Weakening the tight coupling between geometry and simulation in isogeometric analysis

In the standard paradigm of isogeometric analysis, the geometry and the simulation spaces are tightly
integrated, i.e. the same non-uniform rational B-splines (NURBS) space, which is used for the geometry
representation of the domain, is employed for the numerical solution of the problem over the domain.
However, there are situations where this tight integration is a bane rather than a boon. Such situations arise
where, e.g.,
(1) the geometry of the domain is simple enough to be represented by low order NURBS, whereas the
unknown (exact) solution of the problem is sufficiently regular, and thus, the numerical solution can be
obtained with improved accuracy by using NURBS of order higher than that required for the geometry,
(2) the constraint of using the same space for the geometry and the numerical solution is particularly
undesirable, such as in the shape and topology optimization, and
(3) the solution of the problem has low regularity but for the curved boundary of the domain one can employ
higher order NURBS.
Therefore, we propose to weaken this constraint. An extensive study of 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 will be discussed. It will be shown, with concrete reasoning, that why
patch test fails in certain cases, and that those cases should be avoided in practice. Thereafter, selective
numerical examples will be presented to address some of the above-mentioned situations, and it will be
shown that weakening the tight coupling between geometry and simulation offers more flexibility in
choosing the numerical solution spaces, and thus, improved accuracy of the numerical solution.
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