A finite element defines a number of variables, typically a dependent variable and its derivatives. Such variables are called shape functions variables because they are computed directly from shape functions and the degrees of freedom.

For example, replace u with the names of the dependent variables in the individual Component, 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 Component contains a deformed mesh or the displacements control the spatial frame (in solid mechanics, for example), you can replace the symbols x, y, z with either the spatial coordinates (x, y, and z by default) or the material/reference coordinates (X, Y, and Z by default).

For the Lagrange element, which is the element type used by most physics interfaces, Table 5-10 lists the available variable names, assuming you gave the name u as the argument to the shape function, and that the names x, y, and z are provided for the independent variables.

For example, with a fluid flow physics interface, you get the set of variables indicated in Table 5-10 for u, v, w, and p, respectively.

If the model uses a deformed mesh, each finite element is associated with a certain frame (the spatial frame or the material frame). The frame determines the names of the variables generated by the finite element. For instance, if the spatial frame is used, the Lagrange element computes derivatives with respect to the spatial coordinates, ux, uy, and uz. If the material frame is used, the Lagrange element computes derivatives with respect to the material coordinates uX, uY, and uZ.

The variable ut is the time derivative of the dependent variable u. You can also form mixed space-time derivatives as uxit, for example, uxt,

If the model contains a deformed mesh, there is, in addition to the usual time derivative ut, the mesh time derivative uTIME. This also holds for mixed space-time derivatives.

On boundaries, edges, and points you also have access to tangential derivative variables. They have names such as uTx, uTy, and uTz. Using these variables, it is possible to create models with phenomena on boundaries, edges, or points as described with PDEs.

In this equation, (∇u)T is the tangential gradient, which consists of the tangential derivatives in each spatial direction, I is the unity tensor, n is the outward unit normal vector, and ∇u is the gradient of u.

If weak constraints are activated for boundary conditions that are constraints (Dirichlet boundary conditions), COMSOL Multiphysics adds variables for the Lagrange multipliers (one for each dependent variable) by adding _lm as a suffix to the dependent variable name. For example, for a dependent variable u, the corresponding Lagrange variable is u_lm. The Lagrange multipliers are available on boundaries, and you can also evaluate them on edges (in 3D) and points (in 2D and 3D).

On boundaries, edges, and points, gradients and second derivatives of the shape functions are available by inheritance; that is, the average of the values of the variables from the adjacent domains are computed. This process can progress for several levels.

For example, ux is the average on a boundary from the adjacent domains, then the average on an edge from the adjacent boundaries, and finally, the average at the points from the adjacent edges.

If possible, avoid using variable inheritance for gradients and second derivatives in a model. Instead, use the tangential derivative variables for equation-based modeling on boundaries, or use the up(expr), down(expr), and side(dom,expr) as appropriate to move the evaluation to the adjacent domain.

For computations of integrals of reaction forces and fluxes, use the reacf operator.

For high accuracy reaction forces and fluxes in other circumstances, use weak constraints and Lagrange multipliers on boundaries instead of directly accessing the gradient through inheritance (see Computing Accurate Fluxes).