geometric implementation of hypercubic lattices with noninteger dimensionality by use of low lacunarity fractal lattices it is claimed that the abstract analytic continuation of hypercubic lattices to noninteger dimensionalities can be implemented explicitly by certain fractal lattices of low lacunarity these lattices are special examples of sierpinski carpets their being of low lacunarity means that they are arbitrarily close to being translationally invariant the claim is substantiated for the ising model in d 1 ensuremath epsilon dimensions and for resistor network models with 1ldl2

numerical investigation on the stability of singular driven cavity flow by applying the singularity subtraction technique to the unsteady driven cavity problem the stability of the impulsively started flow is investigated without smoothing the corner singularity a second order spectral projection method allows localization of the critical reynolds number for the first hopf bifurcation in the interval 8017 6 8018 8

critical exponents of the ising model on low dimensional fractal media abstract the critical behavior of the ising model on fractal substrates with noninteger hausdorff dimension d h 2 and infinite ramification order is studied by means of the short time critical dynamic scaling approach our determinations of the critical temperatures and critical exponents and are compared to the predictions of the wilson fisher expansion the wallace zia expansion the transfer matrix method and more recent monte carlo simulations using finite size scaling analysis we also determined the effective dimension d e f which plays the role of the euclidean dimension in the formulation of the dynamic scaling and in the hyperscaling relationship d e f 2 furthermore we obtained the dynamic exponent z of the nonequilibrium correlation length and the exponent that governs the initial increase of the magnetization our results are consistent with the convergence of the lower critical dimension towards d 1 for fractal substrates and suggest that the hausdorff dimension may be different from the effective dimension

accurate three dimensional lid driven cavity flow a chebyshev collocation method in space is introduced which allows an accurate calculation of three dimensional lid driven cavity flows the time integration is carried out by an adams bashforth backward euler scheme the accuracy of the method relies on the representation of the solution as a superposition of stationary local asymptotic solutions and a residual flow field this way the most severe discontinuities in the boundary conditions which arise along the lines where moving and stationary walls meet are taken care of analytically and thus do not spoil the numerical part of the solution calculations are carried out for no slip boundary conditions at the cavity end walls as well as for periodic end wall conditions in general the spatial accuracy is better than fifth order for rigid end wall conditions the accuracy is reduced near the end walls to o n 3 2 but recovers in the bulk tabulated data are provided for the most interesting flow properties

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

a spectral solver for the navier stokes equations in the velocity vorticity formulation for flows with two nonperiodic directions a novel pseudospectral solution procedure for the navier stokes equations in the velocity vorticity formulation suitable to simulate flows with two nonperiodic directions is proposed an influence matrix method including a tau correction procedure has been employed to elicit an algorithmic substitute for thea priorilacking boundary conditions for the vorticity following o daube j comput phys 103 402 1992 the influence matrix is built by enforcing either the definition of the vorticity or the continuity equation at the boundary of the domain the influence matrix is nonsingular in both cases the order of the influence matrix is twice larger than one would expect from analogous solution methods based on finite differences or finite elements the spatial discretization is based on a 2d chebyshev expansion on anonstaggeredgrid of collocation points and the boundary conditions are imposed via the lanczos tau procedure the time marching scheme is adams bashforth for the advection term and crank nicolson for the viscous term the proposed scheme yields machine accurate divergence free flow fields and the definition of the vorticity is satisfied within machine accuracy

free energy of magnetic model systems at the fractal to euclidian crossover abstract we prove that the free energy per spin of the nearest neighbour classical n vector spin model situated on the sierpinski gasket sg family of fractals tends to the free energy per spin of the same model situated on the standard triangular lattice when the linear size b of the sg fractal generator tends to infinity this finding can be regarded as the first step in the analytical investigation of the spin system thermodynamics at the fractal to euclidean crossover

spectral dimension of a wire network by a rigorous analysis for diffusion on a wire network the spectral dimension of a fractal is shown to be independent of the local structure it is discussed that diffusion does not occur when the lattice spacing tends to zero on such a fractal that the spectral dimension is less than the hausdorff dimension if a free field on the fractal exhibits a certain long distance behaviour for a sierpinski carpet the spectral dimension is evaluated within bond moving approximation migdal kadanov renormalisation as a result the author obtains a value smaller than the hausdorff dimension

scaling exponents for random walks on sierpinski carpets and number of distinct sites visited a new algorithm for infinite fractal lattices the scaling exponent for the mean square distance covered in a random walk dw and the average number of distinct sites visited dn are determined for a family of sierpinski carpet patterns we suggest a new random walk algorithm to generate walks on an effectively infinite deterministic fractal lattice the algorithm is applied to several sierpinski carpet patterns with the same hausdorff dimension we show that the systems have a quite different scaling exponent dw and further that the generally accepted result dn ds does not hold for all of these where ds is the spectral dimension

on error estimates of some higher order projection and penalty projection methods for navier stokes equations this paper is a continuation of our previous work 10 on projection methods we study first existing higher order projection schemes in the semidiscretized form for the navier stokes equations one error analysis suggests that the precision of these schemes is most likely plagued by the inconsistent neumann boundary condition satisfied by the pressure approximations we then propose a penalty projection scheme for which we obtain improved error estimates

