标题： 二阶兼容应变混合有限元在二维可压缩非线性弹性力学中的应用 显示英文标题

标题： Second-Order Compatible-Strain Mixed Finite Elements for 2D Compressible Nonlinear Elasticity

摘要： 近年来，一类新的混合有限元方法——协调应变混合有限元（CSMFEs）——出现，它利用了非线性弹性问题的微分复形。它们在基准问题中的出色表现，例如在近不可压缩固体中建模大变形时的数值稳定性，使其成为解决工程问题的有前景选择。一阶CSMFEs的各类形状函数存在显式形式。相比之下，现有的二阶CSMFEs通过数值积分来评估形状函数。本文提出了具有显式位移梯度和应力张量形状函数的二阶CSMFEs。介绍了源于外微积分的矢量微积分概念，并用于提供自然坐标系下形状函数的有效形式。随后应用协变和逆变Piola变换将形状函数转换到物理空间。在单元边界上使用中间节点和伪节点来强制位移梯度和应力张量的连续性约束。详细讨论了所提出的二阶CSMFEs的公式化及其实施的技术细节。通过多个基准问题验证了CSMFEs与一阶CSMFEs以及其他依赖于数值积分的二阶元素的性能比较。结果表明，所提出的CSMFEs在有限应变范围内对近不可压缩固体建模具有数值稳定性。 显示英文摘要

摘要： In recent years, a new class of mixed finite elements -- compatible-strain mixed finite elements (CSMFEs) -- has emerged that uses the differential complex of nonlinear elasticity. Their excellent performance in benchmark problems, such as numerical stability for modeling large deformations in near-incompressible solids, makes them a promising choice for solving engineering problems. Explicit forms exist for various shape functions of first-order CSMFEs. In contrast, existing second-order CSMFEs evaluate shape functions using numerical integration. In this paper, we formulate second-order CSMFEs with explicit shape functions for the displacement gradient and stress tensor. Concepts of vector calculus that stem from exterior calculus are presented and used to provide efficient forms for shape functions in the natural coordinate system. Covariant and contravariant Piola transformations are then applied to transform the shape functions to the physical space. Mid-nodes and pseudo-nodes are used to enforce the continuity constraints for the displacement gradient and stress tensor over the boundaries of elements. The formulation of the proposed second-order CSMFEs and technical aspects regarding their implementation are discussed in detail. Several benchmark problems are solved to compare the performance of CSMFEs with first-order CSMFEs and other second-order elements that rely on numerical integration. It is shown that the proposed CSMFEs are numerically stable for modeling near-incompressible solids in the finite strain regime.