Cells and organelles are surrounded by a biological membrane which separates inside from the outside environment. This membrane consisting of two monolayers of lipid molecules called lipid bilayer. Lipids are amphiphilic molecules and hence the lipid bilayer is a polar structure which is selectively permeable to ions and molecules. Shape transformations in biological membranes are crucial in a variety of cellular processes such as traort in the Golgi apparatus and endoplasmic reticulum, shaping the cell organelles and signaling in neuronal synapses. Dynamic analysis of lipid bilayer membranes is popular among researchers since valuable information about cell functions can be retrieved. There are several limitations in experimental tests and simulations such as computational and implementation cost. However, in theoretical studies, different phenomena can be modeled and the effect of each parameter can be investigated. In this study, a mathematical model based on the mechanical properties of lipid bilayer is utilized. Elastic energies and dissipation functions are considered for fluid bilayer membranes. An energy approach is used to obtain the governing equations of an enclosed lipid bilayer membrane. The governing equations representing the dynamics of the bilayer are discretized by B-spline basis function and then solved numerically to obtain the shape of the membrane as a function of time. The stationary shape of the vesicles for different values of reduced volume and reduced area difference is obtained to explore the phase diagram and to verify the governing equations. Next, the density asymmetry in bilayers caused by the change in the density or the equilibrium density of the outer monolayer is studied. This can be observed in the recruitment of proteins to the outer monolayer or pH gradients of the environment of a vesicle leading to the formation of buds, tubules, and pearls. The effect of density difference and curvature on creation and growth of tubules are investigated. Tethers and tubules are involved in many cellular processes such as inter or intracellular traort and cell-cell adhesion. In the next step, the procedure of tether extrusion of a bilayer membrane is studied. The effect of surface tension and the pulling rate of the tether on the dynamic behavior of the tether are investigated. There is a critical pulling rate at which the membrane and the tether become unstable. This pulling rate is dependent on the material parameters of the membrane such as bending and stretching moduli. At the end, the effect of external source of mass on the outer monolayer in the tether formation is investigated. The external density added to the membrane has a great effect on the dynamic behavior of the tether. Tether may undergo pearling during extension of the membrane. It also affects the critical pulling rate. Keywords : Lipid bilayer membranes; Mathematical modeling; Finite element method; Stationary shapes; Density asymmetry; Tether extrusion .