on supremum of a set in a multi dimensional space on donne une definition exacte du supremum d un ensemble dans l espace euclidien a n dimensions etendu et on montre qu elle est presque equivalente a celle proposee par kawasaki 1982 on etudie certaines proprietes de ce supremum ce qui pourra etre utile pour developper une theorie de dualite conjuguee en optimisation vectorielle

study on gas kinetic unified algorithm for flows from rarefied transition to continuum the modified bgk equation adapted to various flow regimes can be presented by the aid of the basic characteristics on molecular movement and collision approaching to equilibrium the discrete velocity ordinate method is developed and applied to the velocity distribution function to remove its continuous dependency on the velocity space and then the velocity distribution function equation is cast into hyperbolic conservation law form with nonlinear source terms based on the unsteady time splitting method and the non oscillatory containing no free parameters and dissipative nnd scheme the gas kinetic finite difference second order scheme is constructed for the computation of the discrete velocity distribution functions the mathematical model on the interaction of molecules with solid surface is studied and used in the numerical method four types of numerical quadrature rules such as the modified gauss hermite formula the composite newton cotes integration method the gauss legendre numerical quadrature rule and the golden section number theoretic integral method are developed and applied to the discretized velocity space to evaluate the macroscopic flow parameters at each point in the physical space as a result a unified simplified gas kinetic algorithm is established for the flows from rarefied transition to continuum regime based on analyzing the inner parallel degree of the unified algorithm the parallel strategy adapted to the gas kinetic numerical algorithm is studied and then the hpf parallel processing software for the unified algorithm is developed to test the present method the one dimensional shock tube problems the flows past two dimensional circular cylinder and the flows around three dimensional sphere and spacecraft shape with various knudsen numbers are simulated the computational results are found in high resolution of the flow fields and good agreement with the theoretical dsmc n s and experimental results

saddle points and gap functions for vector equilibrium problems via conjugate duality in vector optimization in this paper two conjugate dual problems based on weak efficiency to a constrained vector optimization problem are introduced some inclusion relations between the dual objective mappings and the properties of the lagrangian maps and their saddle points for primal problem are discussed gap functions for a vector equilibrium problem are established by using the weak and strong duality

the transient responses of a special non homogeneous magneto electro elastic hollow cylinder for axisymmetric plane strain problem by virtue of the introduction of new dependent variable and the separation of variables technique the transient responses of a special non homogeneous magneto electro elastic hollow cylinder are transformed to two volterra integral equations of the second kind of about two functions with respect to time these integral equations can be solved successfully by means of the interpolation method then the complete solutions of displacements stresses electric potential electric displacements magnetic potential and magnetic inductions are obtained the present method is suitable for a magneto electro elastic hollow cylinder with an arbitrary thickness subjected to arbitrary axisymmetric mechanical and electromagnetic loads numerical results are finally presented

dynamic asymmetry of piezoelectric shell structures abstract the non symmetry and asymmetric dynamic characteristics of piezoelectric shell structures are studied analytically and numerically the basic formulations of piezoelectric structures are firstly given in the tensor form the differential equations for displacements and electric potential are derived from the formulations which validate that the description in curvilinear coordinates with the electrical and mechanical coupling induces the non symmetry in generalized stiffness and then the dynamic asymmetry of piezoelectric structures also expanding the structural displacements and electric potential with various spatial characteristics can induce the non symmetry in generalized stiffness matrix of the converted ordinary differential equations then for the multi degree of freedom system with asymmetric stiffness matrix the conventional right modes or eigenvectors are proved to have not the auto orthogonality relations with the mass or stiffness weight however the left and right eigenvectors have the cross orthogonality relations with the mass or stiffness weight which can be used for uncoupling the asymmetric systems furthermore the non symmetric stiffness matrix is divided into the corresponding symmetric and anti symmetric stiffness matrices the algebraic equations for eigenvalues and singular values of the asymmetric symmetric and anti symmetric systems are given it is obtained that the eigenvalues are equal to the corresponding singular values for the symmetric system the singular values are equal to the products of the unit imaginary number and corresponding eigenvalues for the anti symmetric system and the differences of the eigenvalues and corresponding singular values for the asymmetric system depend on the anti symmetric stiffness also the upper limits of the absolute and relative differences of the singular values of the asymmetric system to the corresponding eigenvalues of the symmetric system and the upper and lower limits of the singular values of the asymmetric system are obtained respectively finally the spherically symmetric piezoelectric shell described in spherical coordinates is studied in detail to show the asymmetric dynamic characteristics the differential equations for the radial displacement and electric potential of the shell structure are obtained to illustrate the non symmetric generalized stiffness involving elastic and piezoelectric constants eliminating the electric potential expanding the displacement in space and using the galerkin method yield the ordinary differential equations which represent a multi degree of freedom dynamic system with the asymmetric generalized stiffness matrix numerical results are given to illustrate the eigenvalues and modes of the asymmetric system different from those of the corresponding symmetric system the relative differences of the eigenvalues of the asymmetric to symmetric system for different piezoelectric constants and geometric parameters and the non orthogonality of the left or right eigenvectors for the asymmetric system the analytical and numerical results on the non symmetric dynamics of piezoelectric shell structures are useful for accurate analysis and design

