Investigation On SBFEM Based Numerical Methods For Deterministic/Uncertain Viscoelastic Problems

A number of natural and artificial materials exhibit viscoelastic property,and the viscoelastic analysis relates to various pratical engineering aspects.Due to the time-dependent constitutive relationship and complex boundary shapes/conditions,the analytical solutions of viscoelastic problems are usually difficult to obtain,thus effective numerical methods are substantially demaned.Due to the advantages of the scaled boundary finite element method(SBFEM)for molelling spatial problems,it is combimed with a temporally piecewise adaptive algorithm(TPAA)to solve the forward/inverse deterministic/uncertain viscoelastic problems in this thesis,such that the accuracy and convenience of spatial numerical analysis can be improved,particularly for the problems with stress singularity and unbounded domain,and the computational accuracy in the time domain can be guaranted using a TPAA.Although SBFEM has been successfully applied in the elastic static/dynamic analysis,heat transfer analysis,electrostatic field analysis,etc.there are very few application on viscoelastic problems,especially for the inverse and the uncertain viscoelastic problems.A disadvantage SBFEM is of lower computing efficiency because an eigenvalue problem with double DOF needs to solve in establishing the SBFEM stiffness matrix,and aggravates the computational burden in the step by step/recusive process of temporal viscoelastic analysis,particularly for the inverse and uncertain viscoelastic analysis which usually need considerable amount of repeated deterministic solutions.In addition,nearly no concern is given to SBFEM modelling with the Third Kind of Boundary Conditions(TKBC)which often relats to some interaction problems in practical engineering.With the above considerations,this thesis focuses on(1)Reducting computational cost relevant to SBFEM.(2)Modelling inverse and uncertain viscoelastic problems under the framework of SBFEM based sensitivity analysis(3)Modelling the Third Kind of Boundary Conditions using SBFEM.The major contributions of this thesis include:(1)A SBFE-TPAA based partitioning algorithm is presented for 2-D viscoelastic structure with cyclic symmetry.Both the eigenvalue and system equations are partitioned into a number of smaller independent problems,so that the solution scale is reduced and the computing efficiency is improved.(2)Three SBFEM based numerical algorithms are developed to deal with TKBC problems.For the linear TKBC,a SBFE equation with a additional stiffness matrix is derived.When the linear TKBC is cyclicly symmetric,the additional stiffness matrix is proved to be block-circulant,and a partitioning algorithm is developed to reduce the computational cost.For the non-smoothed bilinear TKBC,a SBFEM and sensitivity analyisis based numerical algorithm is developed using a smoothing technique.For the time-dependent TKBC,a SBFEM-TPAA based algorithm is presented.(3)A SBFEM-TPAA based recursive self-adaptive algorithm is proposed for computing derivatives which is required for the sensitivity analysis based inverse and uncertain viscoelastic analysis.On the basis of SBFEM-TPAA,a sensitivity analysis based two-step strategy is presented to deal with the deterministic multi-variables inverse viscoelastic problems with regional inhomogeneity.(4)The SBFEM-TPAA and sensitivity analysis based numerical models are presented for the forward/inverse viscoelastic problems with interval uncertainties.The interval relations between displacements/stresses and constitutive parameters are established,with which the upper and lower bounds of displacements are determined via Taylor serie expansion and interval arithmetic.In the inverse analysis,a two-step strategy is developed.When the experimentally determined information of displacements is of interval uncertainty,the viscoelastic constitutive parameters can be identified in terms of center values and radii of their intervals.(5)SBFEM-TPAA and sensitivity analysis based numerical algorithms are presented for solving viscoelastic problems with probabilistic uncertainties.When constitutive parameters are random variables,a Mean-Value First Order Second Moment and sensitivity analysis based algorithm is proposed to estimate the mean values and standard deviations of displacements,and a two-step strategy is developed for the inverse problem of identifying the mean values and standard deviations of constitutive parameters.When constitutive parameters are random fields,by utilizing SBFEM-TPAA and K-L expansion,a perturbation and sensitivity analysis based algorithm is presented to estimate mean values and standard deviations of displacements.Numerical examples are given to illustrate the effectiveness of the proposed algorithms.An extension of SBFEM to viscoelastic analysis is realized in this thesis,which provides new effective numerical methods for solving deterministic/uncertain forward/inverse viscoelastic problems,particularly in terms of the accuracy and convenience of spatial numerical analysis.The proposed approches are hopefully applied to deal with practicl engineering problem via the further improvement and development.