Donor Search

For each acceptor cell, donor cells must be found. The set of donor cells depends on the interpolation option that is chosen and on the number of active cells in the donor region around the acceptor cell centroid.

The interpolation options are:

• Distance-weighted interpolation, where the interpolation factors are inversely proportional to the distance from acceptor to donor cell center, resulting in the closest cell giving the largest contribution. This involves 3 donor cells (in 2D) or 4 (in 3D). The complete overset assembly process is performed at each time-step.
• Linear interpolation using shape functions spanning a triangle (in 2D) or a tetrahedron (in 3D) defined by centroids of the donor cells.

  This option is more accurate but also more expensive. It is important in simulations involving moving meshes as it ensures that interpolation elements do not overlap. The choice of donor cells is not unique since the available donor cell centroids that enclose the acceptor cell centroid can define more than one triangle or tetrahedron. This way the interpolation is continuous as the acceptor cell centroid passes from one interpolation element to the next.

  For moving mesh simulations, when the movements of the overset region compared to the cell size during one time-step are small, the donor triangle (in 2D) or the donor tetrahedron (in 3D) can still be valid. If this is the case, only the interpolation factors are re-calculated instead of executing the entire overset assembly process. For such situations, the linear interpolation provides an optimization regarding runtime. For overset motion setups involving DFBI, the probability of Simcenter STAR-CCM+ applying this optimization is high, because of typically small motions being performed during each inner DFBI iteration.

• Least squares interpolation.

  For a description of the least squares interpolation method, see Least Squares Interpolation.

  The interpolation function is built directly into the coefficient matrix of the algebraic equation system. This approach ensures implicit coupling of the overset meshes. In the image below, two acceptor cells are shown using dashed lines, one in the background mesh and one in the overset mesh.

  
  
  

  The fluxes through the cell face between the last active cell and the acceptor cell are approximated in the same way as between two active cells. However, whenever the variable value at the acceptor cell centroid (marked by the open symbols in the above figure) is referenced, the weighted variable values at the donor cells are substituted:

  $${\mathrm{\Phi }}_{\text{acceptor}}={\displaystyle \sum _{}^{}{\alpha }_i{\mathrm{\Phi }}_i}$$

  (109)

  In this equation, ${\alpha }_i$ are the interpolation weighting factors, ${\Phi }_i$ are the values of the dependent variable $\Phi$ at donor cells ${\text{N}}_i$ and subscript $i$ runs over all donor nodes of an interpolation element (denoted by the green triangles in the figure). This way, the algebraic equation for the cell “C” in the above figure involves three neighbor cells from the same mesh ( ${\text{N}}_1$ to ${\text{N}}_3$ ) and three cells from the overlapping mesh ( ${\text{N}}_4$ to ${\text{N}}_6$ ). The coefficient matrix of each equation solved (both for the segregated and coupled solution method) is updated accordingly to ensure that equations can be solved up to the round-off level of residuals.