Towards a gradient flow for microstructure
Towards a gradient flow for microstructure
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走向微观结构的梯度流
DOI：
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
S. Ta'asan
中科院分区：
文献类型：
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作者：
Patrick Bardsley;K. Barmak;Eva Eggeling;Y. Epshteyn;D. Kinderlehrer;S. Ta'asan
- 微结构的中心问题是开发能够产生多晶材料的排列或有序的技术，根据介观参数，如几何形状和晶体学，适合于给定的应用。首先，有这样的命令吗？我们的目标是描述晶界特征分布（GBCD）的出现，这是一个最近发现的详细纹理演变的统计数据，并说明为什么它应该被认为是一种材料属性。对于GBCD统计，我们已经发展了一个依赖于质量输运和熵的理论。本文的重点是它作为De Giorgi意义上的梯度流的识别，如Ambrosio，Gigli和Savaré所示。通过这种方式，经验织构统计被揭示为Fokker-Planck型方程的解，该方程的演化由弱拓扑动力学决定，并且其极限行为是玻尔兹曼分布。通过我们的方法识别为梯度流相当于将收获的统计量展示为JKO隐式方案中的迭代。这需要一些新的想法。这一发展暴露了一个问题，即如何理解在何种情况下，所获得的经验统计数据是基本过程的一个属性。
— A central problem of microstructure is to develop technologies capable of producing an arrangement, or ordering, of a polycrystalline material, in terms of mesoscopic parameters, like geometry and crystallography, appropriate for a given application. Is there such an order in the first place? Our goal is to describe the emergence of the grain boundary character distribution (GBCD), a statistic that details texture evolution discovered recently, and to illustrate why it should be considered a material property. For the GBCD statistic, we have developed a theory that relies on mass transport and entropy. The focus of this paper is its identification as a gradient flow in the sense of De Giorgi, as illustrated by Ambrosio, Gigli, and Savaré. In this way, the empirical texture statistic is revealed as a solution of a Fokker–Planck type equation whose evolution is determined by weak topology kinetics and whose limit behavior is a Boltzmann distribution. The identification as a gradient flow by our method is tantamount to exhibiting the harvested statistic as the iterates in a JKO implicit scheme. This requires several new ideas. The development exposes the question of how to understand the circumstances under which a harvested empirical statistic is a property of the underlying process.