Utility class that provides methods for calculating distributions of points on different cells. More...

#include <Intrepid_PointTools.hpp>

Static Public Member Functions
static int	getLatticeSize (const shards::CellTopology &cellType, const int order, const int offset=0)
	Computes the number of points in a lattice of a given order on a simplex (currently disabled for other cell types). If offset == 0, the lattice will include only include the vertex points if order == 1, and will include edge midpoints if order == 2, and so on. In particular, this is the dimension of polynomials of degree "order" on the given simplex. The offset argument is used to indicate that the layer of points on the boundary is omitted (if offset == 1). For greater offsets, more layers are omitteed. More...

template<class Scalar , class ArrayType >
static void	getLattice (ArrayType &pts, const shards::CellTopology &cellType, const int order, const int offset=0, const EPointType pointType=POINTTYPE_EQUISPACED)
	Computes a lattice of points of a given order on a reference simplex (currently disabled for other cell types). The output array is (P,D), where. More...

template<class Scalar , class ArrayType >
static void	getGaussPoints (ArrayType &pts, const int order)

Static Private Member Functions
template<class Scalar , class ArrayTypeOut , class ArrayTypeIn1 , class ArrayTypeIn2 >
static void	cartToBaryTriangle (ArrayTypeOut &baryValues, const ArrayTypeIn1 &cartValues, const ArrayTypeIn2 &vertices)
	Converts Cartesian coordinates to barycentric coordinates on a batch of triangles. The input array cartValues is (C,P,2) The output array baryValues is (C,P,3). The input array vertices is (C,3,2), where. More...

template<class Scalar , class ArrayTypeOut , class ArrayTypeIn1 , class ArrayTypeIn2 >
static void	baryToCartTriangle (ArrayTypeOut &cartValues, const ArrayTypeIn1 &baryValues, const ArrayTypeIn2 &vertices)
	Converts barycentric coordinates to Cartesian coordinates on a batch of triangles. The input array baryValues is (C,P,3) The output array cartValues is (C,P,2). The input array vertices is (C,3,2), where. More...

template<class Scalar , class ArrayTypeOut , class ArrayTypeIn1 , class ArrayTypeIn2 >
static void	cartToBaryTetrahedron (ArrayTypeOut &baryValues, const ArrayTypeIn1 &cartValues, const ArrayTypeIn2 &vertices)
	Converts Cartesian coordinates to barycentric coordinates on a batch of tetrahedra. The input array cartValues is (C,P,3) The output array baryValues is (C,P,4). The input array vertices is (C,4,3), where. More...

template<class Scalar , class ArrayTypeOut , class ArrayTypeIn1 , class ArrayTypeIn2 >
static void	baryToCartTetrahedron (ArrayTypeOut &cartValues, const ArrayTypeIn1 &baryValues, const ArrayTypeIn2 &vertices)
	Converts barycentric coordinates to Cartesian coordinates on a batch of tetrahedra. The input array baryValues is (C,P,4) The output array cartValues is (C,P,3). The input array vertices is (C,4,3), where. More...

template<class Scalar , class ArrayType >
static void	getEquispacedLattice (const shards::CellTopology &cellType, ArrayType &points, const int order, const int offset=0)
	Computes an equispaced lattice of a given order on a reference simplex (currently disabled for other cell types). The output array is (P,D), where. More...

template<class Scalar , class ArrayType >
static void	getWarpBlendLattice (const shards::CellTopology &cellType, ArrayType &points, const int order, const int offset=0)
	Computes a warped lattice (ala Warburton's warp-blend points of a given order on a reference simplex (currently disabled for other cell types). The output array is (P,D), where. More...

template<class Scalar , class ArrayType >
static void	getEquispacedLatticeLine (ArrayType &points, const int order, const int offset=0)
	Computes an equispaced lattice of a given order on the reference line [-1,1]. The output array is (P,1), where. More...

template<class Scalar , class ArrayType >
static void	getEquispacedLatticeTriangle (ArrayType &points, const int order, const int offset=0)
	Computes an equispaced lattice of a given order on the reference triangle. The output array is (P,2), where. More...

template<class Scalar , class ArrayType >
static void	getEquispacedLatticeTetrahedron (ArrayType &points, const int order, const int offset=0)
	Computes an equispaced lattice of a given order on the reference tetrahedron. The output array is (P,3), where. More...

template<class Scalar , class ArrayType >
static void	getWarpBlendLatticeLine (ArrayType &points, const int order, const int offset=0)
	Returns the Gauss-Lobatto points of a given order on the reference line [-1,1]. The output array is (P,1), where. More...

