A class to assist in the construction of nodal simplex and hybrid elements in two and three dimensions. More...

#include <NodalUtil.h>

Inheritance diagram for Nektar::LibUtilities::NodalUtil:

Public Member Functions
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
	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
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

A class to assist in the construction of nodal simplex and hybrid elements in two and three dimensions.

The NodalUtil class and its subclasses are designed to take care of some common issues that arise when considering triangles, tetrahedra and prismatic elements that are equipped with a nodal Lagrangian basis, defined using a set of nodal points \( \xi_i \) that we store in the array NodalUtil::m_xi. Since one cannot write this basis analytically, we instead construct the Vandermonde matrix

\[ \mathbf{V}_{ij} = \psi_j(\xi_i) \]

where \( \psi_j \) is a basis that spans the polynomial space of the element. The Vandermonde matrix can then be used to construct the integration weights, derivative and interpolation matrices. Although this can be any basis, such as the monomial basis \( x^i y^j z^k \), in practice this is numerically unstable at high polynomial orders. Elements are therefore expected to use the 'traditional' modal orthogonal basis. See Sherwin & Karniadakis or Hesthaven & Warburton for further details of this basis and the surrounding numerical issues.

This class therefore contains the generic logic needed to construct various matrices, and subclasses override virtual functions that define the orthogonal basis and its derivatives for a particular element type.

Definition at line 77 of file NodalUtil.h.

Constructor & Destructor Documentation

◆ ~NodalUtil()

virtual Nektar::LibUtilities::NodalUtil::~NodalUtil ( )

virtualdefault

◆ NodalUtil()

Nektar::LibUtilities::NodalUtil::NodalUtil	(	size_t	degree,
		size_t	dim
	)

inlineprotected

Set up the NodalUtil object.

Parameters

dim	Dimension of the element.
degree	Polynomial degree of the element.

Definition at line 95 of file NodalUtil.h.

        : m_dim(dim), m_degree(degree), m_xi(dim)
    {
    }

Member Function Documentation

◆ GetDerivMatrix()

SharedMatrix Nektar::LibUtilities::NodalUtil::GetDerivMatrix ( size_t dir )

Return the derivative matrix for the nodal distribution.

This routine constructs and returns the derivative matrices \(\mathbf{D}_d\) for coordinate directions \( 0 \leq d \leq 2 \), which can be used to evaluate the derivative of a nodal expansion at the points defined by NodalUtil::m_xi. These are calculated as \( \mathbf{D}_d = \mathbf{V}_d \mathbf{V}^{-1} \), where \( \mathbf{V}_d \) is the derivative Vandermonde matrix and \( \mathbf{V} \) is the Vandermonde matrix.

Parameters

dir	Coordinate direction in which to evaluate the derivative.

Returns: The derivative matrix for direction dir.

Definition at line 163 of file NodalUtil.cpp.

{
    SharedMatrix V  = GetVandermonde();
    SharedMatrix Vd = GetVandermondeForDeriv(dir);
    SharedMatrix D  = MemoryManager<NekMatrix<NekDouble>>::AllocateSharedPtr(
        V->GetRows(), V->GetColumns(), 0.0);
 
    V->Invert();
 
    *D = (*Vd) * (*V);
 
    return D;
}

References GetVandermonde(), and GetVandermondeForDeriv().

◆ GetInterpolationMatrix()

SharedMatrix Nektar::LibUtilities::NodalUtil::GetInterpolationMatrix ( Array< OneD, Array< OneD, NekDouble > > & xi )

Construct the interpolation matrix used to evaluate the basis at the points xi inside the element.

This routine returns a matrix \( \mathbf{I}(\mathbf{a}) \) that can be used to evaluate the nodal basis at the points defined by the parameter xi, which is denoted by \( \mathbf{a} = (a_1, \dots, a_N) \) and \( N \) is the number of points in xi.

In particular, if the array \( \mathbf{u} \) with components \( \mathbf{u}_i = u^\delta(\xi_i) \) represents the polynomial approximation of a function \( u \) evaluated at the nodal points NodalUtil::m_xi, then the evaluation of \( u^\delta \) evaluated at the input points \( \mathbf{a} \) is given by \( \mathbf{I}(\mathbf{a})\mathbf{u} \).

Parameters

xi	An array of first size number of spatial dimensions \( d \) and secondary size the number of points to interpolate.

Returns: The interpolation matrix for the points xi.

Definition at line 197 of file NodalUtil.cpp.

