Triphasic mixture modeling and meshless numerical computations

Abstract

There are two major themes contained in this thesis. The first theme is devoted to the development and applications of meshless numerical computation method based on the radial basis functions (RBFs). To tackle the ill-conditioning problem resulted from using the global RBFs, overlapping domain decomposition method (DDM) with both multiplicative and additive Schwarz iterative techniques is developed. By this method, the bottle-neck of RBFs method for large scale problems is efficiently circumvented. Another attractive advantage of DDM is its great potential for parallel computations. The effectiveness and robustness of the method are demonstrated by performing numerical experiments with both a regular elliptic problem and a singularly perturbed elliptic problem. Applications of the RBFs method to solve free boundary problems have been carried out. Taking the advantages of the truly meshless RBFs method, a robust searching algorithm for the unknown optimal exercise boundary in the valuation of the American option pricing model is devised. A comparison on using various global RBFs and compactly supported RBFs for solving the option pricing models is presented. Combined with the Lagrangian-Eulerian hybrid scheme, a meshless algorithm is devised to solve the shallow water model under moving boundary condition. The advantages of the meshless method have been demonstrated in the computations of both the dam break problem and the run-up problem. The second theme of the thesis is placed on the development of triphasic mixture model. From the first law of thermodynamics for irreversible thermo-dynamical system, a new set of governing equations for mechano-electrochemical triphasic mixture theory is derived. It is revealed that both the gradient of fluid pressure and that of electrochemical potential are the driving forces for the movement of ions in the triphasic mixture. The chemical-expansion stress in the solid phase can be embedded in the constitutive property by modifying the constitutive parameters of isotropic elastic material. The gradient of electric potential is theoretically added in the deiving forces for ions. So the triphasic model is extended to be applicable to the cases under electric field. In the case of no external electric field, a new biphasic model is derived by combining the ion phase with the water phase. Based on the RBFs method, a meshless numerical algorithm, in which the domain decomposition method is successfully applied, is developed to solve two-dimensional triphasic models for charged and hydrated soft tissues. Three cases of a plane strain problem are analyzed and compared. An axisymmetric model for a real synovia joint is set and computed to simulate the mechanical responses of the articular cartilage. Unlike finite element method which requires complicated variational formulation of the original governing partial differential equations, the proposed numerical method can be applied directly by collocation to solve the triphasic model which is comprised of continuity, momentum, and constitutive equations. The commonly used penalty technique for handling incompressibility is not required in the proposed method and hence conditions on stability can be avoided. A numerical model based on the triphasic theory is developed to study the mechanism of the deformation response of a hydrogel strip immersed into an acidic solution under an external electric field. The complexly coupled nonlinear governing equations are numerically solved by using the meshless RBFs method. The factors which significantly influence the deformation are investigated and analyzed. Numerical results show good agreement with the experimental measurements.

Date of Award	14 Feb 2003
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Yiu Chung Benny HON (Supervisor)