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, 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.
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. They therefore lock their dependent variables to 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 a dependent variable u, there are typically two possibilities:
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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:
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The mesh time derivative, which is taken for a fixed point in the mesh:
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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.
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.