Analytical and numerical methods of symplectic system for Stokes flow in two-dimensional rectangular domain

In this paper, a new analytical method of symplectic system, Hamiltonian system, is introduced for solving the problem of the Stokes flow in a two-dimensional rectangular domain. In the system, the fundamental problem is reduced to an eigenvalue and eigensolution problem. The solution and boundary conditions can be expanded by eigensolutions using adjoint relationships of the symplectic ortho-normalization between the eigensolutions. A closed method of the symplectic eigensolution is presented based on completeness of the symplectic eigensolution space. The results show that fundamental flows can be described by zero eigenvalue eigensolutions, and local effects by nonzero eigenvalue eigensolutions. Numerical examples give various flows in a rectangular domain and show effectiveness of the method for solving a variety of problems. Meanwhile, the method can be used in solving other problems.     

Dynamic response and reliability analysis of random structures

The finite element method determining dynamic response of random structuressubjected to random dynamic load is studied,the formulas for dynamic response ofrandom structures and reliability based on random resistibility are proposed in thispaper.     

SEMI-ANALYTICAL AND SEMI-NUMERICAL METHOD FOR DYNAMIC ANALYSIS OF FOUNDATION

A semi-analytical and semi-numerical method is proposed for the dynamic analysis of foundations. The Lamb's solution and the approximate formulae were used to establish the relation of the contact force and deflection between the foundation and soil. Therefore, the foundation can be separated from soil and analyzed by FEM as for the static cases. The plate can be treated as that the known forces are acting on the upper surface, and the contact pressure from soil can be represented as the deflection. So that only the plate needs to be divided into elements in the analysis. By this method, a series of vibration problems, including various shapes and rigidities of foundations, different excitation frequencies, were analyzed. Furthermore, it can be used for the embedded foundation. The numerical examples show that this method has simplicity, highly accurate and versatile. It is an effective method for the dynamic analysis of foundations.     

DYNAMIC BEHAVIORS OF CONDUCTIVE CIRCULAR PLATE IN TIME-VARYING MAGNETIC FIELDS

In this study, we proposed an analytical solution for eddy currents as well as electromagnetic forces of a conductive circular plate in a time varying magnetic field. Specifically, an analytical series solution for eddy currents in a circular plate subjected to an axisymmetrie time varying magnetic field has been proposed based on the T-method that has been widely used in the eddy current analysis of conductive and superconductive structures. Accordingly, the dynamic response, the dynamic instability and the magnetic damping of a circular plate in a transverse transient magnetic field as well as a stationary in-plane magnetic field have also been obtained. The analytical series solution proposed in this work as well as the subsequent numerical analysis not only confirmed the emergence of dynamic instability of a circular plate in a strong transverse magnetic field, but also demonstrated the existence of magneto-damping of a circular conductive plate in an in-plane magnetic field. The method developed in this paper provides a potential new possible way by which the analysis of the electromagnetic coupling problems of conductive structures can be simplified.     

The Scheme to Determine the Convergence Term of the Galerkin Method for Dynamic Analysis of Sandwich Plates on Nonlinear Foundations

The vibration of a plate resting on elastic foundations under a moving load is of great significance in the design of many engineering fields,such as the vehicle-pavement system and the aircraft-runway system.Pavements or runways are always laminated structures.The Galerkin truncation method is widely used in the research of vibration.The number of truncation terms directly affects the convergence and accuracy of the response results.However,the selection of the number of truncation terms has not been clearly stated.A nonlinear viscoelastic foundation model under a moving load is established.Based on the natural frequency of linear undisturbed derivative systems,the truncation terms are used to determine the convergence of vibration response.The criterion for the convergence of the Galerkin truncation term is presented.The scheme is related to the natural frequency with high efficiency and practicability.Through the dynamic response of the sandwich beam under a moving load,the feasibility of the scheme is verified.The effects of different system parameters on the scheme and the truncation convergence of dynamic response are presented.The research in this paper can be used as a reference for the study of the vibration of elastic foundation plates.Especially,the model established and the truncation analysis method proposed are helpful for studying the vibration of vehicle-pavement system and related systems.     

