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.

在本文中，我们使用等几何分析（IGA）来解决约束的Willmore问题和电场，以模拟受到静电场作用的囊泡的形态演变。该模型包括两个阶段，脂质双层和电解质。两阶段问题是使用相界面法（扩散界面模型的子类）建模的。使用相场参数重新制定模型的弯曲，挠性和介电能。使用改进的增强拉格朗日方法来满足约束条件，同时保持数值稳定性和相对较大的时间步长。这种方法保证了在整个时域上每个时间步的约束满足。结果表明，等几何分析在解决高阶微分算子方面的优势，而无需其他中间方程式即可解决基于经典网格的有限连续性问题。在物理方面，即使考虑生物膜的柔电响应，也可以使用IGA精确模拟囊泡的形态演变，这会给系统增加另一层数值复杂性。通过几个3D数值示例研究了柔电性，电导率和问题的其他方面的影响。即使考虑生物膜的柔电响应，这也会增加系统的另一层数值复杂性。通过几个3D数值示例研究了柔电性，电导率和问题的其他方面的影响。即使在考虑生物膜的柔电响应时，这也会给系统增加另一层数值复杂性。通过几个3D数值示例研究了柔电性，电导率和问题的其他方面的影响。