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V_TetMetric.cpp
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/*=========================================================================
Module: V_TetMetric.cpp
Copyright 2003,2006,2019 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.
=========================================================================*/
/*
*
* TetMetric.cpp contains quality calculations for Tets
*
* This file is part of VERDICT
*
*/
#include "V_GaussIntegration.hpp"
#include "VerdictVector.hpp"
#include "verdict.h"
#include "verdict_defines.hpp"
#include <math.h>
namespace VERDICT_NAMESPACE
{
static constexpr double three_times_1plussqrt3 = 3.0 * (1 + sqrt3);
static constexpr double normal_coeff = 180. * .3183098861837906715377675267450287;
static constexpr double aspect_ratio_normal_coeff = sqrt6 / 12.;
VERDICT_HOST_DEVICE double tet10_characteristic_length(const double coordinates[][3]);
VERDICT_HOST_DEVICE static const int* tet10_subtet_conn(int i)
{
static constexpr int stet10_subtet_conn[12][4] = { { 0, 4, 6, 7 }, { 1, 5, 4, 8 }, { 2, 6, 5, 9 },
{ 3, 8, 7, 9 }, { 4, 8, 5, 10 }, { 5, 8, 9, 10 }, { 9, 8, 7, 10 }, { 7, 8, 4, 10 },
{ 4, 5, 6, 10 }, { 5, 9, 6, 10 }, { 9, 7, 6, 10 }, { 7, 4, 6, 10 } };
return stet10_subtet_conn[i];
}
VERDICT_HOST_DEVICE static double fix_range(double v)
{
if (isnan(v))
{
return VERDICT_DBL_MAX;
}
if (v >= VERDICT_DBL_MAX)
{
return VERDICT_DBL_MAX;
}
if (v <= -VERDICT_DBL_MAX)
{
return -VERDICT_DBL_MAX;
}
return v;
}
VERDICT_HOST_DEVICE double tet_equiangle_skew(int /*num_nodes*/, const double coordinates[][3])
{
VerdictVector ab, ac, bc, bd, ad, cd;
ab.set(coordinates[1][0] - coordinates[0][0], coordinates[1][1] - coordinates[0][1],
coordinates[1][2] - coordinates[0][2]);
ab.normalize();
ac.set(coordinates[2][0] - coordinates[0][0], coordinates[2][1] - coordinates[0][1],
coordinates[2][2] - coordinates[0][2]);
ac.normalize();
ad.set(coordinates[3][0] - coordinates[0][0], coordinates[3][1] - coordinates[0][1],
coordinates[3][2] - coordinates[0][2]);
ad.normalize();
bc.set(coordinates[2][0] - coordinates[1][0], coordinates[2][1] - coordinates[1][1],
coordinates[2][2] - coordinates[1][2]);
bc.normalize();
bd.set(coordinates[3][0] - coordinates[1][0], coordinates[3][1] - coordinates[1][1],
coordinates[3][2] - coordinates[1][2]);
bd.normalize();
cd.set(coordinates[3][0] - coordinates[2][0], coordinates[3][1] - coordinates[2][1],
coordinates[3][2] - coordinates[2][2]);
cd.normalize();
VerdictVector abc = bc * ab;
abc.normalize();
VerdictVector abd = ab * ad;
abd.normalize();
VerdictVector acd = cd * ad;
acd.normalize();
VerdictVector bcd = bc * cd;
bcd.normalize();
double alpha = acos(-(abc % abd));
double beta = acos(-(abc % acd));
double gamma = acos(-(abc % bcd));
double delta = acos(-(abd % acd));
double epsilon = acos(-(abd % bcd));
double zeta = acos(-(acd % bcd));
double min_angle = alpha;
min_angle = min_angle < beta ? min_angle : beta;
min_angle = min_angle < gamma ? min_angle : gamma;
min_angle = min_angle < delta ? min_angle : delta;
min_angle = min_angle < epsilon ? min_angle : epsilon;
min_angle = min_angle < zeta ? min_angle : zeta;
min_angle *= normal_coeff;
double max_angle = alpha;
max_angle = max_angle > beta ? max_angle : beta;
max_angle = max_angle > gamma ? max_angle : gamma;
max_angle = max_angle > delta ? max_angle : delta;
max_angle = max_angle > epsilon ? max_angle : epsilon;
max_angle = max_angle > zeta ? max_angle : zeta;
max_angle *= normal_coeff;
double theta = acos(1 / 3.0) * normal_coeff; // 70.528779365509308630754000660038;
double dihedral_skew_max = (max_angle - theta) / (180 - theta);
double dihedral_skew_min = (theta - min_angle) / theta;
min_angle = 360.0;
max_angle = 0.0;
double angles[12];
angles[0] = acos(-(ab % bc));
angles[1] = acos((bc % ac));
angles[2] = acos((ac % ab));
angles[3] = acos(-(ab % bd));
angles[4] = acos((bd % ad));
angles[5] = acos((ad % ab));
angles[6] = acos(-(bc % cd));
angles[7] = acos((cd % bd));
angles[8] = acos((bd % bc));
angles[9] = acos((ad % cd));
angles[10] = acos(-(cd % ac));
angles[11] = acos((ac % ad));
for (int a = 0; a < 12; a++)
{
if (angles[a] < min_angle)
{
min_angle = angles[a];
}
if (angles[a] > max_angle)
{
max_angle = angles[a];
}
}
max_angle *= normal_coeff;
min_angle *= normal_coeff;
double skew_max = (max_angle - 60.0) / 120.0;
double skew_min = (60.0 - min_angle) / 60.0;
double max_skew = dihedral_skew_min;
max_skew = max_skew > dihedral_skew_max ? max_skew : dihedral_skew_max;
max_skew = max_skew > skew_min ? max_skew : skew_min;
max_skew = max_skew > skew_max ? max_skew : skew_max;
return max_skew;
}
/*!
