Isogeometric analysis is a computational approach that offers the possibility of integrating finite element analysis (FEA) into conventional NURBS-based CAD design tools.
Currently, it is necessary to convert data between CAD and FEA packages to analyse new designs during development, a difficult task since the two computational geometric approaches are different.
Isogeometric analysis employs complex NURBS geometry (the basis of most CAD packages) in the FEA application directly.
This allows models to be designed, tested and adjusted in one go, using a common data set.
[1] The pioneers of this technique are Tom Hughes and his group at The University of Texas at Austin.
A reference free software implementation of some isogeometric analysis methods is GeoPDEs.
For instance, PetIGA[4] is an open framework for high performance isogeometric analysis heavily based on PETSc.
In particular, FEAP[6] is a finite element analysis program which includes an Isogeometric analysis library FEAP IsoGeometric (Version FEAP84 & Version FEAP85).
Isogeometric analysis presents two main advantages with respect to the finite element method:[1][7] In the framework of IGA, the notions of both control mesh and physical mesh are defined.
Control points play also the role of degrees of freedom (DOFs).
[1] The physical mesh lays directly on the geometry and it consists of patches and knot spans.
According to the number of patches that are used in a specific physical mesh, a single-patch or a multi-patch approach is effectively employed.
A patch is mapped from a reference rectangle in two dimensions and from a reference cuboid in three dimensions: it can be seen as the entire computational domain or a smaller portion of it.
Each patch can be decomposed into knot spans, which are points, lines and surfaces in 1D, 2D and 3D, respectively.
[1][7] Once a definition of knot vector is provided, several types of basis functions can be introduced in this context, such as B-splines, NURBS and T-splines.
[1] B-splines can be derived recursively from a piecewise constant function with
Using De Boor's algorithm, it is possible to generate B-splines of arbitrary order
B-splines that are generated in this way own both the partition of unity and positivity properties, i.e.:[1]
An extension to the two dimensional case can be easily obtained from B-splines curves.
In IGA basis functions are also employed to develop the computational domain and not only for representing the numerical solution.
For this reason they should have all the properties that permit to represent the geometry in an exact way.
B-splines, due to their intrinsic structure, are not able to generate properly circular shapes for instance.
[1] In order to circumvent this issue, non-uniform rational B-splines, also known as NURBS, are introduced in the following way:[1]
Following the idea developed in the subsection about B-splines, NURBS curve are generated as follows:[1]
The extension of NURBS basis functions to manifolds of higher dimensions (for instance 2 and 3) is given by:[1]
There are three techniques in IGA that permit to enlarge the space of basis functions without touching the geometry and its parametrization.
[1] The first one is known as knot insertion (or h-refinement in the FEA framework), where
with the addition of more knots, which implies an increment of both the number of basis functions and control points.
[1] The second one is called degree elevation (or p-refinement in the FEA context), which permits to increase the polynomial order of the basis functions.
[1] Finally the third method, known as k-refinement (without a counterpart in FEA), derives from the preceding two techniques, i.e. combines the order elevation with the insertion of a unique knot in