Polydim::Utilities::GBasis_3D Class Reference

#include <GBasis_3D.hpp>

Public Member Functions
Polydim::Utilities::GBasis_Data	Compute (const unsigned int polynomial_degree)
	Build the vector polynomial G-basis of a given degree in 3D.
std::vector< Eigen::MatrixXd >	VanderGBigOPlus (const Polydim::Utilities::GBasis_Data &data, const Eigen::MatrixXd &vander) const
	Evaluate the Vandermonde matrix of the \(\mathcal{G}_k^\oplus\) block.
std::vector< std::vector< Eigen::MatrixXd > >	VectorDecomposition (const Polydim::Utilities::GBasis_Data &data) const
	Assemble the vector decomposition into a two-block ( \(\nabla\), \(\oplus\)) form.

Member Function Documentation

Compute()

GBasis_Data Polydim::Utilities::GBasis_3D::Compute

(

const unsigned int

polynomial_degree

)

Build the vector polynomial G-basis of a given degree in 3D.

Assembles all quantities needed to represent the vector polynomial space \([\mathbb{P}_k]^3\) through its decomposition \(\mathcal{G}_k^\nabla \oplus \mathcal{G}_k^\oplus\), where \(\mathcal{G}_k^\nabla = \nabla \mathbb{P}_{k+1}\) and \(\mathcal{G}_k^\oplus = \boldsymbol{x} \times [\mathbb{P}_{k-1}]^3\). The \(\mathcal{G}^\oplus\) block is stored in three parts of widths DimFirstBasis, Nkm1 and Nkm1 (total \(\dim\mathcal{G}_k^\oplus = \) DimFirstBasis + 2*Nkm1).

Parameters

polynomial_degree

Maximum polynomial degree \(k\) of the basis.

Returns

A fully populated Polydim::Utilities::GBasis_Data structure (space dimensions, exponent matrix, and the four-block vector-decomposition maps).

VanderGBigOPlus()

vector< MatrixXd > Polydim::Utilities::GBasis_3D::VanderGBigOPlus	(	const Polydim::Utilities::GBasis_Data &	data,
		const Eigen::MatrixXd &	vander
	)		const

Evaluate the Vandermonde matrix of the \(\mathcal{G}_k^\oplus\) block.

Builds the three Cartesian components (x, y, z) of the Vandermonde matrix of \(\mathcal{G}_k^\oplus = \boldsymbol{x} \times [\mathbb{P}_{k-1}]^3\), evaluated at the points whose scalar monomial values are provided in vander.

Parameters

data	Precomputed G-basis data (DimFirstBasis and Nkm1 are used here).
vander	Vandermonde matrix of the scalar monomial basis evaluated at the evaluation points (one row per point), from which the linear monomials \(x_1, x_2, x_3\) and the \(\mathbb{P}_{k-1}\) monomials are extracted.

Returns

A vector of three Eigen::MatrixXd (x-, y- and z-components), each of size vander.rows() \(\times\) (data.DimFirstBasis + 2*data.Nkm1), forming the Vandermonde matrix of \(\mathcal{G}_k^\oplus\).

VectorDecomposition()

std::vector< std::vector< Eigen::MatrixXd > > Polydim::Utilities::GBasis_3D::VectorDecomposition ( const Polydim::Utilities::GBasis_Data &  data ) const

inline

Assemble the vector decomposition into a two-block ( \(\nabla\), \(\oplus\)) form.

Repackages the internal four-block storage of data into a compact per-component representation. For each of the three Cartesian components, the result holds two matrices: the gradient block \(\mathcal{G}_k^\nabla\) (size \(N_k \times N_{k+1}\)) and the full \(\mathcal{G}_k^\oplus\) block, obtained by horizontally concatenating the three stored sub-blocks into a single \(N_k \times (\text{DimFirstBasis} + 2N_{k-1})\) matrix.

Parameters

data	Precomputed G-basis data containing the four-block VectorDecomposition.

Returns

A std::vector of three components, each a pair of Eigen::MatrixXd: index 0 is the \(\nabla\) block, index 1 is the assembled \(\oplus\) block.

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The documentation for this class was generated from the following files:

• GBasis_3D.hpp
• GBasis_3D.cpp