A numerical treatment of the periodic solutions of non-linear vibration systems

Direct numerical integration can be used to find the periodicsolutions for the equations of motion of nonlinear vibrationsystems.The initial conditions are iterated so that theycoincide With the terminal conditions.The time interval ofthe integration(i.e.,the period)and certain parameters ofthe equations of motion can be included in the iterations.Theintegration method has a variable stoplength.This Sbooting method can produce periodic solutions witha shorter computex time.The only error occurs in the numeri-cal integration and it can therefore be estimated and madesmall enough.Using this method one can treat a variety ofvibration problems.such as free conservative.forced.para-meter-excited and self-sustained vibrations with one or se-veral degrees-of-freedom.Unstable solutions and those Whichare sensitive to parameter Changes can also be calculated.Thestability of the solutions is investigated based on the thecryof differential equations with periodic coefficients.The ex-trapolation method and the proc     

A DISCRETE NONLINEAR STEPPING OPTIMIZATION METHOD AND ITS APPLICATION

Most of the practical design variables should always be discrete quantity within engineering optimization design problems. To obtain the true optimization solution, a discrete optimization method must be used. In this paper, a new method called step optimization search method is presented to solve the discrete quantity mathematic programming problems. The basic idea oft his method is to find out an initialfeasible point and then to search the optimum point step by step in the neighbouring region of this point so as to obtain an improved new discrete point. Respectively, the new point can be taken us initial one, and the whole process can be carried out once more until the optimum solution of the problem is obtained. Some results of numerical examples of practical problems show that this new method can solve problems quickly and simply and can be applied in a lot of engineering design problems.     

Vertical vibrations of elastic foundation resting on saturated half-space

This paper is mainly concerned with the dynamic response of an elastic foun- dation of finite height bounded to the surface of a saturated half-space.The foundation is subjected to time-harmonic vertical loadings.First,the transform solutions for the governing equations of the saturated media are obtained.Then,based on the assumption that the contact between the foundation and the half-space is fully relaxed and the half- space is completely pervious or impervious,this dynamic mixed boundary-value problem can lead to dual integral equations,which can be further reduced to the Fredholm integral equations of the second kind and solved by numerical procedures.In the numerical exam- ples,the dynamic compliances,displacements and pore pressure are developed for a wide range of frequencies and material/geometrical properties of the saturated soil-foundation system.In most of the cases,the dynamic behavior of an elastic foundation resting on the saturated media significantly differs from that of a rigid disc on the saturated half-space. The solutions obtained can be used to study a variety of wave propagation problems and dynamic soil-structure interactions.     

General analytic solution of dynamic response of beams with nonhomogeneity and variable cross-section

In this paper, a new method, the step-reduction method, is proposed to investigate the dynamic response of the Bernoulli-Euler beams with arbitrary nonhomogeneity and arbitrary variable cross-section under arbitrary loads. Both free vibration and forced vibration of such beams are studied. The new method requires to discretize the space domain into a number of elements. Each element can be treated as a homogeneous one with uniform thickness. Therefore, the general analytical solution of homogeneous beams with uniform cross-section can be used in each element. Then, the general analytic solution of the whole beam in terms of initial parameters can be obtained by satisfying the physical and geometric continuity conditions at the adjacent elements. In the case of free vibration, the frequency equation in analytic form can be obtained, and in the case of forced vibration, a final solution in analytical form can also be obtained which is involved in solving a set of simultaneous algebraic equations with only     

DYNAMIC CHARACTERISTIC ANALYSIS OF FUZZY- STOCHASTIC TRUSS STRUCTURES BASED ON FUZZY FACTOR METHOD AND RANDOM FACTOR METHOD

A new fuzzy stochastic finite element method based on the fuzzy factor method and random factor method is given and the analysis of structural dynamic characteristic for fuzzy stochastic truss structures is presented. Considering the fuzzy randomness of the structural physical parameters and geometric dimensions simultaneously, the structural stiffness and mass matrices are constructed based on the fuzzy factor method and random factor method; from the Rayleigh's quotient of structural vibration, the structural fuzzy random dynamic characteristic is obtained by means of the interval arithmetic; the fuzzy numeric characteristics of dynamic characteristic are then derived by using the random variable's moment function method and algebra synthesis method. Two examples are used to illustrate the validity and rationality of the method given. The advantage of this method is that the effect of the fuzzy randomness of one of the structural parameters on the fuzzy randomness of the dynamic characteristic can be reflected expediently and objectively.     

