developing numerical techniques for solving low mach number fluid acoustic problems conservation equations applicable to this regime are derived by applying four simplifying assumptions to the fully compressible form of the navier stokes and energy equations isentropic state changes adiabatic flow negligible density variation inertial terms and stokes hypothesis the resulting equations termed acoustic compressible are similar to the traditional pseudocompressible equations except for two distinctions pseudocompressible solutions use artificially high mach numbers for computational convenience and neglect the pressure convection term in the mass conservation equation

a direct one step pressure actualization for incompressible flow with presssure neumann condition we develop a velocity pressure algorithm in primitive variables and finite differences for incompressible viscous flow with a neumann pressure boundary condition the pressure field is initialized by least squares and up dated from the poisson equation in one step without iteration simulations with the square cavity problem are made for several reynolds numbers we obtain the expected displacement of the central vortex and the appearance of secondary and tertiary eddies different geometry ratios and a 3d cavity simulation are also considered

a stream function implicit finite difference scheme for 2d incompressible flows of newtonian fluids the development of a new algorithm to solve the navier stokes equations by an implicit formulation for the finite difference method is presented that can be used to solve two dimensional incompressible flows by formulating the problem in terms of only one variable the stream function two algebraic equations with 11 unknowns are obtained from the discretized mathematical model through the adi method an original algorithm is developed which allows a reduction from the original 11 unknowns to five and the use of the pentadiagonal matrix algorithm pdma in each one of the equations an iterative cycle of calculations is implemented to assess the accuracy and speed of convergence of the algorithm the relaxation parameter required is analytically obtained in terms of the size of the grid and the value of the reynolds number by imposing the diagonal dominancy condition in the resulting pentadiagonal matrixes the algorithm developed is tested by solving two classical steady fluid mechanics problems cavity driven flow with re 100 400 and 1000 and flow in a sudden expansion with expansion ratio h h 2 and re 50 100 and 200 the results obtained for the stream function are compared with values obtained by different available numerical methods to evaluate the accuracy and the cpu time required by the proposed algorithm copyright 2002 john wiley sons ltd

a divergence free condition compensated method for incompressible navier stokes equations the present study aims to develop a new formulation to effectively calculate the incompressible navier stokes solutions in non staggered grids the distinguished feature of the proposed method which avoids directly solving the divergence free equation is to add a rigorously derived source term to the momentum equation to ensure satisfaction of the fluid incompressibility for the sake of numerical accuracy dispersion relation preserving upwind scheme developed within the two dimensional context was employed to approximate the convection terms the validity of the proposed mass preserving navier stokes method is justified by solving two benchmark problems at high reynolds and rayleigh numbers based on the simulated navier stokes solutions the proposed formulation is shown to outperform the conventional segregated method in terms of the reduction of cpu time 2007 elsevier b v all rights reserved

on the role of the helmholtz decomposition in mixed methods for incompressible flows and a new variational crime abstract according to the helmholtz decomposition the irrotational parts of the momentum balance equations of the incompressible navier stokes equations are balanced by the pressure gradient unfortunately nearly all mixed methods for incompressible flows violate this fundamental property resulting in the well known numerical instability of poor mass conservation the origin of this problem is the lack of l 2 orthogonality between discretely divergence free velocities and irrotational vector fields therefore a new variational crime for the nonconforming crouzeix raviart element is proposed where divergence free lowest order raviart thomas velocity reconstructions reestablish l 2 orthogonality this approach allows to construct a cheap flow discretization for general 2d and 3d simplex meshes that possesses the same advantageous robustness properties like divergence free flow solvers in the stokes case optimal a priori error estimates for the velocity gradients and the pressure are derived moreover the discrete velocity is independent of the continuous pressure several detailed linear and nonlinear numerical examples illustrate the theoretical findings

boundary domain integral method for high reynolds viscous fluid flows in complex planar geometries abstract the article presents new developments in boundary domain integral method bdim for computation of viscous fluid flows governed by the navier stokes equations the bdim algorithm uses velocity vorticity formulation and is based on poisson velocity equation for flow kinematics this results in accurate determination of boundary vorticity values a crucial step in constructing an accurate numerical algorithm for computation of flows in complex geometries i e geometries with sharp corners the domain velocity computations are done by the segmentation technique using large segments after solving the kinematics equation the vorticity transport equation is solved using macro element approach this enables the use of macro element based diffusion convection fundamental solution a key factor in assuring accuracy of computations for high reynolds value laminar flows the versatility and accuracy of the proposed numerical algorithm is shown for several test problems including the standard driven cavity together with the driven cavity flow in an l shaped cavity and flow in a z shaped channel the values of reynolds number reach 10 000 for driven cavity and 7500 for l shaped driven cavity whereas the z shaped channel flow is computed up to re 400 the comparison of computational results shows that the developed algorithm is capable of accurate resolution of flow fields in complex geometries

an improved unsteady unstructured artificial compressibility finite volume scheme for viscous incompressible flows part ii application in part i of this paper a preconditioned artificial compressibility scheme was developed for modelling laminar steady state and transient incompressible flows for a wide range of reynolds and rayleigh numbers in this part several examples of laminar incompressible problems are solved and discussed the influence of various ac parameters on robustness and convergence rates are assessed for a complex category of problems it is shown that the scheme developed in part i is an accurate robust and easy to use method for solving incompressible laminar flow problems over a wide range of flow regimes copyright 2002 john wiley sons ltd

a deferred correction multigrid algorithm based on a new smoother for the navier stokes equations abstract an algorithm which brings together the techniques of multigrid and deferred correction through their common relationship with imperfect newton iteration manages to combine the ease of calculation of a low order with the accuracy of a high order difference approximation of any given differential equation problem a stable explicit gauss seidel relaxation algorithm for the navier stokes equations based on an appropriate kind of upwinding of as well as derivatives especially developed for use as a multigrid smoother in this context is presented and the complete algorithm is tested on the standard conservative second order discretization of the driven cavity problem

orthogonal matrix polynomials and applications orthogonal matrix polynomials on the real line or on the unit circle have properties which are natural generalizations of properties of scalar orthogonal polynomials appropriately modified for the matrix calculus we show that orthogonal matrix polynomials both on the real line and on the unit circle appear at various places and we describe some of them the spectral theory of doubly infinite jacobi matrices can be described using orthogonal 2 x 2 matrix polynomials on the real line scalar orthogonal polynomials with a sobolev inner product containing a finite number of derivatives can be studied using matrix orthogonal polynomials on the real line orthogonal matrix polynomials on the unit circle are related to unitary block hessenberg matrices and are very useful in multivariate time series analysis and multichannel signal processing finally we show how orthogonal matrix polynomials can be used for gaussian quadrature of matrix valued functions parallel algorithms for this purpose have been implemented using pvm and some examples are worked out

perturbations in the nevai matrix class of orthogonal matrix polynomials in this paper we study a jacobi block matrix and the behavior of the limit of its entries when a perturbation of its spectral matrix measure by the addition of a dirac delta matrix measure is introduced 2001 elsevier science inc all rights reserved

