applications of field theoretic renormalization group methods to reaction diffusion problems we review the application of field theoretic renormalization group rg methods to the study of fluctuations in reaction diffusion problems we first investigate the physical origin of universality in these systems before comparing rg methods to other available analytic techniques including exact solutions and smoluchowski type approximations starting from the microscopic reaction diffusion master equation we then pedagogically detail the mapping to a field theory for the single species reaction ka a k we employ this particularly simple but non trivial system to introduce the field theoretic rg tools including the diagrammatic perturbation expansion renormalization and callan symanzik rg flow equation we demonstrate how these techniques permit the calculation of universal quantities such as density decay exponents and amplitudes via perturbative dc d expansions with respect to the upper critical dimension dc with these basics established we then provide an overview of more sophisticated applications to multiple species reactions disorder effects l evy flights persistence problems and the influence of spatial boundaries we also analyse field theoretic approaches to non equilibrium phase transitions separating active from absorbing states we focus particularly on the generic directed percolation universality class as well as on the most prominent exception to this class even offspring branching and annihilating random walks finally we summarize the state of the field and present our perspective on outstanding problems for the future

classification of phase transitions in reaction diffusion models equilibrium phase transitions are associated with rearrangements of minima of a lagrangian potential treatment of nonequilibrium systems requires doubling of degrees of freedom which may be often interpreted as a transition from the coordinate to the phase space representation as a result one has to deal with the hamiltonian formulation of the field theory instead of the lagrangian one we suggest a classification scheme of phase transitions in reaction diffusion models based on the topology of the phase portraits of corresponding hamiltonians in models with an absorbing state such a topology is fully determined by intersecting curves of zero energy we identify four families of topologically distinct classes of phase portraits stable upon renormalization group transformations

a finite difference galerkin formulation for the incompressible navier stokes equations abstract the development of a new computational method for solving the incompressible navier stokes equations in primitive variable form is presented it is found that certain finite difference approximations for these equations can be transformed into an equivalent system which efficiently determines the discrete velocity field and which completely eliminates the pressure two such difference schemes for two dimensional problems are examined and some preliminary numerical results are discussed for the steady driven cavity problem

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

an unsplit higher order godunov method for scalar conservation laws in multiple dimensions abstract in this paper we develop an unsplit higher order godunov method for scalar conservation laws in two dimensions the method represents an extension for the special case being considered of methods developed by colella and by van leer in our method we begin with a piecewise bilinear representation of the solution on each grid cell a new piecewise bilinear representation at the next time level is then obtained from a two step procedure in the first step a conservative predictor corrector scheme derived from the integral form of the differential equation and characteristic considerations is used to obtain average values of the solution over grid cells at the new time level next these new average values are then used to construct a limited piecewise bilinear profile for each cell at the new time level the resulting method is shown to satisfy a maximum principle for constant coefficient linear advection computational results are presented comparing the new method to colella s method for linear advection the method is also applied to two model problems from porous media flow miscible and immiscible displacement the new scheme provides accurate resolution of sharp fronts without any significant distortion

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

stable and accurate pressure approximation for unsteady incompressible viscous flow how to properly specify boundary conditions for pressure is a longstanding problem for the incompressible navier stokes equations with no slip boundary conditions an analytical resolution of this issue stems from a recently developed formula for the pressure in terms of the commutator of the laplacian and leray projection operators here we make use of this formula to a improve the accuracy of computing pressure in two kinds of existing time discrete projection methods implicit in viscosity only and b devise new higher order accurate time discrete projection methods that extend a slip correction idea behind the well known finite difference scheme of kim and moin we test these schemes for stability and accuracy using various combinations of c 0 finite elements for all three kinds of time discretization one can obtain third order accuracy for both pressure and velocity without a time step stability restriction of diffusive type furthermore two kinds of projection methods are found stable using piecewise linear elements for both velocity and pressure

a quadratic spline based interface quasi reconstruction algorithm for accurate tracking of two phase flows a new quadratic spline based interface quasi reconstruction algorithm is presented which provides an accurate and continuous representation of the interface in a multiphase domain and facilitates the direct estimation of local interfacial curvature the fluid interface in each of the mixed cells is represented by piecewise parabolic curves and an initial discontinuous plic approximation of the interface is progressively converted into a smooth quadratic spline made of these parabolic curves the conversion is achieved by a sequence of predictor corrector operations enforcing function c 0 and derivative c 1 continuity at the cell boundaries using simple analytical expressions for the continuity requirements the efficacy and accuracy of the current algorithm has been demonstrated using standard test cases involving reconstruction of known static interface shapes and dynamically evolving interfaces in prescribed flow situations these benchmark studies illustrate that the present algorithm performs excellently as compared to the other interface reconstruction methods available in literature quadratic rate of error reduction with respect to grid size has been observed in all the cases with curved interface shapes only in situations where the interface geometry is primarily flat the rate of convergence becomes linear with the mesh size the flow algorithm implemented in the current work is designed to accurately balance the pressure gradients with the surface tension force at any location as a consequence it is able to minimize spurious flow currents arising from imperfect normal stress balance at the interface this has been demonstrated through the standard test problem of an inviscid droplet placed in a quiescent medium finally the direct curvature estimation ability of the current algorithm is illustrated through the coupled multiphase flow problem of a deformable air bubble rising through a column of water