get the weights based on the average size
of a tet
*/
VERDICT_HOST_DEVICE static int tet_get_weight(
VerdictVector& w1, VerdictVector& w2, VerdictVector& w3, double average_tet_volume)
{
w1.set(1, 0, 0);
w2.set(0.5, 0.5 * sqrt3, 0);
w3.set(0.5, sqrt3 / 6.0, sqrt2 / sqrt3);
double scale = pow(6. * average_tet_volume / determinant(w1, w2, w3), one_third);
w1 *= scale;
w2 *= scale;
w3 *= scale;
return 1;
}
/*!
the edge ratio of a tet
NB (P. Pebay 01/22/07):
Hmax / Hmin where Hmax and Hmin are respectively the maximum and the
minimum edge lengths
*/
VERDICT_HOST_DEVICE double tet_edge_ratio(int /*num_nodes*/, const double coordinates[][3])
{
VerdictVector a, b, c, d, e, f;
a.set(coordinates[1][0] - coordinates[0][0], coordinates[1][1] - coordinates[0][1],
coordinates[1][2] - coordinates[0][2]);
b.set(coordinates[2][0] - coordinates[1][0], coordinates[2][1] - coordinates[1][1],
coordinates[2][2] - coordinates[1][2]);
c.set(coordinates[0][0] - coordinates[2][0], coordinates[0][1] - coordinates[2][1],
coordinates[0][2] - coordinates[2][2]);
d.set(coordinates[3][0] - coordinates[0][0], coordinates[3][1] - coordinates[0][1],
coordinates[3][2] - coordinates[0][2]);
e.set(coordinates[3][0] - coordinates[1][0], coordinates[3][1] - coordinates[1][1],
coordinates[3][2] - coordinates[1][2]);
f.set(coordinates[3][0] - coordinates[2][0], coordinates[3][1] - coordinates[2][1],
coordinates[3][2] - coordinates[2][2]);
double a2 = a.length_squared();
double b2 = b.length_squared();
double c2 = c.length_squared();
double d2 = d.length_squared();
double e2 = e.length_squared();
double f2 = f.length_squared();
double m2, M2, mab, mcd, mef, Mab, Mcd, Mef;
if (a2 < b2)
{
mab = a2;
Mab = b2;
}
else // b2 <= a2
{
mab = b2;
Mab = a2;
}
if (c2 < d2)
{
mcd = c2;
Mcd = d2;
}
else // d2 <= c2
{
mcd = d2;
Mcd = c2;
}
if (e2 < f2)
{
mef = e2;
Mef = f2;
}
else // f2 <= e2
{
mef = f2;
Mef = e2;
}
m2 = mab < mcd ? mab : mcd;
m2 = m2 < mef ? m2 : mef;
if (m2 < VERDICT_DBL_MIN)
{
return (double)VERDICT_DBL_MAX;
}
M2 = Mab > Mcd ? Mab : Mcd;
M2 = M2 > Mef ? M2 : Mef;
const double edge_ratio = sqrt(M2 / m2);
return fix_range(edge_ratio);
}
/*!
the scaled jacobian of a tet
minimum of the jacobian divided by the lengths of 3 edge vectors
*/
template <typename CoordsContainerType>
VERDICT_HOST_DEVICE static double tet_scaled_jacobian_impl(int /*num_nodes*/, const CoordsContainerType coordinates)
{
const VerdictVector side0{coordinates[0], coordinates[1]};
const VerdictVector side1{coordinates[1], coordinates[2]};
const VerdictVector side2{coordinates[2], coordinates[0]};
const VerdictVector side3{coordinates[0], coordinates[3]};
const VerdictVector side4{coordinates[1], coordinates[3]};
const VerdictVector side5{coordinates[2], coordinates[3]};
const double jacobi = side3 % (side2 * side0);
// products of lengths squared of each edge attached to a node.
