Conditional Integration of a Linear Function on a Tetrahedron

@inproceedings{RoyerRM2005,
 abstract = { In reservoir modelling or mining applications, it is often needed to calculate the volume of a region defined conditionally by one or several cut-off values on one or several functions. Moreover, it can be generalized to the evaluation of the integral of a function on a volume defined by several cut-offs (NTG, Volume in place). This can be easily done in GOCAD using a script applied on a fine structure grid on which properties are defined. The smaller are the cells of the grid, the more accurate is the result. However, the accuracy of such a technique applied to unstructured grids with non uniform cell size, is very poor. In the unstructured grid case, it is therefore important to improve the estimation of volumes regardless to the cell size. This paper proposes linear formulas defined on tetrahedra to calculate the volume defined conditionally by a cut-off on a linear function. It is extended to the evaluation of the intersection volume of a tetrahedron with a cut-off tangent plane that approximates an iso-value function ϕ = ϕc = cst. When the function varies linearly in the intersected volume, its integral is evaluated. As an application, the resulting formula lead to an exact computation of the NTG and of the volume of a reservoir in the framework of the Geochron model. },
 author = { Royer, Jean-Jacques },
 booktitle = { 25th gOcad Meeting },
 month = { "june" },
 publisher = { ASGA },
 title = { Conditional Integration of a Linear Function on a Tetrahedron },
 year = { 2005 }
}