generalized finite element analysis of three dimensional heat transfer problems exhibiting sharp thermal gradients abstract in this paper heat transfer problems exhibiting sharp thermal gradients are analyzed using the classical and generalized finite element methods the effect of solution roughness on the ability of the methods to obtain accurate approximations is investigated convergence studies show that low order linear and quadratic elements require strongly refined meshes for acceptable accuracy we propose a generalized fem with global local enrichments for the class of problems investigated in the paper in this procedure a global solution space defined on a coarse mesh is enriched through the partition of unity framework of the generalized fem with solutions of local boundary value problems the local problems are defined using the same procedure as in the global local fem where boundary conditions are provided by a coarse scale global solution coarse uniform global meshes are acceptable even at regions with thermal spikes that are orders of magnitude smaller than the element size convergence on these discretizations was achieved even when no or limited convergence was observed in the local problems two approaches are proposed to improve the boundary conditions prescribed on local problems and their convergence the use of the corresponding improved local solutions as enrichments for the global problem extends the range of target error level for the enriched global problem the two way information transfer provided by the proposed generalized fem is appealing to several classes of problems especially those involving multiple spatial scales the proposed methodology brings the benefits of generalized fem to problems where limited or no information about the solution is known a priori

adaptive variational multiscale methods for incompressible flow based on two local gauss integrations we consider variational multiscale vms methods with h adaptive technique for the stationary incompressible navier stokes equations the natural combination of vms with adaptive strategy retains the best features of both methods and overcomes many of their deficits a reliable a posteriori projection error estimator is derived which can be computed by two local gauss integrations at the element level finally some numerical tests are presented to illustrate the method s efficiency

a unified gas kinetic scheme for continuum and rarefied flows iv fluid dynamic equations are valid in their respective modeling scales such as the particle mean free path scale of the boltzmann equation and the hydrodynamic scale of the navier stokes ns equations with a variation of the modeling scales theoretically there should have a continuous spectrum of fluid dynamic equations even though the boltzmann equation is claimed to be valid in all scales many boltzmann solvers including direct simulation monte carlo method require the cell resolution to the order of particle mean free path scale therefore they are still single scale methods in order to study multiscale flow evolution efficiently the dynamics in the computational fluid has to be changed with the scales a direct modeling of flow physics with a changeable scale may become an appropriate approach the unified gas kinetic scheme ugks is a direct modeling method in the mesh size scale and its underlying flow physics depends on the resolution of the cell size relative to the particle mean free path the cell size of ugks is not limited by the particle mean free path with the variation of the ratio between the numerical cell size and local particle mean free path the ugks recovers the flow dynamics from the particle transport and collision in the kinetic scale to the wave propagation in the hydrodynamic scale the previous ugks is mostly constructed from the evolution solution of kinetic model equations even though the ugks is very accurate and effective in the low transition and continuum flow regimes with the time step being much larger than the particle mean free time it still has space to develop more accurate flow solver in the region where the time step is comparable with the local particle mean free time in such a scale there is dynamic difference from the full boltzmann collision term and the model equations this work is about the further development of the ugks with the implementation of the full boltzmann collision term in the region where it is needed the central ingredient of the ugks is the coupled treatment of particle transport and collision in the flux evaluation across a cell interface where a continuous flow dynamics from kinetic to hydrodynamic scales is modeled the newly developed ugks has the asymptotic preserving ap property of recovering the ns solutions in the continuum flow regime and the full boltzmann solution in the rarefied regime in the mostly unexplored transition regime the ugks itself provides a valuable tool for the non equilibrium flow study the mathematical properties of the scheme such as stability accuracy and the asymptotic preserving will be analyzed in this paper as well

a high order generalized fem for through the thickness branched cracks this paper presents high order implementations of a generalized finite element method for through the thickness three dimensional branched cracks this approach can accurately represent discontinuities such as triple joints in polycrystalline materials and branched cracks independently of the background finite element mesh representative problems are investigated to illustrate the accuracy of the method in combination with various discretizations and refinement strategies the combination of local refinement at crack fronts and high order continuous and discontinuous enrichments proves to be an excellent combination which can deliver convergence rates close to that of problems with smooth solutions copyright 2007 john wiley sons ltd

calculation of incompressible viscous flows by an unconditionally stable projection fem this paper investigates the numerical performance of a finite element implementation of a new incremental fractional step method to compute steady and unsteady incompressible viscous flows under general boundary conditions and using unstructured meshes a variational framework is adopted which accommodates two different spaces for representing and approximating the velocity fields calculated respectively in the viscous and inviscid phases of the method but which leads to a very simple numerical scheme in terms of only one discrete velocity field an unconditionally stable semi implicit approximation of the nonlinear term is used to eliminate any time step restriction as far as the numerical stability is concerned numerical results for five test problems in two dimensions are reported to illustrate the flexibility of the proposed method

a compact monotonic discretization scheme for solving second order vorticity velocity equations abstract this paper presents a numerical method for solving the steady state navier stokes equations for incompressible fluid flows using velocities and vorticity as working variables the method involves solving a second order differential equation for the velocity and a convection diffusion equation for the vorticity in cartesian grids the key to the success of the numerical simulation of this class of flow equations depends largely on proper simulation of vorticity transport equation subject to proper boundary vorticity in this paper we present a monotonic advection diffusion multi dimensional scheme and a theoretically rigorous implementation of vorticity boundary conditions while the derivation of the proposed integral vorticity boundary condition is more elaborate and is more difficult to solve than conventional local approaches the present approach offers significant advantages in this study both lid driven and backward facing step problems have been selected for comparison and validation purposes

on the role of polynomials in rbf fd approximations ii numerical solution of elliptic pdes rbf generated finite differences rbf fd have in the last decade emerged as a very powerful and flexible numerical approach for solving a wide range of pdes we find in the present study that combining polyharmonic splines phs with multivariate polynomials offers an outstanding combination of simplicity accuracy and geometric flexibility when solving elliptic equations in irregular or regular regions in particular the drawbacks on accuracy and stability due to runge s phenomenon are overcome once the rbf stencils exceed a certain size due to an underlying minimization property test problems include the classical 2 d driven cavity and also a 3 d global electric circuit problem with the earth s irregular topography as its bottom boundary the results we find are fully consistent with previous results for data interpolation

analysis and applications of a generalized finite element method with global local enrichment functions abstract this paper presents a procedure to build enrichment functions for partition of unity methods like the generalized finite element method and the hp cloud method the procedure combines classical global local finite element method concepts with the partition of unity approach it involves the solution of local boundary value problems using boundary conditions from a global problem defined on a coarse discretization the local solutions are in turn used to enrich the global space using the partition of unity framework the computations at local problems can be parallelized without difficulty allowing the solution of large problems very efficiently the effectiveness of the approach in terms of convergence rates and computational cost is investigated in this paper we also analyze the effect of inexact boundary conditions applied to local problems and the size of the local domains on the accuracy of the enriched global solution key aspects of the computational implementation in particular the numerical integration of generalized fem approximations built with global local enrichment functions are presented the method is applied to fracture mechanics problems with multiple cracks in the domain our numerical experiments show that even on a serial computer the method is very effective and allows the solution of complex problems our analysis also demonstrates that the accuracy of a global problem defined on a coarse mesh can be controlled using a fixed number of global degrees of freedom and the proposed global local enrichment functions

a hybrid element formulation for the three dimensional penalty finite element analysis of incompressible fluids this work presents a hybrid element formulation for the three dimensional penalty finite element analysis of incompressible newtonian fluids the formulation is based on a mixed variational statement in which velocity and stresses are treated as independent field variables the main advantage of this formulation is that it bypasses the use of ad hoc techniques such as selective reduced integration that are commonly used in penalty based finite element formulations and directly yields high accuracy for the velocity and stress fields without the need to carry out smoothing in addition since the stress degrees of freedom are condensed out at an element level the cost of solving for the global degrees of freedom is the same as in a standard penalty finite element method although the gain in accuracy for both the velocity and stress including the pressure fields is quite significant

