#include <ExplicitVectorOperators_fwd.hpp>

Detailed Description

template<class shape_storage_type>
class mm::ExplicitVectorOperators< shape_storage_type >

A class for evaluating typical operators needed in spatial discretization.

This class represents vector differential operators, discretely approximated with shape functions that are stored in given shape storage. These shape functions are used to construct approximations of common operators such as gradient, Laplacian, divergence, ... that can be applied to vector fields at given points.

Note: The scalar types (and sometimes even dimensions) of the given fields and the shapes can be different. We support computing the gradient of a 3d complex-valued field over 2d domain with double shapes: the result is a 3x2 complex matrix.

Template Parameters

shape_storage_type Storage class for shapes. Must satisfy the Shape storage concept.

Usage example:

    BoxShape<Vec2d> box({0.0, 0.0}, {1.0, 1.0});
    DomainDiscretization<Vec2d> d = box.discretizeWithStep(0.02);
    d.findSupport(FindClosest(9));
    WLS<Monomials<Vec2d>, NoWeight<Vec2d>, ScaleToFarthest> wls(2);
    auto storage = d.computeShapes(wls);
    ExplicitVectorOperators<decltype(storage)> op(storage);
    std::cout << op << std::endl;
    VectorField2d v(d.size());  // constant scalar field, value 1.34 at each point
    v = Vec2d(12.4, -3.4);
    Vec2d val = op.lap(v, 2);  // laplace at point d.pos(2);
    Eigen::Matrix2d grad = op.grad(v, 4);  // gradient at point d.pos(4);
    double div = op.div(v, 3);  // divergence at points d.pos(3);
 
    VectorField<std::complex<double>, 3> complex_field(d.size());  // 3d complex field
    complex_field.setConstant(std::complex<double>(2.5, -7.1));
    Eigen::Matrix<std::complex<double>, 3, 2> cgrad = op.grad(complex_field, 1);

See also: ExplicitOperators

Definition at line 38 of file ExplicitVectorOperators_fwd.hpp.

Public Member Functions
	ExplicitVectorOperators ()
	Construct empty explicit vector operators. More...

	ExplicitVectorOperators (const shape_storage_t &ss)
	Construct explicit vector operators over given shape storage. More...

void	setShapes (const shape_storage_t &shape_storage_)
	Sets a new shape storage from which this operators are generated. More...

bool	hasShapes () const
	Returns `true` if operators have a non-null pointer to storage and `false` otherwise. More...

template<class vector_field_t >
vector_type< vector_field_t >::type	lap (const vector_field_t &u, int node) const
	Returns Laplacian of vector field `u` in node with index `node`. More...

template<typename op_family_t , typename vector_field_t >
vector_type< vector_field_t >::type	apply (const vector_field_t &u, int node, typename op_family_t::operator_t o) const
	Returns an approximation of applying the requested operator to each component of field `u` at node `node`. More...

template<typename op_family_t , typename vector_field_t >
vector_type< vector_field_t >::type	apply (const vector_field_t &u, int node) const
	Overload for default constructible operator. More...

template<class vector_field_t >
Eigen::Matrix< typename vector_type< vector_field_t >::type::scalar_t, vector_type< vector_field_t >::type::dim, dim >	grad (const vector_field_t &u, int node) const
	Returns gradient of vector field `u` in node with index `node`. More...

template<class vector_field_t >
vector_type< vector_field_t >::type::scalar_t	div (const vector_field_t &u, int node) const
	Returns divergence of a vector field `u` in in node with index `node`. More...

template<class vector_field_t >
vector_type< vector_field_t >::type	curl (const vector_field_t &u, int node) const
	Returns curl of a vector field `u` in node with index `node`. More...

template<class vector_field_t >
vector_type< vector_field_t >::type	graddiv (const vector_field_t &u, int node) const
	Returns gradient of divergence of a vector field `u` in in node with index `node`. More...

template<class vector_field_t >
vector_type< vector_field_t >::type	neumann (const vector_field_t &u, int node, const vector_t &normal, typename vector_type< vector_field_t >::type val) const
	This is a vectorised version of ExplicitOperators::neumann. More...

