Analytic Common namespace. More...

Classes
class	ConstantFunction
	Constant Analytic function. More...

class	ConstantVectorFunction
	Constant vector valued Analytic function. More...

class	CornerSingularity2D
	Singularity function for a 2D reentry corner with homogeneous boundary conditions. More...

class	CornerSingularity2DRadial
	radial version of the CornerSingularity function... should be used with the PolarCoordinateWrapper More...

class	CosineTensorStatic
	Cosine-Tensor Static function. More...

class	DFG95SteadyInflowFunction
	Steady-State inflow function used in DFG95 flow-around-a-cylinder benchmarks. More...

class	ExpBubbleScalarStatic
	Exponential-Bubble scalar Static function. More...

class	ExpScalarStatic

class	ExpScalarStatic< Half >

class	Frankes3DVariantFunction
	3D Version of Franke's Function More...

class	FrankesFunction
	Saddle-Point-Gauss type interpolation test function. More...

class	GuermondStokesSol
	Time dependent divergence free velocity field. More...

class	GuermondStokesSolPressure
	Pressure to GuermondStokesSol. More...

class	GuermondStokesSolRhs
	Time dependent divergence free velocity field. More...

class	HarmonicShellFunction
	Harmonic shell function. More...

class	MinOfTwoFunctions
	Function representing the minimum of two analytic functions. More...

class	ParProfileBase
	2D Parabolic Profile function base-class More...

class	ParProfileScalar
	2D Scalar Parabolic Profile function More...

class	ParProfileVector
	2D Vector-Valued Parabolic Profile function More...

class	PoiseuillePipeFlow
	Parabolic Poiseuille Pipe-Flow velocity field. More...

class	PolynomialFunction1D
	1D Polynomial function class template More...

class	Q2BubbleScalarStatic
	Q2-bubble scalar Static function. More...

class	RigidBodyVortexPres2D
	Rigid-Body Vortex pressure function. More...

class	RigidBodyVortexVelo2D
	Rigid-Body Vortex velocity field. More...

class	SineRingVortexPres2D
	Sine-Vortex pressure function on the 2D ring domain with distance [1/2,1] around (0,0) More...

class	SineRingVortexRHS2D
	Sine-Vortex RHS field on the 2D ring domain with distance [1/2,1] around (0,0) More...

class	SineRingVortexVelo2D
	Sine-Vortex velocity field on the 2D ring domain with distance [1/2,1] around (0,0) More...

class	SineTensorStatic
	Sine-Tensor Static function. More...

class	SinYT0
	Time dependent divergence free velocity field. More...

class	SinYT0StokesRhs
	Time dependent divergence free velocity field. More...

class	SphereCapFunction
	Sphere-Cap Function on the 2D unit circle / 3D unit ball. More...

class	SphereSinBubbleFunction
	Sphere-Normalized Sine-Bubble Function. More...

class	StandingVortexFunction2D
	Standing-Vortex function in 2D. More...

class	TaylorGreenVortexPres2D
	Taylor-Green Vortex pressure function. More...

class	TaylorGreenVortexVelo2D
	Taylor-Green Vortex velocity field. More...

class	TensorStatic
	Wrapper class for Tensor-Product scalar static functions. More...

class	XYPlaneRotation
	Velocity field for a rigid body rotation in the x,y plane. More...

class	YZPlaneParabolic
	Velocity field for a parabolic profile in the y,z plane, constant in x. More...

Typedefs
using	BazaraaShettyFunction = StaticWrapperFunction< 2, BazaraaShettyStatic, true, true, true >
	Bazaraa/Shetty function. More...

template<typename DT_ >
using	CornerSingluratity2DSimple = PolarCoordinate< CornerSingularity2DRadial< DT_ > >

template<int dim_>
using	CosineWaveFunction = StaticWrapperFunction< dim_, CosineWaveStatic, true, true, true >
	*‍/ More...

template<int dim_>
using	ExpBubbleFunction = StaticWrapperFunction< dim_, ExpBubbleStatic, true, true, true >
	*‍/ More...

template<int dim_>
using	ExpFunction = Analytic::StaticWrapperFunction< dim_, ExpStatic, true, true, true >
	*‍/ More...

using	GoldsteinPriceFunction = StaticWrapperFunction< 2, GoldsteinPriceStatic, true, true, true >
	*‍/ More...

template<int dim_>
using	HeavisideFunction = StaticWrapperFunction< dim_, HeavisideStatic, true, false, false >
	*‍/ More...

template<int dim_>
using	HeavisideRegFunction = StaticWrapperFunction< dim_, HeavisideRegStatic, true, false, false >
	*‍/ More...

using	HimmelblauFunction = StaticWrapperFunction< 2, HimmelblauStatic, true, true, true >
	*‍/ More...

template<int dim_>
using	Q2BubbleFunction = StaticWrapperFunction< dim_, Q2BubbleStatic, true, true, true >
	*‍/ More...

using	RosenbrockFunction = StaticWrapperFunction< 2, RosenbrockStatic, true, true, true >
	*‍/ More...

template<int dim_>
using	SineBubbleFunction = StaticWrapperFunction< dim_, SineBubbleStatic, true, true, true >
	*‍/ More...

