GridFunction module

Defines GridFunctions to be used in operators, boundary-conditions, interpolation and integration. More...

Classes
class	AnalyticGridFunction< Function, GridView >
	A Gridfunction that evaluates a function with global coordinates. More...
class	ComposerGridFunction< Sig, EntitySet, Functor, GridFunctions >
	A Gridfunction that applies a functor to the evaluated Gridfunctions. More...
class	ConstantGridFunction< T, GridView >
	Gridfunction returning a constant value. More...
class	DerivativeGridFunction< GridFunction, Type >
	A Gridfunction that returns the derivative when calling localFunction. More...
class	DiscreteFunction< Coeff, GB, TreePath, R >
	A mutable view on the subspace of a DOFVector,. More...
class	ElementGridFunction< GridView, Container >
	Gridfunction returning a constant value per element. More...

Functions
template<class Expr >
auto	gradientOf (Expr const &expr)
template<class PreGridFct , class GridView >
decltype(auto)	makeGridFunction (PreGridFct const &preGridFct, GridView const &gridView)
	Generator for Gridfunctions from Expressions (PreGridfunctions) More...
template<class T , REQUIRES(Concepts::AnyGridFunction< T >) >
auto	abs (T &&value)
template<class Functor , class... PreGridFcts>
auto	invokeAtQP (Functor const &f, PreGridFcts &&... gridFcts)
	Generator function for ComposerGridFunction. More...
template<class Lhs , REQUIRES(Concepts::AnyGridFunction< Lhs >) >
auto	operator- (Lhs &&lhs)
	Applies Operation::Negate to GridFunctions.
template<class Lhs , class Rhs , REQUIRES(Concepts::AnyGridFunction< Lhs, Rhs >) >
auto	operator+ (Lhs &&lhs, Rhs &&rhs)
	Applies Operation::Plus to GridFunctions.
template<class Lhs , class Rhs , REQUIRES(Concepts::AnyGridFunction< Lhs, Rhs >) >
auto	operator- (Lhs &&lhs, Rhs &&rhs)
	Applies Operation::Minus to GridFunctions.
template<class Lhs , class Rhs , REQUIRES(Concepts::AnyGridFunction< Lhs, Rhs >) >
auto	operator* (Lhs &&lhs, Rhs &&rhs)
	Applies Operation::Multiplies to GridFunctions.
template<class Lhs , class Rhs , REQUIRES(Concepts::AnyGridFunction< Lhs, Rhs >) >
auto	operator/ (Lhs &&lhs, Rhs &&rhs)
	Applies Operation::Divides to GridFunctions.
template<class Lhs , class Rhs , REQUIRES(Concepts::AnyGridFunction< Lhs, Rhs >) >
auto	max (Lhs &&lhs, Rhs &&rhs)
	Applies Operation::Max to GridFunctions.
template<class Lhs , class Rhs , REQUIRES(Concepts::AnyGridFunction< Lhs, Rhs >) >
auto	min (Lhs &&lhs, Rhs &&rhs)
	Applies Operation::Min to GridFunctions.
template<class Lhs , class Rhs , REQUIRES(Concepts::AnyGridFunction< Lhs, Rhs >) >
auto	abs_max (Lhs &&lhs, Rhs &&rhs)
	Applies Operation::AbsMax to GridFunctions.
template<class Lhs , class Rhs , REQUIRES(Concepts::AnyGridFunction< Lhs, Rhs >) >
auto	abs_min (Lhs &&lhs, Rhs &&rhs)
	Applies Operation::AbsMin to GridFunctions.
template<class V , class T , REQUIRES(Concepts::AnyGridFunction< V >) >
auto	clamp (V &&v, T const &lo, T const &hi)
	Applies Operation::Clamp to GridFunction.
template<class T , REQUIRES(Concepts::AnyGridFunction< T >) >
auto	tanh (T &&value)
	Applies Operation::Tanh to GridFunctions.
template<class T , REQUIRES(Concepts::AnyGridFunction< T >) >