DYNAMIC OPTIMIZATION FOR UNCERTAIN STRUCTURES USING INTERVAL METHOD

An interval optimization method for the dynamic response of structures with interval parameters is presented. The matrices of structures with interval parameters are given. Combining the interval extension with the perturbation, the method for interval dynamic response analysis is derived. The interval optimization problem is transformed into a corresponding deterministic one. Because the mean values and the uncertainties of the interval parameters can be elected design variables, more information of the optimization results can be obtained by the present method than that obtained by the deterministic one. The present method is implemented for a truss structure. The numerical results show that the method is effective.     

Local Petrov-Galerkin method for a thin plate

The meshless local Petrov-Galerkin (MLPG) method for solving the bendingproblem of the thin plate were presented and discussed. The method used the moving least-squares approximation to interpolate the solution variables, and employed a local symmetricweak form. The present method was a truly meshless one as it did not need a finite elementor boundary element mesh, either for purpose of interpolation of the solution, or for theintegration of the energy. All integrals could be easily evaluated over regularly shapeddomains ( in general, spheres in three-dimensional problems ) and their boundaries. Theessential boundary conditions were enforced by the penalty method. Several numericalexamples were presented to illustrate the implementation and performance of the presentmethod. The numerical examples presented show that high accuracy can be achieved forarbitrary grid geometries for clamped and simply-supported edge conditions. No postprocessing procedure is required to computer the strain and stress, since the originalsolution from the present method, using the moving least squares approximation, is already smooth enough.     

NONSTATIONARY PANDOM VIBRATION ANALYSIS OF LINEAR ELASTIC STRUCTURES WITH FINITE ELEMENT METHOD

At present,the finite element method is an efficient method for an-alyzing structural dynamic problems.When the physical quantitiessuch as displacements and stresses are resolved in the spectra andthe dynamic matrices are obtained in spectral resolving form,the re-lative equations cannot be solved by the vibration mode resolvingmethod as usual.For solving such problems,a general method is putforward in this paper.The excitations considered with respect tononstationary processes are as follows:P(t) ={P_i(t)}, P_i(t)=a_i(t)P_i~o(t),a_i(t) is a time function already known.We make Fourier transforma-tion for the discretized equations obtained by finite element method,and by utilizing the behaviour of orthcgonal increment of spectralquantities in random process,some formulas of relations about thespectra of excitation and response are derived.The cross power spec-tral denisty matrices of responses can be found by these formulas,then the structrual safety analysis can be made.When a_(i)(t)=1(i=1,2,…,n),the.m     

ANALYTICAL SOLUTIONS FOR RESPONSE OF COLLISION OF PARTICLE WITH CONICAL ROD CAUSED BY LONGITUDINAL VIBRATION

The objective is to present exact analytical solutions of longitudinal impact analysis for slender conical rods struck by a particle and a new method is proposed for conical rod-particle impact analysis, in which the superposition method is used and the response of the rod is presented. These analytical results are exact and can be used to validate the numerical methods or other analytical results. The numerical example shows that one of the advantages of the present method is that the analytical form is very simple. The result is that mass ratio and some variables describing the geometrical shape of rods such as taper, length and radius play an important role in impact dynamic system.     

Topology optimization of 3D structures with design-dependent loads

Topology optimization of continuum structures with design-dependent loads has long been a challenge. In this paper, the topology optimization of 3D structures subjected to design-dependent loads is investigated. A boundary search scheme is proposed for 3D problems, by means of which the load surface can be identified effectively and efficiently, and the difficulties arising in other approaches can be overcome. The load surfaces are made up of the boundaries of finite elements and the loads can be directly applied to corresponding element nodes, which leads to great convenience in the application of this method. Finally, the effectiveness and efficiency of the proposed method is validated by several numerical examples.     

