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This is the second part of a two-part paper developing the module theory, and a theory of complexity and varieties for modules, in the context of Kropholler's class of groupsLHF. The numbering is a continuation of the numbering for the first part4.

The first paper on isogeometric analysis appeared only five years ago [1], and the first book appeared last year [2]. Progress has been rapid. Isogeometric analysis has been applied to a wide variety of problems in solids, fluids and fluid-structure interactions. Superior accuracy to traditional finite elements has been demonstrated in all cases. In addition, it has been shown that isogeometric analysis provides an efficient framework for the design through analysis paradigm. In this contribution we present two examples which illustrate the potential of T-splines, a powerful generalization of NURBS, the CAD industry standard. The first example utilizes T-splines for a problem of propagating cracks. Smoothly turning discontinuities can be introduced in T-splines through local knot insertion. The possibility of enhancing a T-spline basis with discontinuities makes isogeometric finite elements suitable for the capturing of discontinuities, in particular, cracks. From an implementation point of view, the concept of Bézier extraction [3] allows for a unified approach to NURBS and T-splines, and will allow this approach to be extended to T-splines of arbitrary topology. The second example demonstrates the enabling power of T-splines in a design through analysis framework. A trimless T-spline automotive bumper is generated and then used directly in an LS-DYNA vibrations analysis [4]. T-splines can describe shapes of arbitrary topological complexity while maintaining a smooth analysis-suitable basis. This makes them an attractive alternative to classical finite elements for many problems of engineering interest. For such problems, the isogeometric paradigm can be invoked thus eliminating the analysis costs associated with geometry clean-up, defeaturing and mesh generation. The two numerical simulations considered in this work demonstrate the capability of the T-spline basis to meet the stringent needs of analysis while providing the flexibility to overcome design through analysis bottlenecks. Specifically, T-splines were found to give efficient discretizations in terms of the number of degrees of freedom and the required computational effort. We anticipate that T-splines and isogeometric analysis will continue to provide efficient solutions to many important problems in computational mechanics.