on convergence of block centered finite differences for elliptic problems we consider linear selfadjoint elliptic problems with neumann boundary conditions in rectangular domains we demonstrate that with sufficiently smooth data the discrete l 2 norm errors for tensor product block centered finite differences in both the approximate solution and its first derivatives are second order for all nonuniform grids extensions to nonselfadjoint and parabolic problems are discussed

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

predictive information in a nonequilibrium critical model we propose predictive information that is information between a long past of duration t and the entire infinitely long future of a time series as a general order parameter to study phase transitions in physical systems independently of the underlying dynamics it can be used in particular to study nonequilibrium transitions and other exotic transitions where a simpler order parameter cannot be identified using traditional symmetry arguments as an example we calculate predictive information for a stochastic nonequilibrium dynamics problem that forms an absorbing state under a continuous change of a parameter the information at the transition point diverges as logt and we calculate the expression for a smooth crossover to t0 away from the transition

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

a cartesian grid embedded boundary method for hyperbolic conservation laws we present a second order godunov algorithm to solve time dependent hyperbolic systems of conservation laws on irregular domains our approach is based on a formally consistent discretization of the conservation laws on a finite volume grid obtained from intersecting the domain with a cartesian grid we address the small cell stability problem associated with such methods by hybridizing our conservative discretization with a stable nonconservative discretization at irregular control volumes and redistributing the difference in the mass increments to nearby cells in a way that preserves stability and local conservation the resulting method is second order accurate in l 1 for smooth problems and is robust in the presence of large amplitude discontinuities intersecting the irregular boundary

a new family of 5 5 cc 4oc schemes applicable for unsteady navier stokes equations in this paper a new family of implicit compact finite difference schemes for computation of unsteady convection diffusion equation with variable convection coefficient is proposed the schemes which are fourth order accurate in space and second or lower order accurate in time depending on the choice of weighted time average parameter are then applied to unsteady navier stokes system the proposed schemes where transport variable and its first derivatives are carried as the unknowns combine virtues of compact discretization and pade scheme for spatial derivative these schemes which are based on a five point stencil with constant coefficients named as 5 5 constant coefficient 4th order compact 5 5 cc 4oc give rise to a diagonally dominant system of equations and shows higher accuracy and better phase and amplitude error characteristics than some of the standard methods these schemes are capable of using a grid aspect ratio other than unity and are unconditionally stable they efficiently capture both transient and steady solutions of linear and nonlinear convection diffusion equations with dirichlet as well as neumann boundary conditions subsequently the proposed schemes are applied to problems governed by the incompressible navier stokes equations the results obtained are in excellent agreement with analytical and available numerical results in all cases establishing efficiency and accuracy of the proposed schemes

properties of discrete delta functions and local convergence of the immersed boundary method many problems involving internal interfaces can be formulated as partial differential equations with singular source terms numerical approximation to such problems on a regular grid necessitates suitable regularizations of delta functions we study the convergence properties of such discretizations for constant coefficient elliptic problems using the immersed boundary method as an example we show how the order of the differential operator order of the finite difference discretization and properties of the discrete delta function all influence the local convergence behavior in particular we show how a recently introduced property of discrete delta functions the smoothing order is important in the determination of local convergence rates