a new cloud based hp finite element method a hybrid computational method for solving boundary value problems is introduced which combines features of the meshless hp cloud methods with features of conventional finite elements the method admits straightforward nonuniform hp type approximations easy implementation of essential boundary conditions is robust under severe distortions of the mesh and can deliver exponential rates of convergence results of numerical experiments are presented

matrix chebyshev polynomials and continued fractions in the first part we expose the notion of continued fractions in the matrix case in this paper we are interested in their connection with matrix orthogonal polynomials in the second part matrix continued fractions are used to develop the notion of matrix chebyshev polynomials in the case of hermitian coefficients in the recurrence formula we give the explicit formula for the stieltjes transform the support of the orthogonality measure and its density as a corollary we get the extension of the matrix version of the blumenthal theorem proved in j approx theory 84 1 1996 96 the third part contains examples of matrix orthogonal polynomials

high order compact computation and nonuniform grids for streamfunction vorticity equations abstract fourth order compact difference schemes for steady streamfunction vorticity formulation of 2d incompressible navier stokes equations on nonuniform grids are derived using maple software package especially we deduce fourth order compact difference scheme of the first partial derivative terms in order to resolve boundary layers grid transformation techniques are used which maps a nonuniform grid onto a uniform one for use with the fourth order compact difference scheme a krylov subspace iterative method with an ilut preconditioning technique is employed to solve the resulting linear system the proposed high accuracy computation method is applied to two model problems computational results are compared with results computed by other schemes in the literature

a staggered space time discontinuous galerkin method for the three dimensional incompressible navier stokes equations on unstructured tetrahedral meshes in this paper we propose a novel arbitrary high order accurate semi implicit space time discontinuous galerkin method for the solution of the three dimensional incompressible navier stokes equations on staggered unstructured curved tetrahedral meshes as is typical for space time dg schemes the discrete solution is represented in terms of space time basis functions this allows to achieve very high order of accuracy also in time which is not easy to obtain for the incompressible navier stokes equations similarly to staggered finite difference schemes in our approach the discrete pressure is defined on the primary tetrahedral grid while the discrete velocity is defined on a face based staggered dual grid while staggered meshes are state of the art in classical finite difference schemes for the incompressible navier stokes equations their use in high order dg schemes is still quite rare a very simple and efficient picard iteration is used in order to derive a space time pressure correction algorithm that achieves also high order of accuracy in time and that avoids the direct solution of global nonlinear systems formal substitution of the discrete momentum equation on the dual grid into the discrete continuity equation on the primary grid yields a very sparse five point block system for the scalar pressure which is conveniently solved with a matrix free gmres algorithm from numerical experiments we find that the linear system seems to be reasonably well conditioned since all simulations shown in this paper could be run without the use of any preconditioner even up to very high polynomial degrees for a piecewise constant polynomial approximation in time and if pressure boundary conditions are specified at least in one point the resulting system is in addition symmetric and positive definite this allows us to use even faster iterative solvers like the conjugate gradient method the flexibility and accuracy of high order space time dg methods on curved unstructured meshes allows to discretize even complex physical domains with very coarse grids in both space and time the proposed method is verified for approximation polynomials of degree up to four in space and time by solving a series of typical 3d test problems and by comparing the obtained numerical results with available exact analytical solutions or with other numerical or experimental reference data to the knowledge of the authors this is the first time that a space time discontinuous galerkin finite element method is presented for the three dimensional incompressible navier stokes equations on staggered unstructured tetrahedral grids high order staggered space time dg scheme for the 3d incompressible navier stokes equations primal grid based on curved tetrahedra dual grid based on non standard five point hexahedra new discretization of the viscous terms on the main grid using lifting operators on the dual grid very simple space time pressure correction algorithm providing also high order in time all linear systems have maximum sparsity and use the smallest possible optimal five point stencil

a multiscale extended finite element method for crack propagation in this paper we propose a multiscale strategy for crack propagation which enables one to use a refined mesh only in the crack s vicinity where it is required two techniques are used in synergy a multiscale strategy based on a domain decomposition method to account for the crack s global and local effects efficiently and a local enrichment technique the x fem to describe the geometry of the crack independently of the mesh the focus of this study is the avoidance of meshing difficulties and the choice of an appropriate scale separation to make the strategy efficient we show that the introduction of the crack s discontinuity both on the microscale and on the macroscale is essential for the numerical scalability of the domain decomposition method to remain unaffected by the presence of a crack thus the convergence rate of the iterative solver is the same throughout the crack s propagation

a multi level computational model for multi scale damage analysis in composite and porous materials abstract an adaptive multi level methodology is developed in this paper to create a hierarchy of computational sub domains with varying resolution for multiple scale problems it is intended to concurrently predict evolution of variables at the structural and microstructural scales as well as to track the incidence and propagation of microstructural damage in composite and porous materials the microstructural analysis is conducted with the voronoi cell finite element model vcfem while a conventional displacement based fem code executes the macroscopic analysis the model introduces three levels in the computational domain which include macro macro micro and microscopic analysis it differentiates between non critical and critical regions and ranges from macroscopic computations using continuum constitutive relations to zooming in at hotspots for pure microscopic simulations coupling between the scales in regions of periodic microstructure is accomplished through asymptotic homogenization an adaptive process significantly increases the efficiency while retaining appropriate level of accuracy for each region numerical examples are conducted for composite and porous materials with a variety of microscopic architectures to demonstrate the potential of the model

a singularity avoiding moving least squares scheme for two dimensional unstructured meshes moving least squares interpolation schemes are in widespread use as a tool for numerical analysis on scattered data in particular they are often employed when solving partial differential equations on unstructured meshes which are typically needed when the geometry defining the domain is complex it is known that such schemes can be singular if the data points in the stencil happen to be in certain special geometric arrangements however little research has specifically addressed this issue in this paper a moving least squares scheme is presented which is an appropriate tool for use when solving partial differential equations in two dimensions and the precise conditions under which singularities occur are identified the theory is used to develop a stencil building algorithm which automatically detects singular stencils and corrects them in an efficient manner while attempting to maintain stencil symmetry as closely as possible finally the scheme is applied in a convection diffusion equation solver and an incompressible navier stokes solver and the results are shown to compare favourably with known analytical solutions and previously published results

special finite element methods for a class of second order elliptic problems with rough coefficients in this paper the approximate solution of a class of second order elliptic equations with rough coefficients is considered problems of the type considered arise in the analysis of unidirectional composites where the coefficients represent the properties of the material several methods for this class of problems are presented and it is shown that they have the same accuracy as usual methods have for problems with smooth coefficients the methods are referred to as special finite element methods because they are of finite element type but employ special shape functions chosen to accurately model the unknown solution

fracture analysis in plane structures with the two scale g xfem method abstract generalized or extended finite element method g xfem uses enrichment functions that holds a priori knowledge about the problem solution these enrichment functions are mostly limited to two dimensional problems a well established solution for problems without any specific types of analytically derived enrichment functions is to use numerically build functions in which they are called global local enrichment functions these functions are extracted from the solution of boundary value problems defined around the region of interest discretized by a fine mesh such solution is used to enrich the global solution space through the partition of unity framework of the g xfem here it is presented a two scale global local g xfem approach to model crack propagation in plane stress strain and reissner mindlin plate problems discontinuous functions along with the asymptotic crack tip displacement fields are used to represent the crack without explicitly represent its geometries under the linear elastic fracture mechanics approach the stress intensity factor obtained from a domain based interaction energy integral can be used to either determine the crack propagation direction or propagation status i e the crack can start to propagate or not the proposed approach is presented in detail and validated by solving several linear elastic fracture mechanics problems for both plane stress strain and reissner mindlin plate cases to demonstrate its the robustness and accuracy