template<class Scalar , class ArrayType >
static void	warpFactor (const int order, const ArrayType &xnodes, const ArrayType &xout, ArrayType &warp)
	interpolates Warburton's warp function on the line More...

template<class Scalar , class ArrayType >
static void	getWarpBlendLatticeTriangle (ArrayType &points, const int order, const int offset=0)
	Returns Warburton's warp-blend points of a given order on the reference triangle. The output array is (P,2), where. More...

template<class Scalar , class ArrayType >
static void	getWarpBlendLatticeTetrahedron (ArrayType &points, const int order, const int offset=0)
	Returns Warburton's warp-blend points of a given order on the reference tetrahedron. The output array is (P,3), where. More...

template<class Scalar , class ArrayType >
static void	warpShiftFace3D (const int order, const Scalar pval, const ArrayType &L1, const ArrayType &L2, const ArrayType &L3, const ArrayType &L4, ArrayType &dxy)
	This is used internally to compute the tetrahedral warp-blend points one each face. More...

template<class Scalar , class ArrayType >
static void	evalshift (const int order, const Scalar pval, const ArrayType &L1, const ArrayType &L2, const ArrayType &L3, ArrayType &dxy)
	Used internally to evaluate the point shift for warp-blend points on faces of tets. More...

template<class Scalar , class ArrayType >
static void	evalwarp (ArrayType &warp, const int order, const ArrayType &xnodes, const ArrayType &xout)
	Used internally to compute the warp on edges of a triangle in warp-blend points. More...

Detailed Description

Utility class that provides methods for calculating distributions of points on different cells.

Simplicial lattices in PointTools are sets of points with certain ordering properties. They are used for defining degrees of freedom for higher order finite elements.

Each lattice has an "order". In general, this is the same as the cardinality of the polynomial space of degree "order". In terms of binomial coefficients, this is binomial(order+d,order) for the simplex in d dimensions. On the line, the size is order+1. On the triangle and tetrahedron, there are (order+1)(order+2)/2 and (order+1)(order+2)(order+3)/6, respectively.

The points are ordered lexicographically from low to high in increasing spatial dimension. For example, the line lattice of order 3 looks like:

x--x--x--x

where "x" denotes a point location. These are ordered from left to right, so that the points are labeled:

0--1--2--3

The triangular lattice of order 3 is

x
|\
| \
x  x
|   \
|    \
x  x  x
|      \
|       \
x--x--x--x

The ordering starts in the bottom left and increases first from left to right. The ordering is

9
|\
| \
8  7
|   \
|    \
4  5  6
|      \
|       \
0--1--2--3

Tetrahedral lattices are similar but difficult to draw with ASCII art.

Each lattice also has an "offset", which indicates a number of layers of points on the bounary taken away. All of the lattices above have a 0 offest. In Intrepid, typically only offset = 0 or 1 will be used. The offset=1 case is used to generate sets of points properly inside a given simplex. These are used, for example, to construct points internal to an edge or face for H(curl) and H(div) finite elements.

For example, for a line lattice with order = 3 and offset = 1, the points will look like

---x--x---

and a triangle with order=3 and offset=1 will contain a single point

.
|\
| \
|  \
|   \
|    \
|  x  \
|      \
|       \
|--------\

When points on lattices with nonzero offset are numbered, the are numbered contiguously from 0, so that the line and triangle above are respectively

---0--1---

.
|\
| \
|  \
|   \
|    \
|  0  \
|      \
|       \
|--------\

Additionally, two types of point distributions are currently support. The points may be on an equispaced lattice, which is easy to compute but can lead to numerical ill-conditioning in finite element bases and stiffness matrices. Alternatively, the warp-blend points of Warburton are provided on each lattice (which are just the Gauss-Lobatto points on the line).

Definition at line 195 of file Intrepid_PointTools.hpp.

Member Function Documentation

template<class Scalar , class ArrayTypeOut , class ArrayTypeIn1 , class ArrayTypeIn2 >

static void Intrepid::PointTools::baryToCartTetrahedron	(	ArrayTypeOut &	cartValues,
		const ArrayTypeIn1 &	baryValues,
		const ArrayTypeIn2 &	vertices
	)

staticprivate

Converts barycentric coordinates to Cartesian coordinates on a batch of tetrahedra. The input array baryValues is (C,P,4) The output array cartValues is (C,P,3). The input array vertices is (C,4,3), where.

C - num. integration domains
P - number of points per cell
D - is the spatial dimension

Parameters

baryValues	[out] - Output array of barycentric coords
cartValues	[in] - Input array of Cartesian coords
vertices	[out] - Vertices of each cell.

template<class Scalar , class ArrayTypeOut , class ArrayTypeIn1 , class ArrayTypeIn2 >

static void Intrepid::PointTools::baryToCartTriangle	(	ArrayTypeOut &	cartValues,
		const ArrayTypeIn1 &	baryValues,
		const ArrayTypeIn2 &	vertices
	)

staticprivate

Converts barycentric coordinates to Cartesian coordinates on a batch of triangles. The input array baryValues is (C,P,3) The output array cartValues is (C,P,2). The input array vertices is (C,3,2), where.

C - num. integration domains
P - number of points per cell
D - is the spatial dimension

Parameters

baryValues	[out] - Output array of barycentric coords
cartValues	[in] - Input array of Cartesian coords
vertices	[out] - Vertices of each cell.

template<class Scalar , class ArrayTypeOut , class ArrayTypeIn1 , class ArrayTypeIn2 >

static void Intrepid::PointTools::cartToBaryTetrahedron	(	ArrayTypeOut &	baryValues,
		const ArrayTypeIn1 &	cartValues,
		const ArrayTypeIn2 &	vertices
	)

staticprivate

Converts Cartesian coordinates to barycentric coordinates on a batch of tetrahedra. The input array cartValues is (C,P,3) The output array baryValues is (C,P,4). The input array vertices is (C,4,3), where.

C - num. integration domains
P - number of points per cell
D - is the spatial dimension

Parameters

baryValues	[out] - Output array of barycentric coords
cartValues	[in] - Input array of Cartesian coords
vertices	[out] - Vertices of each cell.

template<class Scalar , class ArrayTypeOut , class ArrayTypeIn1 , class ArrayTypeIn2 >

static void Intrepid::PointTools::cartToBaryTriangle	(	ArrayTypeOut &	baryValues,
		const ArrayTypeIn1 &	cartValues,
		const ArrayTypeIn2 &	vertices
	)

staticprivate

Converts Cartesian coordinates to barycentric coordinates on a batch of triangles. The input array cartValues is (C,P,2) The output array baryValues is (C,P,3). The input array vertices is (C,3,2), where.

C - num. integration domains

P - number of points per cell

Parameters

baryValues	[out] - Output array of barycentric coords
cartValues	[in] - Input array of Cartesian coords
vertices	[out] - Vertices of each cell.

template<class Scalar , class ArrayType >

void Intrepid::PointTools::evalshift	(	const int	order,
		const Scalar	pval,
		const ArrayType &	L1,
		const ArrayType &	L2,
		const ArrayType &	L3,
		ArrayType &	dxy
	)

staticprivate

Used internally to evaluate the point shift for warp-blend points on faces of tets.

Parameters

order	[in] - the order of lattice
pval	[in] - the "alpha" term in the warping function
L1	[in] - the first barycentric coordinate of the input points
L2	[in] - the second barycentric coordinate of the input points
L3	[in] - the third barycentric coordinate of the input points
dxy	[out] - contains the amount to shift each point in the x and y direction

Definition at line 487 of file Intrepid_PointToolsDef.hpp.

template<class Scalar , class ArrayType >

void Intrepid::PointTools::evalwarp	(	ArrayType &	warp,
		const int	order,
		const ArrayType &	xnodes,
		const ArrayType &	xout
	)

staticprivate

Used internally to compute the warp on edges of a triangle in warp-blend points.

Parameters

warp	[out] - a 1d array containing the amount to move each point
order	[in] - the order of the lattice
xnodes	[in] - the points to warp to, typically the Gauss-Lobatto points
xout	[in] - the equispaced points on the edge

Definition at line 566 of file Intrepid_PointToolsDef.hpp.

References Intrepid::FieldContainer< Scalar, ArrayTypeId >::initialize().

template<class Scalar , class ArrayType >

void Intrepid::PointTools::getEquispacedLattice	(	const shards::CellTopology &	cellType,
		ArrayType &	points,
		const int	order,
		const int	offset = `0`
	)

staticprivate

Computes an equispaced lattice of a given order on a reference simplex (currently disabled for other cell types). The output array is (P,D), where.

P - number of points per cell

D - is the spatial dimension

Parameters

points	[out] - Output array of point coords
order	[in] - number of points per side, plus 1
offset	[in] - Number of points on boundary to skip
cellType	[in] - type of reference cell (currently only supports the simplex)

Definition at line 93 of file Intrepid_PointToolsDef.hpp.

References getLatticeSize().

template<class Scalar , class ArrayType >

void Intrepid::PointTools::getEquispacedLatticeLine	(	ArrayType &	points,
		const int	order,
		const int	offset = `0`
	)

staticprivate

Computes an equispaced lattice of a given order on the reference line [-1,1]. The output array is (P,1), where.

