Lipid membranes are interesting subjects of biological studies. In this talk, we report some recent works done by the Penn State group on the phase field modeling and simulations of the vesicle membrane deformation under elastic bending energy and the interaction with background fluid flows. Connections are made to the well$-$known Willmore problem in differential geometry and the Gamma convergence of nonlinear functionals in the calculus of variation. We also discuss how to effectively retrieve topological information within the phase field framework which may have broad applications