Nonlinear analysis and numerical investigations of orientational effects in Swift-Hohenberg dynamics

• Ano de defesa: 2020
• Autor(a) principal: Coelho, Daniel Lessa
• Tipo de documento: Dissertação
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Universidade do Estado do Rio de Janeiro; Centro de Tecnologia e Ciências::Faculdade de Engenharia; Brasil; UERJ; Programa de Pós-Graduação em Engenharia Mecânica
• Palavras-chave em Português: Mechanical engineering; Pattern formation (Physical Sciences); Numerical analysis; Finite element method; Swift-Hohenberg equation; Pseudo-spectral methods; Engenharia mecânica; Formação de padrões (Ciências físicas); Análise numérica; Método dos elementos finitos; Equação de Swift-Hohenberg; Métodos pseudo-espectrais; ENGENHARIAS::ENGENHARIA MECANICA::FENOMENOS DE TRANSPORTE
• Link de acesso: http://www.bdtd.uerj.br/handle/1/16778

Spatio-temporal pattern formation in natural systems originates from rich nonlinear dynamics, which may lead to the emergence of periodic nonequilibrium structures. One of the most successful equations currently available for theoretically investigating the behavior of these structures is the Swift-Hohenberg (SH), which contains a bifurcation parameter (forcing) that controls the dynamics by changing the energy landscape of the system. Though a large part of the literature on pattern formation addresses uniformly forced systems, nonuniform forcings are also observed in several natural systems, for instance, in developmental biology and in soft matter applications. In these cases, an orientation effect due to a forcing gradient is a new factor playing a role in the development of patterns, particularly in the class of stripe patterns, which we investigate through the nonuniformly forced SH dynamics. The present work addresses the stability of stripes orientation, and the competition between the orientation effect of the gradient and other bulk, boundary, and geometric effects taking part in the selection of the emerging patterns. A weakly nonlinear analysis shows that stripes tend to align with the gradient, and become unstable when perpendicular to the preferred direction. This analysis is complemented by a numerical work that accounts for other competing effects. The adopted numerical approach consists of a semi-implicit finite-difference scheme with second order accuracy in both time and space, which is successfully reviewed and extended for the quadratic-cubic (SH23) and cubic-quintic (SH35) equations. Simulations show that stripes align, or even reorient from preexisting conditions. However, we observe that this orientation effect does not always prevail in face of further competing effects.