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V_WedgeMetric.cpp
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/*=========================================================================
Module: V_WedgeMetric.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.
=========================================================================*/
/*
*
* WedgeMetric.cpp contains quality calculations for wedges
*
* This file is part of VERDICT
*
*/
#include "verdict.h"
#include "VerdictVector.hpp"
#include "verdict_defines.hpp"
#include <math.h>
namespace VERDICT_NAMESPACE
{
VERDICT_HOST_DEVICE extern double tri_equiangle_skew(int num_nodes, const double coordinates[][3]);
VERDICT_HOST_DEVICE extern double quad_equiangle_skew(int num_nodes, const double coordinates[][3]);
// local methods
VERDICT_HOST_DEVICE void make_wedge_faces(const double coordinates[][3], double tri1[][3], double tri2[][3],
double quad1[][3], double quad2[][3], double quad3[][3]);
/*
the wedge element
5
^
/ \
/ | \
/ /2\ \
6/_______\4
| / \ |
|/_____\|
3 1
*/
VERDICT_HOST_DEVICE static const double* WEDGE21_node_local_coord(int i)
{
static constexpr double sWEDGE21_node_local_coord[21][3] = { { 0, 0, -1 }, { 1.0, 0, -1 },
{ 0, 1.0, -1 }, { 0, 0, 1.0 }, { 1.0, 0, 1.0 }, { 0, 1.0, 1.0 }, { 0.5, 0, -1 }, { 0.5, 0.5, -1 },
{ 0, 0.5, -1 }, { 0.0, 0.0, 0 }, { 1.0, 0, 0 }, { 0, 1.0, 0 }, { 0.5, 0, 1.0 }, { 0.5, 0.5, 1.0 },
{ 0, 0.5, 1.0 }, { one_third, one_third, 0 }, { one_third, one_third, -1 },
{ one_third, one_third, 1.0 }, { 0.5, 0.5, 0 }, { 0, 0.5, 0 }, { 0.5, 0, 0 } };
return sWEDGE21_node_local_coord[i];
}
VERDICT_HOST_DEVICE static void WEDGE21_gradients_of_the_shape_functions_for_RST(
const double rst[3], double dhdr[21], double dhds[21], double dhdt[21])
{
double RSM = 1.0 - rst[0] - rst[1];
double RR = rst[0] * rst[0];
double RS = rst[0] * rst[1];
double SS = rst[1] * rst[1];
double TP = 1.0 + rst[2];
double TM = 1.0 - rst[2];
double T2P = 1.0 + 2.0 * rst[2];
double T2M = 1.0 - 2.0 * rst[2];
dhdr[0] = -0.5 * rst[2] * TM * (4.0 * rst[0] + 7.0 * rst[1] - 3.0 - 6.0 * RS - 3.0 * SS);
dhds[0] = -0.5 * rst[2] * TM * (7.0 * rst[0] + 4.0 * rst[1] - 3.0 - 6.0 * RS - 3.0 * RR);
dhdt[0] = -0.5 * T2M * RSM * (1.0 - 2.0 * (rst[0] + rst[1]) + 3.0 * RS);
dhdr[1] = -0.5 * rst[2] * TM * (4.0 * rst[0] - 1.0 + 3.0 * rst[1] - 6.0 * RS - 3.0 * SS);
dhds[1] = -0.5 * rst[2] * TM * (3.0 * rst[0] - 6.0 * RS - 3.0 * RR);
dhdt[1] = -0.5 * T2M * (rst[0] - 2.0 * (RSM * rst[0] + RS) + 3.0 * RSM * RS);
