The underlying ideas for this algorithm are presented in the book Computational Nonlinear Mechanics in Aerospace Engineering, American Institute of Aeronautics and Astronautics, edited by S. Atluri, ISBN 1563470446, Chapter 5, Fast Projection Algorithm for Unstructured Meshes by K. Jansen, F. Shakib, and T.
Hughes, 1992. Specifically, the implementation of the Smart Bucket Algorithm as described in the chapter stated above is used in system coupling. This algorithm generates weights that are ideal for transferring the profiles of non-conserved quantities like stresses, displacement, temperature, and heat transfer coefficient from a source mesh to a target mesh. Since a complete description of the algorithm is available in the reference quoted above, only a brief overview of it is presented below.
The first step in the process of computing the mapping weights using the Smart Bucket Algorithm is to divide the mapping source mesh into an imaginary structured grid, with each grid section called a
“bucket.” A 2D bucket is used to demonstrate this concept in Figure 9: Example of a Bucket Grid on a 2D Source Mesh (p. 48). Similarly, a 3D bucket grid is generated for a 3D mesh, and this is what is used in System Coupling.
Next, each node on the data transfer regions of the target mesh is initially associated with a bucket. In System Coupling, data transfer regions consist of element faces from the 3D mesh. Two cases arise:
buckets associated with the target node are either empty (without even one source element in it) or non-empty. For example, bucket A shown in Figure 9: Example of a Bucket Grid on a 2D Source
Mesh (p. 48) is empty. Each case (empty and non-empty buckets) is discussed separately in the sections below.
Figure 9: Example of a Bucket Grid on a 2D Source Mesh
Case 1: The bucket associated with a target node is non-empty
If the bucket associated to a given target node is non-empty, the mapping algorithm attempts to match each of the target nodes to one source element in the bucket.
First, each target node is checked to see if it is in the domain of any of the source elements. This is done by looping through all the source elements in that bucket and checking to see if the target node is within their domain. For each source element in the bucket, the vector element–local (or natural) coordinates (corresponding to the vector of global coordinates of the target node, ) is found by solving the set of nonlinear equations given by the isoparametric mapping below:
(4) where is the matrix of linear shape functions associated with the source element and is the vector of global coordinates of element–local node . It is then checked to see if lies within the domain of the source element based on certain criteria discussed next.
For a four-noded quadrilateral source element, if the natural coordinates corresponding to a target node satisfy the conditions in Equation 5 (p. 49) below, the target node is said to be exactly within the domain of the element.
(5)
where and are the components of the vector of natural coordinates . However, if the natural coordinates do not satisfy the conditions in Equation 5 (p. 49) but do satisfy the ones in Equation 6 (p. 49) below, then the target node is in the domain of the source element but only within the specified toler-ance (also known as element edge tolerance). The value of tolerance is exposed in the System Coupling UI as one of the expert settings. See the description of ProfMapEdgeTol in the section Expert Settings Related to Mapping in Expert Settings (p. 24).
(6)
This concept is explained with the help of Figure 10: A Quadrilateral Source Element in the Natural Co-ordinate Space (p. 50) wherein a quadrilateral source element is shown along with two different target nodes, one of which satisfies Equation 5 (p. 49), and other that satisfies Equation 6 (p. 49).
Data Transfers
Figure 10: A Quadrilateral Source Element in the Natural Coordinate Space
Similarly, for a three-noded triangle element, the conditions listed in Equation 7 (p. 50) below, are used to check if a target node is exactly within the domain of the element:
(7)
And the conditions in Equation 8 (p. 50) below will determine if the target node is within the domain but up to a tolerance .
(8)
Now that target nodes are determined to be in the domain of specific source elements, each node must be paired with only one source element. In both of the cases above (four-noded quadrilateral and three-noded triangle), it is possible that a target node occurs (either exactly or within a tolerance) in more than one source element’s domain. The finite element interpolation of the nodal solution requires each target node to be paired with only one source element.
To satisfy this requirement, the target node is consequently paired with that source element for which the gap is minimized. The gap is defined as the Euclidean distance between the target node and its projection onto/into a source element. In some cases, such as when candidate source elements are co-planar, the gap values may be identical and an alternate approach is required to pair the target node with one source element. Under these conditions, only the source elements with identical (and minimized) gaps are considered. The target node may be exactly in the domain of any of these source element, or
it will be in their domain within a tolerance. Preference is given to the last source element for which the target node is exactly in its domain. If the target node is only in the different domains within a tolerance, then the last candidate source element is used.
Once the target node is paired with a source element, mapping weights are computed by evaluating the finite element shape functions associated with the paired source element at the target node.
If no target node-to-source element match is found in a non-empty bucket, then the target node is re-ported as being unmapped. It is important to note, however, that mapping weights are still evaluated for such nodes using the Bucket Surface Algorithm. Specifically, all unmapped target nodes are simply mapped to the nearest source node in the bucket and the target node is assigned the solution value corresponding to that source node.
Note
Significant ‘gap’ distances between successfully-mapped target nodes and source elements may occur. For information about how to have mapped nodes with ‘gap’ distances larger
than a specified tolerance be reported as unmapped, see Expert Settings Related to Mapping in Expert Settings (p. 24).
Case 2: The bucket associated with a target node is empty
If the bucket initially found for the target node is empty, then the closest non-empty bucket is found and the same procedure as highlighted in Case 1 is followed so that each target node is mapped to one source element and mapping weights are calculated.
Unmapped Nodes
With the Bucket Surface algorithm, there are two types of target nodes that can be reported as un-mapped: nodes that do not fall within a bucket (these are “unmatched nodes”), and nodes that do fall within a bucket, but that do not meet the gap tolerance (these are "gap nodes”). Unmatched nodes are mapped to the nearest source node in the bucket and the target node is simply assigned the solution value corresponding to that source node. Unmatched nodes are always reported as unmapped in the SCL file. Gap nodes are within a bucket, and so are mapped to the source nodes like all of the other mapped nodes and given a value accordingly. Gap nodes are reported as mapped in the SCI file. The gap tolerance and the reporting of gap nodes in the SCI file can be modified using Expert Settings (p. 24).