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element_class.hh
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/**
* @file element_class.hh
*
* @author Guillaume Anciaux <guillaume.anciaux@epfl.ch>
* @author Nicolas Richart <nicolas.richart@epfl.ch>
*
* @date Fri Jun 18 11:47:19 2010
*
* @brief element class definition
*
* @section LICENSE
*
* Copyright (©) 2010-2011 EPFL (Ecole Polytechnique Fédérale de Lausanne)
* Laboratory (LSMS - Laboratoire de Simulation en Mécanique des Solides)
*
* Akantu is free software: you can redistribute it and/or modify it under the
* terms of the GNU Lesser General Public License as published by the Free
* Software Foundation, either version 3 of the License, or (at your option) any
* later version.
*
* Akantu is distributed in the hope that it will be useful, but WITHOUT ANY
* WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR
* A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more
* details.
*
* You should have received a copy of the GNU Lesser General Public License
* along with Akantu. If not, see <http://www.gnu.org/licenses/>.
*
*/
/* -------------------------------------------------------------------------- */
/* -------------------------------------------------------------------------- */
#ifndef __AKANTU_ELEMENT_CLASS_HH__
#define __AKANTU_ELEMENT_CLASS_HH__
/* -------------------------------------------------------------------------- */
#include "aka_common.hh"
#include "aka_math.hh"
#include "aka_types.hh"
/* -------------------------------------------------------------------------- */
__BEGIN_AKANTU__
/**
* Class describing the different type of element for mesh or finite element
* purpose
*
* @tparam type the element type for the specialization of the element class
*/
template<ElementType type> class ElementClass {
/* ------------------------------------------------------------------------ */
/* Constructors/Destructors */
/* ------------------------------------------------------------------------ */
public:
/* ------------------------------------------------------------------------ */
/* Methods */
/* ------------------------------------------------------------------------ */
public:
/**
* compute the shape functions, the shape functions derivatives and the
* jacobians
* @param[in] coord coordinates of the nodes
* @param[out] shape shape functions [nb_quad*node_per_elem]
* @param[out] shape_deriv shape functions derivatives [nb_quad*node_per_elem*spatial_dim]
* @param[out] jacobian jacobians * integration weights [nb_quad]
*/
inline static void preComputeStandards(const Real * coord,
const UInt dimension,
Real * shape,
Real * shape_deriv,
Real * jacobian);
/// compute the shape values for a point given in natural coordinates
inline static void computeShapes(const Real * natural_coords, Real * shapes);
/// compute the shape values for a set of points given in natural coordinates
inline static void computeShapes(const Real * natural_coords,
const UInt nb_points,
Real * shapes);
inline static void computeShapes(const Real * natural_coords,
const UInt nb_points,
Real * shapes,
const Real * local_coord,
UInt id = 0);
inline static void computeShapes(const Real * natural_coords,
Real * shapes,
const Real * local_coord,
UInt id = 0);
/**
* compute dxds the variation of real coordinates along with
* variation of natural coordinates on a given point in natural
* coordinates
*/
inline static void computeDXDS(const Real * dnds,
const Real * node_coords,
const UInt dimension,
Real * dxds);
/**
* compute dxds the variation of real coordinates along with
* variation of natural coordinates on a given set of points in
* natural coordinates
*/
inline static void computeDXDS(const Real * dnds,
const UInt nb_points,
const Real * node_coords,
const UInt dimension, Real * dxds);
/**
* compute dnds the variation of real shape functions along with
* variation of natural coordinates on a given point in natural
* coordinates
*/
inline static void computeDNDS(const Real * natural_coords,
Real * dnds);
/**
* compute dnds the variation of shape functions along with
* variation of natural coordinates on a given set of points in
* natural coordinates
*/
inline static void computeDNDS(const Real * natural_coords,
const UInt nb_points,
Real * dnds);
/// compute jacobian (or integration variable change factor) for a set of points
inline static void computeJacobian(const Real * dxds,
const UInt nb_points,
const UInt dimension,
Real * jac);
/// compute jacobian (or integration variable change factor) for a given point
inline static void computeJacobian(const Real * dxds,
const UInt dimension,
Real & jac);
/// compute shape derivatives (input is dxds) for a set of points
inline static void computeShapeDerivatives(const Real * dxds,
const Real * dnds,
const UInt nb_points,
const UInt dimension,
Real * shape_deriv);
/// compute shape derivatives (input is dxds) for a given point
inline static void computeShapeDerivatives(const Real * dxds,
const Real * dnds,
Real * shape_deriv);
inline static void computeShapeDerivatives(const Real * natural_coords,
const UInt nb_points,
const UInt dimension,
Real * shape_deriv,
const Real * local_coord,
UInt id = 0);
inline static void computeShapeDerivatives(const Real * natural_coords,
Real * shape_deriv,
const Real * local_coord,
UInt id);
/// compute normals on quad points
inline static void computeNormalsOnQuadPoint(const Real * dxds,
const UInt dimension,
Real * normals);
/// interpolate a field given (arbitrary) natural coordinates
inline static void interpolateOnNaturalCoordinates(const Real * natural_coords,
const Real * nodal_values,
UInt dimension,
Real * interpolated);
/// inverse map: get natural coordinates from real coordinates
/**
* In the non linear cases we need to iterate to find the natural coordinates @f$\xi@f$
* provided real coordinates @f$x@f$.
