triangle_felippa_rule.c

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# include <stdlib.h>
# include <stdio.h>
# include <time.h>
# include <std::string.h>
# include <math.h>
 
# include "triangle_felippa_rule.h"
 
/******************************************************************************/
 
void comp_next ( int n, int k, int a[], int *more, int *h,  int *t )
 
/******************************************************************************/
/*
  Purpose:
 
    COMP_NEXT computes the compositions of the integer N into K parts.
 
  Discussion:
 
    A composition of the integer N into K parts is an ordered sequence
    of K nonnegative integers which sum to N.  The compositions (1,2,1)
    and (1,1,2) are considered to be distinct.
 
    The routine computes one composition on each call until there are no more.
    For instance, one composition of 6 into 3 parts is
    3+2+1, another would be 6+0+0.
 
    On the first call to this routine, set MORE = FALSE.  The routine
    will compute the first element in the sequence of compositions, and
    return it, as well as setting MORE = TRUE.  If more compositions
    are desired, call again, and again.  Each time, the routine will
    return with a new composition.
 
    However, when the LAST composition in the sequence is computed
    and returned, the routine will reset MORE to FALSE, signaling that
    the end of the sequence has been reached.
 
    This routine originally used a STATICE statement to maintain the
    variables H and T.  I have decided (based on an wasting an
    entire morning trying to track down a problem) that it is safer
    to pass these variables as arguments, even though the user should
    never alter them.  This allows this routine to safely shuffle
    between several ongoing calculations.
 
 
    There are 28 compositions of 6 into three parts.  This routine will
    produce those compositions in the following order:
 
     I         A
     -     ---------
     1     6   0   0
     2     5   1   0
     3     4   2   0
     4     3   3   0
     5     2   4   0
     6     1   5   0
     7     0   6   0
     8     5   0   1
     9     4   1   1
    10     3   2   1
    11     2   3   1
    12     1   4   1
    13     0   5   1
    14     4   0   2
    15     3   1   2
    16     2   2   2
    17     1   3   2
    18     0   4   2
    19     3   0   3
    20     2   1   3
    21     1   2   3
    22     0   3   3
    23     2   0   4
    24     1   1   4
    25     0   2   4
    26     1   0   5
    27     0   1   5
    28     0   0   6
 
  Licensing:
 
    This code is distributed under the GNU LGPL license.
 
  Modified:
 
    02 July 2008
 
  Author:
 
    Original FORTRAN77 version by Albert Nijenhuis, Herbert Wilf.
    C version by John Burkardt.
 
  Reference:
 
    Albert Nijenhuis, Herbert Wilf,
    Combinatorial Algorithms for Computers and Calculators,
    Second Edition,
    Academic Press, 1978,
    ISBN: 0-12-519260-6,
    LC: QA164.N54.
 
  Parameters:
 
    Input, int N, the integer whose compositions are desired.
 
    Input, int K, the number of parts in the composition.
 
    Input/output, int A[K], the parts of the composition.
 
    Input/output, int *MORE.
    Set MORE = FALSE on first call.  It will be reset to TRUE on return
    with a new composition.  Each new call returns another composition until
    MORE is set to FALSE when the last composition has been computed
    and returned.
 
    Input/output, int *H, *T, two internal parameters needed for the
    computation.  The user should allocate space for these in the calling
    program, include them in the calling sequence, but never alter them!
*/
{
  int i;
 
  if ( !( *more ) )
  {
    *t = n;
    *h = 0;
    a[0] = n;
    for ( i = 1; i < k; i++ )
    {
       a[i] = 0;
    }
  }
  else
  {
    if ( 1 < *t )
    {
      *h = 0;
    }
    *h = *h + 1;
    *t = a[*h-1];
    a[*h-1] = 0;
    a[0] = *t - 1;
    a[*h] = a[*h] + 1;
  }
 
  *more = ( a[k-1] != n );
 
  return;
}
/******************************************************************************/
 
double *monomial_value ( int m, int n, int e[], double x[] )
 
/******************************************************************************/
/*
  Purpose:
 
    MONOMIAL_VALUE evaluates a monomial.
 
