V_TriMetric.cpp

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/*=========================================================================
 
  Module:    $RCSfile: V_TriMetric.cpp,v $
 
  Copyright (c) 2007 Sandia Corporation.
  All rights reserved.
  See Copyright.txt or http://www.kitware.com/Copyright.htm for details.
 
     This software is distributed WITHOUT ANY WARRANTY; without even
     the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR
     PURPOSE.  See the above copyright notice for more information.
 
=========================================================================*/
 
/*
 *
 * TriMetric.cpp contains quality calculations for Tris
 *
 * This file is part of VERDICT
 *
 */
 
#define VERDICT_EXPORTS
 
#include "moab/verdict.h"
#include "verdict_defines.hpp"
#include "V_GaussIntegration.hpp"
#include "VerdictVector.hpp"
#include <memory.h>
#include <cstddef>
 
// the average area of a tri
static double verdict_tri_size      = 0;
static ComputeNormal compute_normal = NULL;
 
/*!
  get weights based on the average area of a set of
  tris
*/
static int v_tri_get_weight( double& m11, double& m21, double& m12, double& m22 )
{
    static const double rootOf3 = sqrt( 3.0 );
    m11                         = 1;
    m21                         = 0;
    m12                         = 0.5;
    m22                         = 0.5 * rootOf3;
    double scale                = sqrt( 2.0 * verdict_tri_size / ( m11 * m22 - m21 * m12 ) );
 
    m11 *= scale;
    m21 *= scale;
    m12 *= scale;
    m22 *= scale;
 
    return 1;
}
 
/*! sets the average area of a tri */
C_FUNC_DEF void v_set_tri_size( double size )
{
    verdict_tri_size = size;
}
 
C_FUNC_DEF void v_set_tri_normal_func( ComputeNormal func )
{
    compute_normal = func;
}
 
/*!
   the edge ratio of a triangle
 
   NB (P. Pebay 01/14/07):
     Hmax / Hmin where Hmax and Hmin are respectively the maximum and the
     minimum edge lengths
 
*/
C_FUNC_DEF double v_tri_edge_ratio( int /*num_nodes*/, double coordinates[][3] )
{
 
    // three vectors for each side
    VerdictVector a( coordinates[1][0] - coordinates[0][0], coordinates[1][1] - coordinates[0][1],
                     coordinates[1][2] - coordinates[0][2] );
 
    VerdictVector b( coordinates[2][0] - coordinates[1][0], coordinates[2][1] - coordinates[1][1],
                     coordinates[2][2] - coordinates[1][2] );
 
    VerdictVector c( coordinates[0][0] - coordinates[2][0], coordinates[0][1] - coordinates[2][1],
                     coordinates[0][2] - coordinates[2][2] );
 
    double a2 = a.length_squared();
    double b2 = b.length_squared();
    double c2 = c.length_squared();
 
    double m2, M2;
    if( a2 < b2 )
    {
        if( b2 < c2 )
        {
            m2 = a2;
            M2 = c2;
        }
        else  // b2 <= a2
        {
            if( a2 < c2 )
            {
                m2 = a2;
                M2 = b2;
            }
            else  // c2 <= a2
            {
                m2 = c2;
                M2 = b2;
            }
        }
    }
    else  // b2 <= a2
    {
        if( a2 < c2 )
        {
            m2 = b2;
            M2 = c2;
        }
        else  // c2 <= a2
        {
            if( b2 < c2 )
            {
                m2 = b2;
                M2 = a2;
            }
            else  // c2 <= b2
            {
                m2 = c2;
                M2 = a2;
            }
        }
    }
 
    if( m2 < VERDICT_DBL_MIN )
        return (double)VERDICT_DBL_MAX;
    else
    {
        double edge_ratio;
        edge_ratio = sqrt( M2 / m2 );
 
        if( edge_ratio > 0 ) return (double)VERDICT_MIN( edge_ratio, VERDICT_DBL_MAX );
        return (double)VERDICT_MAX( edge_ratio, -VERDICT_DBL_MAX );
    }
}
 
/*!
   the aspect ratio of a triangle
 
   NB (P. Pebay 01/14/07):
     Hmax / ( 2.0 * sqrt(3.0) * IR) where Hmax is the maximum edge length
     and IR is the inradius
 
     note that previous incarnations of verdict used "v_tri_aspect_ratio" to denote
     what is now called "v_tri_aspect_frobenius"
 
*/
C_FUNC_DEF double v_tri_aspect_ratio( int /*num_nodes*/, double coordinates[][3] )
{
    static const double normal_coeff = sqrt( 3. ) / 6.;
 
    // three vectors for each side
    VerdictVector a( coordinates[1][0] - coordinates[0][0], coordinates[1][1] - coordinates[0][1],
                     coordinates[1][2] - coordinates[0][2] );
 
    VerdictVector b( coordinates[2][0] - coordinates[1][0], coordinates[2][1] - coordinates[1][1],
                     coordinates[2][2] - coordinates[1][2] );
 
    VerdictVector c( coordinates[0][0] - coordinates[2][0], coordinates[0][1] - coordinates[2][1],
                     coordinates[0][2] - coordinates[2][2] );
 
    double a1 = a.length();
    double b1 = b.length();
    double c1 = c.length();
 
    double hm = a1 > b1 ? a1 : b1;
    hm        = hm > c1 ? hm : c1;
 
