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geom::SimplexJac

geom::SimplexJac< N, T > Class Template Reference

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

#include <SimplexJac.h>

List of all members.

Public Types

typedef T Number
 The number type.
typedef ads::FixedArray< N,
Number > 		Vertex
 The class for a vertex.
typedef 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 geom::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 geom::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 geom::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:
Generated on Fri Aug 24 12:56:01 2007 for Computational Geometry Package by  doxygen 1.4.7