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| | PolynomialError (const Parameters &par, const std::vector< Polynomials::Polynomial< double >> potentials_1d) |
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| virtual void | cell (DoFInfo< dim > &dinfo, IntegrationInfo< dim > &info) const |
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| virtual void | boundary (DoFInfo< dim > &dinfo, IntegrationInfo< dim > &info) const |
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| virtual void | face (DoFInfo< dim > &dinfo1, DoFInfo< dim > &dinfo2, IntegrationInfo< dim > &info1, IntegrationInfo< dim > &info2) const |
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 Public Member Functions inherited from AmandusIntegrator< dim > |
| | AmandusIntegrator () |
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| virtual void | extract_data (const dealii::AnyData &data) |
| | Extract data independent of the cell. More...
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| unsigned int | n_errors () const |
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| unsigned int | error_type (unsigned int i) const |
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| std::string | error_name (unsigned int i) const |
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| dealii::UpdateFlags | update_flags () const |
| | Returns the update flags to be used. More...
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| dealii::UpdateFlags | update_flags_face () const |
| | Returns the update flags to be used on boundary and interior faces. More...
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| void | add_flags (const dealii::UpdateFlags flags) |
| | Add update flags on all objects. More...
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| void | add_flags_face (const dealii::UpdateFlags flags) |
| | Add update flags on boundary and internal faces. More...
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template<int dim>
class Elasticity::PolynomialError< dim >
Computes the error between a numerical solution and a known exact polynomial solution of a Elasticity problem.
Since we are planning to use this even in the nearly incompressible case, we use Helmholtz decomposition and represent the solution as the sum of the gradient of one polynomial and the curl of either one (id 2D) or three (in 3D) polynomials. These are in the vector of polynomials $$ given to the constructor, such that the gradient potential is first.
\begin{alignat*}{2} \mathbf u &= \nabla \phi_0 + \nabla\times \phi_1 & \text{2D} \\ \mathbf u &= \nabla \phi_0 + \nabla\times (\phi_1,\dots,\phi_3)^T & \text{3D} \\ \end{alignat*}
The according right hand sides of the Elasticity equations and the residuals are integrated by the functions of the classes PolynomialRHS.
Here, we use the formulas
\begin{gather*} 2 \nabla \cdot \epsilon(u) = 2 \nabla \cdot \begin{pmatrix} \partial_1 u_1 & \tfrac{\partial_2 u_1 + \partial_1 u_2}{2}\\ \tfrac{\partial_2 u_1 + \partial_1 u_2}{2} & \partial_2 u_2 \end{pmatrix} = \begin{pmatrix} 2 \partial_{11} u_1 + \partial_{22} u_1 + \partial_{12} u_2 \\ 2 \partial_{22} u_2 + \partial_{11} u_2 + \partial_{12} u_1 \end{pmatrix} \end{gather*}
- Author
- Guido Kanschat
- Date
- 2014
1.8.11