    VerdictVector ab   = a * b;
    double denominator = ab.length();
 
    if( denominator < VERDICT_DBL_MIN )
        return (double)VERDICT_DBL_MAX;
    else
    {
        double aspect_ratio;
        aspect_ratio = normal_coeff * hm * ( a1 + b1 + c1 ) / denominator;
 
        if( aspect_ratio > 0 ) return (double)VERDICT_MIN( aspect_ratio, VERDICT_DBL_MAX );
        return (double)VERDICT_MAX( aspect_ratio, -VERDICT_DBL_MAX );
    }
}
 
/*!
   the radius ratio of a triangle
 
   NB (P. Pebay 01/13/07):
     CR / (3.0*IR) where CR is the circumradius and IR is the inradius
 
     this quality metric is also known to VERDICT, for tetrahedral elements only,
     a the "aspect beta"
 
*/
C_FUNC_DEF double v_tri_radius_ratio( int /*num_nodes*/, double coordinates[][3] )
{
 
    // three vectors for each side
    VerdictVector a( coordinates[1][0] - coordinates[0][0], coordinates[1][1] - coordinates[0][1],
                     coordinates[1][2] - coordinates[0][2] );
 
    VerdictVector b( coordinates[2][0] - coordinates[1][0], coordinates[2][1] - coordinates[1][1],
                     coordinates[2][2] - coordinates[1][2] );
 
    VerdictVector c( coordinates[0][0] - coordinates[2][0], coordinates[0][1] - coordinates[2][1],
                     coordinates[0][2] - coordinates[2][2] );
 
    double a2 = a.length_squared();
    double b2 = b.length_squared();
    double c2 = c.length_squared();
 
    VerdictVector ab   = a * b;
    double denominator = ab.length_squared();
 
    if( denominator < VERDICT_DBL_MIN ) return (double)VERDICT_DBL_MAX;
 
    double radius_ratio;
    radius_ratio = .25 * a2 * b2 * c2 * ( a2 + b2 + c2 ) / denominator;
 
    if( radius_ratio > 0 ) return (double)VERDICT_MIN( radius_ratio, VERDICT_DBL_MAX );
    return (double)VERDICT_MAX( radius_ratio, -VERDICT_DBL_MAX );
}
 
/*!
   the Frobenius aspect of a tri
 
   srms^2/(2 * sqrt(3.0) * area)
   where srms^2 is sum of the lengths squared
 
   NB (P. Pebay 01/14/07):
     this method was called "aspect ratio" in earlier incarnations of VERDICT
 
*/
 
C_FUNC_DEF double v_tri_aspect_frobenius( int /*num_nodes*/, double coordinates[][3] )
{
    static const double two_times_root_of_3 = 2 * sqrt( 3.0 );
 
    // three vectors for each side
    VerdictVector side1( coordinates[1][0] - coordinates[0][0], coordinates[1][1] - coordinates[0][1],
                         coordinates[1][2] - coordinates[0][2] );
 
    VerdictVector side2( coordinates[2][0] - coordinates[1][0], coordinates[2][1] - coordinates[1][1],
                         coordinates[2][2] - coordinates[1][2] );
 
    VerdictVector side3( coordinates[0][0] - coordinates[2][0], coordinates[0][1] - coordinates[2][1],
                         coordinates[0][2] - coordinates[2][2] );
 
    // sum the lengths squared of each side
    double srms = ( side1.length_squared() + side2.length_squared() + side3.length_squared() );
 
    // find two times the area of the triangle by cross product
    double areaX2 = ( ( side1 * ( -side3 ) ).length() );
 
    if( areaX2 == 0.0 ) return (double)VERDICT_DBL_MAX;
 
    double aspect = (double)( srms / ( two_times_root_of_3 * ( areaX2 ) ) );
    if( aspect > 0 ) return (double)VERDICT_MIN( aspect, VERDICT_DBL_MAX );
    return (double)VERDICT_MAX( aspect, -VERDICT_DBL_MAX );
}
 
/*!
  The area of a tri
 
  0.5 * jacobian at a node
*/
C_FUNC_DEF double v_tri_area( int /*num_nodes*/, double coordinates[][3] )
{
    // two vectors for two sides
    VerdictVector side1( coordinates[1][0] - coordinates[0][0], coordinates[1][1] - coordinates[0][1],
                         coordinates[1][2] - coordinates[0][2] );
 
    VerdictVector side3( coordinates[2][0] - coordinates[0][0], coordinates[2][1] - coordinates[0][1],
                         coordinates[2][2] - coordinates[0][2] );
 
    // the cross product of the two vectors representing two sides of the
    // triangle
    VerdictVector tmp = side1 * side3;
 
    // return the magnitude of the vector divided by two
    double area = 0.5 * tmp.length();
    if( area > 0 ) return (double)VERDICT_MIN( area, VERDICT_DBL_MAX );
    return (double)VERDICT_MAX( area, -VERDICT_DBL_MAX );
}
 
/*!
  The minimum angle of a tri
 
  The minimum angle of a tri is the minimum angle between
  two adjacents sides out of all three corners of the triangle.
*/
C_FUNC_DEF double v_tri_minimum_angle( int /*num_nodes*/, double coordinates[][3] )
{
 
