sldfesq_maindoc_page.cc

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/**\defgroup LuaSLDFESQ Simplified LDFESQ
 * \ingroup LuaQuadrature
 *
 * This quadrature is a simplified implementation of the quadrature
 * defined in the paper by Cheuk Lau and Marvin Adams:
 * "Discrete Ordinates Quadratures Based on Linear
 * and Quadratic Discontinuous Finite Elements over
 * Spherical Quadrilaterals", Nuclear Science and Engineering, 185:1,
 * pages 36-52, 2017.
 *
 * \image html "SLDFESQn40.png" "LDFE Spherical Quadrilaterals at a refinement level of n=40" width=800px
 *
 *
 * # Synopsis
 * - The initial refinement is generated for the octant where all the
 *   direction cosines are positive. We will call this the first octant.
 * - A cube of side length \f$a=\frac{1}{\sqrt{3}}\f$ is inscribed within
 *   the unit sphere and the origin planes. This cube has three faces
 *   that project onto the unit sphere in the first octant.
 *   \image html "SLDFESQFig2.png" "The inscribed cube with one of the faces subdivided" width=800px
 * - Each face of the cube can be described by a local coordinate system,
 *   \f$\tilde{x}\f$, \f$ \tilde{y}\f$. In this reference frame each face
 *   is identically subdivided into an orthogonal grid. The generated
 *   grid-cells are called sub-squares.
 * - A face and its associated sub-squares, in the local reference frame,
 *   can be rotated and translated
 *   onto the cube faces, for which we will call the reference frame the
 *   \f$xyz\f$-prime reference frame.
 * - The sub-squares on each face project onto
 *   Spherical Quadrilaterals (SQs) on the unit sphere, a space which we
 *   shall refer to as the \f$xyz\f$ reference frame. Or in the paper
 *   mu-eta-xi.
 *   \image html "SLDFESQFig3.png" width=400px
 * - Once the SQs are generated we have numerous choices on where to
 *   define quadrature points and associated shape functions. In the LDFE
 *   formulation we need 4 quadrature points and, for simplicity, we restrict
 *   the placement of these points in a certain manner. We form radii, in
 *   the xy-tilde reference, from the sub-square centroids to each corner
 *   of the sub-square (essentially producing sub-sub-squares) and then
 *   chose to place our quadrature points only on these points.
 * - The placement of the quadrature points can then be projected to
 *   xyz after which the shape function coefficients can be determined.
 * - The weight of each quadrature point is the spherical surface area
 *   integral of its shape function. This integral is done with a high order
 *   Guass-Legendre quadrature and a Jacobian \f$ \frac{a}{r^3} \f$.
 * - The original Lau and Adams paper used a Multi-variate secant method to
 *   determine the placement of the quadrature points along the sub-sub-square
 *   radii.
 *
 * # Specific notes for the simplified version
 *
 * ## Initial sub-division of the inscribed cube face
 * The orthogonal grid in the xy-tilde reference frame is dependent on a
 * a diagonal spacing of grid points. The paper by Lau and Adams did not
 * specify exactly how to generate this spacing but only stated that their
 * spacing minimizes the max/avg and max/min ratio of projected SQ to
 * 1.1 and 1.3 respectively. We found that the application of a weighting
 * function, \f$ \alpha (\cos \beta \frac{\pi}{2} x - \cos \beta \frac{\pi}{2})\f$
 * produces the optimal spacing. See the code for additional clarification.
 *
 * ## Placement of the quadrature points
 * Instead of using a multi-variate secant method to place the quadrature
 * points we found that placing the points on the 4 Gauss-Legendre points
 * for a quadrilateral (\f$[\pm \frac{1}{\sqrt{3}},\pm \frac{1}{\sqrt{3}} ]\f$)still produced 4th order convergence and added a
 * lot of speed to the algorithm.
 *
 * ## Integration of basis functions
 * The determinant of the Jacobian can be hard to derive. It can be done
 * in the two ways:
 * [using a cross-product](../../HTMLimages/SLDFESQ_JacobianA.jpg)
 * or in [angle-space](../../HTMLimages/SLDFESQ_JacobianB.jpg).
 *
 *
 * ## Local refinement
 * The data structures employed allows the SQs to easily be refined in certain
 * regions.
 *
 *
 * */