Use the Size Expression () attribute to specify the size of mesh elements as an expression evaluated on a grid (as shown in Figure 8-101), a computed solution, or a temporary solution corresponding to the initial value of a study step.

It is possible to add a Size Expression node both as a global node and as a local node. Free Triangular, Free Tetrahedral, and Free Quad operations that follow below the Size Expression node in the meshing sequence take the size expression into account.

For some types of adaptation and error estimation in the study settings (using mesh initialization), the adaptive solver add a Size Expression node to the meshing sequences that represent the adaptive mesh generation, but you can also add a Size Expression node to set up a mesh size expression of your choice. Right-click a Mesh node to add a Size Expression node from the context menu.

To add Size Expression () as a global node, choose one of the following:

To add Size Expression () as a local attribute, right-click a mesh operation and select Size Expression.

Enter the properties for Size Expression using the following sections:

In this section you define the geometric entities where you want to specify a size expression. Choose the level of the geometry from the Geometric entity level list:

From the Evaluate on list, choose Grid (the default), Solution, or Initial expression to specify whether the evaluation is done on a grid, a solution, or an expression evaluated using initial values.

Use the Size expression field to enter an expression for the absolute element size as a function of x, y, and z (depending on the space dimension). You can also use parameters and global functions, variables, and materials in the expression, but no features from the component are available. The actual size used in a coordinate is the minimum of the specified size expression and the result of the other size parameters (such as the maximum element size and the curvature factor), but this minimum is cut off by the specified minimum element size on a given entity, which acts as a lower bound. If the expression evaluates to something smaller than the minimum element size, you get a warning that the value was cut off.

To control the grid, you can specify the Grid type, which can be Resolution, where you specify the number of isotropic cells per dimension in the grid in the Number of cells per dimension field (which only accepts positive integer constant values), or Cell size, where you specify the side length of a grid cell in the Side length field.

From the Solution list, select the solution to use for evaluation.

From the Solution selection list, select which solution to use for evaluate error estimates:

In the Weights field (only available when Solution selection is Manual or All), enter weights as a space-separated list of positive (relative) weights so that the error estimate is a weighted sum of the error estimates for the various solutions (eigenmodes). The default value is 1, which means that all the weight is put on the first solution (eigenmode). That is, any omitted weight components are treated as zero weight.

From the Type of expression list, choose Error indicator (the default) or Absolute size.

If you select Error indicator, the following settings appear:

The Error expression list, where you can add expressions for the error, for example, in terms of the dependent variables (including the variables for error estimation). You can use any expression, including field variables and their derivatives, defined in the domain. Use the Active column to control which error expressions that are active by selecting or clearing the corresponding check box.

The Error order column is available when Element selection is set to Rough global minimum and forms an array of h-exponents for the decrease of the error expression. This array has the same indexing as the error expression’s indexing (for an error expression with two expressions, such as errexpr1, errexpr2, and error orders = 2, 3, it means that errexpr1 = O(h2) and errexpr2 = O(h3)). When the adaptation method is used, these numbers are filled in automatically based on the Residual order and Error estimate settings (on the adaptation study). The shape functions can also influence this order. Note that these order numbers need to be positive to generate additional elements (element growth).

Use the Move Up (), Move Down (), Add ()and Delete () buttons as needed. If you use several expressions, such as one for each solution component, the error expressions are added to form the sum of all of them. Or right-click a table cell and select Move Up, Move Down, Add, or Delete.

In 2D, use the Refinement method list to control how to refine mesh elements. Select:

Use the Element selection list to specify how the element refinement vector is determined from the error expression. Select:

If you select Absolute size, the following setting appears:

In the Size expression field, type an expression for the element size (in terms of variables defined by the solution).

With this option, you can use a size expression that contains variable values in physics, material data, and settings made in the study (such as a frequency), with the same evaluation as you get when running Get Initial Value for Step on a study step. A typical application could be to adapt the mesh size to a (nonconstant) wavelength.

From the Mesh list, select a mesh (on the same geometry as the Size Expression node) on which the interpolation is done. If Automatic (the default) is chosen, a mesh will be created automatically.

From the Study step list, choose a study step (for which Get Initial Value for Step will be run). If you choose None (the default), an empty study is created. The mesh that was specified above is used in the referenced or empty study step that you choose. These actions take place in a copy of the model and are therefore not visible in the Model Builder tree.

Use the Size expression field to enter an expression for the absolute element size as a function of x, y, and z (depending on the space dimension). You can also use variables and materials from the physics, parameters, and global functions and materials in the expression. The actual size used in a coordinate is the minimum of the specified size expression and the result of the other size parameters (such as the maximum element size and the curvature factor), but this minimum is cut off by the specified minimum element size on a given entity, which acts as a lower bound. If the expression evaluates to something smaller than the minimum element size, you get a warning that the value was cut off.

The following settings are available under External changes:

There is a caching scheme that caches various data required to compute the mesh size field. If you have changed something in the model, which has not been picked up by the code and thus invalidated the cache, you can clear the cache by clicking Reevaluate with Updated Model.

The selection in the Update when parameter is changed list provides the possibility to have the cache cleared whenever the values of the selected parameter are changed.

In this section, you can control the amount of smoothing of the mesh size expression. The Maximum size field growth rate (default: 1.4) controls how fast the size field can grow. It is a factor that must be larger than 1.0. Use a higher value to make the mesh coarser closer to areas with a small mesh size, which results in fewer mesh elements.