const double side0_length_squared = side0.length_squared();
const double side1_length_squared = side1.length_squared();
const double side2_length_squared = side2.length_squared();
const double side3_length_squared = side3.length_squared();
const double side4_length_squared = side4.length_squared();
const double side5_length_squared = side5.length_squared();
const double length_squared[4] = {
side0_length_squared * side2_length_squared * side3_length_squared,
side0_length_squared * side1_length_squared * side4_length_squared,
side1_length_squared * side2_length_squared * side5_length_squared,
side3_length_squared * side4_length_squared * side5_length_squared };
int which_node = 0;
if (length_squared[1] > length_squared[which_node])
{
which_node = 1;
}
if (length_squared[2] > length_squared[which_node])
{
which_node = 2;
}
if (length_squared[3] > length_squared[which_node])
{
which_node = 3;
}
double length_product = sqrt(length_squared[which_node]);
if (length_product < fabs(jacobi))
{
length_product = fabs(jacobi);
}
if (length_product < VERDICT_DBL_MIN)
{
return VERDICT_DBL_MAX;
}
return sqrt2 * jacobi / length_product;
}
VERDICT_HOST_DEVICE double tet_scaled_jacobian(int num_nodes, const double coordinates[][3])
{
return tet_scaled_jacobian_impl(num_nodes, coordinates);
}
VERDICT_HOST_DEVICE double tet_scaled_jacobian_from_loc_ptrs(int num_nodes, const double * const * coordinates)
{
return tet_scaled_jacobian_impl(num_nodes, coordinates);
}
/*!
The radius ratio of a tet
NB (P. Pebay 04/16/07):
CR / (3.0 * IR) where CR is the circumsphere radius and IR is the inscribed
sphere radius.
Note that this function is similar to the aspect beta of a tet, except that
it does not return VERDICT_DBL_MAX if the element has negative orientation.
*/
VERDICT_HOST_DEVICE double tet_radius_ratio(int /*num_nodes*/, const double coordinates[][3])
{
// Determine side vectors
VerdictVector side[6];
side[0].set(coordinates[1][0] - coordinates[0][0], coordinates[1][1] - coordinates[0][1],
coordinates[1][2] - coordinates[0][2]);
side[1].set(coordinates[2][0] - coordinates[1][0], coordinates[2][1] - coordinates[1][1],
coordinates[2][2] - coordinates[1][2]);
side[2].set(coordinates[0][0] - coordinates[2][0], coordinates[0][1] - coordinates[2][1],
coordinates[0][2] - coordinates[2][2]);
side[3].set(coordinates[3][0] - coordinates[0][0], coordinates[3][1] - coordinates[0][1],
coordinates[3][2] - coordinates[0][2]);
side[4].set(coordinates[3][0] - coordinates[1][0], coordinates[3][1] - coordinates[1][1],
coordinates[3][2] - coordinates[1][2]);
side[5].set(coordinates[3][0] - coordinates[2][0], coordinates[3][1] - coordinates[2][1],
coordinates[3][2] - coordinates[2][2]);
VerdictVector numerator = side[3].length_squared() * (side[2] * side[0]) +
side[2].length_squared() * (side[3] * side[0]) + side[0].length_squared() * (side[3] * side[2]);
double area_sum;
area_sum = ((side[2] * side[0]).length() + (side[3] * side[0]).length() +
(side[4] * side[1]).length() + (side[3] * side[2]).length()) *
0.5;
double volume = tet_volume(4, coordinates);
if (fabs(volume) < VERDICT_DBL_MIN)
{
return (double)VERDICT_DBL_MAX;
}
else
{
const double radius_ratio = numerator.length() * area_sum / (108 * volume * volume);
return fix_range(radius_ratio);
}
}
/*!
The aspect ratio of a tet
NB (P. Pebay 01/22/07):
Hmax / (2 sqrt(6) r) where Hmax and r respectively denote the greatest edge
length and the inradius of the tetrahedron
NB (J. Pouderoux 01/27/15)
This will return VERDICT_DBL_MAX when the volume of the tetrahedron is ill-
conditioned. Previously, this would only happen when the volume was small
and positive, but now ill-conditioned inverted tetrahedra are also included.
*/
template <typename CoordsContainerType>
VERDICT_HOST_DEVICE static double tet_aspect_ratio_impl(int /*num_nodes*/, const CoordsContainerType coordinates)
{
// Determine side vectors
const VerdictVector ab{coordinates[0], coordinates[1]};
const VerdictVector ac{coordinates[0], coordinates[2]};
const VerdictVector ad{coordinates[0], coordinates[3]};
double detTet = ab % (ac * ad);
if (fabs(detTet) < VERDICT_DBL_MIN)
{
return (double)VERDICT_DBL_MAX;
}
VerdictVector bc{coordinates[1], coordinates[2]};
VerdictVector bd{coordinates[1], coordinates[3]};
VerdictVector cd{coordinates[2], coordinates[3]};
const double ab2 = ab.length_squared();
const double bc2 = bc.length_squared();
const double ac2 = ac.length_squared();
const double ad2 = ad.length_squared();
const double bd2 = bd.length_squared();
const double cd2 = cd.length_squared();
double A = ab2 > bc2 ? ab2 : bc2;
double B = ac2 > ad2 ? ac2 : ad2;
double C = bd2 > cd2 ? bd2 : cd2;
double D = A > B ? A : B;
const double hm = D > C ? sqrt(D) : sqrt(C);
bd = ab * bc;
A = bd.length();
bd = ab * ad;
B = bd.length();
bd = ac * ad;
C = bd.length();
bd = bc * cd;
D = bd.length();
const double aspect_ratio = aspect_ratio_normal_coeff * hm * (A + B + C + D) / fabs(detTet);
return fix_range(aspect_ratio);
}
VERDICT_HOST_DEVICE double tet_aspect_ratio(int num_nodes, const double coordinates[][3])
{
return tet_aspect_ratio_impl(num_nodes, coordinates);
}
VERDICT_HOST_DEVICE double tet_aspect_ratio_from_loc_ptrs(int num_nodes, const double * const *coordinates)
{
return tet_aspect_ratio_impl(num_nodes, coordinates);
}
/*!
