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

#include <NodalUtil.h>

Inheritance diagram for Nektar::LibUtilities::NodalUtilTriangle:

Public Member Functions
	NodalUtilTriangle (size_t degree, Array< OneD, NekDouble > r, Array< OneD, NekDouble > s)
	Construct the nodal utility class for a triangle. More...

	~NodalUtilTriangle () override

Public Member Functions inherited from Nektar::LibUtilities::NodalUtil
virtual	~NodalUtil ()=default

NekVector< NekDouble >	GetWeights ()
	Obtain the integration weights for the given nodal distribution. More...

SharedMatrix	GetVandermonde ()
	Return the Vandermonde matrix for the nodal distribution. More...

SharedMatrix	GetVandermondeForDeriv (size_t dir)
	Return the Vandermonde matrix of the derivative of the basis functions for the nodal distribution. More...

SharedMatrix	GetDerivMatrix (size_t 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
NekVector< NekDouble >	v_OrthoBasis (const size_t mode) override
	Return the value of the modal functions for the triangular element at the nodal points m_xi for a given mode. More...

NekVector< NekDouble >	v_OrthoBasisDeriv (const size_t dir, const size_t mode) override
	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...

std::shared_ptr< NodalUtil >	v_CreateUtil (Array< OneD, Array< OneD, NekDouble > > &xi) override
	Construct a NodalUtil object of the appropriate element type for a given set of points. More...

NekDouble	v_ModeZeroIntegral () override
	Return the value of the integral of the zero-th mode for this element. More...

size_t	v_NumModes () override
	Calculate the number of degrees of freedom for this element. More...

Protected Member Functions inherited from Nektar::LibUtilities::NodalUtil
	NodalUtil (size_t degree, size_t dim)
	Set up the NodalUtil object. More...

virtual NekVector< NekDouble >	v_OrthoBasis (const size_t mode)=0
	Return the values of the orthogonal basis at the nodal points for a given mode. More...

virtual NekVector< NekDouble >	v_OrthoBasisDeriv (const size_t dir, const size_t mode)=0
	Return the values of the derivative of the orthogonal basis at the nodal points for a given mode. More...

virtual std::shared_ptr< NodalUtil >	v_CreateUtil (Array< OneD, Array< OneD, NekDouble > > &xi)=0
	Construct a NodalUtil object of the appropriate element type for a given set of points. More...

virtual NekDouble	v_ModeZeroIntegral ()=0
	Return the value of the integral of the zero-th mode for this element. More...

virtual size_t	v_NumModes ()=0
	Calculate the number of degrees of freedom for this element. 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
size_t	m_dim
	Dimension of the nodal element. More...

size_t	m_degree
	Degree of the nodal element. More...

size_t	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 165 of file NodalUtil.h.

Constructor & Destructor Documentation

◆ NodalUtilTriangle()

Nektar::LibUtilities::NodalUtilTriangle::NodalUtilTriangle	(	size_t	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

degree	Polynomial 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 234 of file NodalUtil.cpp.

    : NodalUtil(degree, 2), m_eta(2)
{
    // Set up parent variables.
    m_numPoints = r.size();
    m_xi[0]     = r;
    m_xi[1]     = s;
 
    // Construct a mapping (i,j) -> m from the triangular tensor product space
    // (i,j) to a single ordering m.
    for (size_t i = 0; i <= m_degree; ++i)
    {
        for (size_t j = 0; j <= m_degree - i; ++j)
        {
            m_ordering.push_back(std::make_pair(i, j));
        }
    }
 
    // Calculate collapsed coordinates from r/s values
    m_eta[0] = Array<OneD, NekDouble>(m_numPoints);
    m_eta[1] = Array<OneD, NekDouble>(m_numPoints);
 
    for (size_t i = 0; i < m_numPoints; ++i)
    {
        if (fabs(m_xi[1][i] - 1.0) < NekConstants::kNekZeroTol)
        {
            m_eta[0][i] = -1.0;
            m_eta[1][i] = 1.0;
        }
        else
        {
            m_eta[0][i] = 2 * (1 + m_xi[0][i]) / (1 - m_xi[1][i]) - 1.0;
            m_eta[1][i] = m_xi[1][i];
        }
    }
}

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

◆ ~NodalUtilTriangle()

Nektar::LibUtilities::NodalUtilTriangle::~NodalUtilTriangle ( )

inlineoverride

Definition at line 172 of file NodalUtil.h.

173 {

174 }

Member Function Documentation

◆ v_CreateUtil()

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

inlineoverrideprotectedvirtual

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

xi	Distribution of nodal points to create utility with.

Implements Nektar::LibUtilities::NodalUtil.

Definition at line 188 of file NodalUtil.h.

    {
        return MemoryManager<NodalUtilTriangle>::AllocateSharedPtr(
            m_degree, xi[0], xi[1]);
    }

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

◆ v_ModeZeroIntegral()

NekDouble Nektar::LibUtilities::NodalUtilTriangle::v_ModeZeroIntegral ( )

inlineoverrideprotectedvirtual

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 195 of file NodalUtil.h.

    {
        return 2.0 * sqrt(2.0);
    }

References tinysimd::sqrt().

◆ v_NumModes()

size_t Nektar::LibUtilities::NodalUtilTriangle::v_NumModes ( )

inlineoverrideprotectedvirtual

Calculate the number of degrees of freedom for this element.

