5

,

MESHING A MODEL IN HOBBIES

5.0 SUMMARY

Generating a mesh is the process by which a Higher Order Basis Based Integral Equation Solver (HOBBIES) mesh is generated from the geometry definition. In HOBBIES, all the properties assigned to geometric entities, such as domain, loading, and excitation (see Chapter 6) and so on, will be transferred to the nodes and elements of the mesh. The mesh will be used for the electromagnetic (EM) analysis at a later stage.

The meshing process is an indispensible operation before running a simulation in HOBBIES. For ease of use, HOBBIES provides automatic methods for creating meshes for structures such as Non-Uniform Rational B-Spline (NURBS) curves or surfaces. Tools to generate a user-configured mesh are also provided in HOBBIES. The mesh generation does not depend on whether layers are ON or OFF at the moment of generation, but frozen layers are not meshed if the preference (see Appendix A, the Meshing Tab under the Preferences) No mesh frozen layers is selected. Every node and element will be assigned to the layer in which the original geometrical entity was defined.

The defaults are as follows:

• An entity is meshed if it does not belong to a higher level entity.
• A line mesh is made of two-node elements, and a surface mesh is made of unstructured quadrilateral elements. The default for structured meshes is also quadrilateral elements.

All default elements use linear interpolations for the unknown variables. The line and quadrilateral element meshes are used for the higher order method of moments (MoM) kernel in the current version of HOBBIES.

When the geometry of the model is composed of quadrilateral patches, the mesh created by HOBBIES, is exactly equal to the geometry of the model, with the “Use geometry as meshing elements” option checked. Note that, if necessary, the MoM kernel in HOBBIES can automatically refine the meshes for structures with straight wires and quadrilateral patches in the simulation. Therefore, very fine meshes from the graphical user interface (GUI) are not needed to perform an efficient simulation.

5.1 UNSTRUCTURED MESH

An unstructured mesh has an irregular connectivity of the nodes. Each node can have a different number of elements around it.

• Assign sizes on points, lines, surfaces:

It is possible to assign different sizes to different entities of the mesh. This means that in the vicinity of these entities, the generated elements will be approximately of that size. All the entities that do not have an assigned size when meshing take the default size. Points do not take any size if none is given.

Assigning the size 0.0 to an entity is the same as setting the default size. The transition between different sizes is controlled by a parameter in Preferences (see PreferencesMeshing).

• Sizes by chordal error:

The Sizes by chordal error option contains fields for chordal error (the maximum distance between the generated element and the real geometry) and minimum and maximum size limits. The Assign sizes by chordal error window are shown in Figure 5.1. HOBBIES assigns the corresponding sizes to all the entities to satisfy this condition. It will only change the current sizes if the new one is smaller than the one defined previously. In structured surfaces, stretching is permitted. This means that, if necessary, elements can have very different sizes along the two principal directions.

In the window (Figure 5.1), one can view the effect of different chordal error values on the number of elements that will be created over a line. Any line of the geometry can be chosen to be used through the icon in this window. An example of unstructured meshes of a circular plate with different chordal errors but the same maximum and minimum meshing sizes is shown in Figure 5.2.

Figure 5.1. Assign sizes by chordal error displayed in the window.

Figure 5.2. Unstructured meshes of a circular plate with different chordal errors: (a) chordal error is 0.01, (b) chordal error is 0.001.

• Correct sizes:

When the Correct sizes option is selected, a window appears, as shown in Figure 5.3. In this window, it is possible to enter a minimum and a maximum mesh size.

Figure 5.3. Correct meshing sizes window.

If the By geometry option is activated, sizes are assigned to all the entities depending on the shape of the geometry. This means that smaller surfaces will have smaller elements.

If the Correct incompatible sizes option is activated, some sizes are reduced to ensure that the transitions between sizes in close entities are not too fast. This option will only change the existing sizes if the new size is smaller than that previously defined.

An example of unstructured meshes of a circular plate with different meshing preferences but the same maximum and minimum meshing sizes is shown in Figure 5.4.

Figure 5.4. Unstructured meshes of a circle with different Automatic correct sizes options: (a) Automatic correct sizes option is set to normal, (b) Automatic correct sizes option is set to none.

