论文标题
在几何非线性弹性动力学及其结构保存离散化的速度压力公式上
On the velocity-stress formulation for geometrically nonlinear elastodynamics and its structure-preserving discretization
论文作者
论文摘要
我们考虑弹性连续体的动力学在大变形下但小应变。这种系统可以通过几何非线性弹性动力学的方程与圣威ant-Kirchhoff材料定律结合使用。事实证明，问题的速度应力表述具有正式的哈米尔顿港结构。与线性情况相反，问题的运算符是由位移场调节的，可以将其作为被动变量处理并与速度一起集成。得出了该问题的薄弱表述，并通过拉格朗日乘数纳入必要的边界条件。这种变分的公式明确编码了内部和跨边界的动力学和势能之间的传递，从而导致了全球功率平衡并确保系统的消极性。弱制剂的特定几何结构可以通过适当的混合有限元元素在盖林近似下保存。另外，可以通过适当的时间离散化获得完全离散的功率平衡。从理论上显示了系统及其离散化的主要特性，并通过数值测试证明。
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.
