## Volume Fraction Calculation

Mesh based analysis programs, such as deterministic transport codes, require the material composition in each mesh cell. When the mesh overlies a combinatorial geometry model such as MCNP and solid body geometries, the Moritz volume fraction feature computes the relative amount, or fraction, of each combinatorial material in a mesh cell. These fractions can then be converted (outside of Moritz) to the material input syntax required by the mesh based code.

The volume fraction calculation is a numerical integration based on ray tracing across the mesh cell. It works in both rectangular and cylindrical aligned meshes. For rectangular meshes, the rays are parallel to the X, Y, and/or Z axes; 1, 2, or 3 directions can be chosen. For cylindrical meshes, only the radial direction is used for calculating the fractions. The volume fraction calculations are controlled from the Volume Fraction dialog.

The faction of material M for rays in one direction is

where the sum is over all rays, L

_{Mj}is the path length in material M for ray j, J is the total number of rays, and the ray length D is the same for all rays in the same direction. When multiple directions are used, the fractions are averaged over directions with equal weighting.

Volume Fraction Rays. The rays are started at the center of a uniform N x N grid across a mesh cell face. The figure at the right shows the rays used for N = 6 in each direction in a 3D orthographic projection. The mesh cell is outlined in green. The rays orthogonal to the page cross the page at the intersection of the visible rays.

For cylindrical mesh cells, a N x N grid is created in a rectangle perpendicular to the mesh axis that tightly bounds the axial face of the mesh cell. Each grid position is tested for intersection with the cell. The fractions of grid positions that intersect the cell is used to define a new grid spacing N’ so that approximately N^{2}rays intersect the mesh cell.

Single Material Test. One of three methods can be used, optionally preceded by a single material test. The single material test also computes fractions by ray tracing, usually with a smaller grid size N than is used in the main fraction calculation. The ray directions are independent of the directions for the main calculation. For rectangular meshes, any combination of X, Y, and Z directions can be used. For cylindrical meshes, the rays can be in the radial direction as well as the axial direction. Because the radial rays are divergent, fractions based on them are incorrect unless the fraction is 1 or 0. If the single material test finds only 1 material with a positive fraction = 1, those fractions are used for the mesh cell and no further calculations are made for that cell. If not, the fractions calculated for the test are not used in the main calculation.

Calculation Methods. The main volume fraction calculation uses one of three methods. The Single N method uses a single value of N in each of the chosen directions. It is the speediest method, but the user must be confident that N is large enough for the desired accuracy.

The N and N+1 method uses two grids, one of size N and other of size N+1. The difference between the fractions calculated on the two grids gives an estimate of the stability of the calculation expressed in terms of a percent relative error for material M

,where F

_{M}^{K}is the fraction for grid size K. If both fractions are 0, . If F_{M}^{N+1}= 0, F_{M}^{N}is used in the denominator. The final fraction is the average

.The Iteration method computes fractions on grids of increasing spacing n until the fractions converge for all materials or until n = the user specified limit N. The iteration starts with n = 1 or the grid spacing used in the single material test is the test is used. n increases by 1 on each iteration. Convergence is reached if

where the default convergence parameter . Both N and can be modified on the dialog.

Calculating the Fractions. The volume fraction calculations can be performed on the entire mesh or on a selected range of mesh cells. While the calculation is underway, the dialog shows the indices of the mesh cell under analysis. The output can be directed to a Moritz window and/or a file. In addition to the mesh grid and volume fractions, the output can optionally contain absolute material volumes, uncertainties, parameter values, and calculation time.

Pressing the <Escape> key terminates the calculation.

It is the user’s responsibility to ensure that the various parameters and settings are sufficient to reach the desired accuracy. The uncertainties are a gauge of how much the results vary with changing grid size N; they should not be interpreted as the absolute error of the results. That being said, small errors coupled with a large N is a good indication that the fractions are accurate. The calculated fractions should be compared to fractions calculated analytically where possible.

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