.. _sec-lsfit:

Beam reference lines and reference configuration
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

The use of LSFEs as our beam spatial discretization presents
opportunities and challenges for the geometric representation of slender
structures like turbine blades. Turbine blades can be curved, swept,
and/or twisted along their length. In this section we describe how beams
are user defined and how the beams are discretized in the reference
configuration.

Kynema-FMB beams are assumed to be defined as a set of points in the
global coordinate system 
:math:`(\widehat{x}^\mathrm{g}, \widehat{y}^\mathrm{g},\widehat{z}^\mathrm{g})`
with the blade root positioned on the :math:`\widehat{x}^\mathrm{g}` axis and oriented such that the
pitch axis, or its primary-length direction, is pointing in the
:math:`x` direction. For turbine blades, the :math:`y` direction is the
direction of the trailing edge and defines a zero pitch/twist. Note that
this differs from the standard turbine-blade definition, which can be
easily rotated into our system.

The user must provide :math:`n^\mathrm{geom} \ge 2` points defining the
reference line with the data
:math:`(\eta_i^\mathrm{geom},\underline{x}_i^\mathrm{geom},\tau_i^\mathrm{geom})`,
:math:`i \in \{ 1, 2, \ldots, n^\mathrm{geom}\}`, where
:math:`\eta_i^\mathrm{geom}\in[0,1]` defines the nondimensional position along
the beam reference line where the associated reference line position is
:math:`\underline{x}_i^\mathrm{geom} \in \mathbb{R}^3` and twist about the
reference line is :math:`\tau_i^\mathrm{geom}`.   Note that Kynema-FMB convention
requires that :math:`\underline{x}_1^\mathrm{geom} = (x_1, 0, 0)^T`.

The user must also provide (through quadrature input choices) the
:math:`n^Q` quadrature locations :math:`\xi_k^Q \in [-1,1]` along the reference
axis. 

Finally, the user must provide the translation and rotation,
:math:`\underline{u}^\mathrm{move}` and
:math:`\underline{\underline{R}}^\mathrm{move}`, respectively, that moves the
beam from its setup postion/orientation to its reference position/orientation.
:math:`\underline{x}^\mathrm{r}` and
:math:`\underline{\underline{R}}^\mathrm{r}`, respectively.
In other words, each
beam is defined in the global system and it is then
translated and rotated into its reference position. While we present a single 
:math:`\underline{u}^\mathrm{move}` and a single
:math:`\underline{\underline{R}}^\mathrm{move}`, in practice those are implemented in multiple steps.

Our approach is to represent the geometry with a single spectral element
with relatively few points, which we describe here. Given
:math:`n^\mathrm{geom}` points describing the beam as above, we wish to
represent that beam with a :math:`P`-point Legendre spectral finite
element, where :math:`P` is a user input. 

We first perform a least-squares fit to the data using a LSFE polynomial with
:math:`\mathrm{min}(P,n^\mathrm{geom})` points, where the endpoints of the
fitting polynomial are constrained to coincide with the reference data at
:math:`\eta_1^\mathrm{geom}` and :math:`\eta^\mathrm{geom}_{n^\mathrm{geom}}`.
If :math:`P \le n^\mathrm{geom}`, that fit defines the :math:`P` nodal
locations of the LSFE. If :math:`P > n^\mathrm{geom}`, then a new LSFE is
constructed with :math:`P`-nodes that lie on the lower-order
:math:`n^\mathrm{geom}`-point least-squares fit. The result are nodal values
:math:`\underline{x}^\mathrm{fit}_\ell \in \mathbb{R}^3`, for :math:`\ell \in
\{1, \ldots, P\}`. The geometry of the reference line is given (in the element
natural coordinates) as

.. math::

   \begin{aligned}
    \underline{x}^\mathrm{geom}(\xi) = \sum_{\ell=1}^P \underline{x}^\mathrm{fit}_\ell \phi_\ell(\xi)
   \end{aligned}

for :math:`\xi\in[-1,1]`.

Figure :numref:`fig-lsfit-example` shows the results of the fitting
process for the IEA 15-MW reference turbine, which is defined by 50
points. Shown are LSFE fits with 3, 7, and 10 nodes. Clearly the 10-node
fit provides an excellent geometric approximation of the discrete data.
For reference, also shown is 10-node representation with linear
segments, which emphasizes the advantage of the LSFE representation for
a given number of nodes.

.. _fig-lsfit-example:

.. figure:: images/doc-lsfit.png
   :alt: A plot of data
   :width: 80%
   :align: center

   Discrete representation of the IEA 15-MW reference-turbine blade reference line with 50 points, and four continuous-function representations of the beam reference line.

.. figure:: images/doc-lsfit-zoom.png
   :alt: A plot of data
   :width: 80%
   :align: center

   Close up of the data shown in Figure :numref:`fig-lsfit-example`.