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

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

nonequilibrium critical behavior in unidirectionally coupled stochastic processes phase transitions from an active into an absorbing inactive state are generically described by the critical exponents of directed percolation dp with upper critical dimension d sub c 4 in the framework of single species reaction diffusion systems this universality class is realized by the combined processes a r arrow a a a a r arrow a and a r arrow 0 we study a hierarchy of such dp processes for particle species a b hor ellipsis ital unidirectionally coupled via the reactions a r arrow b hor ellipsis with rates mu sub ab hor ellipsis when the dp critical points at all levels coincide ital multicritical behavior emerges with density exponents beta sub i which are markedly reduced at each hierarchy level i ge 2 this scenario can be understood on the basis of the mean field rate equations which yield beta sub i 1 2 sup i minus 1 at the multicritical point using field theoretic renormalization group techniques in d 4 minus epsilon dimensions we identify a new crossover exponent phi and compute phi 1 o epsilon sup 2 in the multicritical regime for small mu sub ab of the second hierarchy level in the active phase we calculate the fluctuation correction to the density exponent on the second hierarchy level beta sub 2 1 2 minus epsilon 8 o epsilon sup 2 outside the multicritial region we discuss the crossover to ordinary dp behavior with the density exponent beta sub 1 1 minus epsilon 6 o epsilon sup 2 monte carlo simulations are then employed to confirm the crossovermore scenario and to determine the values for the new scaling exponents in dimensions d le 3 including the critical initial slip exponent our theory is connected to specific classes of growth processes and to certain cellular automata and the above ideas are also applied to unidirectionally coupled pair annihilation processes we also discuss some technical as well as conceptual problems of the loop expansion and suggest some possible interpretations of these difficulties copyright ital 1999 ital the american physical society less

efficient implementation of thinc scheme in this paper we improve the multi dimensional thinc tangent of hyperbola for interface capturing scheme f xiao y honma t kono a simple algebraic interface capturing scheme using hyperbolic tangent function int j numer meth fluid 48 2005 1023 the thinc scheme is a vof volume of fluid type method in the original thinc scheme one dimensional thinc scheme was straightforwardly used for multi dimensional cases in this paper we propose the wlic weighed line interface calculation method to extend the thinc scheme to multi dimension in the wlic method the interface is reconstructed by taking an average of line interfaces along x y and z coordinates with the weights calculated from surface normal the wlic method can extend the thinc scheme to multi dimension while maintaining simplicity of implementation and achieve a higher accuracy than the original thinc scheme the wlic method can readily extend the thinc scheme to three dimension

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

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

new cartesian grid methods for interface problems using the finite element formulation new finite element methods based on cartesian triangulations are presented for two dimensional elliptic interface problems involving discontinuities in the coefficients the triangulations in these methods do not need to fit the interfaces the basis functions in these methods are constructed to satisfy the interface jump conditions either exactly or approximately both non conforming and conforming finite element spaces are considered corresponding interpolation functions are proved to be second order accurate in the maximum norm the conforming finite element method has been shown to be convergent with cartesian triangulations these new methods can be used as finite difference methods numerical examples are provided to support the methods and the theoretical analysis

a nurbs based immersed methodology for fluid structure interaction abstract we introduce an isogeometric immersed and fully implicit formulation for fluid structure interaction fsi the method focuses on viscous incompressible flows and nonlinear hyperelastic incompressible solids which are a common case in various fields such as for example biomechanics in our fsi method we utilize an eulerian mesh on the whole domain and a lagrangian mesh on the solid domain the lagrangian mesh is arbitrarily located on top of the eulerian mesh in a non conforming fashion due to the formulation of our problem based on the immersed finite element method ifem we do not need mesh update or remeshing algorithms the fluid structure interface is the boundary of the lagrangian mesh but cuts arbitrarily the eulerian mesh the generalized method is used for time discretization and nurbs based isogeometric analysis is employed for the spatial discretization on both fluid and solid domains the information transfer between the two meshes is carried out using the nurbs functions which avoids the use of the so called discretized delta functions the higher order and especially the higher continuity of nurbs functions allow us to deal with severe mesh distortion in the lagrangian mesh in comparison with classical c 0 linear piecewise functions as we prove numerically our numerical solutions attain good agreement with theoretical data for free falling objects in two and three dimensions which confirms the feasibility of our methodology

a high order discontinuous galerkin method for 2d incompressible flows in this paper we introduce a high order discontinuous galerkin method for two dimensional incompressible flow in the vorticity stream function formulation the momentum equation is treated explicitly utilizing the efficiency of the discontinuous galerkin method the stream function is obtained by a standard poisson solver using continuous finite elements there is a natural matching between these two finite element spaces since the normal component of the velocity field is continuous across element boundaries this allows for a correct upwinding gluing in the discontinuous galerkin framework while still maintaining total energy conservation with no numerical dissipation and total enstrophy stability the method is efficient for inviscid or high reynolds number flows optimal error estimates are proved and verified by numerical experiments

l p convergence of the immersed boundary method for stationary stokes problems in this paper we analyze the convergence of the immersed boundary method as applied to a static stokes flow problem using estimates obtained in y liu and y mori siam j numer anal 50 2012 pp 2986 3015 we consider a problem in which a d dimensional structure is immersed in an n dimensional domain and prove error estimates for both the pressure and the velocity field in the l p 1 leq p leq infty norm one interesting consequence of our analysis is that the asymptotic error rates in the l 1 norm do not depend on either d or n and in the l p p 1 norm they only depend on n d the resulting estimates are checked numerically for optimality

