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Petr Kahánek, Alexej Kolcun of this deformation is a structured mesh. It is obvious (and important) that it is topologically equivalent to the original grid. Structured triangulation is a structured mesh, whose elements (quadrilaterals) were triangulated, i.e. for every element, one of the two possible diagonals was chosen. There is one great advantage of a structured mesh – we can obtain very efficient form of the incidence matrix of structured mesh. The incidence matrix of nodes plays a significant role for most kinds of simulations. It depends on the manner of node indexing. Fig.4 shows incidence matrix of the regular grid, where nodes are indexed a) randomly, b) in the “spiral way from the boundary to the interior” c) in the “natural way – row by row”. We can see that the last method gives us the best form of the incidence matrix, where all nonzero elements are concentrated in the regular structure near the main diagonal. As a result, use of a structured mesh allows us to solve much larger problems (in terms of number of points), because effective methods for storage and manipulations with incidence matrix are offered in this case [2, 3]. Figure 3: Incidence matrices for different indexing of nodes On the other hand, there is a disadvantage, too: structured meshes are not suitable for refinement. However, the technique of nested meshes can solve this problem (Multigrid methods which use this idea are used for PDE solving) [3]. 4 Structured triangulations with defined geometry Before any calculations can be done, it is neccesary to deform the mesh so that it respects the geometry inherent to the given problem. In general, such geometry can be very complicated; it is, however, a common task to model some basic shapes, i.e. curves in 2D, surfaces in 3D. Here, we will deal with the simplest curve type in two dimensions, line segment. 98
BRESENHAM'S REGULAR MESH DEFORMATION AND... Start in I 1 Until we have reached I 2 Find indices x, y of the next node (via Bresenham) Move it so that it lies on the line If both x and y have changed (in Bresenham) Force corresponding diagonal Algorithm 1: Local mesh deformation There are different ways of mesh deformation [2, 3]. Usually, mesh generators use some kind of parametrization; they, in fact, divide the mesh into several smaller meshes, which are "glued" together with the modelled boundary. Let us suppose that we have a line segment, whose ends are fixed in nodes of the original (not yet deformed) mesh, say I 1 and I 2 . For the process of deformation, we will use an adaptation of the well-known Bresenham's algorithm. We will use it to find out which nodes should be moved during deformation. See Algorithm 1. It is important to say that we move nodes only in direction of one axis (x or y). This is determined by slope α of the given line: for 0 ≤ α ≤ π /4, we will move the nodes only vertically, for π /4
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BRESENHAM'S REGULAR MESH DEFORMATION AND...<br />
Start in I 1<br />
Until we have reached I 2<br />
Find indices x, y of the next node (via<br />
Bresenham)<br />
Move it so that it lies on the line<br />
If both x and y have changed (in Bresenham)<br />
Force corresponding diagonal<br />
Algorithm 1: Local mesh deformation<br />
There are different ways of mesh deformation [2, 3]. Usually, mesh<br />
generators use some kind of parametrization; they, in fact, divide the mesh<br />
into several smaller meshes, which are "glued" together with the modelled<br />
boundary.<br />
Let us suppose that we have a line segment, whose ends are fixed in<br />
nodes of the original (not yet deformed) mesh, say I 1 and I 2 . For the process<br />
of deformation, we will use an adaptation of the well-known Bresenham's<br />
algorithm. We will use it to find out which nodes should be moved during<br />
deformation. See Algorithm 1.<br />
It is important to say that we move nodes only in direction of one axis (x<br />
or y). This is determined by slope α of the given line: for 0 ≤ α ≤ π /4, we<br />
will move the nodes only vertically, for π /4
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