two scale approach to predict multi site cracking potential in 3 d structures using the generalized finite element method this research utilized the novel computational framework of the generalized finite element method gfem to predict the potential for crack propagation in concrete slabs a two scale approach using the global local concept within the gfem framework gfemg l is applied to multi site cracking problems msc where different crack geometries are placed simultaneously at different positions in a three dimensional airfield slab loaded by new generation aircraft gears the gfemg l approach efficiently simulated multiple cracks without discretization in the global mesh but only in the local domain the gfemg l enrichment functions allow the displacements of the local problem to be represented in the global domain through enrichment functions from the local problems rather than explicitly modeling each crack discretely in the global domain the main contribution of this work was extension of the gfemg l approach to a class of three dimensional msc problems involving realistic boundaries conditions and existence of multiple cracks spanning different orders of magnitude in size scales within the domain for the linear elastic structure bottom and surface initiated cracks with small dimensions were considered in conjunction with a larger macro crack unlike traditional numerical methods the proposed gfemg l made it possible to tackle this class of problems by avoiding refined crack front meshes in the global domain as well as numerical round off errors furthermore the two scale approach significantly reduces the computational cost for large scale 3 d msc problems

on the roles of minimization and linearization in least squares finite element models of nonlinear boundary value problems in this paper we examine the roles of minimization and linearization in the least squares finite element formulations of nonlinear boundary values problems the least squares principle is based upon the minimization of the least squares functional constructed via the sum of the squares of appropriate norms of the residuals of the partial differential equations in the present case we consider l 2 norms since the least squares method is independent of the discretization procedure and the solution scheme the least squares principle suggests that minimization should be performed prior to linearization where linearization is employed in the context of either the picard or newton iterative solution procedures however in the least squares finite element analysis of nonlinear boundary value problems it has become common practice in the literature to exchange the sequence of application of the minimization and linearization operations the main purpose of this study is to provide a detailed assessment on how the finite element solution is affected when the order of application of these operators is interchanged the assessment is performed mathematically through an examination of the variational setting for the least squares formulation of an abstract nonlinear boundary value problem and also computationally through the numerical simulation of the least squares finite element solutions of both a nonlinear form of the poisson equation and also the incompressible navier stokes equations the assessment suggests that although the least squares principle indicates that minimization should be performed prior to linearization such an approach is often impractical and not necessary

relative asymptotics for orthogonal matrix polynomials in this paper we study sequences of matrix polynomials that satisfy a non symmetric recurrence relation to study this kind of sequences we use a vector interpretation of the matrix orthogonality in the context of these sequences of matrix polynomials we introduce the concept of the generalized matrix nevai class and we give the ratio asymptotics between two consecutive polynomials belonging to this class we study the generalized matrix chebyshev polynomials and we deduce its explicit expression as well as we show some illustrative examples the concept of a dirac delta functional is introduced we show how the vector model that includes a dirac delta functional is a representation of a discrete sobolev inner product it also allows to reinterpret such perturbations in the usual matrix nevai class finally the relative asymptotics between a polynomial in the generalized matrix nevai class and a polynomial that is orthogonal to a modification of the corresponding matrix measure by the addition of a dirac delta functional is deduced

meshfree iterative stabilized taylor galerkin and characteristic based split cbs algorithms for incompressible n s equations abstract this paper presents two stabilized schemes of the fractional step algorithm using characteristic based split and taylor galerkin like based split respectively in the meshfree context an iterative procedure which makes the convective term satisfy the momentum conservation equation in an implicit sense is introduced into the two schemes to allow much larger time step sizes a modified version of finite increment calculus process for incompressibility stabilization is developed to avoid the calculation of high order spatial derivatives numerical experiments demonstrate the capability and stability of the proposed schemes of the meshfree based fractional step method in the simulation of incompressible flows

orthogonal matrix polynomials and higher order recurrence relations abstract it is well known that orthogonal polynomials on the real line satisfy a three term recurrence relation and conversely every system of polynomials satisfying a three term recurrence relation is orthogonal with respect to some positive borel measure on the real line we extend this result and show that every system of polynomials satisfying some 2 n 1 term recurrence relation can be expressed in terms of orthonormal matrix polynomials for which the coefficients are n n matrices we apply this result to polynomials orthogonal with respect to a discrete sobolev inner product and other inner products in the linear space of polynomials as an application we give a short proof of krein s characterization of orthogonal polynomials with a spectrum having a finite number of accumulation points

vorticity boundary condition and related issues for finite difference schemes this paper discusses three basic issues related to the design of finite difference schemes forunsteadyviscous incompressible flows using vorticity formulations the boundary condition for vorticity an efficient time stepping procedure and the relation between these schemes and the ones based on velocity pressure formulation we show that many of the newly developed global vorticity boundary conditions can actually be written as some local formulas derived earlier we also show that if we couple a standard centered difference scheme with third or fourth order explicit runge kutta methods the resulting schemes havenocell reynolds number constraints for high reynolds number flows these schemes are stable under the cfl condition given by the convective terms finally we show that the classical mac scheme is the same as thom s formula coupled with second order centered differences in the interior in the sense that one can define discrete vorticity in a natural way for the mac scheme and get the same values as the ones computed from thom s formula we use this to derive an efficient fourth order runge kutta time discretization for the mac scheme from the one for thom s formula we present numerical results for driven cavity flow at high reynolds number 105

analysis of three dimensional fracture mechanics problems a two scale approach using coarse generalized fem meshes this paper presents a generalized finite element method gfem based on the solution of interdependent global structural and local crack scale problems the local problems focus on the resolution of fine scale features of the solution in the vicinity of three dimensional cracks while the global problem addresses the macro scale structural behavior the local solutions are embedded into the solution space for the global problem using the partition of unity method the local problems are accurately solved using an hp gfem and thus the proposed method does not rely on analytical solutions the proposed methodology enables accurate modeling of three dimensional cracks on meshes with elements that are orders of magnitude larger than the process zone along crack fronts the boundary conditions for the local problems are provided by the coarse global mesh solution and can be of dirichlet neumann or cauchy type the effect of the type of local boundary conditions on the performance of the proposed gfem is analyzed several three dimensional fracture mechanics problems aimed at investigating the accuracy of the method and its computational performance both in terms of problem size and cpu time are presented copyright 2009 john wiley sons ltd

stabilized finite element schemes with lbb stable elements for incompressible flows we study stabilized fe approximations of supg type to the incompressible navier stokes problem revisiting the analysis for the linearized model we show that for conforming lbb stable elements the design of the stabilization parameters for many practical flows differs from that commonly suggested in literature and initially designed for the case of equal order approximation then we analyze a reduced supg scheme often used in practice for lbb stable elements to provide the reduced scheme with appropriate stability estimates we introduce a modified lbb condition which is proved for a family of fe approximations the analysis is given for the linearized equations numerical experiments for some linear and nonlinear benchmark problems support the theoretical results

a multiscale projection method for macro microcrack simulations we present a new multiscale method for crack simulations this approach is based on a two scale decomposition of the displacements and a projection to the coarse scale by using coarse scale test functions the extended finite element method xfem is used to take into account macrocracks as well as microcracks accurately the transition of the field variables between the different scales and the role of the microfield in the coarse scale formulation are emphasized the method is designed so that the fine scale computation can be done independently of the coarse scale computation which is very efficient and ideal for parallelization several examples involving microcracks and macrocracks are given it is shown that the effect of crack shielding and amplification for crack growth analyses can be captured efficiently copyright 2007 john wiley sons ltd