    // vectors for all the sides
    VerdictVector sides[4];
    sides[0].set( coordinates[1][0] - coordinates[0][0], coordinates[1][1] - coordinates[0][1],
                  coordinates[1][2] - coordinates[0][2] );
    sides[1].set( coordinates[2][0] - coordinates[1][0], coordinates[2][1] - coordinates[1][1],
                  coordinates[2][2] - coordinates[1][2] );
    sides[2].set( coordinates[2][0] - coordinates[0][0], coordinates[2][1] - coordinates[0][1],
                  coordinates[2][2] - coordinates[0][2] );
 
    // in case we need to find the interior angle
    // between sides 0 and 1
    sides[3] = -sides[1];
 
    // calculate the lengths squared of the sides
    double sides_lengths[3];
    sides_lengths[0] = sides[0].length_squared();
    sides_lengths[1] = sides[1].length_squared();
    sides_lengths[2] = sides[2].length_squared();
 
    if( sides_lengths[0] == 0.0 || sides_lengths[1] == 0.0 || sides_lengths[2] == 0.0 ) return 0.0;
 
    // using the law of sines, we know that the minimum
    // angle is opposite of the shortest side
 
    // find the shortest side
    int short_side = 0;
    if( sides_lengths[1] < sides_lengths[0] ) short_side = 1;
    if( sides_lengths[2] < sides_lengths[short_side] ) short_side = 2;
 
    // from the shortest side, calculate the angle of the
    // opposite angle
    double min_angle;
    if( short_side == 0 )
    {
        min_angle = sides[2].interior_angle( sides[1] );
    }
    else if( short_side == 1 )
    {
        min_angle = sides[0].interior_angle( sides[2] );
    }
    else
    {
        min_angle = sides[0].interior_angle( sides[3] );
    }
 
    if( min_angle > 0 ) return (double)VERDICT_MIN( min_angle, VERDICT_DBL_MAX );
    return (double)VERDICT_MAX( min_angle, -VERDICT_DBL_MAX );
}
 
/*!
  The maximum angle of a tri
 
  The maximum angle of a tri is the maximum angle between
  two adjacents sides out of all three corners of the triangle.
*/
C_FUNC_DEF double v_tri_maximum_angle( int /*num_nodes*/, double coordinates[][3] )
{
 
    // vectors for all the sides
    VerdictVector sides[4];
    sides[0].set( coordinates[1][0] - coordinates[0][0], coordinates[1][1] - coordinates[0][1],
                  coordinates[1][2] - coordinates[0][2] );
    sides[1].set( coordinates[2][0] - coordinates[1][0], coordinates[2][1] - coordinates[1][1],
                  coordinates[2][2] - coordinates[1][2] );
    sides[2].set( coordinates[2][0] - coordinates[0][0], coordinates[2][1] - coordinates[0][1],
                  coordinates[2][2] - coordinates[0][2] );
 
    // in case we need to find the interior angle
    // between sides 0 and 1
    sides[3] = -sides[1];
 
    // calculate the lengths squared of the sides
    double sides_lengths[3];
    sides_lengths[0] = sides[0].length_squared();
    sides_lengths[1] = sides[1].length_squared();
    sides_lengths[2] = sides[2].length_squared();
 
    if( sides_lengths[0] == 0.0 || sides_lengths[1] == 0.0 || sides_lengths[2] == 0.0 )
    {
        return 0.0;
    }
 
    // using the law of sines, we know that the maximum
    // angle is opposite of the longest side
 
    // find the longest side
    int short_side = 0;
    if( sides_lengths[1] > sides_lengths[0] ) short_side = 1;
    if( sides_lengths[2] > sides_lengths[short_side] ) short_side = 2;
 
    // from the longest side, calculate the angle of the
    // opposite angle
    double max_angle;
    if( short_side == 0 )
    {
        max_angle = sides[2].interior_angle( sides[1] );
    }
    else if( short_side == 1 )
    {
        max_angle = sides[0].interior_angle( sides[2] );
    }
    else
    {
        max_angle = sides[0].interior_angle( sides[3] );
    }
 
    if( max_angle > 0 ) return (double)VERDICT_MIN( max_angle, VERDICT_DBL_MAX );
    return (double)VERDICT_MAX( max_angle, -VERDICT_DBL_MAX );
}
 
/*!
  The condition of a tri
 
  Condition number of the jacobian matrix at any corner
*/
C_FUNC_DEF double v_tri_condition( int /*num_nodes*/, double coordinates[][3] )
{
    static const double rt3 = sqrt( 3.0 );
 
    VerdictVector v1( coordinates[1][0] - coordinates[0][0], coordinates[1][1] - coordinates[0][1],
                      coordinates[1][2] - coordinates[0][2] );
 
    VerdictVector v2( coordinates[2][0] - coordinates[0][0], coordinates[2][1] - coordinates[0][1],
                      coordinates[2][2] - coordinates[0][2] );
 
    VerdictVector tri_normal = v1 * v2;
    double areax2            = tri_normal.length();
 
    if( areax2 == 0.0 ) return (double)VERDICT_DBL_MAX;
 
    double condition = (double)( ( ( v1 % v1 ) + ( v2 % v2 ) - ( v1 % v2 ) ) / ( areax2 * rt3 ) );
 
