Stochastic Spline Fictitious Boundary Element Method For Elastodynamic Problems With KL Expansion Of Random Fields

Dynamic analysis of a structure is one of the major issues in structural analysis.In conventional dynamic analysis,it is generally based on deterministic structural models,in which the effects of the inherent randomness in structural systems have been completely neglected.However,it has been found that the structural uncertainties have a certain influence on the dynamic responses of mechanical systems.Therefore,it is more appropriate to take into account these uncertainties at the modeling level in structural dynamic analysis.In the field of stochastic computational mechanics,numerical schemes in the framework of finite element method(FEM)have been well developed.However,the accuracy and efficiency of such schemes are inevitably affected by the inherent defects of FEM.In this study,the stochastic spline fictitious boundary element method(SFBEM)is proposed for random analysis of elastodynamic problems with both structural and loading uncertainties,which are modeled as random fields and random processes,respectively,and are represented by Karhunen-Loeve(KL)expansion.The work in this dissertation is described as follows:(1)The basic concepts and main methods of stochastic analysis are briefly stated,and the statistical properties and discretization methods for random fields are introduced with more attention paid to the KL expansion in conjunction with the Galerkin projection for representation of random fields.The development and application of stochastic FEM and stochastic BEM are summarized systematically in this paper,and their advantages and disadvantages are also pointed out.(2)The formulation of deterministic SFBEM for elasticity problems is presented,and the numerical stability and error estimates are discussed.Numerical examples are used to illustrate the choice principle of computional parameters in SFBEM.(3)The stochastic SFBEM is proposed for modal analysis of plane elastic problems with structural parameters modeled as random fields.Two sets of governing differential equations with respect to the means and deviations of displacement modes are derived by including the first order terms of deviations.These equations are in similar forms to those of deterministic plane elastostatic problems,and can be solved using deterministic elastostatic fundamental solutions,resulting in the means and covariances of the eigenvalues and mode shapes.(4)The stochastic SFBEM is proposed for random vibration analysis of plane elastic problems with both structural and loading uncertainties modeled as random fields and random processes,respectively.Two sets of governing differential equations with respect to the means and deviations of dynamic responses are derived by including the first order terms of deviations.As these equations are in similar forms to those of deterministic plane elastostatic problems,they can be solved by the traditional procedure of SFBEM with deterministic fundamental solutions,leading to the mean and covariance solutions to dynamic responses.(5)For the effective treatment of the domain integrals involved in the deviation solution,the random fields considered are represented by KL expansion,leading to lesser random variables for description of the statistical properties of the random fields.Closed-form derivatives of the random fields can also be derived analytically in terms of the given basis functions used in KL expansion in conjunction with the Galerkin projection,which leads to higher accuracy for the covariance solution of the problem.(6)Numerical examples are given to show the accuracy and efficiency of the proposed method,and the effects on the analytic results of various factors,including the correlation types,correlation lengths,and the coefficients of variation,are investigated and certain conclusions are obtained regarding the influence of the above factors.In addition,to investigate the feasibility and effectiveness of the present method for problems with complex domains,numerical examples about the perforated plate with semi-circular notches have been analyzed.The main innovative work in this dissertation is that the stochastic SFBEM based on KL expansion technique is proposed for elastodynamic problems with both structural and loading uncertainties modeled as random fields and random processes,respectively.Numerical examples show that the proposed method is of good accuracy and high efficiency.