# Finite element mesh

# Generate

This sophisticated multi-phased mesh generating tool will consider the defined minimum division numbers and the average element sizes and will generate the most balanced mesh. The tool generates a so-called unstructured mesh. After selecting the regions where the mesh will be generated, the tool splits the regions into sub-regions and performs the multi-phased mesh generation.

The phases of mesh generation are:

- Defining the vertices of the elements.
- Creating a triangle mesh using the vertices.
- Converting the triangle mesh to mixed quadrate-triangle mesh.
- Optimizing the coordinates of the nodes in the mesh (smoothing the mesh).
- Setting the middle points of the element sides.

**Defining the vertices of the elements**

The vertices will be evenly placed to a distance of the average element size from each other along the lines, which are parallel with the longest edge of the sub-re- gion and are at the average element size distance from each other.

**Creating a triangle mesh using the vertices**

The triangles are created using the well-known Delaunay triangulation techni-que which uses the Voronai domains.

**Converting the triangle mesh to mixed quadrate-triangle mesh**

The function used to convert the triangle mesh to a mixed quadrate-triangle mesh is capable of creating the mesh with the globally optimal shape. This in-volves the solving of a linear programming problem known in the mathematics as assignment problem. Our mathematicians have developed a new procedure to find the optimum for the linear programming problem. This problem is similar to the distribution method procedure.

**Optimizing the co-ordinates of the nodes in the mesh (smoothing the mesh)**

The optimization of the mesh is based on Dr. István Kirchner’s new procedure, which was published in [10]. This procedure places the nodes of the triangle elements in such a way, that the area of the triangles will be balanced. In achieving the most balanced area of the triangles an iteration technique is used.

**Setting the middle points of the element sides**

In the present version the sides of the elements are straight lines. Nodes are placed in the middle of the element sides.

The automatic mesh generator has some other unique and special automatic feature. Some of the most important features are:

- If the need arises the mesh will automatically be thicker around some local effects. This is solved by placing new nodes in the critical places. If the generator has found a place on the structure, where the mesh needs to be thick- er, in the third phase of the generation the number of the iterations will be greater than 1. If the required thickness couldn’t be achieved during the maximal iterations, the user will be notified with a warning. In this case, the geo- metrical structure possibly has some serious geometrical anomalies. The mesh around the places, where the geometrical anomalies are present will be very dense.
- If the calculated or user-set average element size is too big, than the genera- tor will automatically recalculate and reduce it with the statistical analysis of the current mesh. In this case the mesh generator will restart the first phase of the generation after finishing the second. It is possible for the generator to execute the reduction of element size as many times as it is needed.
- During the mesh generation on the actual sub-region the generator takes into account the division number of the sub-region borders which belong to an- other sub-region. If the division number of a border, which belongs to an- other sub-region too is altered, the generator automatically regenerates the mesh on the other sub-region too. The visible sign of this automatic recalculation is that in the progress window the original number of the total sub-regions increases. If the automatic recalculation of the sub-regions is needed too many times, it may suggest some serious geometrical and statical problems on the structure. The critical places on the structure are marked by their unusual density of the mesh. In order to minimise the number of the automatic recalculation the generator first resolves the sub-region with the smallest average element size and than proceeds in increase order.

# Refine

This tool is used to increase the thickness of the balanced mesh generated automatically by the program. Using the dialog box the user can easily define where the mesh should be thicker. Because of numerical reasons it is needed to refine the mesh around the effects, which are in a point or along a line. These effects are for example point and line supports and loads, the places where there is a drastically change in the value of a surface load or the borders of two regions which have different material. It can be useful to refine the mesh along the free edges of the structure too.

The Refine function basically consists of two phases. In the first phase the user selects all the elements, which are to be divided. The second phase automatically splits the selected elements in the suitable way. In this second phase the pro- gram uses the principles published by Dr. István Kirchner in [11]. The dialog box makes the selection of the elements comfortable for the user.

# Optimal rebuild

This option rebuilds the mesh according to the global optimum. The nodes are not moved during the process. In the first phase of the process the program builds a triangle mesh using the principles of Delaunay triangulation technique based on the Voronai domains. The second phase converts these triangles to quadrates corresponding to the global optimum for the selected regions. During the converting process the program uses the unique function, which is capable of creating the global optimum of the mesh. This involves the solving of a linear programming problem known in the mathematics as assignment problem. Our mathematicians have developed a new procedure to find the optimum for the li- near programming problem. This problem is similar to the distribution method procedure.

# Smooth

This option calculates the optimal coordinates for the corner nodes of the elements. The optimization of the mesh is based on Dr. István Kirchner’s new procedure, which was published in [10]. This procedure places the nodes of the triangle elements in such a way, that the area of the triangles will be balanced. In achieving the most balanced area of the triangles an iteration technique is used.