论文标题
具有非凸能的速率独立系统的自适应时间步进方案
An Adaptive Time Stepping Scheme for Rate-Independent Systems with Non-Convex Energy
论文作者
论文摘要
我们研究了一个局部的增量固定方案，用于与速率无关系统的数值解。这样的系统的特征是（可能）非凸的能量和耗散电位，这是一级均匀的。由于能量的非转化性，系统通常不承认及时的解决方案。为了解决这些潜在的不连续性，该算法会产生一系列状态变量和物理时间点作为曲线参数的函数。与现有方法相比，我们的方法的主要新颖性是，根据所谓的局部稳定性条件的互补关系，根据对残基的规定公差以及在互补性关系中对曲线参数更新的自适应选择。事实证明，对于倾向于零的耐受性，算法收敛（弱）生成的分段仿射近似值（弱）与所谓的$ \ mathbb {v} $ - 参数化平衡粘度解决方案。数值实验说明了理论发现，并表明，步骤尺寸的自适应选择确实会导致在粘附和粘性跳跃过程中的步长显着增加。
We investigate a local incremental stationary scheme for the numerical solution of rate-independent systems. Such systems are characterized by a (possibly) non-convex energy and a dissipation potential, which is positively homogeneous of degree one. Due to the non-convexity of the energy, the system does in general not admit a time-continuous solution. In order to resolve these potential discontinuities, the algorithm produces a sequence of state variables and physical time points as functions of a curve parameter. The main novelty of our approach in comparison to existing methods is an adaptive choice of the step size for the update of the curve parameter depending on a prescribed tolerance for the residua in the energy-dissipation balance and in a complementarity relation concerning the so-called local stability condition. It is proven that, for tolerance tending to zero, the piecewise affine approximations generated by the algorithm converge (weakly) to a so-called $\mathbb{V}$-parametrized balanced viscosity solution. Numerical experiments illustrate the theoretical findings and show that an adaptive choice of the step size indeed pays off as they lead to a significant increase of the step size during sticking and in viscous jumps.