pressure stability in fractional step finite element methods for incompressible flows abstract the objective of this paper is to analyze the pressure stability of fractional step finite element methods for incompressible flows that use a pressure poisson equation for the classical first order projection method it is shown that there is a pressure control which depends on the time step size and therefore there is a lower bound for this time step for stability reasons the situation is much worse for a second order scheme in which part of the pressure gradient is kept in the momentum equation the pressure stability in this case is extremely weak to overcome these shortcomings a stabilized fractional step finite element method is also considered and its stability is analyzed some simple numerical examples are presented to support the theoretical results

remarks on parameter identification ii convergence of the algorithm on demontre un theoreme de convergence pour l algorithme d identification du coefficient a x de l equation parabolique u t a x u x x f x t et on donne des resultats de stabilite

lacunarity and universality lacunarity as a criterion of universality in phase transitions on sierpinski carpets is studied the approximate theoretical and numerical results suggest that systems of different lacunarity may belong to a common universality class there seems to be a lack of solutions to the complete set of universality criteria on sierpinski carpets

potts model on sierpinski carpets and their relation to hypercubic lattices of non integer dimensionality the authors study the critical frontiers and the high and low temperature asymptotic behavior of the potts model two site correlation function for several sierpinski carpets with fractal dimension lying between one and two they simulate the sierpinski carpets by means of appropriate hierarchical lattices obtaining from their basic cells the two graphs among others that arise from the anisotropic bond moving schemes they correspond to the asymptotic high and low temperature dominant terms for the hierarchical lattice two site correlation function exact result by taking the infinite limit size of the basic cell these asymptotic behaviors tends to those of the sierpinski carpets it is then possible to compare them with similar results obtained from analytical extension to d 2 of the hypercubic lattice their results indicate that these asymptotic behaviors do not always coincide suggesting that possibly they may not identify the sierpinski carpets exhibited by gefen et al 1984 as a correct implementation of the analytical extension of hypercubic lattices

phase transitions on fractals iii infinitely ramified lattices for pt ii see ibid vol 17 p 435 1984 in the first two papers of this series the authors considered self similar fractal lattices with a finite order of ramification r in the present paper they study physical models defined on a family of fractals with r infinity in order to characterise the geometry of these systems they need the connectivity q and the lacunarity l in addition to the fractal dimensionality d it is found that discrete symmetry spin models on these lattices undergo a phase transition at tc 0 an approximate renormalisation group scheme is constructed and used to find the dependence of tc and the critical exponents on the geometrical factors they also solve the problem of resistor networks on these fractals and discuss its consequences concerning spin models with continuous symmetry

a numerical simulation of the transition to turbulence in a two dimensional flow abstract a numerical simulation of the transition to turbulence is performed using a finite element method the unsteady navier stokes equations are discretized using a standard galerkin approximation and a loading strategy for increasing the reynolds number the numerical results are then analysed at different reynolds numbers showing a transition from a steady state solution to a weakly chaotic one phase space diagrams are presented showing the presence of strange attractors the dimension and lyapunov s exponents of these attractors are computed and compared with existing results in the literature

a pure compact scheme for the streamfunction formulation of navier stokes equations a pure streamfunction formulation is introduced for the numerical simulation of the two dimensional incompressible navier stokes equations the idea is to replace the vorticity in the vorticity streamfunction evolution equation by the laplacian of the streamfunction the resulting formulation includes the streamfunction only thus no inter function relations need to be invoked a compact numerical scheme which interpolates streamfunction values as well as its first order derivatives is presented and analyzed a number of numerical experiments are presented including driven and double driven cavities where the reynolds numbers are sufficiently large leading to symmetry breaking of asymptotic solutions

high order direct stokes solvers with or without temporal splitting numerical investigations of their comparative properties a recently proposed direct stokes solver which decouples the velocity and pressure operators without calling for a temporal scheme is numerically analyzed in comparison first with the splitting scheme proposed by g karniadakis m israeli and s orszag in j comput phys 97 1991 pp 414 443 and with the unique grid bbb p n bbb p n 2 uzawa approach for the space accuracy and computational costs the chebyshev collocation approximation is used to analyze the spectra of the continuous temporal evolution operators and their discrete time versions from the first to the fourth order an explicit boundary condition is also involved in the proposed stokes solver and it is numerically shown that the trace of bf delta bf u on the boundary must be evaluated through its bf nabla times bf nabla times contribution only otherwise the ellipticity is lost before proceeding to the time discretization the explicit evaluation of the rotational boundary term does not prevent the first and second order in time schemes from being unconditionally stable while the schemes built at the next two higher orders are limited by the usual explicit cal o n 4 stability criterion on the time step the proposed bf u p decoupling gives this limitation a less restrictive coefficient and supplies the expected temporal orders on the whole explored range of time step sizes the effective accuracy obtained for the navier stokes 2d solutions has been measured for the re 1000 lid driven cavity problem and has been found equivalent to what is supplied by the much more expensive uzawa decoupling method

analysis of a sequential regularization method for the unsteady navier stokes equations the incompressibility constraint makes navier stokes equations difficult a reformulation to a better posed problem is needed before solving it numerically the sequential regularization method srm is a reformulation which combines the penalty method with a stabilization method in the context of constrained dynamical systems and has the benefit of both methods in the paper we study the existence and uniqueness for the solution of the srm and provide a simple proof of the convergence of the solution of the srm to the solution of the navier stokes equations we also give error estimates for the time discretized srm formulation