P - number of points per cell

Parameters

points	[out] - Output array of point coords
order	[in] - The lattice has order + 1 points, minus any skipped by offset
offset	[in] - Number of points on boundary to skip coming in per boundary

Definition at line 179 of file Intrepid_PointToolsDef.hpp.

template<class Scalar , class ArrayType >

void Intrepid::PointTools::getEquispacedLatticeTetrahedron	(	ArrayType &	points,
		const int	order,
		const int	offset = `0`
	)

staticprivate

Computes an equispaced lattice of a given order on the reference tetrahedron. The output array is (P,3), where.

P - number of points, which is

Parameters

points	[out] - Output array of point coords
order	[in] - The lattice has order + 1 points, minus any skipped by offset
offset	[in] - Number of points on boundary to skip coming in from boundary

Definition at line 224 of file Intrepid_PointToolsDef.hpp.

template<class Scalar , class ArrayType >

void Intrepid::PointTools::getEquispacedLatticeTriangle	(	ArrayType &	points,
		const int	order,
		const int	offset = `0`
	)

staticprivate

Computes an equispaced lattice of a given order on the reference triangle. The output array is (P,2), where.

P - number of points, which is

Parameters

points	[out] - Output array of point coords
order	[in] - The lattice has order + 1 points, minus any skipped by offset
offset	[in] - Number of points on boundary to skip coming in from boundary

Definition at line 201 of file Intrepid_PointToolsDef.hpp.

template<class Scalar , class ArrayType >

void Intrepid::PointTools::getGaussPoints	(	ArrayType &	pts,
		const int	order
	)

static

Retrieves the Gauss-Legendre points from PolyLib, but lets us do it in an arbitrary ArrayType.

Parameters

pts	[out] - Output array of point coords (P,)
order	[out] - number of Gauss points - 1

Definition at line 75 of file Intrepid_PointToolsDef.hpp.

References Intrepid::IntrepidPolylib::zwgj().

template<class Scalar , class ArrayType >

void Intrepid::PointTools::getLattice	(	ArrayType &	pts,
		const shards::CellTopology &	cellType,
		const int	order,
		const int	offset = `0`,
		const EPointType	pointType = `POINTTYPE_EQUISPACED`
	)

static

Computes a lattice of points of a given order on a reference simplex (currently disabled for other cell types). The output array is (P,D), where.

P - number of points per cell

D - is the spatial dimension

Parameters

pts	[out] - Output array of point coords
cellType	[in] - type of reference cell (currently only supports the simplex)
order	[in] - number of points per side, plus 1
pointType	[in] - flag for point distribution. Currently equispaced and warp/blend points are supported
offset	[in] - Number of points on boundary to skip

Definition at line 52 of file Intrepid_PointToolsDef.hpp.

static int Intrepid::PointTools::getLatticeSize	(	const shards::CellTopology &	cellType,
		const int	order,
		const int	offset = `0`
	)

inlinestatic

Computes the number of points in a lattice of a given order on a simplex (currently disabled for other cell types). If offset == 0, the lattice will include only include the vertex points if order == 1, and will include edge midpoints if order == 2, and so on. In particular, this is the dimension of polynomials of degree "order" on the given simplex. The offset argument is used to indicate that the layer of points on the boundary is omitted (if offset == 1). For greater offsets, more layers are omitteed.

Parameters

cellType	[in] - type of reference cell (currently only supports the simplex)
order	[in] - order of the lattice
offset	[in] - the number of boundary layers to omit

Definition at line 214 of file Intrepid_PointTools.hpp.

Referenced by Intrepid::Basis_HCURL_TET_In_FEM< Scalar, ArrayScalar >::Basis_HCURL_TET_In_FEM(), Intrepid::Basis_HCURL_TRI_In_FEM< Scalar, ArrayScalar >::Basis_HCURL_TRI_In_FEM(), Intrepid::Basis_HDIV_TET_In_FEM< Scalar, ArrayScalar >::Basis_HDIV_TET_In_FEM(), Intrepid::Basis_HDIV_TRI_In_FEM< Scalar, ArrayScalar >::Basis_HDIV_TRI_In_FEM(), getEquispacedLattice(), getWarpBlendLattice(), Intrepid::Basis_HGRAD_TET_Cn_FEM< Scalar, ArrayScalar >::initializeTags(), Intrepid::Basis_HCURL_TET_In_FEM< Scalar, ArrayScalar >::initializeTags(), and main().

template<class Scalar , class ArrayType >

void Intrepid::PointTools::getWarpBlendLattice	(	const shards::CellTopology &	cellType,
		ArrayType &	points,
		const int	order,
		const int	offset = `0`
	)

staticprivate

Computes a warped lattice (ala Warburton's warp-blend points of a given order on a reference simplex (currently disabled for other cell types). The output array is (P,D), where.