    // check for inverted if we have access to the normal
    if( compute_normal )
    {
        // center of tri
        double point[3], surf_normal[3];
        point[0] = ( coordinates[0][0] + coordinates[1][0] + coordinates[2][0] ) / 3;
        point[1] = ( coordinates[0][1] + coordinates[1][1] + coordinates[2][1] ) / 3;
        point[2] = ( coordinates[0][2] + coordinates[1][2] + coordinates[2][2] ) / 3;
 
        // dot product
        compute_normal( point, surf_normal );
        if( ( tri_normal.x() * surf_normal[0] + tri_normal.y() * surf_normal[1] + tri_normal.z() * surf_normal[2] ) <
            0 )
            return (double)VERDICT_DBL_MAX;
    }
    return (double)VERDICT_MIN( condition, VERDICT_DBL_MAX );
}
 
/*!
  The scaled jacobian of a tri
 
  minimum of the jacobian divided by the lengths of 2 edge vectors
*/
C_FUNC_DEF double v_tri_scaled_jacobian( int /*num_nodes*/, double coordinates[][3] )
{
    static const double detw = 2. / sqrt( 3.0 );
    VerdictVector first, second;
    double jacobian;
 
    VerdictVector edge[3];
    edge[0].set( coordinates[1][0] - coordinates[0][0], coordinates[1][1] - coordinates[0][1],
                 coordinates[1][2] - coordinates[0][2] );
 
    edge[1].set( coordinates[2][0] - coordinates[0][0], coordinates[2][1] - coordinates[0][1],
                 coordinates[2][2] - coordinates[0][2] );
 
    edge[2].set( coordinates[2][0] - coordinates[1][0], coordinates[2][1] - coordinates[1][1],
                 coordinates[2][2] - coordinates[1][2] );
    first  = edge[1] - edge[0];
    second = edge[2] - edge[0];
 
    VerdictVector cross = first * second;
    jacobian            = cross.length();
 
    double max_edge_length_product;
    max_edge_length_product =
        VERDICT_MAX( edge[0].length() * edge[1].length(),
                     VERDICT_MAX( edge[1].length() * edge[2].length(), edge[0].length() * edge[2].length() ) );
 
    if( max_edge_length_product < VERDICT_DBL_MIN ) return (double)0.0;
 
    jacobian *= detw;
    jacobian /= max_edge_length_product;
 
    if( compute_normal )
    {
        // center of tri
        double point[3], surf_normal[3];
        point[0] = ( coordinates[0][0] + coordinates[1][0] + coordinates[2][0] ) / 3;
        point[1] = ( coordinates[0][1] + coordinates[1][1] + coordinates[2][1] ) / 3;
        point[2] = ( coordinates[0][2] + coordinates[1][2] + coordinates[2][2] ) / 3;
 
        // dot product
        compute_normal( point, surf_normal );
        if( ( cross.x() * surf_normal[0] + cross.y() * surf_normal[1] + cross.z() * surf_normal[2] ) < 0 )
            jacobian *= -1;
    }
 
    if( jacobian > 0 ) return (double)VERDICT_MIN( jacobian, VERDICT_DBL_MAX );
    return (double)VERDICT_MAX( jacobian, -VERDICT_DBL_MAX );
}
 
/*!
  The shape of a tri
 
  2 / condition number of weighted jacobian matrix
*/
C_FUNC_DEF double v_tri_shape( int num_nodes, double coordinates[][3] )
{
    double condition = v_tri_condition( num_nodes, coordinates );
 
    double shape;
    if( condition <= VERDICT_DBL_MIN )
        shape = VERDICT_DBL_MAX;
    else
        shape = ( 1 / condition );
 
    if( shape > 0 ) return (double)VERDICT_MIN( shape, VERDICT_DBL_MAX );
    return (double)VERDICT_MAX( shape, -VERDICT_DBL_MAX );
}
 
/*!
  The relative size of a tri
 
  Min(J,1/J) where J is the determinant of the weighted jacobian matrix.
*/
C_FUNC_DEF double v_tri_relative_size_squared( int /*num_nodes*/, double coordinates[][3] )
{
    double w11, w21, w12, w22;
 
    VerdictVector xxi, xet, tri_normal;
 
    v_tri_get_weight( w11, w21, w12, w22 );
 
    double detw = determinant( w11, w21, w12, w22 );
 
    if( detw == 0.0 ) return 0.0;
 
    xxi.set( coordinates[0][0] - coordinates[1][0], coordinates[0][1] - coordinates[1][1],
             coordinates[0][2] - coordinates[1][2] );
 
    xet.set( coordinates[0][0] - coordinates[2][0], coordinates[0][1] - coordinates[2][1],
             coordinates[0][2] - coordinates[2][2] );
 
    tri_normal = xxi * xet;
 
    double deta = tri_normal.length();
    if( deta == 0.0 || detw == 0.0 ) return 0.0;
 
    double size = pow( deta / detw, 2 );
 
    double rel_size = VERDICT_MIN( size, 1.0 / size );
 
    if( rel_size > 0 ) return (double)VERDICT_MIN( rel_size, VERDICT_DBL_MAX );
    return (double)VERDICT_MAX( rel_size, -VERDICT_DBL_MAX );
}
 
