在2D网格中查找节点/顶点的邻居

Question

I have a 2D mesh defined by nodes and elements.

Structure of a node: Node ID, X position, Y position

Structure of an element: Element ID, Node 1, Node 2, Node 3, Node 4

Example of a 2x2 elements mesh:

Nodes:

 ID  X  Y
  1  0  0
  2  0  1
  3  0  2
  4  1  0
  5  1  1
  6  1  2
  7  2  0
  8  2  1
  9  2  2

Elements:

 ID N1 N2 N3 N4
  1  1  2  4  5
  2  2  3  5  6  
  3  4  5  7  8
  4  5  6  8  9

N7-----N8-----N9
|      |      |
|  E3  |  E4  |
|      |      |
N4-----N5-----N6
|      |      |
|  E1  |  E2  |
|      |      |
N1-----N2-----N3

I'm storing both nodes and elements in linked lists.

My question: How can I find the neighbors (nodes) for an arbitrary selected node?

The neighbors of N5, for example, would be N2, N4, N6 and N8.

*Note: This 2x2 element mesh simplified example for explanation proposes, the meshes I'm dealing with may contain several thousands of nodes and elements. I also have been looking at some concepts of graph theory, but I'm not sure which may be the right way to go.

Answer 1

It would be good to have element's vertices ordered in a way that they make closed polygon. Vertices [1, 2, 4, 5] do not uniquely define first element. From your description it can be seen that you mean that is a polygon with four vertices in order (1, 2, 5, 4). But without picture it can be also degenerated quad (1, 2, 4, 5).

Like:

Elements:

 ID N1 N2 N3 N4
  1  1  2  5  4
  2  2  3  6  5
  3  4  5  8  7
  4  5  6  9  8

If you are not sure about vertices order, than you have to check about element self-intersection, and reorder vertices to resolve intersections.

With that kind of data it is easy to find all neighbours of given node. Pass through all elements, if element contains given node, than there are two neighbours in that element, vertex before and after in a list.

For node 5, in first element there are neighbours 2 and 4, in second element there are neighbours 6 and 2, ...

If there will be lot of inquires of this kind, than it is better to make extract connectivity information in separate structure. That can be map that maps node to set of it's neighbours. To make it, pass through all elements, and for each element vertex add two neighbours in node's list.