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

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

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.

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

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.

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.

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

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

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

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

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

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

Questions?

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 275-284 [4] D. Juric, G. Tryggvason, A Front-Tracking Method for Dendritic Solidification, Journal of Computational Physics, Volume 123, Issue 1, 1996, Pages 127-148 [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 535-574 [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

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 875-898 [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 https://www.youtube.com/watch?v=GmUUub54Je4 [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 1633-1646,