolic pdes in 1 d our approach is a direct infinite dimensional extension of the finite dimensional feedback linearization backstepping approaches and employs spatial volterra series nonlinear operators both in the transformation to a stable linear pde and in the feedback law the control law design consists of solving a recursive sequence of linear hyperbolic pdes for the gain kernels of the spatial volterra nonlinear control operator these pdes evolve on domains t n of increasing dimensions n 1 and with a domain shape in the form of a hyper pyramid 0 x n x n 1 x 1 x 1 we illustrate our design method with several examples one of the examples is analytical while in the remaining two examples the controller is numerically approximated for all the examples we include simulations showing blow up in open loop and stabilization for large initial conditions in closed loop in a companion paper we give a theoretical study of the properties of the transformation showing global convergence of the transformation and of the control law nonlinear volterra operators and explicitly constructing the inverse of the feedback linearizing volterra transformation this in turn allows us to prove l 2 and h 1 local exponential stability with an estimate of the region of attraction where possible and explicitly construct the exponentially decaying closed loop solutions

the geometry of real sine gordon wavetrains the characterization ofreal n phase quasiperiodic solutions of the sine gordon equation has been an open problem in this paper we achieve this result employing techniques of classical algebraic geometry which have not previously been exploited in the soliton literature a significant by product of this approach is a naturalalgebraic representation of the full complex isospectral manifolds and an understanding of how the real isospectral manifolds are embedded by placing the problem in this general context these methods apply directly to all soliton equations whose multiphase solutions are related to hyperelliptic functions

topological charge of the real periodic finite gap sine gordon solutions an effective description of the inverse spectral data corresponding to the real periodic and quasi periodic finite gap algebro geometrical solutions for the sine gordon equation is obtained in particular the explicit formula for the so called topological charge of the solutions is found and proven as was already understood 20 years ago it is very hard to extract any formula for this quantity from the functional expressions a new method was developed by the authors for this goal in appendix c an analogue of the fourier laurent integral transform on the riemann surface is defined based on the constructions of soliton theory it is a natural continuous analogue of the discrete krichever novikov bases developed in the late 1980s for the needs of the operator quantization of string theory 2003 wiley periodicals inc

a volterra series representation for a class of nonlinear infinite dimensional systems with periodic boundary conditions abstract this paper proves the existence of a volterra series representation for the mild solutions of a class of nonlinear infinite dimensional systems more specifically given the evolutionary system operator u t s 0 s t associated with a semilinear evolution equation u t 2 u x 2 f u u 0 u 0 x with periodic boundary conditions it is proved that under suitable conditions the unique mild solution u t u t 0 u 0 t 0 can be expanded by a volterra series a recursive algorithm is given to construct the volterra kernels series terms and a nonlinear heat equation is discussed to illustrate the proposed method

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

a fast robust and non stiff immersed boundary method we propose a fast and non stiff approach for the solutions of the immersed boundary method for newtonian incompressible flows in two or three dimensions the proposed methodology is built on a robust semi implicit discretization introduced by peskin in the late 70s which is solved efficiently through the novel use of a fast treecode strategy to compute flow structure interactions optimal multipole type expansions are performed numerically by solving a least squares problem with a new fast iterative algorithm the new immersed boundary method is particularly well suited for three dimensional applications and or for problems where the number of immersed boundary points is large we demonstrate the efficacy and superiority of the method over existing approaches with two simple but illustrative examples in 3d

truncated newton methods and the modeling of complex immersed elastic structures truncated newton minimization methods are combined with peskin s immersed boundary method to facilitate investigation of the dynamic interaction between a viscous incompressible fluid and immersed elastic objects of complex structure applications to aquatic animal locomotion and platelet aggregation during blood clotting are presented 1993 john wiley sons inc

guaranteed cost design for controlling semilinear parabolic pde systems with mobile collocated actuators and sensors this paper investigates the guaranteed cost design problem for controlling a class of semilinear parabolic partial differential equation pde systems using mobile collocated actuators and sensors

