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Advanced Scientific Computing

ITAPS: Frontier-Lite
X. Li, Z. Xu, and B. Fix

A general purpose software package for geometry and interface dynamics has been extracted from the FronTier code and is now publicly available.

This package, called FronTier-Lite [1], is designed for users with little training in the algorithms used in the front tracking method and yet deciding to apply the high quality front tracking method to various scientific problems with dynamic front propagation. This code is downloadable from the web, and is accompanied by a web-based testing and evaluation site and extensive web-based documentation.

The software is organized in three levels. The first level deals with the static manifolds and geometry. This includes functions to initialize the interface, optimize an existing interface, and calculate geometry-dependent variables.

The dynamic front package provides a set of functions for the propagation of the interface in a given velocity field and with geometry-dependent velocity functions. The driver function advance_front() contains preset pointers to functions based on dimension and algorithms selected by the user. This function will also detect the geometrical and topological correctness of the interface after the propagation. When necessary, it will perform a sequence of operations to guarantee the soundness and quality of the interface before the next propagation step.

This front tracking method has been compared with other interface methods on some benchmark problems. In Figure 1, we compare with the level set method. We used the fifth order WENO scheme for the convection of the level set function, while for the front tracking code, we used the fourth order Runge-Kutta method for the point propagation. After 13 revolutions, the fourth order Runge-Kutta method appears to be extremely accurate in the front tracking simulation, while the level set computation begins to show edge smoothing after the second rotation. At the end of the 13th circulation, the slot is closed at the top, resulting in a topologically incorrect bifurcation.

Comparison with the volume of fluid method also showed high quality for the front tracking code. One of the benchmark tests is the three-dimensional deformation velocity field described by the velocity functions

u(x,y,z) = 2sin² (∏x) sin(2∏y) sin(2∏z) cos(∏t/T)                     (1)
v(x,y,z) = -sin(2∏x) sin²(∏y) sin(2∏z) cos(∏t/T)                      (2)
w(x,y,z) = -sin(2∏x) sin(2∏y) sin²(∏z) cos(∏t/T).                    (3)


Figure 1. Comparison of slotted disk simulation using high order methods. The upper sequence shows the result of the level set method using the fifth order WENO scheme, and the lower sequence shows the result of front tracking using the fourth order Runge-Kutta method.	Figure 2. Reversal test of a 3D interface in deformation velocity field with CFL = 0.5. The sequence above has the mesh of 64³, and the sequence below has the mesh of 128³. From left to right are t=0, 1.5, 3 respectively.

The interface evolves dynamically from an initial sphere of radius 0.15 centered at (0.35, 0.35, 0.35) to t=1.5. The velocity field will then reverse its direction. At t=3.0, the interface comes back to its initial state. The error comparison with the two PLIC methods is given in Table 1, and shows superior performance for LGB Front Tracking.

Mesh	LGB	Order	CVTNA	Youngs
32³	5.72 x 10^-3	3.72	7.41 x 10^-3	7.71 x 10^-3
64³	4.33 x 10^-4	1.82	1.99 x 10^-3	2.78 x 10^-3
128³	1.23 x 10^-4	N/A	3.09 x 10^-4	7.58 x 10^-4

Table 1. L₁ norms at t = .3 for the LGB method in the three-dimensional deformation simulation compared to the two interface methods used in [8] with CFL = 0.5.

The third level of the front tracking code includes applications to the physical problems, especially the CFD code. The code has recently been used to achieve agreement to experimental value of α, the mixing rates for the Rayleigh-Taylor instability [2,3]. The code can interoperate with the LLNL code Overture, which provides adaptive mesh refinement (AMR).

The locally-grid based tracking (LGB) provides robust and accurate resolution for the interface geometry. Conservative tracking [4,5,6] has been developed in research code and will be included in a future release. It gives higher accuracy for both the interior and the front.

References

[1] Du, J., Fix, B., Glimm, J., Li, X., Li, Y., and Wu, L. A simple package for front tracking. J. Comp. Phys, 213(2): 613-628 (2006). Stony Brook University preprint SUNYSB-AMS-05-02.
[2] George, E. and Glimm, J. Self similarity of Rayleigh-Taylor mixing rates. Phys. Fluids 7: 054101-1 – 054101-13 (2005). Stony Brook University preprint SUNYSB-AMS-04-05.
[3] George, E., Glimm, J., Li, X.L., Li, Y.H., and Liu, X.F. The influence of scale-breaking phenomena on turbulent mixing rates. Phys. Rev. E 73: 016304-1 – 016304-5, 2006. Stony Brook University preprint SUNYSB-AMS-05-11.
[4] Glimm, J., Li, X.L., and Liu, Y.J. Conservative front tracking with improved accuracy. SIAM J. Numerical Analysis 41, 1926-1947, 2003.
[5] Liovic, P., Rudman, M., Liow, J.-L., Lakehal, D., and Kothe, D. A 3d unsplit-advection volume tracking algorithm with planarity-preserving interface reconstruction. Computers and Fluids, submitted, 2005.
[6] Liu, J.-J., Glimm, J., and Li, X.L. A conservative front-tracking method. In Proceedings of the Tenth International Conference on Hyperbolic Problems: Theory, Numerics, and Applications, Yokohama Publishers, Osaka, Japan, in press, 2005.

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Last Modified: January 31, 2008
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