  Discussion:
 
    This routine evaluates a monomial of the form
 
      product ( 1 <= i <= m ) x(i)^e(i)
 
    The combination 0.0^0 is encountered is treated as 1.0.
 
  Licensing:
 
    This code is distributed under the GNU LGPL license.
 
  Modified:
 
    17 August 2014
 
  Author:
 
    John Burkardt
 
  Parameters:
 
    Input, int M, the spatial dimension.
 
    Input, int N, the number of evaluation points.
 
    Input, int E[M], the exponents.
 
    Input, double X[M*N], the point coordinates.
 
    Output, double MONOMIAL_VALUE[N], the monomial values.
*/
{
  int i;
  int j;
  double *v;
 
  v = ( double * ) malloc ( n * sizeof ( double ) );
 
  for ( j = 0; j < n; j++ )
  {
    v[j] = 1.0;
  }
 
  for ( i = 0; i < m; i++ )
  {
    if ( 0 != e[i] )
    {
      for ( j = 0; j < n; j++ )
      {
        v[j] = v[j] * pow ( x[i+j*m], e[i] );
      }
    }
  }
 
  return v;
}
/******************************************************************************/
 
void r8vec_copy ( int n, double a1[], double a2[] )
 
/******************************************************************************/
/*
  Purpose:
 
    R8VEC_COPY copies an R8VEC.
 
  Discussion:
 
    An R8VEC is a vector of R8's.
 
  Licensing:
 
    This code is distributed under the GNU LGPL license.
 
  Modified:
 
    03 July 2005
 
  Author:
 
    John Burkardt
 
  Parameters:
 
    Input, int N, the number of entries in the vectors.
 
    Input, double A1[N], the vector to be copied.
 
    Input, double A2[N], the copy of A1.
*/
{
  int i;
 
  for ( i = 0; i < n; i++ )
  {
    a2[i] = a1[i];
  }
  return;
}
/******************************************************************************/
 
double r8vec_dot_product ( int n, double a1[], double a2[] )
 
/******************************************************************************/
/*
  Purpose:
 
    R8VEC_DOT_PRODUCT computes the dot product of a pair of R8VEC's.
 
  Licensing:
 
    This code is distributed under the GNU LGPL license.
 
  Modified:
 
    26 July 2007
 
  Author:
 
    John Burkardt
 
  Parameters:
 
    Input, int N, the number of entries in the vectors.
 
    Input, double A1[N], A2[N], the two vectors to be considered.
 
    Output, double R8VEC_DOT_PRODUCT, the dot product of the vectors.
*/
{
  int i;
  double value;
 
  value = 0.0;
  for ( i = 0; i < n; i++ )
  {
    value = value + a1[i] * a2[i];
  }
  return value;
}
/******************************************************************************/
 
void subcomp_next ( int n, int k, int a[], int *more, int *h, int *t )
 
/******************************************************************************/
/*
  Purpose:
 
    SUBCOMP_NEXT computes the next subcomposition of N into K parts.
 
  Discussion:
 
    A composition of the integer N into K parts is an ordered sequence
    of K nonnegative integers which sum to a value of N.
 
    A subcomposition of the integer N into K parts is a composition
    of M into K parts, where 0 <= M <= N.
 
  Licensing:
 
    This code is distributed under the GNU LGPL license.
 
  Modified:
 
    27 March 2009
 
  Author:
 
    Original FORTRAN77 version by Albert Nijenhuis, Herbert Wilf.
    C version by John Burkardt.
 
  Reference:
 
    Albert Nijenhuis, Herbert Wilf,
    Combinatorial Algorithms for Computers and Calculators,
    Second Edition,
    Academic Press, 1978,
    ISBN: 0-12-519260-6,
    LC: QA164.N54.
 
  Parameters:
 
    Input, int N, the integer whose subcompositions are desired.
 
