Jacobian.h file

Jacobian matrix operator for vector-valued functions.

This file defines the Jacobian class, which computes the Jacobian matrix (matrix of all first-order partial derivatives) of vector-valued functions in variational formulations.

Mathematical Foundation

For a vector-valued function , the Jacobian matrix is defined as:

resulting in an matrix.

Special Cases

When , the determinant appears in change of variables
For , related to deformation gradient in mechanics
Transpose gives the gradient of components

Applications

Nonlinear elasticity: deformation gradient tensor
Fluid dynamics: velocity gradient tensor
Differential geometry: metric tensor computations
Coordinate transformations

Usage Example

// Displacement gradient for elasticity
P1 Vh(mesh, mesh.getSpaceDimension());
GridFunction<P1> u(Vh);
auto F = Jacobian(u);  // Deformation gradient F = \nabla u

Namespaces

namespace Rodin: The Rodin library for finite element methods and shape optimization.
namespace Rodin::Variational: Namespace Variational.
namespace Rodin::FormLanguage: Namespace FormLanguage.

Classes

template<class FES, class Data, class Derived> class Rodin::Variational::JacobianBase<GridFunction<FES, Data>, Derived>: Jacobian of a P1 GridFunction.
template<class FES, class Data> struct Rodin::FormLanguage::Traits<Variational::Jacobian<Variational::GridFunction<FES, Data>>>: Type traits for the matrix or tensor expression specialization.
template<class Derived, class FES, Variational::ShapeFunctionSpaceType Space> struct Rodin::FormLanguage::Traits<Variational::Jacobian<Variational::ShapeFunction<Derived, FES, Space>>>: Type traits for the matrix or tensor expression specialization.
template<class FES, class Data> class Rodin::Variational::Jacobian<GridFunction<FES, Data>>: Matrix-field jacobian, .
template<class Derived, class FES, ShapeFunctionSpaceType Space> class Rodin::Variational::Jacobian<ShapeFunction<Derived, FES, Space>>: Matrix-basis jacobian, .