Geometric Variables, Mesh Variables, and Variables Created by Frames

The variables that characterize geometric properties and the mesh are listed in Table 5-9, with detailed descriptions for some of the variables following the table.

Table 5-9: Geometric Variables and Mesh Variables
Variable	Description
curv curv1, curv2	The curvature of a boundary in 2D is called curv. A boundary in 3D has two principal curvatures corresponding to the minimal and maximal normal curvatures. They are called curv1 and curv2, respectively. See Curvature Variables for details.
dGeomChar	The diagonal size of a bounding box containing all of the geometry.
dnx, dny, dnz, dnxmesh, dnymesh, dnzmesh	Normal vector components pointing toward the upside of a boundary. SeeDirection of the Normal Component on Interior Boundaries.
dom	The domain number, the boundary number, the edge number, or the vertex (point) number (all are integer values). The variable sd also exists as an alias but is considered obsolete.
dvol	The volume scale factor variable, dvol, is the determinant of the Jacobian matrix for the mapping from local (element) coordinates to global coordinates. For 3D domains, this is the factor that the software multiplies volumes by when moving from local coordinates to global coordinates. On surfaces in 3D and domains in 2D, it is an area scaling factor. On edges in 3D and 2D, and in 1D domains, it is a length scaling factor. In points of all dimensions, dvol is always equal to 1 by definition. If a moving mesh is used, dvol is the mesh element scale factor for the material frame mesh. The corresponding factor for the spatial mesh is named dvol_spatial. Similarly, for geometry frame and mesh frame, the factors are named dvol_geometry and dvol_mesh, respectively.
emetric2	Squared norm of the element metric emetric(vector). See emetric for details.
geomapproxdist	Indicate, for each element, how far a node point in each element was moved from the geometry. See Avoiding Inverted Mesh Elements for more information.
h	Available on all geometric entities, the variable h represents the mesh element size in the material/reference frame (that is, the length of the longest edge of the element).
linearizedelem	In some calculations, COMSOL Multiphysics forces mesh elements to become linear. This variable returns one inside such an element and zero otherwise. Note that the faces of the linearized mesh elements are not considered to be linearized themselves. You can use this variable to identify mesh elements with linearized elements.
meshtype	The element type index for the mesh element. This is the number of edges in the element.
meshelement	The mesh element number for each element type. The values range from 1 to the number of elements of the corresponding element type.
meshelementall	A unique numbering for all elements in the mesh, canonized on the meshtype (vtx, edg, tri, quad, tet, pyr, prism, and hex).
meshvol	Volume of the (linearized) mesh element.
nx, ny, nz	See Normal Variables.
nxc, nyc, nzc	Continuous mesh-based normals. See Normal Vector Continuous Variables.
nxmesh, nymesh, nzmesh	Mesh-based normals. See Normal Vector Variables Representing Element Surface Normals.
qualskewness, qualmaxangle, qual, qualvollength, qualcondition, qualgrowth, qualcurvedskewness	Mesh quality measures. See Mesh Element Quality. See also Frame Variables.
reldetjac reldetjacmin	The determinant of the Jacobian matrix for the mapping from the straight mesh element to the possibly curved element used when solving. Use this variable to measure the difference in shape between a curved element and the corresponding straight element. The variable reldetjacmin is a scalar for each element defined as the minimum value of the reldetjac variable for the corresponding element. A reldetjacmin value less than zero for an element means that the element is wrapped inside-out; that is, the element is an inverted mesh element.
s, s1, s2	See Parameterization Variables.
tcurvx, tcurvy (2D) tcurv1x, tcurv1y, tcurv1z, tcurv2x, tcurv2y, tcurv2z (3D)	Tangential directions for the corresponding curvatures. See Curvature Variables for more information.
tremetric	Trace of the element metric emetric(vector). See emetric for details.
tx and ty (2D) t1x, t1y, t1z (3D edges and boundaries) t2x, t2y, t2z (3D boundaries)	See Tangent Variables.
unx, uny, unz, unxmesh, unymesh, unzmesh	Normal vector components pointing toward the downside of a boundary. See Direction of the Normal Component on Interior Boundaries.
x, y, z r, z	See Spatial Coordinate Variables.
xi1, xi2, xi3	Local (barycentric) coordinates ξ i in each mesh element; see the section Finite Elements in the Elements and Shape Functions chapter.


	When entering the spatial coordinate, parameterization, tangent, and normal geometric variables, replace the letters highlighted below in an italic font with the actual names for the dependent variables (solution components) and independent variables (spatial coordinates) for the Component node.

For example, replace u with the names of the dependent variables in the model, and replace x, y, and z with the first, second, and third spatial coordinate variable, respectively. xi represents the ith spatial coordinate variable. If the model contains a deformed mesh or the displacements control the spatial frame (in solid mechanics, for example), you can replace the symbols x, y, and z with either the spatial coordinates (x, y, and z by default) or the material (reference) coordinates (X, Y, and Z by default).