*
* We want to solve: @f$ x- \phi(\xi) = 0@f$ with @f$\phi(\xi) = \sum_I N_I(\xi) x_I@f$
* the mapping function which uses the nodal coordinates @f$x_I@f$.
*
* To that end we use the Newton method and the following series:
*
* @f$ \frac{\partial \phi(x_k)}{\partial \xi} \left( \xi_{k+1} - \xi_k \right) = x - \phi(x_k)@f$
*
* When we consider elements embedded in a dimension higher than them (2D triangle in a 3D space for example)
* @f$ J = \frac{\partial \phi(\xi_k)}{\partial \xi}@f$ is of dimension @f$dim_{space} \times dim_{elem}@f$ which
* is not invertible in most cases. Rather we can solve the problem:
*
* @f$ J^T J \left( \xi_{k+1} - \xi_k \right) = J^T \left( x - \phi(\xi_k) \right) @f$
*
* So that
*
* @f$ d\xi = \xi_{k+1} - \xi_k = (J^T J)^{-1} J^T \left( x - \phi(\xi_k) \right) @f$
*
* So that if the series converges we have:
*
* @f$ 0 = J^T \left( \phi(\xi_\infty) - x \right) @f$
*
* And we see that this is ill-posed only if @f$ J^T x = 0@f$ which means that the vector provided
* is normal to any tangent which means it is outside of the element itself.
*
*
* @param real_coords: the real coordinates the natural coordinates are sought for
* @param node_coords: the coordinates of the nodes forming the element
* @param natural_coords: output->the sought natural coordinates
* @param spatial_dimension: spatial dimension of the problem
**/
inline static void inverseMap(const types::RVector & real_coords,
const types::RMatrix & node_coords,
UInt spatial_dimension,
types::RVector & natural_coords,
Real tolerance = 1e-8);
//! return true if the provided natural coordinates are with the element. False otherwise
inline static bool contains(const types::RVector & natural_coords);
/// function to print the containt of the class
virtual void printself(std::ostream & stream, int indent = 0) const {};
/* ------------------------------------------------------------------------ */
/* Accessors */
/* ------------------------------------------------------------------------ */
public:
static AKANTU_GET_MACRO_NOT_CONST(Kind, kind, ElementKind);
static AKANTU_GET_MACRO_NOT_CONST(NbNodesPerElement, nb_nodes_per_element, UInt);
static AKANTU_GET_MACRO_NOT_CONST(P1ElementType, p1_element_type, ElementType);
static AKANTU_GET_MACRO_NOT_CONST(NbQuadraturePoints, nb_quadrature_points, UInt);
static AKANTU_GET_MACRO_NOT_CONST(SpatialDimension, spatial_dimension, UInt);
static AKANTU_GET_MACRO_NOT_CONST(FacetElementType, facet_type, const ElementType &);
static AKANTU_GET_MACRO_NOT_CONST(NbFacetsPerElement, nb_facets, UInt);
static AKANTU_GET_MACRO_NOT_CONST(FacetLocalConnectivityPerElement, facet_connectivity, UInt**);
static AKANTU_GET_MACRO_NOT_CONST(NbShapeFunctions, nb_shape_functions, UInt);
static inline Real * getQuadraturePoints();
static inline UInt getShapeSize();
static inline UInt getShapeDerivativesSize();
/// compute the in-radius
static inline Real getInradius(const Real * coord);
static inline Real * getGaussIntegrationWeights();
/* ------------------------------------------------------------------------ */
/* Class Members */
/* ------------------------------------------------------------------------ */
public:
/// Number of nodes per element
static UInt nb_nodes_per_element;
private:
/// Kind of element
static ElementKind kind;
/// Number of quadrature points per element
static UInt nb_quadrature_points;
/// Dimension of the element
static UInt spatial_dimension;
/// Type of the facet elements
static ElementType facet_type;
/// number of facets for element
static UInt nb_facets;
/// local connectivity of facets
static UInt * facet_connectivity[];
/// vectorial connectivity of facets
static UInt vec_facet_connectivity[];
/// type of element P1 associated
static ElementType p1_element_type;
/// quadrature points in natural coordinates
static Real quad[];
/// Number of shape functions
static UInt nb_shape_functions;
/// weights for the Gauss integration
static Real gauss_integration_weights[];
};
/* -------------------------------------------------------------------------- */
/* inline functions */
/* -------------------------------------------------------------------------- */
#if defined (AKANTU_INCLUDE_INLINE_IMPL)
# include "element_class_inline_impl.cc"
#endif
/* -------------------------------------------------------------------------- */
__END_AKANTU__
#endif /* __AKANTU_ELEMENT_CLASS_HH__ */

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