Detailed Description

Analytic Common namespace.

This namespace encapsulated commonly used functions, which are often used in standard benchmark problems.

Typedef Documentation

◆ BazaraaShettyFunction

using FEAT::Analytic::Common::BazaraaShettyFunction = typedef StaticWrapperFunction<2, BazaraaShettyStatic, true, true, true>

Bazaraa/Shetty function.

This class implements the AnalyticFunction interface representing the function

\[ u(x,y) = (x - 2)^4 + (x - 2 y)^2 \]

This function has a global miminum in \( x_0 = (2, 1)^T, u(x_0) = 0 \) and is frequently used for testing optimization algorithms because the hessian at the minimal point is singular.

Author: Jordi Paul

Definition at line 1541 of file common.hpp.

◆ CornerSingluratity2DSimple

template<typename DT_ >

using FEAT::Analytic::Common::CornerSingluratity2DSimple = typedef PolarCoordinate<CornerSingularity2DRadial<DT_> >

Definition at line 4273 of file common.hpp.

◆ CosineWaveFunction

template<int dim_>

using FEAT::Analytic::Common::CosineWaveFunction = typedef StaticWrapperFunction<dim_, CosineWaveStatic, true, true, true>

*‍/

Cosine-Wave Analytic function

This class implements the AnalyticFunction interface representing the function

1D: u(x) = cos(pi*x)
2D: u(x,y) = cos(pi*x) * cos(pi*y)
3D: u(x,y,z) = cos(pi*x) * cos(pi*y) * cos(pi*z)

This class supports function values, gradient and hessians for all dimensions.

This function fulfills homogeneous Neumann boundary conditions and has vanishing integral mean on the unit-cube domain.

Author: Peter Zajac

Definition at line 567 of file common.hpp.

◆ ExpBubbleFunction

template<int dim_>

using FEAT::Analytic::Common::ExpBubbleFunction = typedef StaticWrapperFunction<dim_, ExpBubbleStatic, true, true, true>

*‍/

Exponential-Bubble Analytic function

This class implements the AnalyticFunction interface representing the function

1D: u(x) = (exp(-(2*x - 1)^2) - exp(-1)) / (1 - exp(-1))
2D: u(x,y) = u(x) * u(y)
3D: u(x,y,z) = u(x) * u(y) * u(z)

This class supports function values, gradient and hessians for all dimensions.

This function fulfills homogeneous Dirichlet boundary conditions on the unit-cube domain.

Author: Peter Zajac

Definition at line 622 of file common.hpp.

◆ ExpFunction

template<int dim_>

using FEAT::Analytic::Common::ExpFunction = typedef Analytic::StaticWrapperFunction<dim_, ExpStatic, true, true, true>

*‍/

Exponential Analytic function

This class implements the AnalyticFunction interface representing the function

1D: u(x) = (exp(10) - exp(10*x^2)) / (exp(10) - 1)
2D: u(x,y) = u(x)*u(y)
3D: u(x,y,z) = u(x)*u(y)*u(z)

This class supports function values, gradient and hessians for all dimensions.

Definition at line 2056 of file common.hpp.

◆ GoldsteinPriceFunction

using FEAT::Analytic::Common::GoldsteinPriceFunction = typedef StaticWrapperFunction<2, GoldsteinPriceStatic, true, true, true>

*‍/

Goldstein-Price function

Template Parameters

DataType_ Floating point precision

This class implements the StaticFunction interface representing the function

\[ u(x,y) =(1+(1+x+y)^2 * (19 - 14*x+3*x^2-14*y+6*x*y+3*y^2)) *(30 + (2*x-3*y)^2 * (18 - 32*x +12*x^2 + 48*y - 36*x*y + 27*y^2)) \]

This function has a global miminum in \( x_0 = (0, -1)^T, u(x_0) = 3 \).

Author: Jordi Paul

Definition at line 1527 of file common.hpp.

◆ HeavisideFunction

template<int dim_>

using FEAT::Analytic::Common::HeavisideFunction = typedef StaticWrapperFunction<dim_, HeavisideStatic, true, false, false>

*‍/

Heaviside function

This class implements the AnalyticFunction interface representing the function

\[ u(x) = H(x) = \begin{cases} 0, & x < 0 \\ 1, & x \geq 0 \end{cases} \]

in \( 1D \) and

\[ u(x,y) = H(x) H(y), u(x,y,z) = H(x) H(y) H(z) \]

in \( 2D, 3D \) respectively.