{
    std::shared_ptr<NodalUtil> subUtil = v_CreateUtil(xi);
    SharedMatrix matS                  = GetVandermonde();
    SharedMatrix matT                  = subUtil->GetVandermonde();
    SharedMatrix D = MemoryManager<NekMatrix<NekDouble>>::AllocateSharedPtr(
        matT->GetRows(), matS->GetColumns(), 0.0);
 
    matS->Invert();
 
    *D = (*matT) * (*matS);
    return D;
}

References GetVandermonde(), and v_CreateUtil().

◆ GetVandermonde()

SharedMatrix Nektar::LibUtilities::NodalUtil::GetVandermonde ( )

Return the Vandermonde matrix for the nodal distribution.

This routine constructs and returns the Vandermonde matrix \(\mathbf{V}\), with each entry as \(\mathbf{V}_{ij} = (\psi_i(\xi_j))\) where \( \psi_i \) is the orthogonal basis obtained through the abstract function NodalUtil::v_OrthoBasis.

Returns: The Vandermonde matrix.

Definition at line 97 of file NodalUtil.cpp.

{
    size_t rows = m_numPoints, cols = v_NumModes();
    SharedMatrix matV =
        MemoryManager<NekMatrix<NekDouble>>::AllocateSharedPtr(rows, cols, 0.0);
 
    for (size_t j = 0; j < cols; ++j)
    {
        NekVector<NekDouble> col = v_OrthoBasis(j);
        for (size_t i = 0; i < rows; ++i)
        {
            (*matV)(i, j) = col[i];
        }
    }
 
    return matV;
}

References m_numPoints, v_NumModes(), and v_OrthoBasis().

Referenced by GetDerivMatrix(), GetInterpolationMatrix(), and GetWeights().

◆ GetVandermondeForDeriv()

SharedMatrix Nektar::LibUtilities::NodalUtil::GetVandermondeForDeriv ( size_t dir )

Return the Vandermonde matrix of the derivative of the basis functions for the nodal distribution.

This routine constructs and returns the Vandermonde matrix for the derivative of the basis functions \(\mathbf{V}_d\) for coordinate directions \( 0 \leq d \leq 2 \), with each entry as \(\mathbf{V}_{ij} = (\partial_d \psi_i(\xi_j))\) where \( \partial_d\psi_i \) is the derivative of the orthogonal basis obtained through the abstract function NodalUtil::v_OrthoBasisDeriv.

Parameters

dir	Direction of derivative in the standard element.

Returns: Vandermonde matrix corresponding with derivative of the basis functions in direction dir.

Definition at line 130 of file NodalUtil.cpp.

{
    size_t rows = m_numPoints, cols = v_NumModes();
    SharedMatrix matV =
        MemoryManager<NekMatrix<NekDouble>>::AllocateSharedPtr(rows, cols, 0.0);
 
    for (size_t j = 0; j < cols; ++j)
    {
        NekVector<NekDouble> col = v_OrthoBasisDeriv(dir, j);
        for (size_t i = 0; i < rows; ++i)
        {
            (*matV)(i, j) = col[i];
        }
    }
 
    return matV;
}

References m_numPoints, v_NumModes(), and v_OrthoBasisDeriv().

Referenced by GetDerivMatrix().

◆ GetWeights()

NekVector< NekDouble > Nektar::LibUtilities::NodalUtil::GetWeights ( )

Obtain the integration weights for the given nodal distribution.

This routine constructs the integration weights for the given nodal distribution inside NodalUtil::m_xi. To do this we solve the linear system \( \mathbf{V}^\top \mathbf{w} = \mathbf{g} \) where \( \mathbf{g}_i = \int_\Omega \psi_i(x)\, dx \), and we use the fact that under the definition of the orthogonal basis for each element type, \( \mathbf{g}_i = 0 \) for \( i > 0 \). We use NodalUtil::v_ModeZeroIntegral to return the analytic value of \( \mathbf{g}_0 \).

Returns: Vector of integration weights for the integration points.

Definition at line 61 of file NodalUtil.cpp.

{
    // Get number of modes in orthogonal basis
    size_t numModes = v_NumModes();
 
    // If we have the same number of nodes as modes, then we can solve the
    // linear system V^T w = (1,0,...)
    if (numModes == m_numPoints)
    {
        NekVector<NekDouble> g(m_numPoints, 0.0);
        g(0) = v_ModeZeroIntegral();
 
        SharedMatrix V = GetVandermonde();
 
        // Solve the system V^T w = g to obtain weights.
        LinearSystem matL(V);
        return matL.SolveTranspose(g);
    }
    else
    {
        // System is either over- or under-determined. Need to do least squares
        // here using SVD.
        return NekVector<NekDouble>();
    }
}

References GetVandermonde(), m_numPoints, Nektar::LinearSystem::SolveTranspose(), v_ModeZeroIntegral(), and v_NumModes().

