Uncertain Analysis in Finite Elements Models

Considering the influence of random factors on the structure, three stochastic finite element methods for general nonlinear problems are proposed. They are Taylor expansion method, perturbation method and Neumann expansion method. The mean value of displacement is obtained by the tangent stiffness method or the initial stress method of nonlinear finite elements. Nonlinear stochastic finite element is transformed into linear stochastic finite element. The mean values of displacement and stress are obtained by the incremental tangent stiffness method and the initial stress method of the finite element of elastic-plastic problems. The stochastic finite element of elastic- plastic problems can be calculated by the linear stochastic finite element method.

The stochastic finite element third-order perturbation method for linear static problems is formulated. The stress-strength interference model, Monte Carlo simulation and a new iterative method (NIM) of reliability calculation for the linear static problem and linear vibration are proposed. Reliability calculation methods using modified iteration formulas by the homotopy perturbation method (MIHPD) and second- order reliability method for a nonlinear static problem and nonlinear vibration are proposed. 

Based on the stochastic finite element, the fuzzy reliability calculation of structure with static problems, linear vibration, nonlinear problems and nonlinear vibration is studied. The mean and variance of stress are obtained by the stochastic finite element method. The normal membership function is generally selected as the membership function of engineering problems. The fuzzy reliability of structure can be obtained by using the calculation formula of fuzzy reliability.

Four methods of interval finite element for static analysis are proposed. Using the second-order and third-order Taylor expansion , interval finite element for static analysis is addressed. Neumann expansion of interval finite element for static analysis is formulated. Interval finite element using Sherman -Morrison-Woodbury expansion is presented. A new iterative method (NIM) is used for interval finite element calculation. Four methods can calculate the upper and lower bounds of node displacement and element stress.

Interval variables have an effect on linear vibration. The linear vibration is transformed into a static problem by Newmark method. The perturbation method, Neumann expansion method, Taylor expansion method, Sherman Morrison Woodbury expansion method and a new iterative method of interval finite element for linear vibration are proposed. The detailed derivation processes are explored. 

Nonlinear structures in engineering are affected by uncertain parameters. Firstly, the displacement when the interval variable takes the midpoint value is obtained, and the nonlinear problem is transformed into a linear problem. Five calculation methods of nonlinear interval finite element for general nonlinear problems and elastoplastic problems are proposed. According to the perturbation technique, a perturbation method is proposed. According to Taylor expansion, Taylor expansion method is proposed. Neumann expansion, Sherman Morrison Woodbury expansion and a new iterative method are proposed. 

 For the influence of non-probabilistic parameters on nonlinear vibration, the nonlinear vibration analysis of interval finite element is proposed. Using the Newmark method, nonlinear vibration is transformed into nonlinear equations. The midpoint values of interval variables are substituted into the nonlinear equations to calculate the displacement. The displacement value is substituted into the nonlinear equations, and the nonlinear equations become linear equations. Five calculation methods of interval finite element for linear vibration are extended to nonlinear vibration.

Material properties are assumed to be random parameters, interval parameters and fuzzy parameters. If the variation range is large, they are not assumed to be constants. Two improved methods of the random field are developed. The midpoint method, local average method, interpolation method and improved interpolation method of interval field are addressed. The midpoint method, local average method, interpolation method and improved interpolation method of the fuzzy field are presented. The calculation method of mixed field is discussed and the calculation formula is proposed. 

The parameters of the structure contain random variables and interval variables. The Taylor expansion method and Neumann expansion method of random interval finite element are proposed. The parameters of the structure are random and fuzzy. Taylor expansion method and Neumann expansion method of the random fuzzy finite element are illustrated. The parameters of the structure are random, fuzzy and non-probabilistic. The mixed finite element calculation should be carried out using Taylor expansion and Neumann expansion.