The Cahn-Hilliard Equations: Modeling Approaches
(2024) In Master's Theses in Mathematical Sciences NUMM03 20241Mathematics (Faculty of Engineering)
Mathematical Statistics
Mathematics (Faculty of Sciences)
Centre for Mathematical Sciences
- Abstract
- In this thesis, we study modeling approaches for the Cahn-Hilliard equations utilizing the Eyre approximation for first and second order in time accuracy. We first considered the Cahn-Hilliard equations and then expand our study by adding the Navier-Stokes coupling. We begin with a study concerning preconditioners, grid size options, mobility functions, energy density discretization, and time-stepping methods for the Cahn-Hilliard equations. The performance evaluation is determined by computation time and graphical representations of simulations. Here, we found that a simple Eyre approximation with a constant mobility and AMG preconditioning was sufficient. Then, we generalize the approach for a diffusive binary droplet problem for usage... (More)
- In this thesis, we study modeling approaches for the Cahn-Hilliard equations utilizing the Eyre approximation for first and second order in time accuracy. We first considered the Cahn-Hilliard equations and then expand our study by adding the Navier-Stokes coupling. We begin with a study concerning preconditioners, grid size options, mobility functions, energy density discretization, and time-stepping methods for the Cahn-Hilliard equations. The performance evaluation is determined by computation time and graphical representations of simulations. Here, we found that a simple Eyre approximation with a constant mobility and AMG preconditioning was sufficient. Then, we generalize the approach for a diffusive binary droplet problem for usage through a coupling scheme with the Navier-Stokes equations. A visual and CPU time comparison is then performed from a problem in the literature. We find that the Eyre approximation with first order in time accuracy performs better in computation time, but that there are some minor graphical differences in the simulation. Overall the simple Eyre approximation with implicit Euler time-stepping performed better in computation time. We also conclude that further research needs to be considered on the implicit-explicit discretization formulation for the energy density discretization for proper optimization. (Less)
- Popular Abstract
- Imagine a situation where you are at a party and pour some food coloring into a glass of water. You will likely see that the liquids do not initially mix. If you take a photo of that instance and zoom in close enough to where they separate, you will likely observe that there is a small interface. By zooming in sufficiently closely, you will see a continuous interface with water on one side, a mixture in the middle, and food coloring on the other. You may see a perfect mixture by taking another photo of the glass after some time has passed. To model this process mathematically, we refer to some specific equations to describe this process. The so-called Navier-Stokes equations may describe the flow of the liquids. In contrast, the... (More)
- Imagine a situation where you are at a party and pour some food coloring into a glass of water. You will likely see that the liquids do not initially mix. If you take a photo of that instance and zoom in close enough to where they separate, you will likely observe that there is a small interface. By zooming in sufficiently closely, you will see a continuous interface with water on one side, a mixture in the middle, and food coloring on the other. You may see a perfect mixture by taking another photo of the glass after some time has passed. To model this process mathematically, we refer to some specific equations to describe this process. The so-called Navier-Stokes equations may describe the flow of the liquids. In contrast, the Cahn-Hilliard equations can describe the interface's evolution.
Suppose you wish to tell a friend that the mixture will look a particular way after a specific time as a party trick. One way would be to implement this problem with a numerical model. Then, it would be best to have some modeling assumptions to describe the mixture's evolution. But to model this quickly and accurately on a computer, you must also consider how to do so optimally. As such, you need appropriate modeling approaches to tackle this issue. But how do you go about it when there are so many options?
Perhaps you search the Internet for a solution or learn the best way to do it from experience. For the latter, you write a general script on your computer that encompasses some solver and allows some free parameters to be inserted to model any problem from some governing equations. If you do this for a long time and document the test cases, you can eventually create a reference manual to guide you through further party tricks. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/student-papers/record/9174960
- author
- Gunnarsson, Jimmy LU
- supervisor
- organization
- course
- NUMM03 20241
- year
- 2024
- type
- H2 - Master's Degree (Two Years)
- subject
- keywords
- Cahn-Hilliard, Preconditioners, Numerical modeling, Eyre approximation, Cahn-Hilliard-Navier-Stokes
- publication/series
- Master's Theses in Mathematical Sciences
- report number
- LUNFNA-3043-2024
- ISSN
- 1404-6342
- other publication id
- 2024:E69
- language
- English
- id
- 9174960
- date added to LUP
- 2025-04-02 18:10:50
- date last changed
- 2025-04-02 18:10:50
@misc{9174960,
abstract = {{In this thesis, we study modeling approaches for the Cahn-Hilliard equations utilizing the Eyre approximation for first and second order in time accuracy. We first considered the Cahn-Hilliard equations and then expand our study by adding the Navier-Stokes coupling. We begin with a study concerning preconditioners, grid size options, mobility functions, energy density discretization, and time-stepping methods for the Cahn-Hilliard equations. The performance evaluation is determined by computation time and graphical representations of simulations. Here, we found that a simple Eyre approximation with a constant mobility and AMG preconditioning was sufficient. Then, we generalize the approach for a diffusive binary droplet problem for usage through a coupling scheme with the Navier-Stokes equations. A visual and CPU time comparison is then performed from a problem in the literature. We find that the Eyre approximation with first order in time accuracy performs better in computation time, but that there are some minor graphical differences in the simulation. Overall the simple Eyre approximation with implicit Euler time-stepping performed better in computation time. We also conclude that further research needs to be considered on the implicit-explicit discretization formulation for the energy density discretization for proper optimization.}},
author = {{Gunnarsson, Jimmy}},
issn = {{1404-6342}},
language = {{eng}},
note = {{Student Paper}},
series = {{Master's Theses in Mathematical Sciences}},
title = {{The Cahn-Hilliard Equations: Modeling Approaches}},
year = {{2024}},
}