/*!
  The shape and size of a tri
 
  Product of the Shape and Relative Size
*/
C_FUNC_DEF double v_tri_shape_and_size( int num_nodes, double coordinates[][3] )
{
    double size, shape;
 
    size  = v_tri_relative_size_squared( num_nodes, coordinates );
    shape = v_tri_shape( num_nodes, coordinates );
 
    double shape_and_size = size * shape;
 
    if( shape_and_size > 0 ) return (double)VERDICT_MIN( shape_and_size, VERDICT_DBL_MAX );
    return (double)VERDICT_MAX( shape_and_size, -VERDICT_DBL_MAX );
}
 
/*!
  The distortion of a tri
 
TODO:  make a short definition of the distortion and comment below
*/
C_FUNC_DEF double v_tri_distortion( int num_nodes, double coordinates[][3] )
{
 
    double distortion;
    int total_number_of_gauss_points = 0;
    VerdictVector aa, bb, cc, normal_at_point, xin;
    double element_area = 0.;
 
    aa.set( coordinates[1][0] - coordinates[0][0], coordinates[1][1] - coordinates[0][1],
            coordinates[1][2] - coordinates[0][2] );
 
    bb.set( coordinates[2][0] - coordinates[0][0], coordinates[2][1] - coordinates[0][1],
            coordinates[2][2] - coordinates[0][2] );
 
    VerdictVector tri_normal = aa * bb;
 
    int number_of_gauss_points = 0;
    if( num_nodes == 3 )
    {
        distortion = 1.0;
        return (double)distortion;
    }
 
    else if( num_nodes == 6 )
    {
        total_number_of_gauss_points = 6;
        number_of_gauss_points       = 6;
    }
 
    distortion = VERDICT_DBL_MAX;
    double shape_function[maxTotalNumberGaussPoints][maxNumberNodes];
    double dndy1[maxTotalNumberGaussPoints][maxNumberNodes];
    double dndy2[maxTotalNumberGaussPoints][maxNumberNodes];
    double weight[maxTotalNumberGaussPoints];
 
    // create an object of GaussIntegration
    int number_dims = 2;
    int is_tri      = 1;
    GaussIntegration::initialize( number_of_gauss_points, num_nodes, number_dims, is_tri );
    GaussIntegration::calculate_shape_function_2d_tri();
    GaussIntegration::get_shape_func( shape_function[0], dndy1[0], dndy2[0], weight );
 
    // calculate element area
    int ife, ja;
    for( ife = 0; ife < total_number_of_gauss_points; ife++ )
    {
        aa.set( 0.0, 0.0, 0.0 );
        bb.set( 0.0, 0.0, 0.0 );
 
        for( ja = 0; ja < num_nodes; ja++ )
        {
            xin.set( coordinates[ja][0], coordinates[ja][1], coordinates[ja][2] );
            aa += dndy1[ife][ja] * xin;
            bb += dndy2[ife][ja] * xin;
        }
        normal_at_point = aa * bb;
        double jacobian = normal_at_point.length();
        element_area += weight[ife] * jacobian;
    }
 
    element_area *= 0.8660254;
    double dndy1_at_node[maxNumberNodes][maxNumberNodes];
    double dndy2_at_node[maxNumberNodes][maxNumberNodes];
 
    GaussIntegration::calculate_derivative_at_nodes_2d_tri( dndy1_at_node, dndy2_at_node );
 
    VerdictVector normal_at_nodes[7];
 
    // evaluate normal at nodes and distortion values at nodes
    int jai = 0;
    for( ja = 0; ja < num_nodes; ja++ )
    {
        aa.set( 0.0, 0.0, 0.0 );
        bb.set( 0.0, 0.0, 0.0 );
        for( jai = 0; jai < num_nodes; jai++ )
        {
            xin.set( coordinates[jai][0], coordinates[jai][1], coordinates[jai][2] );
            aa += dndy1_at_node[ja][jai] * xin;
            bb += dndy2_at_node[ja][jai] * xin;
        }
        normal_at_nodes[ja] = aa * bb;
        normal_at_nodes[ja].normalize();
    }
 
    // determine if element is flat
    bool flat_element = true;
    double dot_product;
 
    for( ja = 0; ja < num_nodes; ja++ )
    {
        dot_product = normal_at_nodes[0] % normal_at_nodes[ja];
        if( fabs( dot_product ) < 0.99 )
        {
            flat_element = false;
            break;
        }
    }
 
    // take into consideration of the thickness of the element
    double thickness, thickness_gauss;
    double distrt;
    // get_tri_thickness(tri, element_area, thickness );
    thickness = 0.001 * sqrt( element_area );
 
    // set thickness gauss point location
    double zl = 0.5773502691896;
    if( flat_element ) zl = 0.0;
 
    int no_gauss_pts_z = ( flat_element ) ? 1 : 2;
    double thickness_z;
 