auto	signum (T &&value)
	Applies Operation::Signum to GridFunctions.
template<class T , REQUIRES(Concepts::AnyGridFunction< T >) >
auto	sqr (T &&value)
	Applies Operation::Sqr to GridFunction.
template<int p, class T , REQUIRES(Concepts::AnyGridFunction< T >) >
auto	pow (T &&value)
	Applies Operation::Pow. More...
template<class T , REQUIRES(Concepts::AnyGridFunction< T >) >
auto	pow (T &&value, double p)
	Applies Operation::Power to GridFunction.
template<std::size_t I, class T , REQUIRES(Concepts::AnyGridFunction< T >) >
auto	get (T &&value)
	Applies Operation::Get<I> to GridFunction.
template<class T , REQUIRES(Concepts::AnyGridFunction< T >) >
auto	get (T &&value, std::size_t i)
	Applies Operation::Get_ to GridFunction.
template<class T >
auto	constant (T const &value)
	Return a reference-wrapper as GridFunction.
template<class T >
auto	reference (T &value)
	Return a reference-wrapper as GridFunction.
template<class Vec , REQUIRES(Concepts::AnyGridFunction< Vec >) >
auto	sum (Vec &&vec)
	Applies a sum() functor to a vector-valued GridFunction.
template<class Vec , REQUIRES(Concepts::AnyGridFunction< Vec >) >
auto	unary_dot (Vec &&vec)
	Applies Operation::UnaryDot to a vector-valued GridFunction.
template<class Vec , REQUIRES(Concepts::AnyGridFunction< Vec >) >
auto	one_norm (Vec &&vec)
	Applies a one_norm() functor to a vector-valued GridFunction.
template<class Vec , REQUIRES(Concepts::AnyGridFunction< Vec >) >
auto	two_norm (Vec &&vec)
	Applies Operation::TwoNorm to a vector-valued GridFunction.
template<int p, class Vec , REQUIRES(Concepts::AnyGridFunction< Vec >) >
auto	p_norm (Vec &&vec)
	Applies a p_norm() functor to a vector-valued GridFunction.
template<class Vec , REQUIRES(Concepts::AnyGridFunction< Vec >) >
auto	infty_norm (Vec &&vec)
	Applies a infty_norm() functor to a vector-valued GridFunction.
template<class Mat , REQUIRES(Concepts::AnyGridFunction< Mat >) >
auto	trans (Mat &&mat)
	Applies Operation::Trans to a matrix-valued GridFunction.
template<class Lhs , class Rhs , REQUIRES(Concepts::AnyGridFunction< Lhs, Rhs >) >
auto	dot (Lhs &&lhs, Rhs &&rhs)
	Applies Operation::Dot to two vector-valued GridFunctions.
template<class Lhs , class Rhs , REQUIRES(Concepts::AnyGridFunction< Lhs, Rhs >) >
auto	distance (Lhs &&lhs, Rhs &&rhs)
	Applies a distance-functor to two vector-valued GridFunctions.
template<class Lhs , class Rhs , REQUIRES(Concepts::AnyGridFunction< Lhs, Rhs >) >
auto	outer (Lhs &&lhs, Rhs &&rhs)
	Applies an outer() functor to two vector-valued GridFunctions.
template<class Lhs , class Rhs , REQUIRES(Concepts::AnyGridFunction< Lhs, Rhs >) >
auto	cross (Lhs &&lhs, Rhs &&rhs)
	Applies a cross() functor to two vector-valued GridFunctions.

Detailed Description

Defines GridFunctions to be used in operators, boundary-conditions, interpolation and integration.

GridFunctions are expressions build up of some elementary terms and can be used to construct a GridFunctionOperator, can be interpolated to a DOFVector, and can be integrated over a GridView.

Thus, GridFunctions are an important incredient to formulate the bilinear and linear forms und to postprocess the solutions.

