Nonlinear analysis of polymeric gels using moving least-squares based element-free methods
Student thesis: Doctoral Thesis
Related Research Unit(s)
Ever since the element-free Galerkin (EFG) method based on moving least-squares (MLS) approximation has been proposed in 1994, the meshless (or element-free) methods have boomed in the field of computational mechanics. In the decades that followed, efforts were devoted to the development of novel meshless (or element-free) methods and their application to a variety of science and engineering problems. Constructing efficient and stable meshless (or element-free) shape functions is one of the key issues for developing novel meshless (or element-free) methods. The improved complex variable moving least-squares (ICVMLS) approximation employs the idea of vector-form approximation with complex basis functions to reduce the number of unknown coefficients in trial functions. The best mean-square approximation for both real and imaginary parts is simultaneously achieved with a well defined error norm function. Thus, the ICVMLS approximation facilitates good computing efficiency as well as accuracy. The improved moving least squares (IMLS) approximation uses the weighted orthogonal function system as a basis function to address singularity in the MLS and improve computational efficiency. Nonlinear deformation is a significant problem encountered in many engineering problems. Its solution is always difficult unless numerical methods are used. Based on nodes rather than elements, the meshless (or element-free) methods naturally can resolve problems that arise from re-meshing and are potentially more effective for nonlinear solids. As a kind of nonlinear material, polymeric gels are receiving much attention currently. Polymeric gels are comprised of a kind of aggregate formed by three-dimensional cross-linked networks of long polymers imbibed in a solvent. The deformation of gel is a complicated multi-physical process involving the diffusion of solvent and the mechanical stretch of polymeric networks. This process leads to mechanical and diffusional equilibrium after a certain period of evolution. Based on ICVMLS, IMLS and two kinds of discretization approaches, this thesis focuses on the development and applications of two types of improvement of MLSbased element-free methods, which are, respectively, termed the improved complex variable element-free Galerkin (ICVEFG) method and the element-free improved moving least-squares Ritz (IMLS-Ritz) method. This study comprises three main parts. The first part presents the construction of the shape functions of two types of improvement of MLS approximation. The ICVMLS approximation is explained thoroughly, for greater understanding and for comparing with the former complex variable moving least-squares (CVMLS) approximation. The complex-form trail function with complex basis functions in the ICVMLS approximation is inherited from the CVMLS approximation. However, the strict error norm function in ICVMLS can ensure that the errors of both real and imaginary parts can be simultaneously minimized whilst determining coefficient terms. With the ICVMLS approximation, the number of unknown coefficients in the trial function is less than that of the MLS approximation. We can, thus, select fewer nodes in the ICVMLS approximation than in the MLS approximation without loss of precision. With IMLS, the weighted orthogonal function system is used as the basis function. The algebra equations system in the IMLS approximation will not be illconditioned, and can be solved without having to derive the inverse matrix. In the second part of the study, an ICVEFG method is presented for twodimensional nonlinear solid mechanical problems. The Galerkin weak form is employed to obtain the equations system. The ICVEFG method is found to offer greater computational precision and efficiency compared with the EFG and complex variable element-free Galerkin (CVEFG) methods. Under the same node distribution, the ICVEFG method offers greater precision than both EFG and CVEFG; and, under similar numerical precision, the ICVEFG method offers greater computational efficiency than the EFG method. The final part of the thesis deals with the numerical analysis of polymeric gel. Firstly, a numerical framework based on the ICVEFG method is developed for the large deformation analysis of the inhomogeneous swelling of gels. A multiplication decomposition of the deformation gradient is adopted and, by taking the swelled state at a certain chemical potential to be the reference configuration, a free-energy function relevant to the mechanical deformation gradient is derived. This new decomposed free-energy function avoids the difficulty of treating the chemical potential as a temperature-like variable by changing the chemical potential load into a mechanical load. Then, based on this theory, a three-dimensional approach is also developed with the element-free IMLS-Ritz methods while an energy formulation for the system is derived and a set of discrete equations is obtained through the Ritz procedure. Lastly, to reveal the multi-physical mechanism of polymeric gel, the element-free IMLS-Ritz method for the three-dimensional transient analysis of coupled mechanical-diffusion induced nonlinear deformation in polymeric gel is developed. Numerical implementation issues and parameter studies for all three approaches are described. A number of numerical experiments are conducted to demonstrate the validity and efficiency of the proposed element-free approaches. Many important results are compared with analytical solutions, experimental results and results obtained by other numerical methods. By developing and applying the two improvements of the MLS-based elementfree methods, it can be concluded that the presented methods are powerful and efficient in application. The main conclusions and possible directions for further research are also highlighted at the end of the thesis.
- Polymer colloids, Meshfree methods (Numerical analysis), Nonlinear theories