Derivatives of Dependent Variables
When solving for some physical quantity, u, COMSOL Multiphysics always stores the solution for a fixed set of mesh nodes. That is, the dependent variable u is treated internally as a function of the mesh coordinates and possibly time, u(Xm, Ym, t). The essence of the ALE system is that it allows treating the physical quantities as functions of the material or spatial coordinates, u(X, Y, t) or u(x, y, t), instead. This transformation is possible only if the mappings given by Equation 18-1 and Equation 18-2 are invertible.
Differentiation in Space
With respect to spatial differentiation, each dependent variable is treated as a function of one or more of the frames present in the model. Most physics interfaces are based on a formulation which is either Eulerian or Lagrangian. Their equations therefore contain derivatives of the dependent variables with respect to either the spatial or the material frame, respectively. A few physics interfaces can formulate their equations in either material or spatial frame, as set by the Frame setting found under Discretization in the physics interface node’s settings.
For compactness, physics interface equations are written using derivative variables. For a dependent variable u, there are two basic possibilities:
•
The variable is defined on the spatial frame and its derivatives with respect to the spatial coordinates are variables denoted ux and uy in the software.
•
The variable is defined on the material frame and its derivatives with respect to the material coordinates are variables denoted uX and uY in the software.
But in many cases both sets of derivative variables exist even if they are not used by the physics interface. In addition, the built-in differentiation operator d(expr,var) can always compute derivatives with respect to any set of coordinates, internally using the chain rule. For example, the first component of the spatial, material, and geometry frame gradients of u are, respectively d(u,x), d(u,X), and d(u,Xg).
Differentiation in Time
When using ALE, the software defines two kinds of time derivatives:
•
The common frame time derivative, valid for a fixed point in the frame on which the variable is defined. This derivative is always denoted ut in the software. For example, for u defined on the spatial frame:
•
The mesh time derivative, which is taken for a fixed point in the mesh:
This derivative is denoted uTIME in the software. Since internally, everything is formulated on the mesh frame, the mesh time derivative is the one computed by the solvers and stored in the solution vector.
The two derivatives are related by the chain rule:
where (xTIME, yTIME) is the mesh velocity. The mesh time derivative is often less important from the user point of view because its value depends on the mesh movement, which in itself has no physical significance. However, for the special case when the mesh follows the material’s motion, the mesh time derivative is physically significant and is also called the material time derivative.