Modeling Phase Changes in Finite Element Analysis

By Colin Smith, Robin Roberts, and Will Horwath

Introduction

In many applications, it is desirable to mathematically predict or describe a material

changing phase. Manufacturing processes like casting may be optimized by understanding the

solidification process, and heat-treating metals could be more accurately predicted without

extensive testing beforehand. However, these problems are mathematically complex systems

using partial differential equations with initial value problems that are difficult or impossible to

solve analytically. This necessitates the creation of numerical methods to solve these problems.

Traditional methods in finite element analysis are unable to predict the behavior of systems with

phase changes, so special methods must be formulated.

As with most fields of numerical methods, there are many approaches to take in

formulating the governing equations and relationships that describe the system. While there are

no strict delineations between methods, they can be broadly separated into the categories of

front tracking methods, enthalpy methods, and phase field methods. Front tracking methods

rely on mathematically controlling and tracking the transition front between the two phases.

Enthalpy methods eschew the front tracking to track enthalpy on a fixed grid. This simplifies the

calculations but is not as flexible as the other two methods. Phase field methods can be used

for many different types of problems and are based around minimizing free energy in a system.

Front tracking

Front tracking is a method for simulating the interaction between two regions with

different bulk material properties with a sharp boundary between the two.  It can handle a wide

range of situations but can be complex and computationally expensive. Unlike in the other

methods discussed, front-tracking methods use the current state conditions to move the front

before updating the global grid, rather capturing the information of grid to determine the location

of the boundary after the calculations.

At its most basic, the procedures for a front tracking model are:

1) using the current state, move the boundary to a new location

2) update the states of the local front tracking mesh based off the new front location

3) update the global mesh

Each of these steps can be extremely complicated on its own, and in some variations of the

method there will be multiple iterations within this procedure before moving on to the next time

step.

Whether front tracking is used to track a difference in substance or phase, the equations

used to locate the front are defined using a Heaviside step function composed of a delta

function for each dimension the problem is defined over. For example, the density can be

defined as

∇ ρ(x , y , t)=ρiH (x , y , t)+ρo(1−H (x , y , t))

Where H is the Heaviside function defining the front, ρiis the density inside the boundary and ρ0

is the density outside the boundary [1].

In the context of a sharp boundary, the Navier-Stokes equations governing the model

are as follows and are valid across the whole region.

∂ ρu∂t

+∇ ∙ ρuu=−∇P+ρf +∇ ∙ μ (∇u+∇T u )+∫σ κ ' n'δ β (x−x' )d s '

Whereρ=density field, μ=viscosity field, u=velocity vector, f= body force vectorδ β=beta dimension function, κ=curvature, n=unit normal vector, x=point where evaluated, Primed variables are points of the front

From the region wide, continuous Navier-Stokes equations, various methods are used to

discretize the equations for use on the fixed background grid and to the local area which directly

affects the front.

Figure 1 Visualization of a front on a fixed grid [1]

The front itself is defined by a series of points arranged into elements of 1 less

dimension than the simulation. All elements on a given front have a defined inside and outside

and must be align in the same direction. In each time step of the model, the front is moved

based on the conditions of the current state, and then the properties of the fixed grid are

updated based on the new grid position. For the most part, information passes from the front to

the fixed grid, and then only to the fixed grid points closest to the front [2]

During the course of moving the front, the elements will change size and over time can

leave the optimal range of size or orientation. In general, 2d elements should be around ¼ to ½

of the size of the fixed grid, and 3d elements should have an edge length of 1/3 to 1x the grid

size. If the elements are too large, detail will be lost, however if they are too small relative to the

grid the transfer of information between the grid and front becomes less accurate and can result

in spurious “wiggles” in the front [1]

The method describe so far can be used for any general front-tracking problem. For

tracking phase changes specifically, the velocity of the front is controlled both by the velocity of

the fluids and the forces induced by the phase change. In a simulation involving transitions

between liquid and gas, both the liquid and the vapor are considered incompressible in the local

region, and the change in volume and pressure that results from transition are accounted for in

changes in the velocity of the boundary movement. Special care must be taken for regions

where 2 or more fronts near each other either by folding or elongation, specific code is need to

handle how the fronts will interact, combine, or split [3].

Figure 2: Simulation of a liquid boiling and forming an unstable bubble, which eventually detaches and rises through the liquid column [1]

Enthalpy Methods

For phase change problems, sometimes it is best to track enthalpy. Unlike front

tracking, there is no need to track a moving front across which there are two models for the two

separated phases, and there is no need to track the heat flux across that front. Enthalpy

methods use a fixed grid, which can greatly simplify a problem, depending on the application.

The whole problem is posed in terms of how the energy balance between nodes on the grid

evolves over time in terms of enthalpy. Enthalpy methods introduce a smooth phase change

region instead of the sharp interface used in front tracking. This region contains a liquid mass

fraction which is calculated based on the enthalpy at the node. The material properties of the

phases are assumed to change continuously and differentiably over the phase change region.

The result is that things like the latent heat of fusion and the actual location of the phase change

front are accounted for in the calculations, so there is no need to specifically balance heat flux

across a boundary. Additionally, the phase change front does not have to be specifically

tracked, so the movement of the front is a result of the calculations instead of an integral part of

them [13].

