10. Three-Dimensional Macroelements

This chapter deals with three-dimensional macroelements (large solid bricks) based on several CAD-based interpolation formulas. First, three alternative expressions are derived for the boundary-only Coons interpolation; the first of them is complete, whereas the next two cover particular cases. It will be shown that the classical eight-node trilinear and the twenty-node triquadratic solid elements are the simplest ones of the Coons family. Second, Gordon interpolation in conjunction with internal nodes is fully explained, and it is shown that the classical 27-node tensor-product Lagrangian element is the simplest element of this class. Based on the two aforementioned CAD interpolations (i) simplified edge-only Coons macroelements will be developed, which are hierarchically improved (ii) through enhanced boundary-only formulation (based on the entire faces) as well as (iii) full tensor-product Coons–Gordon macroelements. Nonrational Bézier elements as well as elements based on B-splines will be presented. For the sake of brevity, numerical results reduce to the acoustic analysis of rectangular and spherical cavities, where all formulations are compared to the closed-form exact solution and are thoroughly criticized. The reader is also referred to previously published numerical examples that include potential (Laplace and Poisson’s equations) and elasticity problems in bodies of cuboidal (parallelepiped with six rectangular faces), cylindrical and spherical shape, using a single macroelement.