◆ v_CreateUtil()

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

protectedpure virtual

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.

Implemented in Nektar::LibUtilities::NodalUtilTriangle, Nektar::LibUtilities::NodalUtilTetrahedron, Nektar::LibUtilities::NodalUtilPrism, Nektar::LibUtilities::NodalUtilQuad, and Nektar::LibUtilities::NodalUtilHex.

Referenced by GetInterpolationMatrix().

◆ v_ModeZeroIntegral()

virtual NekDouble Nektar::LibUtilities::NodalUtil::v_ModeZeroIntegral ( )

protectedpure virtual

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.

Implemented in Nektar::LibUtilities::NodalUtilTriangle, Nektar::LibUtilities::NodalUtilTetrahedron, Nektar::LibUtilities::NodalUtilPrism, Nektar::LibUtilities::NodalUtilQuad, and Nektar::LibUtilities::NodalUtilHex.

Referenced by GetWeights().

◆ v_NumModes()

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

protectedpure virtual

Calculate the number of degrees of freedom for this element.

Implemented in Nektar::LibUtilities::NodalUtilTriangle, Nektar::LibUtilities::NodalUtilTetrahedron, Nektar::LibUtilities::NodalUtilPrism, Nektar::LibUtilities::NodalUtilQuad, and Nektar::LibUtilities::NodalUtilHex.

Referenced by GetVandermonde(), GetVandermondeForDeriv(), and GetWeights().

◆ v_OrthoBasis()

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

protectedpure virtual

Return the values of the orthogonal basis at the nodal points for a given mode.

Parameters

mode	Mode number, which is between 0 and NodalUtil::v_NumModes() 1.

Returns: Orthogonal mode mode evaluated at the nodal points.

Implemented in Nektar::LibUtilities::NodalUtilTriangle, Nektar::LibUtilities::NodalUtilTetrahedron, Nektar::LibUtilities::NodalUtilPrism, Nektar::LibUtilities::NodalUtilQuad, and Nektar::LibUtilities::NodalUtilHex.

Referenced by GetVandermonde().

◆ v_OrthoBasisDeriv()

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

protectedpure virtual

Return the values of the derivative of the orthogonal basis at the nodal points for a given mode.

Parameters

dir	Coordinate direction of derivative.
mode	Mode number, which is between 0 and NodalUtil::v_NumModes() 1.

Implemented in Nektar::LibUtilities::NodalUtilTriangle, Nektar::LibUtilities::NodalUtilTetrahedron, Nektar::LibUtilities::NodalUtilPrism, Nektar::LibUtilities::NodalUtilQuad, and Nektar::LibUtilities::NodalUtilHex.

Referenced by GetVandermondeForDeriv().

Member Data Documentation

◆ m_degree

size_t Nektar::LibUtilities::NodalUtil::m_degree

protected

Degree of the nodal element.

Definition at line 103 of file NodalUtil.h.

◆ m_dim

size_t Nektar::LibUtilities::NodalUtil::m_dim

protected

Dimension of the nodal element.

Definition at line 101 of file NodalUtil.h.

◆ m_numPoints

size_t Nektar::LibUtilities::NodalUtil::m_numPoints

protected

Total number of nodal points.

Definition at line 105 of file NodalUtil.h.

◆ m_xi

Array<OneD, Array<OneD, NekDouble> > Nektar::LibUtilities::NodalUtil::m_xi

protected

Coordinates of the nodal points defining the basis.

Definition at line 107 of file NodalUtil.h.

Public Member Functions

Protected Member Functions

Protected Attributes

Detailed Description

Constructor & Destructor Documentation

◆ ~NodalUtil()

◆ NodalUtil()

Member Function Documentation

◆ GetDerivMatrix()

◆ GetInterpolationMatrix()

◆ GetVandermonde()

◆ GetVandermondeForDeriv()

◆ GetWeights()

◆ v_CreateUtil()

◆ v_ModeZeroIntegral()

◆ v_NumModes()

◆ v_OrthoBasis()

◆ v_OrthoBasisDeriv()

Member Data Documentation

◆ m_degree

◆ m_dim

◆ m_numPoints

◆ m_xi