the aspect gamma of a tet
srms^3 / (8.48528137423857*V) where srms = sqrt(sum(Si^2)/6), where Si is the edge length
*/
VERDICT_HOST_DEVICE double tet_aspect_gamma(int /*num_nodes*/, const double coordinates[][3])
{
// Determine side vectors
VerdictVector side0, side1, side2, side3, side4, side5;
side0.set(coordinates[1][0] - coordinates[0][0], coordinates[1][1] - coordinates[0][1],
coordinates[1][2] - coordinates[0][2]);
side1.set(coordinates[2][0] - coordinates[1][0], coordinates[2][1] - coordinates[1][1],
coordinates[2][2] - coordinates[1][2]);
side2.set(coordinates[0][0] - coordinates[2][0], coordinates[0][1] - coordinates[2][1],
coordinates[0][2] - coordinates[2][2]);
side3.set(coordinates[3][0] - coordinates[0][0], coordinates[3][1] - coordinates[0][1],
coordinates[3][2] - coordinates[0][2]);
side4.set(coordinates[3][0] - coordinates[1][0], coordinates[3][1] - coordinates[1][1],
coordinates[3][2] - coordinates[1][2]);
side5.set(coordinates[3][0] - coordinates[2][0], coordinates[3][1] - coordinates[2][1],
coordinates[3][2] - coordinates[2][2]);
double volume = fabs(tet_volume(4, coordinates));
if (volume < VERDICT_DBL_MIN)
{
return (double)VERDICT_DBL_MAX;
}
else
{
double srms =
sqrt((side0.length_squared() + side1.length_squared() + side2.length_squared() +
side3.length_squared() + side4.length_squared() + side5.length_squared()) /
6.0);
double aspect_ratio_gamma = (srms * srms * srms) / (8.48528137423857 * volume);
return (double)aspect_ratio_gamma;
}
}
/*!
The aspect frobenius of a tet
NB (P. Pebay 01/22/07):
Frobenius condition number when the reference element is regular
*/
VERDICT_HOST_DEVICE double tet_aspect_frobenius(int /*num_nodes*/, const double coordinates[][3])
{
VerdictVector ab, ac, ad;
ab.set(coordinates[1][0] - coordinates[0][0], coordinates[1][1] - coordinates[0][1],
coordinates[1][2] - coordinates[0][2]);
ac.set(coordinates[2][0] - coordinates[0][0], coordinates[2][1] - coordinates[0][1],
coordinates[2][2] - coordinates[0][2]);
ad.set(coordinates[3][0] - coordinates[0][0], coordinates[3][1] - coordinates[0][1],
coordinates[3][2] - coordinates[0][2]);
double denominator = ab % (ac * ad);
denominator *= denominator;
denominator *= 2.;
denominator = 3. * pow(denominator, one_third);
if (denominator < VERDICT_DBL_MIN)
{
return (double)VERDICT_DBL_MAX;
}
double u[3];
ab.get_xyz(u);
double v[3];
ac.get_xyz(v);
double w[3];
ad.get_xyz(w);
double numerator = u[0] * u[0] + u[1] * u[1] + u[2] * u[2];
numerator += v[0] * v[0] + v[1] * v[1] + v[2] * v[2];
numerator += w[0] * w[0] + w[1] * w[1] + w[2] * w[2];
numerator *= 1.5;
numerator -= v[0] * u[0] + v[1] * u[1] + v[2] * u[2];
numerator -= w[0] * u[0] + w[1] * u[1] + w[2] * u[2];
numerator -= w[0] * v[0] + w[1] * v[1] + w[2] * v[2];
double aspect_frobenius = numerator / denominator;
return fix_range(aspect_frobenius);
}
/*!
The minimum angle of a tet
NB (P. Pebay 01/22/07):
minimum nonoriented dihedral angle
*/
VERDICT_HOST_DEVICE double tet_minimum_angle(int /*num_nodes*/, const double coordinates[][3])
{
// Determine side vectors
VerdictVector ab, bc, ad, cd;
ab.set(coordinates[1][0] - coordinates[0][0], coordinates[1][1] - coordinates[0][1],
coordinates[1][2] - coordinates[0][2]);
ad.set(coordinates[3][0] - coordinates[0][0], coordinates[3][1] - coordinates[0][1],
coordinates[3][2] - coordinates[0][2]);
bc.set(coordinates[2][0] - coordinates[1][0], coordinates[2][1] - coordinates[1][1],
coordinates[2][2] - coordinates[1][2]);
cd.set(coordinates[3][0] - coordinates[2][0], coordinates[3][1] - coordinates[2][1],
coordinates[3][2] - coordinates[2][2]);
VerdictVector abc = ab * bc;
double nabc = abc.length();
VerdictVector abd = ab * ad;
double nabd = abd.length();
VerdictVector acd = ad * cd;
double nacd = acd.length();
VerdictVector bcd = bc * cd;
double nbcd = bcd.length();
double alpha = acos((abc % abd) / (nabc * nabd));
double beta = acos((abc % acd) / (nabc * nacd));
double gamma = acos((abc % bcd) / (nabc * nbcd));
double delta = acos((abd % acd) / (nabd * nacd));
double epsilon = acos((abd % bcd) / (nabd * nbcd));
double zeta = acos((acd % bcd) / (nacd * nbcd));
alpha = alpha < beta ? alpha : beta;
alpha = alpha < gamma ? alpha : gamma;
alpha = alpha < delta ? alpha : delta;
alpha = alpha < epsilon ? alpha : epsilon;
alpha = alpha < zeta ? alpha : zeta;
alpha *= normal_coeff;
return fix_range(alpha);
}
/*!
