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// vi: set et ts=4 sw=2 sts=2:
#ifndef DUNE_PYRAMID_P2_LOCALBASIS_HH
#define DUNE_PYRAMID_P2_LOCALBASIS_HH
#include <dune/common/fmatrix.hh>
#include <dune/localfunctions/common/localbasis.hh>
namespace Dune
{
/**@ingroup LocalBasisImplementation
\brief Quadratic Lagrange shape functions on the pyramid.
Defines the quadratic shape functions on pyramid.
Taken from
Liping Liu, Kevin B. Davies, Michal Krizek, and Li Guan:
On Higher Order Pyramidal Finite Elements
Adv. Appl. Math. Mech., Vol. 3, No. 2, pp. 131-140, 2011
\tparam D Type to represent the field in the domain.
\tparam R Type to represent the field in the range.
\nosubgrouping
*/
template<class D, class R>
class PyramidP2LocalBasis
{
public:
//! \brief export type traits for function signature
typedef LocalBasisTraits<D,3,Dune::FieldVector<D,3>,R,1,Dune::FieldVector<R,1>,
Dune::FieldMatrix<R,1,3> > Traits;
//! \brief number of shape functions
unsigned int size () const
{
return 14;
}
//! \brief Evaluate all shape functions
inline void evaluateFunction (const typename Traits::DomainType& in,
std::vector<typename Traits::RangeType>& out) const
{
out.resize(14);
// transform to reference element with base [-1,1]^2
const R x = 2.0*in[0] + in[2] - 1.0;
const R y = 2.0*in[1] + in[2] - 1.0;
const R z = in[2];
if (x > y)
{
// vertices
out[0] = 0.25*(x + z)*(x + z - 1)*(y - z - 1)*(y - z);
out[1] = -0.25*(x + z)*(y - z)*((x + z + 1)*(-y + z + 1) - 4*z) - z*(x - y);
out[2] = 0.25*(x + z)*(y - z)*(y - z + 1)*(x + z - 1);
out[3] = 0.25*(y - z)*(x + z)*(y - z + 1)*(x + z + 1);
out[4] = z*(2*z - 1);
// lower edges
out[5] = -0.5*(y - z + 1)*(x + z - 1)*((y - 1)*(x + 1) + z*(x - y + z + 1));
out[6] = -0.5*(y - z + 1)*(((x + z + 1)*(y - 1)*x - z) + z*(2*y + 1));
out[7] = -0.5*(x + z - 1)*(((y - z - 1)*(x + 1)*y - z) + z*(2*x + 1));
out[8] = -0.5*(y - z + 1)*(x + z - 1)*(x + 1)*y;
// upper edges
out[9] = z*(x + z - 1)*(y - z - 1);
out[10] = -z*((x + z + 1)*(y - z - 1) + 4*z);
out[11] = -z*(y - z + 1)*(x + z - 1);
out[12] = z*(y - z + 1)*(x + z + 1);
// base face
out[13] = (y - z + 1)*(x + z - 1)*((y - 1)*(x + 1) + z*(x - y + z + 1));
}
else
{
// vertices
out[0] = 0.25*(y + z)*(y + z - 1)*(x - z - 1)*(x - z);
out[1] = -0.25*(x - z)*(y + z)*(x - z + 1)*(-y - z + 1);
out[2] = 0.25*(x - z)*(y + z)*((x - z - 1)*(y + z + 1) + 4*z) + z*(x - y);
out[3] = 0.25*(y + z)*(x - z)*(x - z + 1)*(y + z + 1);
out[4] = z*(2*z - 1);
// lower edges
out[5] = -0.5*(y + z - 1)*(((x - z - 1)*(y + 1)*x - z) + z*(2*y + 1));
out[6] = -0.5*(x - z + 1)*(y + z - 1)*(y + 1)*x;
out[7] = -0.5*(x - z + 1)*(y + z - 1)*(x - 1)*y;
out[8] = -0.5*(x - z + 1)*(((y + z + 1)*(x - 1)*y - z) + z*(2*x + 1));
// upper edges
out[9] = z*(y + z - 1)*(x - z - 1);
out[10] = -z*(x - z + 1)*(y + z - 1);
out[11] = -z*((y + z + 1)*(x - z - 1) + 4*z);
out[12] = z*(x - z + 1)*(y + z + 1);
// base face
out[13] = (x - z + 1)*(y + z - 1)*((y + 1)*(x - 1) - z*(x - y - z - 1));
}
}
//! \brief Evaluate Jacobian of all shape functions
inline void
evaluateJacobian (const typename Traits::DomainType& in, // position
std::vector<typename Traits::JacobianType>& out) const // return value
{
out.resize(14);
// transform to reference element with base [-1,1]^2
const R x = 2.0*in[0] + in[2] - 1.0;
const R y = 2.0*in[1] + in[2] - 1.0;
const R z = in[2];
// transformation of the gradient leads to a multiplication
// with the Jacobian [2 0 0; 0 2 0; 1 1 1]
if (x > y)
{
// vertices
out[0][0][0] = 0.5*(y - z - 1)*(y - z)*(2*x + 2*z - 1);
out[0][0][1] = 0.5*(x + z)*(x + z - 1)*(2*y - 2*z - 1);
out[0][0][2] = 0.5*(out[0][0][0] + out[0][0][1])
+ 0.25*((2*x + 2*z - 1)*(y - z - 1)*(y - z)
+ (x + z)*(x + z - 1)*(-2*y + 2*z + 1));
out[1][0][0] = 2*(-0.25*((y - z)*((x + z + 1)*(-y + z + 1) - 4*z)
+ (x + z)*(y - z)*(-y + z + 1)) - z);
out[1][0][1] = 2*(-0.25*((x + z)*((x + z + 1)*(-y + z + 1) - 4*z)
+ (x + z)*(y - z)*(-(x + z + 1))) + z);
out[1][0][2] = 0.5*(out[1][0][0] + out[1][0][1])
- 0.25*((y - z)*((x + z + 1)*(-y + z + 1) - 4*z)
- (x + z)*((x + z + 1)*(-y + z + 1) - 4*z)
+ (x + z)*(y - z)*(x - y + 2*z - 2))
- (x - y);
out[2][0][0] = 0.5*(y - z)*(y - z + 1)*(2*x + 2*z - 1);
out[2][0][1] = 0.5*(x + z)*(2*y - 2*z + 1)*(x + z - 1);
out[2][0][2] = 0.5*(out[2][0][0] + out[2][0][1])
+ 0.25*((y - x - 2*z)*(y - z + 1)*(x + z - 1)
+ (x + z)*(y - z)*(y - x - 2*z + 2));
out[3][0][0] = 0.5*(y - z)*(2*x + 2*z + 1)*(y - z + 1);
out[3][0][1] = 0.5*(2*y - 2*z + 1)*(x + z)*(x + z + 1);
out[3][0][2] = 0.5*(out[3][0][0] + out[3][0][1])
+ 0.25*((y - x - 2*z)*(y - z + 1)*(x + z + 1)
+ (y - z)*(x + z)*(y - x - 2*z));
out[4][0][0] = 0;
out[4][0][1] = 0;
out[4][0][2] = 4*z - 1;
// lower edges
out[5][0][0] = -((y - z + 1)*((y - 1)*(x + 1) + z*(x - y + z + 1))
+ (y - z + 1)*(x + z - 1)*((y - 1) + z));
out[5][0][1] = -((x + z - 1)*((y - 1)*(x + 1) + z*(x - y + z + 1))
+ (y - z + 1)*(x + z - 1)*((x + 1) - z));
out[5][0][2] = 0.5*(out[5][0][0] + out[5][0][1])
- 0.5*((-x + y - 2*z + 2)*((y - 1)*(x + 1) + z*(x - y + z + 1))
+ (y - z + 1)*(x + z - 1)*(x - y + 2*z + 1));
out[6][0][0] = -(y - z + 1)*(2*x + z + 1)*(y - 1);
