Γ-limits and relaxations for rate-independent evolutionary problems
Γ-limits and relaxations for rate-independent evolutionary problems
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DOI：
10.1007/s00526-007-0119-4
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发表时间：
2008-03
影响因子：
2.1
通讯作者：
A. Mielke;T. Roubíček;U. Stefanelli
中科院分区：
文献类型：
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作者：
A. Mielke;T. Roubíček;U. Stefanelli
这项工作使用基于储能泛函和耗散距离的速率无关系统的能量公式。对于序列and，我们解决在哪些条件下解的极限q∞满足具有极限泛函and的适当极限问题的问题，其是相应的Γ-极限。我们推导了一个充分条件，称为稳定集的条件上半连续性，它对于保证 q∞ 解决极限问题至关重要。特别是，如果存在某些关节恢复序列，则该条件成立。此外，我们表明时间增量最小化问题可以用来近似解决方案。第一个例子涉及使用有限元空间的泛函的数值近似。第二个例子表明，如果产量集在 Mosco 意义上收敛，则停止和播放运算符会收敛。第三个例子涉及在极限k→ ∞ 下发展微观结构的问题，该问题在极限下可以用有效的宏观模型来描述。
This work uses the energetic formulation of rate-independent systems that is based on the stored-energy functionalsand the dissipation distance. For sequencesandwe address the question under which conditions the limitsq∞of solutionssatisfy a suitable limit problem with limit functionalsand, which are the corresponding Γ-limits. We derive a sufficient condition, calledconditional upper semi-continuity of the stable sets, which is essential to guarantee thatq∞solves the limit problem. In particular, this condition holds if certainjoint recovery sequencesexist. Moreover, we show that time-incremental minimization problems can be used to approximate the solutions. A first example involves the numerical approximation of functionals using finite-element spaces. A second example shows that the stop and the play operator converge if the yield sets converge in the sense of Mosco. The third example deals with a problem developing microstructure in the limitk→ ∞, which in the limit can be described by an effective macroscopic model.