We present a structure-preserving scheme based on a recently-proposed mixed formulation for incompressible hyperelasticity formulated in principal stretches. Although there exist Hamiltonians introduced for quasi-incompressible elastodynamics based on different variational formulations, the one in the fully incompressible regime has yet been identified in the literature. The adopted mixed formulation naturally provides a new Hamiltonian for fully incompressible elastodynamics. Invoking the discrete gradient formula, we are able to design fully-discrete schemes that preserve the Hamiltonian and momenta. The scaled mid-point formula, another popular option for constructing algorithmic stresses, is analyzed and demonstrated to be non-robust numerically. The generalized Taylor-Hood element based on the spline technology conveniently provides a higher-order, robust, and inf-sup stable spatial discretization option for finite strain analysis. To enhance the element performance in volume conservation, the grad-div stabilization, a technique initially developed in computational fluid dynamics, is introduced here for elastodynamics. It is shown that the stabilization term does not impose additional restrictions for the algorithmic stress to respect the invariants, leading to an energy-decaying and momentum-conserving fully discrete scheme. A set of numerical examples is provided to justify the claimed properties. The grad-div stabilization is found to enhance the discrete mass conservation effectively. Furthermore, in contrast to conventional algorithms based on Cardano's formula and perturbation techniques, the spectral decomposition algorithm developed by Scherzinger and Dohrmann is robust and accurate to ensure the discrete conservation laws and is thus recommended for stretch-based material modeling.

翻译：我们提出了一种基于近期提出的主伸长率公式的不可压缩超弹性力学混合公式的保结构方案。尽管基于不同变分公式的准不可压缩弹性动力学已经引入了哈密顿量，但完全不可压缩情况下的哈密顿量尚未在文献中被识别。所采用的混合公式自然地提供了一个完全不可压缩弹性动力学的新哈密顿量。借助离散梯度公式，我们能够设计出保持哈密顿量和动量的全离散格式。另一种构建算法应力的流行选项——比例中点公式，经过分析被证明在数值上不稳健。基于样条技术的广义泰勒-胡德单元为有限应变分析提供了高阶、稳健且满足inf-sup条件的空间离散化选择。为增强单元在体积守恒方面的性能，我们引入了一种最初在计算流体动力学中开发的梯度散度稳定化技术。研究表明，稳定化项不会对算法应力满足不变量施加额外限制，从而得到了一个能量衰减且动量守恒的全离散格式。通过一系列数值算例验证了所声称的性质。发现梯度散度稳定化能有效增强离散质量守恒。此外，与基于Cardano公式和摄动技术的传统算法相比，Scherzinger和Dohrmann开发的谱分解算法在确保离散守恒律方面稳健且精确，因此被推荐用于基于伸长率的材料建模。