# Phase Function Renormalization Method#

Phase Function Renormalization Method = <char_string>


## Description / Usage#

This card indicates the method to be used to renormalize the phase function fields during the course of the computation.

{char_string}

A character string which specifies the type of method for renormalization. Choices for this string are: Huygens, Huygens_Constrained, Correction.

Huygens

In this method a set of m points P is constructed:

$$\mathbf{P} = \left\{ \left( x_i, y_i, z_i \right), \, i = 1,2, \ldots m | \quad \phi_j \left( x_i, y_i, z_i \right) = 0 \right\}$$

which in a sense represent a discretization of the interface location. The finite element interpolation functions are used to find exact locations for these points. For each mesh node $$j$$, a minimum distance $$D_j$$, can be found to this set of points. Renormalization is accomplished by replacing the phase field value at this node $$\phi_j$$ with $$D_j$$ multiplied by the sign of the previous value for the phase field function. This method is fast and robust and reasonably accurate given sufficiently refined meshes using high order phase field interpolation. However, this method is prone to losing material if low order phase field interpolation is employed.

Huygens_Constrained

This method renormalizes the function in much the same way as the Huygens method, except it employs a Lagrange multiplier to enforce a global integrated constraint that requires the volume occupied by the “negative” phase to remain unchanged before and after renormalization. This requirement makes this method better at conserving mass. However, since it enforces a global constraint, it is possible that material might be moved nonphysically around the computational domain.

## Examples#

This is a sample card:

Phase Function Renormalization Method = Huygens_Constrained


## Technical Discussion#

Renormalization is an operation particular to phase function (and level set) embedded interface tracking. The phase function fields are defined originally as distanes from a known curve or surface. This type of function offers benefits in terms of smoothness of representation and the easy with which interfacial physics can be included. However, typically we are evolving these functions using the commonplace advection operator:

$\frac{D \phi_j}{D t} = 0$

which does not necessarily perpetuate the phase field as a distance function. Sharp gradients or flat regions in the function may therefore appear near the interface which have various detrimental effects on the accuracy of the solution. The solution that is most often used is to periodically construct the interfaces from the phase function field and renormalize the phase function fields, i.e. reevaluated them so that they return to being distance functions from the interface. In general, this is a satisfactory solution if the frequency of renormalization is not too great. To set the criteria for determining when to renormalize the phase functions see the Phase Function Renormalization Tolerance card.