    // loop on integration points
    int igz;
    for( ife = 0; ife < total_number_of_gauss_points; ife++ )
    {
        // loop on the thickness direction gauss points
        for( igz = 0; igz < no_gauss_pts_z; igz++ )
        {
            zl          = -zl;
            thickness_z = zl * thickness / 2.0;
 
            aa.set( 0.0, 0.0, 0.0 );
            bb.set( 0.0, 0.0, 0.0 );
            cc.set( 0.0, 0.0, 0.0 );
 
            for( ja = 0; ja < num_nodes; ja++ )
            {
                xin.set( coordinates[jai][0], coordinates[jai][1], coordinates[jai][2] );
                xin += thickness_z * normal_at_nodes[ja];
                aa += dndy1[ife][ja] * xin;
                bb += dndy2[ife][ja] * xin;
                thickness_gauss = shape_function[ife][ja] * thickness / 2.0;
                cc += thickness_gauss * normal_at_nodes[ja];
            }
 
            normal_at_point = aa * bb;
            distrt          = cc % normal_at_point;
            if( distrt < distortion ) distortion = distrt;
        }
    }
 
    // loop through nodal points
    for( ja = 0; ja < num_nodes; ja++ )
    {
        for( igz = 0; igz < no_gauss_pts_z; igz++ )
        {
            zl          = -zl;
            thickness_z = zl * thickness / 2.0;
 
            aa.set( 0.0, 0.0, 0.0 );
            bb.set( 0.0, 0.0, 0.0 );
            cc.set( 0.0, 0.0, 0.0 );
 
            for( jai = 0; jai < num_nodes; jai++ )
            {
                xin.set( coordinates[jai][0], coordinates[jai][1], coordinates[jai][2] );
                xin += thickness_z * normal_at_nodes[ja];
                aa += dndy1_at_node[ja][jai] * xin;
                bb += dndy2_at_node[ja][jai] * xin;
                if( jai == ja )
                    thickness_gauss = thickness / 2.0;
                else
                    thickness_gauss = 0.;
                cc += thickness_gauss * normal_at_nodes[jai];
            }
        }
 
        normal_at_point      = aa * bb;
        double sign_jacobian = ( tri_normal % normal_at_point ) > 0 ? 1. : -1.;
        distrt               = sign_jacobian * ( cc % normal_at_point );
 
        if( distrt < distortion ) distortion = distrt;
    }
    if( element_area * thickness != 0 )
        distortion *= 1. / ( element_area * thickness );
    else
        distortion *= 1.;
 
    if( distortion > 0 ) return (double)VERDICT_MIN( distortion, VERDICT_DBL_MAX );
    return (double)VERDICT_MAX( distortion, -VERDICT_DBL_MAX );
}
 
/*!
  tri_quality for calculating multiple tri metrics at once
 
  using this method is generally faster than using the individual
  method multiple times.
 
*/
C_FUNC_DEF void v_tri_quality( int num_nodes,
                               double coordinates[][3],
                               unsigned int metrics_request_flag,
                               TriMetricVals* metric_vals )
{
 
    memset( metric_vals, 0, sizeof( TriMetricVals ) );
 
    // for starts, lets set up some basic and common information
 
    /*  node numbers and side numbers used below
 
               2
               ++
              /  \
           2 /    \ 1
            /      \
           /        \
         0 ---------+ 1
               0
    */
 
    // vectors for each side
    VerdictVector sides[3];
    sides[0].set( coordinates[1][0] - coordinates[0][0], coordinates[1][1] - coordinates[0][1],
                  coordinates[1][2] - coordinates[0][2] );
    sides[1].set( coordinates[2][0] - coordinates[1][0], coordinates[2][1] - coordinates[1][1],
                  coordinates[2][2] - coordinates[1][2] );
    sides[2].set( coordinates[2][0] - coordinates[0][0], coordinates[2][1] - coordinates[0][1],
                  coordinates[2][2] - coordinates[0][2] );
    VerdictVector tri_normal = sides[0] * sides[2];
    // if we have access to normal information, check to see if the
    // element is inverted.  If we don't have the normal information
    // that we need for this, assume the element is not inverted.
    // This flag will be used for condition number, jacobian, shape,
    // and size and shape.
    bool is_inverted = false;
    if( compute_normal )
    {
        // center of tri
        double point[3], surf_normal[3];
        point[0] = ( coordinates[0][0] + coordinates[1][0] + coordinates[2][0] ) / 3;
        point[1] = ( coordinates[0][1] + coordinates[1][1] + coordinates[2][1] ) / 3;
        point[2] = ( coordinates[0][2] + coordinates[1][2] + coordinates[2][2] ) / 3;
        // dot product
        compute_normal( point, surf_normal );
        if( ( tri_normal.x() * surf_normal[0] + tri_normal.y() * surf_normal[1] + tri_normal.z() * surf_normal[2] ) <
            0 )
            is_inverted = true;
    }
 
    // lengths squared of each side
    double sides_lengths_squared[3];
    sides_lengths_squared[0] = sides[0].length_squared();
    sides_lengths_squared[1] = sides[1].length_squared();
    sides_lengths_squared[2] = sides[2].length_squared();
 
    // if we are doing angle calcuations
    if( metrics_request_flag & ( V_TRI_MINIMUM_ANGLE | V_TRI_MAXIMUM_ANGLE ) )
    {
        // which is short and long side
        int short_side = 0, long_side = 0;
 
        if( sides_lengths_squared[1] < sides_lengths_squared[0] ) short_side = 1;
        if( sides_lengths_squared[2] < sides_lengths_squared[short_side] ) short_side = 2;
 
        if( sides_lengths_squared[1] > sides_lengths_squared[0] ) long_side = 1;
        if( sides_lengths_squared[2] > sides_lengths_squared[long_side] ) long_side = 2;
 
