About Frames
The COMSOL Multiphysics software refers to the spatial, material/reference, geometry, and mesh coordinate systems described above as spatial frame, material frame (reference frame), geometry frame, and mesh frame, respectively. Physics can be formulated in the spatial frame or in the material frame, depending on whether it is more convenient to interpret the equations as Eulerian or Lagrangian, respectively. It is not possible to use the geometry and mesh frames and their associated coordinates to formulate physics because they are neither connected to the material nor to the true Euclidean space.
Conceptually, all four frames always exist, with their own separate coordinate names:
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The spatial frame coordinates are by default x, y, z, or r, phi, z in an axisymmetric geometry.
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The material frame coordinates are by default X, Y, Z, or R, PHI, Z in an axisymmetric geometry.
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The geometry frame coordinates are by default Xg, Yg, Zg, or Rg, PHIg, Zg in an axisymmetric geometry.
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The mesh frame coordinates are by default Xm, Ym, Zm or Rm, PHIm, Zm in an axisymmetric geometry.
You can change these names in the Settings window of a Component. Initially, all frames coincide and their coordinates evaluate to the same value at any given point in the mesh; technically, the spatial, material and geometry coordinates are all aliases for the mesh coordinates.
When a Moving Mesh feature is added to a component, or a structural mechanics interface is added and Include geometric nonlinearity is switched on in a study step, the spatial frame is separated from the material frame. From this point, spatial and material coordinates will evaluate differently at a given point in the mesh. Eulerian and Lagrangian formulations behave differently because they, among other things, define derivatives with respect to different sets of independent variables.
The geometry frame and the material frame coordinates coincide until a Deformed Geometry feature is added, or you enable shape optimization (available with the Optimization Module). From that point, the geometry frame coordinates refer to the geometry as it is represented by the Geometry Sequence, while the material frame coordinates represent the geometry seen by physics interfaces. By inserting a nontrivial transformation from geometry coordinates to material coordinates, the shape of the geometry can be effectively changed without having to create a new mesh. This can be useful as a means of parameterizing the geometry, for example, before performing optimization or sensitivity analysis.
Using Deformed Geometry features affects both Eulerian and Lagrangian physics in the same way. The reason is that the Deformed Geometry interface controls the material frame in relation to the geometry frame. Unless there is also a Moving Mesh or Solid Mechanics interface present, the material frame and spatial frame coordinates still coincide, so the spatial mesh is deformed in the same way as the material mesh. The three frames refer to three different sets of coordinates only when there is both some Deformed Geometry feature and some Moving Mesh feature active in the Component.
The geometry frame and the mesh frame coincide until a manual or automatic remeshing operation is performed. At that point, a new mesh is created in the original geometry, and mesh frame coordinates are associated with this new mesh. The original geometry coordinates are mapped and stored together with the new mesh such that any Deformed Geometry interface can still define the material frame relative to the original geometry shape.
To avoid confusion, note that:
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The spatial frame is the usual, fixed, global, Euclidean coordinate system with the spatial coordinates (x, y). In the ALE context, the spatial coordinate system as such is fixed, while the spatial coordinates (x, y) of each material point and mesh node can be functions of time. Therefore, it is correct to refer to the model as having a moving mesh.
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The material frame is a coordinate system that identifies material points by their spatial coordinates (X, Y) in some — actual or imagined — reference configuration. Think of the material coordinate system as having been printed on the material in the reference configuration such that it follows it during deformation. It is therefore in general curvilinear and cannot be used directly to measure true distances and angles. See also Figure 18-1 and Figure 18-2.
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The geometry frame is a coordinate system that identifies points by their spatial coordinates (Xg, Yg) in the original geometry. It is often natural to use the original geometry also as reference state to define material coordinates. Therefore, the geometry frame and material frame usually coincide. The only exception is when some Deformed Geometry feature is used to deform or parameterize the original geometry.
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The mesh frame is a coordinate system used internally by the finite element method. It identifies mesh points by their spatial coordinates (Xm, Ym) at the time the mesh was created. The original mesh is always created based on the original geometry. Therefore, the mesh frame coincides with the geometry frame until a new mesh is created in the — then current — deformed configuration.
Figure 18-1: An undeformed mesh. In the reference configuration, which can be the actual configuration at a reference time or a hypothetical state, the spatial frame (x, y) and the material frame (X, Y) coincide.
Figure 18-2: After deformation of the material, the spatial frame (x, y) remains the same, while the material coordinate system (X, Y) has been deformed, following the material. Meanwhile, the material coordinates of each material point remain the same but its spatial coordinates have changed.