Defines the Geometry interface and provides a number of classes that implement this interface + additional geometry related helper routines. More...

Namespaces
namespace	test_utils
	Defines the Geometry::test_utils module and provides a number of test functions to check geometry objects.

Classes
class	Geometry
	Interface class for shape information on a mesh cell in the spirit of parametric finite element methods More...

class	Parallelogram
	Affine quadrilateral = parallelogram. More...

class	Point

class	QuadO1
	Bilinear quadrilateral element shape. More...

class	QuadO2
	A second-order quadrilateral in the plane or in 3D space. More...

class	RefinementPattern
	Abstract interface class for encoding topological local refinement More...

class	SegmentO1
	A straight edge defined by the location of its two endpoints. More...

class	SegmentO2
	A second-order segment in the plane or in 3D space. More...

class	TriaO1
	An affine triangle in the plane or in 3D space. More...

class	TriaO2
	A second-order triangle in the plane or in 3D space. More...

Functions
std::unique_ptr< Geometry >	Compose (std::unique_ptr< Geometry > &&a, std::unique_ptr< Geometry > &&b)
	Create a new geometry object that is the composition of \( \Phi_a \circ \Phi_b \). More...

double	Volume (const Geometry &geo)
	Compute the (approximate) volume (area) of a shape. More...

Eigen::MatrixXd	Corners (const Geometry &geo)
	The corners of a shape with piecewise smooth boundary. More...

void	PrintInfo (std::ostream &o, const Geometry &geom, int output_ctrl)

std::ostream &	operator<< (std::ostream &stream, const Geometry &geom)

bool	assertNonDegenerateQuad (const Eigen::Matrix< double, Eigen::Dynamic, 4 > &coords, double tol=1.0E-8)
	Asserting a non-degenerate bilinear quadrilateral. More...

bool	isParallelogram (const Eigen::Matrix< int, Eigen::Dynamic, Eigen::Dynamic > &polygon)
	Test whether a lattice polygon describes a logical parallelogram. More...

bool	assertNonDegenerateTriangle (const Eigen::Matrix< double, Eigen::Dynamic, 3 > &coords, double tol=1.0E-8)
	Asserting non-degenerate shape of a flat triangle. More...

Detailed Description

Defines the Geometry interface and provides a number of classes that implement this interface + additional geometry related helper routines.

Function Documentation

◆ assertNonDegenerateQuad()

bool lf::geometry::assertNonDegenerateQuad	(	const Eigen::Matrix< double, Eigen::Dynamic, 4 > &	coords,
		double	tol = `1.0E-8`
	)

Asserting a non-degenerate bilinear quadrilateral.

Parameters

coords	w x 4 Eigen matrix whose columns contain the vertex coordinates of the quadrilateral, w = world dimension
tol	relative tolerance for numerical tests of equality with zero

Terminates execution in case degenerate shape is detected.

Definition at line 12 of file quad_o1.cc.

Referenced by lf::geometry::Parallelogram::Parallelogram(), and lf::geometry::QuadO1::QuadO1().

◆ assertNonDegenerateTriangle()

bool lf::geometry::assertNonDegenerateTriangle	(	const Eigen::Matrix< double, Eigen::Dynamic, 3 > &	coords,
		double	tol = `1.0E-8`
	)

Asserting non-degenerate shape of a flat triangle.

Parameters

coords	w x 3 Eigen matrix whose columns contain the vertex coordinates of the quadrilateral, w = world dimension
tol	relative tolerance for numerical tests of equality with zero

Terminates execution in case degenerate shape is detected.

Definition at line 10 of file tria_o1.cc.

Referenced by lf::geometry::TriaO1::TriaO1().

◆ Compose()

std::unique_ptr< Geometry > lf::geometry::Compose	(	std::unique_ptr< Geometry > &&	a,
		std::unique_ptr< Geometry > &&	b
	)

Create a new geometry object that is the composition of \( \Phi_a \circ \Phi_b \).

