K-d tree

In structured meshes, neighborhood relations are implicitly determined by the mapping from the physical to the logical space. In unstructured mesh based approaches, support domains can be determined from the mesh topology. However, in meshless methods, the nearest neighboring nodes in $\Omega_S$ are determined with various algorithms and specialized data structures.

Even though the support domain of a node $\Omega_{Si}$ can be in principle of arbitrary shape, most of the known meshless approaches are based on a circular or elliptical $\Omega_S$. We will limit our further description to circular $\Omega_S$ with radius $r_S$. We are aware that an arbitrary nodal distribution, which should be common in meshless discretization, implies that the nodal density changes throughout the domain. There are different strategies for determining support nodes in $\Omega_S$.

A constant radius $r_S$ can be selected. However, in this case, $n_S$ varies significantly, which influences the accuracy and robustness of the methodology. Alternatively, the nodal density can be smoothed \cite{liu_book03}, which yields to an interpolated $r_S$ and slightly varying $n_S$. But this strategy is also not adequate when the density of nodes varies highly. Also, special treatments for the construction of $\Omega_S$ near boundaries are needed.

The input to Algorithm \ref{alg_constNs} is a list of $(n_S +1)$ nearest nodes to ${\bf x}$. A naive implementation of finding these nodes could become quite complex if implemented with classical sorting of all nodal distances. Regarding the number of nodes $N$, it approaches the quadratic computational complexity. However, by using efficient data structures, e.g. quadtree \cite{finding_nodes_quadtree}, R tree \cite{Agarwal_Rtree_2003} or kD tree \cite{k-D_tree} (2D tree for 2D domains), the problem becomes tractable. The strategy is to build the data structure only once, before the solution procedure. During the solution, the support nodes of the desired points will then be found much faster. \index{kD tree example}

Let us illustrate the whole procedure with a simple example of a 2D tree for eleven nodes, with node numbers and coordinates listed in the first and the second column of Table \ref{tab:kD_nodes} and attached to the corresponding dots on the unit square in the left part of Figure \ref{fig.2D_treegraph}.

{\begin{table}[h] \centering \begin{tabular}{|c|c|c|c|c|c|} \hline Node number&Unsorted & Sorted by '"`UNIQ-MathJax16-QINU`"' & Sorted by '"`UNIQ-MathJax17-QINU`"' & Sorted by '"`UNIQ-MathJax18-QINU`"' & Bucket\\ \hline 1&(0,0)&(0,0)&(0,0)&(0,0)& \bf (0,0) \\ \hline 2&(0.6,0)&(0,0.4)&(0,0.4)&\bf (0,0.4)& \\ \hline 3&(1,0)&(0,1)&\bf (0.24,0.6)& & \\ \hline 4&(0,0.4)&(0.24,0.6)&(0,1)&(0,1)& \bf (0,1) \\ \hline 5&(0.6,0.3)&(0.47,1)&(0.47,1)&\bf (0.47,1)&\\ \hline 6&(1,0.5)&\bf (0.6,0)& & &\\ \hline 7&(0.24,0.6)&(0.6,0.3)&(1,0)&(0.6,0.3)&\bf (0.6,0.3)\\ \hline 8&(0.76,0.8)&(0.76,0.8)&(0.6,0.3)&\bf (1,0)&\\ \hline 9&(0,1)&(1,0)&\bf (1,0.5)& &\\ \hline 10&(0.47,1)&(1,0.5)&(0.76,0.8)&(0.76,0.8)&\bf (0.76,0.8)\\ \hline 11&(1,1)&(1,1)&(1,1)&\bf (1,1)&\\ \hline \end{tabular} \caption{The list of eleven nodes (1st column) determined with coordinates (2nd column), after sorting by $x$ (3rd column), after sorting sub lists by $y$ (4th column) and after sorting sub sub lists by $x$ again (5th column). The nodes nearest to medians are shown in bold.} \label{tab:kD_nodes} \end{table}

In the first step of 2D tree construction, the list of nodes is sorted by their $x$ coordinate, which is shown in third column of Table \ref{tab:kD_nodes}. Then a node with median coordinate, $x = 0.6$ in our case (shown in bold), is selected as the root of the first level of the 2D tree. If there is more than one such node, any one can be selected. The sorted set in column 3 is split into two parts, the one for $x$ below the median, i.e. $x < 0.6$, and the one for $x$ above or equal the median, i.e. $x\geq 0.6$. The two sub sets of nodes are shown in the left part of Figure \ref{fig.2D_treegraph} within two distinct rectangles, and on the right side of Figure \ref{fig.2D_treegraph} as the left and the right part of the 2D tree.

In the second step, the two sub lists of nodes are sorted by their $y$ coordinate, which is shown in fourth column of Table \ref{tab:kD_nodes}. The median coordinates $y$ are $0.6$ and $0.5$, respectively. The corresponding nodes $(0.24,0.6)$ and $(1,0.5)$ are taken as roots for the second level of 2D tree and are shown in bold and used to further split the tree. The resulting four sub sub sets of nodes are shown in the right side of Figure \ref{fig.2D_treegraph} as nodes on the lower two levels of the 2D tree.

Finally, the sub sub lists are sorted again by their $x$ coordinate, with the result shown in fifth column. Four roots are obtained with the coordinate $x$ nearest to medians, namely the nodes $(0,0.4)$, $(0.47,1)$, $(1,0)$, and $(1,1)$. The remaining nodes of the last level of the 2D tree, also termed the bucket, are $(0,0)$, $(0,1)$, $(0.6,0.3)$, and $(0.76,0.8)$. In practical cases, the refinement of the tree stops sooner, when its leaves are represented by list of several nodes, because such a fine grained distribution of leaves as in the presented example is often not beneficial from the computational efficiency point of view.