lf::geometry Namespace Reference

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 \( \Phi_a \circ \Phi_b \).
double	Volume (const Geometry &geo)
	Compute the (approximate) volume (area) of a shape.
Eigen::MatrixXd	Corners (const Geometry &geo)
	The corners of a shape with piecewise smooth boundary.
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.
bool	isParallelogram (const Eigen::Matrix< int, Eigen::Dynamic, Eigen::Dynamic > &polygon)
	Test whether a lattice polygon describes a logical parallelogram.
bool	assertNonDegenerateTriangle (const Eigen::Matrix< double, Eigen::Dynamic, 3 > &coords, double tol=1.0E-8)
	Asserting non-degenerate shape of a flat triangle.

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(), 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().

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

1. it has four vertices
2. 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 )

related

Definition at line 26 of file print_info.cc.

PrintInfo()

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

related

Definition at line 5 of file print_info.cc.

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().

Referenced by lf::uscalfe::LinearFELocalLoadVector< SCALAR, FUNCTOR >::Eval().