generating relations for reducing matrices i ordinary representations auxiliary groups are constructed that make it possible to reduce the multiplicity problem and to derive consistent generating relations for the elements of reducing matrices three examples are worked out to illustrate the general scheme

a jacobian free based iim for incompressible flows involving moving interfaces with dirichlet boundary conditions in this paper a finite difference marker and cell mac scheme is presented for the steady stokes equations with moving interfaces and dirichlet boundary condition the moving interfaces are represented by lagrangian control points and their position is updated implicitly using a jacobian free approach within each time step the forces at the moving interfaces are calculated from the position of the interfaces and interpolated using cubic splines and then applied to the fluid through the related jump conditions the proposed jacobian free newton generalized minimum residual gmres method avoids the need to form and store the matrix explicitly in the computation of the inverse of the jacobian and betters numerical stability the stokes equations are discretized on a mac grid via a second order finite difference scheme with the incorporation of jump contributions and the resulting saddle point system is solved by the conjugate gradient uzawa type method numerical results demonstrate very well the accuracy and effectiveness of the proposed method the present algorithm has been applied to solve incompressible navier stokes flows with moving interfaces copyright 2010 john wiley sons ltd

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 stable and convergent scheme for viscoelastic flow in contraction channels we present a new algorithm to simulate unsteady viscoelastic flows in abrupt contraction channels in our approach we split the viscoelastic terms of the oldroyd b constitutive equation using duhamel s formula and discretize the resulting pdes using a semi implicit finite difference method based on a lax wendroff method for hyperbolic terms in particular we leave a small residual elastic term in the viscous limit by design to make the hyperbolic piece well posed a projection method is used to impose the incompressibility constraint we are able to compute the full range of unsteady elastic flows in an abrupt contraction channel from the viscous limit to the elastic limit in a stable and convergent manner we demonstrate the range of our method for unsteady flow of a maxwell fluid with and without viscosity in planar contraction channels we also demonstrate stable and convergent results for benchmark high weissenberg number problems at we 1 and we 10

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

an immersed interface method for solving incompressible viscous flows with piecewise constant viscosity across a moving elastic membrane this paper presents an implementation of the second order accurate immersed interface method to simulate the motion of the flexible elastic membrane immersed in two viscous incompressible fluids with different viscosities which further develops the work reported in tan et al z j tan d v le k m lim b c khoo an immersed interface method for the incompressible navier stokes equations with discontinuous viscosity across the interface submitted for publication focussing mainly on the fixed interface problems in this work we introduce the velocity components at the membrane as two augmented unknown interface variables to decouple the originally coupled jump conditions for the velocity and pressure three forms of augmented equation are derived to determine the augmented variables to satisfy the continuous condition of the velocity the velocity at the membrane which determine the motion of the membrane is then solved by the gmres iterative method the forces calculated from the configuration of the flexible elastic membrane and the augmented variables are interpolated using cubic splines and applied to the fluid through the jump conditions the position of the flexible elastic membrane is updated implicitly using a quasi newton method bfgs within each time step the navier stokes equations are solved on a staggered cartesian grid using a second order accurate projection method with the incorporation of spatial and temporal jump conditions in addition we also show that the inclusion of the temporal jump contributions has non negligible effect on the simulation results when the grids are crossed by the membrane using the above method we assess the effect of different viscosities on the flow solution and membrane motion

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

discrete exterior calculus discretization of incompressible navier stokes equations over surface simplicial meshes a conservative discretization of incompressible navier stokes equations is developed based on discrete exterior calculus dec a distinguishing feature of our method is the use of an algebraic discretization of the interior product operator and a combinatorial discretization of the wedge product the governing equations are first rewritten using the exterior calculus notation replacing vector calculus differential operators by the exterior derivative hodge star and wedge product operators the discretization is then carried out by substituting with the corresponding discrete operators based on the dec framework numerical experiments for flows over surfaces reveal a second order accuracy for the developed scheme when using structured triangular meshes and first order accuracy for otherwise unstructured meshes by construction the method is conservative in that both mass and vorticity are conserved up to machine precision the relative error in kinetic energy for inviscid flow test cases converges in a second order fashion with both the mesh size and the time step

generating relations for reducing matrices ii corepresentations the auxiliary group approach developed in paper 1 r dirl p kasperkovitz m i aroyo j n kotzev and m angelova tjurkedjieva j math phys 27 37 1986 is generalized for the case of corepresentations of antiunitary groups it allows us to reduce the multiplicity problem and to derive consistent generating relations for the elements of the reducing matrices for coreps two examples are worked out to illustrate the general scheme

