This article discusses a mixed FE technique for 3D nonlinear elasticity using a Hu-Washizu (HW) type variational principle. Here, the deformed configuration and sections from its cotangent bundle are taken as additional input arguments. The critical points of the HW functional enforce compatibility of these sections with the configuration, in addition to mechanical equilibrium and constitutive relations. The present FE approximation distinguishes a vector from a 1-from, a feature not commonly found in FE approximations. This point of view permits us to construct finite elements with vastly superior performance. Discrete approximations for the differential forms appearing in the variational principle are constructed with ideas borrowed from finite element exterior calculus. The discrete equations describing mechanical equilibrium, compatibility and constitutive rule, are obtained by extemizing the discrete functional with respect to appropriate DoF, which are then solved using the Newton's method. This mixed FE technique is then applied to benchmark problems wherein conventional displacement based approximations encounter locking and checker boarding.

中文翻译：

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一种使用有限元外部微积分的 3D 非线性弹性混合方法

本文讨论了使用 Hu-Washizu (HW) 型变分原理的 3D 非线性弹性混合有限元技术。在这里，变形的配置和来自其余切丛的部分被作为额外的输入参数。除了机械平衡和本构关系之外，硬件功能的关键点还强制这些部分与配置兼容。当前的 FE 近似将向量与 1-from 区分开来，这是 FE 近似中不常见的特征。这种观点使我们能够构造具有极其优越性能的有限元。变分原理中出现的微分形式的离散近似是用从有限元外部微积分中借来的思想构造的。描述机械平衡的离散方程，兼容性和本构规则，是通过将离散泛函相对于适当的 DoF 来获得，然后使用牛顿方法求解。然后将这种混合有限元技术应用于基准问题，其中传统的基于位移的近似会遇到锁定和棋盘格。