computational study of incompressible turbulent flows with method of characteristics the current study presents numerical investigation of incompressible turbulent flow momentum and continuity equations are coupled by the artificial compressibility method acm the one equation spalart allmaras turbulence model along with rans reynolds averaged navier stokes equations is used to calculate reynolds stresses in 2 d problems for convective fluxes a characteristic based cb finite volume solution has been developed also a revised jameson averaging avr method was implemented in comparison the proposed cb upwind scheme demonstrated very accurate results high stability and faster convergence for time discretization the fifth order runge kutta scheme is used as the convergence acceleration techniques the local time stepping and implicit residual smoothing were applied considering flow behavior suitable boundary conditions have been implemented cross flow around a horizontal circular cylinder at high reynolds numbers has been studied as a benchmark the results obtained using the new proposed scheme are in good agreement with the standard available solutions in the literature

analysis of robust multigrid methods for steady viscous low mach number flows a theoretical and numerical analysis of different implicit methods in multigrid form is carried out in order to study their behaviour at various flow conditions the fourier analysis for scalar convection diffusion equations is extended to the coupled set of laminar navier stokes equations for first and second order discretized navier stokes equations the analysis of point line and block methods is in good agreement with the numerical results the influence on the convergence rate of different second order implementations in the multigrid method is also considered the combination of multigrid line methods and mixed discretization enables to tackle stiff problems in a cheap and robust way the comparison is further extended to reynolds averaged navier stokes equations with the low reynoldsk equations describing the turbulence when line and block methods are used it is advantageous not to incorporate the turbulence equations into the multigrid cycle

an improved unsteady unstructured artificial compressibility finite volume scheme for viscous incompressible flows part i theory and implementation a robust artificial compressibility scheme has been developed for modelling laminar steady state and transient incompressible flows over a wide range of reynolds and rayleigh numbers artificial compressibility is applied in a consistant manner resulting in a system of preconditioned governing equations a locally generalized preconditioner is introduced designed to be robust and offer good convergence rates free artificial compressibility parameters in the equations are automated to allow ease of use while facilitating improved or comparable convergence rates as compared with the standard artificial compressibility scheme memory efficiency is achieved through a multistage pseudo time explicit time marching solution procedure a node centred dual cell edge based finite volume discretization technique suitable for unstructured grids is used due to its computational efficiency and high resolution spatial accuracy in the interest of computational efficiency and ease of implementation stabilization is achieved via a scalar valued artificial dissipation scheme temporal accuracy is facilitated by employing a second order accurate dual time stepping method in this part of the paper the theory and implementation details are discussed in part 11 the scheme will be applied to a number of example problems to solve flows over a wide range of reynolds and rayleigh numbers

wu s method and its application to perspective viewing abstract an algebraic method for proving a class of geometry theorems recently proposed by wu is informally discussed key concepts relevant to the method are explained an application of this method to perspective viewing in image understanding is discussed finally it is outlined how the method can be helpful to prove geometry theorems involving inequalities to capture the relation of a point being in between two given points this is also illustrated using an example from image understanding

artificial compressibility characteristics based schemes for variable density incompressible multispecies flows part ii multigrid implementation and numerical tests the paper presents an investigation of the accuracy and efficiency of artificial compressibility characteristics based cb schemes for variable density incompressible flows the cb schemes have been implemented in conjunction with a multigrid method for accelerating numerical convergence and a fourth order explicit runge kutta method for the integration of the governing equations in time the implementation of the cb schemes is obtained in conjunction with first second and third order interpolation formulas for calculating the variables at the cell faces of the computational volume the accuracy and efficiency of the schemes are examined against analytical and experimental results for diffusion broadening in two and three dimensional microfluidic channels a problem that has motivated the development of the present methods moreover unsteady inviscid simulations have been performed for variable density mixing layer the computations revealed that accuracy and efficiency depend on the cb scheme design the best multigrid convergence rates were exhibited by the conservative cb scheme which is obtained by the fully conservative formulation of the variable density incompressible equations

artificial compressibility characteristics based schemes for variable density incompressible multi species flows part i derivation of different formulations and constant density limit the paper presents various formulations of characteristics based schemes in the framework of the artificial compressibility method for variable density incompressible flows in contrast to constant density incompressible flows where the characteristics based variables reconstruction leads to a single formulation in the case of variable density flows three different schemes can be obtained henceforth labeled as transport conservative and hybrid schemes the conservative scheme results in pseudo compressibility terms in the multi species density reconstruction it is shown that in the limit of constant density the transport scheme becomes the original characteristics based scheme for incompressible flows but the conservative and hybrid schemes lead to a new characteristics based variant for constant density flows the characteristics based schemes are combined with second and third order interpolation for increasing the computational accuracy locally at the cell faces of the control volume numerical experiments for constant density flows reveal that all the characteristics based schemes result in the same flow solution but they exhibit different convergence behavior the multigrid implementation and numerical studies for variable density flows are presented in part ii of this study

