pbat::math::polynomial::DivergenceFreeBasis< Dims, Order > Struct Template Reference

Divergence-free polynomial basis \( \left\{ \mathbf{P}_i(X) \; \text{s.t.} \; \nabla_X \mathbf{P}_i = 0 \right\} \) in dimensions \( d \) and order \( p \). More...

#include <Basis.h>

Inheritance diagram for pbat::math::polynomial::DivergenceFreeBasis< Dims, Order >:

Public Types
using	BaseType = typename detail::DivergenceFreeBasis<Dims, Order>
	Base type.

Public Member Functions
Matrix< kSize, kDims >	eval (Vector< kDims > const &X) const

Static Public Attributes
static constexpr int	kDims = BaseType::kDims
	Spatial dimensions.
static constexpr int	kOrder = BaseType::kOrder
	Polynomial order.
static constexpr int	kSize = BaseType::kSize
	Number of basis functions.

Detailed Description

template<int Dims, int Order>
struct pbat::math::polynomial::DivergenceFreeBasis< Dims, Order >

Divergence-free polynomial basis \( \left\{ \mathbf{P}_i(X) \; \text{s.t.} \; \nabla_X \mathbf{P}_i = 0 \right\} \) in dimensions \( d \) and order \( p \).

The basis satisfies \( \nabla_X \cdot \mathbf{P}_i = 0 \) on the reference simplex in dimensions \( d \), e.g.

• the line segment \( 0,1 \) in 1D,
• the triangle \(\begin{pmatrix} 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \end{pmatrix} \) in 2D, and
• the tetrahedron \(\begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ 1\end{pmatrix} \) in 3D.

See [10]

Template Parameters

Dims	Spatial dimensions
Order	Polynomial order

Member Function Documentation

eval()

template<int Dims, int Order>

Matrix< kSize, kDims > pbat::math::polynomial::DivergenceFreeBasis< Dims, Order >::eval ( Vector< kDims > const & X ) const

inline

Parameters

X

Returns

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The documentation for this struct was generated from the following file:

• C:/git/PhysicsBasedAnimationToolkit/source/pbat/math/polynomial/Basis.h