#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	AssemblePA (const FiniteElementSpace &fes)
	Method defining partial assembly. More...

virtual void	AssemblePA (const FiniteElementSpace &trial_fes, const FiniteElementSpace &test_fes)

virtual void	AssemblePAInteriorFaces (const FiniteElementSpace &fes)

virtual void	AssemblePABoundaryFaces (const FiniteElementSpace &fes)

virtual void	AssembleDiagonalPA (Vector &diag)
	Assemble diagonal and add it to Vector diag. More...

virtual void	AssembleDiagonalPA_ADAt (const Vector &D, Vector &diag)
	Assemble diagonal of ADA^T (A is this integrator) and add it to diag. More...

virtual void	AddMultPA (const Vector &x, Vector &y) const
	Method for partially assembled action. More...

virtual void	AddMultTransposePA (const Vector &x, Vector &y) const
	Method for partially assembled transposed action. More...

virtual void	AssembleEA (const FiniteElementSpace &fes, Vector &emat, const bool add=true)
	Method defining element assembly. More...

virtual void	AssembleMF (const FiniteElementSpace &fes)
	Method defining matrix-free assembly. More...

virtual void	AddMultMF (const Vector &x, Vector &y) const

virtual void	AddMultTransposeMF (const Vector &x, Vector &y) const

virtual void	AssembleDiagonalMF (Vector &diag)
	Assemble diagonal and add it to Vector diag. More...

virtual void	AssembleEAInteriorFaces (const FiniteElementSpace &fes, Vector &ea_data_int, Vector &ea_data_ext, const bool add=true)

virtual void	AssembleEABoundaryFaces (const FiniteElementSpace &fes, Vector &ea_data_bdr, const bool add=true)

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. Note that the default implementation in the base class is general but not efficient. More...

virtual void	AssembleFaceVector (const FiniteElement &el1, const FiniteElement &el2, FaceElementTransformations &Tr, const Vector &elfun, Vector &elvect)
	Perform the local action of the BilinearFormIntegrator resulting from a face integral term. Note that the default implementation in the base class is general but not efficient. More...

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

virtual void	AssembleFaceGrad (const FiniteElement &el1, const FiniteElement &el2, FaceElementTransformations &Tr, const Vector &elfun, DenseMatrix &elmat)
	Assemble the local action of the gradient of the NonlinearFormIntegrator resulting from a face integral term. More...

virtual void	ComputeElementFlux (const FiniteElement &el, ElementTransformation &Trans, Vector &u, const FiniteElement &fluxelem, Vector &flux, bool with_coef=true)
	Virtual method required for Zienkiewicz-Zhu type error estimators. More...

virtual double	ComputeFluxEnergy (const FiniteElement &fluxelem, ElementTransformation &Trans, Vector &flux, Vector *d_energy=NULL)
	Virtual method required for Zienkiewicz-Zhu type error estimators. More...

virtual	~BilinearFormIntegrator ()

Public Member Functions inherited from mfem::NonlinearFormIntegrator
virtual void	SetIntRule (const IntegrationRule *ir)
	Prescribe a fixed IntegrationRule to use (when ir != NULL) or let the integrator choose (when ir == NULL). More...

void	SetIntegrationRule (const IntegrationRule &ir)
	Prescribe a fixed IntegrationRule to use. More...

void	SetPAMemoryType (MemoryType mt)

const IntegrationRule *	GetIntegrationRule () const
	Get the integration rule of the integrator (possibly NULL). More...

virtual double	GetElementEnergy (const FiniteElement &el, ElementTransformation &Tr, const Vector &elfun)
	Compute the local energy. More...

virtual void	AssembleGradPA (const Vector &x, const FiniteElementSpace &fes)
	Prepare the integrator for partial assembly (PA) gradient evaluations on the given FE space fes at the state x. More...

virtual double	GetLocalStateEnergyPA (const Vector &x) const
	Compute the local (to the MPI rank) energy with partial assembly. More...

virtual void	AddMultGradPA (const Vector &x, Vector &y) const
	Method for partially assembled gradient action. More...

virtual void	AssembleGradDiagonalPA (Vector &diag) const
	Method for computing the diagonal of the gradient with partial assembly. More...

virtual bool	SupportsCeed () const
	Indicates whether this integrator can use a Ceed backend. More...

ceed::Operator &	GetCeedOp ()

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::NonlinearFormIntegrator
const IntegrationRule *	IntRule

ceed::Operator *	ceedOp

MemoryType	pa_mt = MemoryType::DEFAULT

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

Protected Member Functions inherited from mfem::NonlinearFormIntegrator
	NonlinearFormIntegrator (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 3165 of file bilininteg.hpp.

Constructor & Destructor Documentation

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

inline

Definition at line 3168 of file bilininteg.hpp.

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

inline

Definition at line 3171 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 3389 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 3432 of file bilininteg.cpp.

Member Data Documentation

DenseMatrix mfem::DGElasticityIntegrator::adjJ

protected

Definition at line 3193 of file bilininteg.hpp.

double mfem::DGElasticityIntegrator::alpha

protected

Definition at line 3183 of file bilininteg.hpp.

DenseMatrix mfem::DGElasticityIntegrator::dshape1

protected

Definition at line 3191 of file bilininteg.hpp.

Vector mfem::DGElasticityIntegrator::dshape1_dnM

protected

Definition at line 3201 of file bilininteg.hpp.

DenseMatrix mfem::DGElasticityIntegrator::dshape1_ps

protected

Definition at line 3197 of file bilininteg.hpp.

DenseMatrix mfem::DGElasticityIntegrator::dshape2

protected

Definition at line 3191 of file bilininteg.hpp.

Vector mfem::DGElasticityIntegrator::dshape2_dnM

protected

Definition at line 3201 of file bilininteg.hpp.

DenseMatrix mfem::DGElasticityIntegrator::dshape2_ps

protected

Definition at line 3197 of file bilininteg.hpp.

DenseMatrix mfem::DGElasticityIntegrator::jmat

protected

Definition at line 3203 of file bilininteg.hpp.

double mfem::DGElasticityIntegrator::kappa

protected

Definition at line 3183 of file bilininteg.hpp.

Coefficient* mfem::DGElasticityIntegrator::lambda

protected

Definition at line 3182 of file bilininteg.hpp.

Coefficient * mfem::DGElasticityIntegrator::mu

protected

Definition at line 3182 of file bilininteg.hpp.

Vector mfem::DGElasticityIntegrator::nL1

protected

Definition at line 3199 of file bilininteg.hpp.

Vector mfem::DGElasticityIntegrator::nL2

protected

Definition at line 3199 of file bilininteg.hpp.

Vector mfem::DGElasticityIntegrator::nM1

protected

Definition at line 3200 of file bilininteg.hpp.

Vector mfem::DGElasticityIntegrator::nM2

protected

Definition at line 3200 of file bilininteg.hpp.

Vector mfem::DGElasticityIntegrator::nor

protected

Definition at line 3198 of file bilininteg.hpp.

Vector mfem::DGElasticityIntegrator::shape1

protected

Definition at line 3188 of file bilininteg.hpp.

Vector mfem::DGElasticityIntegrator::shape2

protected

Definition at line 3188 of file bilininteg.hpp.

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

fem/bilininteg.hpp
fem/bilininteg.cpp

Public Member Functions

Static Protected Member Functions

Protected Attributes

Additional Inherited Members

Detailed Description

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation