qfel.c

The Free Finite Element Package is a library which contains numerical methods required when working

/*--------------------------------------------------------------------------

Free Finite Element Package                             
Copyright (c) 2002-2006 by Joerg Frochte
All rights reserved.

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"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
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A PARTICULAR PURPOSE ARE DISCLAIMED.  IN NO EVENT SHALL THE COPYRIGHT OWNER OR
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NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
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----------------------------------------------------------------------------*/

#include "qfel.h"

/** 
 * \file 
 * \brief Implementation of finite elements with quadratic base functions including FEM-Solvers like Multigrid Methods
 * \author Joerg Frochte 
*/

static inline MEML_FLOAT Q1 (MEML_FLOAT x, MEML_FLOAT y)
{
  return (1 - x - y);
}
static inline double Q2 (MEML_FLOAT x, MEML_FLOAT y)
{
  y = y;
  return (x);
}

static inline double Q3 (MEML_FLOAT x, MEML_FLOAT y)
{
  x = x;
  return (y);
}

/*
static inline MEML_FLOAT Q1(MEML_FLOAT x, MEML_FLOAT y)
{
  return( (1-x-y)*(1-2*x-2*y) );
}
static inline double Q2(MEML_FLOAT x,MEML_FLOAT y)
{
  return( x*(2*x-1) );
}
static inline double Q3(MEML_FLOAT x,MEML_FLOAT y)
{
  return( y*(2*y-1) );
}
*/

static inline MEML_FLOAT Q4 (MEML_FLOAT x, MEML_FLOAT y)
{
  return ((1 - x - y) * x * 4);
}
static inline double Q5 (MEML_FLOAT x, MEML_FLOAT y)
{
  return (4 * x * y);
}
static inline double Q6 (MEML_FLOAT x, MEML_FLOAT y)
{
  return ((1 - x - y) * y * 4);
}

static void qfel_22mul (MEML_FLOAT A[2][2], MEML_FLOAT v[2], MEML_FLOAT u[2])
{
  u[0] = A[0][0] * v[0] + A[0][1] * v[1];
  u[1] = A[1][0] * v[0] + A[1][1] * v[1];
}

static inline void qfel_invtrans_22_Matrix (MEML_FLOAT A[2][2],
					    MEML_FLOAT I[2][2])
{
  I[0][0] = A[1][1];
  I[1][1] = A[0][0];

  I[0][1] = -A[1][0];
  I[1][0] = -A[0][1];
}


/**
 * \~german 
 *  \brief Assembliert die Massen-Matrix/den Reaktionsanteil
 *   fuer das Galerkinverfahren
 *
 *   Die Matrix A muss vor dem Aufruf dieser Funktion erzeugt worden sein.
 *   Bei groesseren Gitter sollte A eine Sparse-Matrix-Typ sein.
 *  
 *  \param r Reaktionsparameter der partiellen Differentialgleichung
 *  \param A die Matrix zu der die Steifigkeitsmatrix addiert werden soll
 */

int qfel_assemb_reaction (const MEML_FLOAT r, ME_MATRIX * A, MESH * mesh)
{
  INDEXARRAY *triangles = mesh->triangles;
  VECTOR *points = mesh->points;
  TRANSFORMATION *transformation = mesh->transformation;

  /* reine quadratische * 1/360 ist noch noetig */
  /*
     static const MEML_FLOAT S4[6][6]= 
     {
     { 6 ,-1 ,-1 ,  0 ,-4  , 0}, 
     {-1 , 6 ,-1 ,  0 , 0  ,-4}, 
     {-1 ,-1 , 6 , -4 , 0  , 0}, 
     { 0 , 0 ,-4 , 32 ,16  ,16}, 
     {-4 , 0 , 0 , 16 ,32  ,16}, 
     { 0 ,-4 , 0 , 16 ,16  ,32}
     };
   */
  /* hierachische Basis */
  static const MEML_FLOAT S4[6][6] = {
    {2 / 24.0, 1 / 24.0, 1 / 24.0, 1.0 / 15, 1.0 / 30, 1.0 / 15},
    {1 / 24.0, 2 / 24.0, 1 / 24.0, 1.0 / 15, 1.0 / 15, 1.0 / 30},
    {1 / 24.0, 1 / 24.0, 2 / 24.0, 1.0 / 30, 1.0 / 15, 1.0 / 15},
    {1.0 / 15, 1.0 / 15, 1.0 / 30, 4.0 / 45, 2.0 / 45, 2.0 / 45},
    {1.0 / 30, 1.0 / 15, 1.0 / 15, 2.0 / 45, 4.0 / 45, 2.0 / 45},
    {1.0 / 15, 1.0 / 30, 1.0 / 15, 2.0 / 45, 2.0 / 45, 4.0 / 45}
  };

