| Recently, we begin to systematically study the shape deformation of vesicle membranes by numerical simulations, sometimes under external fluid fields, using a unified energetic variational formulation with phase field methods based on the minimization of elastic bending energy with volume and surface area constraints ([21, 19, 20, 22, 23, 24]). Analysis and numerical methods in both static and dynamic are developed to solve the phase field models.; Phase field approach is a global method, allowing topological changes of the interface. And complex interfaces may be described as a relatively simple phase function within the phase field approach. Compare to other numerical methods, phase field method is more unified, global, and relatively easier for implementation.; First we build the phase field theory for finding the equilibrium vesicle shapes. Theoretically, we build the phase field model for the biological elastic bending energy model. And the consistency of our phase field model with the general sharp interface model is verified. Further, we develop a serial methods including the Euler-Lagrange and penalty constraints methods to solving the phase field model. In guiding the numerical simulations, we prove the convergence of our numerical simulation results to the analytical phase field energy minimizers.; Many simulations are carried out to find the equilibrium shapes of vesicle membranes in the axial symmetrical and the truly 3D cases. Different energetic bifurcation phenomena are discussed. We also plot a relatively complete energy diagram. The effect of the spontaneous curvature is also discussed for both constant and variable cases. In the 3D non-symmetrical case, some non-symmetrical examples are found and compared with biological experiments.; The study of the vesicle transformations within fluid fields is another important contribution of this work. We successfully couple the phase field transformation with the fluid dynamics. Theoretically analysis of the extra stress term caused by the membrane to the fluids is carried out and further compared with the Euler-Lagrangian equation of Willmore's problem. Energy laws within the coupled systems ensure the similar asymptotic limit of the phase field formulation to the equilibrium system. Extensive three dimensional numerical simulations are carried out guiding by a set of numerical schemes for both the phase field transformation and fluid dynamics.; The last contribution of this work is that a series of new formulae is used in detecting the topological changes in vesicle membrane transformations. More important, some of the formulae are developed in a very general frame work and can be applied to other problems and potentially can be used in controlling the structure of vesicle membranes. Numerical simulations are carried out to check those formulae in all kinds of cases involving the topological events.; For biology, this work gives the mathematical simulation to study the physics of vesicle membranes. For mathematics, this work verifies the power of phase field method and further develops this methods by combining with the fluid mechanics and topology. |