fast multigrid method for solving incompressible hydrodynamic problems with free surfaces we develop of a finite volume multigrid euler scheme for solving three dimensional fully nonlinear ship wave problems the flowfield and the a priori unknown free surface location are calculated by coupling the free surface kinematic and dynamic equations with the equations of motion for the bulk flow the evolution of the free surface boundary condition is linked to the evolution of the bulk flow via a novel iteration strategy that allows temporary leakage through the surface before the solution is converged the method of artificial compressibility is used to enforce the incompressibility constraint for the bulk flow a multigrid algorithm is used to accelerate convergence to a steady state

embedded minimal surfaces exotic spheres and manifolds with positive ricci curvature let n be a three dimensional riemannian manifold let e be a closed embedded surface in n then it is a question of basic interest to see whether one can deform in its isotopy class to some canonical embedded surface from the point of view of geometry a natural canonical surface will be the extremal surface of some functional defined on the space of embedded surfaces the simplest functional is the area functional the extremal surface of the area functional is called the minimal surface such minimal surfaces were used extensively by meeks yau my21 in studying group actions on three dimensional manifolds in my2 the theory of minimal surfaces was used to simplify and strengthen the classical dehn s lemma loop theorem and the sphere theorem in the setting there one minimizes area among all immersed surfaces and proves that the extremal object is embedded in this paper we minimize area among all embedded surfaces isotopic to a fixed embedded surface in the category of these surfaces we prove a general existence theorem theorem 1 a particular consequence of this theorem is that for irreducible manifolds an embedded incompressible surface is isotopic to an embedded incompressible surface with minimal area we also prove that there exists an embedded sphere of least area enclosing a fake cell provided the complementary volume is not a standard ball and provided there exists no embedded one sided rp2 by making use of the last result and a cutting and pasting argument we are able to settle a well known problem in the theory of three dimensional manifolds we prove that the covering space of any irreducible orientable three dimensional manifold is irreducible it is possible to exploit our existence theorem to study

solution of the euler equations for two dimensional transonic flow by a multigrid method the assumption of potential o w implies that the o w is irrotational this is not strictly correct when shock waves are present an exact description of transonic inviscid o w requires the solution of the euler equations the numerical solution of the euler equations for steady transonic o ws is therefore a problem of great interest to the aeronautical community it also presents a testing challenge to applied mathematicians and numerical analysts

multigrid navier stokes calculations for three dimensional cascades a vertex based finite volume method for solving the three dimensional compressible reynolds averaged navier stokes equations is presented for calculating turbomachinery cascade flows a discretization scheme for the viscous terms is proposed the baldwin lomax algebraic turbulence model is used the scheme is verified against laminar and turbulent flows over a flat plate two and three dimensional computations were carried out for a low pressure turbine cascade at design and off design conditions results are compared with inviscid solutions and experimental data

acceleration of multigrid flow computations through dynamic adaptation of the smoothing procedure abstract the paper presents the development and investigation of an adaptive smoothing as procedure in conjunction with the full multigrid full approximation storage method the latter has been previously implemented by the authors 1 for the incompressible navier stokes equations in conjunction with the artificial compressibility method and forms the basis for investigating the current as approach the principle of adaptive smoothing is to exploit the nonuniform convergence behavior of the numerical solution during the iterations to reduce the size of the computational domain and subsequently to reduce the total computing time the implementation of the as approach is investigated in conjunction with three different adaptivity criteria for two and three dimensional incompressible flows furthermore a dynamic procedure henceforth labeled dynamic adaptivity for defining variably the as parameters during the computation is also proposed and its performance is investigated in contrast to as with constant parameters henceforth labeled static adaptivity both static and dynamic adaptivity can provide similar acceleration but the latter additionally provides more stable numerical solutions and also requires less experimentation for optimizing the performance of the method numerical experiments are presented for attached and separated flows around airfoils as well as for three dimensional flow in a curved channel for external flows the as performs better when it is applied in all grid levels of the multigrid method while for internal flows the best performance is achieved when as is applied in the finest grid only overall the results show that substantial acceleration of multigrid computations can be achieved by using adaptive smoothing