Author: Jordi Paul

Definition at line 1048 of file common.hpp.

◆ HeavisideRegFunction

template<int dim_>

using FEAT::Analytic::Common::HeavisideRegFunction = typedef StaticWrapperFunction<dim_, HeavisideRegStatic, true, false, false>

*‍/

Regularized Heaviside static function

This class implements the AnalyticFunction interface representing the function

\[ u(x) = H(x) = \begin{cases} 0, & x < 0 \\ 1, & 2 (\cosh(x) - 1) \geq 0 \end{cases} \]

in \( 1D \) and

\[ u(x,y) = H(x)*H(y), u(x,y,z) = H(x)*H(y)*H(z) \]

in \( 2D, 3D \) respectively.

Author: Jordi Paul

Definition at line 1238 of file common.hpp.

◆ HimmelblauFunction

using FEAT::Analytic::Common::HimmelblauFunction = typedef StaticWrapperFunction<2, HimmelblauStatic, true, true, true>

*‍/

Himmelblau function

This class implements the AnalyticFunction interface representing the function

\[ u(x,y) = (x^2 + y^2 - 11)^2 + (x + y^2 - 7)^2 \]

The function is nonconvex and has 4 local minima at

\begin{align*} x_0 & \approx (-3.77931025337774689189076584129, -3.28318599128616941226600051437)^T \\ x_1 & \approx (-2.80511808695274485305357239809, 3.13131251825057296580430072341)^T \\ x_2 & = (3, 2)^T \\ x_3 & \approx (3.58442834033049174494433823938, -1.84812652696440355353830020904)^T. \end{align*}

with \( \forall i = 0, \dots, 3: u(x_i) = 0 \).

It is often used for testing optimization algorithms because of the nonconvexity and existence of a local maximum in \( x_4 \approx ( -0.270845, -0.93039)^T \) and several saddle points.

Author: Jordi Paul

Definition at line 1639 of file common.hpp.

◆ Q2BubbleFunction

template<int dim_>

using FEAT::Analytic::Common::Q2BubbleFunction = typedef StaticWrapperFunction<dim_, Q2BubbleStatic, true, true, true>

*‍/

Q2-Bubble Analytic function

This class implements the AnalyticFunction interface representing the function

1D: u(x) = 4*x*(1-x)
2D: u(x,y) = 16*x*(1-x)*y*(1-y)
3D: u(x,y,z) = 64*x*(1-x)*y*(1-y)*z*(1-z)

This class supports function values, gradient and hessians for all dimensions.

This function fulfills homogeneous Dirichlet boundary conditions on the unit-cube domain.

Author: Peter Zajac

Definition at line 677 of file common.hpp.

◆ RosenbrockFunction

using FEAT::Analytic::Common::RosenbrockFunction = typedef StaticWrapperFunction<2, RosenbrockStatic, true, true, true>

*‍/

Rosenbrock function

This class implements the AnalyticFunction interface representing the function

\[ u(x,y) = 100(y-x^2)^2 + (1-x)^2. \]

The function has a global minimum in \( x_0 = (1, 1)^T\) and a "steep valley" along the parabola \( y = x^2 \). This is a great challenge to descend-based optimization algorithms like steepest descent or nonlinear CG and the reason it is frequently used as a target function to test such algorithms.

Author: Jordi Paul

Definition at line 1705 of file common.hpp.

◆ SineBubbleFunction

template<int dim_>

using FEAT::Analytic::Common::SineBubbleFunction = typedef StaticWrapperFunction<dim_, SineBubbleStatic, true, true, true>

*‍/

Sine-Bubble Analytic function

This class implements the AnalyticFunction interface representing the function

1D: u(x) = sin(pi*x)
2D: u(x,y) = sin(pi*x) * sin(pi*y)
3D: u(x,y,z) = sin(pi*x) * sin(pi*y) * sin(pi*z)

This class supports function values, gradient and hessians for all dimensions.

This function fulfills homogeneous Dirichlet boundary conditions on the unit-cube domain.

Author: Peter Zajac

Definition at line 376 of file common.hpp.

Classes

Typedefs

Detailed Description

Typedef Documentation

◆ BazaraaShettyFunction

◆ CornerSingluratity2DSimple

◆ CosineWaveFunction

◆ ExpBubbleFunction

◆ ExpFunction

◆ GoldsteinPriceFunction

◆ HeavisideFunction

◆ HeavisideRegFunction

◆ HimmelblauFunction

◆ Q2BubbleFunction

◆ RosenbrockFunction

◆ SineBubbleFunction