periodic solutions of the kdv equation in this paper we construct a large family of special solutions of the kdv equation which are periodic in x and almost periodic in t these solutions lie on n dimensional tori very likely they are dense among all solutions the special solutions are characterized variationally they minimize fn u subject to the constraints fj u aj j 1 n 1 here fj denote the remarkable sequence of conserved functionals discovered by kruskal and zabusky the above minimum problem was originally suggested by them in exploring the manifold of solutions of this minimum problem we make essential use of gardner s discovery that these functionals are in involution with respect to a suitable poisson bracket gardner greene kruskal and miura have shown that the eigenvalues of the schrodinger operator are conserved functionals if the potential is a function of t and satisfies the kdv equation in section 6 a new set of conserved quantities is constructed which serve as a link between the eigenvalues of the schrodinger operator and the fj another result in section 6 is a slight sharpening of an earlier result of the author and j moser for the special solutions constructed above all but 2n 1 eigenvalues of the schrodinger operator are double the simplest class of special solutions n 1 are cnoidal waves in an appendix m hyman describes the results of computing numerically the next simplest case n 2 these calculations show that the shape of these solutions recurs exactly after a finite time in a shifted position the theory verifies this fact

hamiltonian structure for the modulation equations of a sine gordon wavetrain on etudie l equation sinus gordon e 2 t 2 x 2 u sin u 0 ou e est un petit parametre on construit une solution asymptotique sous la forme d un train d onde n phase variant lentement

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

boundary feedback stabilization of the schlogl system the schlogl system is governed by a nonlinear reaction diffusion partial differential equation with a cubic nonlinearity that determines three constant equilibrium states it is a classical example of a chemical reaction system that is bistable the constant equilibrium that is enclosed by the other two constant equilibrium points is unstable in this paper robin boundary feedback laws are presented that stabilize the system in a given stationary state or more generally in a given time dependent desired system orbit the exponential stability of the closed loop system with respect to the l 2 norm is proved in particular it is shown that with the boundary feedback law the unstable constant equilibrium point can be stabilized

a poroelastic immersed boundary method with applications to cell biology the immersed boundary method is a widely used mixed eulerian lagrangian framework for simulating the motion of elastic structures immersed in viscous fluids in the traditional immersed boundary method the fluid and structure move with the same velocity field in this work a model based on the immersed boundary method is presented for simulating poroelastic media in which the fluid permeates a porous elastic structure of small volume fraction that moves with its own velocity field two distinct methods for calculating elastic stresses are presented and compared the methods are validated on a radially symmetric test problem by comparing with a finite difference solution of the classical equations of poroelasticity finally two applications of the modeling framework to cell biology are provided cellular blebbing and cell crawling it is shown that in both examples poroelastic effects are necessary to explain the relevant mechanics

a cartesian treecode for screened coulomb interactions a treecode algorithm is presented for evaluating electrostatic potentials in a charged particle system undergoing screened coulomb interactions in 3d the method uses a far field taylor expansion in cartesian coordinates to compute particle cluster interactions the taylor coefficients are evaluated using new recurrence relations which permit efficient computation of high order approximations two types of clusters are considered uniform cubes and adapted rectangular boxes the treecode error cpu time and memory usage are reported and compared with direct summation for randomly distributed particles inside a cube on the surface of a sphere and on an 8 sphere configuration for a given order of taylor approximation the treecode cpu time scales as o n log n and the memory usage scales as o n where n is the number of particles results show that the treecode is well suited for non homogeneous particle distributions as in the sphere and 8 sphere test cases

a spectral element formulation of the immersed boundary method for newtonian fluids a spectral element formulation of the immersed boundary method ibm is presented the spectral element formulation se ibm is a generalisation of the finite element immersed boundary method fe ibm based on high order approximations of the fluid variables several schemes for tracking the movement of the immersed boundary are considered and a semi implicit euler scheme is shown to offer advantages in terms of accuracy and efficiency high order spectral element approximations provide improved area conservation properties of the ibm due to the incompressibility constraint being more accurately satisfied superior orders of convergence are obtained for se ibm compared with fe ibm in both l2 and h1 norms the area conservation and convergence properties of the scheme are demonstrated on a series of benchmark problems

unconditionally stable discretizations of the immersed boundary equations the immersed boundary ib method is known to require small timesteps to maintain stability when solved with an explicit or approximately implicit method many implicit methods have been proposed to try to mitigate this timestep restriction but none are known to be unconditionally stable and the observed instability of even some of the fully implicit methods is not well understood in this paper we prove that particular backward euler and crank nicolson like discretizations of the nonlinear immersed boundary terms of the ib equations in conjunction with unsteady stokes flow can yield unconditionally stable methods we also show that the position at which the spreading and interpolation operators are evaluated is not relevant to stability so as long as both operators are evaluated at the same location in time and space we further demonstrate through computational tests that approximate projection methods which do not provide a discretely divergence free velocity field appear to have a stabilizing influence for these problems and that the implicit methods of this paper when used with the full navier stokes equations are no longer subject to such a strict timestep restriction and can be run up to the cfl constraint of the advection terms

