Routines for the evalutation of pointwise errors. More...

Detailed Description

Routines for the evalutation of pointwise errors.

Function Documentation

◆ calculateFirstCoefficient()

double Bembel::calculateFirstCoefficient	(	Eigen::VectorXd	cs,
		unsigned int	deg,
		Eigen::MatrixXd	ps_l,
		Eigen::MatrixXd	ps_f,
		Eigen::MatrixXd	ps_b
	)

Calculates the first coefficient such that the mean of the kernel vanishes.

Parameters

cs	The already calculated other coefficients
deg	The degree
ps_l	Gauss quadrature points on the left side of the cube
ps_f	Gauss quadrature points on the front side of the cube
ps_b	Gauss quadrature points on the bottom side of the cube

Definition at line 346 of file Coefficients.hpp.

◆ double_to_string()

std::string Bembel::double_to_string	(	double	d,
		const int	precision
	)

Convert a double to a string with given precision.

Parameters

d	double to be converted.
precision	Precision of the conversion.

Definition at line 253 of file GeometryIGS.hpp.

◆ evaluate_dsolid_sphericals()

Eigen::Vector3d Bembel::evaluate_dsolid_sphericals	(	Eigen::Vector3d	x,
		Eigen::VectorXd	cs,
		unsigned int	deg
	)

Evaluates the series \( \sum_{n = 0}^{\rm deg} \sum_{m = -n}^n |x|^{n-3} c_m^n (\nabla Y_m^n)(\frac{x}{|x|}) \) for real coefficients.

Parameters

x	The point of evaluation
cs	The coefficients stored in the order \( [(0, 0), (1, -1), (1, 0), (1, 1) (2, -2), (2, -1), ... (n, n)] \)
deg	The degree

Definition at line 302 of file Sphericals.hpp.

◆ evaluate_dsphericals()

Eigen::Vector3d Bembel::evaluate_dsphericals	(	Eigen::Vector3d	x,
		Eigen::VectorXd	cs,
		unsigned int	deg
	)

Evaluates the series \( \sum_{n = 0}^{\rm deg} \sum_{m = -n}^n c_m^n \nabla Y_m^n(x) \) for real coefficients.

Parameters

x	The point of evaluation, a vector with length 1
cs	The coefficients stored in the order \( [(0, 0), (1, -1), (1, 0), (1, 1) (2, -2), (2, -1), ... (n, n)] \)
deg	The degree

Definition at line 213 of file Sphericals.hpp.

◆ evaluate_solid_sphericals()

double Bembel::evaluate_solid_sphericals	(	Eigen::Vector3d	x,
		Eigen::VectorXd	cs,
		unsigned int	deg,
		bool	grad
	)

Evaluates the series \( \sum_{n = 0}^{\rm deg} \sum_{m = -n}^n |x|^n c_m^n Y_m^n( \frac{x}{|x|}) \) for real coefficients cs if grad is false, and \( \sum_{n = 0}^{\rm deg} \sum_{m = -n}^n n |x|^{n-1} c_m^n Y_m^n(\frac{x}{|x|}) \) for real coefficients cs if grad is true.

Parameters

x	The point of evaluation
cs	The coefficients stored in the order \( [(0, 0), (1, -1), (1, 0), (1, 1) (2, -2), (2, -1), ... (n, n)] \)
deg	The degree
grad	distinguish the two cases.

Definition at line 128 of file Sphericals.hpp.

◆ evaluate_sphericals()

double Bembel::evaluate_sphericals	(	Eigen::Vector3d	x,
		Eigen::VectorXd	cs,
		unsigned int	deg
	)

Evaluates the series \( \sum_{n = 0}^{\rm deg} \sum_{m = -n}^n c_m^n Y_m^n(x) \) for real coefficients, with the convenction that \( Y_{-m}^n := \overline{Y_m^n} \).

See also: K. Giebermann. Schnelle Summationsverfahren zur numerischen Lösung von Integralgleichungen für Streuprobleme im \( \mathbb{R}^3 \). PhD thesis, Universität Karlsruhe (TH), Germany, 1997.; R. von Rickenbach. Boundary Element Methods for Shape Optimisation in Homogenisation. MSc thesis, Universität Basel, Switzerland, 2021.

Parameters

x	The point of evaluation, a vector with length 1
cs	The coefficients stored in the order \( [(0, 0), (1, -1), (1, 0), (1, 1) (2, -2), (2, -1), ... (n, n)] \)
deg	The degree

Definition at line 68 of file Sphericals.hpp.

◆ getCoefficients()

Eigen::VectorXd Bembel::getCoefficients ( double precision )

Calculates the coefficients for the solid harmonics expansion of the periodic kernel \( k(x) = \sum_{m \in \{-1, 0, 1\}^3} \frac{1}{4 \pi |x-m|} + \frac{|x|^2}{6} + \sum_{n = 0}^{N} \sum_{m = -n}^n c_m^n \phi_n^m(x) \).

Parameters

precision The desired mean error from periodicity.

