Symmetry and periodicity

Symmetric and periodic problems can basically be solved in the PMD system in three ways.

For static and dynamic problems, the displacements of nodes lying in the plane of symmetry are fixed in the name.i2 file, in the direction perpendicular to this plane. Since zero displacements can only be prescribed in the directions of the axes of the global Cartesian coordinate system, this approach can only be used when the planes of symmetry are parallel to the coordinate planes.

For heat transfer problems, it is sufficient to prescribe a zero heat flux in the name.iB file, in the direction perpendicular to the faces of the elements forming the plane of symmetry. For heat transfer problems, this approach is completely general.

For static and dynamic problems, very stiff springs or a Winkler's foundation are prescribed in the name.i2 file, preventing the displacement of nodes in the plane of symmetry in the chosen direction. Unlike the previous case, this approach is not tied to the global coordinate system, since the axes of the springs can be arbitrary. The stiffness of the springs should be approximately 6 orders of magnitude greater than the local stiffness of the body at the given location (which can at least be estimated by order of magnitude).

For heat transfer problems and linear elastostatic problems, a symmetric or periodic segment is declared. During the subsequent factorization of the system matrix, the corresponding boundary conditions are automatically taken into account. The advantage of this approach is its simplicity and the possibility of using general symmetry or periodicity. The disadvantage is that it is limited to heat transfer problems and linear elastostatic problems.

The mesh is generated in the usual way. In the name.i1 file, on the IP line, $\mathtt{KPER}=0$ and $\mathtt{NPSN}=N$ are specified, where $N$ is the number of nodes on the plane of symmetry. On the RP line, the value of $\mathtt{ALPHA}$ does not matter; the angle $\alpha$ is calculated from the coordinates of node $1$. For accurate determination of the angle $\alpha$, node $1$ must not be chosen near the center/axis of rotation.

Formal conditions for a symmetric segment are:

• In 2D problems, the center of rotation must coincide with the origin of the global Cartesian coordinate system.
• In 3D problems, the axis of rotation must coincide with the $z$ axis of the global Cartesian coordinate system.
• Nodes on the plane of symmetry must be numbered from $1$ to $N$. Nodes with zero radius must not occur on the plane of symmetry.
• The opposite plane of symmetry, which coincides with a coordinate axis/plane, is specified in the usual way according to the first method.

The mesh is generated in the usual way. In the name.i1 file, on the IP line, $\mathtt{KPER}=1$ and $\mathtt{NPSN}=N$ are specified, where $N$ is the number of nodes on the periodicity surface. On the RP line, $\mathtt{ALPHA}=\alpha$ is specified, where $\alpha$ is the angle by which the periodicity surface (master) is mapped onto the opposite surface (slave). An integer multiple of the angle $\alpha$ does not necessarily give $360^\circ$, see the figure.

The formal conditions for a periodic segment are:

• In 2D problems, the center of rotation must coincide with the origin of the global Cartesian coordinate system.
• In 3D problems, the axis of rotation must coincide with the $z$ axis of the global Cartesian coordinate system.
• Nodes on the master surface must be numbered from $1$ to $N$. If midside nodes of semi-loof elements occur on the periodicity surface, they must be numbered last.
• Periodically associated nodes on the slave surface must be numbered in the same order, from $\mathtt{J1}=\mathtt{NNOD}-N+1$ to $\mathtt{JN}=\mathtt{NNOD}$.
• The mesh must not contain any element that has one face on the master surface and another face on the slave surface. In such a case, the nodes of one face of the element would be renumbered to the node numbers of the other face, and the element would contain different nodes with identical numbers.

Note
The coordinates of the periodically associated nodes on the slave surface are automatically calculated by the program from the coordinates of the nodes on the master surface and the angle $\alpha$. The coordinates of the periodically associated nodes specified in the name.i1 file are therefore irrelevant.