critical behavior of self avoiding walks on fractals using a new graph counting technique suitable for self similar fractals exact 18th order series expansions for saws on some sierpinski carpets are generated from them the critical fugacityxc and critical exponents saw and saw are obtained the results show a linear dependence of the critical fugacity with the average number of bonds per site of the lattices studied we find for some carpets with low lacunarity that saw saw d for saws on the fractals which are embedded in ad dimensional euclidean space

exact renormalization group approach to critical phenomena in an ising system with long range interactions on the sierpinski gasket lattice abstract an ising spin system with two point long range interactions on the sierpinski gasket lattice is considered the interactions are assigned along edges of all equilateral triangles being part of the self similar structure in addition the long range couplings i e all couplings besides nearest neighbor interactions k are assumed to be independent of the distance i e q1 q l 1 2 where q1 denotes an interaction between the two spins on the distance 2 a with a being a lattice constant and q a positive parameter using an exact renormalization group method based upon the decimation transformation the system is shown to display a line of nontrivial fixed points k k q each point of this line corresponds to a phase transition of a rather high order the critical behavior of the system near every nontrivial fixed point is claimed to be characterized by two dimensions viz the fractal dimension and the so called critical dimension which reflects some topological properties of the system the critical dimension is defined through considering the decay of an effective two point correlation function and depends on q

critical behaviour of the bond diluted potts model on sierpinski carpets a four parameter bond moving renormalisation is used to study the critical properties of the bond diluted q state potts model on sierpinski carpets fixed points critical exponents and a phase diagram are obtained in the authors results there is a borderline value q qt for q qt there is crossover to a new diluted fixed point this behaviour is similar to that of the diluted system on a regular lattice with translational invariance they give some values of qt for carpets with different b l where b and l are structure parameters of carpets

spectrally accurate numerical incompressible flows with deformed boundaries a finite volume preconditioning method spectrally accurate solutions of the stokes problem have been computed with finite volume preconditioning of a chebyshev collocation method it has been validated on two analytical test cases and applied to the resolution of the navier stokes equations in a deformed driven cavity configuration this numerical tool is ready for application to flows with deformed free surface

the ideal chain problem in infinitely ramified self similar structures abstract series expansions for the ideal chain problem in sierpinski carpets were calculated and critical exponents 1 2 were obtained with good accuracy from the scaling properties of the probability of the chain returning to the starting site it is shown that the ideal chain has asymptotic behaviour different from the random walk problem in those lattices

the treatment of spurious pressure modes in spectral incompressible flow calculations algorithms for the transient and steady state simulation of incompressible newtonian and non newtonian flows are described for the primitive variable formulation of the governing equations spectral approximations are used for the spatial discretization attention is given to the satisfaction of the incompressibility constraint and the determination of the pressure spurious pressure modes are removed by means of a singular value decomposition the corresponding velocity field is divergence free at all the collocation points numerical results are presented for newtonian flow in a regularized driven square cavity and for non newtonian flow in a planar channel and in a journal bearing for a realistic range of material parameters

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

derivation of projection methods from formal integration of the navier stokes equations projection methods constitute a class of numerical methods for solving the incompressible navier stokes equations these methods operate using a two step procedure in which the zero divergence constraint on the velocity is first relaxed while the velocity evolves then after a certain period of time the resulting velocity field is projected onto a divergence free subspace although these methods can be quite efficient there have been certain concerns regarding their formulation in this paper we show how a formal integration of the navier stokes equations leads to a new and general procedure for the derivation of projection methods by following this procedure we show how each of three practical projection methods approximates a system of equations that is equivalent to the navier stokes equations we also show how the auxiliary boundary conditions required in projection methods are related to the physical boundary conditions these results should allay the concerns regarding the legitimacy of projection methods and may assist in their future development

critical properties of the s4 model on diamond type hierarchial lattices we study the critical properties of the s 4 model on diamond type hierarchical lattices in the presence of an external magnetic field it is assumed that for this type of inhomogenous fractal lattice the gaussian distribution constant the four spins interaction parameter and the external magnetic field in the s4 model depend on the coordination number of the site on the fractal lattices by combining the real space renormalization group scheme with the cumulative expansion method we obtain the critical points and further calculate critical exponents according to the scaling theory the results show that on diamond type hierarchical lattices with branches m 4 of 2 bonds the critical point of the s4 model is just the gaussian fixed point and therefore critical exponents are in full agreement with those of the gaussian model and that on those with m 4 the wilson fisher fixed point as well as the gaussian fixed point is obtained the wilson fisher fixed point has a decisive influence on the critical behavior of the system and critical exponents are related to the fractal dimensionality we found that the s4 model on the fractal lattices and that on the translation symmetric lattices show similar behaviors in the dependence of the critical properties on the dimensionality c 2005 elsevier b v all rights reserved

fractals their transfer matrices and their eigen dimensional sequences this paper defines and studies a general algorithm for constructing new families of fractals in euclidean space this algorithm involves a sequence of linear interscale transformations that proceed from large to small scales the authors find that the fractals obtained in this fashion decompose in intrinsic fashion into linear combinations of a variable number of addend fractals the addends relative weights and fractal dimensionalities are obtained explicitly through an interscale matrix which they call the transfer matrix of the fractal tmf the authors first demonstrate by a series of examples then prove rigorously that the eigenvalues of our tmfs are real and positive and that the fractal dimensions of the addend fractals are the logarithms of the eigenvalues of their tmf they say that these dimensions form the overall fractal s eigendimensional sequence the eigenvalues of their tmf are integers in the non random variants of the construction but are non integer in the random variants a geometrical interpretation of the eigenvalues and the eigenvectors is given their tmf have other striking and very special properties that deserve additional attention

remarks on parameter identification i in this paper a method for constructing a spatially varying diffusion coefficient for a parabolic partial differential equation is given this function is obtained as the limit of a sequence of functions which are obtained by solving a sequence of finite dimensional optimization problems

an efficient spectral projection method for the navier stokes equations in cylindrical geometries an efficient and accurate numerical scheme is presented for the three dimensional navier stokes equations in primitive variables in a cylinder the scheme is based on a spectral galerkin approximation for the space variables and a second order projection scheme for time the new spectral projection scheme is implemented to simulate unsteady incompressible flows in a cylinder

phase transitions on fractals ii sierpinski gaskets for pt i see ibid vol 16 p 1267 1983 the authors construct and investigate a family of fractals which are generalisations of the sierpinski gaskets sgs to all euclidean dimensionalities these fractal lattices have a finite order of ramification and can be considered marginal between one dimensional and higher dimensional geometries physical models defined on them are exactly solvable the authors argue that short range spin models on the sg show no finite temperature phase transitions as examples they solve a few spin models and study the resistor network and percolation problems on these lattices

a semi lagrangian high order method for navier stokes equations we present a semi lagrangian method for advection diffusion and incompressible navier stokes equations the focus is on constructing stable schemes of second order temporal accuracy as this is a crucial element for the successful application of semi lagrangian methods to turbulence simulations we implement the method in the context of unstructured spectral hp element discretization which allows for efficient search interpolation procedures as well as for illumination of the nonmonotonic behavior of the temporal advection error of the form o tk xp 1 t we present numerical results that validate this error estimate for the advection diffusion equation and we document that such estimate is also valid for the navier stokes equations at moderate or high reynolds number two and three dimensional laminar and transitional flow simulations suggest that semi lagrangian schemes are more efficient than their eulerian counterparts for high order discretizations on nonuniform grids

dilute ising model on fractal lattices the authors study dilute ising models on a family of infinitely ramified exact fractals whose fractal dimension varies from 1 to 2 they calculate the phase diagram and critical exponents as a function of the fractal dimensionality within migdal s approximation 1975

approximation of the incompressible navier stokes equations using orthogonal subscale stabilization and pressure segregation on anisotropic finite element meshes abstract this paper describes a finite element model to solve the incompressible navier stokes equations based on the stabilization with orthogonal subscales and a pressure segregation the former consists of adding a least square form of the component orthogonal to the finite element space of the convective and pressure gradient terms this allows to deal with convection dominated flows and to use equal velocity pressure interpolation the pressure segregation is inspired in fractional step schemes although the converged solution corresponds to that of a monolithic time integration likewise we put special emphasis on the use of anisotropic grids in particular we describe some possible choices for the calculation of the element length that appears in the stabilization parameters and check their behavior in two classical numerical examples the preconditioning strategy used to solve the resulting algebraic system for very anisotropic meshes is also briefly described

cavity flow dynamics at higher reynolds number and higher aspect ratio abstract a primitive variable unsteady viscous incompressible navier stokes flow simulation in a higher aspect ratio a depth width 2 driven cavity at a higher reynold s number re 10 000 exhibits dynamical features not apparent in previous studies of stationary solutions or at lower reynold s number or in the more usual unit cavity investigations the initial dynamics e g between t 0 sec and t 70 sec reveals transient bifurcations between states including two three and four interacting vortices in an anterior separation region following this there is an intermediate interval e g between t 70 sec and t 140 sec characterized by a vertical oscillation of the primary vortex related to the general activities of the secondary features of the flow the long time behavior e g between t 140 seconds and t 360 seconds is one of qualitative smoothing of all secondary features except a persistent oscillation indicating a hopf bifurcation

studies of the spectral dimension for branched koch curves the scaling behaviour of koch curves with a rather general branched structure is studied by means of the exact renormalisation method and the general formula of spectral dimension is obtained it is proved that the spectral dimension is only dependent on the topology of the curve the einstein relation between the random walk dimension fractal dimension and size scaling exponent of dc conductivity is also checked

the component consistent pressure correction projection method for the incompressible navier stokes equations abstract in this paper we propose the component consistent pressure correction projection method for the numerical solution of the incompressible navier stokes equations this projection preserves a discrete form of the component consistent condition between components of the solution at every time step we also propose in particular the cnmt2 ccpc method and the rkmt ccpc method both involving one pressure poisson solution per time step we show that they are both of o t 2 for the velocity and o t for the pressure on fixed meshes and finite time intervals numerical tests on flow simulation support our claim that the component consistent pressure correction projection method solves the deviation problem encountered sometimes by the original pressure correction projection method

