11. Global Collocation Using Macroelements

The CAD-based global approximation of the approximate solution within a patch (or a volume block) leads to large matrices, and therefore, high computer effort is required. The involved matrices may be either fully populated (as happens when using Lagrange and Bernstein–Bézier polynomials) or partially populated (thanks to the compact support of B-splines and NURBS). This fact is the motivation for preserving the global basis functions but replacing the Galerkin–Ritz with a collocation method which is here called the “global collocation method.” In the latter method, each element of the large matrices can be calculated without performing domain integration, since only a substitution of the basis functions into the partial differential operator is needed. Nevertheless, the numerical solution is highly influenced by the location of the so-called collocation points, and this is an open topic for research. Through a number of test problems, it will be shown that the global collocation method performs equally well in 1D, 2D, and 3D, static and dynamic, problems. Numerical results are presented for a broad spectrum of test problems, such as eigenvalue analysis of rods and beams, transient thermal analysis, plane elasticity, rectangular and circular acoustic cavities and plates.