Geometry-Independent Field approximaTion: CAD-Analysis Integration, geometrical exactness and adaptivity

In isogeometric analysis (IGA), the same spline representation is employed for both the geometry
of the domain and approximation of the unknown fields over this domain. This identity of
the geometry and field approximation spaces was put forward in the now classic 2005 paper [20] as
a key advantage on the way to the integration of Computer Aided Design (CAD) and subsequent
analysis in Computer Aided Engineering (CAE). [20] claims indeed that any change to the geometry
of the domain is automatically inherited by the approximation of the field variables, without
requiring the regeneration of the mesh at each change of the domain geometry. Yet, in Finite Element
versions of IGA, a parameterization of the interior of the domain must still be constructed,
since CAD only provides information about the boundary. The identity of the boundary and field
representation decreases the flexibility in which this parameterization can be generated and somewhat
constrains the modeling and simulation process, because an approximation able to represent
the domain geometry accurately need not be adequate to also approximate the field variables accurately,
in particular when the solution is not smooth. We propose here a new paradigm called
Geometry-Independent Field approximaTion (GIFT) where the spline spaces used for the geometry
and the field variables can be chosen and adapted independently while preserving geometric
exactness and tight CAD integration. GIFT has the following features: (1) It is possible to flexibly choose between different spline spaces with different properties to better represent the solution of
the problem, e.g. the continuity of the solution field, boundary layers, singularities, whilst retaining
geometrical exactness of the domain boundary. (2) For multi-patch analysis, where the domain is
composed of several spline patches, the continuity condition between neighboring patches on the
solution field can be automatically guaranteed without additional constraints in the variational
form. (3) Refinement operations by knot insertion and degree elevation are performed directly
on the spline space of the solution field, independently of the spline space of the geometry of the
domain, which makes the method versatile. GIFT with PHT-spline solution spaces and NURBS
geometries is used to show the effectiveness of the proposed approach. Keywords : Super-parametric methods, Isogeometric analysis (IGA), Geometry-independent
Spline Space, PHT-splines, local refinement, adaptivity