a refutational approach to geometry theorem proving abstract a refutational method for proving universally quantified formulae in algebraic geometry is proposed a geometry statement to be proved is usually stated as a finite set of hypotheses implying a conclusion a hypothesis is either a polynomial equation expressing a geometric relation or a polynomial inequation the negation of a polynomial equation expressing a subsidiary condition that rules out degenerate cases and perhaps some general cases a conclusion is a polynomial equation expressing a geometry relation to be derived instead of showing that the conclusion directly follows from the hypothesis equations and inequations the proof by contradiction technique is employed it is checked whether the negation of the conclusion is inconsistent with the hypotheses this can be done by converting the hypotheses and the negation of the conclusion into a finite set of polynomial equations and checking that they do not have a common solution there exist many methods for this check thus giving a complete decision procedure for such geometry statements this approach has been recently employed to automatically prove a number of interesting theorems in plane euclidean geometry a grobner basis algorithm is used to check whether a finite set of polynomial equations does not have a solution two other formulations of geometry problems are also discussed and complete methods for solving them using the grobner basis computations are given

design of a high performance unsteady naiver stokes solver using a flexible cycle additive correction multigrid technique a multigrid technique based on a flexible cycle additive correction multigrid fcac mg scheme is utilized to design a high performance solver for the unsteady incompressible navier stokes n s equations the unsteady incompressible n s solver discussed here incorporates the fractional step method and finite volume discretization over a staggered cartesian grid the current research indicates that the fcac mg acceleration technique is highly efficient reliable and robust which makes it feasible for cpu intensive computations such as large eddy simulation les and direct numerical simulation dns the high efficiency and the robustness of the solver are achieved through developing the fcac mg acceleration technique as well as flow physics oriented solving strategies the flow solver based on the fcac mg technique is applied to both temporal and spatial turbulence simulations using both les and dns the residuals of the large scale algebraic equation system are guaranteed to be continuously driven down to the level of the computer machine round off error on each time marching step which warrants strong conservations of mass and momentum satisfied over all the control volumes in addition the flow solver developed in the current research possesses the potential capability for handling complex geometry flows since the performance of the solver does not rely on the use of special type of operations on the large scale algebraic equation system unlike the cyclic reduction algorithm but instead it is driven by a flow physics oriented solving strategy les for flows in a square duct a square annular duct and a confined square coaxial jet are performed to present the advantages of the fcac mg technique and the flow physics oriented solving strategy the analyses of the simulation results for these three flow configurations provide strong evidence that the flow solver based on fcac mg method is capable of capturing the major characteristics of turbulence physics and correctly predicting the relevant turbulent flow phenomena

application of grobner bases to problems of movement of a particle abstract the classical method of grobner bases for multivariate polynomials in computer algebra and the related buchberger s algorithm and its modifications for the computation of such bases are applied to some elementary problems of kinematics as well as to the classical kepler newton problem in celestial mechanics where beyond the variables in the polynomials the differential operator d appears as well the popular computer algebra system maple v and the related standard package were used for this purpose and several possibilities of using grobner bases for the proof and or the derivation of formulae in mechanics are illustrated the present results generalize well known results for the proof derivation of geometric theorems by using classical grobner bases and related techniques and they illustrate the power of commercial computer algebra systems in the aforementioned tasks in kinematics modifications and generalizations of the present approach are also possible

efficient and non reflecting far field boundary conditions for incompressible flow calculations abstract traditionally the artificial compressibility ac method of chorin is used for simulation of low speed and incompressible flows in this method the difficulty of continuity and momentum equations decoupling is removed by adding an artificial time derivative of pressure to the continuity equation for the first time a fully two dimensional upwind scheme was presented for ac equations by using characteristic structure of equations by the authors in this paper a new remote boundary calculation method is presented by using the idea of characteristics for these equations instead of simple boundary conditions which usually are employed for incompressible flows the flow quantities at the far field boundaries are evaluated by compatibility relations of characteristic equations this is implemented by assuming a row of ghost cells outside of the far field boundary and using the flux calculation method based on characteristics similar to the inside of computational domain the idea in conjunction with multidimensional characteristic based scheme was tested for incompressible flow around circular cylinder in comparison with conventional far field boundary condition and showed good improvements in the terms of accuracy and convergence speed

multigrid solution of the navier stokes equations at low speeds with large temperature variations multigrid methods for the navier stokes equations at low speeds and large temperature variations are investigated the compressible equations with time derivative preconditioning and preconditioned flux difference splitting of the inviscid terms are used three implicit smoothers have been incorporated into a common multigrid procedure both full coarsening and semi coarsening with directional fine grid defect correction have been studied the resulting methods have been tested on four 2d laminar problems over a range of reynolds numbers on both uniform and highly stretched grids two of the three methods show efficient and robust performance over the entire range of conditions in addition none of the methods has any difficulty with the large temperature variations

