a canonical transformation relating the lagrangian and eulerian description of ideal hydrodynamics abstract we derive by means of a canonical transformation the variational principle of seliger and whitham for the eulerian description of ideal hydrodynamics from the more familiar variational principle that yields the equations of motion of an ideal fluid in the lagrangian description

direct simulations of turbulent flow using finite difference schemes abstract this paper presents finite difference solutions to the evolution of small amplitude disturbances and incompressible fully developed turbulent channel flow the main objective of the paper is to provide a comprehensive comparison between the results obtained using finite difference and spectral methods an advantage of finite difference schemes over the highly accurate spectral methods lies in the ease with which they can be applied to complex geometries the finite difference methods used include a kinetic energy conserving type of central difference scheme and a high order accurate upwind difference scheme unlike the central difference scheme the upwind difference scheme was found not to require a kinetic energy conservation property to control aliasing error the dissipative nature of the upwind scheme results in a damping of the higher frequency content as a result very little energy is aliased back the computed data including first and second order statistics for the turbulent channel flow case are found to compare well with experimental data and earlier spectral simulations it appears that the high order accurate upwind scheme is a good candidate for direct simulations of turbulent flows over complex geometries

symmetry transformations isovectors and conservation laws a system of second order partial differential equations is considered it is shown that a conservation law may be associated with any pair consisting of i a symmetry transformation ii a symmetry transformation if the system is self adjoint a solution to the adjoint of the equations of variation otherwise such conservation laws continue to hold when symmetry transformations are replaced by isovectors it is also proved that isovectors identify additional conservation laws by deformation of a given one whence it follows that there exists a natural conserved current associated with every isovector also an application is made to find conserved currents for the navier stokes equations

multidimensional optimization of finite difference schemes for computational aeroacoustics because of the long propagation distances computational aeroacoustics schemes must propagate the waves at the correct wave speeds and lower the isotropy error as much as possible the spatial differencing schemes are most frequently analyzed and optimized for one dimensional test cases therefore in multidimensional problems such optimized schemes may not have isotropic behavior in this work optimized finite difference schemes for multidimensional computational aeroacoustics are derived which are designed to have improved isotropy compared to existing schemes the derivation is performed based on both taylor series expansion and fourier analysis various explicit centered finite difference schemes and the associated boundary stencils have been derived and analyzed the isotropy corrector factor a parameter of the schemes can be determined by minimizing the integrated error between the phase or group velocities on different spatial directions the order of accuracy of the optimized schemes is the same as that of the classical schemes the advantage being in reducing the isotropy error the present schemes are restricted to equally spaced cartesian grids so the generalized curvilinear transformation method and cartesian grid methods are good candidates the optimized schemes are tested by solving various multidimensional problems of aeroacoustics

the inverse problem of the calculus of variations applied to continuum physics necessary and sufficient conditions for a differential system of equations to admit a variational formulation are established by having recourse to vainberg s theorem which provides also a systematic method for producing the sought functional an application of the method to the lagrangian description of fluid dynamics leads to a new variational principle which while being fully general reveals a hierarchy between variational approaches to fluid dynamics next the method is applied in an attempt to obtain new variational formulations in various areas of research pertaining to continuum physics water wave models elasticity heat conduction in solids dynamics of anharmonic crystals and electromagnetism owing to the power of the method relevant variational formulations are found whenever the given system allows them the paper places particular emphasis on equations which have or are supposed to have soliton solutions

the stability of numerical boundary treatments for compact high order finite difference schemes the stability characteristics of various compact fourth and sixth order spatial operators are assessed with the theory of gustafsson kreiss and sundstrom g k s for the semidiscrete initial boundary value problem these results are generalized to the fully discrete case with a recently developed theory of kreiss and wu in all cases favorable comparisons are obtained between g k s theory eigenvalue determination and numerical simulation the conventional definition of stability then is sharpened to include only those spatial discretizations that are asymptotically stable bounded left half plane eigenvalues many of the higher order schemes that are g k s stable are not asymptotically stable a series of compact fourth and sixth order schemes that are both asymptotically and g k s stable for the scalar case are then developed

