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Biomembrane modeling with isogeometric analysis

机译：具有等几何分析的生物膜建模

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We consider the numerical approximation of lipid biomembranes at equilibrium described by the Canham-Helfrich model, according to which the bending energy is minimized under area and volume constraints. Energy minimization is performed via L-2-gradient flow of the Canham-Helfrich energy using two Lagrange multipliers to weakly enforce the constraints. This yields a highly nonlinear, high order, time dependent geometric Partial Differential Equation (PDE). We represent the biomembranes as single-patch NURBS closed surfaces. We discretize the geometric PDEs in space with NURBS-based Isogeometric Analysis and in time with Backward Differentiation Formulas. We tackle the nonlinearity in our formulation through a semi-implicit approach by extrapolating, at each time level, the geometric quantities of interest from previous time steps. We report the numerical results of the approximation of the Canham-Helfrich problem on ellipsoids of different aspect ratio, which leads to the classical biconcave shape of lipid vesicles at equilibrium. We show that this framework permits an accurate approximation of the Canham-Helfrich problem, while being computationally efficient. (C) 2019 Elsevier B.Y. All rights reserved.
机译：我们考虑了由Canham-Helfrich模型描述的处于平衡状态的脂质生物膜的数值近似，根据该近似值，在面积和体积约束下，弯曲能被最小化。通过使用两个拉格朗日乘数来弱执行约束，通过Canham-Helfrich能量的L-2梯度流执行能量最小化。这产生了高度非线性，高阶，时间相关的几何偏微分方程（PDE）。我们将生物膜表示为单补丁NURBS封闭表面。我们使用基于NURBS的等几何分析离散化了空间中的几何PDE，并及时使用了向后差分公式离散化了几何PDE。我们通过半隐式方法来解决配方中的非线性问题，方法是在每个时间级别外推先前时间步长中感兴趣的几何量。我们报告了在不同纵横比的椭球上Canham-Helfrich问题的近似值的数值结果，这导致了脂质小泡在平衡时的经典双凹形状。我们表明，该框架允许在计算效率高的同时，精确地逼近Canham-Helfrich问题。 （C）2019 Elsevier B.Y.版权所有。

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