generating relations for reducing matrices iii kronecker products in the first two of this series of papers r dirl et al j math phys 27 37 1986 m i aroyo et al ibid 27 2236 1986 a systematic method to calculate the elements of reducing matrices has been developed in this paper this auxiliary group approach is adapted to multiple kronecker products three examples are worked out to illustrate the efficiency of generating relations and the proposed reduction of the multiplicity problem

renormalization group study of thea b diffusion limited reaction thea b diffusion limited reaction with equal initial densitiesa 0 b 0 n0 is studied by means of a field theoretic renormalization group formulation of the problem for dimensiond 2 an effective theory is derived from which the density and correlation functions call be calculated we find the density decays in time as a b sim c sqrt delta dt d 4 ford 4 with n0 c n 0 d 2 wherec is a universal constant andc is nonuniversal the calculation is extended to the case of unequal diffusion constantsd a d b resulting in a new amplitude but the same exponent ford 2 a controlled calculation is not possible but a heuristic argument is presented that the results above give at least the leading term in an e 2 d expansion finally we address reaction zones formed in the steady state by opposing currents ofa andb particles and derive scaling properties

physically motivated domain decomposition for singularly perturbed equations a domain decomposition algorithm suitable for the efficient and accurate solution of a parabolic reaction convection diffusion equation with small parameter multiplying the diffusion term is presented convergence is established via maximum principle arguments the equation arises in the modeling of laminar transonic flow decomposition into subdomains is accomplished via singular perturbation analysis which dictates regions where certain reduced equations may be solved in place of the full equation effectively preconditioning the problem this paper concentrates on the theoretical basis of the method establishing local and global a priori error bounds

violation of scaling in the contact process with quenched disorder we study the two dimensional contact process cp with quenched disorder dcp and determine the static critical exponents beta and nu perp the dynamic behavior is incompatible with scaling as applied to models such as the pure cp that have a continuous phase transition to an absorbing state we find that the survival probability starting with all sites occupied for a finite size system at critical decays according to a power law as does the off critical density autocorrelation function thus the critical exponent nu parallle which governs the relaxation time is undefined since the characteristic relaxation time is itself undefined the logarithmic time dependence found in recent simulations of the critical dcp moreira and dickman phys rev e54 r3090 1996 is further evidence of violation of scaling a simple argument based on percolation cluster statistics yields a similar logarithmic evolution

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

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

an immersed discontinuous finite element method for the stokes problem with a moving interface abstract we present a discontinuous immersed finite element ife method for incompressible interfacial flows that are governed by the stokes equations the method is based on a cartesian mesh with elements cut by the moving interface on this fixed unfitted mesh we employ an immersed q 1 q 0 finite element space constructed according to the location of the interface and pertinent interface jump conditions as such the smearing of solution across the interface is greatly reduced the interface represented by a sequence of marker points is advected on the fixed background mesh by the local fluid velocity the mesh is locally refined near the interface to further improve accuracy compared with the phase field method on adaptive meshes our method can achieve the same level of accuracy with much less degrees of freedoms we present some numerical examples to validate and demonstrate the capability of the proposed method

parametric resonance in immersed elastic boundaries in this paper we investigate the stability of a fluid structure interaction problem in which a flexible elastic membrane immersed in a fluid is excited via periodic variations in the elastic stiffness parameter this model can be viewed as a prototype for active biological tissues such as the basilar membrane in the inner ear or heart muscle fibers immersed in blood problems such as this in which the system is subjected to internal forcing through a parameter can give rise to parametric resonance we formulate the equations of motion in two dimensions using the immersed boundary formulation assuming small amplitude motions we can apply floquet theory to the linearized equations and derive an eigenvalue problem whose solution defines the marginal stability boundaries in parameter space the eigenvalue equation is solved numerically to determine values of fiber stiffness and fluid viscosity for which the problem is linearly unstable we present direct numerical simulations of the fluid structure interaction problem using the immersed boundary method that verify the existence of the parametric resonances suggested by our analysis

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

field theoretic treatment of an effective action for a model of catalyzed autoamplification reaction diffusion models can exhibit continuous phase transitions in behaviors and their dynamics at criticality often exhibit scalings with key parameters that can be characterized by exponents while models with only a single field that transitions between absorbing and nonabsorbing states are well characterized and typically fall in the directed percolation universality class the effects of coupling multiple fields remain poorly understood we recently introduced a model which has three fields one of which relaxes exponentially one of which displays critical behavior and one of which has purely diffusive dynamics but exerts an influence on the critical field tchernookov j chem phys 130 134906 2009 simulations suggested that this model is in a universality class distinct from other reaction diffusion systems studied previously although the three fields give rise to interesting physics they complicate analysis of the model with renormalization group methods here we show how to systematically simplify the action for this model such that analytical expressions for the exponents of this universality class can be obtained by standard means we expect the approach taken here to be of general applicability in reaction diffusion systems with coupled order parameters that display qualitatively different behaviors close to criticality