        // calculate the minimum angle of the tri
        if( metrics_request_flag & V_TRI_MINIMUM_ANGLE )
        {
            if( sides_lengths_squared[0] == 0.0 || sides_lengths_squared[1] == 0.0 || sides_lengths_squared[2] == 0.0 )
            {
                metric_vals->minimum_angle = 0.0;
            }
            else if( short_side == 0 )
                metric_vals->minimum_angle = (double)sides[2].interior_angle( sides[1] );
            else if( short_side == 1 )
                metric_vals->minimum_angle = (double)sides[0].interior_angle( sides[2] );
            else
                metric_vals->minimum_angle = (double)sides[0].interior_angle( -sides[1] );
        }
 
        // calculate the maximum angle of the tri
        if( metrics_request_flag & V_TRI_MAXIMUM_ANGLE )
        {
            if( sides_lengths_squared[0] == 0.0 || sides_lengths_squared[1] == 0.0 || sides_lengths_squared[2] == 0.0 )
            {
                metric_vals->maximum_angle = 0.0;
            }
            else if( long_side == 0 )
                metric_vals->maximum_angle = (double)sides[2].interior_angle( sides[1] );
            else if( long_side == 1 )
                metric_vals->maximum_angle = (double)sides[0].interior_angle( sides[2] );
            else
                metric_vals->maximum_angle = (double)sides[0].interior_angle( -sides[1] );
        }
    }
 
    // calculate the area of the tri
    // the following metrics depend on area
    if( metrics_request_flag &
        ( V_TRI_AREA | V_TRI_SCALED_JACOBIAN | V_TRI_SHAPE | V_TRI_RELATIVE_SIZE_SQUARED | V_TRI_SHAPE_AND_SIZE ) )
    {
        metric_vals->area = (double)( ( sides[0] * sides[2] ).length() * 0.5 );
    }
 
    // calculate the aspect ratio
    if( metrics_request_flag & V_TRI_ASPECT_FROBENIUS )
    {
        // sum the lengths squared
        double srms = sides_lengths_squared[0] + sides_lengths_squared[1] + sides_lengths_squared[2];
 
        // calculate once and reuse
        static const double twoTimesRootOf3 = 2 * sqrt( 3.0 );
 
        double div = ( twoTimesRootOf3 * ( ( sides[0] * sides[2] ).length() ) );
 
        if( div == 0.0 )
            metric_vals->aspect_frobenius = (double)VERDICT_DBL_MAX;
        else
            metric_vals->aspect_frobenius = (double)( srms / div );
    }
 
    // calculate the scaled jacobian
    if( metrics_request_flag & V_TRI_SCALED_JACOBIAN )
    {
        // calculate once and reuse
        static const double twoOverRootOf3 = 2 / sqrt( 3.0 );
        // use the area from above
 
        double tmp = tri_normal.length() * twoOverRootOf3;
 
        // now scale it by the lengths of the sides
        double min_scaled_jac = VERDICT_DBL_MAX;
        double temp_scaled_jac;
        for( int i = 0; i < 3; i++ )
        {
            if( sides_lengths_squared[i % 3] == 0.0 || sides_lengths_squared[( i + 2 ) % 3] == 0.0 )
                temp_scaled_jac = 0.0;
            else
                temp_scaled_jac =
                    tmp / sqrt( sides_lengths_squared[i % 3] ) / sqrt( sides_lengths_squared[( i + 2 ) % 3] );
            if( temp_scaled_jac < min_scaled_jac ) min_scaled_jac = temp_scaled_jac;
        }
        // multiply by -1 if the normals are in opposite directions
        if( is_inverted )
        {
            min_scaled_jac *= -1;
        }
        metric_vals->scaled_jacobian = (double)min_scaled_jac;
    }
 
    // calculate the condition number
    if( metrics_request_flag & V_TRI_CONDITION )
    {
        // calculate once and reuse
        static const double rootOf3 = sqrt( 3.0 );
        // if it is inverted, the condition number is considered to be infinity.
        if( is_inverted )
        {
            metric_vals->condition = VERDICT_DBL_MAX;
        }
        else
        {
            double area2x = ( sides[0] * sides[2] ).length();
            if( area2x == 0.0 )
                metric_vals->condition = (double)( VERDICT_DBL_MAX );
            else
                metric_vals->condition = (double)( ( sides[0] % sides[0] + sides[2] % sides[2] - sides[0] % sides[2] ) /
                                                   ( area2x * rootOf3 ) );
        }
    }
 
    // calculate the shape
    if( metrics_request_flag & V_TRI_SHAPE || metrics_request_flag & V_TRI_SHAPE_AND_SIZE )
    {
        // if element is inverted, shape is zero.  We don't need to
        // calculate anything.
        if( is_inverted )
        {
            metric_vals->shape = 0.0;
        }
        else
        {  // otherwise, we calculate the shape
            // calculate once and reuse
            static const double rootOf3 = sqrt( 3.0 );
            // reuse area from before
            double area2x = metric_vals->area * 2;
            // dot products
            double dots[3] = { sides[0] % sides[0], sides[2] % sides[2], sides[0] % sides[2] };
 
            // add the dots
            double sum_dots = dots[0] + dots[1] - dots[2];
 
