Magnetic microstructures - a paradigm of multiscale problems
Magnetic microstructures - a paradigm of multiscale problems
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磁性微结构——多尺度问题的范例
DOI：
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发表时间：
1999
期刊：
影响因子：
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通讯作者：
F. Otto
中科院分区：
文献类型：
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作者：
A. DeSimone;R. Kohn;S. Müller;F. Otto
铁磁材料在m到几nm的尺度上表现出复杂的畴壁、布洛赫线和奇异点的微观结构，了解这些结构的形成和整体效果对关键技术应用至关重要，同时丰富的实验数据来源和简单的数学公式使磁性微观结构的分析成为一个很好的模型问题，可以开发新的数学工具来理解多尺度在本文中，我们描述了一些基本的数学问题，并报告了在三个领域的最新分析进展：严格的标度定律，分支和降维理论。能量功能铁磁材料表现出复杂的微观结构，磁畴，壁，布洛赫线和奇点范围从m到几nm甚至更大，见图同时，丰富的实验数据来源和简单的数学公式使磁性微观结构的分析成为一个极好的模型问题，可以开发新的数学工具来理解科学中普遍存在的多尺度问题。有些令人惊讶的是，磁性结构的巨大多样性通常可以通过最小化一个简单的能量泛函来理解，它只涉及两种材料设R表示磁性体，m R表示单位体积的磁化强度，在无量纲变量中，与磁化强度相关的能量为Br HS E m w Z jrmj Q Z m Z
Ferromagnetic materials display a complex microstructure of domains walls Bloch lines and singular points on scales ranging from m down to a few nm Understanding the formation and overall e ects of these structures is crucial for key technological applications At the same time the rich source of experimental data and the simple mathematical formulation makes the analysis of magnetic microstructure an excellent model problem to develop new mathematical tools for the understanding of multiscale problems which are ubiquitous in science In this paper we describe some basic mathematical problems and re port on recent analytical progress in three areas rigorous scaling laws branching and dimensionally reduced theories for thin lms The energy functional Ferromagnetic materials display a complex microstructure of magnetic domains walls Bloch lines and singular points ranging from m down to a few nm and beyond see Fig Understanding the formation and the overall e ects of these structures is crucial for a number of key technology applications At the same time the rich source of experimental data and the simple mathematical formulation makes the analysis of magnetic microstructures an excellent model problem to develop new mathematical tools for the understanding of multiscale problems which are ubiquitous in science Somewhat surprisingly the huge variety of magnetic structures can often be understood through minimisation of a simple energy functional which only involves two material parameters Let R represent a magnetic body and let m R denote its magnetisation per unit volume In non dimensional variables the energy associated to such a magnetisation is given as Br HS E m w Z jrmj Q Z m Z