Geometric Variables and Mesh Variables

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.
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. In 2D and 1D domains, it is an area scaling factor and length scaling factor, respectively. If a moving mesh is used, dvol is the mesh element scale factor for the undeformed mesh. The corresponding factor for the deformed mesh is named dvol_spatial.
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 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 mesh type index for the mesh element. This is the number of edges in the element.
meshelement	The mesh element number.
meshvol	Volume of the (linearized) mesh element.
nx, ny, nz	See Normal Variables.
qual	A mesh quality measure.
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.
tx and ty (2D) t1x, t1y, t1z (3D edges and boundaries) t2x, t2y, t2z (3D boundaries)	See Tangent Variables.
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 COMSOL Multiphysics Programming Reference Manual.

When entering the spatial coordinate, parameterization, tangent, and normal geometric variables, replace the letters highlighted below in 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 mesh have a suffix _mesh (for example, h_mesh).

Spatial Coordinate Variables

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

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

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

•	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.

•	If the model includes a deformed mesh, the variables xTIME, yTIME, 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).

•	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 which 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 Spatial Derivatives.

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

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

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

In 3D, nx, ny, and nz define a normal vector pointing outward relative to the 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: • unx, the outward unit normal seen from the upper domain • dnx, the outward unit normal seen from the lower domain

In 2D: • unx and uny for the up direction • 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: • unx, uny, and unz for the up direction • 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.

Normal Vector Continuous Variables

The names are 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). 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 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) 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.