The variables curv, dvol, h, qual, reldetjac, and reldetjacmin are based on the mesh viewed in the material (reference) frame. If you have a moving mesh, the corresponding variables for the mesh viewed in the spatial frame have a suffix _spatial (that is, curv_spatial, dvol_spatial, and so on). If you use a deformed geometry, the corresponding variables for the original, undeformed geometry have a suffix _geometry (for example dvol_geometry). After automatic remeshing, variables referring to the current mesh have a suffix _mesh (for example, h_mesh).

Spatial Coordinate Variables

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The spatial coordinate variables (independent variables) are available for all domain types.

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For a Cartesian geometry the default names for the spatial coordinates are x, y, and z (for the x, y, and z coordinates).

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For axisymmetric geometries the default names for the spatial coordinates are r, phi, and z (for the r,

, and z coordinates).

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If a deformed mesh is used, x, y, z can be both the spatial coordinates (x, y, z) and the material/reference coordinates (X, Y, Z); see Mathematical Description of the Mesh Movement.

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If the model includes a deformed mesh, the variables xTIME, yTIME, and zTIME represent the mesh velocity. To access these variables, replace x, y, and z with the names of the spatial coordinates in the model (x, y, and z).

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If the model includes a deformed geometry, the default names for the spatial coordinates in the geometry frame and the mesh frame are Xg, Yg, and Zg (for the xg, yg, and zg coordinates).


	Most physics interfaces are based on a formulation that is either Eulerian or Lagrangian. They therefore lock their dependent variables to the spatial or the material frame, and the spatial derivatives are then defined with respect to x, y, and z (spatial frame) or X, Y, and Z (material frame), when using the default names for the spatial coordinates. See Differentiation in Space.

Parameterization Variables

The surface-boundary parameterization variables can be useful for defining distributed loads and constraints such as a parabolic velocity profile. The available parameterization variables are:


	The curve parameter s (or s1) in 2D. Use a line plot to visualize the range of the parameter, to see if the relationship between x and y (the spatial coordinates) and s is nonlinear, and to see if the curve parameterization is aligned with the direction of the corresponding boundary. In most cases it runs from 0 to 1 in the direction indicated by the arrows shown on the edges when in the boundary or edge selection mode and if you have selected the Show edge direction arrows check box in the Settings window for View (). You can use s on boundaries in 2D when specifying boundary conditions.


	The arc length parameter s1 available on edges in 3D. It is approximately equivalent to the arc length of the edge. Use a line plot to visualize the values of s1. The surface parameters s1 and s2 in 3D are available on boundaries (faces). They can be difficult to use because the relationship between x, y, and z (the spatial coordinates) and s1 and s2 is nonlinear. Often it is more convenient to use expressions with x, y, and z for specifying distributed boundary conditions. To see the values of s1 and s2, plot them using a surface plot.

Tangent and Normal Variables

The tangent and normal variables are components of the tangential and normal unit vectors.

Tangent Variables


	In 2D, tx and ty define the curve tangent vector associated with the direction of the boundary.


	In 3D, the tangent variables t1x, t1y, and t1z are defined on edges. The tangent variables t1x, t1y, t1z, t2x, t2y, and t2z are defined on surfaces according to These most often define two orthogonal vectors on a surface, but the orthogonality can be ruined by scaling geometry objects. The vectors are normalized; ki is a normalizing parameter in the expression just given.

If a deformed mesh is used, the tangent variables are available both for the deformed configuration and for the undeformed configuration. In the first case, replace x, y, and z with the spatial coordinate names (x, y, and z by default). In the second case, replace x, y, and z with the material/reference coordinate names (X, Y, and Z by default).

Normal Variables

The variables components nx, ny and nz make up a vector that normally points from the downside toward the upside of a boundary. Boundaries between meshed and unmeshed domains constitute an exception where the normal vector instead points from the meshed domain toward the unmeshed domain. On interior boundaries, which side is up and which side is down is arbitrary, but the void surrounding a geometry is always on the upside, so on boundaries exterior to the geometry, the normal points away from the geometry.

For frame coordinates x, y, z which are identical to the mesh frame coordinates, the normal is evaluated based on the geometry (if a geometry is available). Otherwise, the components are evaluated based on the mesh element shapes, and are identical to the corresponding nxmesh, nymesh, nzmesh variables. See Normal Vector Variables Representing Element Surface Normals. This, for example, happens for the spatial frame normal vector when there are Moving Mesh features or a Solid Mechanics interface in the model


	In 1D, nx is the outward unit normal pointing out from the meshed domain.


