Study of an alternative phase field model for low interfacial energy in elastic solids / Anke Böttcher

• Record Nr.: UNINA9910557400803321
• Autore: Böttcher Anke
• Pubbl/distr/stampa: Berlin, : Logos Verlag Berlin, 2021; [s.l.] : , : Logos Verlag Berlin, , 2021
• Descrizione fisica: 1 online resource (152 p.)
• Soggetti: Science / Chemistry; Science / Physics; Mathematics; Science
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Sommario/riassunto: In 2005, the hybrid model was published by Prof. H.-D. Alber and Prof. P. Zhu as an alternative to the Allen-Cahn model for the description of phase field transformations. With low interfacial energy, it is more efficient, since the resolution of the diffuse interface is numerically broader for the same solution accuracy and allows coarser meshing. The solutions of both models are associated with energy minimisation and in this work the error terms introduced in the earlier publications are discussed and documented using one and two dimensional numerical simulations. In the last part of this book, phase field problems, initially not coupled with material equations, are combined with linear elasticity and, after simple introductory examples, a growing martensitic inclusion is simulated and compared with literature data. In addition to the confirmed numerical advantage, another phenomenon not previously described in the literature is found: with the hybrid model, in contrast to the examples calculated with the Allen-Cahn model, an inclusion driven mainly by curvature energy does not disappear completely. The opposite problem prevents inclusions from growing from very small initial configurations, but this fact can be ; remedied by a very finely chosen diffuse interface width and by analysing and adjusting the terms that generate the modelling errors. The last example shows that the hybrid model can be used with numerical advantages despite the above mentioned peculiarities.

Geometry Through History : Euclidean, Hyperbolic, and Projective Geometries / / by Meighan I. Dillon

• Record Nr.: UNINA9910300127303321
• Autore: Dillon Meighan I
• Pubbl/distr/stampa: Cham : , : Springer International Publishing : , : Imprint : Springer, , 2018
• ISBN: 3-319-74135-7
• Edizione: [1st ed. 2018.]
• Descrizione fisica: 1 online resource (XII, 350 p. 233 illus. in color.)
• Disciplina: 519.009
• Soggetti: Convex geometry; Discrete geometry; Mathematics; History; Geometry, Hyperbolic; Geometry, Projective; Convex and Discrete Geometry; History of Mathematical Sciences; Hyperbolic Geometry; Projective Geometry
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di contenuto: Preface -- 1. The Elements of Euclid -- 2. Neutral Geometry -- 3. The Hyperbolic Plane -- 4. Hilbert's Grundlagen -- 5. More Euclidean Geometry -- 6. Models for the Hyperbolic Plane -- 7. Affine Geometry -- 8. An Introduction to Projective Geometry -- 9. Algebraic Curves -- 10. Rotations and Quaternions -- Index.
• Sommario/riassunto: Presented as an engaging discourse, this textbook invites readers to delve into the historical origins and uses of geometry. The narrative ; traces the influence of Euclid’s system of geometry, as developed in his classic text The Elements, through the Arabic period, the modern era in the West, and up to twentieth century mathematics. Axioms and proof methods used by mathematicians from those periods are explored alongside the problems in Euclidean geometry that lead to their work. Students cultivate skills applicable to much of modern mathematics through sections that integrate concepts like projective and hyperbolic geometry with representative proof-based exercises. For its sophisticated account of ancient to modern geometries, this text assumes only a year of college mathematics as it builds towards its conclusion with algebraic curves and quaternions. Euclid’s work has affected geometry for thousands of years, so this text has something to offer to anyone who wants to broaden their appreciation for the field.