A. hexahedron element formulation with a new multi-resolution analysis
  A multi-resolution hexahedron element and method is presented with a new multi-resolution analysis (MRA) framework. The MRA framework is formulated out of a mutually nesting displacement subspace sequence, whose basis functions are constructed of scaling and shifting on the element domain of a basic node shape function. The basic node shape function is constructed from extending to other seven quadrants around a specific node of a node isoparametric shape function. The MRA endows the proposed element with the resolution level (RL) to adjust the element node number, thus modulating structural analysis accuracy accordingly. As a result, the traditional 8-node hexahedron element and method is a mono-resolution one and also a special case of the proposed element and method. The accuracy of a structural analysis is actually determined by the RL, not by the mesh. The simplicity and clarity of shape function construction with the Kronecker delta property and the rational MRA enable the proposed element method to be more rational, easier and efficient in its implementation than the conventional mono-resolution solid element method or other MRA methods. The multi-resolution hexahedron element is more adapted to dealing with the accurate computation of structural problems. 
Yi Ming Xia, Shao Lin Chen
physics.comp-ph math.NA