几何非线性弹性动力学的速度-应力公式及其保结构离散化
On the velocity-stress formulation for geometrically nonlinear elastodynamics and its structure-preserving discretization
AI总结：
针对大变形小应变的几何非线性弹性动力学问题，提出具有端口哈密顿结构的速度-应力公式，推导含拉格朗日乘子边界条件的保能量弱形式，通过混合有限元和时间离散实现保结构离散，理论与数值验证了其功率平衡与无源性。
摘要：
我们研究大变形但小应变条件下弹性连续体的动力学行为。这类系统可通过几何非线性弹性动力学方程结合St. Venant-Kirchhoff（圣维南-基尔霍夫）材料本构关系来描述。该问题的速度-应力公式具有形式化的port-Hamiltonian（端口哈密顿）结构。与线性情况不同，该问题的算子受位移场调制，位移场可作为被动变量处理，并与速度一同积分。我们推导了问题的弱形式，并通过Lagrange（拉格朗日）乘子引入本质边界条件。该变分公式明确编码了内部以及跨边界的动能与势能之间的转换，因此可得到全局功率平衡关系，并保证系统的无源性。通过选用合适的混合有限元进行Galerkin（伽辽金）近似，弱形式的特殊几何结构可以得到保留。此外，通过合适的时间离散化方法，可以得到完全离散的功率平衡关系。我们从理论上证明了系统及其离散化的主要性质，并通过数值测试进行了验证。
英文摘要：
We consider the dynamics of an elastic continuum under large deformation but small strain. Such systems can be described by the equations of geometrically nonlinear elastodynamics in combination with the St. Venant-Kirchhoff material law. The velocity-stress formulation of the problem turns out to have a formal port-Hamiltonian structure. In contrast to the linear case, the operators of the problem are modulated by the displacement field which can be handled as a passive variable and integrated along with the velocities. A weak formulation of the problem is derived and essential boundary conditions are incorporated via Lagrange multipliers. This variational formulation explicitly encodes the transfer between kinetic and potential energy in the interior as well as across the boundary, thus leading to a global power balance and ensuring passivity of the system. The particular geometric structure of the weak formulation can be preserved under Galerkin approximation via appropriate mixed finite elements. In addition, a fully discrete power balance can be obtained by appropriate time discretization. The main properties of the system and its discretization are shown theoretically and demonstrated by numerical tests.
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