5.2 STRUCTURED MESH

A structured mesh has a regular connectivity of the nodes. Each node has the same number of elements around it (for some meshes a small number of nodes, such as the nodes on boundary lines, have a different number of elements around them).

The size of the elements is defined in a different way than for an unstructured mesh. In this case, the number of divisions must be the same for all lines that are opposite each other on each four-sided surface.

• Structured mesh by number of cells

Choose this option to create a structured mesh dividing the lines/edge of the four-sided surface in the number of cells desired. After choosing this option, the window where one sets the number of cells is shown. Once the number of cells is selected, the lines/edges must be selected. Note that for surfaces, the user needs to select the surface and press Esc before setting the number of the cells for its edges. This process can be repeated as many times as necessary until all lines have a new value.

• Structured mesh by size

Choose this option to create a structured mesh using the same size for all the elements. After choosing this option, the window where one sets the size of the elements is shown. Once the mesh size is selected, the lines/edges must be selected. Note that for surfaces, the user needs to select the surface and press Esc before setting the size of the elements for its edges. This process can be repeated as many times as necessary until all lines have a new value.

Four-sided structured mesh and unstructured mesh for a NURBS surface are compared in Figure 5.5. By default, the generated elements in four-sided structured meshes are quadrilaterals.

Figure 5.5. Meshes of a four-sided NURBS surface: (a) the NURBS model, (b) the structured mesh, (c) the unstructured mesh.

It is possible to mix some entities with structured meshes and others with unstructured ones.

• Element concentration

By default, all partitions in one structured line have the same approximate length. This command lets the user select one line, which will be shown in the graphical window with an arrow indicating its direction. If the weight assignment is positive, the elements will be concentrated towards the end of the line; if negative, the elements will be repelled from the end of the line. As the magnitude of the weight increases, the difference between element sizes will be greater.

Example

This example will show how to generate the structured mesh with concentrated elements for a square plate. The steps are as follows:

1. Create a square plate with an edge length of 1.0 m (see Chapter 4).

2. Click MeshStructuredLinesConcentrate elements, and select lines to assign a weight to concentrate elements. In this example, one line is selected as denoted by the arrow in Figure 5.6 (i.e., line 1). Press Esc and the Concentrate elements window appears, as shown in Figure 5.7. The default values for the Start Weight and End Weight are 0.0. Set both the values as −0.5 for this example, and click Ok. Press Esc to finish this operation.

Figure 5.6. The line with an arrow.

Figure 5.7. Concentrate elements window.

3. Click MeshStructuredSurfacesAssign number of cells, select the plate, and press Esc. An Enter value window appears as shown in Figure 5.8. The default number of the cells is 2, and if one decided to use a value of 10 then the following needs to be done. Click Assign and select line 1 as well as line 4 in Figure 5.6. Press Esc and click Close to close the Enter value window.

Figure 5.8. Enter value window for the number of cells.

4. Click MeshGenerate mesh to bring up the Mesh generation window. Users do not need to change the values in this window because the mesh parameters have been set in step 3. Click Ok and the mesh is generated. Click View mesh and the structured mesh is shown as in Figure 5.9 (a).

If the Start Weight is −0.5 and the End Weight is 0, the structured mesh is shown in Figure 5.9 (b).

Figure 5.9. Examples of the concentrate elements feature: (a) Start Weight = −0.5 and End Weight = −0.5, (b) Start Weight = −0.5 and End Weight = 0.

5.3 ELEMENT TYPE

The element type for surfaces is quadrilateral, while that for wires is segment.

5.4 MESH CRITERIA

HOBBIES provides different criteria to generate the mesh. The Default option does not mesh the boundaries, that is, lines for surface meshes.

The Mesh option lets the user choose the entities to be meshed, while the No Mesh option does the opposite. Note that the entities chosen for the No Mesh option will not be meshed and thus will not be taken into account for the simulation.

The Skip option forces HOBBIES to skip a geometrical entity when meshing (so the entity will not be meshed), while the No Skip option forces HOBBIES not to skip the geometrical entity (so the entity will be meshed) when the RJump surface mesher is used (see PreferencesMeshing).

