Development of moving kriging technique in the framework of meshless methods for thermo-mechanical modeling and simulation of functionally graded plates
基於移動 Kriging 技術的無網格方法的發展及其在功能梯度板熱力分析中的應用
Student thesis: Doctoral Thesis
Related Research Unit(s)
Meshless or meshfree methods have been applied to a large number of partial differential equation problems in solid and fluid mechanics. Seeking an efficient and stable approach for constructing meshless shape functions is always one of the key issues in the development of meshless methods. In recent years, Kriging interpolation approach has been applied in meshless methods for constructing shape functions. Shape functions are predominantly formed by moving least-squares approximation in meshless methods but shape functions established with Kriging technique possess the Kronecker delta function property which makes it easy to enforce essential boundary conditions. Plate structures, as important structural components, have been fully investigated theoretically and numerically. However, effort to develop efficient, accurate and stable numerical approaches for analyzing plate structures made of multiple phases of material constituents with nonlinear constitutive relationship, having complex geometrical shape and boundary conditions and subjected to multi-field loadings is still meaningful. Due to their distinctive properties and tailor-made capabilities, functionally graded materials are receiving considerable attention, especially when used in high-temperature and large temperature gradient environments. Based on local, global weak-form and strong-form of governing equations, this thesis focuses on the development and applications of three types of Kriging-based meshless methods, which are respectively termed local Kriging meshless method, meshfree Kriging-Ritz, and moving Kriging-DQ method. This study is divided into three main parts. The first part presents the construction of shape functions by Kriging interpolation technique elaborately for better understanding and correcting derivation errors in some related references. The shape functions can be formed based on the second order stationarity and intrinsic hypothesis of a random function, respectively. The forms of these two kinds of shape functions are similar and thus the same shape functions can be generated when identical models are adopted, except that one is used as covariance while the other is used as a semivariogram. The reason is that there exists an interrelation between the covariance and the semivariogram. In the second part, the three types of Kriging-based meshfree methods, local Kriging meshless method, meshfree Kriging-Ritz and Kriging-DQ method are presented. The local Kriging meshless method combines the Kriging-based shape functions with the local Petrov-Galerkin formulation while the meshfree Kriging-Ritz method incorporate the same kind of shape functions into the Ritz technique. These two kinds of meshless methods are accordingly developed from the points of view of local and global domains based on weak-forms of governing equation. The Kriging-DQ method is developed in terms of strong-form of governing equations. It is very meaningful to investigate the performances of the three Kriging-based meshless methods in structural analysis. The last part of the thesis deals with free vibration, nonlinear deflection and buckling analysis of shear deformable functionally graded plates in thermal environment. In these numerical simulations, both the mechanical and thermal loadings are applied in combination. Numerical implementation issues and parameter studies are described. A number of numerical experiments are conducted to demonstrate the validity and efficiency of the proposed meshfree models. Many important results are compared with analytical solutions, experimental results or those obtained by other numerical methods. By comparing the three Kriging-based meshless methods, it can be concluded that Kriging interpolation technique is powerful and efficient in use of the meshless framework for structural analysis. The main conclusions and possible directions for further research are also highlighted at the end of the thesis.
- Kriging, Mathematical models, Meshfree methods (Numerical analysis), Functionally gradient materials