Public Types
enum	{ dim = shape_storage_t::dim }
	Store dimension of the domain. More...

typedef shape_storage_type	shape_storage_t
	Type of shape storage. More...

typedef shape_storage_t::vector_t	vector_t
	Vector type. More...

typedef shape_storage_t::scalar_t	scalar_t
	Scalar type. More...

Friends
template<typename S >
std::ostream &	operator<< (std::ostream &os, const ExplicitVectorOperators< S > &op)
	Output basic info about given operators. More...

Private Attributes
const shape_storage_t *	ss
	Pointer to shape storage, but name is shortened for readability. More...

Member Enumeration Documentation

◆ anonymous enum

template<class shape_storage_type >

anonymous enum

Store dimension of the domain.

Enumerator
dim	Dimensionality of the function domain.

Definition at line 44 of file ExplicitVectorOperators_fwd.hpp.

Constructor & Destructor Documentation

◆ ExplicitVectorOperators() [1/2]

template<class shape_storage_type >

mm::ExplicitVectorOperators< shape_storage_type >::ExplicitVectorOperators ( )

inline

Construct empty explicit vector operators.

Definition at line 52 of file ExplicitVectorOperators_fwd.hpp.

◆ ExplicitVectorOperators() [2/2]

template<class shape_storage_type >

mm::ExplicitVectorOperators< shape_storage_type >::ExplicitVectorOperators ( const shape_storage_t & ss )

inlineexplicit

Construct explicit vector operators over given shape storage.

The pointer to given storage is stored as a non-owning pointer. It is the user's responsibility that given storage is valid throughout the use use of this class.

This class is usually constructed directly from shape storage using the explicitVectorOperators() member function.

Definition at line 61 of file ExplicitVectorOperators_fwd.hpp.

Member Function Documentation

◆ apply() [1/2]

template<class shape_storage_type >

template<typename op_family_t , typename vector_field_t >

vector_type<vector_field_t>::type mm::ExplicitVectorOperators< shape_storage_type >::apply	(	const vector_field_t &	u,
		int	node
	)		const

inline

Overload for default constructible operator.

See also: apply

Definition at line 110 of file ExplicitVectorOperators_fwd.hpp.

◆ apply() [2/2]

template<class shape_storage_type >

template<typename op_family_t , typename vector_field_t >

vector_type<vector_field_t>::type mm::ExplicitVectorOperators< shape_storage_type >::apply	(	const vector_field_t &	u,
		int	node,
		typename op_family_t::operator_t	o
	)		const

Returns an approximation of applying the requested operator to each component of field u at node node.

Parameters

u	Field to which the operator is applied.
node	Node at which the derived field is evaluated.
o	The operator to apply.

Warning: The operator approximation is obtained from shape storage using the op_family_t type. The first weights computed for family of type op_family_t are used.

Usage example:

    struct Dx : Der1<2> { Dx() : Der1(0) {} };
    struct Dy : Der1<2> { Dy() : Der1(1) {} };
 
    auto storage = d.computeShapes<std::tuple<Dx, Dy, Lap<2>>>(wls);
    ExplicitVectorOperators<decltype(storage)> op(storage);
    VectorField<std::complex<double>, 3> field(d.size());
    std::complex<double> c(2.4, -0.3);
    field.setConstant(c);
 
    Vec<std::complex<double>, 3> lap = op.apply<Lap<2>>(field, 52);
    Vec<std::complex<double>, 3> dx = op.apply<Dx>(field, 34);
    Vec<std::complex<double>, 3> dy = op.apply<Dy>(field, 32);

◆ curl()

template<class shape_storage_type >

template<class vector_field_t >

vector_type<vector_field_t>::type mm::ExplicitVectorOperators< shape_storage_type >::curl	(	const vector_field_t &	u,
		int	node
	)		const

Returns curl of a vector field u in node with index node.

Returns: The rotor of u, where dim = 3 and \(\vec{u} = [u, v, w]\). The return value is \( \curl \vec{u} = \[\dpar{w}{y} - \dpar{v}{z}, \dpar{u}{z} - \dpar{w}{x}, \dpar{v}{x} - \dpar{u}{y}\] \) evaluated in point with index node, where derivatives are approximated using shape functions from ss.

