Derivative.h file

Directional derivative operator for scalar functions.

This file defines the Derivative class, which computes directional derivatives of scalar functions. This generalizes the gradient concept to derivatives in specific directions.

Mathematical Foundation

The directional derivative of a function in direction is:

Coordinate Derivatives

Special cases are partial derivatives:

where is the -th coordinate direction.

Applications

Normal derivatives:
Material derivatives in fluid dynamics
Characteristic methods for PDEs
Shape sensitivity analysis

Usage Example

// Normal derivative on boundary
auto n = BoundaryNormal();
auto normal_deriv = Derivative(u, n);  // \partialu/\partialn = \nabla u\cdotn

Namespaces

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

Classes

template<class NestedDerived, class FES, Variational::ShapeFunctionSpaceType Space> struct Rodin::FormLanguage::Traits<Variational::Derivative<Variational::ShapeFunction<NestedDerived, FES, Space>>>: Type traits for Derivative over a shape function: exposes the shape function space, the finite element space, the scalar type and the operand type.
template<class FES, class Data, class Derived> class Rodin::Variational::DerivativeBase<GridFunction<FES, Data>, Derived>: CRTP base for the partial derivative of a grid function.
template<class NestedDerived, class FES, ShapeFunctionSpaceType SpaceType> class Rodin::Variational::Derivative<ShapeFunction<NestedDerived, FES, SpaceType>>: Partial derivative of a shape function.
template<class FES, class Data> struct Rodin::FormLanguage::Traits<Variational::Derivative<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::Derivative<Variational::ShapeFunction<Derived, FES, Space>>>: Type traits for the matrix or tensor expression specialization.
template<class FES, class Data> class Rodin::Variational::Derivative<GridFunction<FES, Data>>: Matrix-field derivative, .
template<class Derived, class FES, ShapeFunctionSpaceType Space> class Rodin::Variational::Derivative<ShapeFunction<Derived, FES, Space>>: Matrix-basis derivative, .