Convergence requirements for displacement functions

For the FE solution to converge to the exact one as the mesh is refined, the assumed displacement function must satisfy these conditions.

Requirements

1. Completeness. The function must be able to represent rigid-body motion (constant terms) and constant strain states. For example, a bar needs $u = a_1 + a_2x$: $a_1$ is rigid-body translation and $a_2$ is constant strain. As elements shrink, the strain in each becomes nearly constant, so this is essential.
2. Compatibility (conformity). Displacements must be continuous across element boundaries so that no gaps or overlaps develop. For second-order problems (bars, plane elasticity) this means $C^0$ continuity; for beams the slope must also be continuous ($C^1$).
3. Rigid-body modes. An element should have zero strain energy under rigid-body movement.
4. Geometric isotropy (invariance). The function should not prefer any direction, so the result should not change if the local axes are rotated. This needs complete polynomial terms of a given order.

Elements that satisfy all the conditions converge monotonically. Elements that satisfy completeness but not compatibility (non-conforming elements) can still converge, provided they pass the patch test.

Pascal's triangle

Order 0:           1
Order 1:         x   y
Order 2:       x2  xy  y2
Order 3:     x3  x2y xy2  y3
Order 4:   x4  x3y x2y2 xy3 y4

• Terms are taken row by row so that the polynomial is complete up to a given degree and has geometric isotropy.
• The number of terms must equal the number of nodal degrees of freedom: 3 terms (1, x, y) for the 3-node triangle; 4 terms (1, x, y, xy) for the 4-node rectangle, where $xy$ is chosen because it is symmetric.
• Terms must be selected symmetrically about the vertical axis of the triangle to maintain isotropy.