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Nektar::LibUtilities::NodalUtilTriangle Class Reference

Specialisation of the NodalUtil class to support nodal triangular elements. More...

#include <NodalUtil.h>

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Public Member Functions

 NodalUtilTriangle (int degree, Array< OneD, NekDouble > r, Array< OneD, NekDouble > s)
 Construct the nodal utility class for a triangle. More...
 
virtual ~NodalUtilTriangle ()
 
- Public Member Functions inherited from Nektar::LibUtilities::NodalUtil
NekVector< NekDoubleGetWeights ()
 Obtain the integration weights for the given nodal distribution. More...
 
SharedMatrix GetVandermonde ()
 Return the Vandermonde matrix for the nodal distribution. More...
 
SharedMatrix GetVandermondeForDeriv (int dir)
 Return the Vandermonde matrix of the derivative of the basis functions for the nodal distribution. More...
 
SharedMatrix GetDerivMatrix (int dir)
 Return the derivative matrix for the nodal distribution. More...
 
SharedMatrix GetInterpolationMatrix (Array< OneD, Array< OneD, NekDouble > > &xi)
 Construct the interpolation matrix used to evaluate the basis at the points xi inside the element. More...
 

Protected Member Functions

virtual NekVector< NekDoublev_OrthoBasis (const int mode)
 Return the value of the modal functions for the triangular element at the nodal points m_xi for a given mode. More...
 
virtual NekVector< NekDoublev_OrthoBasisDeriv (const int dir, const int mode)
 Return the value of the derivative of the modal functions for the triangular element at the nodal points m_xi for a given mode. More...
 
virtual boost::shared_ptr
< NodalUtil
v_CreateUtil (Array< OneD, Array< OneD, NekDouble > > &xi)
 Construct a NodalUtil object of the appropriate element type for a given set of points. More...
 
virtual NekDouble v_ModeZeroIntegral ()
 Return the value of the integral of the zero-th mode for this element. More...
 
virtual int v_NumModes ()
 Calculate the number of degrees of freedom for this element. More...
 
- Protected Member Functions inherited from Nektar::LibUtilities::NodalUtil
 NodalUtil (int degree, int dim)
 Set up the NodalUtil object. More...
 

Protected Attributes

std::vector< std::pair< int,
int > > 
m_ordering
 Mapping from the $ (i,j) $ indexing of the basis to a continuous ordering. More...
 
Array< OneD, Array< OneD,
NekDouble > > 
m_eta
 Collapsed coordinates $ (\eta_1, \eta_2) $ of the nodal points. More...
 
- Protected Attributes inherited from Nektar::LibUtilities::NodalUtil
int m_dim
 Dimension of the nodal element. More...
 
int m_degree
 Degree of the nodal element. More...
 
int m_numPoints
 Total number of nodal points. More...
 
Array< OneD, Array< OneD,
NekDouble > > 
m_xi
 Coordinates of the nodal points defining the basis. More...
 

Detailed Description

Specialisation of the NodalUtil class to support nodal triangular elements.

Definition at line 170 of file NodalUtil.h.

Constructor & Destructor Documentation

Nektar::LibUtilities::NodalUtilTriangle::NodalUtilTriangle ( int  degree,
Array< OneD, NekDouble r,
Array< OneD, NekDouble s 
)

Construct the nodal utility class for a triangle.

The constructor of this class sets up two member variables used in the evaluation of the orthogonal basis:

  • NodalUtilTriangle::m_eta is used to construct the collapsed coordinate locations of the nodal points $ (\eta_1, \eta_2) $ inside the square $[-1,1]^2$ on which the orthogonal basis functions are defined.
  • NodalUtilTriangle::m_ordering constructs a mapping from the index set $ I = \{ (i,j)\ |\ 0\leq i,j \leq P, i+j \leq P \}$ to an ordering $ 0 \leq m(ij) \leq (P+1)(P+2)/2 $ that defines the monomials $ \xi_1^i \xi_2^j $ that span the triangular space. This is then used to calculate which $ (i,j) $ pair corresponding to a column of the Vandermonde matrix when calculating the orthogonal polynomials.
Parameters
degreePolynomial order of this nodal triangle.
r$ \xi_1 $-coordinates of nodal points in the standard element.
s$ \xi_2 $-coordinates of nodal points in the standard element.

