LocalIntegrators::GradDiv Namespace Reference

Functions
template<int dim>
void	cell_matrix (FullMatrix< double > &M, const FEValuesBase< dim > &fe, double factor=1.)
template<int dim, typename number >
void	cell_residual (Vector< number > &result, const FEValuesBase< dim > &fetest, const ArrayView< const std::vector< Tensor< 1, dim > > > &input, const double factor=1.)
template<int dim>
void	nitsche_matrix (FullMatrix< double > &M, const FEValuesBase< dim > &fe, double penalty, double factor=1.)
template<int dim>
void	nitsche_residual (Vector< double > &result, const FEValuesBase< dim > &fe, const ArrayView< const std::vector< double > > &input, const ArrayView< const std::vector< Tensor< 1, dim > > > &Dinput, const ArrayView< const std::vector< double > > &data, double penalty, double factor=1.)
template<int dim>
void	ip_matrix (FullMatrix< double > &M11, FullMatrix< double > &M12, FullMatrix< double > &M21, FullMatrix< double > &M22, const FEValuesBase< dim > &fe1, const FEValuesBase< dim > &fe2, double penalty, double factor1=1., double factor2=-1.)
template<int dim>
void	ip_residual (Vector< double > &result1, Vector< double > &result2, const FEValuesBase< dim > &fe1, const FEValuesBase< dim > &fe2, const ArrayView< const std::vector< double > > &input1, const ArrayView< const std::vector< Tensor< 1, dim > > > &Dinput1, const ArrayView< const std::vector< double > > &input2, const ArrayView< const std::vector< Tensor< 1, dim > > > &Dinput2, double pen, double int_factor=1., double ext_factor=-1.)

Detailed Description

Local integrators related to the grad-div operator and its boundary traces

Function Documentation

cell_matrix()

template<int dim>

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

The weak form of the grad-div operator penalizing volume changes

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

Definition at line 51 of file grad_div.h.

cell_residual()

template<int dim, typename number >

void LocalIntegrators::GradDiv::cell_residual	(	Vector< number > &	result,
		const FEValuesBase< dim > &	fetest,
		const ArrayView< const std::vector< Tensor< 1, dim > > > &	input,
		const double	factor = 1. )

The weak form of the grad-div residual

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

Definition at line 85 of file grad_div.h.

nitsche_matrix()

template<int dim>

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

inline

The matrix for the weak boundary condition of Nitsche type for linear elasticity:

\[ \int_F \Bigl(\gamma (u \cdot n)(v \cdot n) - \nabla\cdot u v\cdot n - u \cdot n \nabla \cdot v \Bigr)\;ds. \]

Definition at line 121 of file grad_div.h.

nitsche_residual()

template<int dim>

void LocalIntegrators::GradDiv::nitsche_residual	(	Vector< double > &	result,
		const FEValuesBase< dim > &	fe,
		const ArrayView< const std::vector< double > > &	input,
		const ArrayView< const std::vector< Tensor< 1, dim > > > &	Dinput,
		const ArrayView< const std::vector< double > > &	data,
		double	penalty,
		double	factor = 1. )

Weak boundary condition for the Laplace operator by Nitsche, vector valued version, namely on the face F the vector

\[ \int_F \Bigl(\gamma (\mathbf u \cdot \mathbf n- \mathbf g \cdot \mathbf n) (\mathbf v \cdot \mathbf n) - \nabla \cdot \mathbf u (\mathbf v \cdot \mathbf n) - (\mathbf u-\mathbf g) \cdot \mathbf n \nabla \cdot v\Bigr)\;ds. \]

Here, u is the finite element function whose values and gradient are given in the arguments input and Dinput, respectively. g is the inhomogeneous boundary value in the argument data. \(\gamma\) is the usual penalty parameter.

Definition at line 173 of file grad_div.h.

ip_matrix()

template<int dim>

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

The interior penalty flux for the grad-div operator. See ip_residual() for details.

Definition at line 220 of file grad_div.h.

ip_residual()

template<int dim>

void LocalIntegrators::GradDiv::ip_residual	(	Vector< double > &	result1,
		Vector< double > &	result2,
		const FEValuesBase< dim > &	fe1,
		const FEValuesBase< dim > &	fe2,
		const ArrayView< const std::vector< double > > &	input1,
		const ArrayView< const std::vector< Tensor< 1, dim > > > &	Dinput1,
		const ArrayView< const std::vector< double > > &	input2,
		const ArrayView< const std::vector< Tensor< 1, dim > > > &	Dinput2,
		double	pen,
		double	int_factor = 1.,
		double	ext_factor = -1. )

Grad-div residual term for the symmetric interior penalty method:

\[ \int_F \Bigl( \gamma [\mathbf u \cdot\mathbf n] \cdot[\mathbf v \cdot \mathbf n] - \{\nabla \cdot \mathbf u\}[\mathbf v\cdot \mathbf n] - [\mathbf u\times \mathbf n]\{\nabla\cdot \mathbf v\} \Bigr) \; ds. \]

See for instance Hansbo and Larson, 2002

Definition at line 300 of file grad_div.h.