Size Expression
Use a Size Expression node () to specify the size of mesh elements as an expression evaluated on a grid, 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, tetrahedral, and quadrilateral mesh 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.
Only one Size Expression node per meshing sequence is supported.
To add this node as a global node, right-click a Mesh node and select Size Expression. To add this as a subnode, right-click a Mesh Operations node and select Size Expression. Also see Global vs. Local Attribute Nodes.
Geometric Entity Selection
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:
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Choose Entire geometry to specify the size expression for the entire geometry.
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Choose Domain to specify the domains for the specification of the size expression. Choose Manual from the Selection list to select the domains in the Graphics window or choose All domains to select all domains.
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Choose Boundary to specify the boundaries for the specification of the size expression. Choose Manual from the Selection list to select the boundaries in the Graphics window or choose All boundaries to select all boundaries.
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Choose Edge (3D only) to specify the edges for the specification of the size expression. Choose Manual from the Selection list to select the edges in the Graphics window or choose All edges to select all edges. This option is only available in 3D.
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Choose Point to specify the points for the specification of the size expression. Choose Manual from the Selection list to select the points in the Graphics window or choose All points to select all points. This option is only available in 2D and 3D.
Element Size Expression
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.
Evaluation on a Grid
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.
The Size expression field uses the unit system defined by the component. The evaluated value of the expression is interpreted as a length in the geometry’s unit system.
Size expressions that can be evaluated on boundaries only cannot be used for geometries with domains, even if they are applied only on the boundary.
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, or Cell size, where you specify the side length of a grid cell in the Side length field.
Evaluation on a Solution
From the Solution list, select the solution to use for evaluation.
From the Solution selection list, select which solution that should be used to evaluate error estimates:
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Select Use last to use the last solution.
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Select Use first to use the first solution.
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Select All (the default for Eigenvalue studies) to use all solutions from that study.
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Select Manual to use a specific solution number that you specify as solution indices in the Index field.
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.
Error Indicator
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 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:
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Longest to make the solver refine only the longest edge of an element.
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Regular to make the solver refine elements in a regular pattern.
Use the Element selection list to specify how the element refinement vector is determined from the error expression. Select:
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Rough global minimum to minimize the L2 norm of the error by refining a fraction of the elements with the largest error in such a way that the total number of elements increases roughly by a factor greater than 1 specified in the accompanying Element growth rate field. The default value is 1.7, which means that the number of elements increases by roughly 70%. The Error orders field is 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).
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Fraction of worst error to refine elements whose local error indicator is larger than a given fraction of the largest local error indicator. Use the accompanying Element fraction field to specify the fraction. The default value is 0.5, which means that the fraction contains the elements with more than 50% of the largest local error.
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Fraction of elements to refine a given fraction of the elements. Use the accompanying Element fraction field to specify the fraction. The default value is 0.5, which means that the solver refines about 50% of the elements with the largest local error indicator.
Absolute Size
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).
The Size expression field uses the unit system defined by the component. The evaluated value of the expression is interpreted as a length in the geometry’s unit system.
Size expressions that can be evaluated on boundaries only cannot be used for geometries with domains, even if they are applied only on the boundary.
Evaluation Using an Initial Expression
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 should be 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 Size expression field uses the unit system defined by the component. The evaluated value of the expression is interpreted as a length in the geometry’s unit system.
Size expressions that can be evaluated on boundaries only cannot be used for geometries with domains, even if they are applied only on the boundary.
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.
Smoothing
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.
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Meshing Operations and Attributes
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Boundary Layers