nonlinear stabilization of shock like unstable equilibria in the viscous burgers pde we stabilize the unstable ldquoshock likerdquo equilibrium profiles of the viscous burgers equation using control at the boundaries these equilibria are not stabilizable even locally using the standard ldquoradiation feedback boundary conditions rdquo using a nonlinear spatially scaled transformation that employs three ingredients of which one is the hopf cole nonlinear integral transformation and linear backstepping we design an explicit nonlinear full state control law that achieves exponential stability with a region of attraction for which we give an estimate the region of attraction is not the entire state space since the burgers pde is known not to be globally controllable

efficient solutions to robust semi implicit discretizations of the immersed boundary method the immersed boundary method is a versatile tool for the investigation of flow structure interaction in a large number of applications the immersed boundaries or structures are very stiff and strong tangential forces on these interfaces induce a well known severe time step restriction for explicit discretizations this excessive stability constraint can be removed with fully implicit or suitable semi implicit schemes but at a seemingly prohibitive computational cost while economical alternatives have been proposed recently for some special cases there is a practical need for a computationally efficient approach that can be applied more broadly in this context we revisit a robust semi implicit discretization introduced by peskin in the late 1970s which has received renewed attention recently this discretization in which the spreading and interpolation operators are lagged leads to a linear system of equations for the interface configuration at the future time when the interfacial force is linear however this linear system is large and dense and thus it is challenging to streamline its solution moreover while the same linear system or one of similar structure could potentially be used in newton type iterations nonlinear and highly stiff immersed structures pose additional challenges to iterative methods in this work we address these problems and propose cost effective computational strategies for solving peskin s lagged operators type of discretization we do this by first constructing a sufficiently accurate approximation to the system s matrix and we obtain a rigorous estimate for this approximation this matrix is expeditiously computed by using a combination of pre calculated values and interpolation the availability of a matrix allows for more efficient matrix vector products and facilitates the design of effective iterative schemes we propose efficient iterative approaches to deal with both linear and nonlinear interfacial forces and simple or complex immersed structures with tethered or untethered points one of these iterative approaches employs a splitting in which we first solve a linear problem for the interfacial force and then we use a nonlinear iteration to find the interface configuration corresponding to this force we demonstrate that the proposed approach is several orders of magnitude more efficient than the standard explicit method in addition to considering the standard elliptical drop test case we show both the robustness and efficacy of the proposed methodology with a 2d model of a heart valve

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

analysis of a one dimensional model for the immersed boundary method numerical methods are studied for the one dimensional heat equation with a singular forcing term u t u xx c t delta x alpha t the delta function delta x is replaced by a discrete approximation d h x and the resulting equation is solved by a crank nicolson method on a uniform grid the accuracy of this method is analyzed for various choices of d h the case where c t is specified and also the case where c is determined implicitly by a constraint on the solution at the point a are studied these problems serve as a model for the immersed boundary method of peskin for incompressible flow problems in irregular regions some insight is gained into the accuracy that can be achieved and the importance of choosing appropriate discrete delta functions

simulations of the whirling instability by the immersed boundary method when an elastic filament spins in a viscous incompressible fluid it may undergo a whirling instability as studied asymptotically by wolgemuth powers and goldstein phys rev lett 84 2000 pp 16 23 we use the immersed boundary ib method to study the interaction between the elastic filament and the surrounding viscous fluid as governed by the incompressible navier stokes equations this allows the study of the whirling motion when the shape of the filament is very different from the unperturbed straight state

computational methods for continuum models of platelet aggregation platelet aggregation plays an important role in blood clotting robust numerical methods for simulating the behavior of fogelson s continuum models of platelet aggregation have been developed which in particular involve a hybrid finite difference and spectral method for the models link evolution equation this partial differential equation involvesfourspatial dimensions and time the new methods are used to begin investigating the influence of new chemically induced activation link formation and shear induced link breaking in determining when aggregates develop sufficient strength to remain intact and when they are broken apart by fluid stresses