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

tether force constraints in stokes flow by the immersed boundary method on a periodic domain the immersed boundary method is an algorithm for simulating the interaction of immersed elastic bodies or boundaries with a viscous incompressible fluid the immersed elastic material is represented in the fluid equations by a system or field of applied forces the particular case of stokes flow with applied forces on a periodic domain involves two related mathematical complications one of these is that an arbitrary constant vector may be added to the fluid velocity and the other is the constraint that the integral of the applied force must be zero typically forces defined on a freely floating elastic immersed boundary or body satisfy this constraint but there are many important classes of forces that do not for example the so called tether forces that are used to prescribe the simulated configuration of an immersed boundary possibly in a time dependent manner typically do not sum to zero another type of force that does not have zero integral is a uniform force density that may be used to simulate an overall pressure gradient driving flow through a system we present a method for periodic stokes flow that when used with tether points admits the use of all forces irrespective of their integral over the domain a byproduct of this method is that the additive constant velocity associated with periodic stokes flow is uniquely determined indeed the additive constant is chosen at each time step so that the sum of the tether forces balances the sum of any other forces that may be applied

the immersed interface method for elliptic equations with discontinuous coefficients and singular sources the authors develop finite difference methods for elliptic equations of the form nabla cdot beta x nabla u x kappa x u x f x in a region omega in one or two space dimension

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

wilson renormalization of a reaction diffusion process abstract healthy and sick individuals a and b particles diffuse independently with diffusion constants da and db sick individuals upon encounter infect healthy ones at rate k but may also spontaneously recover at rate 1 the propagation of the epidemic therefore couples to the fluctuations in the total population density global extinction occurs below a critical value c of the spatially averaged total density the epidemic evolves as the diffusion reaction decay process a b 2b b a for which we write down the field theory the stationary state properties of this theory when da db were obtained by kree et al the critical behavior for da db remains unsolved

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

infinite randomness critical point in the two dimensional disordered contact process we study the nonequilibrium phase transition in the two dimensional contact process on a randomly diluted lattice by means of large scale monte carlo simulations for times up to 10 10 and system sizes up to 8000 x8000 sites our data provide strong evidence for the transition being controlled by an exotic infinite randomness critical point with activated exponential dynamical scaling we calculate the critical exponents of the transition and find them to be universal i e independent of disorder strength the griffiths region between the clean and the dirty critical points exhibits power law dynamical scaling with continuously varying exponents we discuss the generality of our findings and relate them to a broader theory of rare region effects at phase transitions with quenched disorder our results are of importance beyond absorbing state transitions because according to a strong disorder renormalization group analysis our transition belongs to the universality class of the two dimensional random transverse field ising model

a cartesian grid method for solving the two dimensional streamfunction vorticity equations in irregular regions we describe a method for solving the two dimensional navier stokes equations in irregular physical domains our method is based on an underlying uniform cartesian grid and second order finite difference finite volume discretizations of the streamfunction vorticity equations geometry representing stationary solid obstacles in the flow domain is embedded in the cartesian grid and special discretizations near the embedded boundary ensure the accuracy of the solution in the cut cells along the embedded boundary we determine a distribution of vorticity sources needed to impose the no slip flow conditions this distribution appears as a right hand side term in the discretized fluid equations and so we can use fast solvers to solve the linear systems that arise to handle the advective terms we use the high resolution algorithms in clawpack we show that our stokes solver is second order accurate for steady state solutions and that our full navier stokes solver is between first and second order accurate and reproduces results from well studied benchmark problems in viscous fluid flow finally we demonstrate the robustness of our code on flow in a complex domain

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

2 d parachute simulation by the immersed boundary method parachute aerodynamics involves an interaction between the flexible elastic porous parachute canopy and the high speed airflow relative to the parachute through which the parachute falls computer simulation of parachute dynamics typically simplifies the problem in various ways e g by considering the parachute as a rigid bluff body here we avoid such simplification by using the immersed boundary ib method to study the full fluid structure interaction the ib method is generalized to handle porous immersed boundaries and the generalized method is used to study the influence of porosity on parachute stability

gerris a tree based adaptive solver for the incompressible euler equations in complex geometries an adaptive mesh projection method for the time dependent incompressible euler equations is presented the domain is spatially discretised using quad octrees and a multilevel poisson solver is used to obtain the pressure complex solid boundaries are represented using a volume of fluid approach second order convergence in space and time is demonstrated on regular statically and dynamically refined grids the quad octree discretisation proves to be very flexible and allows accurate and efficient tracking of flow features the source code of the method implementation is freely available