out[6][0][1] = -(((x + z + 1)*(y - 1)*x - z) + z*(2*y + 1)
+ (y - z + 1)*((x + z + 1)*x + 2*z));
out[6][0][2] = 0.5*(out[6][0][0] + out[6][0][1])
- 0.5*(-(((x + z + 1)*(y - 1)*x - z) + z*(2*y + 1))
+ (y - z + 1)*(((y - 1)*x - 1) + 2*y + 1));
out[7][0][0] = -(((y - z - 1)*(x + 1)*y - z) + z*(2*x + 1)
+ (x + z - 1)*((y - z - 1)*y + 2*z));
out[7][0][1] = -(x + z - 1)*(2*y - z - 1)*(x + 1);
out[7][0][2] = 0.5*(out[7][0][0] + out[7][0][1])
- 0.5*(((y - z - 1)*(x + 1)*y - z) + z*(2*x + 1)
+ (x + z - 1)*((-(x + 1)*y - 1) + 2*x + 1));
out[8][0][0] = -(y - z + 1)*(2*x + z)*y;
out[8][0][1] = -(2*y - z + 1)*(x + z - 1)*(x + 1);
out[8][0][2] = 0.5*(out[8][0][0] + out[8][0][1])
- 0.5*(-x + y - 2*z + 2)*(x + 1)*y;
// upper edges
out[9][0][0] = 2*z*(y - z - 1);
out[9][0][1] = 2*z*(x + z - 1);
out[9][0][2] = 0.5*(out[9][0][0] + out[9][0][1])
+ (x + z - 1)*(y - z - 1) + z*(-x + y - 2*z);
out[10][0][0] = -2*z*(y - z - 1);
out[10][0][1] = -2*z*(x + z + 1);
out[10][0][2] = 0.5*(out[10][0][0] + out[10][0][1])
- ((x + z + 1)*(y - z - 1) + 4*z)
- z*(-x + y - 2*z + 2);
out[11][0][0] = -2*z*(y - z + 1);
out[11][0][1] = -2*z*(x + z - 1);
out[11][0][2] = 0.5*(out[11][0][0] + out[11][0][1])
- (y - z + 1)*(x + z - 1) - z*(-x + y - 2*z + 2);
out[12][0][0] = 2*z*(y - z + 1);
out[12][0][1] = 2*z*(x + z + 1);
out[12][0][2] = 0.5*(out[12][0][0] + out[12][0][1])
+ (y - z + 1)*(x + z + 1) + z*(-x + y - 2*z);
// base face
out[13][0][0] = 2*((y - z + 1)*((y - 1)*(x + 1) + z*(x - y + z + 1))
+ (y - z + 1)*(x + z - 1)*(y - 1 + z));
out[13][0][1] = 2*((x + z - 1)*((y - 1)*(x + 1) + z*(x - y + z + 1))
+ (y - z + 1)*(x + z - 1)*(x + 1 - z));
out[13][0][2] = 0.5*(out[13][0][0] + out[13][0][1])
+ ((-x + y - 2*z + 2)*((y - 1)*(x + 1) + z*(x - y + z + 1))
+ (y - z + 1)*(x + z - 1)*(x - y + 2*z + 1));
}
else
{
// vertices
out[0][0][0] = 0.5*(y + z)*(y + z - 1)*(2*x - 2*z - 1);
out[0][0][1] = 0.5*(2*y + 2*z - 1)*(x - z - 1)*(x - z);
out[0][0][2] = 0.5*(out[0][0][0] + out[0][0][1])
+ 0.25*((2*y + 2*z - 1)*(x - z - 1)*(x - z)
+ (y + z)*(y + z - 1)*(-2*x + 2*z + 1));
out[1][0][0] = -0.5*(y + z)*(2*x - 2*z + 1)*(-y - z + 1);
out[1][0][1] = -0.5*(x - z)*(x - z + 1)*(-2*y - 2*z + 1);
out[1][0][2] = 0.5*(out[1][0][0] + out[1][0][1])
- 0.25*((x - y - 2*z)*(x - z + 1)*(-y - z + 1)
+ (x - z)*(y + z)*(-x + y + 2*z - 2));
out[2][0][0] = 0.5*((y + z)*((x - z - 1)*(y + z + 1) + 4*z)
+ (x - z)*(y + z)*(y + z + 1) + 4*z);
out[2][0][1] = 0.5*((x - z)*((x - z - 1)*(y + z + 1) + 4*z)
+ (x - z)*(y + z)*(x - z - 1) - 4*z);
out[2][0][2] = 0.5*(out[2][0][0] + out[2][0][1])
+ 0.25*((x - y - 2*z)*((x - z - 1)*(y + z + 1) + 4*z)