a computational model of the cochlea using the immersed boundary method this thesis describes a two dimensional computational model of the cochlea inner ear solved using a modified form of peskin s immersed boundary method this method is an extension of peskin s method that was originally applied to solving a model of the heart 1972 this method solves the time dependent incompressible navier stokes equations in the presence of immersed boundaries the fluid equations are specified on a fixed eulerian grid while the immersed boundaries are specified on a moving lagrangian grid the immersed boundaries are used to represent the structures of the cochlea the rigid bone the actively moving stapes and the flexible membranes the immersed boundaries exert forces locally on the fluid these local forces are seen by the fluid as external forces that are added to the other forces pressure and viscous acting on the fluid the computational method divides into three separate parts first the forces exerted by the points of the immersed boundary are calculated and extended to the fluid grid second the fluid equations are solved and the velocities are restricted to the immersed boundary points finally the immersed boundary points are moved to their new locations at the new local velocities the immersed boundary forces are either specified such that the boundary moves at a given velocity zero or non zero or they are specified such that the forces obey some given force law for example hooke s law these forces can be calculated either explicitly or implicitly or a combination of both for the fluid the navier stokes equations are solved on a doubly periodic rectangular grid in a second order accurate manner using a projection method developed by bell colella and glaz 1988 the extension and restriction of the boundary forces and fluid velocities to and from the fluid grid are also done in a second order accurate manner the cochlea is modelled as two fluid chambers separated by a flexible partition whose stiffness varies exponentially along its length the stapes is represented by a moving piston and the bony outer walls are allowed to be either straight or tapered a travelling wave propagates along the flexible partition under the influence of the moving piston and the dependence of this wave on cochlea geometry and partition stiffness is studied also included are studies of bone conduction and transient signal analysis finally comparisons of the model results are made to experimental data and asymptotic results

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

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

a semi implicit level set method for multiphase flows and fluid structure interaction problems in this paper we present a novel semi implicit time discretization of the level set method introduced in 8 for fluid structure interaction problems the idea stems from a linear stability analysis derived on a simplified one dimensional problem the semi implicit scheme relies on a simple filter operating as a pre processing on the level set function it applies to multiphase flows driven by surface tension as well as to fluid structure interaction problems the semi implicit scheme avoids the stability constraints that explicit scheme need to satisfy and reduces significantly the computational cost it is validated through comparisons with the original explicit scheme and refinement studies on two dimensional benchmarks

a front tracking method for viscous incompressible multi fluid flows abstract a method to simulate unsteady multi fluid flows in which a sharp interface or a front separates incompressible fluids of different density and viscosity is described the flow field is discretized by a conservative finite difference approximation on a stationary grid and the interface is explicitly represented by a separate unstructured grid that moves through the stationary grid since the interface deforms continuously it is necessary to restructure its grid as the calculations proceed in addition to keeping the density and viscosity stratification sharp the tracked interface provides a natural way to include surface tension effects both two and three dimensional full numerical simulations of bubble motion are presented

analysis of stiffness in the immersed boundary method and implications for time stepping schemes the immersed boundary method is known to exhibit a high degree of numerical stiffness associated with the interaction of immersed elastic fibres with the surrounding fluid we perform a linear analysis of the underlying equations of motion for immersed fibres and identify a discrete set of fibre modes which are associated solely with the presence of the fibre this work generalises our results in a previous paper 1995 siam j appl math 55 1577 by incorporating the effect of spreading the singular fibre force over a finite smoothing radius corresponding to the approximate delta function used in the immersed boundary method we investigate the stability of the fibre modes their stiffness and their dependence on the problem parameters focusing on the influence of smoothing we then extend the analytical results to include the effect of time discretisation and draw conclusions about the time step restrictions on various explicit schemes as well as the convergence rates for an iterative semi implicit method we draw comparisons with computations and show how the results can be applied to help in choosing alternate time stepping schemes that are specially tailored to handle the stiffness in immersed fibres in particular we present numerical results that show how fully explicit runge kutta schemes perform in comparison with the best of the semi implicit schemes currently in use

improved volume conservation in the computation of flows with immersed elastic boundaries this paper introduces a recipe for the construction of a finite difference divergence operator the coefficients of which are completely determined once an interpolation scheme has been chosen substitution of this divergence operator for the previously used divergence based on central differences makes a dramatic improvement in the overall volume conservation that is observed in an immersed boundary computation this improvement is particularly important for computations in which an elastic boundary separates chambers containing fluid at substantially different pressures a situation which is prominent in cardiac fluid dynamics during the contraction of the ventricles

boundary feedback stabilization of an unstable heat equation in this paper we study the problem of boundary feedback stabilization for the unstable heat equation ut x t uxx x t a x u x t this equation can be viewed as a model of a heat conducting rod in which not only is the heat being diffused mathematically due to the diffusive term uxx but also the destabilizing heat is generating mathematically due to the term a u with a 0 we show that for any given continuously differentiable function a and any given positive constant l we can explicitly construct a boundary feedback control law such that the solution of the equation with the control law converges to zero exponentially at the rate of l this is a continuation of the recent work of boskovic krstic and liu ieee trans automat control 46 2001 pp 2022 2028 and balogh and krstic european j control 8 2002 pp 165 176