Exceptions

Static-assertion fails if u is not three dimensional.

◆ div()

template<class shape_storage_type >

template<class vector_field_t >

vector_type<vector_field_t>::type::scalar_t mm::ExplicitVectorOperators< shape_storage_type >::div	(	const vector_field_t &	u,
		int	node
	)		const

Returns divergence of a vector field u in in node with index node.

Returns: The divergence of u, e.g. when dim = 3 and \(\vec{u} = [u, v, w]\), the returned value is \( \nabla\cdot \vec{u} = \dpar{u}{x} + \dpar{v}{y} + \dpar{w}{z} \) evaluated in point with index node, where derivatives are approximated using shape functions from ss.

Exceptions

Static-assertion fails if the dimension of u is not equal to domain dimension.

◆ grad()

template<class shape_storage_type >

template<class vector_field_t >

Eigen::Matrix<typename vector_type<vector_field_t>::type::scalar_t, vector_type<vector_field_t>::type::dim, dim> mm::ExplicitVectorOperators< shape_storage_type >::grad	(	const vector_field_t &	u,
		int	node
	)		const

Returns gradient of vector field u in node with index node.

The field u and the domain can have different dimensions.

Returns: The gradient (Jacobi matrix) of u, e.g. when dim(u) = 3 and and \(\vec{u} = [u, v, w]\) and the domain dimension is 2, the returned value is
\[ \nabla \vec{u} = \begin{bmatrix} \dpar{u}{x}, \dpar{u}{y} \\ \dpar{v}{x}, \dpar{v}{y} \\ \dpar{w}{x}, \dpar{w}{y} \end{bmatrix} \]
evaluated in point with index node, where derivatives are approximated using shape functions from ss.

◆ graddiv()

template<class shape_storage_type >

template<class vector_field_t >

vector_type<vector_field_t>::type mm::ExplicitVectorOperators< shape_storage_type >::graddiv	(	const vector_field_t &	u,
		int	node
	)		const

Returns gradient of divergence of a vector field u in in node with index node.

Returns: The gradient of divergence of u, e.g. when dim = 3 and \(\vec{u} = [u, v, w]\), the returned value is
\[ \nabla(\nabla\cdot \vec{u}) = \begin{bmatrix} \dpar{^2u}{x^2} + \dpar{^2v}{x\partial y} + \dpar{^2w}{x\partial z} \\ \dpar{^2u}{x\partial y} + \dpar{^2v}{^2y} + \dpar{^2w}{y\partial z} \\ \dpar{^2u}{x\partial z} + \dpar{^2v}{y\partial z} + \dpar{^2w}{^2z} \end{bmatrix} \]
evaluated in point with index node, where derivatives are approximated using shape functions from ss.

Exceptions

Static-assertion fails if the dimension of u is not equal to domain dimension.

◆ hasShapes()

template<class shape_storage_type >

bool mm::ExplicitVectorOperators< shape_storage_type >::hasShapes ( ) const

inline

Returns true if operators have a non-null pointer to storage and false otherwise.

Definition at line 72 of file ExplicitVectorOperators_fwd.hpp.

◆ lap()

template<class shape_storage_type >

template<class vector_field_t >

vector_type<vector_field_t>::type mm::ExplicitVectorOperators< shape_storage_type >::lap	(	const vector_field_t &	u,
		int	node
	)		const

inline

Returns Laplacian of vector field u in node with index node.

Returns: The Laplacian of u, e.g. when dim = 3 and \(\vec{u} = [u, v, w]\), the returned value is \( \nabla^2 \vec{u} = [ \dpar{^2 u}{x^2} + \dpar{^2 u}{y^2} + \dpar{^2 u}{z^2}, \dpar{^2 v}{x^2} + \dpar{^2 v}{y^2} + \dpar{^2 v}{z^2}, \dpar{^2 w}{x^2} + \dpar{^2 w}{y^2} + \dpar{^2 w}{z^2} ]^T\) evaluated in point with index node where derivatives are approximated using shape functions from ss.

Exceptions

Static-assertion fails if the dimension of u is not equal to domain dimension.

Definition at line 87 of file ExplicitVectorOperators_fwd.hpp.