efficient kinetic schemes for steady and unsteady flow simulations on unstructured meshes this paper presents efficient second order kinetic schemes on unstructured meshes for both compressible unsteady and incompressible steady flows for compressible unsteady flows a time dependent gas distribution function with a discontinuous particle velocity space at a cell interface is constructed and used for the evaluations of both numerical fluxes and conservative flow variables as a result a compact scheme on the unstructured meshes is developed for incompressible steady flows a continuous second order gas kinetic bgk type scheme is presented for which the time dependent gas distribution function with a continuous particle velocity is used on unstructured meshes the efficiency of the schemes lies in the fact that the slopes of the flow variables inside each cell can be constructed using values of the flow variables within that cell only without involving neighboring cells therefore even with the stencil of a first order scheme a high resolution method is constructed numerical examples are presented which are compared with the benchmark solutions and the experimental measurements

conjugate duality in set valued vector optimization abstract in this paper conjugate duality results for convexlike set valued vector optimization problems are presented under closedness or boundedness hypotheses some properties of the value mapping of a set valued vector optimization problem are studied a conjugate duality result is also proved for a convex set valued vector optimization problem without the requirements of closedness and boundedness

stochastic response analysis of piezoelectric axisymmetric hollow cylinders the stochastic response of a piezoelectric thick axisymmetric hollow cylinder in plane strain under boundary stochastic excitations is analyzed and calculated the stochastic stress and electric potential boundary conditions of the piezoelectric hollow cylinder are converted into homogeneous boundary conditions by transformations that yields the electrical and mechanical coupling partial differential equations of motion with damping and stochastic excitations the equation for electric potential is integrated radially to obtain the electric potential as a function of displacement and the displacement is expanded as a series in terms of the legendre polynomials the partial differential equation for displacement is further converted into ordinary differential equations by using the galerkin method which represent a stochastic multi degree of freedom system with asymmetric stiffness matrix due to the asymmetric electrical and mechanical coupling and the transformed boundary conditions the frequency response function matrix and correlation function matrix of the system response are derived from these equations based on the theory of random vibration the expressions of mean square displacement and electric potential of the piezoelectric hollow cylinder are finally obtained and illustrated by numerical results for non white stochastic excitations the frequency response characteristics and electrical and mechanical coupling properties are explored

numerical hydrodynamics from gas kinetic theory a new high resolution numerical hydrodynamic scheme is developed from considerations of gas kinetic theory this boltzmann type scheme uses the particle distribution function and follows its evolution to evaluate the numerical fluxes the intrinsic multidimensional property in the evolution stage of gas distribution function distinguishes it from any other high order upwind scheme from approximate boltzmann equation the navier stokes equations can be derived and this new numerical method uses this advantage to approximate the solutions of the navier stokes equations where the advection and dissipation effects are coupled in the time dependent gas distribution function several well defined numerical test cases are presented in one and two dimensions and it is shown that the resolution in the numerical results is competitive with those of any of the current high resolution schemes without special treatment of the discontinuities for strong rarefaction waves our scheme gives better results in this thesis we try to separate clearly numerical and physical effects and we also give a concise explanation of the necessity of including numerical viscosities implicitly or explicitly in high resolution godunov type and central difference schemes even for the navier stokes equations the main part of this thesis will be published in two papers 27 and 45 in the journal of computational physics

the transient responses of piezoelectric hollow cylinders for axisymmetric plane strain problems by virtue of the separation of variables technique the axisymmetric plane strain electroelastic dynamic problem of hollow cylinder is transferred to an integral equation about a function with respect to time which can be solved successfully by means of the interpolation method then the solution of the displacements stresses electric displacements and electric potentials are finally obtained the present method is suitable for the hollow cylinder with arbitrary thickness subjected to arbitrary mechanical and electrical loads numerical results are also presented

