# NURBS Miniapps

These miniapps demonstrate the use of NURBS-based Isogeometric analysis1,2.

### NURBS Ex 1: Laplace problem

This example code solves a simple Laplace problem \begin{align} -\Delta u = 1 \end{align} with homogeneous Dirichlet boundary conditions. For implementation see miniapps/nurbs/nurbs__ex1.

### NURBS Ex 3: Maxwell problem

This example code solves a simple 3D electromagnetic diffusion problem corresponding to the second order definite Maxwell equation \begin{align} \nabla\times\nabla\times\, E + E = f \end{align} with boundary condition $E \times n$ = "given tangential field". Here, we use a given exact solution $E$ and compute the corresponding r.h.s. $f$. We discretize with Nedelec finite elements in 2D or 3D.

The example demonstrates the use of $H(curl)$ finite element spaces with the curl-curl and the (vector finite element) mass bilinear form, as well as the computation of discretization error when the exact solution is known. Static condensation is also illustrated. For implementation see miniapps/nurbs/nurbs__ex1.

### NURBS Ex 5: Darcy problem

This example code solves a simple 2D/3D mixed Darcy problem corresponding to the saddle point system \begin{align} \begin{array}{rcl} k\,{\bf u} + {\rm grad}\,p &=& f \\ -{\rm div}\,{\bf u} &=& g \end{array} \end{align} with natural boundary condition $-p =$ "given pressure". Here we use a given exact solution $({\bf u},p)$ and compute the corresponding right hand side $(f, g)$. We discretize with Raviart-Thomas finite elements (velocity $\bf u$) and piecewise discontinuous polynomials (pressure $p$). For implementation see miniapps/nurbs/nurbs__ex5.

1. T.J.R. Hughes, J.A. Cottrell, Y. Bazilevs: "Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement", Computer Methods in Applied Mechanics and Engineering, Elsevier, 2005, 194 (39-41), pp.4135-4195.

2. T.J.R. Hughes, J.A. Cottrell, Y. Bazilevs: "Isogeometric analysis: toward integration of CAD and FEA", Wiley&Sons 2009