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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 | AssemblePA (const FiniteElementSpace &fes) |
| | Method defining partial assembly. More...
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| virtual void | AssemblePA (const FiniteElementSpace &trial_fes, const FiniteElementSpace &test_fes) |
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| virtual void | AssemblePAInteriorFaces (const FiniteElementSpace &fes) |
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| virtual void | AssemblePABoundaryFaces (const FiniteElementSpace &fes) |
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| virtual void | AssembleDiagonalPA (Vector &diag) |
| | Assemble diagonal and add it to Vector diag. More...
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| virtual void | AssembleDiagonalPA_ADAt (const Vector &D, Vector &diag) |
| | Assemble diagonal of ADA^T (A is this integrator) and add it to diag. More...
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| virtual void | AddMultPA (const Vector &x, Vector &y) const |
| | Method for partially assembled action. More...
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| virtual void | AddMultTransposePA (const Vector &x, Vector &y) const |
| | Method for partially assembled transposed action. More...
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| virtual void | AssembleEA (const FiniteElementSpace &fes, Vector &emat, const bool add=true) |
| | Method defining element assembly. More...
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| virtual void | AssembleMF (const FiniteElementSpace &fes) |
| | Method defining matrix-free assembly. More...
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| virtual void | AddMultMF (const Vector &x, Vector &y) const |
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| virtual void | AddMultTransposeMF (const Vector &x, Vector &y) const |
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| virtual void | AssembleDiagonalMF (Vector &diag) |
| | Assemble diagonal and add it to Vector diag. More...
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| virtual void | AssembleEAInteriorFaces (const FiniteElementSpace &fes, Vector &ea_data_int, Vector &ea_data_ext, const bool add=true) |
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| virtual void | AssembleEABoundaryFaces (const FiniteElementSpace &fes, Vector &ea_data_bdr, const bool add=true) |
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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. Note that the default implementation in the base class is general but not efficient. 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 BilinearFormIntegrator resulting from a face integral term. Note that the default implementation in the base class is general but not efficient. 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, bool with_coef=true) |
| | Virtual method required for Zienkiewicz-Zhu type error estimators. More...
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| virtual double | ComputeFluxEnergy (const FiniteElement &fluxelem, ElementTransformation &Trans, Vector &flux, Vector *d_energy=NULL) |
| | Virtual method required for Zienkiewicz-Zhu type error estimators. More...
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| virtual | ~BilinearFormIntegrator () |
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| virtual 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 &ir) |
| | Prescribe a fixed IntegrationRule to use. More...
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| void | SetPAMemoryType (MemoryType mt) |
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| const IntegrationRule * | GetIntegrationRule () const |
| | Get the integration rule of the integrator (possibly NULL). 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 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...
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| virtual double | GetLocalStateEnergyPA (const Vector &x) const |
| | Compute the local (to the MPI rank) energy with partial assembly. More...
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| virtual void | AddMultGradPA (const Vector &x, Vector &y) const |
| | Method for partially assembled gradient action. More...
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| virtual void | AssembleGradDiagonalPA (Vector &diag) const |
| | Method for computing the diagonal of the gradient with partial assembly. More...
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| virtual bool | SupportsCeed () const |
| | Indicates whether this integrator can use a Ceed backend. More...
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| ceed::Operator & | GetCeedOp () |
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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 3165 of file bilininteg.hpp.
Public Member Functions inherited from mfem::BilinearFormIntegrator
1.8.5