  MEML_INT i, row, col;
  MEML_INT x, y;
  MEML_FLOAT det, gamma, temp;
  TRIANGLE the_triangle;
  MEML_INT number_of_triangles = (triangles->dim) / 3;
  MEML_INT *triangle = triangles->data;
  M_TYPE type;
  MEML_INT knoten[6];

  meml_matrix_property_get (A, &col, &row, &type);

  if ((row < points->dim / 2) || (col < points->dim / 2))
    {
      fprintf (stderr, "The size of the matrix is not big enough "
	       "so store the data!\n");
      return (EXIT_FAILURE);
    }

  for (i = 0; i < number_of_triangles; i++)
    {

      the_triangle = smfl_get_triangle (i, triangle);

      det = fabs (transformation[i].det);

      gamma = det * r;

      /* Warnung. MatLab nutzt 1,1 als obersten Eintrag
         C hingegen 0,0. Das ist ein gut moeglichkeit
         speicherzugriffsfehler zu produzieren */

      for (x = 0; x < 3; x++)
	knoten[x] = the_triangle.p[x] - 1;

      for (x = 0; x < 3; x++)
	knoten[x + 3] = mesh->triangles_adof[i][x] + mesh->number_of_points;

      for (x = 0; x < 6; x++)
	for (y = 0; y < 6; y++)
	  {
	    temp = gamma * S4[x][y];
	    meml_matrix_element_add (A, knoten[x], knoten[y], temp);
	  }
    }

  return (EXIT_SUCCESS);
}


int qfel_assemb_diffusion (const MEML_FLOAT eps, ME_MATRIX * A, MESH * mesh)
{
  INDEXARRAY *triangles = mesh->triangles;
  VECTOR *points = mesh->points;
  TRANSFORMATION *transformation = mesh->transformation;

  /* reine quadratische * 1/6 ist noch noetig */
  /*
     static const MEML_FLOAT S1[6][6]= 
     {
     { 3 , 1 , 0 , -4 , 0  , 0}, 
     { 1 , 3 , 0 , -4 , 0  , 0}, 
     { 0 , 0 , 0 ,  0 , 0  , 0}, 
     {-4 ,-4 , 0 ,  8 , 0  , 0}, 
     { 0 , 0 , 0 ,  0 , 8  ,-8}, 
     { 0 , 0 , 0 ,  0 ,-8  , 8}
     }; 
     static const MEML_FLOAT S2[6][6] = 
     { 
     { 6 , 1 , 1 , -4 , 0  ,-4}, 
     { 1 , 0 ,-1 , -4 , 4  , 0}, 
     { 1 ,-1 , 0 ,  0 , 4  ,-4}, 
     {-4 ,-4 , 0 ,  8 ,-8  , 8}, 
     { 0 , 4 , 4 , -8 , 8  ,-8}, 
     {-4 , 0 ,-4 ,  8 ,-8  , 8}
     };
     static const MEML_FLOAT S3[6][6] = 
     {
     { 3 , 0 , 1 ,  0 , 0  ,-4}, 
     { 0 , 0 , 0 ,  0 , 0  , 0}, 
     { 1 , 0 , 3 ,  0 , 0  ,-4}, 
     { 0 , 0 , 0 ,  8 ,-8  , 8}, 
     { 0 , 0 , 0 , -8 , 8  , 0}, 
     {-4 , 0 ,-4 ,  0 , 0  , 8}
     };
   */
  /* hirachsiche Basis */