computer based manipulation of systems of equations in elasticity problems with grobner bases abstract the classical method of buchberger s grobner bases for multivariate polynomial equations with a variety of applications in computer algebra is applied to three problems in elasticity concerning i a simple elastic truss ii an elementary plane elasticity problem in rectangular coordinates and iii the derivation of a compatibility partial differential equation for the stress components and the proof of biharmonicity of the same components also in plane elasticity the present results aim at the practical illustration of the possibility of deriving computer generated formulae in elasticity problems by using grobner bases as has been done in the case of geometric problems and the popular and modern computer algebra system maple v equipped with a related package has been used further generalizations of the present results are quite possible and some of these are discussed in brief

improved convergence to the steady state of the euler equations by enhanced wave propagation the convergence to the numerical solution of the stationary euler equations in two and three dimensions is studied the basic iterative algorithms are runge kutta time stepping gmres and a modified gmres method convergence acceleration is achieved by two preconditioning techniques residual smoothing and multigrid iteration the preconditioners are such that they increase the propagation of smooth error modes out of the computational domain runge kutta time stepping and the modified gmres method guarantee this wave propagation the results from a number of numerical experiments are reported

kahler einstein metrics on complex surfaces with c 1 0 various estimates of the lower bound of the holomorphic invariant m defined in t are given here by using branched coverings potential estimates and lelong numbers of positive d closed 1 1 currents of certain type etc these estimates are then applied to produce kahler einstein metrics on complex surfaces withc1 0 in particular we prove that there are kahler einstein structures withc1 0 on any manifold of differential type cp 2 overline ncp 2 3 leqq n leqq 8

a nonlinear multigrid method for the three dimensional incompressible navier stokes equations a nonlinear multigrid method is developed for solving the three dimensional navier stokes equations in conjunction with the artificial compressibility formulation the method is based on the full multigrid fmg full approximation storage fas algorithm and is realized via an unsteady type procedure according to which the equations are not solved exactly on the coarsest grid but some pseudo time iterations are performed on the finer grids and some on the coarsest grid the multigrid method is implemented in conjunction with a third order upwind characteristics based scheme for the discretization of the convection terms and the fourth order runge kutta scheme for time integration the performance of the method is investigated for three dimensional flows in straight and curved channels as well as flow in a cubic cavity the multigrid acceleration is assessed in contrast to the single grid and mesh sequencing algorithms the effects of various multigrid components on the convergence acceleration such as prolongation operators as well as pre and postrelaxation iterations are also investigated

short note on the characteristics based acm for incompressible flows in this paper the revised characteristics based cb method for incompressible flows recently derived by neofytou p neofytou revision of the characteristic based scheme for incompressible flows j comput phys 222 2007 475 484 has been further investigated we have derived all the formulas for pressure and velocities from this revised cb method which is based on the artificial compressibility method acm a j chorin a numerical solution for solving incompressible viscous flow problems j comput phys 2 1967 12 then we analyze the formulations of the original cb method d drikakis p a govatsos d e papatonis a characteristic based method for incompressible flows int j numer meth fluids 19 1994 667 685 e shapiro d drikakis non conservative and conservative formulations of characteristics numerical reconstructions for incompressible flows int j numer meth eng 66 2006 1466 1482 d drikakis p k smolarkiewicz on spurious vortical structures j comput phys 172 2001 309 325 f mallinger d drikakis instability in three dimensional unsteady stenotic flows int j heat fluid flow 23 2002 657 663 e shapiro d drikakis artificial compressibility characteristics based schemes for variable density incompressible multi species flows parts i derivation of different formulations and constant density limit j comput phys 210 2005 584 607 y zhao b zhang a high order characteristics upwind fv method for incompressible flow and heat transfer simulation on unstructured grids comput meth appl mech eng 190 5 7 2000 733 756 to investigate their consistency with the governing flow equations after convergence has been achieved furthermore we have implemented both formulations in an unstructured grid finite volume solver y zhao b zhang a high order characteristics upwind fv method for incompressible flow and heat transfer simulation on unstructured grids comput meth appl mech eng 190 5 7 2000 733 756 detailed numerical experiments show that both methods give almost identical solutions and convergence rates both can generate solutions which agree well with published results and experimental measurements we thus conclude that both methods being upwind schemes designed for the acm have the same performances in terms of accuracy and convergence speed even though the revised method is more complex with less stringent assumptions made while the original cb method is simpler due to the use of extra simplifying assumptions