There are many ways to formulate an enthalpy-based solution depending on the specific

application, but many of them stem from a basic heat conduction or energy equation. A simple

one-dimensional formulation is a useful example. Kim et. al. [8] performed the derivation

beginning with the heat conduction equation:

δδt

(ρh)= δδx

(k δTδx

)+S (1)

where h = specific enthalpy, T = temperature, ρ = density, k = thermal conductivity, S = internal

heat source. The equation is subject to some initial condition

T (x , t=0)=T (x ) (2)

Boundary conditions, like a forced temperature or heat flux on a boundary, are applied based on

the specifics of the problem. To determine the specific enthalpy in (1), the following is used:

h=hs+ fL+c(T−T sat ) (3)

where hs = saturated enthalpy of the solid, Tsat = saturated temperature, L = latent heat, c =

specific heat, f = mass fraction of liquid phase. The mass fraction is calculated as

f={0if h<hs:h−hsLif hs<h<hs+L:1 if hs+L<h (4)

Substituting (3) into (1) obtains the governing equation for the enthalpy model:

δδt

(ρcT )= δδx

(k δTδx

)+S− δHδt (5)

where

H= ρ(hs+ fL−cT sat ) (6)

These equations are then analyzed node-by-node as is typical for finite element methods.

Below are the results of this method in a one-dimensional example where a slab of length 5.0m

has two phase change fronts. The slab starts in purely liquid form (f=1 for all x) and is

exposed to two different cold heat reservoirs. At one end (x = 0m) the slab is exposed to a

reservoir with a temperature of 1.0°C less than the saturation temperature. On the other end (x

= 5.0m) it is exposed to cold flow at 10.0° less than the saturation temperature such that

convective heat transfer occurs.

Figure 3: The slab's temperature in time [8].

Figure 4: The slab's liquid mass fraction in time [8].

These results show the expected outcome; the slab cools much faster due to convection

at the lower temperature, so the solidification happens faster on that side.

Voller et. al. [9] provide an in-depth method of expanding the one-dimensional problem

into two and three dimensions.

Phase field

Like boundary tracking methods phase field models can be used for many different

systems, not just phase changes and are not tied to finite element analysis but can be used with

finite element analysis. Alan Turing [10] used an early phase field model to model animal stirp

and spot growth. When combined with phase field models combined finite element analysis can

model crack propagation [11]. Phase field models are useful for modeling discontinuities

because they impose artificial continuity [12].

How it works

The crux of a phase field model is the phase field parameter and how that parameter

related to the free energy equation of the system. This relation is facilitated by a system of two

nonlinear PDEs. One equation is the balance equation that comes from the physics of the

system and the other that relates the parameter to the state of the system. These equations

can be solved simultaneously or iteratively.

In “Phase-field modeling of brittle fracture with multi-level hp-FEM and the Finite Cell

Method” [11] the parameter represents full cracked and undamaged and ranges from 0

(cracked) to 1 (undamaged). Truex [13] assigned the parameter to represent solid or liquid and

it ranges from -1 (solid) to 1 (liquid). Fig b shows the how the free energy density function

related to the field parameter. Fig c is an example of a pair of equations that was used by Truex

[13]. The top equation is the physics of the phase change and the bottom equation relates the

free energy and the phase parameter.

Figure 5: An example of a phase field parameter [13].

Figure 6: The free energy in relation to the phase field parameter [13].

Figure 7: Example equations for a pure material [13].

Applications

Phase field models have the advantage that you don’t need to keep track of the

boundary between the two phases. However due to the nature of the free energy equation

having two minimums a phase field model can’t initiate a new interface, only work off an existing

one. Additionally, you must have knowledge on the free energy density function of your

application. “Phase-field simulation of solidification morphology in laser powder deposition of Ti-

Nb alloys” [15] used both a liquid-solid phase field and a concentration phase field to model

dendritic growth in Ti-Nb powder welding.

Figure 8: Experimental vs. numerical dendritic structures [14].

Conclusion

The front-tracking, enthalpy, and phase field methods have been presented. Because of

their relative strengths and weaknesses, each method is best used for different applications.

Front-tracking works well for applications with advective flow and sharp transitions in material

properties, but must be tailored to each specific situation, is computationally costly, and can only

be used in simulations where the boundary is modeled as an interface with no thickness.

Enthalpy methods are fast and simple in comparison. The equations used in these methods are

applied over the entire grid regardless of material properties and the grid stays constant through

the whole simulation. The phase front is determined retroactively from the results rather than

being an integral part of the solution, and the front can be a region of mixed phase rather than a

sharp transition. These methods are best used in pure phase change problems with no flow,

stress, or velocity. They are relatively inflexible, though, and cannot handle more complex

systems. Phase field methods have several advantages. The phase interface can evolve on its

own, and its methods are somewhat versatile. Multiple phase fields can be used at once. In the

general sense, outside of phase change, phase fields are good at modeling discontinuities by

forcing a continuous function. They have the disadvantage that they cannot generate interfaces

spontaneously. They also struggle with moving fluids.

Each method is best suited to a set of situations, so selecting which method to use

depends on a complete understanding of each method’s strengths and the conditions of the

problem.

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

[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,