    Input, int K, the number of parts in the subcomposition.
 
    Input/output, int A[K], the parts of the subcomposition.
 
    Input/output, int *MORE, set by the user to start the computation,
    and by the routine to terminate it.
 
    Input/output, int *H, *T, two internal parameters needed for the
    computation.  The user should allocate space for these in the calling
    program, include them in the calling sequence, but never alter them!
*/
{
  int i;
  static int more2 = 0;
  static int n2 = 0;
/*
  The first computation.
*/
  if ( !( *more ) )
  {
    n2 = 0;
 
    for ( i = 0; i < k; i++ )
    {
      a[i] = 0;
    }
    more2 = 0;
    *h = 0;
    *t = 0;
 
    *more = 1;
  }
/*
  Do the next element at the current value of N.
*/
  else if ( more2 )
  {
    comp_next ( n2, k, a, &more2, h, t );
  }
  else
  {
    more2 = 0;
    n2 = n2 + 1;
 
    comp_next ( n2, k, a, &more2, h, t );
  }
/*
  Termination occurs if MORE2 = FALSE and N2 = N.
*/
  if ( !more2 && n2 == n )
  {
    *more = 0;
  }
 
  return;
}
/******************************************************************************/
 
//void timestamp ( ) //NOTE: it is defined in gm_rule.c
 
/******************************************************************************/
/*
  Purpose:
 
    TIMESTAMP prints the current YMDHMS date as a time stamp.
 
  Example:
 
    31 May 2001 09:45:54 AM
 
  Licensing:
 
    This code is distributed under the GNU LGPL license.
 
  Modified:
 
    24 September 2003
 
  Author:
 
    John Burkardt
 
  Parameters:
 
    None
*/
/*{
# define TIME_SIZE 40
 
  static char time_buffer[TIME_SIZE];
  const struct tm *tm;
  size_t len;
  time_t now;
 
  now = time ( NULL );
  tm = localtime ( &now );
 
  len = strftime ( time_buffer, TIME_SIZE, "%d %B %Y %I:%M:%S %p", tm );
 
  fprintf ( stdout, "%s\n", time_buffer );
 
  return;
# undef TIME_SIZE
}*/
 
/******************************************************************************/
 
double triangle_unit_monomial ( int expon[2] )
 
/******************************************************************************/
/*
  Purpose:
 
    TRIANGLE_UNIT_MONOMIAL integrates a monomial over the unit triangle.
 
  Discussion:
 
    This routine integrates a monomial of the form
 
      product ( 1 <= dim <= 2 ) x(dim)^expon(dim)
 
    where the exponents are nonnegative integers.  Note that
    if the combination 0^0 is encountered, it should be treated
    as 1.
 
    Integral ( over unit triangle ) x^m y^n dx dy = m! * n! / ( m + n + 2 )!
 
    The integration region is:
 
      0 <= X
      0 <= Y
      X + Y <= 1.
 
  Licensing:
 
    This code is distributed under the GNU LGPL license.
 
  Modified:
 
    18 April 2009
 
  Author:
 
    John Burkardt
 
  Parameters:
 
    Input, int EXPON[2], the exponents.
 
    Output, double TRIANGLE_UNIT_MONOMIAL, the integral of the monomial.
*/
{
  int i;
  int k;
  double value;
/*
  The first computation ends with VALUE = 1.0;
*/
  value = 1.0;
 
/* k = 0;
 
  The first loop simply computes 1 so we short circuit it!
 
 for ( i = 1; i <= expon[0]; i++ )
 {
   k = k + 1;
   value = value * ( double ) ( i ) / ( double ) ( k );
 }*/
 
  k = expon[0];
 
  for ( i = 1; i <= expon[1]; i++ )
  {
    k = k + 1;
    value = value * ( double ) ( i ) / ( double ) ( k );
  }
 
  k = k + 1;
  value = value / ( double ) ( k );
 
  k = k + 1;
  value = value / ( double ) ( k );
 
  return value;
}
/******************************************************************************/
 
void triangle_unit_o01 ( double w[], double xy[] )
 
/******************************************************************************/
/*
  Purpose:
 
    TRIANGLE_UNIT_O01 returns a 1 point quadrature rule for the unit triangle.
 
