FEA Periodic Boundaries

Theory

Periodic boundary conditions means to run calculations that are periodic in strain in just the x direction, just the y direction, or in both the x and y directions. For axisymmetric problems the calculations can only be periodic in the z direction. Many researchers use symmetry-based displacement conditions for periodic calculations (thinking they are periodic), but that approach cannot handle problems where plane sections do not remain plane even though the structure and strains are periodic. The periodic conditions in NairnFEA are more advanced than simple symmetry conditions and they essential for some problems such as shear strain applied to a periodic composite material.

The figure shows a mesh of width Δx and height Δx in which the left and right edges are parallel to the y axis and each node on the left edge corresponds to a node on the right edge at the same <yy>y coordinate. If this structure is an element on a large object and periodic in the x direction, it means the the strains are periodic or that:

      [math]\displaystyle{ \varepsilon_{xx}(x+\Delta x,y) = \varepsilon_{xx}(x,y), \qquad \varepsilon_{yy}(x+\Delta x,y) = \varepsilon_{yy}(x,y), \qquad {\rm and} \qquad \gamma_{yx}(x+\Delta x,y) = \gamma_{xy}(x,y) }[/math]

These conditions are satisfied, in the most general sense, if the x and y displacements (u(x,y) and v(x,y)) have the form:

      [math]\displaystyle{ u(x+\Delta x,y) = u(x,y) + k_1 + k_2 y \qquad {\rm and} \qquad v(x+\Delta x,y) = v(x,y) + k_3 - k_2 x }[/math]

If there are n nodes on each edge, these conditions are satisfied if the problem is solved subjected to the following constraints:

      [math]\displaystyle{ u_i(\Delta x,y_i) = u_i(0,y+i) + k_1 + k_2 y_i \qquad {\rm for}\ i=1,\ {\rm to}\ n }[/math]

      [math]\displaystyle{ v_i(\Delta x,y_i) = v_i(0,y+i) + v_1(\Delta x,y_1) - v_1(0,y_1) \qquad {\rm for}\ i=2,\ {\rm to}\ n }[/math]

These conditions are solved by using FEA method of Lagrangian multipliers. All x displacements on the right are replaced by constraints as are all y (except v_1(Δx,y_1), which remains a degree of freedom. In addition, k₁ and k₂ are introduced as two new degrees of freedom. Physically, k₁ is the jump in displacement across the modeled object and give the average x direction strain:

      [math]\displaystyle{ \left\langle \varepsilon_{xx}\right\rangle = {k_1\over \Delta x} }[/math]

The dof attribute defines the periodic direction. It must be x (or 1) or y (or 2) or z (or 3). The delta, slope, and shear attributes let you specify an average displacement jump, rotation, and/or shear between the two ends in the periodic direction(s). If any are left unspecified, they will be degrees of freedom. When specified, they correspond to globally applied strains or rotation. The meanings of the parameters are slightly different depending on whether just one direction or both directions are periodic: