|  | v0.15.0 | 
#include "src/finite_elements/adJoint.hpp"
| Public Types | |
| template<int SPACE_DIM> | |
| using | OpGetCoFactor = OpGetCoFactorImpl< SPACE_DIM, I, EleOp > | 
| Operator to compute cofactor matrix derivatives for higher-order geometry. | |
Definition at line 16 of file adJoint.hpp.
| using MoFEM::AdJoint< EleOp >::Integration< I >::OpGetCoFactor = OpGetCoFactorImpl<SPACE_DIM, I, EleOp> | 
Operator to compute cofactor matrix derivatives for higher-order geometry.
The cofactor matrix \(\text{cof}(\mathbf{J})\) of the Jacobian matrix \(\mathbf{J}\) is defined as:
\[ \text{cof}(\mathbf{J}) = \det(\mathbf{J}) \cdot \mathbf{J}^{-T} \]
where \(\mathbf{J}^{-T} = (\mathbf{J}^{-1})^T\) is the transpose of the inverse Jacobian.
For a 2D case with Jacobian matrix:
\[ \mathbf{J} = \begin{pmatrix} J_{11} & J_{12} \\ J_{21} & J_{22} \end{pmatrix} \]
The cofactor matrix is:
\[ \text{cof}(\mathbf{J}) = \begin{pmatrix} J_{22} & -J_{12} \\ -J_{21} & J_{11} \end{pmatrix} \]
This operator computes the derivative of the cofactor with respect to geometry parameters, which is essential for higher-order geometry sensitivity analysis. The computed quantity is:
\[ \frac{d}{d\alpha}\text{cof}(\mathbf{J}) = \frac{d}{d\alpha}(\det(\mathbf{J}) \cdot \mathbf{J}^{-T}) \]
| SPACE_DIM | Spatial dimension (2D or 3D) | 
Definition at line 54 of file adJoint.hpp.