The Automatic skip option lets HOBBIES decide whether the geometrical entity should be skipped or not skipped when the RJump surface mesher is used. This decision is made according to the tangency between entities: Those entities that are close to a tangent will be skipped when meshing.

By using the Skip by option (Figure 5.10), users can apply some of the following criteria to select the entities to be skipped, not to be skipped, or to be skipped automatically for efficient simulation.

• Tangency limit angle: This criterion selects the lines between surfaces in which angles between normals to the surfaces are lower than a specific value.
• Take care layers: If this is set, lines between surfaces that are in different layers will not be selected.
• Take care curvatures: This ratio means the difference between curvatures in the surfaces that share a line. If the ratio between the curvatures exceeds this value, the shared line is not considered to be selected. As an example, if this value is 0.1 (10%), the only lines that will be selected will be the ones in which the two surfaces that share the line have almost the same curvature (only a 10% difference is accepted). As this number increases, more lines are selected.
• Lines of surfaces smaller than: All the lines of surfaces that have some lines smaller than this value will be selected.

By clicking Select by criteria, all the selected criteria are used to select the lines. Modify selection allows one to modify manually the selection made.

When Apply is clicked, the marked option (Skip, No skip, or Automatic skip) is applied to the selected lines.

The Force points to option forces the surface mesh to pass through the selected points from geometry, even though they do not necessarily belong to the surface. An example is demonstrated in Figure 5.11.

Figure 5.10. Window to assign skip criteria to lines.

Figure 5.11. Example of the Force points to option: (a) a point is forced at the origin, (b) no point is forced.

5.5 RESET MESH DATA

This command resets all the mesh parameters including the sizes and the mesh criteria assigned to entities. This means that all of them will be unassigned.

To reset the mesh size of certain entities as default, assign the size 0.0 (see Unstructured in Section 5.1) to the entities.

5.6 DRAW

This option is used to draw meshing properties in geometrical entities.

• Sizes

When sizes are assigned to points, lines, or surfaces using the assign unstructured sizes option, it is possible to draw the different assigned sizes in different colors (see Unstructured in Section 5.1).

Example: In the following example, some different sizes are assigned to surfaces, as shown in Figure 5.12. Sizes of 0.1, 0.2, and 0.3 are assigned depending on the surface.

After choosing the Draw Sizes option (over surfaces), we get the following result (Figure 5.13).

Figure 5.12. Surfaces with different mesh sizes.

Figure 5.13. Surfaces with different colors.

• Num of divisions

With this option, one can see the number of divisions assigned to the structured lines of the model (see Structured in Section 5.2).

• Element type

With this option, one can see which element types have been assigned to each geometric entity (see ViewModeMesh).

• Mesh / No mesh

With this option, one can see the entities that HOBBIES has been either forced to mesh or forced not to mesh (see Mesh Criteria in Section 5.4). If the meshing criteria have not been assigned to an entity, it is shown as Default (see ViewModeMesh).

• Structured Type

With this option, one can see what kind of mesh will be generated for geometrical entities (i.e., Unstructured or Structured). If no level of structure has been assigned, it is shown as Default (see ViewModeMesh).

• Skip entities (Rjump)

With this option, one can see which lines and points will be skipped when meshing using the meshing preferences set at that moment (see PreferencesMeshing), and the lines and points set as Skip or No Skip mesh criteria (see Mesh Criteria in Section 5.4).

• Force points to

With this option, one can see the number of points forced to be in the surface mesh (see Mesh Criteria in Section 5.4).

5.7 GENERATE MESH

When geometric models and mesh parameters are ready for mesh generation, select this command. If there is a previously generated mesh, HOBBIES asks whether this should be erased. It will be lost from the memory, but it will remain on the disk until the project is next saved.

The unstructured mesher for surfaces can be chosen in Preferences (see Preferences in Appendix A).

To mesh, it is first necessary to determine element sizes that will be applied to lines and surfaces (see Unstructured in Section 5.1). HOBBIES automatically calculates the default mesh sizes to define a coarse mesh. The default sizes for lines and surfaces are calculated using the wavelength of the stop frequency selected for the simulation.

Sometimes the wavelength calculated from the stop frequency could be larger than the geometry of the model, which may occur when electrically smaller structures are encountered. In this case, HOBBIES is able to use the shape of the geometry to calculate the default sizes to define a coarse mesh.