modeling water transport across elastic boundaries using an explicit jump method a mathematical model is presented to simulate water and solute transport in a highly viscous fluid with a water permeable elastic immersed membrane in this model fluid motion is described by stokes flow whereas water fluxes across the membrane are driven by transmural pressure and solute concentration differences the elastic forces arising from the membrane being distorted from its relaxed configuration and the transmembrane water fluxes introduce into model solutions discontinuities across the membrane such discontinuities are faithfully captured using a second order explicit jump method a mayo siam j numer anal 21 1984 pp 285 299 in which jumps in the solution and its derivatives are incorporated into a finite difference scheme numerical results suggest that the method exhibits desirable volume accuracy and mass conservation

the auxiliary group approach to the reduction of kronecker products bijective mappings of matrix representations are used to derive generating relations for clebsch gordon coefficients and to reduce the multiplicity problem

an efficient numerical method for the two fluid stokes equations with a moving immersed boundary we consider the immersed boundary problem in which the boundary separates two very viscous fluids with differing viscosities the moving elastic boundary may exert a force on the local fluid the model solution is obtained using the immersed interface method which computes second order accurate approximations by incorporating known jumps in the solution or its derivatives into a finite difference method these jump conditions become coupled when the fluid viscosity has a jump across the boundary and this coupling renders the application of the immersed interface method challenging we present a method that first uses boundary integral equations to reduce the two fluid stokes problem to the single fluid case and then solves the single fluid problem using the immersed interface method using this method we assess through two numerical examples how the fluid dynamics are affected by differing viscosities in the two fluid regions we also propose an implicit algorithm and a fractional step algorithm for advancing the boundary position because both algorithms make use of the integral form of the solution neither one requires the solution of a large system of coupled nonlinear equations as is traditionally the case numerical results suggest that for sufficiently stiff problems the fractional time stepping algorithm is the most efficient in the sense that it allows the largest time interval between subsequent updates of global model solutions

improving the speed and accuracy of projection type incompressible flow solvers abstract superseding so called first generation incompressible flow solvers of the projection type based on taylor galerkin advection second order pressure damping and element based data structures the current second generation solvers based on high order upwind advection fourth order pressure damping and edge based data structures have now been in use for half a decade and have proven remarkably robust and efficient for many large scale problems in order to achieve higher accuracy and speed these solvers have recently been enhanced in a variety of ways a substepping for advection b implicit treatment of advective terms via sgs and gmres lu sgs iterative solvers c fully implicit time accurate advancement of pressure and velocities and d linelet preconditioning for the pressure poisson equation the combined effect of these third generation improvements leads to speedups of the order of o 1 5 1 10 with similar or even better temporal accuracy as demonstrated on a variety of academic and industrial problems

a new algorithm for solving some mechanical problems this paper explores the utility of a discrete singular convolution algorithm for solving certain mechanical problems benchmark mechanical systems including plate vibrations and incompressible flows are employed to illustrate the robustness and to test accuracy of the present algorithm numerical results indicate that the present approach is very accurate efficient and reliable for solving the aforementioned problems

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

simulations of valveless pumping in an open elastic tube mathematical models and numerical simulations of flows driven by pumping without valves valveless pumping are presented this work has been originally motivated by a biomedical objective to explain the complicated valveless blood flow mechanism in the circulation for instance the fetal circulation before the formation of valves and blood circulation during cardiopulmonary resuscitation a mathematical model of valveless pumping either in a closed loop system or in an open system consists of a couple of tubes with different elasticities or radii in this work we develop new models which are not necessarily connected by two different materials only one open elastic tube and an open elastic tube contained in a closed almost rigid tank although only one soft material is used or the soft and rigid materials are not connected we have observed the existence of a net flow driven by the periodic compress and release action and the important features of valveless pumping that have been observed in earlier models or experiments we have also shown that there exist mostly one directional net flows inside the elastic tube if a long and thin tube is considered and the pumping is applied to the short portion near the edge of the tube this might explain why most earlier reports on the physical experiments could not easily observe the existence of both directional net flows depending on the parameters such as frequency another important result is that the direction and the magnitude of a net flow can be explained by the sign and the amount of power which is work done on the fluid by the fluid pressure and the elastic wall over one period respectively a new feature in this work is that only one elastic material is sufficient to show the existence of a net flow in a valveless pump system because of a simple structure in our new model it is much easier to construct a valveless pump system applicable to real world applications such as a microelectromechanical system device

