#include <bilininteg.hpp>

Inheritance diagram for mfem::DGElasticityIntegrator:

Collaboration diagram for mfem::DGElasticityIntegrator:

Public Member Functions
	DGElasticityIntegrator (double alpha_, double kappa_)

	DGElasticityIntegrator (Coefficient &lambda_, Coefficient &mu_, double alpha_, double kappa_)

virtual void	AssembleFaceMatrix (const FiniteElement &el1, const FiniteElement &el2, FaceElementTransformations &Trans, DenseMatrix &elmat)

Public Member Functions inherited from mfem::BilinearFormIntegrator
virtual void	AssembleElementMatrix (const FiniteElement &el, ElementTransformation &Trans, DenseMatrix &elmat)
	Given a particular Finite Element computes the element matrix elmat. More...

virtual void	AssembleElementMatrix2 (const FiniteElement &trial_fe, const FiniteElement &test_fe, ElementTransformation &Trans, DenseMatrix &elmat)

virtual void	AssembleFaceMatrix (const FiniteElement &trial_face_fe, const FiniteElement &test_fe1, const FiniteElement &test_fe2, FaceElementTransformations &Trans, DenseMatrix &elmat)

virtual void	AssembleElementVector (const FiniteElement &el, ElementTransformation &Tr, const Vector &elfun, Vector &elvect)
	Perform the local action of the BilinearFormIntegrator. More...

virtual void	AssembleElementGrad (const FiniteElement &el, ElementTransformation &Tr, const Vector &elfun, DenseMatrix &elmat)
	Assemble the local gradient matrix. More...

virtual void	ComputeElementFlux (const FiniteElement &el, ElementTransformation &Trans, Vector &u, const FiniteElement &fluxelem, Vector &flux, int with_coef=1)

virtual double	ComputeFluxEnergy (const FiniteElement &fluxelem, ElementTransformation &Trans, Vector &flux, Vector *d_energy=NULL)

void	SetIntRule (const IntegrationRule *ir)

virtual	~BilinearFormIntegrator ()

Public Member Functions inherited from mfem::NonlinearFormIntegrator
virtual double	GetElementEnergy (const FiniteElement &el, ElementTransformation &Tr, const Vector &elfun)
	Compute the local energy. More...

virtual	~NonlinearFormIntegrator ()

Static Protected Member Functions
static void	AssembleBlock (const int dim, const int row_ndofs, const int col_ndofs, const int row_offset, const int col_offset, const double jmatcoef, const Vector &col_nL, const Vector &col_nM, const Vector &row_shape, const Vector &col_shape, const Vector &col_dshape_dnM, const DenseMatrix &col_dshape, DenseMatrix &elmat, DenseMatrix &jmat)

Protected Attributes
Coefficient *	lambda

Coefficient *	mu

double	alpha

double	kappa

Vector	shape1

Vector	shape2

DenseMatrix	dshape1

DenseMatrix	dshape2

DenseMatrix	adjJ

DenseMatrix	dshape1_ps

DenseMatrix	dshape2_ps

Vector	nor

Vector	nL1

Vector	nL2

Vector	nM1

Vector	nM2

Vector	dshape1_dnM

Vector	dshape2_dnM

DenseMatrix	jmat

Protected Attributes inherited from mfem::BilinearFormIntegrator
const IntegrationRule *	IntRule

Additional Inherited Members
Protected Member Functions inherited from mfem::BilinearFormIntegrator
	BilinearFormIntegrator (const IntegrationRule *ir=NULL)

Detailed Description

Integrator for the DG elasticity form, for the formulations see:

PhD Thesis of Jonas De Basabe, High-Order Finite Element Methods for Seismic Wave Propagation, UT Austin, 2009, p. 23, and references therein
Peter Hansbo and Mats G. Larson, Discontinuous Galerkin and the Crouzeix-Raviart Element: Application to Elasticity, PREPRINT 2000-09, p.3

$- \left< \{ \tau(u) \}, [v] \right> + \alpha \left< \{ \tau(v) \}, [u] \right> + \kappa \left< h^{-1} \{ \lambda + 2 \mu \} [u], [v] \right>$

where $\left<u, v\right> = \int_{F} u \cdot v$ , and $ F $ is a face which is either a boundary face $ F_b $ of an element $ K $ or an interior face $ F_i $ separating elements $ K_1 $ and $ K_2 $ .

In the bilinear form above $\tau(u)$ is traction, and it's also $\tau(u) = \sigma(u) \cdot \vec{n}$ , where $\sigma(u)$ is stress, and $\vec{n}$ is the unit normal vector w.r.t. to $ F $ .

In other words, we have

$- \left< \{ \sigma(u) \cdot \vec{n} \}, [v] \right> + \alpha \left< \{ \sigma(v) \cdot \vec{n} \}, [u] \right> + \kappa \left< h^{-1} \{ \lambda + 2 \mu \} [u], [v] \right>$

For isotropic media

$\begin{split} \sigma(u) &= \lambda \nabla \cdot u I + 2 \mu \varepsilon(u) \\ &= \lambda \nabla \cdot u I + 2 \mu \frac{1}{2} (\nabla u + \nabla u^T) \\ &= \lambda \nabla \cdot u I + \mu (\nabla u + \nabla u^T) \end{split}$

where $ I $ is identity matrix, $\lambda$ and $\mu$ are Lame coefficients (see ElasticityIntegrator), $ u, v $ are the trial and test functions, respectively.

