Generalizing the isogeometric concept: weakening the tight coupling between geometry and simulation in IGA

In the standard paradigm of isogeometric analysis [2, 1], the geometry and the simulation
spaces are tightly integrated, i.e. the non-uniform rational B-splines (NURBS) space, which
is used for the geometry representation of the domain, is also employed for the numerical
solution of the problem over the domain. However, in certain situations, such as, when the
geometry of the domain can be represented by low order NURBS but the numerical solution
can be obtained with improved accuracy by using NURBS of order higher than that required
for the geometry; or in the shape and topology optimization where the constraint of using the
same space for the geometry and the numerical solution is not favorable, this tight coupling is
disadvantageous.
Therefore, we study the effect of decoupling the spaces for the geometry representation and
the numerical solution, though still using the prevalent functions in CAD/CAGD. To begin
with, we perform the patch tests on various combinations of polynomial degree, geometry
type, and various cases of varying degrees and control variables between the geometry and
the numerical solution. This shows that certain cases, perhaps intuitive, should be avoided in
practice because patch test fails. The above-mentioned situations are further explored with
some numerical examples, which shows that weakening the tight coupling between geometry
and simulation offers more flexibility in choosing the numerical solution spaces.