The collapse ratio of a tet
NB (J. Pouderoux 01/27/15)
This will return VERDICT_DBL_MAX when the volume of the tetrahedron is ill-
conditioned. Previously, this would only happen when the volume was small
and positive, but now ill-conditioned inverted tetrahedra are also included.
*/
VERDICT_HOST_DEVICE double tet_collapse_ratio(int /*num_nodes*/, const double coordinates[][3])
{
// Determine side vectors
VerdictVector e01, e02, e03, e12, e13, e23;
e01.set(coordinates[1][0] - coordinates[0][0], coordinates[1][1] - coordinates[0][1],
coordinates[1][2] - coordinates[0][2]);
e02.set(coordinates[2][0] - coordinates[0][0], coordinates[2][1] - coordinates[0][1],
coordinates[2][2] - coordinates[0][2]);
e03.set(coordinates[3][0] - coordinates[0][0], coordinates[3][1] - coordinates[0][1],
coordinates[3][2] - coordinates[0][2]);
e12.set(coordinates[2][0] - coordinates[1][0], coordinates[2][1] - coordinates[1][1],
coordinates[2][2] - coordinates[1][2]);
e13.set(coordinates[3][0] - coordinates[1][0], coordinates[3][1] - coordinates[1][1],
coordinates[3][2] - coordinates[1][2]);
e23.set(coordinates[3][0] - coordinates[2][0], coordinates[3][1] - coordinates[2][1],
coordinates[3][2] - coordinates[2][2]);
double l[6];
l[0] = e01.length();
l[1] = e02.length();
l[2] = e03.length();
l[3] = e12.length();
l[4] = e13.length();
l[5] = e23.length();
// Find longest edge for each bounding triangle of tetrahedron
double l012 = l[4] > l[0] ? l[4] : l[0];
l012 = l[1] > l012 ? l[1] : l012;
double l031 = l[0] > l[2] ? l[0] : l[2];
l031 = l[3] > l031 ? l[3] : l031;
double l023 = l[2] > l[1] ? l[2] : l[1];
l023 = l[5] > l023 ? l[5] : l023;
double l132 = l[4] > l[3] ? l[4] : l[3];
l132 = l[5] > l132 ? l[5] : l132;
// Compute collapse ratio for each vertex/triangle pair
VerdictVector N;
double h, magN;
double cr;
double crMin;
N = e01 * e02;
magN = N.length();
h = (e03 % N) / magN; // height of vertex 3 above 0-1-2
crMin = h / l012; // ratio of height to longest edge of 0-1-2
N = e03 * e01;
magN = N.length();
h = (e02 % N) / magN; // height of vertex 2 above 0-3-1
cr = h / l031; // ratio of height to longest edge of 0-3-1
if (cr < crMin)
{
crMin = cr;
}
N = e02 * e03;
magN = N.length();
h = (e01 % N) / magN; // height of vertex 1 above 0-2-3
cr = h / l023; // ratio of height to longest edge of 0-2-3
if (cr < crMin)
{
crMin = cr;
}
N = e12 * e13;
magN = N.length();
h = (e01 % N) / magN; // height of vertex 0 above 1-3-2
cr = h / l132; // ratio of height to longest edge of 1-3-2
if (cr < crMin)
{
crMin = cr;
}
return fix_range(crMin);
}
VERDICT_HOST_DEVICE double tet_equivolume_skew(int num_nodes, const double coordinates[][3])
{
//- Find the vectors from the origin to each of the nodes on the tet.