◆ neumann()

template<class shape_storage_type >

template<class vector_field_t >

vector_type<vector_field_t>::type mm::ExplicitVectorOperators< shape_storage_type >::neumann	(	const vector_field_t &	u,
		int	node,
		const vector_t &	normal,
		typename vector_type< vector_field_t >::type	val
	)		const

This is a vectorised version of ExplicitOperators::neumann.

Calculates a new field value a for u at node, such that the Neumann boundary condition du/dn(node) = val holds. Denote given index node by \(i\) and val by \(\vec v\). The function returns such \(\vec a\), that by setting \( \vec{u}(i) = \vec{a}\) we satisfy the condition \( \dpar{\vec u}{\vec n}(i) = \vec v\). This is done using the following formula:

\[ \vec{a} = \frac{ \vec{v} - \sum\limits_{i=1}^{s} \sum\limits_{d=1}^{dim} n_d \chi_{\dpar{}{ x_d}}(i)\ \vec{u}(i) }{\sum\limits_{d=1}^{dim} n_d \chi_{\dpar{}{ x_d}}(0)}, \]

where \(\chi_{\dpar{}{ x_d}}\) represents the formula for the shape function of first derivative wrt. \(d\).

Note

A few assumptions must hold for the support domain.

Each node should be its own first first support node.
For symmetrical supports the central node may not affect the calculated derivation and therefore this function is unable to give a value and will crash.

Parameters

u	Scalar field to evaluate.
node	Index of the node, whose value should be changed to satisfy the boundary condition.
normal	Vector \(\vec{n}\) with the same dimensionality as the problem domain giving the direction of the derivation. Usually the unit normal vector.
val	The value that \(\dpar{ \vec u}{ \vec n}\) should be equal to.

Returns: A new value \(\vec a\), such that \(\dpar{\vec u}{\vec n} = \vec v\) after setting \(\vec u(i) = \vec a\).

Exceptions

Assertion	fails if `node`'s first support node is not `node`.
Assertion	fails if `node`'s value has negligible effect on the derivation, because no value could change the value of \(\dpar{\vec u}{\vec n}\).
Static-assertion	fails if the dimension of `u` is not equal to domain dimension.

See also: ExplicitOperators::neumann

◆ setShapes()

template<class shape_storage_type >

void mm::ExplicitVectorOperators< shape_storage_type >::setShapes ( const shape_storage_t & shape_storage_ )

inline

Sets a new shape storage from which this operators are generated.

As in the constructor, the pointer to given storage is stored as a non-owning pointer. It is the user's responsibility that given storage is valid throughout the use use of this class.

Parameters

shape_storage_ New shape storage.

Definition at line 69 of file ExplicitVectorOperators_fwd.hpp.

Friends And Related Function Documentation

◆ operator<<

template<class shape_storage_type >

template<typename S >

std::ostream& operator<<	(	std::ostream &	os,
		const ExplicitVectorOperators< S > &	op
	)

friend

Output basic info about given operators.

Definition at line 151 of file ExplicitVectorOperators.hpp.

Member Data Documentation

◆ ss

template<class shape_storage_type >

const shape_storage_t* mm::ExplicitVectorOperators< shape_storage_type >::ss

private

Pointer to shape storage, but name is shortened for readability.

Definition at line 48 of file ExplicitVectorOperators_fwd.hpp.

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

include/medusa/bits/operators/ExplicitVectorOperators_fwd.hpp

Detailed Description

template<class shape_storage_type> class mm::ExplicitVectorOperators< shape_storage_type >

Public Member Functions

Public Types

Friends

Private Attributes

Member Enumeration Documentation

◆ anonymous enum

Constructor & Destructor Documentation

◆ ExplicitVectorOperators() [1/2]

◆ ExplicitVectorOperators() [2/2]

Member Function Documentation

◆ apply() [1/2]

◆ apply() [2/2]

◆ curl()

◆ div()

◆ grad()

◆ graddiv()

◆ hasShapes()

◆ lap()

◆ neumann()

◆ setShapes()

Friends And Related Function Documentation

◆ operator<<

Member Data Documentation

◆ ss

template<class shape_storage_type>
class mm::ExplicitVectorOperators< shape_storage_type >