P - number of points per cell

D - is the spatial dimension

Parameters

points	[out] - Output array of point coords
order	[in] - number of points per side, plus 1
offset	[in] - Number of points on boundary to skip
cellType	[in] - type of reference cell (currently only supports the simplex)

Definition at line 134 of file Intrepid_PointToolsDef.hpp.

References getLatticeSize().

template<class Scalar , class ArrayType >

void Intrepid::PointTools::getWarpBlendLatticeLine	(	ArrayType &	points,
		const int	order,
		const int	offset = `0`
	)

staticprivate

Returns the Gauss-Lobatto points of a given order on the reference line [-1,1]. The output array is (P,1), where.

P - number of points

Parameters

points	[out] - Output array of point coords
order	[in] - The lattice has order + 1 points, minus any skipped by offset
offset	[in] - Number of points on boundary to skip coming in per boundary

Definition at line 250 of file Intrepid_PointToolsDef.hpp.

References Intrepid::IntrepidPolylib::zwglj().

template<class Scalar , class ArrayType >

void Intrepid::PointTools::getWarpBlendLatticeTetrahedron	(	ArrayType &	points,
		const int	order,
		const int	offset = `0`
	)

staticprivate

Returns Warburton's warp-blend points of a given order on the reference tetrahedron. The output array is (P,3), where.

P - number of points

Parameters

points	[out] - Output array of point coords
order	[in] - The lattice has order + 1 points, minus any skipped by offset
offset	[in] - Number of points on boundary to skip coming in per boundary

Definition at line 612 of file Intrepid_PointToolsDef.hpp.

References Intrepid::FieldContainer< Scalar, ArrayTypeId >::initialize(), Intrepid::CellTools< Scalar >::mapToReferenceFrame(), and Intrepid::FieldContainer< Scalar, ArrayTypeId >::resize().

template<class Scalar , class ArrayType >

void Intrepid::PointTools::getWarpBlendLatticeTriangle	(	ArrayType &	points,
		const int	order,
		const int	offset = `0`
	)

staticprivate

Returns Warburton's warp-blend points of a given order on the reference triangle. The output array is (P,2), where.

P - number of points

Parameters

points	[out] - Output array of point coords
order	[in] - The lattice has order + 1 points, minus any skipped by offset
offset	[in] - Number of points on boundary to skip coming in per boundary

Definition at line 332 of file Intrepid_PointToolsDef.hpp.

References Intrepid::CellTools< Scalar >::mapToReferenceFrame().

template<class Scalar , class ArrayType >

void Intrepid::PointTools::warpFactor	(	const int	order,
		const ArrayType &	xnodes,
		const ArrayType &	xout,
		ArrayType &	warp
	)

staticprivate

interpolates Warburton's warp function on the line

Parameters

order	[in] - The polynomial order
xnodes	[in] - vector of node locations to interpolate
xout	[in] - warpfunction at xout, +/- 1 roots deflated
warp	[out] - the amount to warp each point

Definition at line 272 of file Intrepid_PointToolsDef.hpp.

References Intrepid::FieldContainer< Scalar, ArrayTypeId >::initialize().

template<class Scalar , class ArrayType >

void Intrepid::PointTools::warpShiftFace3D	(	const int	order,
		const Scalar	pval,
		const ArrayType &	L1,
		const ArrayType &	L2,
		const ArrayType &	L3,
		const ArrayType &	L4,
		ArrayType &	dxy
	)

staticprivate

This is used internally to compute the tetrahedral warp-blend points one each face.

Parameters

order	[in] - the order of lattice
pval	[in] - the "alpha" term in the warping function
L1	[in] - the first barycentric coordinate of the input points
L2	[in] - the second barycentric coordinate of the input points
L3	[in] - the third barycentric coordinate of the input points
L4	[in] - the fourth barycentric coordinate of the input points
dxy	[out] - contains the amount to shift each point in the x and y direction

Definition at line 474 of file Intrepid_PointToolsDef.hpp.

The documentation for this class was generated from the following files:

http://docs.trilinos.org/dev/packages/intrepid/src/Shared/Intrepid_PointTools.hpp
http://docs.trilinos.org/dev/packages/intrepid/src/Shared/Intrepid_PointToolsDef.hpp

Static Public Member Functions

Static Private Member Functions

Detailed Description

Member Function Documentation