Definition at line 239 of file NodalUtil.cpp.

References Nektar::NekConstants::kNekZeroTol, Nektar::LibUtilities::NodalUtil::m_degree, m_eta, Nektar::LibUtilities::NodalUtil::m_numPoints, m_ordering, and Nektar::LibUtilities::NodalUtil::m_xi.

242  : NodalUtil(degree, 2), m_eta(2)
243 {
244  // Set up parent variables.
245  m_numPoints = r.num_elements();
246  m_xi[0] = r;
247  m_xi[1] = s;
248 
249  // Construct a mapping (i,j) -> m from the triangular tensor product space
250  // (i,j) to a single ordering m.
251  for (int i = 0; i <= m_degree; ++i)
252  {
253  for (int j = 0; j <= m_degree - i; ++j)
254  {
255  m_ordering.push_back(std::make_pair(i,j));
256  }
257  }
258 
259  // Calculate collapsed coordinates from r/s values
260  m_eta[0] = Array<OneD, NekDouble>(m_numPoints);
261  m_eta[1] = Array<OneD, NekDouble>(m_numPoints);
262 
263  for (int i = 0; i < m_numPoints; ++i)
264  {
265  if (fabs(m_xi[1][i]-1.0) < NekConstants::kNekZeroTol)
266  {
267  m_eta[0][i] = -1.0;
268  m_eta[1][i] = 1.0;
269  }
270  else
271  {
272  m_eta[0][i] = 2*(1+m_xi[0][i])/(1-m_xi[1][i])-1.0;
273  m_eta[1][i] = m_xi[1][i];
274  }
275  }
276 }
std::vector< std::pair< int, int > > m_ordering
Mapping from the indexing of the basis to a continuous ordering.
Definition: NodalUtil.h:184
int m_degree
Degree of the nodal element.
Definition: NodalUtil.h:108
static const NekDouble kNekZeroTol
Array< OneD, Array< OneD, NekDouble > > m_eta
Collapsed coordinates of the nodal points.
Definition: NodalUtil.h:187
NodalUtil(int degree, int dim)
Set up the NodalUtil object.
Definition: NodalUtil.h:101
Array< OneD, Array< OneD, NekDouble > > m_xi
Coordinates of the nodal points defining the basis.
Definition: NodalUtil.h:112
int m_numPoints
Total number of nodal points.
Definition: NodalUtil.h:110
virtual Nektar::LibUtilities::NodalUtilTriangle::~NodalUtilTriangle ( )
inlinevirtual

Definition at line 177 of file NodalUtil.h.

178  {
179  }

Member Function Documentation

virtual boost::shared_ptr<NodalUtil> Nektar::LibUtilities::NodalUtilTriangle::v_CreateUtil ( Array< OneD, Array< OneD, NekDouble > > &  xi)
inlineprotectedvirtual

Construct a NodalUtil object of the appropriate element type for a given set of points.

This function is used inside NodalUtil::GetInterpolationMatrix so that the (potentially non-square) Vandermonde matrix can be constructed to create the interpolation matrix at an arbitrary set of points in the domain.

Parameters
xiDistribution of nodal points to create utility with.

Implements Nektar::LibUtilities::NodalUtil.

Reimplemented in Nektar::Utilities::NodalUtilTriMonomial.

Definition at line 193 of file NodalUtil.h.

References Nektar::MemoryManager< DataType >::AllocateSharedPtr(), and Nektar::LibUtilities::NodalUtil::m_degree.

195  {
197  m_degree, xi[0], xi[1]);
198  }
static boost::shared_ptr< DataType > AllocateSharedPtr()
Allocate a shared pointer from the memory pool.
int m_degree
Degree of the nodal element.
Definition: NodalUtil.h:108
virtual NekDouble Nektar::LibUtilities::NodalUtilTriangle::v_ModeZeroIntegral ( )
inlineprotectedvirtual

Return the value of the integral of the zero-th mode for this element.

Note that for the orthogonal basis under consideration, all modes integrate to zero asides from the zero-th mode. This function is used in NodalUtil::GetWeights to determine integration weights.

Implements Nektar::LibUtilities::NodalUtil.

Definition at line 200 of file NodalUtil.h.