The mesh generation window is shown in Figure 5.14, which contains the default mesh sizes for wires and surfaces programmed in HOBBIES. The default sizes can be changed using the entries available in the Mesh Sizes frame. Alternatively, users can change the wavelength factors in the Size Options frame and click the Change buttons to change the mesh sizes. The mesh size is related to the term factor × wavelength. To change the entered sizes back to the default sizes, click the Suggest buttons.

Figure 5.14. Mesh generation window.

Furthermore, HOBBIES offers two options:

• Use geometry as meshing elements. If this option is activated, HOBBIES will ignore the mesh sizes set by an user, and create a mesh that is exactly equal to the quadrilateral patches already present in the geometry. If the element size is larger than two wavelengths, the kernel will subdivide the element automatically (refer to Section 1.4.3).
• Do not subdivide straight wires. If this option is activated, straight wire structures will not be meshed during the meshing process.

During meshing, a progress bar indicates the number of generated surfaces in relation to the total number of surfaces. Meshing can be stopped at any time by clicking the Stop button. It may be necessary to press the button repeatedly or to keep it pressed for a few seconds.

5.8 ERASE MESH

If the model has a mesh, this option erases it.

5.9 EDIT MESH

This option lets one modify a mesh. All modifications will be lost when the mesh is generated again.

• Move node

By using this command, an existing node is selected and moved, as demonstrated in Figure 5.15. The new position is entered in the usual way (see Point Creation in Section 4.2.1).

To know the coordinates of a node, click UtilitiesListNodes and select that node. Press Esc and the List Entities window appears, which will show the coordinates.

Figure 5.15. Movement of a mesh node.

• Smooth Elements

To smooth elements, select them in the usual way, and then press Esc to perform the action. All selected elements must be of the same quadratic type.

• Collapse

The Collapse function converts coincident entities (i.e., entities that are close to each other) into one.

It is possible to collapse edges, nodes, elements, or the whole mesh.

• Collapse mesh collapses all the nodes of the mesh.
• Collapse edges joins nodes that are connected by edges shorter than the Import Tolerance value (see PreferencesExchange).
• Collapse nodes asks one to select some nodes. Nodes closer together than the Import Tolerance value are collapsed.
• Collapse elements asks one to select some elements. Then, the nodes of these elements that are closer together than the Import Tolerance value are collapsed.

• Delete nodes/elements

To delete elements or nodes, click this command, select them in the usual way, and then press Esc to perform the action.

Nodes that no longer belong to any element after the operation are also erased.

5.10 SHOW ERRORS

This option opens the mesh errors window. This window presents a list of the entities that HOBBIES could not mesh, and some information about the problems that occurred during the meshing process. By right-clicking over an item in the list, advice will be displayed about how to solve the meshing problems for each geometrical entity.

5.11 MESH QUALITY

This option opens a window that shows information about the quality of the mesh elements, as shown in Figure 5.16.

Figure 5.16. Mesh quality window.

There are several criteria in the Quality criteria drop-down menu used to measure the quality of the elements:

1. Minimum angle: The quality criterion is the minimum angle in surface elements. This means that elements with smaller angles are considered to be of a worse quality than ones with bigger angles.

2. Maximum angle: This gives the maximum angle for every element. Elements with bigger angles are considered worse. Typically, the Minimum angle criterion is good for qualifying triangles and the Maximum angle criterion is good for quadrilaterals.

3. Element vol: The quality criterion is the size of elements (distance for lines and area for surfaces). Elements with small “volume” are considered worse.

4. Minimum edge: The quality criterion is the size of the smaller edge of each element. Elements with smaller edges are considered worse.

5. Maximum edge: The quality criterion is the size of the largest edge of each element. Elements with bigger edges are considered worse.

6. Shape quality: The quality criterion measures the likeness of the element to the reference one (a square in the case of quadrilaterals). Its value is 1 for a perfect element (the reference one), and it decreases as the element becomes worse. If it reaches a negative value, it means that the element has a negative Jacobian at some point [1].

7. Minimum Jacobian: The quality criterion is the value of the minimum Jacobian between the Jacobians calculated at each element Gauss point. If there are elements with negative Jacobians, problems may be encountered in some calculation processes.

