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首页> 外文会议>IUTAM Symposium on Computational Mechanics of Solid Materials at Large Strains Aug 20-24, 2001 Stuttgart, Germany >Nonconvex Energy Minimization and Relaxation in Computational Material Science
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Nonconvex Energy Minimization and Relaxation in Computational Material Science

机译：计算材料科学中的非凸能量最小化和松弛

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摘要

The mathematical modeling of microstructures has important applications in material science, e.g. advanced materials or phase transitions, micromagnetism, homogenization, and optimization. It is typical in those models that the continuous minimization problem (M) is nonconvex and lacks classical solutions. There exist infimising sequences in (M) that have a weak limit, but non-(quasi-)convexity implies, in general, that the weak limit u is not a solution of the problem (M). In physical and in numerical experiments, we observe oscillations of strains which form a macroscopic or averaged quantity Du. The efficient numerical simulation on the macroscopic level aims to compute the weak limit M as a solution of a related Relaxed Problem. This is in contrast to microscopic mechanisms which are directly approached by a finite element minimization of (M). The two relaxations discussed are based on Young measures (G) and on quasiconvexification (Q). The presentation discusses adapted finite element strategies for relaxed problems such as the 2-well problem in one-dimension, in higher dimensions, in linearized elasticity (with hysteresis) or in related topics in micromagnetism and homogenization problems.
机译：微结构的数学建模在材料科学中具有重要的应用，例如高级材料或相变，微磁性，均质化和优化。在那些模型中，连续最小化问题（M）通常是非凸的，并且缺乏经典的解决方案。 （M）中存在有弱极限的无约束序列，但非（准）凸性通常表示，弱极限u不是问题（M）的解决方案。在物理和数值实验中，我们观察到形成宏观或平均量Du的应变的振荡。宏观层面上的有效数值模拟旨在计算弱极限M，作为相关松弛问题的解决方案。这与通过有限元素最小化（M）直接接近的微观机制相反。讨论的两个松弛是基于杨氏测度（G）和拟凸变（Q）的。该演讲讨论了针对松弛问题的适应性有限元策略，例如一维，高维，线性弹性（带滞后性）或微磁和均质化问题的相关主题中的2井问题。

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