Energy Estimates in Hierarchical Plate Theories

The starting assumption of hierarchical plate theories is a representation of the displacement field in the form u n. x, z.s nks0fk z.b kn. x., where x stands for the in-plane coordinates and z for the transverse coordinate, where the linearly independent functions fk are the first nq1.in a complete system, and where the functions b kn. are determined by solving a minimum problem in an approximation space V n.. For V the function space where the exact solution u is sought to the three-dimensional problem that a given hierarchical plate theory is meant to approximate, we show that the sequence u n.4 converges in energy to a limit element u `.gV , whatever the functions fk; and that, iffk z.sz k, then u and u `. coincide pointwise, provided their difference is smooth.