lower and upper bounds for the anomalous diffusion exponent on sierpinski carpets the anomalous diffusion exponent dw of random walks on a family of sierpinski carpets are studied by analytical and numerical methods we construct an effective bulk resistor and then establish the lower and upper bounds for dw where the lower bound turns out to be the same as that obtained with bond moving renormalization numerical simulations on a family of sierpinski carpets confirm our bounds and show strong dependence of dw on the lacunarity of the carpet

critical properties of ising models with long range interactions on sierpinski gasket lattices in higher euclidean dimensions abstract ising systems with two spin long range interactions are studied on sierpinski gasket lattices generalized to euclidean dimensions d 3 4 5 6 for each lattice the long range interactions are assigned to edges that are part of the appropriate self similar structure in consequence these systems are infinitely ramified and exhibit critical behavior at finite temperatures an analysis of critical properties of the systems in particular the calculation of critical exponents is carried out by using an exact renormalization group method it is shown that there exists some similarity between critical properties of the long range interacting fractal models under study here and critical properties of short range interacting ising models defined on abstract regular lattices interpolated to noninteger dimensionalities

chebyshev collocation spectral lattice boltzmann method for simulation of low speed flows in this study the chebyshev collocation spectral lattice boltzmann method ccslbm is developed and assessed for the computation of low speed flows both steady and unsteady flows are considered here the discrete boltzmann equation with the bhatnagar gross krook approximation based on the pressure distribution function is considered and the space discretization is performed by the chebyshev collocation spectral method to achieve a highly accurate flow solver to provide accurate unsteady solutions the time integration of the temporal term in the lattice boltzmann equation is made by the fourth order runge kutta scheme to achieve numerical stability and accuracy physical boundary conditions based on the spectral solution of the governing equations implemented on the boundaries are used an iterative procedure is applied to provide consistent initial conditions for the distribution function and the pressure field for the simulation of unsteady flows the main advantage of using the ccslbm over other high order accurate lattice boltzmann method lbm based flow solvers is the decay of the error at exponential rather than at polynomial rates note also that the ccslbm applied does not need any numerical dissipation or filtering for the solution to be stable leading to highly accurate solutions three two dimensional 2d test cases are simulated herein that are a regularized cavity the taylor vortex problem and doubly periodic shear layers the results obtained for these test cases are thoroughly compared with the analytical and available numerical results and show excellent agreement the computational efficiency of the proposed solution methodology based on the ccslbm is also examined by comparison with those of the standard streaming collision classical lbm and two finite difference lbm solvers the study indicates that the ccslbm provides more accurate and efficient solutions than these lbm solvers in terms of cpu and memory usage and an exponential convergence is achieved rather than polynomial rates the solution methodology proposed the ccslbm is also extended to three dimensions and a 3d regularized cavity is simulated the corresponding results are presented and validated indications are that the ccslbm developed and applied herein is robust efficient and accurate for computing 2d and 3d low speed flows note also that high accuracy solutions obtained by applying the ccslbm can be used as benchmark solutions for the assessment of other lbm based flow solvers

lacunarity and universality it is shown that the new expression for the lacunarity of sierpinski carpets proposed by taguchi 1987 is not appropriate and that the new expression of lacunarity fractal dimension and connectivity together do not specify uniquely critical exponents of ising models with spins on the carpet

on error estimates of projection methods for navier stokes equations first order schemes in this paper projection methods or fractional step methods are studied in the semi discretized form for the navier stokes equations in a two or three dimensional bounded domain error estimates for the velocity and the pressure of the classical projection scheme are established via the energy method a modified projection scheme which leads to improved error estimates is also proposed

lacunarity and percolation phase transitions the lacunarity is computed for randomly diluted two dimensional lattices in terms of the chemical distance the crossover behaviour and the question of universality are discussed

hopf bifurcation of the unsteady regularized driven cavity flow abstract a numerical simulation of the unsteady incompressible flow in the unit cavity is performed by using a chebyshev tau approximation for the space variables the high accuracy of the spectral methods and the condensed distribution of the chebyshev collocation points near the boundary enable us to obtain reliable results for high reynolds numbers with a moderate number of modes it is found that the flow converges to a stationary state for reynolds numbers re up to 10 000 for reynolds numbers larger than a critical value 10 000

a suggested generalization for the lacunarity index abstract nowadays lacunarity is a well accepted concept to complement fractality studies of complex sets lacunarity features are used to assess the largeness of gaps or holes of complex signals and images as it stands the lacunarity analysis used is an index averaging the behavior of small and large size structures in a given set the aim of this work is to propose a generalized lacunarity concept on the basis of generalized moments in this form the lacunarity index depending on a discrimination parameter is oriented to quantify lacunarity associated with objects of different size textbook and real images are used to illustrate the ability of the generalized lacunarity index for displaying interesting features of real complex sets

critical behavior of the gaussian model on a diamond type hierarchical lattice with periodic and aperiodic interactions abstract the critical behavior of the gaussian models on a diamond type hierarchical lattice with periodic and aperiodic interactions according to a class of substitution sequences are studied by the exact renormalization group approach and spin rescaling for the periodic case with a two letter general fibonacci sequence given by the generating substitutions a ab b ab and the aperiodic case with the generating substitutions a ab b ba the critical behaviors are found unchanged with respect to the uniform case and belong to the same universality of the uniform case for the other aperiodic case with the generating substitutions a ab b aa the geometric fluctuation is relevant by luck s criterion and the critical phenomena of the uniform and periodic cases are found to disappear

