The Canham-Helfrich-Evans models of biomembranes consist of a family of geometric constrained variational problems. In this article, we compare two classes of numerical methods for these variational problems based on piecewise linear (PL) and subdivision surfaces (SS). Since SS methods are based on spline approximation and can be viewed as higher order versions of PL methods, one may expect that the only difference between the two methods is in the accuracy order. In this paper, we prove that a numerical method based on minimizing any one of the `PL Willmore energies' proposed in the literature would fail to converge to a solution of the continuous problem, whereas a method based on minimization of the bona fide Willmore energy, well-defined for SS but not PL surfaces, succeeds. Motivated by this analysis, we propose also a regularization method for the PL method based on techniques from conformal geometry. We address a number of implementation issues crucial for the efficiency of our solver. A software package called Wmincon accompanies this article, provides parallel implementations of all the relevant geometric functionals. When combined with a standard constrained optimization solver, the geometric variational problems can then be solved numerically. To this end, we realize that some of the available optimization algorithms/solvers are capable of preserving symmetry, while others manage to break symmetry; we explore the consequences of this observation.

中文翻译：

---

生物膜的数值方法：符合细分方法与不符合 PL 方法

生物膜的 Canham-Helfrich-Evans 模型由一系列几何约束变分问题组成。在本文中，我们比较了基于分段线性 (PL) 和细分曲面 (SS) 的这些变分问题的两类数值方法。由于 SS 方法基于样条近似，并且可以被视为 PL 方法的高阶版本，人们可能会认为这两种方法之间的唯一区别在于精度顺序。在本文中，我们证明了基于最小化文献中提出的任何一个“PL Willmore 能量”的数值方法将无法收敛到连续问题的解决方案，而基于最小化真正的 Willmore 能量的方法，为 SS 而不是 PL 表面定义良好，成功。受此分析的启发，我们还提出了一种基于共形几何技术的 PL 方法的正则化方法。我们解决了许多对求解器效率至关重要的实现问题。本文随附了一个名为 Wmincon 的软件包，它提供了所有相关几何泛函的并行实现。当与标准的约束优化求解器结合使用时，几何变分问题可以被数值求解。为此，我们意识到一些可用的优化算法/求解器能够保持对称性，而另一些则设法打破对称性；我们探讨了这一观察结果。本文随附了一个名为 Wmincon 的软件包，它提供了所有相关几何泛函的并行实现。当与标准的约束优化求解器结合使用时，几何变分问题可以被数值求解。为此，我们意识到一些可用的优化算法/求解器能够保持对称性，而另一些则设法打破对称性；我们探讨了这一观察结果。本文随附了一个名为 Wmincon 的软件包，它提供了所有相关几何泛函的并行实现。当与标准的约束优化求解器结合使用时，几何变分问题可以被数值求解。为此，我们意识到一些可用的优化算法/求解器能够保持对称性，而另一些则设法打破对称性；我们探讨了这一观察结果。