Variational time integrators in computational solid mechanics
Variational time integrators in computational solid mechanics
复制标题
计算固体力学中的变分时间积分器
DOI：
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发表时间：
2003
期刊：
影响因子：
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通讯作者：
A. Lew
中科院分区：
文献类型：
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作者：
A. Lew
本文发展了计算固体力学问题的变分积分的理论和实现，在一定程度上也适用于流体力学问题。本文简要地回顾了有限维机械系统的变分积分器，并将其作为扩展到连续统系统的基础。后者是通过拉格朗日连续介质力学的时空公式来实现的，该公式将线性动量、能量和构型力的平衡的推导统一起来，所有这些都是扩展汉密尔顿原理的欧拉-拉格朗日方程。在这个公式中，能量守恒和J-积分和l -积分的路径无关性是由诺特定理产生的守恒量。通过模拟这种变分结构，构造了连续介质力学的变分积分器，并给出了适用于一般时空离散化的离散Noether定理。此外，算法是自动（多）辛的，并且（多）辛形式是由理论唯一定义的。例如，在非线性弹性动力学中，只要连续系统保持线性和角动量，算法就能准确地保持线性和角动量。
This thesis develops the theory and implementation of variational integrators for computational solid mechanics problems, and to some extent, for fluid mechanics problems as well. Variational integrators for finite dimensional mechanical systems are succinctly reviewed, and used as the foundations for the extension to continuum systems. The latter is accomplished by way of a space-time formulation for Lagrangian continuum mechanics that unifies the derivation of the balance of linear momentum, energy and configurational forces, all of them as Euler-Lagrange equations of an extended Hamilton's principle. In this formulation, energy conservation and the path independence of the J- and L-integrals are conserved quantities emanating from Noether's theorem. Variational integrators for continuum mechanics are constructed by mimicking this variational structure, and a discrete Noether's theorem for rather general space-time discretizations is presented. Additionally, the algorithms are automatically (multi)symplectic, and the (multi)symplectic form is uniquely defined by the theory. For instance, in nonlinear elastodynamics the algorithms exactly preserve linear and angular momenta, whenever the continuous system does.
A class of variational algorithms is constructed, termed asynchronous variational integrators (AVI), which permit the selection of independent time steps in each element of a finite element mesh, and the local time steps need not bear an integral relation to each other. The conservation properties of both synchronous and asynchronous variational integrators are discussed in detail. In particular, AVI are found to nearly conserve energy both locally and globally, a distinguishing feature of variational integrators. The possibility of adapting the elemental time step to exactly satisfy the local energy balance equation, obtained from the extended variational principle, is analyzed. The AVI are also extended to include dissipative systems. The excellent accuracy, conservation and convergence characteristics of AVI are demonstrated via selected numerical examples, both for conservative and dissipative systems. In these tests AVI are found to result in substantial speedups, at equal accuracy, relative to explicit Newmark.
In elastostatics, the variational structure leads to the formulation of discrete path-independent integrals and a characterization of the configurational forces acting in discrete systems. A notable example is a discrete, path-independent J-integral at the tip of a crack in a finite element mesh.