computation of natural convection in two dimensional cavities with chebyshev polynomials abstract a semi implicit spectral method for the computation of buoyancy induced two dimensional flow of a viscous incompressible fluid is presented the incompressible navier stokes equations expressed in terms of the primitive variables velocity and pressure and an additional equation for the temperature are considered all the variables are expanded in double truncated series of chebyshev polynomials the pressure distribution at the boundary of the computational domain is determined by means of an influence matrix technique in order to satisfy the incompressibility condition everywhere in the field the algorithm is applied to the classical problem of natural convection of a boussinesq fluid within a differentially heated vertical square cavity results show that for a prandtl number of 0 71 corresponding to air and up to rayleigh numbers of 106 very accurate solutions are obtained with a spatial resolution of 33 polynomials in each direction new interesting phenomena are shown for higher values of the rayleigh number

the coastline and lake shores on a fractal island we compute the fractal dimensions of the hulls or external boundary and the boundaries of the internal cavities in several deterministic as well as random fractal structures our conclusion is that the two fractal dimensions are in fact identical the deterministic fractals we study are sierpinski carpets sc in a two dimensional space and the random fractals are percolation clusters at criticality as an intermediate case we present results on some randomized sc in the random structures statistics of the area and perimeters of all internal cavities or holes are taken and the fractal dimension of the hull borderline is computed two different definitions of the borderline are used considering nearest neighbours as well as nearest and second nearest neighbours as connected the conclusion is valid for both cases

hysteresis scaling for ising systems on fractal structures dynamical phase transitions in ising systems on sierpinski carpets and bond percolation lattices at percolation threshold are studied by means of standard monte carlo simulations we find that the area of hysteresis loop a can be scaled with respect to the sweep rate h of a linear driving field however the exponent in the scaling expression a hb is universal only for ising systems on sierpinski carpets we conclude that the hysteresis scaling is universal for the field driven first order phase transitions in ising systems on fractal structures based on scaling hypothesis we derive the expression of finite size effect on the hysteresis the exponent b is obtained by this method in some sierpinski carpets

noninteger dimensional hyper euclidean lattices on sierpinski carpets random walks on sierpinski carpets are studied with bond moving renormalisation using the results the author constructs fractal lattices whose fractal dimension of random walk is two such fractal lattices are called hyper euclidean lattices it turns out that hyper euclidean lattices have a connectivity which is one smaller than the fractal dimension

direct evidence for weak universality on fractal structures abstract we provide a direct quantitative evidence that the critical behavior of the ising model on self similar lattices depends upon the geometrical features of the underlying structure intensive monte carlo simulations enable to show that the values of the ratio of exponents depend upon the lacunarity of the structure which measures the deviation from the translational symmetry finally the topological character of the scaling corrections when the fractal dimension is smaller than 2 is discussed

numerical simulations of a two layer quasi geostrophic equation of the ocean our object in this article is to present some numerical schemes for a two layer quasi geostrophic model of the ocean and to study the stability of these schemes we prove the unconditional stability of our numerical schemes numerical results from a wind driven two layer quasi geostrophic model of the ocean using these schemes are also presented

a suggested lacunarity expression for sierpinski carpets a suggested approximate expression for lacunarity is proposed some numerical investigations show that the old expression fails in describing the homogeneity of elimination for some sierpinski carpets

a central difference scheme for a pure stream function formulation of incompressible viscous flow we present a numerical scheme for incompressible viscous flow formulated as an equation for the stream function the pure stream function formulation obviates the difficulty associated with vorticity boundary conditions the resulting biharmonic equation is discretized with a compact scheme and solved with an algebraic multigrid solver the advection of vorticity is implemented with a high resolution central scheme that remains stable and accurate in the presence of large gradients the accuracy and robustness of the method are demonstrated for high reynolds number flows in a lid driven cavity

lacunarity and universality a new expression for the lacunarity of sierpinski carpets is proposed it turns out that the universality can be described with the new expression the fractal dimension and the connectivity

finite volume calculations of self sustained oscillations in a grooved channel a numerical simulation of unsteady incompressible navier stokes flow in a periodically grooved channel is performed for reynolds numbers r o 103 the numerical scheme is based on finite differences approximation with an explicit quadratic leith type of temporal discretization periodic self sustained oscillatory flow is reported to occur in the investigated flow configuration indicating a hopf bifurcation the threshold for the onset of self sustained flow oscillations is the critical reynolds number 950 rc 1050 the numerical predictions are in good agreement with reported spectral element method solutions demonstrating the accuracy and efficiency of the present method

galerkin spectral method for the vorticity and stream function equations a galerkin legendre spectral method for the solution of the vorticity and stream function equations in uncoupled form under no slip conditions in a square domain is presented which fully exploits the separation of variables in the two elliptic problems benefits from a nonsingular influence matrix and is able to solve the singular driven cavity problem modulo gibbs phenomenon even without regularizing the boundary condition at the corners

