In this paper we propose a novel numerical scheme for the Canham-Helfrich-Evans bending energy based on a three-field lifting procedure of the distributional shape operator to an auxiliary mean curvature field. Together with its energetic conjugate scalar stress field as Lagrange multiplier the resulting fourth order problem is circumvented and reduced to a mixed saddle point problem involving only second order differential operators. Further, we derive its analytical first variation (also called first shape derivative), which is valid for arbitrary polynomial order, and discuss how the arising shape derivatives can be computed automatically in the finite element software NGSolve. We finish the paper with several numerical simulations showing the pertinence of the proposed scheme and method.

翻译：在本文中,我们提出了一个新颖的数字方案,用于Canham-Helfrich-Evans弯曲能量,其依据是分配形状操作员的三维提升程序,将其弯曲成一个辅助平均曲线场。连同其高强度的共振卡路拉压力场作为拉格兰格乘数,由此产生的第四级问题被绕过,并降为只涉及第二级差操作员的混合马鞍点问题。此外,我们从中得出其分析上的第一次变异(也称为第一型衍生物),该变异对任意的多元顺序有效,并讨论如何在有限元素软件NGSolve中自动计算产生的形状衍生物。我们用数个数字模拟来完成论文,以显示拟议办法和方法的适切性。