SimplexJac< N, T > Class Template Reference

Implements operations for the Jacobian matrix of a simplex. More...

#include <SimplexJac.h>

Public Types
typedef T	Number
	The number type.
typedef ads::FixedArray< N, Number >	Vertex
	The class for a vertex.
typedef geom::Simplex< N, Vertex, Number >	Simplex
	The simplex type.
typedef ads::SquareMatrix< N, Number >	Matrix
	An NxN matrix.
Public Member Functions
Constructors etc.

	SimplexJac ()
	Default constructor. Un-initialized memory.
	SimplexJac (const SimplexJac &other)
	Copy constructor.
	SimplexJac (const Simplex &s)
	Construct from a simplex.
SimplexJac &	operator= (const SimplexJac &other)
	Assignment operator.
	~SimplexJac ()
	Trivial destructor.
Accessors

const Matrix &	getMatrix () const
	Return a const reference to the Jacobian matrix.
const ads::FixedArray< N, Matrix > &	getGradientMatrix () const
	Return a const reference to the gradient of the Jacobian matrix.
Number	getDeterminant () const
	Return the determinant of the Jacobian matrix.
const Vertex &	getGradientDeterminant () const
	Return a const reference to the gradient of the determinant of the Jacobian matrix.
Number	computeContent () const
	Return the content (hypervolume) of the simplex.
void	computeGradientContent (Vertex *grad) const
	Calculate the gradient of the content (hypervolume) of the simplex.
Vertex	computeGradientContent () const
	Return the gradient of the content (hypervolume) of the simplex.
Manipulators

void	setFunction (const Simplex &s)
	Set the vertices. Calculate the Jacobian matrix and determinant.
void	set (const Simplex &s)
	Set the vertices. Calculate the Jacobian matrix and the determinant and its gradient.
void	setFunction (const geom::Simplex< N, ads::FixedArray< N+1, Number >, Number > &s)
	Set the vertices. Calculate the Jacobian matrix and determinant.
void	set (const geom::Simplex< N, ads::FixedArray< N+1, Number >, Number > &s)
	Set the vertices. Calculate the Jacobian matrix and the determinant and its gradient.

Detailed Description

template<int N, typename T = double>
class SimplexJac< N, T >

Implements operations for the Jacobian matrix of a simplex.

Parameters:

	N	is the dimension.
	T	is the number type. By default it is double.

The Jacobian Matrix

Consider a simplex $T$ with vertices $\{ \mathbf{x}_0, \ldots \mathbf{x}_{N-1} \}$ . We call this the physical simplex. The identity simplex $T_I$ may be mapped to the physical simplex $T$ by an affine transformation and a translation.

$T = S T_I + \mathbf{x}_0$

$S$ is the Jacobian matrix of the transformation.

In 2-D, the identity triangle has vertices: $(0,0)$ , $(1,0)$ and $(1/2,\sqrt{3}/2)$ . In 3-D, the identity tetrahedron has vertices: $(0,0,0)$ , $(1,0,0)$ , $(1/2,\sqrt{3}/2,0)$ and $(1/2,\sqrt{3}/6,\sqrt{2} / \sqrt{3})$ .

The logical simplex, $T_L$ is the simplex whose vertices are the origin and unit displacements in each coordinate direction. In 2-D, the logical triangle has vertices: $(0,0)$ , $(1,0)$ and $(0,1)$ . In 3-D, the logical tetrahedron has vertices: $(0,0,0)$ , $(1,0,0)$ , $(0,1,0)$ , $(0,0,1)$ . It is easy to map the logical simplex to the physical simplex.

$T = A T_L + \mathbf{x}_0$

The columns of $A$ are the displacements of the vertices from the first vertex $\mathbf{x}_0$ . In 3-D, this is

$A = \left( \begin{array}{ccc} x_1 - x_0 & x_2 - x_0 & x_3 - x_0 \\ y_1 - y_0 & y_2 - y_0 & y_3 - y_0 \\ z_1 - z_0 & z_2 - z_0 & z_3 - z_0 \end{array} \right).$

It is also easy to map the logical simplex to the identity simplex.

$T_I = W T_L$

The columns of $W$ are the vertices of $T_I$ . In 3-D, this is

$W = \left( \begin{array}{ccc} 1 & 1/2 & 1/2 \\ 0 & \sqrt{3} / 2 & \sqrt{3} / 6 \\ 0 & 0 & \sqrt{2} / \sqrt{3} \end{array} \right).$

By combining the former transformation with the inverse of the latter, we can map the identity simplex to the physical simplex.

$T = A W^{-1} T_I + \mathbf{x}_0$

The Jacobian matrix of the transformation is $S = A W^{-1}$ .

Usage

Construct a SimplexJac with the default constructor or from a SimplexJac::Simplex.

  typedef geom::SimplexJac<3> TetJac;
  typedef TetJac::Vertex Vertex;
  typedef TetJac::Simplex Tetrahedron;
  // Default constructor.
  TetJac tet;

  // The identity tetrahedron.
  Tetrahedron t(Vertex(0, 0, 0),
                Vertex(1, 0, 0),
                Vertex(1./2, std::sqrt(3.)/2, 0),
                Vertex(1./2, std::sqrt(3.)/6, std::sqrt(2./3.)));
  TetJac tet(t);

The Simplex constructor calls set() to enable evaluation of the determinant, the content and their gradients.

To evaluate the determinant or the content of the simplex, first call setFunction() to set the Jacobian matrix and then use the getDeterminant() and computeContent() member functions.

  tet.setFunction(t);
  std::cout << "Identity tetrahedron:\n"
            << "determinant = " << tet.getDeterminant()
            << "\nvolume = " << tet.computeContent()
            << '\n';

To evaluate the determinant and content and/or their gradients, first call set() to set the Jacobian matrix and its gradient. Then use the member functions to access the appropriate qantities.

  tet.set(t);
  std::cout << "Identity tetrahedron:\n"
            << "determinant = " << tet.getDeterminant()
            << "\ngrad determinant = " << tet.getGradientDeterminant()
            << "\nvolume = " << tet.computeContent()
            << "\ngrad volume = " << tet.computeGradientContent() 
            << '\n';

Member Function Documentation

template<int N, typename T = double>

void SimplexJac< N, T >::set ( const geom::Simplex< N, ads::FixedArray< N+1, Number >, Number > & s )

Set the vertices. Calculate the Jacobian matrix and the determinant and its gradient.

This first projects the simplex to N-D and then call the above set().

template<int N, typename T = double>

void SimplexJac< N, T >::setFunction ( const geom::Simplex< N, ads::FixedArray< N+1, Number >, Number > & s )

Set the vertices. Calculate the Jacobian matrix and determinant.

This first projects the simplex to N-D and then call the above setFunction().

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

SimplexJac.h

SimplexJac< N, T > Class Template Reference

Public Types

Public Member Functions

Detailed Description

template<int N, typename T = double> class SimplexJac< N, T >

Member Function Documentation

template<int N, typename T = double>
class SimplexJac< N, T >