scale dependent lacunarity and fractal ising models as an alternate way of measuring the lacunarity of sierpinski carpets the authors have computed the site to site fluctuation of the quantity m n namely the total number of lattice sites within the chemical distance of n in long scales of n the resultant lacunarity agrees qualitatively with the results obtained in the past using the covering method in short scales of n however the lacunarity shows a different trend among fractals it varies monotonically with the ising correlation length exponent a trend which was sought before but not found with the previous method this applies only for the carpets of the first kind

exact results for an ising system with long range interactions on the sierpinski gasket lattice thermodynamic properties of an ising model with two point long range interactions on the sierpinski gasket lattice are studied using an exact recursive methodfor calculating the partition function the long range interactions are introduced in a self similar manner reflecting the structure of the lattice and are assumed to be of the form ql q 2 l 1 where q 0 0 are parameters and l 1 2 indicates that ql couples pairs of spins on the distance 2l measured in units of the lattice spacing in the case of constant long range interactions 0 exact analytic expression for the critical temperature and the critical exponent v are derived it is also shown that the recursive method can be applied to analyze the problem of the existence of phase transitions at finite temperatures in the case of 0 i e when the long range coupling decrease with the distance

phase transitions in regularly diluted 2d ising models with a complicated inherent structure of a lattice cell abstract we analyze the critical point and study the nonuniversal peculiarities occuring in the temperature dependence of the specific heat for a sequence of the exactly solvable highly decorated two dimensional triangular type ferromagnetic ising models with a complicated fractal type inherent structure of elementary cell the lattices are indexed by the decoration parameter n 0 1 2 3 the elementary cell for the n lattice is the n th order lattice approximation of the sierpinski gasket fractal at any finite n there is the second order phase transition indicated by the logarithmic singularity in the specific heat the critical temperature decreases as 4 ln 4 n 1 with increasing decoration the specific heat critical amplitude vanishes as 3 n n ln n 2 as n the basic nonuniversal features observable in the specific heat at strong decoration are the anomalous narrowing of the critical interval and the appearance of a rounded peak above the true t c point these effects are closely related and can both be explained by the above t c local ordering in the local lattice inhomogeneities the mechanism of the above t c local ordering is analyzed the models under discussion can as well be interpreted as the regular bond diluted 2d ising ferromagnets with a hierarchy of the nonmagnetic holes at different scales and varying dilution a comparative discussion on the properties of the regularly and randomly diluted 2d ising ferromagnets is given

bubble stabilized spectral methods for the incompressible navier stokes equations abstract we present and investigate a stabilized spectral legendre schemefor approximating the incompressible navier stokes equations the stabilization is obtained by addinglocal bubble functions to the usual global polynomial trial and testvelocities this modification allows the use of equal degree velocities andpressures furthermore in the advection dominated regime it induces an effectsimilar to the supg effect in finite elements the method is validated on a stokes flow and onthe regularized driven cavity flow at different reynolds regimes acomparison with other spectral and finite element schemes is also provided

self similar and translationally invariant structures in a bond diluted ising model a bond diluted ising model is used to simulate a process from a cantor set to a translationally invariant lattice the free energy function shows that there is no singularity which seem to imply that no transition exists in this process

chebyshev pseudospectral multi domain technique for viscous flow calculation abstract a novel chebyshev pseudospectral multi domain technique is introduced for the numerical solution of the navier stokes equations in the primitive variable formulation careful consideration is given to the proper interface condition in the multi domain method which decomposes the solution domain into subdomains by overlapping one grid point a finite difference approximation is used for the time discretization which transforms the system to a set of coupled equations which is then solved by an efficient computational boundary cb method in the cb method the pressure neumann boundary conditions are simply obtained by re arranging the discrete algebraic equations resulting from chebyshev pseudospectral multi domain approximation for the space variables to demonstrate the effectiveness of this technique we present some accurate computational results for calculating the driven cavity flows with re up to 10 000 and for the flow past a circular cylinder with re up to 100 the results for both applications are in good agreement with those of previous researchers

three dimensional viscous flow over rotating periodic structures the three dimensional stokes flow in a periodic domain is examined in this study the problem corresponds closely to the flow inside internal mixers where the flow is driven by the movement of a rotating screw the outer barrel remaining at rest a hybrid spectral finite difference approach is proposed for the general expansion of the flow field and the solution of the expansion coefficients the method is used to determine the flow field between the screw and barrel the regions of elongation and shear are closely examined these are the two mechanisms responsible for mixing besides its practical importance the study also allows the assessment of the validity of the various assumptions usually adopted in mixing and lubrication problems copyright 2003 john wiley sons ltd

lacunarity calculation in the true fractal limit we measure the lacunarity of regular fractals in the limit of an infinite fractal lattice using a suitable graph counting method the calculations are performed for a recent expression proposed for lacunarity that explores the scaling properties of the mass distribution of the fractal set on cells of radius r we present results for various sierpinski carpets and for the sierpinski gasket