wave propagation in a piezoelectric rod of 6mm symmetry abstract the wave propagation in a piezoelectric rod of 6 mm symmetry is investigated by applying a 3 d piezoelectric elastic model a self adjoint method is introduced to solve this problem this method avoids calculating the generalized eigenvalue equation it completely draws the dispersion curves in the forms of quasi p wave quasi sv wave and quasi sh wave under the self adjoint boundary condition and it can evaluate the dispersion curves of all kinds of boundary conditions as an example the dispersion curves of plt 5h are completely drawn we also found the quasi sv wave has standing wave phenomenon in the plt 5h rod in addition the relation of dispersion curves among different boundary conditions is discussed and an experiment method is introduce to decide the dispersion curves for another boundary conditions

on godunov type methods near low densities abstract when the energy of a flow is largely kinetic many conservative differencing schemes may fail by predicting non physical states with negative density or internal energy we describe as positively conservative the subclass of schemes that by contrast always generate physical solutions from physical data and show that the godunov method is positively conservative it is also shown that no scheme whose interface flux derives from a linearised riemann solution can be positively conservative classes of data that will bring about the failure of such schemes are described however the harten lax van leer hlle scheme is positively conservative under certain conditions on the numerical wavespeeds and this observation allows the linearised schemes to be rescued by modifying the wavespeeds employed

free vibrations of piezoelectric cylindrical shells filled with compressible fluid abstract three displacement functions are introduced to represent each mechanical displacement according to the three dimensional theory after expanding these functions and the electric potential with orthogonal series the free vibration equation of piezoelectric cylindrical shells satisfying ss3 edge conditions can be obtained the equation was solved by utilizing bessel functions with complex arguments the effects of compressible fluid on shells are considered and some new phenomena which are exclusive for piezoelectric cylindrical shells are reported results of empty infinite piezoelectric cylindrical shells are compared to those presented in relative references some lowest frequencies that were missed by paul and venkatesan paul h s and venkatesan m 1987 vibrations of hollow circular cylinder of piezoelectric ceramics journal of the acoustic society of america82 852 856 were discovered

numerical navier stokes solutions from gas kinetic theory abstract in order to test the gas kinetic based hydrodynamic scheme given in paper i k h prendergast and k xu j comput phys 109 53 1993 as a navier stokes solver we extend the scheme to two dimensions and exhibit some navier stokes solutions the scheme is a high resolution gas kinetic scheme in both space and time the advective and diffusive fluxes are coupled and solved at the same time by following the time dependent velocity distribution function for particles numerical results for some well defined euler and navier stokes test cases are presented the kolmogorov and the laminar boundary layer problems exhibit the dominance of the real viscosity effects the strong shock interaction test cases show the applicability of the scheme to supersonic gas flow

relative perturbation techniques for singular value problems a technique is presented for deriving bounds on the relative change in the singular values of a real matrix or the eigenvalues of a real symmetric matrix due to a perturbation as well as bounds on the angles between the unperturbed and perturbed singular vectors or eigenvectors the class of perturbations considered consists of all delta b for which b delta b d l bd r for some nonsingular matrices d l and d r this class includes componentwise relative perturbations of a bidiagonal or biacyclic matrix and perturbations that annihilate the off diagonal block in a block triangular matrix many existing relative perturbation and deflation bounds are derived from results for this general class of perturbations also some new relative perturbation and deflation results for the singular values and vectors of biacyclic triangular and shifted triangular matrices are presented

gas kinetic finite volume methods flux vector splitting and artificial diffusion abstract in order to improve our understanding of boltzmann type schemes which have been recently proposed for the euler and navier stokes equations by prendergast and xu we have modified the gas kinetic approach and applied it to simple hyperbolic conservation laws we found that the gas kinetic discretization can be explained in terms of standard central difference and upwind schemes artificial viscosity concepts are reviewed and linked to the grid size and the physical length scale of the discontinuity also a new three dimensional gas kinetic scheme for the numerical navier stokes equations whose solution satisfies the entropy condition is presented two numerical limits of the scheme are obtained the first one is the one step lax wendroff scheme and the second one is the kinetic flux vector splitting scheme a new relaxation scheme for steady state calculations is also formulated and implemented in the multigrid time stepping technique of jameson when applied to the euler equations the resulting method yields high accuracy and fast convergence to a steady state