Examples:

1. Usage of GridFunctions to build Operators:

   ProblemStat<Traits> prob("name");
   prob.initialize(INIT_ALL);
    
   auto opB = makeOperator(BiLinearForm, Expression);
   prob.addMatrixOperator(opB, Row, Col);
    
   auto opL = makeOperator(LinearForm, Expression);
   prob.addVectorOperator(opL, Row);

2. Usage of GridFunctions in BoundaryConditions:

   prob.addDirichletBC(Predicate, Row, Col, Expression);

3. Interpolate a GridFunction to a DOFVector:

   prob.solution(_0).interpol(Expression);

4. Integrate a GridFunction on a GridView:

   auto value = integrate(Expression, prob.gridView());

Remarks:

• An Expression is anything, a GridFunction can be created from, sometimes also called PreGridFunction. It includes constants, functors callable with GlobalCoordinates, and any combination of GridFunctions.

• Anything that needs a quadrature formula, e.g., makeOperator() and integrate(), needs to determine the (approximative) polynomial degree of the GridFunctions. If the Gridfunction builds a polynomial expression, it can be deduced automatically, i.e. if it includes constants, DOFVectors, and arithmetic operator operator+, operator-, or operator*.

  If the polynomial order can not be deduced, the compiler gives an error. Then, these functions accept an additional argument, to provide either the polynomial degree of the expression, or a quadrature rule explicitly.

  Examples:

  • auto op1 = makeOperator(B, 1.0 + pow<2>(prob.solution(_0)));
  • auto op2 = makeOperator(B, sin(X(0)), 4);
  • auto op3 = makeOperator(B, sin(X(0)), Dune::QuadratureRules(Dune::GeometryType::simplex, 4));
  • auto value1 = integrate(sin(X(0)), 4);

Function Documentation

gradientOf()

auto gradientOf

(

Expr const &

expr

)

Generates a Gridfunction representing the gradient of a GridFunction. See DerivativeGridFunction.

Examples:

• gradientOf(prob.solution(_0))
• gradientOf(X(0) + X(1) + prob.solution(_0))

Referenced by NewtonIteration< ProblemType >::oneIteration().

invokeAtQP()

auto invokeAtQP ( Functor const &  f, PreGridFcts &&...  gridFcts  )

related

Generator function for ComposerGridFunction.

Applies the functor f to the grid-functions gridFcts.... See ComposerGridFunction.

Examples:

• invokeAtQP([](Dune::FieldVector<double, 2> const& x) { return two_norm(x); }, X());
• invokeAtQP([](double u, auto const& x) { return u + x[0]; }, 1.0, X());
• invokeAtQP(Operation::Plus{}, X(0), X(1));

References AMDiS::Concepts::Functor.

makeGridFunction()

decltype(auto) makeGridFunction	(	PreGridFct const &	preGridFct,
		GridView const &	gridView
	)

Generator for Gridfunctions from Expressions (PreGridfunctions)

Create an evaluable GridFunction from an expression that itself can not be evaluated. Therefore, it binds the GridFunction to a GridView.

Example:

ProblemStat<Traits> prob("name");
prob.initialize(INIT_ALL);
 
auto gridFct = makeGridFunction(Expression, prob.leafGridView());
 
// eval GridFunction at GlobalCoordinates
auto value = gridFct(Dune::FieldVector<double,2>{1.0, 2.0});
 
auto localFct = localFunction(gridFct);
for (auto const& element : elements(prob.leafGridView())) {
  localFct.bind(element);
  // eval LocalFunction at local coordinates
  auto x = localFct(element.geometry().center());
  localFct.unbind();
}

In contrast to Expressions, GridFunctions can be evaluated and

• have the free-function localFunction() to obtain a LocalFunction
• its LocalFunctions have the free-function order() to obtain the polynomial order of the Expression (if available)
• its LocalFunctions have the free-function derivativeOf() to differentiate the Expression with respect to global Coordinates. A derivative Expression can be created, using gradientOf() that can be converted to a GridFunction afterwards.

Referenced by ProblemStat< Traits >::addDirichletBC(), and DiscreteFunction< Coeff, GB, TreePath, R >::interpolate_noalias().

pow()

auto pow ( T &&  value )

related

Applies Operation::Pow.

to GridFunction.