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V_PyramidMetric.cpp
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/*=========================================================================
Module: V_PyramidMetric.cpp
Copyright 2006 National Technology & Engineering Solutions of Sandia,
LLC (NTESS). Under the terms of Contract DE-NA0003525 with NTESS,
the U.S. Government retains certain rights in this software.
See LICENSE for details.
=========================================================================*/
/*
*
* PyramidMetrics.cpp contains quality calculations for Pyramids
*
* This file is part of VERDICT
*
*/
#include "verdict.h"
#include "VerdictVector.hpp"
#include "verdict_defines.hpp"
#include <memory.h>
#include <vector>
#include <algorithm>
static const double SQRT2_HALVES = sqrt(2.0)/2.0;
extern double quad_equiangle_skew( int num_nodes, double coordinates[][3] );
extern double tri_equiangle_skew( int num_nodes, double coordinates[][3] );
namespace VERDICT_NAMESPACE
{
// local methods
void make_pyramid_tets(double coordinates[][3], double tet1_coords[][3], double tet2_coords[][3],
double tet3_coords[][3], double tet4_coords[][3]);
void make_pyramid_faces(double coordinates[][3], double base[][3], double tri1[][3],
double tri2[][3], double tri3[][3], double tri4[][3]);
void make_pyramid_edges( VerdictVector edges[8], double coordinates[][3] );
double distance_point_to_pyramid_base( int num_nodes, double coordinates[][3], double &cos_angle);
double largest_pyramid_edge( double coordinates[][3] );
/*
the pyramid element
5
^
|\
/| \\_
| \ \
| | \_ \_
/ \/4\ \
| /| \ \_
| / \ \ \
/ | \
1 \_ | _/3
\_ \ _/
\_ | _/
\_/
2
a quadrilateral base and a pointy peak like a pyramid
*/
double pyramid_equiangle_skew( int num_nodes, double coordinates[][3] )
{
double base[4][3];
double tri1[3][3];
double tri2[3][3];
double tri3[3][3];
double tri4[3][3];
make_pyramid_faces(coordinates, base,tri1,tri2,tri3,tri4);
double quad_skew=quad_equiangle_skew( 4, base );
double tri1_skew=tri_equiangle_skew(3,tri1);
double tri2_skew=tri_equiangle_skew(3,tri2);
double tri3_skew=tri_equiangle_skew(3,tri3);
double tri4_skew=tri_equiangle_skew(3,tri4);
double max_skew=quad_skew;
max_skew = max_skew > tri1_skew ? max_skew : tri1_skew;
max_skew = max_skew > tri2_skew ? max_skew : tri2_skew;
max_skew = max_skew > tri3_skew ? max_skew : tri3_skew;
max_skew = max_skew > tri4_skew ? max_skew : tri4_skew;
return max_skew;
}
/*!
the volume of a pyramid
the volume is calculated by dividing the pyramid into
2 tets and summing the volumes of the 2 tets.
*/
double pyramid_volume( int num_nodes, double coordinates[][3] )
{
double center_coords[3];
//calculate the center of the quads
center_coords[0] = (coordinates[0][0] + coordinates[1][0] +
coordinates[2][0] + coordinates[3][0]) / 4;
center_coords[1] = (coordinates[0][1] + coordinates[1][1] +
coordinates[2][1] + coordinates[3][1] ) / 4;
center_coords[2] = (coordinates[0][2] + coordinates[1][2] +
coordinates[2][2] + coordinates[3][2] ) / 4;
double tet_coords[4][4][3];
tet_coords[0][0][0] = coordinates[0][0];
tet_coords[0][0][1] = coordinates[0][1];
tet_coords[0][0][2] = coordinates[0][2];
tet_coords[0][1][0] = coordinates[1][0];
tet_coords[0][1][1] = coordinates[1][1];
tet_coords[0][1][2] = coordinates[1][2];
tet_coords[0][2][0] = center_coords[0];
tet_coords[0][2][1] = center_coords[1];
tet_coords[0][2][2] = center_coords[2];
