Partial Differential Equations — Towards a gradient flow for microstructure

Research output: Contribution to journalArticleResearchpeer-review

Abstract

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.
Original languageEnglish
Pages (from-to)777-805
JournalRendiconti Lincei / Matematica e Applicazioni
Volume28
Issue number4
DOIs
Publication statusPublished - 2017

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Gradient Flow
Statistic
Microstructure
Partial differential equation
Grain Boundary
Texture
Limit Behavior
Mass Transport
Weak Topology
Implicit Scheme
Ludwig Boltzmann
Iterate
Material Properties
Evolution Equation
Arrangement
Kinetics
Entropy

Keywords

    Fields of Expertise

    • Information, Communication & Computing

    Cite this

    Partial Differential Equations — Towards a gradient flow for microstructure. / Eggeling, Eva.

    In: Rendiconti Lincei / Matematica e Applicazioni, Vol. 28, No. 4, 2017, p. 777-805.

    Research output: Contribution to journalArticleResearchpeer-review

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