Nektar::LibUtilities::NodalUtilTetrahedron Class Reference

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

#include <NodalUtil.h>

Inheritance diagram for Nektar::LibUtilities::NodalUtilTetrahedron:

Public Member Functions
	NodalUtilTetrahedron (size_t degree, Array< OneD, NekDouble > r, Array< OneD, NekDouble > s, Array< OneD, NekDouble > t)
	Construct the nodal utility class for a tetrahedron. More...
virtual	~NodalUtilTetrahedron ()
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
virtual NekVector< NekDouble >	v_OrthoBasis (const size_t mode) override
	Return the value of the modal functions for the tetrahedral element at the nodal points m_xi for a given mode. More...
virtual 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 tetrahedral element at the nodal points m_xi for a given mode. More...
virtual 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...
virtual NekDouble	v_ModeZeroIntegral () override
	Return the value of the integral of the zero-th mode for this element. More...
virtual 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< Mode >	m_ordering
	Mapping from the \( (i,j,k) \) indexing of the basis to a continuous ordering. More...
Array< OneD, Array< OneD, NekDouble > >	m_eta
	Collapsed coordinates \( (\eta_1, \eta_2, \eta_3) \) 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...

Private Types
typedef std::tuple< int, int, int >	Mode

Detailed Description

Specialisation of the NodalUtil class to support nodal tetrahedral elements.

Definition at line 212 of file NodalUtil.h.

Member Typedef Documentation

Mode

typedef std::tuple<int, int, int> Nektar::LibUtilities::NodalUtilTetrahedron::Mode

private

Definition at line 214 of file NodalUtil.h.

Constructor & Destructor Documentation

NodalUtilTetrahedron()

Nektar::LibUtilities::NodalUtilTetrahedron::NodalUtilTetrahedron	(	size_t	degree,
		Array< OneD, NekDouble >	r,
		Array< OneD, NekDouble >	s,
		Array< OneD, NekDouble >	t
	)

Construct the nodal utility class for a tetrahedron.

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

• NodalUtilTetrahedron::m_eta is used to construct the collapsed coordinate locations of the nodal points \( (\eta_1, \eta_2, \eta_3) \) inside the cube \([-1,1]^3\) on which the orthogonal basis functions are defined.
• NodalUtilTetrahedron::m_ordering constructs a mapping from the index set \( I = \{ (i,j,k)\ |\ 0\leq i,j,k \leq P, i+j \leq P, i+j+k \leq P \}\) to an ordering \( 0 \leq m(ijk) \leq (P+1)(P+2)(P+3)/6 \) that defines the monomials \( \xi_1^i \xi_2^j \xi_3^k \) that span the tetrahedral space. This is then used to calculate which \( (i,j,k) \) triple (represented as a tuple) corresponding to a column of the Vandermonde matrix when calculating the orthogonal polynomials.

Parameters

degree	Polynomial order of this nodal tetrahedron
r	\( \xi_1 \)-coordinates of nodal points in the standard element.
s	\( \xi_2 \)-coordinates of nodal points in the standard element.
t	\( \xi_3 \)-coordinates of nodal points in the standard element.

Definition at line 412 of file NodalUtil.cpp.

    : NodalUtil(degree, 3), m_eta(3)
{
    m_numPoints = r.size();
    m_xi[0]     = r;
    m_xi[1]     = s;
    m_xi[2]     = t;
 
    for (size_t i = 0; i <= m_degree; ++i)
    {
        for (size_t j = 0; j <= m_degree - i; ++j)
        {
            for (size_t k = 0; k <= m_degree - i - j; ++k)
            {
                m_ordering.push_back(Mode(i, j, k));
            }
        }
    }
 