Parameters

a	The mapping \( \Phi_a \)
b	The mapping \( \Phi_b \)

Returns: The composition of \(\Phi_a \circ \Phi_b\)

◆ Corners()

Eigen::MatrixXd lf::geometry::Corners ( const Geometry & geo )

inline

The corners of a shape with piecewise smooth boundary.

Parameters

geo	The geometry object

Returns: the coordinates vectors for the corners of the shape represented by the geometry object packed into the columns of a dxn-matrix, where d is the dimension of physical space, and n stands for the number of corners.

Note: Only for 2D shapes with all straight edges the positions of the corners will completely define the shape. This is the case with shapes represented by the types lf::geometry::TriaO1 and lf::geometry::QuadO1

Demonstration code [Lecture Document]

(https://www.sam.math.ethz.ch/~grsam/NUMPDEFL/NUMPDE.pdf) Code 2.7.2.23

void PrintGeometryInfo(const lf::mesh::Mesh &mesh, lf::base::dim_t codim) {
  // loop over all entities of the specified codimension
  for (const lf::mesh::Entity *ent : mesh.Entities(codim)) {
    // Number of nodes = number of corner points
    const lf::base::size_type num_nodes = ent->RefEl().NumNodes();
    // Obtain pointer to geometry object associated with entity
    const lf::geometry::Geometry *geo_ptr = ent->Geometry();
    // Fetch coordinates of corner points in packed format \cref{par:coords}
    Eigen::MatrixXd corners = lf::geometry::Corners(*geo_ptr);
    std::cout << ent->RefEl() << "(" << mesh.Index(*ent) << ") pts: ";
    for (int l = 0; l < num_nodes; ++l) {
      std::cout << l << " =[" << corners.col(l).transpose() << "], ";
    }
    std::cout << std::endl;
  }
}

Additional explanations can be found in [Lecture Document] (https://www.sam.math.ethz.ch/~grsam/NUMPDEFL/NUMPDE.pdf) Paragraph 2.7.2.21.

Definition at line 261 of file geometry_interface.h.

References lf::geometry::Geometry::Global(), lf::base::RefEl::NodeCoords(), and lf::geometry::Geometry::RefEl().

Referenced by projects::hldo_sphere::debugging::WhitneyOneCurlTest::Compute(), and projects::dpg::OuterNormals().

◆ isParallelogram()

bool lf::geometry::isParallelogram ( const Eigen::Matrix< int, Eigen::Dynamic, Eigen::Dynamic > & polygon )

Test whether a lattice polygon describes a logical parallelogram.

Parameters

polygon an integer matrix whose column contain the lattice coordinates of the vertices of the polygon.

A polygon passes the parallelogram test, if

it has four vertices
the vectors \(x_1-x_0\) and \(x_2-x_3\) agree.

Definition at line 18 of file refinement_pattern.cc.

Referenced by lf::geometry::Parallelogram::ChildGeometry().

◆ operator<<()

std::ostream & lf::geometry::operator<<	(	std::ostream &	stream,
		const Geometry &	geom
	)

◆ PrintInfo()

void lf::geometry::PrintInfo	(	std::ostream &	o,
		const Geometry &	geom,
		int	output_ctrl
	)

◆ Volume()

double lf::geometry::Volume ( const Geometry & geo )

Compute the (approximate) volume (area) of a shape.

Parameters

geo	The geometry object

Returns: approximate volume

Note: the volume can be computed exactly only for planar affine/bilinear (2D) shapes Otherwise this functions returns a one-point quadrature approximation

Definition at line 11 of file geometry_interface.cc.

References lf::geometry::Geometry::DimLocal(), lf::geometry::Geometry::IntegrationElement(), lf::base::RefEl::kPoint(), lf::base::RefEl::kQuad(), lf::base::RefEl::kSegment(), lf::base::RefEl::kTria(), lf::geometry::Geometry::RefEl(), and lf::base::RefEl::ToString().

Namespaces

Classes

Functions

Detailed Description

Function Documentation

◆ assertNonDegenerateQuad()

◆ assertNonDegenerateTriangle()

◆ Compose()

◆ Corners()

Demonstration code [Lecture Document]

◆ isParallelogram()

◆ operator<<()

◆ PrintInfo()

◆ Volume()