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

critical dynamics of the contact process with quenched disorder we study critical spreading in monte carlo simulations of the two dimensional contact process cp with quenched disorder in the form of random dilution in the pure model spreading from a single particle at the critical point ensuremath lambda c follows power laws with the critical exponents of directed percolation with disorder critical spreading is logarithmic not power law below ensuremath lambda c there is a griffiths phase in which the time dependence is governed by nonuniversal power laws the effects of disorder are also apparent above ensuremath lambda c in the active phase where the relaxation of the survival probability is algebraic rather than exponential as in the pure model our results support the conjecture by bramson durrett and schonmann ann prob 19 960 1991 that in two or more dimensions the disordered cp has only a single phase transition

uncoupled temporally second order accurate implicit solver of incompressible navier stokes equations an efficient method with temporal second order accuracy for solving the incompressible navier stokes equations in generalized coordinate systems is developed efficiency is obtained by extending an existing fractional step semi implicit solver into an implicit one with implicit approximation of the convection terms a novel three level linearization scheme is proposed to decouple the momentum equations from one another in addition to the fractional step approach that decouples the continuity equation from the momentum equations consequently all of the governing equations are uncoupled without degrading temporal accuracy or stability and without the need for iterations to solve the nonlinear system of equations at each time level the proposed decoupling technique can be used for other systems of nonlinear and time dependent partial differential equations moreover only minor modifications are required to apply the method to existing two level schemes several test cases confirm that the proposed decoupled scheme has temporal second order accuracy and allows the use of large time steps

a hybrid variational collocation immersed method for fluid structure interaction using unstructured t splines summary we present a hybrid variational collocation immersed and fully implicit formulation for fluid structure interaction fsi using unstructured t splines in our immersed methodology we define an eulerian mesh on the whole computational domain and a lagrangian mesh on the solid domain which moves arbitrarily on top of the eulerian mesh mathematically the problem reduces to solving three equations namely the linear momentum balance mass conservation and a condition of kinematic compatibility between the lagrangian displacement and the eulerian velocity we use a weighted residual approach for the linear momentum and mass conservation equations but we discretize directly the strong form of the kinematic relation deriving a hybrid variational collocation method we use t splines for both the spatial discretization and the information transfer between the eulerian mesh and the lagrangian mesh t splines offer us two main advantages against non uniform rational b splines they can be locally refined and they are unstructured the generalized method is used for the time discretization we validate our formulation with a common fsi benchmark problem achieving excellent agreement with the theoretical solution an example involving a partially immersed solid is also solved the numerical examples show how the use of t junctions and extraordinary nodes results in an accurate efficient and flexible method copyright 2015 john wiley sons ltd

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

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

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

three dimensional viscous finite element formulation for acoustic fluid structure interaction abstract a three dimensional viscous finite element model is presented in this paper for the analysis of the acoustic fluid structure interaction systems including but not limited to the cochlear based transducers the model consists of a three dimensional viscous acoustic fluid medium interacting with a two dimensional flat structure domain the fluid field is governed by the linearized navier stokes equation with the fluid displacements and the pressure chosen as independent variables the mixed displacement pressure based formulation is used in the fluid field in order to alleviate the locking in the nearly incompressible fluid the structure is modeled as a mindlin plate with or without residual stress the hinton huang s 9 noded lagrangian plate element is chosen in order to be compatible with 27 4 u p fluid elements the results from the full 3d fem model are in good agreement with experimental results and other fem results including beltman s thin film viscoacoustic element w m beltman p j m van der hoogt r m e j spiering h tijdeman implementation and experimental validation of a new viscothermal acoustic finite element for acousto elastic problems j sound vib 216 1 1998 159 185 and two and half dimensional inviscid elements a a parthasarathi k grosh a l nuttall three dimensional numerical modeling for global cochlear dynamics j acoust soc am 107 2000 474 485 although it is computationally expensive it provides a benchmark solution for other numerical models or approximations to compare to besides experiments and it is capable of modeling any irregular geometries and material properties while other numerical models may not be applicable

non equilibrium scaling in the schlogl model from the reaction diffusion master equation formulation of the schlogl model the authors develop a comprehensive description of the non equilibrium dynamics using the poisson transformation and the dynamical renormalisation group they firstly show how the equilibrium limit is recovered and secondly describe the low frequency low momenta scaling behaviour of the concentration fluctuations the approach extends simplifies or corrects previous studies and is easily generalised to treat more complicated systems

a hybrid particle level set method for improved interface capturing in this paper we propose a new numerical method for improving the mass conservation properties of the level set method when the interface is passively advected in a flow field our method uses lagrangian marker particles to rebuild the level set in regions which are underresolved this is often the case for flows undergoing stretching and tearing the overall method maintains a smooth geometrical description of the interface and the implementation simplicity characteristic of the level set method our method compares favorably with volume of fluid methods in the conservation of mass and purely lagrangian schemes for interface resolution the method is presented in three spatial dimensions