Implements Nektar::LibUtilities::NodalUtil.

Definition at line 200 of file NodalUtil.h.

    {
        return (m_degree + 1) * (m_degree + 2) / 2;
    }

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

◆ v_OrthoBasis()

NekVector< NekDouble > Nektar::LibUtilities::NodalUtilTriangle::v_OrthoBasis ( const size_t mode )

overrideprotectedvirtual

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

mode	The mode of the orthogonal basis to evaluate.

Returns: Vector containing orthogonal basis evaluated at the points m_xi.

Implements Nektar::LibUtilities::NodalUtil.

Definition at line 289 of file NodalUtil.cpp.

{
    std::vector<NekDouble> jacobi_i(m_numPoints), jacobi_j(m_numPoints);
    std::pair<int, int> modes = m_ordering[mode];
 
    // Calculate Jacobi polynomials
    Polylib::jacobfd(m_numPoints, &m_eta[0][0], &jacobi_i[0], nullptr,
                     modes.first, 0.0, 0.0);
    Polylib::jacobfd(m_numPoints, &m_eta[1][0], &jacobi_j[0], nullptr,
                     modes.second, 2.0 * modes.first + 1.0, 0.0);
 
    NekVector<NekDouble> ret(m_numPoints);
    NekDouble sqrt2 = sqrt(2.0);
 
    for (size_t i = 0; i < m_numPoints; ++i)
    {
        ret[i] = sqrt2 * jacobi_i[i] * jacobi_j[i] *
                 pow(1.0 - m_eta[1][i], modes.first);
    }
 
    return ret;
}

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

◆ v_OrthoBasisDeriv()

NekVector< NekDouble > Nektar::LibUtilities::NodalUtilTriangle::v_OrthoBasisDeriv	(	const size_t	dir,
		const size_t	mode
	)

overrideprotectedvirtual

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

dir	Coordinate direction in which to evaluate the derivative.
mode	The 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.

Definition at line 326 of file NodalUtil.cpp.

{
    std::vector<NekDouble> jacobi_i(m_numPoints), jacobi_j(m_numPoints);
    std::vector<NekDouble> jacobi_di(m_numPoints), jacobi_dj(m_numPoints);
    std::pair<int, int> modes = m_ordering[mode];
 
    // Calculate Jacobi polynomials and their derivatives. Note that we use both
    // jacobfd and jacobd since jacobfd is only valid for derivatives in the
    // open interval (-1,1).
    Polylib::jacobfd(m_numPoints, &m_eta[0][0], &jacobi_i[0], nullptr,
                     modes.first, 0.0, 0.0);
    Polylib::jacobfd(m_numPoints, &m_eta[1][0], &jacobi_j[0], nullptr,
                     modes.second, 2.0 * modes.first + 1.0, 0.0);
    Polylib::jacobd(m_numPoints, &m_eta[0][0], &jacobi_di[0], modes.first, 0.0,
                    0.0);
    Polylib::jacobd(m_numPoints, &m_eta[1][0], &jacobi_dj[0], modes.second,
                    2.0 * modes.first + 1.0, 0.0);
 
    NekVector<NekDouble> ret(m_numPoints);
    NekDouble sqrt2 = sqrt(2.0);
 
    if (dir == 0)
    {
        // d/d(\xi_1) = 2/(1-\eta_2) d/d(\eta_1)
        for (size_t i = 0; i < m_numPoints; ++i)
        {
            ret[i] = 2.0 * sqrt2 * jacobi_di[i] * jacobi_j[i];
            if (modes.first > 0)
            {
                ret[i] *= pow(1.0 - m_eta[1][i], modes.first - 1.0);
            }
        }
    }
    else
    {
        // d/d(\xi_2) = 2(1+\eta_1)/(1-\eta_2) d/d(\eta_1) + d/d(eta_2)
        for (size_t i = 0; i < m_numPoints; ++i)
        {
            ret[i] = (1 + m_eta[0][i]) * sqrt2 * jacobi_di[i] * jacobi_j[i];
            if (modes.first > 0)
            {
                ret[i] *= pow(1.0 - m_eta[1][i], modes.first - 1.0);
            }
 
            NekDouble tmp = jacobi_dj[i] * pow(1.0 - m_eta[1][i], modes.first);
            if (modes.first > 0)
            {
                tmp -= modes.first * jacobi_j[i] *
                       pow(1.0 - m_eta[1][i], modes.first - 1);
            }
 
            ret[i] += sqrt2 * jacobi_i[i] * tmp;
        }
    }
 
    return ret;
}

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

Member Data Documentation

◆ m_eta

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

protected

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

Definition at line 182 of file NodalUtil.h.

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

◆ m_ordering

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 179 of file NodalUtil.h.

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

Public Member Functions

Protected Member Functions

Protected Attributes

Detailed Description

Constructor & Destructor Documentation

◆ NodalUtilTriangle()

◆ ~NodalUtilTriangle()

Member Function Documentation

◆ v_CreateUtil()

◆ v_ModeZeroIntegral()

◆ v_NumModes()

◆ v_OrthoBasis()

◆ v_OrthoBasisDeriv()

Member Data Documentation

◆ m_eta

◆ m_ordering