tet_coords[0][3][0] = coordinates[4][0];
tet_coords[0][3][1] = coordinates[4][1];
tet_coords[0][3][2] = coordinates[4][2];
tet_coords[1][0][0] = coordinates[1][0];
tet_coords[1][0][1] = coordinates[1][1];
tet_coords[1][0][2] = coordinates[1][2];
tet_coords[1][1][0] = coordinates[2][0];
tet_coords[1][1][1] = coordinates[2][1];
tet_coords[1][1][2] = coordinates[2][2];
tet_coords[1][2][0] = center_coords[0];
tet_coords[1][2][1] = center_coords[1];
tet_coords[1][2][2] = center_coords[2];
tet_coords[1][3][0] = coordinates[4][0];
tet_coords[1][3][1] = coordinates[4][1];
tet_coords[1][3][2] = coordinates[4][2];
tet_coords[2][0][0] = coordinates[2][0];
tet_coords[2][0][1] = coordinates[2][1];
tet_coords[2][0][2] = coordinates[2][2];
tet_coords[2][1][0] = coordinates[3][0];
tet_coords[2][1][1] = coordinates[3][1];
tet_coords[2][1][2] = coordinates[3][2];
tet_coords[2][2][0] = center_coords[0];
tet_coords[2][2][1] = center_coords[1];
tet_coords[2][2][2] = center_coords[2];
tet_coords[2][3][0] = coordinates[4][0];
tet_coords[2][3][1] = coordinates[4][1];
tet_coords[2][3][2] = coordinates[4][2];
tet_coords[3][0][0] = coordinates[3][0];
tet_coords[3][0][1] = coordinates[3][1];
tet_coords[3][0][2] = coordinates[3][2];
tet_coords[3][1][0] = coordinates[0][0];
tet_coords[3][1][1] = coordinates[0][1];
tet_coords[3][1][2] = coordinates[0][2];
tet_coords[3][2][0] = center_coords[0];
tet_coords[3][2][1] = center_coords[1];
tet_coords[3][2][2] = center_coords[2];
tet_coords[3][3][0] = coordinates[4][0];
tet_coords[3][3][1] = coordinates[4][1];
tet_coords[3][3][2] = coordinates[4][2];
double volume=0.0;
for(int t=0;t<4;t++)
{
volume+=tet_volume(4,tet_coords[t]);
}
return (double)volume;
}
double pyramid_jacobian( int num_nodes, double coordinates[][3] )
{
// break the pyramid into four tets return the minimum jacobian of the two tets
double tet1_coords[4][3];
double tet2_coords[4][3];
double tet3_coords[4][3];
double tet4_coords[4][3];
make_pyramid_tets(coordinates, tet1_coords, tet2_coords, tet3_coords, tet4_coords);
double j1 = tet_jacobian(4, tet1_coords);
double j2 = tet_jacobian(4, tet2_coords);
double j3 = tet_jacobian(4, tet3_coords);
double j4 = tet_jacobian(4, tet4_coords);
double p1 = j1 < j2 ? j1 : j2;
double p2 = j3 < j4 ? j3 : j4;
return p1 < p2 ? p1 : p2;
}
double pyramid_scaled_jacobian( int num_nodes, double coordinates[][3] )
{
// break the pyramid into four tets return the minimum scaled jacobian of the tets
double tet1_coords[4][3];
double tet2_coords[4][3];
double tet3_coords[4][3];
double tet4_coords[4][3];
make_pyramid_tets(coordinates, tet1_coords, tet2_coords, tet3_coords, tet4_coords);
std::vector<double> scaled_jacob(4);
scaled_jacob[0] = tet_scaled_jacobian(4, tet1_coords);
scaled_jacob[1] = tet_scaled_jacobian(4, tet2_coords);
scaled_jacob[2] = tet_scaled_jacobian(4, tet3_coords);
scaled_jacob[3] = tet_scaled_jacobian(4, tet4_coords);
std::vector<double>::iterator iter = std::min_element(scaled_jacob.begin(), scaled_jacob.end());
// scale the minimum scaled jacobian so that a perfect pyramid has
// a value of 1 and cap it to make sure it is not > 1.0 or < 0.0
if (*iter <= 0.0)
return 0.0;
double min_jac = (*iter)*2.0/sqrt(2.0);
return min_jac < 1.0 ? min_jac : 1.0 - (min_jac - 1.0);
}
double pyramid_shape( int num_nodes, double coordinates[][3] )
{
// ideally there will be four equilateral triangles and one square.