enhancement of the stability of lattice boltzmann methods by dissipation control artificial dissipation is a well known tool for the improvement of stability of numerical algorithms however the use of this technique affects the accuracy of the computation we analyse various approaches proposed for enhancement of the lattice boltzmann methods lbm stability in addition to some previously known methods the multiple relaxation time mrt models the entropic lattice boltzmann method elbm and filtering including entropic median filtering we develop and analyse new filtering techniques with independent filtering of different modes all these methods affect dissipation in the system and may adversely affect the reproduction of the proper physics to analyse the effect of dissipation on accuracy and to prepare practical recommendations we test the enhanced lbm methods on the standard benchmark the 2d lid driven cavity on a coarse grid 101 101 nodes the accuracy was estimated by the position of the first hopf bifurcation points in these systems we find that two techniques mrt and median filtering succeed in yielding a reasonable value of the reynolds number for the first bifurcation point the newly created limiters which filter the modes independently also pick a reasonable value of the reynolds number for the first bifurcation

renormalization group approach to the multi component site percolation on sierpinski carpets abstract the multi component site percolation problems on a family of sierpinski carpets scs are solved by the real space renormalization group rg method three sorts of sites on an sc lattice are distinguished and three independent kinds of occupation probabilities are assigned to them we develope the usual choice of a cell on the translationally invariant lattices and choose suitable cells to construct the rg transformation scheme based on the connectivity the non trivial percolation threshold values and correlation length exponents on such fractals are obtained

high reynolds number solutions of navier stokes equations using incremental unknowns abstract the unsteady 2d navier stokes equations on an unregularized driven cavity are solved in vorticity streamfunction variables using incremental unknowns periodic asymptotic solutions have been found for re 10000 and re 12500

classification and universal properties of sierpinski carpets the type of critical points for the potts model is found to be dependent on the structure of sierpinski carpets all possible types of structure for sierpinski carpets with two interactions after bond moving are found the author presents general sierpinski carpets and new parameters are proposed and used to describe them the lacunarity expression is revised the author discusses and proposes a universal classification critical points eigenvalues and flow diagrams are presented some fixed points display negative eigenvalues

potts model on sierpinski carpets migdal kadanoff bond moving renormalisation is used to study the q state potts model on sierpinski carpets a general approximate recursion relation including q as a parameter is given the property of bond interchanging invariance is found and used in deriving the recursion relation fixed points and critical exponents for some carpets are presented marginal fixed points instead of unstable ones are found several typical flow diagrams are also shown

hopf bifurcation in the driven cavity abstract incompressible two dimensional calculations are reported for the impulsively started lid driven cavity with aspect ratio two the algorithm is based on the time dependent stream function equation with a crank nicolson differencing scheme for the diffusion terms and with an adams bashforth scheme for the convection terms a multigrid method is used to solve the linear implicit equations at each time step periodic asymptotic solutions have been found for re 10000 and for re 5000 the re 5000 results are validated by grid refinement calculations the solutions are shown to be precisely periodic and care is taken to demonstrate that asymptotic states have been reached a discussion is included about the indicators that are used to show that an asymptotic state has been reached and to show that the asymptotic state is indeed periodic

identification of parameters in distributed parameter systems by regularization identification of spatially varying parameters in distributed parameter systems from noisy data is an ill posed problem the concept of regularization widely used in solving linear fredholm integral equations is developed for the identification of parameters in distributed parameter systems a general regularization identification theory is first presented and then applied to the identification of parabolic systems the performance of the regularization identification method is evaluated by numerical experiments on the identification of a spatially varying diffusivity in the diffusion equation

absence of phase transition in the xy model on menger sponge abstract we have performed a monte carlo study of the classical xy model on a menger sponge with the wolff cluster algorithm u wolff 1989 the menger sponge is a fractal object with infinite order of ramification and fractal dimension d log 20 log 3 2 7268 from the dependence of the helicity modulus on system size and on boundary conditions we conclude that there is no phase transition in the system at any finite temperature

phase transitions on fractals i quasi linear lattices magnetic spin models and resistor networks are studied on certain self similar fractal lattices which are described as quasi linear because they share a significant property of the line finite portions can be isolated from the rest by removal of two points sites in all cases there is no long range order at finite temperature the transition at zero temperature has a discontinuity in the magnetisation and the associated magnetic exponent is equal to the fractal dimensionality d when the lattice reduces to a non branching curve the thermal exponent v 1 y is equal to d when the lattice is a branching curve y is related respectively to the dimensionality of the single channel segments of the curve for the ising model or to the exponent describing the resistivity for models with continuous spin symmetry

an efficient spectral projection method for the navier stokes equations in cylindrical geometries i axisymmetric cases abstract an efficient and accurate numerical scheme is presented for the axisymmetric navier stokes equations in primitive variables in a cylinder the scheme is based on a new spectral galerkin approximation for the space variables and a second order projection scheme for the time variable the new spectral projection scheme is implemented to simulate the unsteady incompressible axisymmetric flow with a singular boundary condition which is approximated to within a desired accuracy by using a smooth boundary condition a sensible comparison is made with a standard second order in time and space finite difference scheme based on a stream function vorticity formulation and with available experimental data the numerical results indicate that both schemes produce very reliable results and that despite the singular boundary condition the spectral projection scheme is still more accurate in terms of a fixed number of unknowns and more efficient in terms of cpu time required for resolving the flow at a fixed reynolds number to within a prescribed accuracy than the finite difference scheme more importantly the spectral projection scheme can be readily extended to three dimensional nonaxisymmetric cases