201  {
202  return 2.0 * sqrt(2.0);
203  }
virtual int Nektar::LibUtilities::NodalUtilTriangle::v_NumModes ( )
inlineprotectedvirtual

Calculate the number of degrees of freedom for this element.

Implements Nektar::LibUtilities::NodalUtil.

Definition at line 205 of file NodalUtil.h.

References Nektar::LibUtilities::NodalUtil::m_degree.

206  {
207  return (m_degree + 1) * (m_degree + 2) / 2;
208  }
int m_degree
Degree of the nodal element.
Definition: NodalUtil.h:108
NekVector< NekDouble > Nektar::LibUtilities::NodalUtilTriangle::v_OrthoBasis ( const int  mode)
protectedvirtual

Return the value of the modal functions for the triangular element at the nodal points m_xi for a given mode.

In a triangle, we use the orthogonal basis

\[ \psi_{m(ij)} = \sqrt{2} P^{(0,0)}_i(\xi_1) P_j^{(2i+1,0)}(\xi_2) (1-\xi_2)^i \]

where $ m(ij) $ is the mapping defined in NodalUtilTriangle::m_ordering and $ J_n^{(\alpha,\beta)}(z) $ denotes the standard Jacobi polynomial.

Parameters
modeThe mode of the orthogonal basis to evaluate.
Returns
Vector containing orthogonal basis evaluated at the points m_xi.

Implements Nektar::LibUtilities::NodalUtil.

Reimplemented in Nektar::Utilities::NodalUtilTriMonomial.

Definition at line 295 of file NodalUtil.cpp.

References Polylib::jacobfd(), m_eta, Nektar::LibUtilities::NodalUtil::m_numPoints, m_ordering, and CG_Iterations::modes.

296 {
297  std::vector<NekDouble> jacobi_i(m_numPoints), jacobi_j(m_numPoints);
298  std::pair<int, int> modes = m_ordering[mode];
299 
300  // Calculate Jacobi polynomials
302  m_numPoints, &m_eta[0][0], &jacobi_i[0], NULL, modes.first, 0.0, 0.0);
304  m_numPoints, &m_eta[1][0], &jacobi_j[0], NULL, modes.second,
305  2.0 * modes.first + 1.0, 0.0);
306 
307  NekVector<NekDouble> ret(m_numPoints);
308  NekDouble sqrt2 = sqrt(2.0);
309 
310  for (int i = 0; i < m_numPoints; ++i)
311  {
312  ret[i] = sqrt2 * jacobi_i[i] * jacobi_j[i] *
313  pow(1.0 - m_eta[1][i], modes.first);
314  }
315 
316  return ret;
317 }
std::vector< std::pair< int, int > > m_ordering
Mapping from the indexing of the basis to a continuous ordering.
Definition: NodalUtil.h:184
Array< OneD, Array< OneD, NekDouble > > m_eta
Collapsed coordinates of the nodal points.
Definition: NodalUtil.h:187
double NekDouble
int m_numPoints
Total number of nodal points.
Definition: NodalUtil.h:110
void jacobfd(const int np, const double *z, double *poly_in, double *polyd, const int n, const double alpha, const double beta)
Routine to calculate Jacobi polynomials, , and their first derivative, .
Definition: Polylib.cpp:1920
NekVector< NekDouble > Nektar::LibUtilities::NodalUtilTriangle::v_OrthoBasisDeriv ( const int  dir,
const int  mode 
)
protectedvirtual

Return the value of the derivative of the modal functions for the triangular element at the nodal points m_xi for a given mode.

Note that this routine must use the chain rule combined with the collapsed coordinate derivatives as described in Sherwin & Karniadakis (2nd edition), pg 150.

Parameters
dirCoordinate direction in which to evaluate the derivative.
modeThe mode of the orthogonal basis to evaluate.
Returns
Vector containing the derivative of the orthogonal basis evaluated at the points m_xi.

Implements Nektar::LibUtilities::NodalUtil.

Reimplemented in Nektar::Utilities::NodalUtilTriMonomial.

Definition at line 333 of file NodalUtil.cpp.

References Polylib::jacobd(), Polylib::jacobfd(), m_eta, Nektar::LibUtilities::NodalUtil::m_numPoints, m_ordering, and CG_Iterations::modes.