  Discussion:
 
    This rule is precise for monomials through degree 1.
 
    The integration region is:
 
      0 <= X
      0 <= Y
      X + Y <= 1.
 
  Licensing:
 
    This code is distributed under the GNU LGPL license.
 
  Modified:
 
    16 April 2009
 
  Author:
 
    John Burkardt
 
  Reference:
 
    Carlos Felippa,
    A compendium of FEM integration formulas for symbolic work,
    Engineering Computation,
    Volume 21, Number 8, 2004, pages 867-890.
 
  Parameters:
 
    Output, double W[1], the weights.
 
    Output, double XY[2*1], the abscissas.
*/
{
  int order = 1;
 
  double w_save[1] = {
    1.0 };
  double xy_save[2*1] = {
    0.33333333333333333333,  0.33333333333333333333 };
 
  r8vec_copy ( order, w_save, w );
  r8vec_copy ( 2*order, xy_save, xy );
 
  return;
}
/******************************************************************************/
 
void triangle_unit_o03 ( double w[], double xy[] )
 
/******************************************************************************/
/*
  Purpose:
 
    TRIANGLE_UNIT_O03 returns a 3 point quadrature rule for the unit triangle.
 
  Discussion:
 
    This rule is precise for monomials through degree 2.
 
    The integration region is:
 
      0 <= X
      0 <= Y
      X + Y <= 1.
 
  Licensing:
 
    This code is distributed under the GNU LGPL license.
 
  Modified:
 
    16 April 2009
 
  Author:
 
    John Burkardt
 
  Reference:
 
    Carlos Felippa,
    A compendium of FEM integration formulas for symbolic work,
    Engineering Computation,
    Volume 21, Number 8, 2004, pages 867-890.
 
  Parameters:
 
    Output, double W[3], the weights.
 
    Output, double XY[2*3], the abscissas.
*/
{
  int order = 3;
 
  double w_save[3] = {
    0.33333333333333333333,
    0.33333333333333333333,
    0.33333333333333333333 };
  double xy_save[2*3] = {
    0.66666666666666666667,  0.16666666666666666667,
    0.16666666666666666667,  0.66666666666666666667,
    0.16666666666666666667,  0.16666666666666666667 };
 
  r8vec_copy ( order, w_save, w );
  r8vec_copy ( 2*order, xy_save, xy );
 
  return;
}
/******************************************************************************/
 
void triangle_unit_o03b ( double w[], double xy[] )
 
/******************************************************************************/
/*
  Purpose:
 
    TRIANGLE_UNIT_O03B returns a 3 point quadrature rule for the unit triangle.
 
  Discussion:
 
    This rule is precise for monomials through degree 2.
 
    The integration region is:
 
      0 <= X
      0 <= Y
      X + Y <= 1.
 
  Licensing:
 
    This code is distributed under the GNU LGPL license.
 
  Modified:
 
    16 April 2009
 
  Author:
 
    John Burkardt
 
  Reference:
 
    Carlos Felippa,
    A compendium of FEM integration formulas for symbolic work,
    Engineering Computation,
    Volume 21, Number 8, 2004, pages 867-890.
 
  Parameters:
 
    Output, double W[3], the weights.
 
    Output, double XY[2*3], the abscissas.
*/
{
  int order = 3;
 
  double w_save[3] = {
    0.33333333333333333333,
    0.33333333333333333333,
    0.33333333333333333333 };
  double xy_save[2*3] = {
    0.0,  0.5,
    0.5,  0.0,
    0.5,  0.5 };
 
  r8vec_copy ( order, w_save, w );
  r8vec_copy ( 2*order, xy_save, xy );
 
  return;
}
/******************************************************************************/
 
void triangle_unit_o06 ( double w[], double xy[] )
 
/******************************************************************************/
/*
  Purpose:
 
    TRIANGLE_UNIT_O06 returns a 6 point quadrature rule for the unit triangle.
 