VerdictVector vectA(coordinates[0][0], coordinates[0][1], coordinates[0][2]);
VerdictVector vectB(coordinates[1][0], coordinates[1][1], coordinates[1][2]);
VerdictVector vectC(coordinates[2][0], coordinates[2][1], coordinates[2][2]);
VerdictVector vectD(coordinates[3][0], coordinates[3][1], coordinates[3][2]);
VerdictVector vectAB = vectB - vectA;
VerdictVector vectAC = vectC - vectA;
VerdictVector vectAD = vectD - vectA;
double sq_lengthAB = vectAB.length_squared();
double sq_lengthAC = vectAC.length_squared();
double sq_lengthAD = vectAD.length_squared();
VerdictVector cpBC = vectAB * vectAC;
VerdictVector cpDB = vectAD * vectAB;
VerdictVector cpCD = vectAC * vectAD;
VerdictVector num = sq_lengthAD * cpBC + sq_lengthAC * cpDB + sq_lengthAB * cpCD;
double den = 2 * vectAB % cpCD;
double circumradius = num.length() / den;
double volume = tet_volume(num_nodes, coordinates);
double optimal_length = circumradius / sqrt(double(3.0) / 8.0);
double optimal_volume = (1.0 / 12.0) * sqrt(double(2.0)) * (optimal_length * optimal_length * optimal_length);
const double eq_v_skew = (optimal_volume - volume) / optimal_volume;
return fix_range(eq_v_skew);
}
VERDICT_HOST_DEVICE double tet_squish_index(int /*num_nodes*/, const double coordinates[][3])
{
VerdictVector vectA(coordinates[0][0], coordinates[0][1], coordinates[0][2]);
VerdictVector vectB(coordinates[1][0], coordinates[1][1], coordinates[1][2]);
VerdictVector vectC(coordinates[2][0], coordinates[2][1], coordinates[2][2]);
VerdictVector vectD(coordinates[3][0], coordinates[3][1], coordinates[3][2]);
VerdictVector tetCenter = vectA + vectB + vectC + vectD;
tetCenter /= 4.0;
/* top view
C
/|\
/ 5 \
2 / D \ 1
/ 3/ \4 \
/_/ \_\
A-----------B
0
*/
VerdictVector side[6];
side[0].set(vectA, vectB);
side[1].set(vectB, vectC);
side[2].set(vectC, vectA);
side[3].set(vectA, vectD);
side[4].set(vectB, vectD);
side[5].set(vectC, vectD);
double maxSquishIndex = 0;
double squishIndex = 0;
VerdictVector faceCenter;
VerdictVector centerCenterVector;
VerdictVector faceAreaVector;
// face 1
faceCenter = (vectA + vectB + vectD) / 3.0;
centerCenterVector = faceCenter - tetCenter;
faceAreaVector = 0.5 * (side[0] * side[4]);
squishIndex = 1 -
(faceAreaVector % centerCenterVector) / (faceAreaVector.length() * centerCenterVector.length());
if (squishIndex > maxSquishIndex)
{
maxSquishIndex = squishIndex;
}
// face 2
faceCenter = (vectB + vectC + vectD) / 3.0;
centerCenterVector = faceCenter - tetCenter;
faceAreaVector = 0.5 * (side[1] * side[5]);
squishIndex = 1 -
(faceAreaVector % centerCenterVector) / (faceAreaVector.length() * centerCenterVector.length());
if (squishIndex > maxSquishIndex)
{
maxSquishIndex = squishIndex;
}
// face 3
faceCenter = (vectA + vectC + vectD) / 3.0;
centerCenterVector = faceCenter - tetCenter;
faceAreaVector = 0.5 * (side[2] * side[3]);
squishIndex = 1 -
(faceAreaVector % centerCenterVector) / (faceAreaVector.length() * centerCenterVector.length());
if (squishIndex > maxSquishIndex)
{
maxSquishIndex = squishIndex;
}
// face 4
faceCenter = (vectA + vectB + vectC) / 3.0;
centerCenterVector = faceCenter - tetCenter;
faceAreaVector = 0.5 * (side[1] * side[0]);
squishIndex = 1 -
(faceAreaVector % centerCenterVector) / (faceAreaVector.length() * centerCenterVector.length());
if (squishIndex > maxSquishIndex)
{
maxSquishIndex = squishIndex;
}
return maxSquishIndex;
}
VERDICT_HOST_DEVICE static const double* TET15_node_local_coord(int i)
{
static const double sTET15_node_local_coord[15][3] = { { 0, 0, 0 }, { 1.0, 0, 0 }, { 0, 1.0, 0 },
{ 0, 0, 1.0 }, { .5, 0, 0 }, { .5, .5, 0 }, { 0, .5, 0 }, { 0, 0, .5 }, { .5, 0, .5 },
