LocalIntegrators::Maxwell Namespace Reference

Local integrators related to curl operators and their traces. More...

Functions
template<int dim>
Tensor< 1, dim >	curl_curl (const Tensor< 2, dim > &h0, const Tensor< 2, dim > &h1, const Tensor< 2, dim > &h2)
template<int dim>
Tensor< 1, dim >	tangential_curl (const Tensor< 1, dim > &g0, const Tensor< 1, dim > &g1, const Tensor< 1, dim > &g2, const Tensor< 1, dim > &normal)
template<int dim>
void	curl_curl_matrix (FullMatrix< double > &M, const FEValuesBase< dim > &fe, const double factor=1.)
template<int dim>
void	curl_matrix (FullMatrix< double > &M, const FEValuesBase< dim > &fe, const FEValuesBase< dim > &fetest, double factor=1.)
template<int dim>
void	nitsche_curl_matrix (FullMatrix< double > &M, const FEValuesBase< dim > &fe, const unsigned int face_no, double penalty, double factor=1.)
template<int dim>
void	tangential_trace_matrix (FullMatrix< double > &M, const FEValuesBase< dim > &fe, double factor=1.)
template<int dim>
void	ip_curl_matrix (FullMatrix< double > &M11, FullMatrix< double > &M12, FullMatrix< double > &M21, FullMatrix< double > &M22, const FEValuesBase< dim > &fe1, const FEValuesBase< dim > &fe2, const double pen, const double factor1=1., const double factor2=-1.)

Detailed Description

Local integrators related to curl operators and their traces.

We use the following conventions for curl operators. First, in three space dimensions

\[ \nabla\times \mathbf u = \begin{pmatrix} \partial_2 u_3 - \partial_3 u_2 \\ \partial_3 u_1 - \partial_1 u_3 \\ \partial_1 u_2 - \partial_2 u_1 \end{pmatrix}. \]

In two space dimensions, the curl is obtained by extending a vector u to \((u_1, u_2, 0)^T\) and a scalar p to \((0,0,p)^T\). Computing the nonzero components, we obtain the scalar curl of a vector function and the vector curl of a scalar function. The current implementation exchanges the sign and we have:

\[ \nabla \times \mathbf u = \partial_1 u_2 - \partial_2 u_1, \qquad \nabla \times p = \begin{pmatrix} \partial_2 p \\ -\partial_1 p \end{pmatrix} \]

Author

Guido Kanschat

Date

2010

Function Documentation

curl_curl()

template<int dim>

Tensor<1, dim> LocalIntegrators::Maxwell::curl_curl	(	const Tensor< 2, dim > &	h0,
		const Tensor< 2, dim > &	h1,
		const Tensor< 2, dim > &	h2
	)

Auxiliary function. Given the tensors of dim second derivatives, compute the curl of the curl of a vector function. The result in two and three dimensions is:

\[ \nabla\times\nabla\times \mathbf u = \begin{pmatrix} \partial_1\partial_2 u_2 - \partial_2^2 u_1 \\ \partial_1\partial_2 u_1 - \partial_1^2 u_2 \end{pmatrix} \nabla\times\nabla\times \mathbf u = \begin{pmatrix} \partial_1\partial_2 u_2 + \partial_1\partial_3 u_3 - (\partial_2^2+\partial_3^2) u_1 \\ \partial_2\partial_3 u_3 + \partial_2\partial_1 u_1 - (\partial_3^2+\partial_1^2) u_2 \\ \partial_3\partial_1 u_1 + \partial_3\partial_2 u_2 - (\partial_1^2+\partial_2^2) u_3 \end{pmatrix} \]

Note

The third tensor argument is not used in two dimensions and can for instance duplicate one of the previous.

Author

Guido Kanschat

Date

2011

Definition at line 97 of file maxwell.h.

tangential_curl()

template<int dim>

Tensor<1, dim> LocalIntegrators::Maxwell::tangential_curl	(	const Tensor< 1, dim > &	g0,
		const Tensor< 1, dim > &	g1,
		const Tensor< 1, dim > &	g2,
		const Tensor< 1, dim > &	normal
	)

Auxiliary function. Given dim tensors of first derivatives and a normal vector, compute the tangential curl

\[ \mathbf n \times \nabla \times u. \]

Note

The third tensor argument is not used in two dimensions and can for instance duplicate one of the previous.

Author

Guido Kanschat

Date

2011

Definition at line 134 of file maxwell.h.

curl_curl_matrix()

template<int dim>

void LocalIntegrators::Maxwell::curl_curl_matrix	(	FullMatrix< double > &	M,
		const FEValuesBase< dim > &	fe,
		const double	factor = 1.
	)

The curl-curl operator

\[ \int_Z \nabla\!\times\! u \cdot \nabla\!\times\! v \,dx \]

in weak form.

Author

Guido Kanschat

Date

2011

Definition at line 174 of file maxwell.h.

curl_matrix()

template<int dim>

void LocalIntegrators::Maxwell::curl_matrix	(	FullMatrix< double > &	M,
		const FEValuesBase< dim > &	fe,
		const FEValuesBase< dim > &	fetest,
		double	factor = 1.
	)

The matrix for the curl operator

\[ \int_Z \nabla\!\times\! u \cdot v \,dx. \]

This is the standard curl operator in 3D and the scalar curl in 2D. The vector curl operator can be obtained by exchanging test and trial functions.

Author

Guido Kanschat

Date

2011

Definition at line 230 of file maxwell.h.

nitsche_curl_matrix()

template<int dim>

void LocalIntegrators::Maxwell::nitsche_curl_matrix	(	FullMatrix< double > &	M,
		const FEValuesBase< dim > &	fe,
		const unsigned int	face_no,
		double	penalty,
		double	factor = 1.
	)

The matrix for weak boundary condition of Nitsche type for the tangential component in Maxwell systems.

\[ \int_F \biggl( 2\gamma (u\times n) (v\times n) - (u\times n)(\nu \nabla\times v) - (v\times n)(\nu \nabla\times u) \biggr) \]

Author

Guido Kanschat

Date

2011

Definition at line 282 of file maxwell.h.

tangential_trace_matrix()

template<int dim>

void LocalIntegrators::Maxwell::tangential_trace_matrix	(	FullMatrix< double > &	M,
		const FEValuesBase< dim > &	fe,
		double	factor = 1.
	)

The product of two tangential traces,

\[ \int_F (u\times n)(v\times n) \, ds. \]

Author

Guido Kanschat

Date

2011

Definition at line 348 of file maxwell.h.

ip_curl_matrix()

template<int dim>

void LocalIntegrators::Maxwell::ip_curl_matrix ( FullMatrix< double > &  M11, FullMatrix< double > &  M12, FullMatrix< double > &  M21, FullMatrix< double > &  M22, const FEValuesBase< dim > &  fe1, const FEValuesBase< dim > &  fe2, const double  pen, const double  factor1 = 1., const double  factor2 = -1.  )

inline

The interior penalty fluxes for Maxwell systems.

\[ \int_F \biggl( \gamma \{u\times n\}\{v\times n\} - \{u\times n\}\{\nu \nabla\times v\}- \{v\times n\}\{\nu \nabla\times u\} \biggr)\;dx \]

Author

Guido Kanschat

Date

2011

Definition at line 408 of file maxwell.h.