    // Calculate collapsed coordinates from r/s values
    m_eta[0] = Array<OneD, NekDouble>(m_numPoints);
    m_eta[1] = Array<OneD, NekDouble>(m_numPoints);
    m_eta[2] = Array<OneD, NekDouble>(m_numPoints);
 
    for (size_t i = 0; i < m_numPoints; ++i)
    {
        if (fabs(m_xi[2][i] - 1.0) < NekConstants::kNekZeroTol)
        {
            // Very top point of the tetrahedron
            m_eta[0][i] = -1.0;
            m_eta[1][i] = -1.0;
            m_eta[2][i] = m_xi[2][i];
        }
        else
        {
            if (fabs(m_xi[1][i] - 1.0) < NekConstants::kNekZeroTol)
            {
                // Distant diagonal edge shared by all eta_x coordinate planes:
                // the xi_y == -xi_z line
                m_eta[0][i] = -1.0;
            }
            else if (fabs(m_xi[1][i] + m_xi[2][i]) < NekConstants::kNekZeroTol)
            {
                m_eta[0][i] = -1.0;
            }
            else
            {
                m_eta[0][i] =
                    2.0 * (1.0 + m_xi[0][i]) / (-m_xi[1][i] - m_xi[2][i]) - 1.0;
            }
            m_eta[1][i] = 2.0 * (1.0 + m_xi[1][i]) / (1.0 - m_xi[2][i]) - 1.0;
            m_eta[2][i] = m_xi[2][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.

~NodalUtilTetrahedron()

virtual Nektar::LibUtilities::NodalUtilTetrahedron::~NodalUtilTetrahedron ( )

inlinevirtual

Definition at line 222 of file NodalUtil.h.

    {
    }

Member Function Documentation

v_CreateUtil()

virtual std::shared_ptr< NodalUtil > Nektar::LibUtilities::NodalUtilTetrahedron::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 239 of file NodalUtil.h.

    {
        return MemoryManager<NodalUtilTetrahedron>::AllocateSharedPtr(
            m_degree, xi[0], xi[1], xi[2]);
    }

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

v_ModeZeroIntegral()

virtual NekDouble Nektar::LibUtilities::NodalUtilTetrahedron::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 246 of file NodalUtil.h.

    {
        return 8.0 * sqrt(2.0) / 3.0;
    }

References tinysimd::sqrt().

v_NumModes()

virtual size_t Nektar::LibUtilities::NodalUtilTetrahedron::v_NumModes ( )

inlineoverrideprotectedvirtual

Calculate the number of degrees of freedom for this element.

Implements Nektar::LibUtilities::NodalUtil.

Definition at line 251 of file NodalUtil.h.

    {
        return (m_degree + 1) * (m_degree + 2) * (m_degree + 3) / 6;
    }

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

v_OrthoBasis()

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

overrideprotectedvirtual

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

In a tetrahedron, we use the orthogonal basis

\[ \psi_{m(ijk)} = \sqrt{8} P^{(0,0)}_i(\xi_1) P_j^{(2i+1,0)}(\xi_2) P_k^{(2i+2j+2,0)}(\xi_3) (1-\xi_2)^i (1-\xi_3)^{i+j} \]

where \( m(ijk) \) is the mapping defined in 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 487 of file NodalUtil.cpp.

{
    std::vector<NekDouble> jacA(m_numPoints), jacB(m_numPoints);
    std::vector<NekDouble> jacC(m_numPoints);
 
    size_t I, J, K;
    std::tie(I, J, K) = m_ordering[mode];
 
    // Calculate Jacobi polynomials
    Polylib::jacobfd(m_numPoints, &m_eta[0][0], &jacA[0], NULL, I, 0.0, 0.0);
    Polylib::jacobfd(m_numPoints, &m_eta[1][0], &jacB[0], NULL, J,
                     2.0 * I + 1.0, 0.0);
    Polylib::jacobfd(m_numPoints, &m_eta[2][0], &jacC[0], NULL, K,
                     2.0 * (I + J) + 2.0, 0.0);
 
    NekVector<NekDouble> ret(m_numPoints);
    NekDouble sqrt8 = sqrt(8.0);
 
    for (size_t i = 0; i < m_numPoints; ++i)
    {
        ret[i] = sqrt8 * jacA[i] * jacB[i] * jacC[i] *
                 pow(1.0 - m_eta[1][i], I) * pow(1.0 - m_eta[2][i], I + J);
    }
 
    return ret;
}

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

v_OrthoBasisDeriv()

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

overrideprotectedvirtual

Return the value of the derivative of the modal functions for the tetrahedral 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 152.