moffatt eddies in the driven cavity a quantification study by an hoc approach abstract in this paper we establish moffatt likeness of the corner vortices in the lid driven square cavity for stokes flow and flow for moderate reynolds numbers in the pre asymptotic regime the flow is computed by using an efficient transient navier stokes n s solver on compact non uniform space grids the quantification of the corner vortices in succession as moffatt vortices follows from them following a fixed geometric ratio in sizes and intensities in the process we also provide more detailed benchmarking results for the corner vortices in the cavity flow the accuracy of the scale resolution of the vortices has been verified by a novel approach to grid independence analysis this approach utilizes the concept of adverse pressure gradients as a tool to validate that the separation zones in the neighborhood of the corners are consistent with the vortices obtained from our computed solution of the n s equations

hp generalized fem and crack surface representation for non planar 3 d cracks a high order generalized finite element method gfem for non planar three dimensional crack surfaces is presented discontinuous p hierarchical enrichment functions are applied to strongly graded tetrahedral meshes automatically created around crack fronts the gfem is able to model a crack arbitrarily located within a finite element fe mesh and thus the proposed method allows fully automated fracture analysis using an existing fe discretization without cracks we also propose a crack surface representation that is independent of the underlying gfem discretization and controlled only by the physics of the problem the representation preserves continuity of the crack surface while being able to represent non planar non smooth crack surfaces inside of elements of any size the proposed representation also provides support for the implementation of accurate robust and computationally efficient numerical integration of the weak form over elements cut by the crack surface numerical simulations using the proposed gfem show high convergence rates of extracted stress intensity factors along non planar curved crack fronts and the robustness of the method copyright 2008 john wiley sons ltd

analysis and improvements of global local enrichments for the generalized finite element method abstract partition of unity methods like the gfem rely on closed form analytical enrichment functions however only limited a priori knowledge about the solution is available for many problems of engineering relevance control of discretization errors in this class of problems requires mesh refinement which offsets some of the advantages associated with these methods this has led to the development of the gfem with global local enrichments gfem gl which uses ideas of the classical global local finite element method to numerically build enrichment functions for the gfem it involves the solution of local boundary value problems using boundary conditions from the solution of the global problem discretized with a coarse mesh these local solutions are in turn used to enrich the solution space of the global problem with the help of the partition of unity framework this paper presents an a priori error estimate accounting for the effect of inexact boundary conditions applied to local problems two strategies to control the error of the gfem gl solution due to these boundary conditions are also presented the first one is based on the use of a buffer or over sampling zone in the local problems the second strategy investigated is based on multiple global local iterations several numerical experiments performed on three dimensional fracture mechanics problems illustrate the effectiveness of these strategies in controlling the error of the gfem gl solution the problems are selected such that the boundary conditions on the local problems range from smooth to singular functions

designing correct fluid hydrodynamics on a rectangular grid using mrt lattice boltzmann approach while the lattice boltzmann method lbm has become a powerful numerical approach for solving complex flows the standard lattice boltzmann method typically uses a square lattice grid in two spatial dimensions and cubic lattice grid in three dimensions for inhomogeneous and anisotropic flows it is desirable to have a lbm model that utilizes a rectangular grid there were two previous attempts to extend the multiple relaxation time mrt lbm to a rectangular lattice grid in 2d however the resulting hydrodynamic momentum equation was not fully consistent with the navier stokes equation due to anisotropy of the transport coefficients in the present work a new mrt model with an additional degree of freedom is developed in order to match precisely the navier stokes equation when a rectangular lattice grid is used we first revisit the previous attempts to understand the origin and nature of anisotropic transport coefficients by conducting an inverse design analysis within the chapman enskog procedure then an additional adjustable parameter that governs the relative orientation in the energy normal stress subspace is introduced it is shown that this adjustable parameter can be used to fully eliminate the anisotropy of transport coefficients thus the exact navier stokes equation can be derived on a rectangular grid our theoretical findings are confirmed by numerical solutions using three two dimension benchmark problems i e the channel flow the cavity flow and the decaying taylor green vortex flow the numerical results demonstrate that the proposed model shows remarkably good performance with appropriate choice of model parameters

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

a simplified finite volume lattice boltzmann method for simulations of fluid flows from laminar to turbulent regime part ii extension towards turbulent flow simulation abstract in this paper the original finite volume lattice boltzmann method fvlbm on an unstructured grid part i of these twin papers is extended to simulate turbulent flows to model the turbulent effect the k sst turbulence model is incorporated into the present fvlbm framework and is also solved by the finite volume method based on the eddy viscosity hypothesis the eddy viscosity is computed from the solution of the k sst model and the total viscosity is modified by adding this eddy viscosity to the laminar kinematic viscosity given in the bhatnagar gross krook collision term because of solving for the collision term with the explicit method in the original fvlbm scheme the computational efficiency is much lower for simulating high reynolds number flow this is due to the fact that the largest time step decided by the stability condition of the collision term which is less than twice the relaxation time is much smaller than that decided by the cfl condition in order to enhance the computational efficiency the three stage second order implicit explicit imex runge kutta method is used for temporal discretization and the time step can be one or two orders of magnitude larger as compared with the explicit euler forward scheme although the computational cost is increased the final computational efficiency is enhanced by about one order of magnitude and good results can also be obtained at a large time step through the test case of a lid driven cavity flow two turbulent flow cases are carried out to validate the present method including flow over a backward facing step and flow around a naca0012 airfoil the numerical results are found to be in agreement with experimental data and numerical solutions demonstrating the applicability of the present fvlbm coupled with the k sst model to accurately predict the incompressible turbulent flows

a discontinuous hp finite element method for convection diffusion problems this paper presents a new method which exhibits the best features of both finite volume and finite element techniques special attention is given to the issues of conservation flexible accuracy and stability the method is elementwise conservative the order of polynomial approximation can be adjusted element by element and the stability is not based on the introduction of artificial diffusion but on the use of a very particular finite element formulation with discontinuous basis functions the method supports h p and hp approximations and can be applied to any type of meshes including non matching grids a priori error estimates and numerical experiments on representative model problems indicate that the method is robust and capable of delivering high accuracy

a generalized finite element method for the simulation of three dimensional dynamic crack propagation this paper is aimed at presenting a partition of unity method for the simulation of three dimensional dynamic crack propagation the method is a variation of the partition of unity finite element method and hp cloud method in the context of crack simulation this method allows for modeling of arbitrary dynamic crack propagation without any remeshing of the domain in the proposed method the approximation spaces are constructed using a partition of unity pu and local enrichment functions the pu is provided by a combination of shepard and finite element partitions of unity this combination of pus allows the inclusion of arbitrary crack geometry in a model without any modification of the initial discretization it also avoids the problems associated with the integration of moving least squares or conventional shepard partitions of unity used in several meshless methods the local enrichment functions can be polynomials or customized functions these functions can efficiently approximate the singular fields around crack fronts the crack propagation is modeled by modifying the partition of unity along the crack surface and does not require continuous remeshings or mappings of solutions between consecutive meshes as the crack propagates in contrast with the boundary element method the proposed method can be applied to any class of problems solvable by the classical finite element method in addition the proposed method can be implemented into most finite element data bases several numerical examples demonstrating the main features and computational efficiency of the proposed method for dynamic crack propagation are presented

gas kinetic scheme with discontinuous derivative for low speed flow computation the present paper concerns the improvement of the gas kinetic scheme gks for low speed flow computation in the modified gks scheme the flow distributions with discontinuous derivatives are used as an initial condition at the cell interface for the flux evaluation this discontinuity is determined by considering both the flow characteristic and grid s resolution compared with gks method with a continuous slope for the flow variables at a cell interface the new scheme is more robust and accurate in the under resolved flow computation the new scheme presents much less numerical oscillation the extension of the current scheme to unstructured mesh is straightforward to validate the method both computations of 2d lid driven cavity flow and 3d flow past a sphere are performed the numerical results validate the current method

an artificial compressibility flux for the discontinuous galerkin solution of the incompressible navier stokes equations discontinuous galerkin dg methods have proved to be well suited for the construction of robust high order numerical schemes on unstructured and possibly nonconforming grids for a variety of problems their application to the incompressible navier stokes ins equations has also been recently considered although the subject is far from being fully explored in this work we propose a new approach for the dg numerical solution of the ins equations written in conservation form the inviscid numerical fluxes both in the continuity and in the momentum equation are computed using the values of velocity and pressure provided by the exact solution of the riemann problem associated with a local artificial compressibility perturbation of the equations unlike in most of the existing methods artificial compressibility is here introduced only at the interface flux level therefore resulting in a consistent discretization of the ins equations irrespectively of the amount of artificial compressibility introduced the discretization of the viscous term follows the well established dg scheme named br2 the performance and the accuracy of the method are demonstrated by computing the kovasznay flow and the two dimensional lid driven cavity flow for a wide range of reynolds numbers and for various degrees of polynomial approximation

the splitting method as a tool for multiple damage analysis the paper presents a reliable and efficient method for solving problems on a large set of domains with different geometries in applications these geometries might be idealizations of a mechanical structure with multiple damage the method is designed and used to analyse large three dimensional problems the present paper formulates the method and presents the theory for a simple model problem i e the laplace equation in two dimensions in the numerical implementation of the splitting scheme the p version of the finite element method and strongly graded meshes are used this approach provides high accuracy and error control the exponential rate of convergence of the method is shown and practically exemplified a numerical example shows the effects of various choices of parameters on the accuracy of the method a complex problem related to aircraft safety which was solved using the present method is used to exemplify the generality and the efficiency of the proposed method

power law fluid flow in driven enclosures with undulation using mrt lattice boltzmann method abstract in this article the flow of non newtonian fluid which is represented by the power law model in two dimensional 2d driven enclosures is studied the enclosure consists of regular rectangular shaped undulations on the bottom wall multiple relaxation time mrt collision model for the lattice boltzmann equation method lbm is employed first numerical validation is performed by comparing the mrt lbm results of power law fluid flow inside the wall driven square enclosure no undulation and flow inside diagonally flipped l shaped enclosure with the literature the strain rate profiles for square enclosure without undulations are compared using various equations available for strain rate calculation in the lbm literature further the effect of different values of the non hydrodynamic relaxation parameters on the flow is examined then for the undulated enclosures flow features and eddy dynamics are analyzed and discussed for the variations in the power law index n to represent shear thinning and shear thickening fluids the effects of various parameters such as reynolds numbers wall undulation heights and wavelength of undulations on the power law fluid flow are analyzed also the variation of viscosity with spatial location for steady state flow and total kinetic energy within the computational domain are presented for various values of power law index

conservative form of interpolated differential operator scheme for compressible and incompressible fluid dynamics the proposed scheme which is a conservative form of the interpolated differential operator scheme ido cf can provide high accurate solutions for both compressible and incompressible fluid equations spatial discretizations with fourth order accuracy are derived from interpolation functions locally constructed by both cell integrated values and point values these values are coupled and time integrated by solving fluid equations in the flux forms for the cell integrated values and in the derivative forms for the point values the ido cf scheme exactly conserves mass momentum and energy retaining the high resolution more than the non conservative form of the ido scheme a direct numerical simulation of turbulence is carried out with comparable accuracy to that of spectral methods benchmark tests of riemann problems and lid driven cavity flows show that the ido cf scheme is immensely promising in compressible and incompressible fluid dynamics studies

a pseudospectral method for solution of the three dimensional incompressible navier stokes equations abstract a new chebyshev pseudospectral technique based on the projection method that was previously applied by the authors to the solution of two dimensional incompressible navier stokes equations in primitive variables for nonperiodic boundary conditions is extended to solve the three dimensional navier stokes equations the crucial point of the method is the requirement that the continuity equation be satisfied everywhere in the domain on the boundaries as well as in the interior the key feature of the work presented in this paper is that the computer storage requirements of the full matrix inversion resulting from direct solution of the pressure poisson equation in three dimensions is greatly reduced by considering an eigenfunction decomposition the method was tested on a two dimensional driven cavity flow and the results were compared with those of the most accurate finite difference calculation the three dimensional driven cavity flow was then calculated at the same reynolds numbers as the two dimensional cases i e re 100 400 and 1000 in the calculated results three dimensional boundary effects were observed in all cases and became more apparent with increasing reynolds number

three time integration methods for incompressible flows with discontinuous galerkin boltzmann method abstract this paper presents three time integration methods for incompressible flows with finite element method in solving the lattice bgk boltzmann equation the space discretization is performed using nodal discontinuous galerkin method which employs unstructured meshes with triangular elements and high order approximation degrees the time discretization is performed using three different kinds of time integration methods namely direct decoupling and splitting from the storage cost temporal accuracy numerical stability and time consumption we systematically compare three time integration methods then benchmark fluid flow simulations are performed to highlight efficient time integration methods numerical results are in good agreement with others or exact solutions

a unified approach to compressible and incompressible flows abstract we present a finite element formulation for solving the compressible navier stokes equations which accommodates the use of any set of variables if primitive variables p u t or entropy variables are used the incompressible limit is well behaved therefore one formulation can be used to solve both compressible and incompressible flows

development of bem for convective viscous flow problems abstract three kinds of boundary element approaches for an unsteady flow problem of incompressible viscous fluid are presented the first approach which is the so called boundary domaintype is based on the use of fundamental solution for the only linear differential operator and the newton raphson iterative procedure the second one is based on the time splitting technique of the governing equations in which the fundamental equations are split into the convection equation and the stokes equations the third one is based on the well known fractional step fs method which is one of the time splitting techniques in order to show the applicability and effectiveness of our approaches numerical results of the driven cavity flow example are demonstrated through a comparison with other numerical results

supg stabilized finite element resolution of the navier stokes equations applications to water treatment engineering in this paper an analysis of the viscous incompressible flow has been carried out from the very definition of the governing equations up to the resolution of some practical problems passing through the comprehensive study of the stabilized finite element techniques used in their resolution as a consequence of this analysis a code based upon a realistic interpretation of the forces has been written which allows for the modelling of the open channel flow with optimum results in the resolution of some benchmark and real flow problems related with the wastewater industry

a global meshless collocation particular solution method for solving the two dimensional navier stokes system of equations the two dimensional navier stokes system of equations for incompressible fluids is solved by the method of approximate particular solutions maps in its global formulation the fluid velocity and pressure fields are approximated by a linear superposition of particular solutions of a stokes non homogeneous system of equations with multiquadric mq radial basis function as the source term the nonlinear convective terms of the momentum equations are linearly approximated by using a guess value of the velocity field and the resulting linear system of equations is solved by a simple direct iterative scheme picard iteration with the velocity guess given by the solution at the previous iteration although the continuity equation is not explicitly imposed in the resulting formulation the scheme is mass conservative because the particular solutions exactly satisfy the mass conservation equation the proposed numerical scheme is validated by comparison of the obtained numerical results with the corresponding analytical solution of the kovasznay flow problem at different reynolds numbers re from this analysis it is observed that the maps results are stable and accurate for a wide range of shape parameter values in addition lid driven cavity flow problems in rectangular and triangular domains up to re 3200 and re 1000 respectively and the backward facing step at re 800 are solved and the results obtained are compared with corresponding benchmark numerical solutions showing excellent agreement