The parameters $\alpha$ and $\kappa$ determine the DG method to use (when this integrator is added to the "broken" ElasticityIntegrator):

IIPG, $\alpha = 0$ , C. Dawson, S. Sun, M. Wheeler, Compatible algorithms for coupled flow and transport, Comp. Meth. Appl. Mech. Eng., 193(23-26), 2565-2580, 2004.
SIPG, $\alpha = -1$ , M. Grote, A. Schneebeli, D. Schotzau, Discontinuous Galerkin Finite Element Method for the Wave Equation, SINUM, 44(6), 2408-2431, 2006.
NIPG, $\alpha = 1$ , B. Riviere, M. Wheeler, V. Girault, A Priori Error Estimates for Finite Element Methods Based on Discontinuous Approximation Spaces for Elliptic Problems, SINUM, 39(3), 902-931, 2001.

This is a 'Vector' integrator, i.e. defined for FE spaces using multiple copies of a scalar FE space.

Definition at line 2160 of file bilininteg.hpp.

Constructor & Destructor Documentation

mfem::DGElasticityIntegrator::DGElasticityIntegrator	(	double	alpha_,
		double	kappa_
	)

inline

Definition at line 2163 of file bilininteg.hpp.

mfem::DGElasticityIntegrator::DGElasticityIntegrator	(	Coefficient &	lambda_,
		Coefficient &	mu_,
		double	alpha_,
		double	kappa_
	)

inline

Definition at line 2166 of file bilininteg.hpp.

Member Function Documentation

void mfem::DGElasticityIntegrator::AssembleBlock	(	const int	dim,
		const int	row_ndofs,
		const int	col_ndofs,
		const int	row_offset,
		const int	col_offset,
		const double	jmatcoef,
		const Vector &	col_nL,
		const Vector &	col_nM,
		const Vector &	row_shape,
		const Vector &	col_shape,
		const Vector &	col_dshape_dnM,
		const DenseMatrix &	col_dshape,
		DenseMatrix &	elmat,
		DenseMatrix &	jmat
	)

staticprotected

Definition at line 2601 of file bilininteg.cpp.

void mfem::DGElasticityIntegrator::AssembleFaceMatrix	(	const FiniteElement &	el1,
		const FiniteElement &	el2,
		FaceElementTransformations &	Trans,
		DenseMatrix &	elmat
	)

virtual

Reimplemented from mfem::BilinearFormIntegrator.

Definition at line 2644 of file bilininteg.cpp.

Member Data Documentation

DenseMatrix mfem::DGElasticityIntegrator::adjJ

protected

Definition at line 2188 of file bilininteg.hpp.

double mfem::DGElasticityIntegrator::alpha

protected

Definition at line 2178 of file bilininteg.hpp.

DenseMatrix mfem::DGElasticityIntegrator::dshape1

protected

Definition at line 2186 of file bilininteg.hpp.

Vector mfem::DGElasticityIntegrator::dshape1_dnM

protected

Definition at line 2196 of file bilininteg.hpp.

DenseMatrix mfem::DGElasticityIntegrator::dshape1_ps

protected

Definition at line 2192 of file bilininteg.hpp.

DenseMatrix mfem::DGElasticityIntegrator::dshape2

protected

Definition at line 2186 of file bilininteg.hpp.

Vector mfem::DGElasticityIntegrator::dshape2_dnM

protected

Definition at line 2196 of file bilininteg.hpp.

DenseMatrix mfem::DGElasticityIntegrator::dshape2_ps

protected

Definition at line 2192 of file bilininteg.hpp.

DenseMatrix mfem::DGElasticityIntegrator::jmat

protected

Definition at line 2198 of file bilininteg.hpp.

double mfem::DGElasticityIntegrator::kappa

protected

Definition at line 2178 of file bilininteg.hpp.

Coefficient* mfem::DGElasticityIntegrator::lambda

protected

Definition at line 2177 of file bilininteg.hpp.

Coefficient * mfem::DGElasticityIntegrator::mu

protected

Definition at line 2177 of file bilininteg.hpp.

Vector mfem::DGElasticityIntegrator::nL1

protected

Definition at line 2194 of file bilininteg.hpp.

Vector mfem::DGElasticityIntegrator::nL2

protected

Definition at line 2194 of file bilininteg.hpp.

Vector mfem::DGElasticityIntegrator::nM1

protected

Definition at line 2195 of file bilininteg.hpp.

Vector mfem::DGElasticityIntegrator::nM2

protected

Definition at line 2195 of file bilininteg.hpp.

Vector mfem::DGElasticityIntegrator::nor

protected

Definition at line 2193 of file bilininteg.hpp.

Vector mfem::DGElasticityIntegrator::shape1

protected

Definition at line 2183 of file bilininteg.hpp.

Vector mfem::DGElasticityIntegrator::shape2

protected

Definition at line 2183 of file bilininteg.hpp.

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

Public Member Functions

Static Protected Member Functions

Protected Attributes

Additional Inherited Members

Detailed Description

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation