Development of Meshless Method in Studies on Mechanical Behaviors of Hydrogels

Abstract

As a rapidly developed numerical method, the meshless method has become increasingly important in mechanical analysis and engineering computing since the element free Galerkin (EFG) method based on the moving least squares (MLS) approximation was proposed. The greatest strength of the meshless method is that the meshless method can eliminate the disadvantages of low accuracy and efficiency caused by the remeshing techniques in the mesh-based numerical methods. To date, the meshless method has extended into a variety of branches due to different ways of constructing shape functions. The core difference between the meshless method and the finite element method (FEM) lies in the form of the shape function. Hence, obtaining an effective and stable shape function is an important topic of study to promote further development of the meshless method.

In this study, an effective meshless method is developed by improving previous complex variable moving least squares methods using an interpolating technique. In the improved interpolating complex variable moving least squares (IICVMLS) method, through an orthogonalization process, a new shape function that possesses the interpolating property is obtained by introducing a complete basis function and a singular weight function. The IICVMLS method is proved through theoretical derivation and numerical examples to have higher computing accuracy compared with the non-interpolating complex variable moving least squares approximations.

The improved interpolating complex variable element free Galerkin (IICVEFG) method for different structural problems is proposed by combining the IICVMLS method with the corresponding Galerkin weak integral forms. In the IICVEFG method, the essential boundary conditions can be applied directly without special techniques, such as the Lagrange multiplier approach and penalty approach. Hence, compared with the non-interpolating complex variable element free Galerkin methods, the final discrete matrix equations in the IICVEFG method are more concise, and thus the IICVEFG has higher computing accuracy and efficiency.

A linear elastic bending problem of Kirchhoff plates is used to examine the performance of the IICVEFG method. In the IICVEFG method for the bending problem of Kirchhoff plates, the Galerkin weak integral form is used to obtain the equilibrium equation, and the IICVMLS method is used to disperse this equation. The continuity and derivative of the new basis function in the IICVMLS method are validated by simulating the bending problems of Kirchhoff plates with different displacement and force boundary conditions. The effects of the node distribution, the scaling parameter and the penalty factor on the computing accuracy and efficiency are also discussed. Compared with the non-interpolating complex variable element free Galerkin methods, the IICVEFG method has higher computing accuracy and efficiency, especially around borders.

As one of the meshless methods, the IICVEFG method is suitable for analyzing nonlinear large deformation problems. As a nonlinear material, the environmentally sensitive hydrogel can sense changes from an external stimulus and produce large deformation, which is attracting more attention in engineering and biomedical aspects. Thus, the IICVEFG method is used in this thesis to investigate the nonlinear mechanical behaviors of hydrogel in multi-fields.

The IICVEFG method is proposed to simulate the two-dimensional large deformation of hydrogel in the steady state by combining the basic equilibrium field theory of hydrogel with the multiplicative decomposition of the deformation gradient. Based on the IICVEFG method, the mechanical deformation behaviors of a square hydrogel with elliptical, circular and square holes are simulated. The effects of the shapes and initial sizes of holes on the pattern transformation, actuating behavior and area reduction of holes are discussed in detail, which offer a reference for designing a square hydrogel with potential applications such as actuation and flow control.

Actually, the swelling of hydrogel is a time-related process that includes the mass transport and the mechanical deformation. Using nonlinear continuum theory while considering thermodynamics and kinetics, the IICVEFG method is developed to analyze the two-dimensional chemical-mechanical coupling behavior of hydrogel in the transient state. The final matrix framework is derived, in which the displacement and chemical potential boundary conditions are applied directly. Then, a constrained hydrogel is simulated, and the numerical results are obtained using the finite difference method as the comparable numerical solution.

In this thesis, it can be concluded that the IICVMLS method and its corresponding IICVEFG method show good performance and feasibility with high computing accuracy and efficiency. The primary conclusions and possible future research directions are presented at the end of this thesis.

Date of Award	27 Aug 2018
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Xiaoqiao HE (Supervisor)