Level Set Renormalization Method

Level Set Renormalization Method = {char_string}

Description / Usage

This card indicates the method to be used to renormalize the level set function during the course of the computation. The syntax of this card is as follows:

{char_string}

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

Each method is described below; see also the Technical Discussion.

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 level set value at this node \(\phi_j\) with \(D_j\) multiplied by the sign of the previous value for the level set function. This method is fast and robust and reasonably accurate given sufficiently refined meshes using high order level set interpolation. However, this method is prone to losing material if low order level set 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 renormalization method input card:

Level Set Renormalization Method = Huygens_Constrained

Technical Discussion

Renormalization is an operation particular to level set embedded interface tracking. The level set function \(\phi\) is usually specified in terms of a signed distance to the interface. This type of function has very nice properties in terms of smoothness and a unitary gradient magnitude in the vicinity of the interface. All of which are beneficial in accurately integrating the function and applying interfacial physics such as surface tension. The difficulty appears because of the velocity field \(\underline{u}\) used to evolve the level set function via the relation:

\[\frac{\partial \phi}{\partial t} + \underline{u} \cdot \nabla \phi = 0\]

There is nothing that requires that this velocity preserve the level set function as a distance function during its evolution. As a result, large gradients in the level set function might appear that would degrade the accuracy of both its time evolution and the accuracy of the interfacial terms related to the level set function. To remedy this problem, periodically the level set function must be reconstructed as a distance function; this process is referred to as renormalization. The criteria for determining when renormalization should occur is discussed under Level Set Renormalization Tolerance.