HIERARCHICAL STOCHASTIC FINITE ELEMENT METHOD FOR STRUCTURAL ANALYSIS

In this paper, the hierarchical approach is adopted for series representation of the stochastic nodal displacement vector using the hierarchical basis vectors, while the Karhunen-Lòeve series expansion technique is employed to discretize the random field into a set of random variables. A set of hierarchical basis vectors are defined to approximate the stochastic response quantities. The stochastic variational principle instead of the projection scheme is adopted to develop a hierarchical stochastic finite element method (HSFEM) for stochastic structures under stochastic loads. Simplified expressions of coefficients of governing equations and the first two statistical moments of the response quantities in the schemes of the HSFEM are developed, so that the time consumed for computation can be greatly reduced. Investigation in this paper suggests that the HSFEM yields a series of stiffness equations with similar dimensionality as the perturbation stochastic finite element method (PSFEM). Two examples are presented for numerical study on the performance of the HSFEM in elastic structural problems with stochastic Young’s Modulus and external loads. Results show that the proposed method can achieve higher accuracy than the PSFEM for cases with large coefficients of variation, and yield results agreeing well with those obtained by the Monte Carlo simulation (MCS).     

STUDY ON PREDICTION METHODS FOR DYNAMIC SYSTEMS OF NONLINEAR CHAOTIC TIME SERIES＊

The prediction methods for nonlinear dynamic systems which are decided by chaotic time series are mainly studied as well as structures of nonlinear self-related chaotic models and their dimensions. By combining neural networks and wavelet theories, the structures of wavelet transform neural networks were studied and also a wavelet neural networks learning method was given. Based on wavelet networks, a new method for parameter identification was suggested, which can be used selectively to extract different scales of frequency and time in time series in order to realize prediction of tendencies or details of original time series. Through pre-treatment and comparison of results before and after the treatment, several useful conclusions are reached:High accurate identification can be guaranteed by applying wavelet networks to identify parameters of self-related chaotic models and more valid prediction of the chaotic time series including noise can be achieved accordingly.     

The Perturbation Parameter in the Problem of Large Deflection of Clamped Circular Plates

In the problems of large deflection of clamped circular plates under uniformly distributed loads,various perturbation parameters relating to load,deflection,slope of deflection,membrane force,etc.are studied.For a general perturbation parameter,the variational principle is used for the solution of such a problem.The applicable range of these perturbation parameters are studied in detail.In the case of uniformly loaded plate,perturbation parameter relating to central deflection seems to be the best among all others.The method of determination of perturbation solution by means of variational principle can be used to treat a variety of problems,including the large deflection problems under combine loads.     

STUDY ON PARAMETERS FOR TOPOLOGICAL VARIABLES FIELD INTERPOLATED BY MOVING LEAST SQUARE APPROXIMATION

This paper presents a new approach to the structural topology optimization of continuum structures. Material-point independent variables are presented to illustrate the existence condition,or inexistence of the material points and their vicinity instead of elements or nodes in popular topology optimization methods. Topological variables field is constructed by moving least square approximation which is used as a shape function in the meshless method. Combined with finite element analyses,not only checkerboard patterns and mesh-dependence phenomena are overcome by this continuous and smooth topological variables field,but also the locations and numbers of topological variables can be arbitrary. Parameters including the number of quadrature points,scaling parameter,weight function and so on upon optimum topological configurations are discussed. Two classic topology optimization problems are solved successfully by the proposed method. The method is found robust and no numerical instabilities are found with proper parameters.     

A SIMPLIFIED CALCULATING METHOD OF NONLINEAR FREQUENCY OF CABLE NET UNDER MEAN WIND LOAD

The cable net supported glass curtain wallas the most advanced technique in dot point supported glass curtain wall, is widely used in China. Because of its large deflection and high nonlinearity under wind load, the dynamic performance of the cable net is greatly different from that of the conventional linear structures. The continuous membrane theory is used to construct the nonlinear vibration differential equation of the cable net, and the harmonic balance method is used to solve the analytic formula of the nonlinear frequency. In order to verify the accuracy of the above analytic formula, the results of the formula and the nonlinear FEM time-history method are compared and found to be in good agreement. Furthermore, the nonlinear vibration differential equation and the nonlinear frequency obtained in this paper are the basis for the wind-induced response analysis of a cable net under fluctuating wind load.