Choosing the Right Space Dimension

Most of the problems solved with COMSOL Multiphysics are three-dimensional (3D) in the real world. In many cases it is sufficient to solve a two-dimensional (2D) or one-dimensional (1D) problem that is close, or equivalent, to the real 3D problem. 2D models are easier to modify and generally solve much faster, so modeling mistakes are easier to find when working in 2D. Once the 2D model is verified, you are in a better position to build a 3D model.

Not all physics interfaces are available in all space dimensions. See the documentation for the physics interfaces in COMSOL and its modules for the supported space dimensions.

1D Models

The following is a guide for some of the common approximations made for 1D models. Remember that modeling in 1D usually represents some 2D or 3D geometry under the assumption that nothing changes in the other dimensions.

Cartesian Coordinates

In a 1D model you view a single straight line that represents the only space dimension where there is spatial (or other) variation.

Axial Symmetry (Cylindrical Coordinates)

In an axially symmetric 1D model you view a straight line that represents the radial direction in an axially symmetric geometry.

2D Models

Cartesian Coordinate Systems

In this case you view a cross section in the xy-plane of the actual 3D geometry. The geometry is mathematically extended to infinity in both directions along the z-axis, assuming no variation along that axis. All the total flows in and out of boundaries are per unit length along the z-axis. A simplified way of looking at this is to assume that the geometry is extruded one unit length from the cross section along the z-axis. The total flow out of each boundary is then from the face created by the extruded boundary (a boundary in 2D is a line).

There are usually two approaches that lead to a 2D cross-sectional view of a model:

•	When there is no variation of the solution in one particular dimension.

•	When there is a problem where the influence of the finite extension in the third dimension can be neglected.

In some applications there are special 2D assumptions, such as the plane strain and plane stress conditions for 2D stress analysis in solid mechanics.

In addition to the unit-depth assumption, some physics interfaces (for solid mechanics and heat transfer, for example) provide the thickness as a user-defined property in 2D models. For heat transfer, the thickness is used when including out-of-plane heat transfer in the model.

Axial Symmetry (Cylindrical Coordinates)

If the 3D geometry can be constructed by revolving a cross section about an axis, and no variations in any variable occur when going around the axis of revolution, an axisymmetric physics interface can be used.

The spatial coordinates are called r and z, where r is the radius. The flow at the boundaries is given per unit length along the third dimension. Because this dimension is a revolution, you have to multiply all flows with αr, where α is the revolution angle (for example, 2π for a full turn). COMSOL Multiphysics provides this as an option during postprocessing.

Geometric Variables and Mesh Variables

3D Models

This section discusses 3D geometry modeling practices.

Although COMSOL fully supports arbitrary 3D geometries, it is important to simplify the problem. This is because 3D models easily get large and require more computer power, memory, and time to solve. The extra time spent on simplifying a problem is probably well spent when solving it.

Is it possible to solve the problem in 2D?

Given that the necessary approximations are small, the solution is more accurate in 2D because a much denser mesh can be used. See 2D Models if this is applicable.

Are there symmetries in the geometry and model?

Many problems have planes where the solution on either side of the plane looks the same. A good way to check this is to flip the geometry around the plane by, for example, turning it upside down around the horizontal plane. You can then remove the geometry below the plane if you do not see any differences between the two cases regarding geometry, materials, and sources. Boundaries created by the cross section between the geometry and this plane need a symmetry boundary condition, which is available in all 3D physics interfaces.

Do you know the dependence in one direction so it can be replaced by an analytical function?

You can use this approach either to convert 3D to 2D or to convert a layer to a boundary condition.

The Coordinate Systems and the Space Dimension

COMSOL uses a global Cartesian or cylindrical (axisymmetric) coordinate system. You select the geometry dimension and coordinate system when creating a new model. The default variable names for the spatial coordinates are x, y, and z for Cartesian coordinates and r, ϕ, and z for cylindrical coordinates. These coordinate variables (together with the variable t for the time in time-dependent models) make up the independent variables in COMSOL models.

The labels assigned to the coordinate system variables vary according to the space dimension:

•	Models that are opened using the space dimensions 1D, 2D, and 3D use the Cartesian coordinate independent variable labels x, y (2D and 3D), and z (3D).

•	In 2D axisymmetric geometries, the x-axis represents the r label, which is the radial coordinate, while the y-axis represents the z label, the height coordinate.

•	In 1D axisymmetric geometries, the default radial coordinate is labeled r, and represented by the x-axis.

For axisymmetric cases the geometry model must fall in the positive half plane (r ≥ 0).

• Creating a New Model • Coordinate Systems • Cylindrical System