Fig. 1. The changes of energy, surface area and volume in preparation for an initial data.

Fig. 2. Cross section views of a starting phase field function and that of the prepared initial phase field along with its 3D and cutting view.

Fig. 3. Effect of different grid sizes: A 3-D view of the cylindrical shaped membrane and the energy plots for two different grids (top). The second row shows the x–y plane cross-sections of the initial phase field ₀ and the final phase field on the 40³ and 64³ grids.

Fig. 4. Comparing different values of : the cross-sections and the energy plots for  = 0.1736 (top) and  = 0.2464 (bottom).

Fig. 5. Plots of Lagrange multipliers λ₁ (top solid line) and λ₂ (dash lines) for different grid sizes and values.

Fig. 6. Computed membranes in a discocyte shape (top) and a gourd shape (bottom) with the corresponding initial shapes (left).

Fig. 7. A flat ellipsoid shaped vesicle pinches off to a torus.

Fig. 8. The initial vesicle (top-left), with a larger preset surface area of 52.04, transforms to a shell (top-right). And with smaller preset surface area values of 45.6 (bottom-left) and 48.30 (bottom-right), it transforms to stomatocyte shapes.

Fig. 9. A non-symmetric torus evolves into a symmetric torus.

Fig. 10. The initial shape (top) and the final non-symmetric shape (bottom).

Fig. 11. A gourd split into three spherical vesicles with C = 3.0.

Fig. 12. Inhomogeneous spontaneous curvature effect: spiculation at some points on the surface. Left: the initial shapes. Right: the equilibrium shapes.

Fig. 13. Left: from top to bottom, merging of three spheres into a torus. Right: merging of six spheres into a cylinder.

Fig. 14. Eight spheres merge into a discocyte.

Fig. 15. Nine spheres merge into a membrane with bumps and holes. Top: the deformation process. Bottom: the 3d view of the final shape, the 3d and cutting views after the corner being shifted to the center using the periodicity.

Fig. 16. The Euler number comes from 5 to 0 after the 5 spheres merge together.

Convergence tests with different time steps