  Discussion:
 
    This rule is precise for monomials through degree 4.
 
    The integration region is:
 
      0 <= X
      0 <= Y
      X + Y <= 1.
 
  Licensing:
 
    This code is distributed under the GNU LGPL license.
 
  Modified:
 
    16 April 2009
 
  Author:
 
    John Burkardt
 
  Reference:
 
    Carlos Felippa,
    A compendium of FEM integration formulas for symbolic work,
    Engineering Computation,
    Volume 21, Number 8, 2004, pages 867-890.
 
  Parameters:
 
    Output, double W[6], the weights.
 
    Output, double XY[2*6], the abscissas.
*/
{
  int order = 6;
 
  double w_save[6] = {
    0.22338158967801146570,
    0.22338158967801146570,
    0.22338158967801146570,
    0.10995174365532186764,
    0.10995174365532186764,
    0.10995174365532186764 };
  double xy_save[2*6] = {
    0.10810301816807022736,  0.44594849091596488632,
    0.44594849091596488632,  0.10810301816807022736,
    0.44594849091596488632,  0.44594849091596488632,
    0.81684757298045851308,  0.091576213509770743460,
    0.091576213509770743460,  0.81684757298045851308,
    0.091576213509770743460,  0.091576213509770743460 };
 
  r8vec_copy ( order, w_save, w );
  r8vec_copy ( 2*order, xy_save, xy );
 
  return;
}
/******************************************************************************/
 
void triangle_unit_o06b ( double w[], double xy[] )
 
/******************************************************************************/
/*
  Purpose:
 
    TRIANGLE_UNIT_O06B returns a 6 point quadrature rule for the unit triangle.
 
  Discussion:
 
    This rule is precise for monomials through degree 3.
 
    The integration region is:
 
      0 <= X
      0 <= Y
      X + Y <= 1.
 
  Licensing:
 
    This code is distributed under the GNU LGPL license.
 
  Modified:
 
    16 April 2009
 
  Author:
 
    John Burkardt
 
  Reference:
 
    Carlos Felippa,
    A compendium of FEM integration formulas for symbolic work,
    Engineering Computation,
    Volume 21, Number 8, 2004, pages 867-890.
 
  Parameters:
 
    Output, double W[6], the weights.
 
    Output, double XY[2*6], the abscissas.
*/
{
  int order = 6;
 
  double w_save[6] = {
    0.30000000000000000000,
    0.30000000000000000000,
    0.30000000000000000000,
    0.033333333333333333333,
    0.033333333333333333333,
    0.033333333333333333333 };
  double xy_save[2*6] = {
    0.66666666666666666667,  0.16666666666666666667,
    0.16666666666666666667,  0.66666666666666666667,
    0.16666666666666666667,  0.16666666666666666667,
    0.0,  0.5,
    0.5,  0.0,
    0.5,  0.5 };
 
  r8vec_copy ( order, w_save, w );
  r8vec_copy ( 2*order, xy_save, xy );
 
  return;
}
/******************************************************************************/
 
void triangle_unit_o07 ( double w[], double xy[] )
 
/******************************************************************************/
/*
  Purpose:
 
    TRIANGLE_UNIT_O07 returns a 7 point quadrature rule for the unit triangle.
 
  Discussion:
 
    This rule is precise for monomials through degree 5.
 
    The integration region is:
 
      0 <= X
      0 <= Y
      X + Y <= 1.
 
  Licensing:
 
    This code is distributed under the GNU LGPL license.
 
  Modified:
 
    16 April 2009
 
  Author:
 
    John Burkardt
 
  Reference:
 
    Carlos Felippa,
    A compendium of FEM integration formulas for symbolic work,
    Engineering Computation,
    Volume 21, Number 8, 2004, pages 867-890.
 
