Investigation On Efficient Numerical Methods For Forward/Inverse And Interval Bi-modular Problems

A number of materials in engineering can be classified into the catalog of bi-modular mate-rial that exhibits different elastic properties in tension and compression,and have been applied in various practical aspects.The solution of the bi-modular problem is mainly challenged by the nonlinearity of the constitutive relationship,analytical solutions are usually difficult to acquire,and efficient numerical methods are in a great demand.However,some aspects concerned with the nonlinear finite element(FE)methods for the deterministic forward bi-modular problems need to be further stressed,particularly in the as-pect of deeper investigations on the FE stress-strain matrix and its impact on the nonlinear FE computation.Meanwhile,the investigations on the inverse and uncertain bi-modular problems are considerably inadequate.The non-differentiable stress-strain relationship makes it difficult to calculate the displacement derivatives which are required for the gradient based numerical methods to solve the inverse bi-modular problem of identifying constitutive parameters.The interval FE analysis for the interval bi-modular problems is challenged by the expensive com-putational cost when the intervals of parameters are large.With regard to the above issues,this dissertation focuses on the following points.1.The stress-strain matrix of bi-modular problems is addressed by emphasizing the coaxial requirement between the principal strain and stress,and is proved to be singular in the coordi-nate transformation without regard to this requirement.A full-scale FE stress-strain matrix is proposed by complementing shear moduli identical with the coaxial requirement,and is em-ployed in the nonlinear FE analysis for 2-D and 3-D bi-modular problems.The difficulty of convergence caused by the singularity in FE analysis is effectively overcome.2.A gradient based algorithms is presented to solve 2-D/3-D bi-modular problems.By virtue of aggregate function,a smoothing function is proposed to smooth the bilinear bi-modular constitutive equations,and the FE stiffness matrix becomes differentiable.Furthermore,a Newton-Raphson method based numerical algorithm for the bi-modular problem is proposed,providing effective means to conduct the sensitivity analysis and sensitivity analysis solutions of non-differentiable bi-modular problems.3.The gradient based algorithms are emphasized in solving inverse bi-modular problems.By utilizing the gradient based algorithm for the forward problem and gradient based optimiza-tion algorithms,a two-level sensitivity analysis based numerical method is put forward to solve 2-D/3-D inverse bi-modular problems of identifying constitutive parameters,and the efficiency in the whole process of identification for the bi-modular constitutive parameters can be effec-tively improved.4.The full-scale interval analysis of interval bi-modular problems are stressed by the sensitivity analysis,Taylor series expansion,interval arithmetic,and optimization techniques.Based on the improved bi-modular FE model,the first/second order derivatives of displacement with respect to constitutive parameters are acquired via a strain/stress status dependent non?linear analysis,by which the interval arithmetic based algorithms are developed for the interval analysis when the interval scales of uncertain parameters are relatively small.When the interval scales are large,an optimization-based algorithm is presented by using the first order derivatives and a global searching technique to provide a rigorous bounds estimation,In addition,the further use of the second order Taylor series approximation results in two algorithms,which can reduce the computational expense in the process of optimization-based bounds estimation.5.The surrogates of FE solutions of deterministic problems are emphasized to reduce the expensive computational cost on interval FE analysis,and an orthogonal polynomial expansion based numerical method is proposed for the interval FE analysis when the interval scales of un-certain parameters are large.In order to alleviate the heavy computational burden of repeated FE analysis in the optimization-based bounds estimation,a Galerkin method based orthogonal polynomials expansion is employed to approximate the deterministic FE solution with parame-ters varying in intervals,and an iterative algorithm is presented to tackle with the non-linearity in the surrogate construction of bi-modular problem.An optimization technique is used to en-sure the accuracy of bounds estimation when the interval scales of uncertain parameters are large.The proposed method is used to solve interval bi-modular and interval/fuzzy convection-diffusion heat transfer problems with multiple interval/fuzzy variables.The effectiveness of the proposed methods is verified via a series of numerical examples,and the impacts of various materials and computing parameters on the computing accuracy and efficiency are discussed.This dissertation provides new effective numerical methods for solving the forward/inverse and interval bi-modular problems.These methods are hopefully to be applied to solve practical engineering problems via a further improvement.