an analytical study of sound transmission through unbounded panels of functionally gradedmaterials the problem of sound transmission and reflection from unbounded panels of functionally graded materials is studied using an analytical approach by means of matrix manipulation and fourier component analysis the three dimensional 3 d governing equations of elastodynamics are converted into a system of ordinary differential equations with variable coefficients in the frequency and wavenumber domain integration of the ordinary differential equation system across the panel thickness leads to a closed form solution for the transfer matrix analytical expressions are then obtained for sound reflection and transmission coefficients for panels of functionally graded materials the present model is used to predict sound transmission losses for various panel examples the results compare well with published data from other methods thereby validating the accuracy of the formulation developed in this study

lagrangians for certain classes of inelastic solids abstract the inverse problem of the calculus of variations is investigated in conjunction with two non linear models of inelastic solids namely those of kelvin voigt and of zener in both cases on appealing to the general theory the most general forms of the constitutive equations compatible with the existence of a variational formulation are determined next the pertinent local in time lagrangian densities are established the main features of the constitutive equations and the lagrangian densities so obtained are also discussed

analysis of a new high resolution upwind compact scheme a new high resolution upwind compact scheme is presented and analyzed along with several previously proposed schemes for numerical dispersion dissipation anisotropy phase damping dispersion relation preservation property and numerical stability the schemes are tested on problems of the propagation of a initially discontinuous wave and of the transport of a sharp scalar cone a benchmark problem of aeroacoustics is solved with the present scheme giving satisfactory resolution the scheme compares well with the existing schemes for the range of properties considered desirable in high resolution schemes

finite difference schemes for long time integration a general method for constructing finite difference schemes for longtime integration problems is presented it is demonstrated for discretizations of first and second space derivatives however the approach is not limited to these cases the schemes are constructed so as to minimize the global truncation error taking into account the initial data the resulting second order compact schemes can be used for integration times fourfold or more longer than previously studied schemes with similar computational complexity a similar approach was used to obtain improved integration schemes

analysis of central and upwind compact schemes central and upwind compact schemes for spatial discretization have been analyzed with respect to accuracy in spectral space numerical stability and dispersion relation preservation a von neumann matrix spectral analysis is developed here to analyze spatial discretization schemes for any explicit and implicit schemes to investigate the full domain simultaneously this allows one to evaluate various boundary closures and their effects on the domain interior the same method can be used for stability analysis performed for the semi discrete initial boundary value problems ibvp this analysis tells one about the stability for every resolved length scale some well known compact schemes that were found to be g k s and time stable are shown here to be unstable for selective length scales by this analysis this is attributed to boundary closure and we suggest special boundary treatment to remove this shortcoming to demonstrate the asymptotic stability of the resultant schemes numerical solution of the wave equation is compared with analytical solution furthermore some of these schemes are used to solve two dimensional navier stokes equation and a computational acoustic problem to check their ability to solve problems for long time it is found that those schemes that were found unstable for the wave equation are unsuitable for solving incompressible navier stokes equation in contrast the proposed compact schemes with improved boundary closure and an explicit higher order upwind scheme produced correct results the numerical solution for the acoustic problem is compared with the exact solution and the quality of the match shows that the used compact scheme has the requisite drp property

the connection between variational principles in eulerian and lagrangian descriptions the question of whether there exists a connection between variational principles in eulerian and lagrangian descriptions is investigated by having recourse to a proper view of the eulerian description it is shown that a variational principle in one description holds whenever a corresponding variational principle in the other description is given this theoretical conclusion is operative in that a precise rule for writing the new lagrangian is exhibited as an application a new lagrangian for fluid dynamics in the eulerian description is determined

compact finite volume schemes on boundary fitted grids the paper focuses on the development of a framework for high order compact finite volume discretization of the three dimensional scalar advection diffusion equation in order to deal with irregular domains a coordinate transformation is applied between a curvilinear non orthogonal grid in the physical space and the computational space advective fluxes are computed by the fifth order upwind scheme introduced by pirozzoli s pirozzoli conservative hybrid compact weno schemes for shock turbulence interaction j comp phys 178 2002 81 while the coupled derivative scheme m h kobayashi on a class of pade finite volume methods j comp phys 156 1999 137 is used for the discretization of the diffusive fluxes numerical tests include unsteady diffusion over a distorted grid linear free surface gravity waves in a irregular domain and the advection of a scalar field the proposed methodology attains high order formal accuracy and shows very favorable resolution characteristics for the simulation of problems with a wide range of length scales