            // then the finale
            if( sum_dots == 0.0 )
                metric_vals->shape = 0.0;
            else
                metric_vals->shape = (double)( rootOf3 * area2x / sum_dots );
        }
    }
 
    // calculate relative size squared
    if( metrics_request_flag & V_TRI_RELATIVE_SIZE_SQUARED || metrics_request_flag & V_TRI_SHAPE_AND_SIZE )
    {
        // get weights
        double w11, w21, w12, w22;
        v_tri_get_weight( w11, w21, w12, w22 );
        // get the determinant
        double detw = determinant( w11, w21, w12, w22 );
        // use the area from above and divide with the determinant
        if( metric_vals->area == 0.0 || detw == 0.0 )
            metric_vals->relative_size_squared = 0.0;
        else
        {
            double size = metric_vals->area * 2.0 / detw;
            // square the size
            size *= size;
            // value ranges between 0 to 1
            metric_vals->relative_size_squared = (double)VERDICT_MIN( size, 1.0 / size );
        }
    }
 
    // calculate shape and size
    if( metrics_request_flag & V_TRI_SHAPE_AND_SIZE )
    {
        metric_vals->shape_and_size = metric_vals->relative_size_squared * metric_vals->shape;
    }
 
    // calculate distortion
    if( metrics_request_flag & V_TRI_DISTORTION ) metric_vals->distortion = v_tri_distortion( num_nodes, coordinates );
 
    // take care of any over-flow problems
    if( metric_vals->aspect_frobenius > 0 )
        metric_vals->aspect_frobenius = (double)VERDICT_MIN( metric_vals->aspect_frobenius, VERDICT_DBL_MAX );
    else
        metric_vals->aspect_frobenius = (double)VERDICT_MAX( metric_vals->aspect_frobenius, -VERDICT_DBL_MAX );
 
    if( metric_vals->area > 0 )
        metric_vals->area = (double)VERDICT_MIN( metric_vals->area, VERDICT_DBL_MAX );
    else
        metric_vals->area = (double)VERDICT_MAX( metric_vals->area, -VERDICT_DBL_MAX );
 
    if( metric_vals->minimum_angle > 0 )
        metric_vals->minimum_angle = (double)VERDICT_MIN( metric_vals->minimum_angle, VERDICT_DBL_MAX );
    else
        metric_vals->minimum_angle = (double)VERDICT_MAX( metric_vals->minimum_angle, -VERDICT_DBL_MAX );
 
    if( metric_vals->maximum_angle > 0 )
        metric_vals->maximum_angle = (double)VERDICT_MIN( metric_vals->maximum_angle, VERDICT_DBL_MAX );
    else
        metric_vals->maximum_angle = (double)VERDICT_MAX( metric_vals->maximum_angle, -VERDICT_DBL_MAX );
 
    if( metric_vals->condition > 0 )
        metric_vals->condition = (double)VERDICT_MIN( metric_vals->condition, VERDICT_DBL_MAX );
    else
        metric_vals->condition = (double)VERDICT_MAX( metric_vals->condition, -VERDICT_DBL_MAX );
 
    if( metric_vals->shape > 0 )
        metric_vals->shape = (double)VERDICT_MIN( metric_vals->shape, VERDICT_DBL_MAX );
    else
        metric_vals->shape = (double)VERDICT_MAX( metric_vals->shape, -VERDICT_DBL_MAX );
 
    if( metric_vals->scaled_jacobian > 0 )
        metric_vals->scaled_jacobian = (double)VERDICT_MIN( metric_vals->scaled_jacobian, VERDICT_DBL_MAX );
    else
        metric_vals->scaled_jacobian = (double)VERDICT_MAX( metric_vals->scaled_jacobian, -VERDICT_DBL_MAX );
 
    if( metric_vals->relative_size_squared > 0 )
        metric_vals->relative_size_squared = (double)VERDICT_MIN( metric_vals->relative_size_squared, VERDICT_DBL_MAX );
    else
        metric_vals->relative_size_squared =
            (double)VERDICT_MAX( metric_vals->relative_size_squared, -VERDICT_DBL_MAX );
 
    if( metric_vals->shape_and_size > 0 )
        metric_vals->shape_and_size = (double)VERDICT_MIN( metric_vals->shape_and_size, VERDICT_DBL_MAX );
    else
        metric_vals->shape_and_size = (double)VERDICT_MAX( metric_vals->shape_and_size, -VERDICT_DBL_MAX );
 
    if( metric_vals->distortion > 0 )
        metric_vals->distortion = (double)VERDICT_MIN( metric_vals->distortion, VERDICT_DBL_MAX );
    else
        metric_vals->distortion = (double)VERDICT_MAX( metric_vals->distortion, -VERDICT_DBL_MAX );
 
    if( metrics_request_flag & V_TRI_EDGE_RATIO )
    {
        metric_vals->edge_ratio = v_tri_edge_ratio( 3, coordinates );
    }
    if( metrics_request_flag & V_TRI_RADIUS_RATIO )
    {
        metric_vals->radius_ratio = v_tri_radius_ratio( 3, coordinates );
    }
    if( metrics_request_flag & V_TRI_ASPECT_FROBENIUS )  // there is no V_TRI_ASPECT_RATIO !
    {
        metric_vals->aspect_ratio = v_tri_radius_ratio( 3, coordinates );
    }
}