// Test each face
double base[4][3];
double tri1[3][3];
double tri2[3][3];
double tri3[3][3];
double tri4[3][3];
make_pyramid_faces(coordinates, base, tri1, tri2, tri3, tri4);
double s1 = quad_shape(4, base);
if (s1 == 0.0)
return 0.0;
double cos_angle;
double dist_to_base = distance_point_to_pyramid_base(num_nodes, coordinates, cos_angle );
if (dist_to_base <= 0 || cos_angle <= 0.0)
return 0.0;
double longest_edge = largest_pyramid_edge(coordinates) * SQRT2_HALVES;
if (dist_to_base < longest_edge)
dist_to_base = dist_to_base/longest_edge;
else
dist_to_base = longest_edge/dist_to_base;
double shape = s1 * cos_angle * dist_to_base;
return shape;
}
void make_pyramid_tets(double coordinates[][3], double tet1_coords[][3], double tet2_coords[][3],
double tet3_coords[][3], double tet4_coords[][3])
{
// tet1
tet1_coords[0][0] = coordinates[0][0];
tet1_coords[0][1] = coordinates[0][1];
tet1_coords[0][2] = coordinates[0][2];
tet1_coords[1][0] = coordinates[1][0];
tet1_coords[1][1] = coordinates[1][1];
tet1_coords[1][2] = coordinates[1][2];
tet1_coords[2][0] = coordinates[2][0];
tet1_coords[2][1] = coordinates[2][1];
tet1_coords[2][2] = coordinates[2][2];
tet1_coords[3][0] = coordinates[4][0];
tet1_coords[3][1] = coordinates[4][1];
tet1_coords[3][2] = coordinates[4][2];
// tet2
tet2_coords[0][0] = coordinates[0][0];
tet2_coords[0][1] = coordinates[0][1];
tet2_coords[0][2] = coordinates[0][2];
tet2_coords[1][0] = coordinates[2][0];
tet2_coords[1][1] = coordinates[2][1];
tet2_coords[1][2] = coordinates[2][2];
tet2_coords[2][0] = coordinates[3][0];
tet2_coords[2][1] = coordinates[3][1];
tet2_coords[2][2] = coordinates[3][2];
tet2_coords[3][0] = coordinates[4][0];
tet2_coords[3][1] = coordinates[4][1];
tet2_coords[3][2] = coordinates[4][2];
// tet3
tet3_coords[0][0] = coordinates[0][0];
tet3_coords[0][1] = coordinates[0][1];
tet3_coords[0][2] = coordinates[0][2];
tet3_coords[1][0] = coordinates[1][0];
tet3_coords[1][1] = coordinates[1][1];
tet3_coords[1][2] = coordinates[1][2];
tet3_coords[2][0] = coordinates[3][0];
tet3_coords[2][1] = coordinates[3][1];
tet3_coords[2][2] = coordinates[3][2];
tet3_coords[3][0] = coordinates[4][0];
tet3_coords[3][1] = coordinates[4][1];
tet3_coords[3][2] = coordinates[4][2];
// tet4
tet4_coords[0][0] = coordinates[1][0];
tet4_coords[0][1] = coordinates[1][1];
tet4_coords[0][2] = coordinates[1][2];
tet4_coords[1][0] = coordinates[2][0];
tet4_coords[1][1] = coordinates[2][1];
tet4_coords[1][2] = coordinates[2][2];
tet4_coords[2][0] = coordinates[3][0];
tet4_coords[2][1] = coordinates[3][1];
tet4_coords[2][2] = coordinates[3][2];
tet4_coords[3][0] = coordinates[4][0];
tet4_coords[3][1] = coordinates[4][1];
tet4_coords[3][2] = coordinates[4][2];
}
void make_pyramid_faces(double coordinates[][3], double base[][3], double tri1[][3],
double tri2[][3], double tri3[][3], double tri4[][3])
{
// base
base[0][0] = coordinates[0][0];
base[0][1] = coordinates[0][1];
base[0][2] = coordinates[0][2];
base[1][0] = coordinates[1][0];
base[1][1] = coordinates[1][1];
base[1][2] = coordinates[1][2];
base[2][0] = coordinates[2][0];
base[2][1] = coordinates[2][1];
base[2][2] = coordinates[2][2];
base[3][0] = coordinates[3][0];
base[3][1] = coordinates[3][1];
base[3][2] = coordinates[3][2];
// tri1
tri1[0][0] = coordinates[0][0];
tri1[0][1] = coordinates[0][1];
tri1[0][2] = coordinates[0][2];
tri1[1][0] = coordinates[1][0];
tri1[1][1] = coordinates[1][1];