  static const MEML_FLOAT S1[6][6] = {
    {0.5, -0.5, 0, 0, -2 / 3.0, 2 / 3.0},
    {-0.5, 0.5, 0, 0, 2 / 3.0, -2 / 3.0},
    {0, 0, 0, 0, 0, 0},
    {0, 0, 0, 4 / 3.0, 0, 0},
    {-2 / 3.0, 2 / 3.0, 0, 0, 4 / 3.0, -4 / 3.0},
    {2 / 3.0, -2 / 3.0, 0, 0, -4 / 3.0, 4 / 3.0}
  };
  static const MEML_FLOAT S2[6][6] = {
    {1, -0.5, -0.5, 2 / 3.0, -4 / 3.0, 2 / 3.0},
    {-0.5, 0, 0.5, -2 / 3.0, 2 / 3.0, 0},
    {-0.5, 0.5, 0, 0, 2 / 3.0, -2 / 3.0},
    {2 / 3.0, -2 / 3.0, 0, 4 / 3.0, -4 / 3.0, 4 / 3.0},
    {-4 / 3.0, 2 / 3.0, 2 / 3.0, -4 / 3.0, 4 / 3.0, -4 / 3.0},
    {2 / 3.0, 0, -2 / 3.0, 4 / 3.0, -4 / 3.0, 4 / 3.0}
  };
  static const MEML_FLOAT S3[6][6] = {
    {0.5, 0, -0.5, 2 / 3.0, -2 / 3.0, 0},
    {0, 0, 0, 0, 0, 0},
    {-0.5, 0, 0.5, -2 / 3.0, 2 / 3.0, 0},
    {2 / 3.0, 0, -2 / 3.0, 4 / 3.0, -4 / 3.0, 0},
    {-2 / 3.0, 0, 2 / 3.0, -4 / 3.0, 4 / 3.0, 0},
    {0, 0, 0, 0, 0, 4 / 3.0}
  };

  MEML_INT i, row, col;
  int x, y;
  MEML_FLOAT det, gamma1, gamma2, gamma3, temp;
  TRIANGLE the_triangle;
  MEML_INT number_of_triangles = (triangles->dim) / 3;
  MEML_INT *triangle = triangles->data;
  M_TYPE type;
  MEML_INT knoten[6];

  meml_matrix_property_get (A, &col, &row, &type);

  if ((row < (points->dim + mesh->additional_degrees_of_freedom->dim) / 2)
      || (col < (points->dim + mesh->additional_degrees_of_freedom->dim) / 2))
    {
      fprintf (stderr, "qfel_assemb_diffusion : "
	       "The size of the matrix is not big enough "
	       "so store the data!\n");
      return (EXIT_FAILURE);
    }

  for (i = 0; i < number_of_triangles; i++)
    {
      the_triangle = smfl_get_triangle (i, triangle);

      det = fabs (transformation[i].det);

      gamma1 = (transformation[i].B[0][1] * transformation[i].B[0][1]
		+
		transformation[i].B[1][1] * transformation[i].B[1][1]) / det;

      gamma2 = -(transformation[i].B[0][0] * transformation[i].B[0][1]
		 +
		 transformation[i].B[1][0] * transformation[i].B[1][1]) / det;

      gamma3 = (transformation[i].B[1][0] * transformation[i].B[1][0] +
		transformation[i].B[0][0] * transformation[i].B[0][0]) / det;

      /* Warnung. MatLab nutzt 1,1 als obersten Eintrag
         C hingegen 0,0. Das ist ein gut moeglichkeit
         speicherzugriffsfehler zu produzieren */

      for (x = 0; x < 3; x++)
	knoten[x] = the_triangle.p[x] - 1;

      for (x = 0; x < 3; x++)
	{
	  knoten[x + 3] = mesh->triangles_adof[i][x] + mesh->number_of_points;
	}

      for (x = 0; x < 6; x++)
	for (y = 0; y < 6; y++)
	  {
	    temp =
	      eps * (gamma1 * S1[x][y] + gamma2 * S2[x][y] +
		     gamma3 * S3[x][y]);
	    meml_matrix_element_add_f (A, knoten[x], knoten[y], temp);
	  }
    }

  return (EXIT_SUCCESS);
}


/**
 * \~german 
 * \brief Assembliert die rechte Seite des LGS fuer das Galerkinverfahren
 */
VECTOR *qfel_assemb_load_vector (xyt_function f, const double time,
				 MESH * mesh)
{
  INDEXARRAY *triangles = mesh->triangles;
  VECTOR *points = mesh->points;
  TRANSFORMATION *transformation = mesh->transformation;

  const MEML_FLOAT gaus_x[7] = { 0, 1, 0, 0.5, 0.5, 0, 1.0 / 3.0 };
  const MEML_FLOAT gaus_y[7] = { 0, 0, 1, 0, 0.5, 0.5, 1.0 / 3.0 };
  const MEML_FLOAT gaus_w[7] = { 1.0 / 40.0, 1.0 / 40.0, 1.0 / 40.0,
    1.0 / 15.0, 1.0 / 15.0, 1.0 / 15.0,
    27.0 / 120.0
  };