compact implicit maccormack type schemes with high accuracy in this work the maccormack methodology is extended to implicit compact differencing schemes a prefactorization method is developed which splits the implicit matrix into two independent upper and lower matrices which are easier to invert using this method a new class of high order accurate compact maccormack type schemes is derived two fourth order schemes are described and results are shown for three linear and nonlinear caa benchmark problems

propagation properties of acoustic waves inside periodic pipelines abstract this paper presents a new method to model and to simulate the propagation properties of the acoustic wave in periodic pipelines first an ideal model of pipeline and its dispersion equations of acoustic waves are constructed and described in detail in order to implement this method subsequently the classical finite difference method is applied to determine the boundary conditions of the pipelines in addition the transfer matrix method is used to simulate the propagation process of the acoustic waves inside the pipelines finally by conducting length limited periodic drill string experiments it is shown that this model and algorithm can be used to obtain the acoustic wave spectrum distribution and impulse response characteristics inside periodic pipelines under operational conditions this method provides a theoretical basis for the fault diagnosis of acoustic wave transmission systems for oil industry applications

the von neumann analysis and modified equation approach for finite difference schemes the von neumann discrete fourier analysis and modified equation technique have been proven to be two effective tools in the design and analysis of finite difference schemes for linear and nonlinear problems the former has merits of simplicity and intuition in practical applications but only restricted to problems of linear equations with constant coefficients and periodic boundary conditions the later pde approach has more extensive potential to nonlinear problems and error analysis despite its kind of relative complexity the dissipation and dispersion properties can be observed directly from the pde point of view even order terms supply dissipation and odd order terms reflect dispersion in this paper we will show rigorously their full equivalence via the construction of modified equation of two level finite difference schemes around any wave number only in terms of the amplification factor used in the von neumann analysis such a conclusion fills in the gap between these two approaches in literatures

wavenumber extended high order upwind biased finite difference schemes for convective scalar transport this paper proposes some new wavenumber extended high order upwind biased schemes the dispersion and dissipation errors of upwind biased finite difference schemes are assessed and compared by means of a fourier analysis of the difference schemes up to 11th order upwind biased schemes are analyzed it is shown that both the upwind biased scheme of order 2n 1 and the corresponding centered differencing scheme of order 2nhave the same dispersion characteristics thus the former can be considered to be the latter plus a correction that reduces the numerical dissipation the new second order wavenumber extended scheme is tested and compared with some well known schemes the range of wavenumbers that are accurately treated by the upwind biased schemes is improved by using additional constraints from the fourier analysis to construct the new schemes the anisotropic behavior of the dispersion and dissipation errors is also analyzed for both the conventional and the new wavenumber extended upwind biased finite difference schemes

composite variational principles and the determination of conservation laws sufficiently regular systems of partial differential equations can always be derived from a composite variational principle via the introduction of a suitable set of adjoint variables by applying an appropriate formulation of noether s theorem to such composite variational principles an operative procedure leading to the determination of the conservation laws of the given system is determined as an application conservation laws for a viscous incompressible fluid and an inviscid barotropic compressible fluid are examined in detail in particular new conservation laws for euler flows are found

a three point sixth order nonuniform combined compact difference scheme a three point nonuniform combined compact difference nccd scheme with sixth order accuracy is proposed for numerical models the nccd scheme is a generalization of the previously proposed combined compact difference ccd scheme with a global hermitan polynomial spline and has major improved features such as error and computational cpu time reduction for nonperiodic boundaries additional sixth or fifth order nonuniform boundary conditions are proposed the nccd scheme with either sixth or fifth order additional boundary conditions can increase the accuracy and decrease the cpu time about 1 2 orders of magnitude compared to the ccd scheme