tri1[1][2] = coordinates[1][2];
tri1[2][0] = coordinates[4][0];
tri1[2][1] = coordinates[4][1];
tri1[2][2] = coordinates[4][2];
// tri2
tri2[0][0] = coordinates[1][0];
tri2[0][1] = coordinates[1][1];
tri2[0][2] = coordinates[1][2];
tri2[1][0] = coordinates[2][0];
tri2[1][1] = coordinates[2][1];
tri2[1][2] = coordinates[2][2];
tri2[2][0] = coordinates[4][0];
tri2[2][1] = coordinates[4][1];
tri2[2][2] = coordinates[4][2];
// tri3
tri3[0][0] = coordinates[2][0];
tri3[0][1] = coordinates[2][1];
tri3[0][2] = coordinates[2][2];
tri3[1][0] = coordinates[3][0];
tri3[1][1] = coordinates[3][1];
tri3[1][2] = coordinates[3][2];
tri3[2][0] = coordinates[4][0];
tri3[2][1] = coordinates[4][1];
tri3[2][2] = coordinates[4][2];
// tri4
tri4[0][0] = coordinates[3][0];
tri4[0][1] = coordinates[3][1];
tri4[0][2] = coordinates[3][2];
tri4[1][0] = coordinates[0][0];
tri4[1][1] = coordinates[0][1];
tri4[1][2] = coordinates[0][2];
tri4[2][0] = coordinates[4][0];
tri4[2][1] = coordinates[4][1];
tri4[2][2] = coordinates[4][2];
}
void make_pyramid_edges( VerdictVector edges[8], double coordinates[][3] )
{
edges[0].set(
coordinates[1][0] - coordinates[0][0],
coordinates[1][1] - coordinates[0][1],
coordinates[1][2] - coordinates[0][2]
);
edges[1].set(
coordinates[2][0] - coordinates[1][0],
coordinates[2][1] - coordinates[1][1],
coordinates[2][2] - coordinates[1][2]
);
edges[2].set(
coordinates[3][0] - coordinates[2][0],
coordinates[3][1] - coordinates[2][1],
coordinates[3][2] - coordinates[2][2]
);
edges[3].set(
coordinates[0][0] - coordinates[3][0],
coordinates[0][1] - coordinates[3][1],
coordinates[0][2] - coordinates[3][2]
);
edges[4].set(
coordinates[4][0] - coordinates[0][0],
coordinates[4][1] - coordinates[0][1],
coordinates[4][2] - coordinates[0][2]
);
edges[5].set(
coordinates[4][0] - coordinates[1][0],
coordinates[4][1] - coordinates[1][1],
coordinates[4][2] - coordinates[1][2]
);
edges[6].set(
coordinates[4][0] - coordinates[2][0],
coordinates[4][1] - coordinates[2][1],
coordinates[4][2] - coordinates[2][2]
);
edges[7].set(
coordinates[4][0] - coordinates[3][0],
coordinates[4][1] - coordinates[3][1],
coordinates[4][2] - coordinates[3][2]
);
}
double largest_pyramid_edge( double coordinates[][3] )
{
VerdictVector edges[8];
make_pyramid_edges(edges, coordinates);
double l0 = edges[0].length_squared();
double l1 = edges[1].length_squared();
double l2 = edges[2].length_squared();
double l3 = edges[3].length_squared();
double l4 = edges[4].length_squared();
double l5 = edges[5].length_squared();
double l6 = edges[6].length_squared();
double l7 = edges[7].length_squared();
double max = std::min(l0, l1);
max = std::max(max, l2);
max = std::max(max, l3);
max = std::max(max, l4);
max = std::max(max, l5);
max = std::max(max, l6);
max = std::max(max, l7);
return sqrt(max);
}
double distance_point_to_pyramid_base( int num_nodes, double coordinates[][3], double &cos_angle )
{
VerdictVector a(coordinates[0]);
VerdictVector b(coordinates[1]);
VerdictVector c(coordinates[2]);
VerdictVector d(coordinates[3]);
VerdictVector peak(coordinates[4]);
VerdictVector centroid = (a + b + c + d)/4.0;
VerdictVector t1 = b - a;
VerdictVector t2 = d - a;
VerdictVector normal = t1 * t2;
double normal_length = normal.length();
VerdictVector pq = peak - centroid;
double distance = (pq % normal)/normal_length;
cos_angle = distance/pq.length();
return distance;
}
} // namespace verdict