There are two visualization modes:

1. Normal: The graph shows the number of elements that have an angle of a certain size.

2. Accumulated: The graph shows the number of elements that have an angle of a given size or smaller.

In the Mesh quality window, if one double clicks on a value, the elements below this value are selected in red. These selected elements can be sent to a layer using the Send To Layer button in the Mesh quality window.

Example: In this example, we are studying the mesh using the minimum angle criterion. We can see that 20 elements of our mesh have an angle of less than 80.9 degrees, as shown in Figure 5.17 (a). If we double-click on the graphic, the 20 elements that have an angle smaller than 80.9 degrees will be selected, as shown in Figure 5.17 (b).

5.12 MESH OPTIONS FROM MODEL

This option loads the meshing preferences of a model, by which the user can view the mesh options in a project. For example, open a HOBBIES project that has a mesh, click this option, and the current meshing preferences will be replaced by those set in that project.

5.13 MESH GENERATION EXAMPLE FOR SURFACES

This section demonstrates the creation of a mesh for the model of a cylinder. The steps of the model creation and the quadrilateral mesh generation are as follows:

1. Create a cylinder using GeometryCreateObjectCylinder (see Section 4.2.5 in Chapter 4). Set the radius to 2.0 m and the height to 6.0 m, for example. The resulting cylinder modeled by four NURBS surfaces—a top, a bottom, and two sides—are shown in Figure 5.18. Note that the model created by the aforementioned command also has the volume entity, which needs to be deleted before meshing (see Section 4.2.4 in Chapter 4).

Figure 5.17. Example of a mesh quality study: (a) Mesh quality window with marked horizontal line and arrow, (b) 20 elements of the mesh that have a minimum angle of less than 80.9 degrees.

2. Select MeshElement typeQuadrilateral and an information window appears (Figure 5.19). Click Ok in the window, and select the surfaces to assign the quadrilateral element type. Press Esc to finish the selection. Note that this step can be skipped because the default element type for surfaces is quadrilateral.

3. Select MeshGenerate mesh to display the Mesh generation window (Figure 5.20). The default Quadrilateral edge length is 0.41 m for the cylinder model. The user may edit the value by entering a new value (e.g., 0.3 m). There are no wire structures in the model; therefore, the wire-related parameters in Figure 5.20 are left as the default.

Figure 5.18. A cylinder created by four NURBS surfaces.

Figure 5.19. Information window for the quadrilateral mesh.

Figure 5.20. Mesh size for a quadrilateral mesh after entering the new value of 0.3.

We have not set the frequency, and the Quadrilateral edge length factor is therefore 0. If the frequency is set, the edge length is typically about one wavelength of the wave in the medium surrounding the model to guarantee an accurate simulation result. For a smooth surface, the edge length can be up to two wavelengths, while for a surface with large curvature, the edge length should be less than one wavelength to maintain the geometric shape of the model and the accuracy of the result. Users can also get a reference value by clicking Suggest after setting the frequency.

4. Click OK in the Mesh generation window. HOBBIES generates the mesh and displays the Progress in meshing window shown in Figure 5.21.

5. When the mesh is generated, a Dialog window appears that includes the number of quadrilateral elements and nodes, as shown in Figure 5.22.

6. Click the View mesh button in the Dialog window and one will see the mesh (Figure 5.23). The quadrilateral mesh is unstructured.

Figure 5.21. Progress in the meshing window.

Figure 5.22. Dialog window of the quadrilateral mesh.

Figure 5.23. Unstructured quadrilateral mesh of the cylinder.

To generate a structured quadrilateral mesh, the user should select the structured property before the third step. Specifically, click MeshStructuredSurfacesAssign size and select the two side surfaces of the cylinder. Press Esc and an Enter value window for the line mesh size appears (Figure 5.24). The default value is 0 m. Enter the value 0.3 m, click Assign, and select all the contour lines of the two surfaces. Press Esc and click Close to close the window. Then execute the operations from step 3 to step 6. The quadrilateral mesh is shown in Figure 5.25. Note that the mesh of the top and bottom surfaces of the cylinder is still unstructured and cannot be generated as the structured one, because both the surfaces have only two contour lines (see Section 5.2). In other words, the cylinder has a mixed structured and unstructured mesh as observed in Figure 5.25.