A beam’s reference line also requires the definition of its orientation.
We construct the orientation from the LSFE line as follows:

#. Calculate the unit tangent at each LSFE node:

   .. math::

      \begin{aligned}
       \widehat{t}^{\mathrm{fit}}_i = 
      \sum_{\ell=1}^P \underline{x}^\mathrm{fit}_\ell 
      \left .  \frac{\partial \phi_\ell}{\partial \xi}\right|_{\xi=\xi_i} / 
      \left \Vert\sum_{i=1}^P \underline{x}^\mathrm{fit}_i 
      \left .  \frac{\partial \phi_i}{\partial \xi}\right|_{\xi=\xi_i}  
      \right\Vert 
      , \quad \forall i \in \{1, \ldots, P\}
      \end{aligned}

#. At each LSFE node, calculate a unit vector,
   :math:`\widehat{n}^\mathrm{fit}_i`, normal to
   :math:`\widehat{t}^\mathrm{fit}_i` with the additional requirement
   that it lies in the local
   :math:`\widehat{x}^\mathrm{g}-\widehat{y}^\mathrm{g}` plane, i.e.,
   :math:`\widehat{n}^\mathrm{fit}_i = (n_1^\mathrm{fit},n_2^\mathrm{fit}, 0 )`
   with the requirements
   :math:`\left|\widehat{n}^\mathrm{fit}_i \right | = 1` and
   :math:`\widehat{n}^\mathrm{fit}_i \cdot \widehat{t}^\mathrm{fit}_i = 0`.

#. The binormal vector is then given by
   :math:`\underline{b}^\mathrm{fit}_i = \widetilde{t^\mathrm{fit}_i} \underline{n}^\mathrm{fit}_i`.

#. The orientation at node :math:`i` is given by

   .. math::

      \begin{aligned}
      \underline{\underline{R}}^\mathrm{fit}_i = 
      \begin{bmatrix}
      \widehat{t}^\mathrm{fit}_i \,\,
      \widehat{n}^\mathrm{fit}_i \,\,
      \widehat{b}^\mathrm{fit}_i
      \end{bmatrix}
      ,\, \forall i \in \{1, \ldots, P\}
      \end{aligned}

   which has the associated quaternions
   :math:`\widehat{q}_i^\mathrm{fit}`.

#. Nodal positions in the reference position, given
   :math:`\underline{u}^\mathrm{move}` and
   :math:`\underline{\underline{R}}^\mathrm{move}`, are calculated as

   .. math::

      \begin{aligned}
      \underline{x}^\mathrm{r}_i = \underline{x}^\mathrm{fit}_i 
      + \underline{u}^\mathrm{move} 
      +\underline{\underline{R}}^\mathrm{move} 
      \left(\underline{x}^\mathrm{fit}_i- \underline{x}^\mathrm{fit}_1\right)
      \quad \forall i \in  \{1,  \ldots, P \}
      \end{aligned}

#. Nodal orientations are calculated as 

   .. math::

      \begin{aligned}
      \underline{\underline{R}}^\mathrm{r}_i = 
      \underline{\underline{R}}^\mathrm{move} \underline{R}^\mathrm{fit}_i
      ,\quad \forall i \in  \{1,  \ldots, P \}
      \end{aligned}

   which has the associated quaternions :math:`\widehat{q}_i^\mathrm{r}`.

#. We also require the reference orientations at quadrature points,
   :math:`\xi^\mathrm{Q}_k`, which are calculated and stored as quaternions:

   .. math::

      \begin{aligned}
      \widehat{q}^{\mathrm{r,Q}}_k = 
      \sum_{\ell=1}^P \widehat{q}^\mathrm{r}_\ell 
       \phi_\ell\left(\xi_k^\mathrm{Q}\right) / 
      \left \Vert\sum_{\ell=1}^P \widehat{q}^\mathrm{r}_\ell 
       \phi_\ell\left(\xi_k^\mathrm{Q} \right) 
      \right \Vert
      , \quad \forall k \in \{1, \ldots, n^\mathrm{Q}\}
      \end{aligned}

#. Finally, regarding the user defined twist, that is applied at initialization to the sectional material matrices, e.g., for the mass matrices:

   .. math::

      \begin{bmatrix} 
      \underline{\underline{R}}(\tau_k) & \underline{\underline{0}}  \\
      \underline{\underline{0}} & \underline{\underline{R}}(\tau_k)  
      \end{bmatrix}
      \underline{\underline{M}}_k^*
      \begin{bmatrix} 
      \underline{\underline{R}}(\tau_k) & \underline{\underline{0}}  \\
      \underline{\underline{0}} & \underline{\underline{R}}(\tau_k)  
      \end{bmatrix}^T \rightarrow \underline{\underline{M}}_k^*

   where :math:`\underline{\underline{M}}_k^*` and :math:`\tau_k` are linearly
   interpolated mass matrix and twist (at each quadrature point), 
   respectively, from nearest neighbors, and

   .. math::

      \underline{\underline{R}}(\tau) = 
      \begin{bmatrix} 
      1 & 0 & 0 \\
      0 & \cos(\tau) & - \sin(\tau)  \\
      0 & \sin(\tau) & \cos(\tau) 
      \end{bmatrix}