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| | DGElasticityIntegrator (double alpha_, double kappa_) |
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| | DGElasticityIntegrator (Coefficient &lambda_, Coefficient &mu_, double alpha_, double kappa_) |
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| virtual void | AssembleFaceMatrix (const FiniteElement &el1, const FiniteElement &el2, FaceElementTransformations &Trans, DenseMatrix &elmat) |
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| virtual void | AssembleElementMatrix (const FiniteElement &el, ElementTransformation &Trans, DenseMatrix &elmat) |
| | Given a particular Finite Element computes the element matrix elmat. More...
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| virtual void | AssembleElementMatrix2 (const FiniteElement &trial_fe, const FiniteElement &test_fe, ElementTransformation &Trans, DenseMatrix &elmat) |
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| virtual void | AssembleFaceMatrix (const FiniteElement &trial_face_fe, const FiniteElement &test_fe1, const FiniteElement &test_fe2, FaceElementTransformations &Trans, DenseMatrix &elmat) |
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| virtual void | AssembleElementVector (const FiniteElement &el, ElementTransformation &Tr, const Vector &elfun, Vector &elvect) |
| | Perform the local action of the BilinearFormIntegrator. More...
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| virtual void | AssembleElementGrad (const FiniteElement &el, ElementTransformation &Tr, const Vector &elfun, DenseMatrix &elmat) |
| | Assemble the local gradient matrix. More...
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| virtual void | ComputeElementFlux (const FiniteElement &el, ElementTransformation &Trans, Vector &u, const FiniteElement &fluxelem, Vector &flux, int with_coef=1) |
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| virtual double | ComputeFluxEnergy (const FiniteElement &fluxelem, ElementTransformation &Trans, Vector &flux, Vector *d_energy=NULL) |
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| void | SetIntRule (const IntegrationRule *ir) |
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| virtual | ~BilinearFormIntegrator () |
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| virtual double | GetElementEnergy (const FiniteElement &el, ElementTransformation &Tr, const Vector &elfun) |
| | Compute the local energy. More...
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| virtual | ~NonlinearFormIntegrator () |
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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
![\[ - \left< \{ \tau(u) \}, [v] \right> + \alpha \left< \{ \tau(v) \}, [u] \right> + \kappa \left< h^{-1} \{ \lambda + 2 \mu \} [u], [v] \right> \]](form_4.png)
where 
, and 
is a face which is either a boundary face 
of an element 
or an interior face 
separating elements 
and 
.
In the bilinear form above 
is traction, and it's also 
, where 
is stress, and 
is the unit normal vector w.r.t. to .
In other words, we have
![\[ - \left< \{ \sigma(u) \cdot \vec{n} \}, [v] \right> + \alpha \left< \{ \sigma(v) \cdot \vec{n} \}, [u] \right> + \kappa \left< h^{-1} \{ \lambda + 2 \mu \} [u], [v] \right> \]](form_16.png)
For isotropic media
![\[ \begin{split} \sigma(u) &= \lambda \nabla \cdot u I + 2 \mu \varepsilon(u) \\ &= \lambda \nabla \cdot u I + 2 \mu \frac{1}{2} (\nabla u + \nabla u^T) \\ &= \lambda \nabla \cdot u I + \mu (\nabla u + \nabla u^T) \end{split} \]](form_17.png)
where 
is identity matrix, 
and 
are Lame coefficients (see ElasticityIntegrator), 
are the trial and test functions, respectively.
The parameters 
and 
determine the DG method to use (when this integrator is added to the "broken" ElasticityIntegrator):
- IIPG,
, 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,
, M. Grote, A. Schneebeli, D. Schotzau, Discontinuous Galerkin Finite Element Method for the Wave Equation, SINUM, 44(6), 2408-2431, 2006.
- NIPG,
, 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.
Public Member Functions inherited from mfem::BilinearFormIntegrator
1.8.5