a class of parallel two level nonlinear schwarz preconditioned inexact newton algorithms we propose and test a new class of two level nonlinear additive schwarz preconditioned inexact newton algorithms aspin the two level aspin combines a local nonlinear additive schwarz preconditioner and a global linear coarse preconditioner this approach is more attractive than the two level method introduced in cai keyes and marcinkowski nonlinear additive schwarz preconditioners and applications in computational fluid dynamics international journal numerical methods in fluids 40 2002 1463 1470 which is nonlinear on both levels since the coarse part of the global function evaluation requires only the solution of a linear coarse system rather than a nonlinear coarse system derived from the discretization of original partial dierential equations the overall computational cost is reduced considerably our parallel numerical results based on an incompressible lid driven flow problem show that the new two level aspin is quite scalable with respect to the number of processors and the fine mesh size when the coarse mesh size is fine enough and in addition the convergence is not sensitive to the reynolds numbers

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

isogeometric boundary element analysis of steady incompressible viscous flow part 1 plane problems abstract a novel approach is presented to the boundary element analysis of steady incompressible flow nurbs basis functions are used for describing the geometry of the problem and for approximating the unknowns in addition the arising volume integrals are treated differently to published work that is volumes are described by bounding nurbs curves instead of cells and a mapping is used the advantage of our approach is that non trivial boundary shapes can be described with very few parameters and that no generation of cells is required for the solution of the non linear equations both classical and modified newton raphson methods are used a comparison of the two methods is made on the classical example of a forced cavity flow where accurate solutions are available in the literature the results obtained agree well with published ones for moderate reynolds numbers using both methods but it is found that the latter requires a relaxation scheme and considerably more iterations to converge finally it is shown on a practical example of an airfoil how more complex boundary shapes can be approximated with few parameters and a solution obtained with a small number of unknowns

reduced navier stokes equations near a flow boundary we derive a hierarchy of pdes for the leading order evolution of wall based quantities such as the skin friction and the wall pressure gradient in two dimensional fluid flows the resulting reduced navier stokes rns equations are defined on the boundary of the flow and hence have reduced spatial dimensionality compared to the navier stokes equations this spatial reduction speeds up numerical computations and makes the equations attractive candidates for flow control design we prove that members of the rns hierarchy are well posed if appended with boundary conditions obtained from wall based sensors we also derive the lowest order rns equations for three dimensional flows for several benchmark problems our numerical simulations show close finite time agreement between the solutions of rns and those of the full navier stokes equations c 2006 elsevier b v all rights reserved

a numerical method for large eddy simulation in complex geometries we discuss the development of a numerical algorithm and solver capable of performing large eddy simulation in the very complex geometries often encountered in industrial applications the algorithm is developed for unstructured hybrid grids is non dissipative yet robust at high reynolds numbers on highly skewed grids simulation results for a variety of flows are presented

an asymptotic fitting finite element method with exponential mesh refinement for accurate computation of corner eddies in viscous flows it is well known that any viscous fluid flow near a corner consists of infinite series of eddies with decreasing size and intensity unless the angle is larger than a certain critical angle h k moffat j fluid mech 18 1964 pp 1 18 the objective of the current work is to simulate such infinite series of eddies occurring in steady flows in domains with corners the problem is approached by high order finite element method with exponential mesh refinement near the corners coupled with analytical asymptotics of the flow near the corners such approach allows one to compute position and intensity of the eddies near the corners in addition to the other main features of the flow the method was tested on the problem of the lid driven cavity flow as well as on the problem of the backward facing step flow the results of computations of the lid driven cavity problem show that the proposed method computes the central eddy with accuracy comparable to the best of existing methods and is more accurate for computing the corner eddies than the existing methods the results also indicate that the relative error of finding the eddies intensity and position decreases uniformly for all the eddies as the mesh is refined i e the relative error in computation of different eddies does not depend on their size

surfaces generated by moving least squares methods an analysis of moving least squares m l s methods for smoothing and interpolating scattered data is presented in particular theorems are proved concerning the smoothness of interpolants and the description of m l s processes as projection methods some properties of compositions of the m l s projector with projectors associated with finiteelement schemes are also considered the analysis is accompanied by examples of univariate and bivariate problems

a staggered grid multidomain spectral method for the compressible navier stokes equations we present a flexible non conforming staggered grid chebyshev spectral multidomain method for the solution of the compressible navier stokes equations in this method subdomain corners are not included in the approximation thereby simplifying the subdomain connectivity to allow for local refinement of the polynomial order or subdomain subdivision non conforming subdomains are treated by a mortar method spectral accuracy is shown in one and two dimensional linear and non linear problems application is made to four compressible flow problems the couette flow a steady boundary layer over a flat plate steady transonic flow in a nozzle and unsteady flow over a cylinder at a reynolds number of 75

a consistently formulated quick scheme for fast and stable convergence using finite volume iterative calculation procedures abstract previous applications of quick for the discretization of convective transport terms in finite volume calculation procedures have failed to employ a rigorous and systematic approach for consistently deriving this finite difference scheme instead earlier formulations have been established numerically by trial and error the new formulation for quick presented here is obtained by requiring that it satisfy four rules that guarantee physically realistic numerical solutions having overall balance careful testing performed for the wall driven square enclosure flow configuration shows that the consistently derived version of quick is more stable and converges faster than any of the formulations previously employed this testing includes the relative evaluation of boundary conditions approximated by second and third order finite difference schemes as well as calculations performed at higher reynolds numbers than previously reported

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

an anisotropic gls stabilized finite element method for incompressible flow problems abstract we consider a stabilized finite element approximation of incompressible flow problems in primitive velocity pressure variables which is based on a gls formulation equal order interpolation for the velocity and the pressure can be employed with this formulation the method introduced here is specially developed to be used on anisotropic finite element meshes with large element aspect ratios rather than using a single stabilization parameter within each element the jacobian matrix of the affine transformation from the reference element is employed for stabilization purposes stability and anisotropic error estimates in a suitable mesh dependent norm are proved for the linear stokes problem numerical results confirm the accuracy and robustness of the proposed method compared to similar previous stabilized schemes

orthogonal polynomials on sobolev spaces old and new directions abstract during the last years orthogonal polynomials on sobolev spaces have attracted considerable attention algebraic properties distribution of their zeros and fourier expansions as well as their relevance in the analysis of spectral methods for partial differential equations provide a very large field to explore and to compare with the standard case in this paper we present an introductory survey about the subject the origin of the problems and their development show the interest and the promising future of this field