+ (x - z)*(y + z)*(x - y - 2*z + 2) + 4*(x - y));
out[3][0][0] = 0.5*(y + z)*(2*x - 2*z + 1)*(y + z + 1);
out[3][0][1] = 0.5*(x - z)*(x - z + 1)*(2*y + 2*z + 1);
out[3][0][2] = 0.5*(out[3][0][0] + out[3][0][1])
+ 0.25*((x - y - 2*z)*(x - z + 1)*(y + z + 1)
+ (y + z)*(x - z)*(x - y - 2*z));
out[4][0][0] = 0;
out[4][0][1] = 0;
out[4][0][2] = 4*z - 1;
// lower edges
out[5][0][0] = -(y + z - 1)*(2*x - z - 1)*(y + 1);
out[5][0][1] = -(((x - z - 1)*(y + 1)*x - z) + z*(2*y + 1)
+ (y + z - 1)*((x - z - 1)*x + 2*z));
out[5][0][2] = 0.5*(out[5][0][0] + out[5][0][1])
- 0.5*((((x - z - 1)*(y + 1)*x - z) + z*(2*y + 1))
+ (y + z - 1)*((-(y + 1)*x - 1) + 2*y + 1));
out[6][0][0] = -(2*x - z + 1)*(y + z - 1)*(y + 1);
out[6][0][1] = -(x - z + 1)*(2*y + z)*x;
out[6][0][2] = 0.5*(out[6][0][0] + out[6][0][1])
- 0.5*(x - y - 2*z + 2)*(y + 1)*x;
out[7][0][0] = -(2*x - z)*(y + z - 1)*y;
out[7][0][1] = -(x - z + 1)*(2*y + z - 1)*(x - 1);
out[7][0][2] = 0.5*(out[7][0][0] + out[7][0][1])
- 0.5*(x - y - 2*z + 2)*(x - 1)*y;
out[8][0][0] = -(((y + z + 1)*(x - 1)*y - z) + z*(2*x + 1)
+ (x - z + 1)*((y + z + 1)*y + 2*z));
out[8][0][1] = -(x - z + 1)*(2*y + z + 1)*(x - 1);
out[8][0][2] = 0.5*(out[8][0][0] + out[8][0][1])
- 0.5*(-(((y + z + 1)*(x - 1)*y - z) + z*(2*x + 1))
+ (x - z + 1)*(((x - 1)*y - 1) + 2*x + 1));
// upper edges
out[9][0][0] = 2*z*(y + z - 1);
out[9][0][1] = 2*z*(x - z - 1);
out[9][0][2] = 0.5*(out[9][0][0] + out[9][0][1])
+ (y + z - 1)*(x - z - 1) + z*(x - y - 2*z);
out[10][0][0] = -2*z*(y + z - 1);
out[10][0][1] = -2*z*(x - z + 1);
out[10][0][2] = 0.5*(out[10][0][0] + out[10][0][1])
- (x - z + 1)*(y + z - 1) - z*(x - y - 2*z + 2);
out[11][0][0] = -2*z*(y + z + 1);
out[11][0][1] = -2*z*(x - z - 1);
out[11][0][2] = 0.5*(out[11][0][0] + out[11][0][1])
- ((y + z + 1)*(x - z - 1) + 4*z) - z*(x - y - 2*z + 2);
out[12][0][0] = 2*z*(y + z + 1);
out[12][0][1] = 2*z*(x - z + 1);
out[12][0][2] = 0.5*(out[12][0][0] + out[12][0][1])
+ (x - z + 1)*(y + z + 1) + z*(x - y - 2*z);
// base face
out[13][0][0] = 2*((y + z - 1)*((y + 1)*(x - 1) - z*(x - y - z - 1))
+ (x - z + 1)*(y + z - 1)*(y + 1 - z));
out[13][0][1] = 2*((x - z + 1)*((y + 1)*(x - 1) - z*(x - y - z - 1))
+ (x - z + 1)*(y + z - 1)*(x - 1 + z));
out[13][0][2] = 0.5*(out[13][0][0] + out[13][0][1])
+ (x - y - 2*z + 2)*((y + 1)*(x - 1) - z*(x - y - z - 1))
+ (x - z + 1)*(y + z - 1)*(-(x - y - 2*z - 1));
}
}
//! \brief Polynomial order of the shape functions
unsigned int order () const
{
return 2;
}
};
}
#endif
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