a novel two grid method for semilinear elliptic equations a new finite element discretization technique based on two coarse and fine subspaces is presented for a semilinear elliptic boundary value problem the solution of a nonlinear system on the fine space is reduced to the solution of two small one linear and one nonlinear systems on the coarse space and a linear system on the fine space it is shown both theoretically and numerically that the coarse space can be extremely coarse and still achieve asymptotically optimal approximation as a result the numerical solution of such a nonlinear equation is not significantly more expensive than the solution of one single linearized equation

parallel multigrid computation of unsteady incompressible viscous flows using a matrix free implicit method and high resolution characteristics based scheme abstract a three dimensional parallel unstructured non nested multigrid solver for solutions of unsteady incompressible viscous flow is developed and validated the finite volume navier stokes solver is based on the artificial compressibility approach with a high resolution method of characteristics based scheme for handling convection terms the unsteady flow is calculated with a matrix free implicit dual time stepping scheme the parallelization of the multigrid solver is achieved by multigrid domain decomposition approach mg dd using single program multiple data spmd and multiple instruction multiple data mimd programming paradigm there are two parallelization strategies proposed in this work first strategy is a one level parallelization strategy using geometric domain decomposition technique alone second strategy is a two level parallelization strategy that consists of a hybrid of both geometric domain decomposition and data decomposition techniques message passing interface mpi and openmp standard are used to communicate data between processors and decompose loop iterations arrays respectively the parallel multigrid code is used to simulate both steady and unsteady incompressible viscous flows over a circular cylinder and a lid driven cavity flow a maximum speedup of 22 5 could be achieved on 32 processors for instance the lid driven cavity flow of re 1000 the results obtained agree well with numerical solutions obtained by other researchers as well as experimental measurements a detailed study of the time step size and number of pseudo sub iterations per time step required for simulating unsteady flow are presented in this paper

a stable and accurate convective modelling procedure based on quadratic upstream interpolation a convective modelling procedure is presented which avoids the stability problems of central differencing while remaining free of the inaccuracies of numerical diffusion associated with upstream differencing for combined convection and diffusion the number of operations at each grid point is comparable to that of standard upstream pluscentral differencing however highly accurate solutions can be obtained with a grid spacing much larger than that required by conventional methods for comparable accuracy with obvious practical advantaged in terms of both speed and storage the algorithm is based on a conservative control volume formulation with cell wall values of each field variable written in terms of a quadratic interpolation using in any one coordinate direction the two adjacent nodal values together with the value at the next upstream node this results in a convective differencing scheme with greater formal accuracy than central differencing while retaining the basic stable convective sensitivity property of upstream weighted schemes the consistent treatment of diffusion terms is equivalent to central differencing with careful modelling numerical boundary conditions are not troublesome some idealized problems are studied showing the practical advantages of the method over other schemes in comparison with exact solutions an application to a complex unsteady two dimensional flow is briefly discussed

geometric multigrid with applications to computational fluid dynamics an overview is given of the development of the geometric multigrid method with emphasis on applications in computational fluid dynamics over the last ten years both compressible and incompressible flow problems and their corresponding multigrid solution methods are discussed the state of the art is described with respect to methods employed in industry as well as the multigrid efficiency obtained in academic applications

multidimensional upwinding for incompressible flows based on characteristics in this paper the multidimensional characteristic based upwind scheme mcb which has been recently introduced by the authors is applied to two another benchmark problems namely flow in a channel with a backward facing step and two dimensional steady and unsteady flows past a circular cylinder extension of mcb scheme for calculating convective fluxes on non cartesian grids is presented here for the flow over backward facing step obtained results were compared against well known experimental data and the results show high accuracy of mcb scheme and faster convergence rate with respect to conventional cb scheme in the case of flow over circular cylinder the flow at steady and transient regimes is investigated by mcb scheme again the results obtained by mcb are compared to the other results in the literature and show good agreement with them also rapid convergence rate of mcb was observed in this case too it is concluded that the genuinely multidimensional characteristic based mcb scheme has been introduced earlier by the authors is a robust and powerful scheme for modeling incompressible viscous flows for achieving the high accuracy and remarkable advantage in convergence rate with respect to conventional characteristic based schemes

on einstein four manifolds with s 1 actions we study closed einstein 4 manifolds which admit s1 actions of a certain type i e warped products in particular we classify them up to isometry when the fixed point of the s1 action satisfies certain natural geometric conditions the proof uses the bochner weitzenbock formula for 1 forms and the theory of minimal surfaces in 3 manifolds