conjugate duality in vector optimization in this paper we develop a conjugate duality theory in vector optimiza tion conjugate duality was fully developed in scalar optimization by rockafellar 3 and provides a unified framework for the duality theory the author and sawaragi extended it to the case of multiobjective optimization by introducing some new concepts such as conjugate maps and subgradients for vector valued set valued maps 4 6 their results are based on the efficiency pareto maximality kawasaki refined their results and obtained a reciprocal duality by introducing the concept of supremum set in an extended euclidean space 1 2 his supremum is based on the weak efficiency weak pareto maximality of the closure of a set in an extended sense the author s newly defined the supremum of a set in the extended multi dimensional euclidean space on the basis of weak ehiciency through several discussions on supremum the definition satisfies some desirable fundamental properties it can be extended to the case of general partially ordered linear topological spaces in a straightforward manner based on this definition of supremum some useful concepts such as conjugate maps and subgradients are introduced for vector valued set valued maps in this paper these concepts enable us to develop the conjugate duality in vector optimization although the results obtained are quite similar to the earlier works by the author and sawaragi 4 or kawasaki 2 our new approach makes the proofs much easier and more understandable and provides a better version of conjugate duality

gas kinetic numerical studies of three dimensional complex flows on spacecraft re entry the gas kinetic numerical algorithm solving the boltzmann model equation is extended and developed to study the three dimensional hypersonic flows of spacecraft re entry into the atmosphere in perfect gas in this study the simplified velocity distribution function equation for various flow regimes is presented on the basis of the kinetic boltzmann shakhov model the discrete velocity ordinate technique and numerical quadrature methods such as the gauss quadrature formulas with the weight function 2 p 1 2exp v 2 and the gauss legendre numerical quadrature rule are studied to resolve the barrier in simulating complex flows from low mach numbers to hypersonic problems specially the gas kinetic finite difference scheme is constructed for the computation of three dimensional flow problems which directly captures the time evolution of the molecular velocity distribution function the gas kinetic boundary conditions and numerical procedures are studied and implemented by directly acting on the velocity distribution function the hpf high performance fortran parallel implementation technique for the gas kinetic numerical method is developed and applied to study the hypersonic flows around three dimensional complex bodies the main purpose of the current research is to provide a way to extend the gas kinetic numerical algorithm to the flow computation of three dimensional complex hypersonic problems with high mach numbers to verify the current method and simulate gas transport phenomena covering various flow regimes the three dimensional hypersonic flows around sphere and spacecraft shape with different knudsen numbers and mach numbers are studied by hpf parallel computing excellent results have been obtained for all examples computed

implicit gas kinetic unified algorithm based on multi block docking grid for multi body reentry flows covering all flow regimes based on the previous researches of the gas kinetic unified algorithm gkua for flows from highly rarefied free molecule transition to continuum a new implicit scheme of cell centered finite volume method is presented for directly solving the unified boltzmann model equation covering various flow regimes in view of the difficulty in generating the single block grid system with high quality for complex irregular bodies a multi block docking grid generation method is designed on the basis of data transmission between blocks and the data structure is constructed for processing arbitrary connection relations between blocks with high efficiency and reliability as a result the gas kinetic unified algorithm with the implicit scheme and multi block docking grid has been firstly established and used to solve the reentry flow problems around the multi bodies covering all flow regimes with the whole range of knudsen numbers from 10 to 3 7e6 the implicit and explicit schemes are applied to computing and analyzing the supersonic flows in near continuum and continuum regimes around a circular cylinder with careful comparison each other it is shown that the present algorithm and modelling possess much higher computational efficiency and faster converging properties the flow problems including two and three side by side cylinders are simulated from highly rarefied to near continuum flow regimes and the present computed results are found in good agreement with the related dsmc simulation and theoretical analysis solutions which verify the good accuracy and reliability of the present method it is observed that the spacing of the multi body is smaller the cylindrical throat obstruction is greater with the flow field of single body asymmetrical more obviously and the normal force coefficient bigger while in the near continuum transitional flow regime of near space flying surroundings the spacing of the multi body increases to six times of the diameter of the single body the interference effects of the multi bodies tend to be negligible the computing practice has confirmed that it is feasible for the present method to compute the aerodynamics and reveal flow mechanism around complex multi body vehicles covering all flow regimes from the gas kinetic point of view of solving the unified boltzmann model velocity distribution function equation

adaptive finite element methods in computational mechanics we present a general approach to adaptivity for finite element methods and give applications to linear elasticity non linear elasto plasticity and nonlinear conservation laws including numerical results