on the inverse problem of the calculus of variations we consider the inverse problem of the calculus of variations for any system by writing its differential equations of motion in first order form we provide a way of constructing infinitely many lagrangians for such a system in terms of its constants of motion using a covariant geometrical approach we present examples of first order lagrangians for systems for which no second order lagrangians exist the hamiltonian theory for first order degenerate lagrangians is constructed using dirac s method for singular lagrangians

elasticity solution for stresses in a sandwich beam with functionally graded core introduction w eight savingsofferedby sandwich constructionsfor structures that require high bending stiffness are signi cant however sandwich constructions have not been fully exploited in structuralapplicationsdue to damage toleranceconcerns the core face sheet delamination is a major concern in sandwich construction the stiffness discontinuityat the face sheet and core interface results in a large increase in shear stresses although the core material itself may be able to withstand very high shear stresses the bond or adhesive layer at the interface could be relativelyweaker resulting in interfacial delamination results from this study indicate that the interfacial shear stresses can be reduced by varying functionallygrading the core properties through the thickness this note presents the displacement and stress elds in a functionallygraded core for a one dimensionalsandwichplate and compares them to that in a uniformcore the objective is to demonstrate the signi cant reduction in shear stresses at the face sheet core interface in the sandwich panel achieved by functionally grading the core properties

exact solution for thermoelastic deformations of functionally graded thick rectangular plates an exact solution is obtained for three dimensional deformations of a simply supported functionally graded rectangular plate subjected to mechanical and thermal loads on its top and or bottom surfaces suitable temperature and displacement functions that identically satisfy boundary conditions at the edges are used to reduce the partial differential equations governing the thermomechanical deformations to a set of coupled ordinary differential equations in the thickness coordinate which are then solved by employing the power series method the exact solution is applicable to both thick and thin plates results are presented for two constituent metal ceramic functionally graded rectangular plates that have a power law through the thickness variation of the volume fractions of the constituents the effective material properties at a point are estimated by either the mori tanaka or the self consistentschemes exact displacementsand stressesatseveral locations for mechanical and thermal loads are used toassess theaccuracyof the classical plate theory thee rst ordershear deformation theory and athird order shear deformation theory for functionally graded plates results are alsocomputed for a functionally graded plate with material properties derived by the mori tanaka method the self consistent scheme and a combination of these two methods

wave propagation of functionally graded layers treated by recursion relations and effective boundary conditions wave propagation through a layer of a material that is inhomogeneous in the thickness direction typically a functionally graded material fgm is investigated the material parameters and the displacement components are expanded in power series in the thickness coordinate leading to recursion relations among the displacement expansion functions these can be used directly in a numerical scheme as a means to get good field representations when applying boundary conditions and this can be done even if the layer is not thin this gives a schema that is much more efficient than the approach of subdividing the layer into many sublayers with constant material properties for thin layers for which the material parameters do not depend on the layer thickness the recursion relations can be used to eliminate all but the lowest order expansion functions employing the boundary conditions this leads to a set of effective boundary conditions relating the displacements and stresses on the two sides of the layer thus completely replacing the layer by these effective boundary conditions numerical examples illustrate the convergence properties of the scheme for fg layers and the influence of different variations of the material parameters in the fg layer

the standard upwind compact difference schemes for incompressible flow simulations compact difference schemes have been used extensively for solving the incompressible navier stokes equations however the earlier formulations of the schemes are of central type called central compact schemes ccs which are dispersive and susceptible to numerical instability to enhance stability of ccs the optimal upwind compact schemes oucs are developed recently by adding high order dissipative terms to ccs in this paper it is found that oucs are essentially not of the upwind type because they do not use upwind biased but central type of stencils furthermore oucs are not the most optimal since orders of accuracy of oucs are at least one order lower than the maximum achievable orders new upwind compact schemes called standard upwind compact schemes sucs are developed in this paper in contrast to oucs sucs are constructed based completely on upwind biased stencils and hence can gain adequate numerical dissipation with no need for introducing optimization calculations furthermore sucs can achieve the maximum achievable orders of accuracy and hence be more compact than oucs more importantly sucs have prominent advantages on combining the stable and high resolution properties which are demonstrated from the global spectral analyses and typical numerical experiments