1. Modeling Phase Changes in Finite Element Analysis
Colin Smith, Robin Roberts, Will Horwath

2. Introduction Why model phase changes numerically? 3 methods
Front-Tracking Methods
Enthalpy Methods
Phase Field Methods

3. Front Tracking
Combines a fixed global grid with local front calculations
Location of the front transition is directly calculated
Local information passed out to global grid after front is moved
[1] G. Tryggvason et. al.

4. Uses the Navier-Stokes equations with a term for the location of the front, valid over the whole domain
Discretized is based off of the type of problem, in our case terms are added to account for latent heat of fusion
Gas expansion, heat flux, and mass transfer rate all influence the front movement, and are then fully account for by the movement of the front

5. Pros Cons Works for applications with bulk movements
Can handled a large number of situations
Cons
Must be tailored for each situation
Computationally expensive
Can only be used for simulations with zero thickness transitions
[1] G. Tryggvason et. al.

6. Enthalpy Methods - Introduction
Fixed grid method
Based on the energy equation in terms of enthalpy
Best for pure phase change problems
Solidification in casting, phases in heat treating, etc.

7. Enthalpy Methods - Introduction
Pros
No need for front-tracking
Relatively simple and fast
Allows “mushy” zone of mixed phases between pure phases
Phase change front is backed out of calculations
Cons
Not as flexible: can’t account for
Liquid flow
Material stress
Crack formation/propagation

8. Enthalpy Methods - Derivation
Simple 1-D example
Method by Kim et. al. [8]
Heat conduction equation and specific enthalpy yield the following equation:
Where:
and: f = mass fraction of liquid
Apply conditions at t = 0
Solid slab of metal, beginning as fully liquid
Left end is exposed to a constant cold temperature
Right end is exposed to constant cold convection

9. Enthalpy Methods - Sample Results
Kim et. al. [8]

10. Phase Field Balances free energy
The transition zone is characterized by a phase parameter
Truex et. al.

11. Phase Field Advantage Can model discontinuities
Interface can evolve on its own
Versatility and multiple fields
Disadvantage
Interfaces cannot spontaneously generate
Must know the free energy density function

12. Phase Field - Applications
Fallah et. al. [14]

13. Questions?

14. References
[1] G. Tryggvason, B. Bunner, A. Esmaeeli, D. Juric, N. Al-Rawahi, W. Tauber, J. Han, S. Nas, Y.-J. Jan, A Front-Tracking Method for the Computations of Multiphase Flow, Journal of Computational Physics, Volume 169, Issue 2, 2001
[2] S. O. Unverdi, G. Tryggvason, A front-tracking method for viscous, incompressible, multi-fluid flows, Journal of Computational Physics, Volume 100, Issue 1, 1992, Pages 25-37
[3] K. Morgan, A numerical analysis of freezing and melting with convection, Computer Methods in Applied Mechanics and Engineering, Volume 28, Issue 3, 1981, Pages
[4] D. Juric, G. Tryggvason, A Front-Tracking Method for Dendritic Solidification, Journal of Computational Physics, Volume 123, Issue 1, 1996, Pages
[5] S. O. Unverdi, G. Tryggvason, Computations of multi-fluid flows, Physica D: Nonlinear Phenomena, Volume 60, Issues 1–4, 1992, Pages 70-83
[6] H.S Udaykumar, R Mittal, Wei Shyy, Computation of Solid–Liquid Phase Fronts in the Sharp Interface Limit on Fixed Grids, Journal of Computational Physics, Volume 153, Issue 2, 1999, Pages
[7] B. Nedjar, An enthalpy-based finite element method for nonlinear heat problems involving phase change, Computers and Structures, Volume 80, 2002, Pages 9-21

15. References
[8] S. Kim, M. C. Kim, W. Chun. A fixed grid finite control volume model for the phase change heat conduction problems with a single-point predictor-corrector algorithm, Korean Journal of Chemical Engineering, Volume 18, Issue 1, 2001 Pages 40-55
[9] V. R. Voller, C. R. Swaminathan, B. G. Thomas, Fixed Grid Techniques for Phase Change Problems: A Review, International Journal for Numerical Methods in Engineering, Volume 30, 1990, Pages
[10] The Chemical Basis of Morphogenesis A. M. Turing Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, Volume 237, No. 641, 1952, Pages 37-72
[11] S. Nagaraja, Phase-field modeling of brittle fracture with multi-level hp-FEM and the Finite Cell Method, Technische Universitaet Braunschweig, 2017, Braunschweig, Germany
[12] M. P. Gururajan, (2018, March 6). Overview of phase field modelling. Lecture. Retrieved from
[13] M. Truex, Numerical Simulation of Liquid-Solid, Solid-Liquid Phase Change Using Finite Element Method in h,p,k Framework with Space-Time Variationally Consistent Integral Forms, University of Kansas, 2010, Lawrence, Kansas, USA
[14] Fallah V., Amoorezaei M., Provatas N., Corbin S. F., Khajepour A., Phase-field simulation of solidification morphology in laser powder deposition of Ti-Nb alloys. Acta Materialia, Volume 60, 2012, Pages ,