See also: A. Barnett and L. Greengard. A new integral representation for quasi-periodic fields and its application to two-dimensional band structure calculations. Journal of Computational Physics, 229(19):6898–6914, 2010.; P. Cazeaux and O. Zahm. A fast boundary element method for the solution of periodic many-inclusion problems via hierarchical matrix techniques. ESAIM: Proceedings and Surveys, 48:156–168, 2015.; R. von Rickenbach. Boundary Element Methods for Shape Optimisation in Homogenisation. MSc thesis, Universität Basel, 2021.

Definition at line 60 of file Coefficients.hpp.

◆ getDegree()

unsigned int Bembel::getDegree ( double precision )

inline

Returns the degree of the sphericals expansion given a precision. Can be extended, use even numbers only!

See also: P. Cazeaux and O. Zahm. A fast boundary element method for the solution of periodic many-inclusion problems via hierarchical matrix techniques. ESAIM: Proceedings and Surveys, 48:156–168, 2015.; R. von Rickenbach. Boundary Element Methods for Shape Optimisation in Homogenisation. MSc thesis, Universität Basel, 2021.

Definition at line 221 of file Coefficients.hpp.

◆ getDisplacement()

Eigen::VectorXd Bembel::getDisplacement	(	Eigen::MatrixXd	ps_l,
		Eigen::MatrixXd	ps_f,
		Eigen::MatrixXd	ps_b
	)

Returns the right-hand side for the homogenised coefficient calculation.

Parameters

ps_l	Points on the left side, i.e. \( \subseteq \{-0.5\} \times [-0.5, 0.5] \times [-0.5, 0.5] \).
ps_f	Points on the front side, i.e. \( \subseteq [-0.5, 0.5] \times \{-0.5\} \times [-0.5, 0.5] \).
ps_b	Points on the bottom side, i.e. \( \subseteq [-0.5, 0.5] \times [-0.5, 0.5] \times \{-0.5\} \).

Definition at line 238 of file Coefficients.hpp.

◆ getFunctionSpaceOutputDimension()

template<unsigned int DF>

constexpr int Bembel::getFunctionSpaceOutputDimension ( )

constexpr

Deprecated:: Use DifferentialFormTraits instead.

Definition at line 75 of file DifferentialFormEnum.hpp.

◆ getFunctionSpaceVectorDimension()

template<unsigned int DF>

constexpr int Bembel::getFunctionSpaceVectorDimension ( )

constexpr

Deprecated:: Use DifferentialFormTraits instead.

Definition at line 67 of file DifferentialFormEnum.hpp.

◆ MakeFile()

void Bembel::MakeFile	(	const std::string &	file_name,
		int	number_of_patches
	)

inlinenoexcept

method to generate textfile for the geometry

Parameters

file_name	name of new textfile
number_of_patches	overall number of patches

Definition at line 130 of file GeometryIO.hpp.

◆ PatchShredder() [1/2]

std::vector<Patch> Bembel::PatchShredder ( const Patch & patch )

inlinenoexcept

This function cuts a patch along internal knots, if any.

Parameters

patch Patch which gets checked and cut if there are internal knots.

Returns: Vector of patches if it got cut, otherwise the vector is of size 1.

Definition at line 386 of file Patch.hpp.

◆ PatchShredder() [2/2]

std::vector<Patch> Bembel::PatchShredder ( const std::vector< Patch > & patches )

inlinenoexcept

This function cuts all patches along internal knots, if any.

Parameters

patches Vector with patches.

Returns: Vector of patches if it got cut.

Definition at line 437 of file Patch.hpp.

◆ spherical_harmonics_full()

Eigen::Matrix< double, Eigen::Dynamic, 2 > Bembel::spherical_harmonics_full	(	Eigen::Vector3d	x,
		unsigned int	N
	)

Calculates the the spherical harmonics \( Y_n^m(\frac{x}{|x|}) \), ordered by \( [Y_0^0, \, Y_1^{-1}, \, Y_1^0, \, Y_1^1, \, Y_2^{-2}, ..., Y_N^N] \).

Parameters

x	the point of evaluation
N	the maximal degree

Definition at line 388 of file Sphericals.hpp.

◆ WritePatch()

void Bembel::WritePatch	(	const std::string &	file_name,
		int	current_patch_number,
		const std::vector< Eigen::MatrixXd > &	xyzw,
		const std::vector< double > &	knt1,
		const std::vector< double > &	knt2
	)

noexcept

method to write Patch information into textfile

Parameters

knt1	knotVector1
knt2	knotVector2
xyzw	Vector with x,y,z,w Matices
file_name	filename
current_patch_number	current patch number

Definition at line 154 of file GeometryIO.hpp.

Variable Documentation

◆ DummyOperator_test_function

std::function<double(const Eigen::Vector2d &, const Eigen::Vector2d &)> Bembel::DummyOperator_test_function

Initial value:

=

[](const Eigen::Vector2d &x, const Eigen::Vector2d &y) { return 1.; }

Definition at line 34 of file DummyOperator.hpp.