  Parameters:
 
    Output, double W[7], the weights.
 
    Output, double XY[2*7], the abscissas.
*/
{
  int order = 7;
 
  double w_save[7] = {
    0.12593918054482715260,
    0.12593918054482715260,
    0.12593918054482715260,
    0.13239415278850618074,
    0.13239415278850618074,
    0.13239415278850618074,
    0.22500000000000000000 };
  double xy_save[2*7] = {
    0.79742698535308732240,  0.10128650732345633880,
    0.10128650732345633880,  0.79742698535308732240,
    0.10128650732345633880,  0.10128650732345633880,
    0.059715871789769820459,  0.47014206410511508977,
    0.47014206410511508977,  0.059715871789769820459,
    0.47014206410511508977,  0.47014206410511508977,
    0.33333333333333333333,  0.33333333333333333333 };
 
  r8vec_copy ( order, w_save, w );
  r8vec_copy ( 2*order, xy_save, xy );
 
  return;
}
/******************************************************************************/
 
void triangle_unit_o12 ( double w[], double xy[] )
 
/******************************************************************************/
/*
  Purpose:
 
    TRIANGLE_UNIT_O12 returns a 12 point quadrature rule for the unit triangle.
 
  Discussion:
 
    This rule is precise for monomials through degree 6.
 
    The integration region is:
 
      0 <= X
      0 <= Y
      X + Y <= 1.
 
  Licensing:
 
    This code is distributed under the GNU LGPL license.
 
  Modified:
 
    19 April 2009
 
  Author:
 
    John Burkardt
 
  Reference:
 
    Carlos Felippa,
    A compendium of FEM integration formulas for symbolic work,
    Engineering Computation,
    Volume 21, Number 8, 2004, pages 867-890.
 
  Parameters:
 
    Output, double W[12], the weights.
 
    Output, double XY[2*12], the abscissas.
*/
{
  int order = 12;
 
  double w_save[12] = {
     0.050844906370206816921,
     0.050844906370206816921,
     0.050844906370206816921,
     0.11678627572637936603,
     0.11678627572637936603,
     0.11678627572637936603,
     0.082851075618373575194,
     0.082851075618373575194,
     0.082851075618373575194,
     0.082851075618373575194,
     0.082851075618373575194,
     0.082851075618373575194 };
  double xy_save[2*12] = {
    0.87382197101699554332,  0.063089014491502228340,
    0.063089014491502228340,  0.87382197101699554332,
    0.063089014491502228340,  0.063089014491502228340,
    0.50142650965817915742,  0.24928674517091042129,
    0.24928674517091042129,  0.50142650965817915742,
    0.24928674517091042129,  0.24928674517091042129,
    0.053145049844816947353,  0.31035245103378440542,
    0.31035245103378440542,  0.053145049844816947353,
    0.053145049844816947353,  0.63650249912139864723,
    0.31035245103378440542,  0.63650249912139864723,
    0.63650249912139864723,  0.053145049844816947353,
    0.63650249912139864723,  0.31035245103378440542 };
 
  r8vec_copy ( order, w_save, w );
  r8vec_copy ( 2*order, xy_save, xy );
 
  return;
}
/******************************************************************************/
 
double triangle_unit_volume ( )
 
/******************************************************************************/
/*
  Purpose:
 
    TRIANGLE_UNIT_VOLUME returns the "volume" of the unit triangle in 2D.
 
  Discussion:
 
    The "volume" of a triangle is usually called its area.
 
    The integration region is:
 
      0 <= X,
      0 <= Y,
      X + Y <= 1.
 
  Licensing:
 
    This code is distributed under the GNU LGPL license.
 
  Modified:
 
    13 March 2008
 
  Author:
 
    John Burkardt
 
  Parameters:
 
    Output, double TRIANGLE_UNIT_VOLUME, the volume.
*/
{
  double volume;
 
  volume = 1.0 / 2.0;
 
  return volume;
}