标题： 使用deal.II库在有限应变下实现表面弹性的一个有限元方法 显示英文标题

标题： A finite element implementation of surface elasticity at finite strains using the deal.II library

摘要： 表面在弹性体整体响应中的潜在重要作用可以使用成熟的表面弹性理论来描述。 本贡献的目的是详细说明基础控制方程（体积和表面）的有限元近似及其使用开源有限元库 deal.II 的求解。 考虑了完全非线性（几何和材料）情况。 使用牛顿-拉夫森过程求解非线性问题，其中体积和表面的切线贡献是精确计算的。 有限元公式是在总体拉格朗日框架内实现的，并采用了体积和表面的 Bubnov-Galerkin 空间离散化。 假设表面是材料的。 使用一个从表面上的自由度到体积边界上的自由度之间的映射，将表面的贡献分配到全局系统矩阵和残差向量中。 deal.II 库大大简化了各种表面算子的计算，使得数值实现能够紧密匹配在一篇配套论文中开发的理论。 通过一系列基准示例问题阐明了该理论和数值实现的关键特征。 提供了完整且有文档说明的源代码。 显示英文摘要

摘要： The potentially significant role of the surface of an elastic body in the overall response of the continuum can be described using the mature theory of surface elasticity. The objective of this contribution is to detail the finite element approximation of the underlying governing equations (both in the volume and on its surface) and their solution using the open-source finite element library deal.II. The fully-nonlinear (geometric and material) setting is considered. The nonlinear problem is solved using a Newton--Raphson procedure wherein the tangent contributions from the volume and surface are computed exactly. The finite element formulation is implemented within the total Lagrangian framework and a Bubnov-Galerkin spatial discretization of the volume and the surface employed. The surface is assumed material. A map between the degrees of freedom on the surface and on the boundary of the volume is used to allocate the contribution from the surface to the global system matrix and residual vector. The deal.II library greatly facilitates the computation of the various surface operators, allowing the numerical implementation to closely match the theory developed in a companion paper. Key features of the theory and the numerical implementation are elucidated using a series of benchmark example problems. The full, documented source code is provided.