335 {
336  std::vector<NekDouble> jacobi_i(m_numPoints), jacobi_j(m_numPoints);
337  std::vector<NekDouble> jacobi_di(m_numPoints), jacobi_dj(m_numPoints);
338  std::pair<int, int> modes = m_ordering[mode];
339 
340  // Calculate Jacobi polynomials and their derivatives. Note that we use both
341  // jacobfd and jacobd since jacobfd is only valid for derivatives in the
342  // open interval (-1,1).
344  m_numPoints, &m_eta[0][0], &jacobi_i[0], NULL, modes.first, 0.0,
345  0.0);
347  m_numPoints, &m_eta[1][0], &jacobi_j[0], NULL, modes.second,
348  2.0*modes.first + 1.0, 0.0);
350  m_numPoints, &m_eta[0][0], &jacobi_di[0], modes.first, 0.0, 0.0);
352  m_numPoints, &m_eta[1][0], &jacobi_dj[0], modes.second,
353  2.0*modes.first + 1.0, 0.0);
354 
355  NekVector<NekDouble> ret(m_numPoints);
356  NekDouble sqrt2 = sqrt(2.0);
357 
358  if (dir == 0)
359  {
360  // d/d(\xi_1) = 2/(1-\eta_2) d/d(\eta_1)
361  for (int i = 0; i < m_numPoints; ++i)
362  {
363  ret[i] = 2.0 * sqrt2 * jacobi_di[i] * jacobi_j[i];
364  if (modes.first > 0)
365  {
366  ret[i] *= pow(1.0 - m_eta[1][i], modes.first - 1.0);
367  }
368  }
369  }
370  else
371  {
372  // d/d(\xi_2) = 2(1+\eta_1)/(1-\eta_2) d/d(\eta_1) + d/d(eta_2)
373  for (int i = 0; i < m_numPoints; ++i)
374  {
375  ret[i] = (1 + m_eta[0][i]) * sqrt2 * jacobi_di[i] * jacobi_j[i];
376  if (modes.first > 0)
377  {
378  ret[i] *= pow(1.0 - m_eta[1][i], modes.first - 1.0);
379  }
380 
381  NekDouble tmp = jacobi_dj[i] * pow(1.0 - m_eta[1][i], modes.first);
382  if (modes.first > 0)
383  {
384  tmp -= modes.first * jacobi_j[i] *
385  pow(1.0 - m_eta[1][i], modes.first-1);
386  }
387 
388  ret[i] += sqrt2 * jacobi_i[i] * tmp;
389  }
390  }
391 
392  return ret;
393 }
std::vector< std::pair< int, int > > m_ordering
Mapping from the indexing of the basis to a continuous ordering.
Definition: NodalUtil.h:184
void jacobd(const int np, const double *z, double *polyd, const int n, const double alpha, const double beta)
Calculate the derivative of Jacobi polynomials.
Definition: Polylib.cpp:2120
Array< OneD, Array< OneD, NekDouble > > m_eta
Collapsed coordinates of the nodal points.
Definition: NodalUtil.h:187
double NekDouble
int m_numPoints
Total number of nodal points.
Definition: NodalUtil.h:110
void jacobfd(const int np, const double *z, double *poly_in, double *polyd, const int n, const double alpha, const double beta)
Routine to calculate Jacobi polynomials, , and their first derivative, .
Definition: Polylib.cpp:1920

Member Data Documentation

Array<OneD, Array<OneD, NekDouble> > Nektar::LibUtilities::NodalUtilTriangle::m_eta
protected

Collapsed coordinates $ (\eta_1, \eta_2) $ of the nodal points.

Definition at line 187 of file NodalUtil.h.

Referenced by NodalUtilTriangle(), v_OrthoBasis(), and v_OrthoBasisDeriv().

std::vector<std::pair<int, int> > Nektar::LibUtilities::NodalUtilTriangle::m_ordering
protected

Mapping from the $ (i,j) $ indexing of the basis to a continuous ordering.

Definition at line 184 of file NodalUtil.h.

Referenced by NodalUtilTriangle(), v_OrthoBasis(), Nektar::Utilities::NodalUtilTriMonomial::v_OrthoBasis(), and v_OrthoBasisDeriv().