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 528 of file NodalUtil.cpp.

{
    std::vector<NekDouble> jacA(m_numPoints), jacB(m_numPoints);
    std::vector<NekDouble> jacC(m_numPoints);
    std::vector<NekDouble> jacDerivA(m_numPoints), jacDerivB(m_numPoints);
    std::vector<NekDouble> jacDerivC(m_numPoints);
 
    size_t I, J, K;
    std::tie(I, J, K) = m_ordering[mode];
 
    // Calculate Jacobi polynomials
    Polylib::jacobfd(m_numPoints, &m_eta[0][0], &jacA[0], NULL, I, 0.0, 0.0);
    Polylib::jacobfd(m_numPoints, &m_eta[1][0], &jacB[0], NULL, J,
                     2.0 * I + 1.0, 0.0);
    Polylib::jacobfd(m_numPoints, &m_eta[2][0], &jacC[0], NULL, K,
                     2.0 * (I + J) + 2.0, 0.0);
    Polylib::jacobd(m_numPoints, &m_eta[0][0], &jacDerivA[0], I, 0.0, 0.0);
    Polylib::jacobd(m_numPoints, &m_eta[1][0], &jacDerivB[0], J, 2.0 * I + 1.0,
                    0.0);
    Polylib::jacobd(m_numPoints, &m_eta[2][0], &jacDerivC[0], K,
                    2.0 * (I + J) + 2.0, 0.0);
 
    NekVector<NekDouble> ret(m_numPoints);
    NekDouble sqrt8 = sqrt(8.0);
 
    // Always compute x-derivative since this term appears in the latter two
    // terms.
    for (size_t i = 0; i < m_numPoints; ++i)
    {
        ret[i] = 4.0 * sqrt8 * jacDerivA[i] * jacB[i] * jacC[i];
 
        if (I > 0)
        {
            ret[i] *= pow(1 - m_eta[1][i], I - 1);
        }
 
        if (I + J > 0)
        {
            ret[i] *= pow(1 - m_eta[2][i], I + J - 1);
        }
    }
 
    if (dir >= 1)
    {
        // Multiply by (1+a)/2
        NekVector<NekDouble> tmp(m_numPoints);
 
        for (size_t i = 0; i < m_numPoints; ++i)
        {
            ret[i] *= 0.5 * (m_eta[0][i] + 1.0);
 
            tmp[i] = 2.0 * sqrt8 * jacA[i] * jacC[i];
            if (I + J > 0)
            {
                tmp[i] *= pow(1.0 - m_eta[2][i], I + J - 1);
            }
 
            NekDouble tmp2 = jacDerivB[i] * pow(1.0 - m_eta[1][i], I);
            if (I > 0)
            {
                tmp2 -= I * jacB[i] * pow(1.0 - m_eta[1][i], I - 1);
            }
 
            tmp[i] *= tmp2;
        }
 
        if (dir == 1)
        {
            ret += tmp;
            return ret;
        }
 
        for (size_t i = 0; i < m_numPoints; ++i)
        {
            ret[i] += 0.5 * (1.0 + m_eta[1][i]) * tmp[i];
 
            NekDouble tmp2 = jacDerivC[i] * pow(1.0 - m_eta[2][i], I + J);
            if (I + J > 0)
            {
                tmp2 -= jacC[i] * (I + J) * pow(1.0 - m_eta[2][i], I + J - 1);
            }
 
            ret[i] +=
                sqrt8 * jacA[i] * jacB[i] * pow(1.0 - m_eta[1][i], I) * tmp2;
        }
    }
 
    return ret;
}

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

Member Data Documentation

m_eta

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

protected

Collapsed coordinates \( (\eta_1, \eta_2, \eta_3) \) of the nodal points.

Definition at line 233 of file NodalUtil.h.

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

m_ordering

std::vector<Mode> Nektar::LibUtilities::NodalUtilTetrahedron::m_ordering

protected

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

Definition at line 229 of file NodalUtil.h.

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