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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 | 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...
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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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| virtual | ~BilinearFormIntegrator () |
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| void | SetIntRule (const IntegrationRule *ir) |
| | Prescribe a fixed IntegrationRule to use (when ir != NULL) or let the integrator choose (when ir == NULL). More...
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| void | SetIntegrationRule (const IntegrationRule &irule) |
| | Prescribe a fixed IntegrationRule to use. More...
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| virtual void | AssembleFaceVector (const FiniteElement &el1, const FiniteElement &el2, FaceElementTransformations &Tr, const Vector &elfun, Vector &elvect) |
| | Perform the local action of the NonlinearFormIntegrator resulting from a face integral term. More...
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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
\[ - \left< \{ \tau(u) \}, [v] \right> + \alpha \left< \{ \tau(v) \}, [u] \right> + \kappa \left< h^{-1} \{ \lambda + 2 \mu \} [u], [v] \right> \]
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 2168 of file bilininteg.hpp.
Public Member Functions inherited from mfem::BilinearFormIntegrator
1.8.5
