Date of Award

2026-05-01

Degree Name

Master of Science

Department

Computational Science

Advisor(s)

Natasha S. Sharma

Abstract

The Phase Field Crystal (PFC) model is a continuum-based framework used to study theevolution of crystalline materials while preserving microscopic structural features over diffusive time scales. It captures complex phenomena such as phase transitions, defect dynamics, and microstructure formation via a nonlinear sixth-order partial differential equation derived from a free-energy functional. In this context, the phrase "sixth-order" indicates that the highest spatial derivative appearing in the equation is of order six. In this work, we study the mathematical formulation and numerical approximation of solutions to the PFC model. Due to the high-order spatial derivatives and nonlinearity in the governing partial differential equation, direct numerical treatment is computationally challenging. In 2007, Backofen et al. introduced an early finite element approach to the PFC equation by recasting it as a system of three coupled second-order equations and employing a semi-implicit time discretization. Although this scheme successfully simulated crystalline nucleation and growth, critical theoretical aspects - including well-posedness and energy dissipation remained unaddressed. This thesis examines these fundamental properties and establishes the specific conditions under which they hold. By detailing rigorous proofs of these key properties, we complete the numerical analysis of this pioneering numerical scheme. Numerical experiments are presented to validate the theoretical results and to demonstrate the effectiveness of the proposed methods in capturing the evolution of crystalline patterns. The results confirm that the schemes accurately reproduce the qualitative and quantitative behavior of the PFC model. Overall, this work contributes to establishing a rigorous foundation for the long-term simulation of complex microstructural evolution in crystalline materials using easy-to-compute, linear, and efficient numerical methods developed for high-order nonlinear parabolic partial differential equations.

Language

en

Provenance

Received from ProQuest

File Size

63 p.

File Format

application/pdf

Rights Holder

Patrick Ameyaw Tabiri

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