skin friction and pressure the footprints of turbulence abstract the problems of exact state reconstruction and approximate state estimation based on wall information in a wall bounded incompressible unsteady flow are addressed it is shown that if in an arbitrarily small neighborhood of time t precise measurements are made of the two components of wall skin friction and the wall pressure all terms in the taylor series expansions of the unsteady flow state near the wall at time t may be determined in the linear setting this determination may be made based on skin friction measurements alone combining this fact with the analyticity of solutions of the nonlinear navier stokes equation and the unique continuation theorem for analytic functions in theory complete reconstruction of a fully developed turbulent flow in a channel at any reynolds number at time t is possible given only information about the unsteady flow available at the wall in a neighborhood of time t without knowledge of the initial conditions of the flow thus skin friction and pressure measurements on the wall in a neighborhood of time t provide a unique footprint of the entire unsteady turbulent flow state no other flow can have the same footprint indeed higher order terms are shown to uniformly improve the correlation of truncated taylor series expansions with the dns of a turbulent flow near the wall however such series extrapolations amplify measurement noise as they require differentiation in both space and time of the measurements and the radius of convergence of the taylor series expansions is less than 10 wall units the so called linear stochastic estimation technique in which the polynomials forming the basis of the series expansion are replaced by well behaved functions such as pod modes on the entire flow domain also demonstrates very poor convergence in light of these limitations on direct extrapolations from measurements in the practical setting an adjoint based algorithm is presented and numerically tested for estimating the state of an entire turbulent channel flow system based on a time history of noisy measurements at the wall this algorithm effectively uses the unsteady nonlinear navier stokes equation itself as a filter to find the flow solution that is most consistent with the available measurements

analytical solutions for heat diffusion beyond fourier law we obtain solutions for differential equations describing a broad range of physical problems by the operational method with recourse to inverse differential operators integral transforms and operational exponent generalized families of orthogonal polynomials and special functions are also employed with recourse to their operational definitions the evolutional type problems for heat transfer in various heat conduction models are studied exact analytical solutions for guyer krumhansl hyperbolic heat equation are obtained and compared with those of fourier and cattaneo equations modelling heat pulse propagation from a laser source is performed in the framework of fourier cattaneo and guyer krumhansl heat transfer models compliance of obtained solutions with the maximum principle is studied

an eulerian description of the streaming process in the lattice boltzmann equation this paper presents a novel strategy for solving discrete boltzmann equation dbe for simulation of fluid flows this strategy splits the solution procedure into streaming and collision steps as in the lattice boltzmann equation lbe method the streaming step can then be carried out by solving pure linear advection equations in an eulerian framework this offers two significant advantages over previous methods first the relationship between the relaxation parameter and the discretization of the collision term developed from the lbe method is directly applicable to the dbe method the resulting dbe collision step remains local and poses no constraint on time step second decoupling of the advection step from the collision step facilitates implicit discretization of the advection equation on arbitrary meshes an implicit unstructured dbe method is constructed based on this strategy and is evaluated using several test cases of flow over a backward facing step lid driven cavity flow and flow past a circular cylinder the speedup of convergence for some cases improves by a factor of about 20

on dyukarev s resolvent matrix for a truncated stieltjes matrix moment problem under the view of orthogonal matrix polynomials abstract the main result of this paper is a new representation of yu m dyukarev s resolvent matrix for the non degenerate truncated matricial version of the classical stieltjes moment tmcsm problem this new representation is based on the use of a quadruple of q q matrix polynomials which are characterized by certain orthogonality properties in this way useful new interrelations between the moment problems of stieltjes and hamburger which are associated with a stieltjes positive definite sequence s j j 0 2 n of complex q q matrices are found this includes an explicit formula connecting the resolvent matrices used by yu m dyukarev and i v kovalishina respectively furthermore the coefficients in the recurrence formulas for the orthogonal polynomials with respect to a stieltjes positive definite sequence are expressed in terms of the dyukarev stieltjes parameters additionally we obtain two different representations for each extremal solution of the tmcsm problem via matrix continued fraction expansions

an implicit boundary approach for viscous compressible high reynolds flows using a hybrid remeshed particle hydrodynamics method abstract we present an implicit boundary particle method with background mesh adaptation we use a brinkman penalisation to represent the boundary of the domain and a remeshed particle method to simulate viscous flow with high reynolds numbers a penalty term is added to the navier stokes equations to impose the boundary conditions the boundary conditions are enforced to a specific precision with no need to modify the numerical method or change the grid achieving an implicit approach for flow around complex boundaries geometries simulation the main idea of the brinkman penalisation method is to model the solid obstacle as a porous medium the governing equations for the compressible fluid and penalised navier stokes for the porous medium are solved simultaneously without need for interface conditions the accuracy of the method is tested for a number of benchmark problems starting with simple cases such as a periodic laminar flow in a channel poiseuille flow to more complex problems such as the lid driven cavity with high reynolds numbers the reflection and propagation of a compressible shock wave and finally swirling flow in a model two stroke diesel engine the remeshed particle mesh method coupled with brinkman penalisation provides a good quality simulation and the results are in agreement with analytical or reference solutions

a low order galerkin finite element method for the navier stokes equations of steady incompressible flow a stabilization issue and iterative methods a galerkin finite element method is considered to approximate the incompressible navier stokes equations together with iterative methods to solve a resulting system of algebraic equations this system couples velocity and pressure unknowns thus requiring a special technique for handling we consider the navier stokes equations in velocity kinematic pressure variables as well as in velocity bernoulli pressure variables the latter leads to the rotation form of nonlinear terms this form of the equations plays an important role in our studies a consistent stabilization method is considered from a new view point theory and numerical results in the paper address both the accuracy of the discrete solutions and the effectiveness of solvers and a mutual interplay between these issues when particular stabilization techniques are applied 2002 published by elsevier science b v

sph with the multiple boundary tangent method in this article we present an improved solid boundary treatment formulation for the smoothed particle hydrodynamics sph method benchmark simulations using previously reported boundary treatments can suffer from particle penetration and may produce results that numerically blow up near solid boundaries as well current sph boundary approaches do not properly treat curved boundaries in complicated flow domains these drawbacks have been remedied in a new boundary treatment method presented in this article called the multiple boundary tangent mbt approach in this article we present two important benchmark problems to validate the developed algorithm and show that the multiple boundary tangent treatment produces results that agree with known numerical and experimental solutions the two benchmark problems chosen are the lid driven cavity problem and flow over a cylinder the sph solutions using the mbt approach and the results from literature are in very good agreement these solutions involved solid boundaries but the approach presented herein should be extendable to time evolving free surface boundaries copyright 2008 john wiley sons ltd

linear stability of incompressible flow using a mixed finite element method we analyzed the linear stability for incompressible flow using the mixed finite element method first by means of the fractional step finite element method the two dimensional base flows were numerically computed over a range of reynolds numbers and then they were perturbed with three dimensional disturbances by using linear stability and normal mode analysis we obtained the partial differential equations governing the evolution of perturbation from linearized navier stokes equations with slight compressibility in terms of the mixed finite element discretization in which six node quadratic lagrange triangular elements with quadratic interpolation for velocities p2 and three node linear lagrange triangular elements for pressure p1 were used a nonsingular generalized eigenproblem was formulated from these equations whose solution gives the dispersion relationship between the complex growth rate and the wave number we examined the stability of lid driven cavity flow in this study the stability results obtained from thep2p1elements were compared with those fromp1p1elements the spatial distribution of the pressure disturbance can be smoothed by using ap2p1element such as that in the simulation of incompressible flows by the mixed finite element method however the choice of elemental type has little influence on the stability results this study presents stability curves to identify the critical reynolds number and the critical wavelength of the neutral mode and discusses the instability mechanism the taylor goertler like vortices in the cavity were obtained by means of the reconstruction of three dimensional flows the stability results and the reconstructed three dimensional flows were in agreement with the observed ones and the other numerical stability results respectively

numerical simulation of incompressible fluid flow using locally solenoidal elements in this paper we present a numerical study of the performance of a discontinuous galerkin formulation for the navier stokes equations this method is characterized by the fact that the velocity field is approximated using piecewise polynomial functions that are totally discontinuous across interelement boundaries and which are pointwise divergence free on each element locally solenoidal in particular numerical results are presented for two well known benchmark problems