  const MEML_FLOAT phi[6][7] = {
    {Q1 (gaus_x[0], gaus_y[0]), Q1 (gaus_x[1], gaus_y[1]),
     Q1 (gaus_x[2], gaus_y[2]), Q1 (gaus_x[3], gaus_y[3]),
     Q1 (gaus_x[4], gaus_y[4]), Q1 (gaus_x[5], gaus_y[5]),
     Q1 (gaus_x[6], gaus_y[6])},
    {Q2 (gaus_x[0], gaus_y[0]), Q2 (gaus_x[1], gaus_y[1]),
     Q2 (gaus_x[2], gaus_y[2]), Q2 (gaus_x[3], gaus_y[3]),
     Q2 (gaus_x[4], gaus_y[4]), Q2 (gaus_x[5], gaus_y[5]),
     Q2 (gaus_x[6], gaus_y[6])},
    {Q3 (gaus_x[0], gaus_y[0]), Q3 (gaus_x[1], gaus_y[1]),
     Q3 (gaus_x[2], gaus_y[2]), Q3 (gaus_x[3], gaus_y[3]),
     Q3 (gaus_x[4], gaus_y[4]), Q3 (gaus_x[5], gaus_y[5]),
     Q3 (gaus_x[6], gaus_y[6])},
    {Q4 (gaus_x[0], gaus_y[0]), Q4 (gaus_x[1], gaus_y[1]),
     Q4 (gaus_x[2], gaus_y[2]), Q4 (gaus_x[3], gaus_y[3]),
     Q4 (gaus_x[4], gaus_y[4]), Q4 (gaus_x[5], gaus_y[5]),
     Q4 (gaus_x[6], gaus_y[6])},
    {Q5 (gaus_x[0], gaus_y[0]), Q5 (gaus_x[1], gaus_y[1]),
     Q5 (gaus_x[2], gaus_y[2]), Q5 (gaus_x[3], gaus_y[3]),
     Q5 (gaus_x[4], gaus_y[4]), Q5 (gaus_x[5], gaus_y[5]),
     Q5 (gaus_x[6], gaus_y[6])},
    {Q6 (gaus_x[0], gaus_y[0]), Q6 (gaus_x[1], gaus_y[1]),
     Q6 (gaus_x[2], gaus_y[2]), Q6 (gaus_x[3], gaus_y[3]),
     Q6 (gaus_x[4], gaus_y[4]), Q6 (gaus_x[5], gaus_y[5]),
     Q6 (gaus_x[6], gaus_y[6])}
  };

  MEML_INT i, x, y;
  MEML_FLOAT temp, det;
  TRIANGLE the_triangle;
  MEML_FLOAT I_temp[6];
  MEML_INT number_of_points = points->dim / 2;
  MEML_INT number_of_triangles = (triangles->dim) / 3;
  MEML_INT *triangle = triangles->data;
  MEML_FLOAT *b;
  VECTOR *temp_vector;
  MEML_FLOAT f_value[7];
  MEML_FLOAT d[2];
  MEML_INT knoten[6];

  temp_vector =
    meml_vector_new (mesh->number_of_adof + mesh->number_of_points);

  b = temp_vector->data;

  for (i = 0; i < number_of_points; i++)
    b[i] = 0;

  for (i = 0; i < number_of_triangles; i++)
    {
      the_triangle = smfl_get_triangle (i, triangle);

      for (x = 0; x < 3; x++)
	knoten[x] = the_triangle.p[x] - 1;

      for (x = 0; x < 3; x++)
	knoten[x + 3] = mesh->triangles_adof[i][x] + mesh->number_of_points;

      for (x = 0; x < 7; x++)
	{
	  d[0] = points->data[(the_triangle.p[0] - 1) * 2];
	  d[1] = points->data[(the_triangle.p[0] - 1) * 2 + 1];

	  f_value[x] = (*f) (transformation[i].B[0][0] * gaus_x[x] +
			     transformation[i].B[0][1] * gaus_y[x] + d[0],
			     transformation[i].B[1][0] * gaus_x[x] +
			     transformation[i].B[1][1] * gaus_y[x] + d[1],
			     time);
	}

      for (x = 0; x < 6; x++)
	I_temp[x] = 0;

      for (x = 0; x < 7; x++)
	{
	  temp = gaus_w[x] * f_value[x];
	  for (y = 0; y < 6; y++)
	    I_temp[y] += temp * phi[y][x];
	}


      det = fabs (transformation[i].det);

      for (x = 0; x < 6; x++)
	{
	  b[knoten[x]] += I_temp[x] * det;
	}
    }

  return (temp_vector);
}


VECTOR *qfel_assemb_right_side (const VECTOR * f, MESH * mesh)
{
  INDEXARRAY *triangles = mesh->triangles;
  TRANSFORMATION *transformation = mesh->transformation;

  /* hierachische Basis */