unified solver for rarefied and continuum flows with adaptive mesh and algorithm refinement this paper describes a unified flow solver ufs for rarefied and continuum gas flows the ufs separates the rarefied and continuum flow domains and selects appropriate solvers to combine the efficiency of continuum models with the accuracy of kinetic models the direct numerical solution of the boltzmann transport equation is used in rarefied regions while kinetic schemes of continuum fluid dynamics are used elsewhere using similar computational techniques for the kinetic and continuum solvers and employing intelligent domain decomposition algorithms attain the efficiency and numerical stability of the ufs solutions of test problems are presented to illustrate the capabilities of the ufs for high and low speed flows it is shown that the ufs can dynamically adapt the computational mesh and automatically introduce and remove kinetic patches to provide significant savings by limiting molecular scale solutions only to the regions where they are needed

a robust and accurate led bgk solver on unstructured adaptive meshes starting from the bgk model of the boltzmann equation we develop a robust and accurate finite volume gas kinetic scheme on unstructured triangular meshes the proposed numerical approach is composed of two steps an initial reconstruction step and a gas evolution step in the initial reconstruction step an unstructured version of the local extremum diminishing interpolation is applied to the conservative variables and to compute left and right states along a node edge in the gas evolution step the local integral solution of the bgk model is used to compute numerical fluxes at a cell interface this approach provides an alternative to riemann solvers and yields numerical schemes which possess many desirable properties that may not be found in godunov type schemes a classich refinement adaptive procedure is implemented to increase the spatial resolution of high speed unsteady flow characteristics such as shock waves contact discontinuities or expansion waves with minimal computational costs and memory overheads this procedure involves mesh enrichment coarsening steps to either insert nodes on an edge center in high gradient regions or delete nodes in over resolved regions numerical results of several test cases for unsteady compressible inviscid flows are presented in order to verify the accuracy and robustness of the current numerical approach the computed results are compared with analytical solutions experimental data the results of structured mesh calculations and the results obtained by widely used flux splitting methods

a robust low diffusive kinetic scheme for the navier stokes euler equations a new kinetic scheme based on the equilibrium flux method efm and modified using osher intermediate states is proposed this new scheme called efmo combines the robustness of the equilibrium flux method and the accuracy of flux difference splitting schemes the original efm scheme is expressed in terms of simple wave decomposition in which only the linearly degenerate subpath is calculated from osher numerical flux while nonlinear waves are still evaluated from the regular efm splitting owing to its capability of withstanding intense nonlinear waves and yet exactly resolving contact discontinuities efmo is particularly well suited for the resolution of the navier stokes equations as demonstrated by a series of severe test cases including the high speed viscous flow around a cone a shock boundary layer interaction problem a vacuum apparition problem the hypersonic flow around a circular cylinder at mach 100 and the forward facing step at mach 3

rkdg finite element method combined with bgk scheme for solving fluid dynamics system this paper presents a new type of second order scheme for solving the system of euler equations which combines the runge kutta discontinuous galerkin rkdg finite element method and the bhatnagar gross krook bgk scheme we first discretize the euler equations in space with the dg method and then the resulting system will be discretized using an rk method finally a second order bgk method is used to construct the numerical flux the proposed scheme keeps the main advantages of the dg finite element method including its flexibility in handling irregular solution domains and in parallelization moreover with the bgk approach the proposed schemes avoid using riemann solvers the efficiency and effectiveness of the proposed method are illustrated by several numerical examples in one and two dimensional spaces

elastic waves in piezoceramic cylinders of sector cross section abstract the propagation of elastic waves in piezoceramic cylindrical waveguides of circular cross sections with sector cut is investigated on the basis of the linear theory of electroelasticity dispersion functions are obtained from boundary conditions in an analytical form of functional determinants for each value of the generalized wave number a selected set of numerical results including real imaginary and complex branches of full dispersion spectrums with various symmetry of wave movements is presented to describe the essential characteristics of the waves leading effects of spectrums transformation by change of waveguide s angular measure are enlightened and wave asymptotic behavior is analyzed the variation of the cross section is considered as a mechanism to control the dispersion characteristics of waveguides

conjugate duality in vector optimization and some applications to the vector variational inequality the aim of this paper is to extend the so called perturbation approach in order to deal with conjugate duality for constrained vector optimization problems to this end we use two conjugacy notions introduced in the past in the literature in the framework of set valued optimization as a particular case we consider a vector variational inequality which we rewrite in the form of a vector optimization problem the conjugate vector duals introduced in the first part allow us to introduce new gap functions for the vector variational inequality the properties in the definition of the gap functions are verified by using the weak and strong duality theorems