On the duality between hybrid and block elements

In hybrid elements, the fields allowed within each element are equilibrated stress fields, the topology is defined by providing for each element the set of its nodes and the interaction between adjacent elements by setting compatibility conditions in terms of the nodal displacements. Hybrid elements are deformable, although rigidly connected along their common boundaries. The approximate solution is obtained by equalizing the displacements at the coinciding nodes of adjacent elements. In block elements, the fields allowed within each element are continuous displacement fields, the topology is defined by providing for each element the set of adjacent elements and the interaction between adjacent elements by equilibrium conditions in terms of the contact forces. Blocks are themselves rigid, although not rigidly connected to the adjacent elements along their common boundaries. The approximate solution is obtained by equilibrating the generalized contact forces associated to the interfaces of adjacent elements. The conclusion is drawn that the decomposition into hybrid elements provides convergence by approximate solutions stiffer than the exact solution; the decomposition into block elements provides convergence by approximate solutions less stiff than the exact solution.