Geometric nonlinearities, material nonlinearities, and volume locking are the notable challenges faced in hyperelastic analysis. Traditional methods in this regard are complex and laborious for implementation as they require linearization and formulation of global matrix equations while simultaneously addressing volumetric locking. A mixed node's residual descent method (NRDM) proposed herein can effectively address the numerical challenges associated with geometric nonlinearities, material nonlinearities, force nonlinearities, and incompressibility. First, the implementation of the NRDM is considerably simplified as unlike traditional incremental–iterative methods, it's a matrix-free iterative method that does not require incremental linear equations. Second, the NRDM addresses the geometric and material nonlinearities with relative ease as it flexibly describes the deformation with the initial configuration or assumed deformed configuration as the reference frame. Third, the NRDM prevents the occurrence of volumetric locking by employing hydrostatic pressure as an independent variable. Furthermore, the NRDM can easily treat the force nonlinearities and boundary nonlinearities by controlling the relation between load and deformation during iteration. Moreover, a notable accuracy of the NRDM is confirmed through numerical verifications, and several critical matters are discussed, including the scheme for adjusting the basic independent variables, treatment of displacement boundaries, scheme for imposing loads, and computational parameter setting.

中文翻译：

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超弹性问题分析的混合节点残差下降法


几何非线性、材料非线性和体积锁定是超弹性分析中面临的显着挑战。这方面的传统方法实施起来复杂且费力，因为它们需要线性化和制定全局矩阵方程，同时解决体积锁定问题。本文提出的混合节点残差下降法（NRDM）可以有效解决与几何非线性、材料非线性、力非线性和不可压缩性相关的数值挑战。首先，NRDM 的实现大大简化，因为与传统的增量迭代方法不同，它是一种无矩阵迭代方法，不需要增量线性方程。其次，NRDM 相对容易地解决几何和材料非线性问题，因为它以初始构型或假设变形构型作为参考系灵活地描述变形。第三，NRDM 通过采用静水压力作为自变量来防止体积锁定的发生。此外，NRDM可以通过控制迭代过程中的载荷和变形之间的关系来轻松处理力非线性和边界非线性。此外，通过数值验证证实了NRDM的显着准确性，并讨论了几个关键问题，包括基本自变量的调整方案、位移边界的处理、施加载荷的方案和计算参数的设置。