Use periodic boundary conditions to make the solution equal on two different (but usually equally shaped) boundaries.

To add a periodic boundary condition, in the Model Builder, right-click a physics interface node and select Periodic Condition. The periodic boundary condition typically implements standard periodicity so that u(x0) = u(x1) (that is, the value of the solution is the same on the periodic boundaries). In most cases you can also choose antiperiodicity so that the solutions have opposing signs: u(x0) = −u(x1). Other options such as Floquet periodicity or cyclic symmetry may be available. The periodicity is implemented in such a way that fluxes become periodic in the same way as the solution itself.

For fluid flow physics interfaces, the Periodic Flow Condition provides a similar periodic boundary condition but without a selection of periodicity. Instead, it allows specifying a pressure difference between the source and destination boundaries.

Typically, the periodic boundary conditions determine the source and destination boundaries automatically, but you can also add Destination Selection subnode to manually split the periodic boundary condition’s selection into source and destination selections.

The periodic condition applies a constraint on the destination selection, constraining the solution at each destination point rdst to be equal to the solution at a corresponding source point rsrc. When the periodic condition is applied on surfaces in 3D or edges in 2D, the source point is computed using a rotation of the position relative to the destination and source centers of mass, r0,dst and r0,src:

where R is a rotation matrix encoding the relative orientation of the source and destination boundaries. It is normally determined automatically from the cross product of the source and destination boundary normal directions. That is, the rotation is performed about an axis perpendicular to the plane spanned by the normal directions, which are evaluated at arbitrary points on each boundary.

The orientation settings appear in an Orientation of Source section in the main periodic condition node and in an Orientation of Destination section in a Destination Selection subnode. To display these settings, first select Advanced Physics Options in the Show More Options dialog box. The Orientation of Destination section is only visible if the source orientation has been chosen manually.

In both sections, there is a Transform to intermediate map list. In the Orientation of Source section in the main periodic condition node, its default value is Automatic. Other possible values represent coordinate systems, including all coordinate system nodes defined in the component as well as the canonical Global coordinate system. The latter is the default choice for the Orientation of Destination section in Destination Selection subnodes.

The chosen source and destination coordinate systems define transformation matrices, Tsrc and Tdst, whose row index refers to local coordinate system components, while the column index refers to global coordinates on the source and destination selections, respectively. A rotation matrix as defined by Equation 3-2 is computed by assuming that the source and destination coordinate system coordinates refer to the same basis: