We provide a fully nonlinear port-Hamiltonian formulation for discrete elastodynamical systems as well as a structure-preserving time discretization. The governing equations are obtained in a variational manner and represent index-1 differential algebraic equations. Performing an index reduction one obtains the port-Hamiltonian state space model, which features the nonlinear strains as an independent state next to position and velocity. Moreover, hyperelastic material behavior is captured in terms of a nonlinear stored energy function. The model exhibits passivity and losslessness and has an underlying symmetry yielding the conservation of angular momentum. We perform temporal discretization using the midpoint discrete gradient, such that the beneficial properties are inherited by the developed time stepping scheme in a discrete sense. The numerical results obtained in a representative example are demonstrated to validate the findings.

翻译：我们针对离散弹性动力系统提出了完全非线性的端口-哈密顿形式化方法，并构建了保持结构的时间离散化方案。控制方程通过变分方法推导，表现为指标为1的微分代数方程。通过指标约简获得端口-哈密顿状态空间模型，该模型将非线性应变作为独立状态变量与位置和速度并列。此外，超弹性材料行为通过非线性存储能量函数进行刻画。该模型具有无源性和无损耗性，其内在对称性保证了角动量守恒。我们采用中点离散梯度进行时间离散化，使得所开发的时间步进方案在离散意义上继承这些优良特性。通过代表性算例的数值结果验证了理论发现的有效性。