a lattice boltzmann based implicit immersed boundary method for fluid structure interaction the immersed boundary method is a practical and effective method for fluid structure interaction problems it has been applied to a variety of problems most of the time stepping schemes used in the method are explicit which suffer a drawback in terms of stability and restriction on the time step we propose a lattice boltzmann based implicit immersed boundary method where the immersed boundary force is computed at the unknown configuration of the structure at each time step the fully nonlinear algebraic system resulting from discretizations is solved by an inexact newton krylov method in a jacobian free manner the test problem of a flexible filament in a flowing viscous fluid is considered numerical results show that the proposed implicit immersed boundary method is much more stable with larger time steps and significantly outperforms the explicit version in terms of computational cost

an efficient parallel immersed boundary algorithm using a pseudo compressible fluid solver we propose an efficient algorithm for the immersed boundary method on distributed memory architectures that has the computational complexity of a completely explicit method and also has excellent parallel scaling the algorithm utilizes the pseudo compressibility method recently proposed by guermond and minev that uses a directional splitting strategy to discretize the incompressible navier stokes equations thereby reducing the linear systems to a series of one dimensional tridiagonal systems we perform numerical simulations of several fluid structure interaction problems in two and three dimensions and study the accuracy and convergence rates of the proposed algorithm we also compare the proposed algorithm with other second order projection based fluid solvers lastly the execution time and scaling properties of the proposed algorithm are investigated and compared to alternate approaches

a fast multipole method for the three dimensional stokes equations many problems in stokes flow and linear elasticity require the evaluation of vector fields defined in terms of sums involving large numbers of fundamental solutions in the fluid mechanics setting these are typically the stokeslet the kernel of the single layer potential or the stresslet the kernel of the double layer potential in this paper we present a simple and efficient method for the rapid evaluation of such fields using a decomposition into a small number of coulombic n body problems following an approach similar to that of fu and rodin y fu g j rodin fast solution methods for three dimensional stokesian many particle problems commun numer meth en 16 2000 145 149 while any fast summation algorithm for coulombic interactions can be employed we present numerical results from a scheme based on the most modern version of the fast multipole method h cheng l greengard v rokhlin a fast adaptive multipole algorithm in three dimensions j comput phys 155 1999 468 498 this approach should be of value in both the solution of boundary integral equations and multiparticle dynamics

a discontinuous galerkin based immersed boundary method a numerical method to approximate partial differential equations on meshes that do not conform to the domain boundaries is introduced the proposed method is conceptually simple and free of user defined parameters starting with a conforming finite element mesh the key ingredient is to switch those elements intersected by the dirichlet boundary to a discontinuous galerkin approximation and impose the dirichlet boundary conditions strongly by virtue of relaxing the continuity constraint at those elements boundary locking is avoided and optimal order convergence is achieved this is shown through numerical experiments in reaction diffusion problems copyright 2008 john wiley sons ltd

a multirate time integrator for regularized stokeslets the method of regularized stokeslets is a numerical approach to approximating solutions of fluid structure interaction problems in the stokes regime regularized stokeslets are fundamental solutions to the stokes equations with a regularized point force term that are used to represent forces generated by a rigid or elastic object interacting with the fluid due to the linearity of the stokes equations the velocity at any point in the fluid can be computed by summing the contributions of regularized stokeslets and the time evolution of positions can be computed using standard methods for ordinary differential equations rigid or elastic objects in the flow are usually treated as immersed boundaries represented by a collection of regularized stokeslets coupled together by virtual springs which determine the forces exerted by the boundary in the fluid for problems with boundaries modeled by springs with large spring constants the resulting ordinary differential equations become stiff and hence the time step for explicit time integration methods is severely constrained unfortunately the use of standard implicit time integration methods for the method of regularized stokeslets requires the solution of dense nonlinear systems of equations for many relevant problems here an alternate strategy using an explicit multirate time integration scheme based on spectral deferred corrections is incorporated that in many cases can significantly decrease the computational cost of the method the multirate methods are higher order methods that treat different portions of the ode explicitly with different time steps depending on the stiffness of each component numerical examples on two nontrivial three dimensional problems demonstrate the increased efficiency of the multi explicit approach with no significant increase in numerical error