Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

A Phase-Field Perspective on Mereotopology

Version 1 : Received: 4 August 2021 / Approved: 5 August 2021 / Online: 5 August 2021 (08:39:00 CEST)

How to cite: Schmitz, G.J. A Phase-Field Perspective on Mereotopology. Preprints 2021, 2021080129 (doi: 10.20944/preprints202108.0129.v1). Schmitz, G.J. A Phase-Field Perspective on Mereotopology. Preprints 2021, 2021080129 (doi: 10.20944/preprints202108.0129.v1).

Abstract

Mereotopology is a concept rooted in analytical philosophy. The phase-field concept is based on mathematical physics and finds applications in materials engineering. The two concepts seem to be disjoint at a first glance. While mereotopology qualitatively describes static relations between things like x isConnected y (topology) or x isPartOf y (mereology) by first order logic and Boolean algebra, the phase-field concept describes the geometric shape of things and its dynamic evolution by drawing on a scalar field. The geometric shape of any thing is defined by its boundaries to one or more neighboring things. The notion and description of boundaries thus provides a bridge between mereotopology and the phase-field concept. The present article aims to relate phase-field expressions describing boundaries and especially triple junctions to their Boolean counterparts in mereotopology and contact algebra. An introductory overview on mereotopology is followed by an introduction to the phase-field concept already indicating first relations to mereo- topology. Mereotopological axioms and definitions are then discussed in detail from a phase-field perspective. A dedicated section introduces and discusses further notions of the isConnected relation emerging from the phase-field perspective like isSpatiallyConnected, isTemporallyConnected, isPhysicallyConnected, isPathConnected and wasConnected. Such relations introduce dynamics and thus physics into mereotopology as transitions from isDisconnected to isPartOf can be described.

Keywords

Region based theory of space; RBTS; Contact algebra; Dyadic and Triadic relations; sequent algebra; boundaries; triple junctions; mereotopology; 4D mereotopology; mereophysics; Region Connect Calculus RCC; invariant spacetime interval; Falaco solitons; phase-field method; intuitionistic logic

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