	In 2D, nx and ny define a normal vector pointing outward relative to the meshed domains.


	In 3D, nx, ny, and nz define a normal vector pointing outward relative to the meshed domains.

Direction of the Normal Component on Interior Boundaries

To get control of the direction of the normal component on interior boundaries, the following variables are available:

In 1D:

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unx, the outward unit normal seen from the upper domain

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dnx, the outward unit normal seen from the lower domain

In 2D:

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unx and uny for the up direction

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dnx and dny for the down direction

The upside is defined as the left side with respect to the direction of the boundary.

In 3D:

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unx, uny, and unz for the up direction

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dnx, dny, and dnz for the down direction

To visualize any of these vector variables use arrow plots on surfaces or lines.

If a deformed mesh is used, the normal variables are available both for the deformed configuration and for the undeformed configuration. In the first case, replace x, y, and z with the spatial coordinate names (x, y, and z by default). In the second case, replace x, y, and z with the material/reference coordinate names (X, Y, and Z by default).

Normal Vector Variables Representing Element Surface Normals

A similar set of variables — nxmesh, unxmesh, and dnxmesh, where x is the name of a spatial coordinate — use the element shape function and are normal to the actual element surfaces rather than to the geometry surfaces. These normal vectors always have unit length, but they are typically not continuous at interelement boundaries.

Normal Vector Continuous Variables

In some situations, it can be necessary to use a normal vector which is based on the mesh element shape functions, but is still continuous. Such normal vector components are defined with names similar to those for the standard normal vector variables, except that they have a c appended at the end: For example, nxc, nyc, and nzc in a 3D model or nrc and nzc in a 2D axisymmetric model. If you have a material frame and a spatial frame in a 3D model, the normal vector continuous variables are nxc, nyc, and nzc for the spatial frame and nXc, nYc, and nZc for the material frame.

These variables are continuous within each boundary (but typically discontinuous where boundaries meet), but they do not necessarily have unit length. It is possible to compute their tangential derivatives with the dtang operator as, for example, dtang(nxc,x). Computing tangential derivatives in this way works only when the normal variable and the coordinate in the second argument of dtang belong to the same frame; dtang(nxc,x) and dtang(nXc,X) both work, but dtang(nXc,x) and dtang(nxc,X) are both 0.

Curvature Variables

The curvature variables are defined on boundaries in 2D and 3D.

In 2D, the curvature is denoted curv. Positive curvature is toward the normal (nx, ny).

In 2D axisymmetry, the curvature variables refer to the curvature of the 2D geometry and use the same naming as in 2D. The second curvature, which is induced by the axisymmetry, is -nr/r (using the default radial space variable name r) and has no predefined variable.

In 3D, there are two principal curvatures named curv1 and curv2, where curv1 is less than curv2 and seen as real numbers. These correspond to the minimal and maximal values for the curvature of a curve you get by intersecting the boundary with a plane in which the normal lies. Positive curvature is toward the normal (nx, ny, nz).

The components of the normalized tangential directions for the corresponding curvatures are called tcurvx, tcurvy in 2D and tcurv1x, tcurv1y, tcurv1z, tcurv2x, tcurv2y, and tcurv2z in 3D. The tangents (tcurv1x,tcurv1y,tcurv1z) and (tcurv2x,tcurv2y,tcurv2z) are orthogonal.

Curvature variables are defined for all separate frames in a model. The names of the curvature variables in the spatial, mesh, and geometry frame are formed by appending the suffix _spatial, _mesh, and _geometry, respectively, to the name curv (in 2D and 2D axisymmetry) or curv1 and curv2 (in 3D). The variables without suffix always refer to the curvature in the material frame. Note that the variables with suffix are defined only if the spatial, mesh, or geometry frame actually is different from the material frame. For more information about frames and deformed mesh configurations, see Deformed Mesh Fundamentals.

In the normalized tangent variable names, replace x, y, and z with the coordinate names in another frame to get the tangents in that frame.

Frame Variables

The following predefined variables are created by the spatial frame and can be of interest, for example, to monitor the quality of the mesh and define a stop criterion for remeshing (see Adding a Stop Condition):

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The local relative element volume, spatial.relVol, is a quantity that measures the local volumetric distortion of the elements. When this measure approaches zero in some part of the mesh, frame transformations become singular causing solvers to fail.

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The minimum relative element volume, spatial.relVolMin, must be > 0; otherwise, the mesh elements are inverted. A suitable stop criterion using this variable is that the minimum relative element volume must be larger than a small positive number.

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The maximum relative element volume, spatial.relVolMax, is a positive scalar number that represents the maximum value of the relative element volume.

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The minimum mesh quality, spatial.minqual, must be > 0; an acceptable mesh quality is typically larger than 0.1 (where the quality measure is a number between 0 and 1).