Reference documentation for deal.II version Git cd5f69ed86 2019-12-07 22:13:35 -0500
\(\newcommand{\dealcoloneq}{\mathrel{\vcenter{:}}=}\)
Functions
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

Author
Guido Kanschat, Timo Heister
Date
2016

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

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

Author
Guido Kanschat
Date
2011

Definition at line 57 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

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

Author
Guido Kanschat
Date
2014

Definition at line 94 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 130 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.

Author
Guido Kanschat
Date
2008, 2009, 2010

Definition at line 185 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.

Author
Guido Kanschat
Date
2016

Definition at line 236 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

Author
Guido Kanschat
Date
2016

Definition at line 319 of file grad_div.h.