Figure 5.24. Enter the mesh size for the structured mesh.

Figure 5.25. Structured quadrilateral mesh of the side surfaces.

5.14 EXAMPLE OF MESH GENERATION FOR A CURVE

Users can create a helix by using the command (see Section 4.1.6.6 in Chapter 4) HOBBIESStructureObjectsParametric Lines. The equation that generates the helix is given by: x = 4*sin(4*pi*t), y = 4*cos(4*pi*t), z = 10*t, with t varying from 0 to 1. The resulting helix is shown in Figure 5.26.

Figure 5.26. A helix.

The steps for the wire mesh generation are as follows:

1. Select MeshGenerate mesh to bring up the Mesh generation window (Figure 5.27). The default Wire segment length is 0.75 m for the helix model. The user can edit the value by entering a new value (e.g., 2.0 m), as shown in Figure 5.27. Typically, the segment length is much larger than the radius of the wire. There is no surface structure in this model. Therefore, the surface-related parameters are left as the default.

Because the frequency has not been set the Wire segment length factor is 0. If the frequency is set, the user can also set the Wire segment length through the factor or get a reference value by clicking Suggest. Note that selecting a segment length that is too large deteriorates the geometric shape of the model and reduces the accuracy of the result.

2. Click Ok in the Mesh generation window to generate the mesh.

3. Once the mesh is generated, a Dialog window appears that includes the number of linear elements and nodes, as shown in Figure 5.28.

4. Click the View mesh button in the Dialog window, and right click the mouse button to select LabelAll in option to view the mesh along with the labels (Figure 5.29). The wire mesh is unstructured.

To generate a structured wire mesh, the user should select the structured property for the helix before the first step. Specifically, click MeshStructuredLinesAssign size and an Enter value window for the wire mesh size appears (Figure 5.30). The default value is 0 m. Enter the value 2.0, click Assign, and then select the helix. Press Esc and click Close to close the window. Then execute the operations from step 1 to step 4. The structured wire mesh along with the labels is shown in Figure 5.31. Note that, after applying the structured mesh, the mesh size assigned in step 1 will not alter the situation.

Figure 5.27. Mesh size for wire mesh elements after entering the new value of 2.0.

Figure 5.28. Dialog window of the wire mesh.

Figure 5.29. Unstructured wire mesh of the helix (segment labels are displayed).

Figure 5.30. Enter the mesh size for the structured wire mesh.

Figure 5.31. Structured wire mesh of the helix (segment labels are displayed).

5.15 ASSIGNING ELEMENT SIZES FOR GENERATING THE MESH

HOBBIES automatically correct element sizes according to the shape of the entity to be meshed and its surrounding entities. This default option may be deactivated and reactivated by going to Preferences (see Appendix A), and then selecting the Meshing tab followed by Automatic correct sizes. Sometimes, however, this type of correction is not sufficient and it is necessary to indicate where a mesh with higher accuracy is needed. In these cases, HOBBIES offers various options and methods allowing sizes to be assigned to elements.

In this example, the mesh is generated for each of the following methods for assigning sizes using RFast mesher:

• Assign sizes using default options.
• Assign sizes on points.
• Assign sizes on lines.
• Assign sizes on surfaces.

In the following subsections, the tail of a Boeing 747 airplane shown in the red color in Figure 5.32 will be used as the model. The element type will be quadrilateral (MeshElement typeQuadrilateral).

Figure 5.32. NURBS model of Boeing 747.

5.15.1 Generating a Mesh Using Default Options

1. Select MeshGenerate Mesh.

2. A window (Figure 5.33) appears showing the quadrilateral element size. Leave this default size unaltered, and click OK.

3. A meshing process window opens. Then another window appears with information about the mesh generated. Click View mesh to visualize the mesh (see Figure 5.34).

Note that in the zone highlighted in Figure 5.34, elements are smaller than in the rest of the model. This is because of the shape of the surface in that region. When all meshing preferences are set to their default levels, as for this example, the RFast surface mesher is used. In this way, geometrical entities are meshed hierarchically: First of all, lines are meshed and then the surfaces. The line elements size depends on the shape of surfaces (as can be seen in this example). Also, users can try using the RJump mesher, where element sizes are distributed differently. (Users can go to Utilities and open Preferences, and then click Meshing to change the mesher. See Appendix A and reference [1] for more details.)