{ 0, .5, .5 }, { one_third, one_third, 0 }, { one_third, 0, one_third },
{ one_third, one_third, one_third }, { 0, one_third, one_third },
{ one_fourth, one_fourth, one_fourth } };
return sTET15_node_local_coord[i];
};
VERDICT_HOST_DEVICE static void TET15_gradients_of_the_shape_functions_for_R_S_T(
const double rst[3], double dhdr[15], double dhds[15], double dhdt[15])
{
// dh/dr;
dhdr[0] = -1.0;
dhdr[1] = 1.0;
dhdr[2] = 0.0;
dhdr[3] = 0.0;
dhdr[4] = 4.0 * (1.0 - 2.0 * rst[0] - rst[1] - rst[2]);
dhdr[5] = 4.0 * rst[1];
dhdr[6] = -4.0 * rst[1];
dhdr[7] = -4.0 * rst[2];
dhdr[8] = 4.0 * rst[2];
dhdr[9] = 0.0;
dhdr[11] = 27.0 * (rst[1] - 2.0 * rst[0] * rst[1] - rst[1] * rst[1] - rst[1] * rst[2]);
dhdr[14] = 27.0 * (rst[2] - 2.0 * rst[0] * rst[2] - rst[1] * rst[2] - rst[2] * rst[2]);
dhdr[12] = 27.0 * rst[1] * rst[2];
dhdr[13] = -27.0 * rst[1] * rst[2];
dhdr[10] = 256.0 *
(rst[1] * rst[2] - 2.0 * rst[0] * rst[1] * rst[2] - rst[1] * rst[1] * rst[2] -
rst[1] * rst[2] * rst[2]);
// dh/ds;
dhds[0] = -1.0;
dhds[1] = 0.0;
dhds[2] = 1.0;
dhds[3] = 0.0;
dhds[4] = -4.0 * rst[0];
dhds[5] = 4.0 * rst[0];
dhds[6] = 4.0 * (1.0 - rst[0] - 2.0 * rst[1] - rst[2]);
dhds[7] = -4.0 * rst[2];
dhds[8] = 0.0;
dhds[9] = 4.0 * rst[2];
dhds[11] = 27.0 * (rst[0] - rst[0] * rst[0] - 2.0 * rst[0] * rst[1] - rst[0] * rst[2]);
dhds[14] = -27.0 * rst[0] * rst[2];
dhds[12] = 27.0 * rst[0] * rst[2];
dhds[13] = 27.0 * (rst[2] - rst[0] * rst[2] - 2.0 * rst[1] * rst[2] - rst[2] * rst[2]);
dhds[10] = 256.0 *
(rst[0] * rst[2] - rst[0] * rst[0] * rst[2] - 2.0 * rst[0] * rst[1] * rst[2] -
rst[0] * rst[2] * rst[2]);
// dh/dt;
dhdt[0] = -1.0;
dhdt[1] = 0.0;
dhdt[2] = 0.0;
dhdt[3] = 1.0;
dhdt[4] = -4.0 * rst[0];
dhdt[5] = 0.0;
dhdt[6] = -4.0 * rst[1];
dhdt[7] = 4.0 * (1.0 - rst[0] - rst[1] - 2.0 * rst[2]);
dhdt[8] = 4.0 * rst[0];
dhdt[9] = 4.0 * rst[1];
dhdt[11] = -27.0 * rst[0] * rst[1];
dhdt[14] = 27.0 * (rst[0] - rst[0] * rst[0] - rst[0] * rst[1] - 2.0 * rst[0] * rst[2]);
dhdt[12] = 27.0 * rst[0] * rst[1];
dhdt[13] = 27.0 * (rst[1] - rst[0] * rst[1] - rst[1] * rst[1] - 2.0 * rst[1] * rst[2]);
dhdt[10] = 256.0 *
(rst[0] * rst[1] - rst[0] * rst[0] * rst[1] - rst[0] * rst[1] * rst[1] -
2.0 * rst[0] * rst[1] * rst[2]);
// ----------------------------------------------;
// ADD CONTRIBUTIONS OF NODES 5-15 TO NODES 1-14;
// ----------------------------------------------;
// dh/dr;
dhdr[11] = dhdr[11] - 108.0 * dhdr[10] / 256.0;
dhdr[14] = dhdr[14] - 108.0 * dhdr[10] / 256.0;
dhdr[12] = dhdr[12] - 108.0 * dhdr[10] / 256.0;
dhdr[13] = dhdr[13] - 108.0 * dhdr[10] / 256.0;
dhdr[4] = dhdr[4] - four_ninths * (dhdr[11] + dhdr[14]) - .25 * dhdr[10];
dhdr[5] = dhdr[5] - four_ninths * (dhdr[11] + dhdr[12]) - .25 * dhdr[10];
dhdr[6] = dhdr[6] - four_ninths * (dhdr[11] + dhdr[13]) - .25 * dhdr[10];
dhdr[7] = dhdr[7] - four_ninths * (dhdr[14] + dhdr[13]) - .25 * dhdr[10];
dhdr[8] = dhdr[8] - four_ninths * (dhdr[14] + dhdr[12]) - .25 * dhdr[10];
dhdr[9] = dhdr[9] - four_ninths * (dhdr[12] + dhdr[13]) - .25 * dhdr[10];
dhdr[0] = dhdr[0] - .5 * (dhdr[4] + dhdr[6] + dhdr[7]) -
one_third * (dhdr[11] + dhdr[14] + dhdr[13]) - .25 * dhdr[10];
dhdr[1] = dhdr[1] - .5 * (dhdr[4] + dhdr[5] + dhdr[8]) -
one_third * (dhdr[11] + dhdr[14] + dhdr[12]) - .25 * dhdr[10];
dhdr[2] = dhdr[2] - .5 * (dhdr[5] + dhdr[6] + dhdr[9]) -
one_third * (dhdr[11] + dhdr[12] + dhdr[13]) - .25 * dhdr[10];
dhdr[3] = dhdr[3] - .5 * (dhdr[7] + dhdr[8] + dhdr[9]) -
one_third * (dhdr[14] + dhdr[12] + dhdr[13]) - .25 * dhdr[10];
// dh/ds;
dhds[11] = dhds[11] - 108.0 * dhds[10] / 256.0;
dhds[14] = dhds[14] - 108.0 * dhds[10] / 256.0;
