Isogeometric analysis for a phase-field constrained optimization problem of morphological evolution of vesicles in electrical fields

Abstract In this paper, we solve a constrained Willmore problem coupled with an electrical field using IsoGeometric Analysis (IGA) to simulate the morphological evolution of vesicles subjected to static electrical fields. The model consist of two phases, the lipid bilayer and the electrolyte. The two-phases problem is modeled using the phase-field method, a subclass of the diffusive interface models. The bending, flexoelectric and dielectric energies of the model are reformulated using the phase-field parameter. A modified Augmented-Lagrangian approach was used to satisfy the constraints while maintaining numerical stability and a relatively large time step. This approach guarantees the satisfaction of the constraints at each time step over the entire temporal domain. The results show the superiority of the isogeometric analysis in solving high-order differential operators without the need for additional intermediate equations to account for classical mesh-based methods limited continuity. On the physical side, the morphological evolution of the vesicles can be simulated accurately using IGA, even when considering the flexoelectric response of the biomembrane, which adds another layer of numerical complexity to the system. The effect of the flexoelectricity, the conductivity ratio and other aspects of the problem are studied through several 3D numerical examples.