[Dune] Isogeometric analysis with Dune

[Nina Pak]

Hello Christian,

thank you for the quick reply.

>> Could somebody confirm our assumption that Nurbs shape functions model 
>> in the dune-localfunctions module will not be compatible with PDELab?

> Together with some additional code, you should be able to use any
shapefunctions defined in dune-localfunctions. In order to estimate the
amount of work, I would have to know a little bit more about the particular
properties of your nurbs approach.

I base on the classical finite element algorithm (see attached), where the
routines in green are replaced by the Nurbs-specific ones.

1. The input file includes the 3 knot vectors (for each direction), control
points and weights, and the polynomial orders.
2. The connectivity information relates the global schemes with the local
entities:
2.1. INC array is such that given a global basis function number and a
parametric direction, it returns the index of the one dimensional basis
function in the specified direction that was used to build the global
function.
2.2. IEN array connects global function numbers to their local ordering on
the element.
3. At each quadrature point the shapefunctions routine evaluates the basis
functions and the necessary derivatives:
3.1. Initialize all variables to zero.
3.2. Determine the Nurbs coordinates using INC and IEN.
3.3. Calculate the parametric coords from the knot vector and the parent
element coords of the quadrature point.
3.4. Calculate the values of basis functions and derivatives w.r.t.
parametric coords:
3.4.1. A subroutine first calculates all non-zero univariate B-spline basis
functions and their parametric derivatives.
3.4.2. The trivariate rational Nurbs functions are calculated as linear
combinations of basis functions and control points.
3.4.3. Calculate derivatives of Nurbs functions w.r.t. parametric coords
with the quotient rule.
3.5. Determine the derivatives w.r.t. physical coords: 
3.5.1. Calculate the gradient of the mapping from parametric to physical
coords and it's inverse (Cramer's rule).
3.5.2. Use parametric derivatives and chain rule.
3.6. Compute Jacobian determinant - for the numerical integration in the
parent element.

Then the local stiffness matrix and the force vector are built, and after we
exit the loop, contributions to the global stiffness matrix and force vector
are added. The assembly uses the connectivity information (2). Then the
system can be solved, and Nurbs-specific output can be written to a file.

I hope the description is clear and elaborate enough. Please write back if
you need further details.

Thank you and best regards
Nina 

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