dhds[12] = dhds[12] - 108.0 * dhds[10] / 256.0;
dhds[13] = dhds[13] - 108.0 * dhds[10] / 256.0;
dhds[4] = dhds[4] - four_ninths * (dhds[11] + dhds[14]) - .25 * dhds[10];
dhds[5] = dhds[5] - four_ninths * (dhds[11] + dhds[12]) - .25 * dhds[10];
dhds[6] = dhds[6] - four_ninths * (dhds[11] + dhds[13]) - .25 * dhds[10];
dhds[7] = dhds[7] - four_ninths * (dhds[14] + dhds[13]) - .25 * dhds[10];
dhds[8] = dhds[8] - four_ninths * (dhds[14] + dhds[12]) - .25 * dhds[10];
dhds[9] = dhds[9] - four_ninths * (dhds[12] + dhds[13]) - .25 * dhds[10];
dhds[0] = dhds[0] - .5 * (dhds[4] + dhds[6] + dhds[7]) -
one_third * (dhds[11] + dhds[14] + dhds[13]) - .25 * dhds[10];
dhds[1] = dhds[1] - .5 * (dhds[4] + dhds[5] + dhds[8]) -
one_third * (dhds[11] + dhds[14] + dhds[12]) - .25 * dhds[10];
dhds[2] = dhds[2] - .5 * (dhds[5] + dhds[6] + dhds[9]) -
one_third * (dhds[11] + dhds[12] + dhds[13]) - .25 * dhds[10];
dhds[3] = dhds[3] - .5 * (dhds[7] + dhds[8] + dhds[9]) -
one_third * (dhds[14] + dhds[12] + dhds[13]) - .25 * dhds[10];
// dh/dt;
dhdt[11] = dhdt[11] - 108.0 * dhdt[10] / 256.0;
dhdt[14] = dhdt[14] - 108.0 * dhdt[10] / 256.0;
dhdt[12] = dhdt[12] - 108.0 * dhdt[10] / 256.0;
dhdt[13] = dhdt[13] - 108.0 * dhdt[10] / 256.0;
dhdt[4] = dhdt[4] - four_ninths * (dhdt[11] + dhdt[14]) - .25 * dhdt[10];
dhdt[5] = dhdt[5] - four_ninths * (dhdt[11] + dhdt[12]) - .25 * dhdt[10];
dhdt[6] = dhdt[6] - four_ninths * (dhdt[11] + dhdt[13]) - .25 * dhdt[10];
dhdt[7] = dhdt[7] - four_ninths * (dhdt[14] + dhdt[13]) - .25 * dhdt[10];
dhdt[8] = dhdt[8] - four_ninths * (dhdt[14] + dhdt[12]) - .25 * dhdt[10];
dhdt[9] = dhdt[9] - four_ninths * (dhdt[12] + dhdt[13]) - .25 * dhdt[10];
dhdt[0] = dhdt[0] - .5 * (dhdt[4] + dhdt[6] + dhdt[7]) -
one_third * (dhdt[11] + dhdt[14] + dhdt[13]) - .25 * dhdt[10];
dhdt[1] = dhdt[1] - .5 * (dhdt[4] + dhdt[5] + dhdt[8]) -
one_third * (dhdt[11] + dhdt[14] + dhdt[12]) - .25 * dhdt[10];
dhdt[2] = dhdt[2] - .5 * (dhdt[5] + dhdt[6] + dhdt[9]) -
one_third * (dhdt[11] + dhdt[12] + dhdt[13]) - .25 * dhdt[10];
dhdt[3] = dhdt[3] - .5 * (dhdt[7] + dhdt[8] + dhdt[9]) -
one_third * (dhdt[14] + dhdt[12] + dhdt[13]) - .25 * dhdt[10];
}
VERDICT_HOST_DEVICE static double calculate_tet_volume_using_sides(
const VerdictVector& side0, const VerdictVector& side2, const VerdictVector& side3)
{
return (double)((side3 % (side2 * side0)) / 6.0);
}
/*!
the volume of a tet
1/6 * jacobian at a corner node
*/
template <typename CoordsContainerType>
VERDICT_HOST_DEVICE static double tet_volume_impl(int num_nodes, const CoordsContainerType coordinates)
{
// Determine side vectors
if (4 == num_nodes)
{
const VerdictVector side2{coordinates[0], coordinates[1]};
const VerdictVector side0{coordinates[0], coordinates[2]};
const VerdictVector side3{coordinates[0], coordinates[3]};
return calculate_tet_volume_using_sides(side0, side2, side3);
}
else
{
VerdictVector tet_pts[15];
VerdictVector side0, side2, side3;
// create a vector for each point
for (int k = 0; k < num_nodes; k++)
{
tet_pts[k].set(coordinates[k][0], coordinates[k][1], coordinates[k][2]);
}
// determine center point of the higher-order nodes
VerdictVector centroid(0.0, 0.0, 0.0);
for (int k = 4; k < num_nodes; k++)
{
centroid += VerdictVector(coordinates[k][0], coordinates[k][1], coordinates[k][2]);
}
centroid /= (num_nodes - 4);
if (8 == num_nodes)
{
double tet_volume = 0;
int tet_face_conn[4][4] = { { 0, 2, 1, 4 }, { 0, 1, 3, 7 }, { 1, 2, 3, 5 }, { 0, 3, 2, 6 } };
for (int i = 0; i < 4; i++)
{
VerdictVector& node0 = tet_pts[tet_face_conn[i][0]];
VerdictVector& node1 = tet_pts[tet_face_conn[i][1]];
VerdictVector& node2 = tet_pts[tet_face_conn[i][2]];