Figure 5.33. Mesh generation window for the airplane tail.

5.15.2 Assignment on Points

1. Select MeshUnstructuredAssign size on points. A window appears (Figure 5.35) in which to enter the element size around the points to be chosen. Enter 0.1 and click Assign.

2. Select the point indicated in Figure 5.36 (a). Press Esc to indicate that the selection of point is finished.

3. Select MeshGenerate Mesh.

4. A window opens asking whether the previous mesh should be eliminated. Click Yes.

5. HOBBIES then prompts the user to enter the element size. Accept the default value by clicking OK. Note that elements around the chosen point are more refined, given the selected size 0.1 of these elements (see Figure 5.36).

Figure 5.34. Tail model and the mesh: (a) NURBS model of the tail, (b) resulting mesh of the tail.

Figure 5.35. Mesh size window.

Figure 5.36. Assign mesh size on points: (a) the point around which the mesh will be refined, (b) the mesh with a refinement of elements around the point.

6. Go to Utilities and open Preferences. Click the Meshing tab. In the window that appears, there is the option Unstructured Size Transitions. This option defines the transition gradient of element sizes (size gradient), whose values are between 0 and 1. The greater the size gradient, the greater the change in space. To test this, set the value 0.1 and click Accept.

7. Again, select MeshGenerate Mesh.

8. A window opens asking whether the previous mesh should be eliminated. Click Yes.

9. HOBBIES then prompts the user to enter the element size. Accept the default value by clicking OK. The mesh is shown in Figure 5.37 (a). Notice that the size gradient 0.1 results in a higher density of elements around the point [see Figure 5.37 (a)].

10. Now go back and enter 0.8 in Unstructured Size Transitions. Click Accept. The resulting mesh is shown in Figure 5.37 (b). It can be seen that a lower density of elements around the point is generated as compared with the mesh in Figure 5.37 (a).

Figure 5.37. Mesh elements resulting from refinement around a point: (a) with a size gradient of 0.1, (b) with a size gradient of 0.8.

5.15.3 Assignment on Lines

1. Select MeshUnstructuredAssign size on lines. In the window that appears, enter the size of the elements around the lines that will be chosen. Enter 0.1 and click Assign.

2. Select the lines (Figure 5.38(a)) and press Esc.

3. Select MeshGenerate Mesh.

4. A window opens asking whether the previous mesh should be eliminated. Click Yes.

5. Another window appears in which one may enter the element size. Accept the default value by clicking OK. This results in a high refinement of elements around the chosen lines, given that the selected element size 0.1 is much smaller than that of the rest of the elements in the model {see Figure 5.38 (b)}.

Figure 5.38. Refining the mesh elements around a line: (a) selected line, (b) resulting mesh.

5.15.4 Assignment on Surfaces

1. Select MeshUnstructuredAssign sizes on surfaces. In the window that appears, enter the size of the elements to be assigned on the surfaces that will be chosen. Enter 0.1 and click Assign.

2. Select the surfaces (Figure 5.39(a)) and press Esc.

3. Select MeshGenerate Mesh.

4. A window opens asking if the previous mesh should be eliminated. Click Yes.

5. Another window appears in which you can enter the element size. Leave the default value unaltered, and click OK. This result in a high level of refinement for the elements on the selected surfaces because the value 0.1 is much smaller than that of the rest of the elements in the model {see Figure 5.39(b)}.

Figure 5.39. Refining the mesh elements on a surface: (a) selected surface, (b) resulting mesh.

5.16 CONCLUSION

Different meshing methodologies are presented in this chapter to mesh wires and/or surfaces of complicated structures into segment/quadrilateral elements. Users can generate unstructured or structured meshes, although the default setting is unstructured. Operations to draw, generate, erase, edit and check the quality of the mesh quality are provided. The detailed meshing examples for arbitrarily shaped surfaces and curved lines are presented at the end of this chapter to help the reader more easily understand functions and options available when creating a mesh.

REFERENCES

[1] http://gid.cimne.upc.es/support/manuals.