dhdr[2] = -0.5 * rst[2] * TM * (3.0 * rst[1] - 6.0 * RS - 3.0 * SS);
dhds[2] = -0.5 * rst[2] * TM * (4.0 * rst[1] - 1.0 + 3.0 * rst[0] - 6.0 * RS - 3.0 * RR);
dhdt[2] = -0.5 * T2M * (rst[1] - 2.0 * (RSM * rst[1] + RS) + 3.0 * RSM * RS);
dhdr[3] = 0.5 * rst[2] * TP * (4.0 * rst[0] + 7.0 * rst[1] - 3.0 - 6.0 * RS - 3.0 * SS);
dhds[3] = 0.5 * rst[2] * TP * (7.0 * rst[0] + 4.0 * rst[1] - 3.0 - 6.0 * RS - 3.0 * RR);
dhdt[3] = 0.5 * T2P * RSM * (1.0 - 2.0 * (rst[0] + rst[1]) + 3.0 * RS);
dhdr[4] = 0.5 * rst[2] * TP * (4.0 * rst[0] - 1.0 + 3.0 * rst[1] - 6.0 * RS - 3.0 * SS);
dhds[4] = 0.5 * rst[2] * TP * (3.0 * rst[0] - 6.0 * RS - 3.0 * RR);
dhdt[4] = 0.5 * T2P * (rst[0] - 2.0 * (RSM * rst[0] + RS) + 3.0 * RSM * RS);
dhdr[5] = 0.5 * rst[2] * TP * (3.0 * rst[1] - 6.0 * RS - 3.0 * SS);
dhds[5] = 0.5 * rst[2] * TP * (4.0 * rst[1] - 1.0 + 3.0 * rst[0] - 6.0 * RS - 3.0 * RR);
dhdt[5] = 0.5 * T2P * (rst[1] - 2.0 * (RSM * rst[1] + RS) + 3.0 * RSM * RS);
dhdr[6] = -0.5 * rst[2] * TM * (4.0 - 8.0 * rst[0] - 16.0 * rst[1] + 24.0 * RS + 12.0 * SS);
dhds[6] = -0.5 * rst[2] * TM * (-16.0 * rst[0] + 12.0 * RR + 24.0 * RS);
dhdt[6] = -0.5 * T2M * RSM * (4.0 * rst[0] - 12.0 * RS);
dhdr[7] = -0.5 * rst[2] * TM * (-8.0 * rst[1] + 24.0 * RS + 12.0 * SS);
dhds[7] = -0.5 * rst[2] * TM * (-8.0 * rst[0] + 12.0 * RR + 24.0 * RS);
dhdt[7] = -0.5 * T2M * (4.0 * RS - 12.0 * RSM * RS);
dhdr[8] = -0.5 * rst[2] * TM * (-16.0 * rst[1] + 24.0 * RS + 12.0 * SS);
dhds[8] = -0.5 * rst[2] * TM * (4.0 - 16.0 * rst[0] - 8.0 * rst[1] + 12.0 * RR + 24.0 * RS);
dhdt[8] = -0.5 * T2M * RSM * (4.0 * rst[1] - 12.0 * RS);
dhdr[12] = 0.5 * rst[2] * TP * (4.0 - 8.0 * rst[0] - 16.0 * rst[1] + 24.0 * RS + 12.0 * SS);
dhds[12] = 0.5 * rst[2] * TP * (-16.0 * rst[0] + 12.0 * RR + 24.0 * RS);
dhdt[12] = 0.5 * T2P * RSM * (4.0 * rst[0] - 12.0 * RS);
dhdr[13] = 0.5 * rst[2] * TP * (-8.0 * rst[1] + 24.0 * RS + 12.0 * SS);
dhds[13] = 0.5 * rst[2] * TP * (-8.0 * rst[0] + 12.0 * RR + 24.0 * RS);
dhdt[13] = 0.5 * T2P * (4.0 * RS - 12.0 * RSM * RS);
dhdr[14] = 0.5 * rst[2] * TP * (-16.0 * rst[1] + 24.0 * RS + 12.0 * SS);
dhds[14] = 0.5 * rst[2] * TP * (4.0 - 16.0 * rst[0] - 8.0 * rst[1] + 12.0 * RR + 24.0 * RS);
dhdt[14] = 0.5 * T2P * RSM * (4.0 * rst[1] - 12.0 * RS);
dhdr[9] = TP * TM * (4.0 * rst[0] + 7.0 * rst[1] - 3.0 - 6.0 * RS - 3.0 * SS);
dhds[9] = TP * TM * (7.0 * rst[0] + 4.0 * rst[1] - 3.0 - 6.0 * RS - 3.0 * RR);
dhdt[9] = -2.0 * rst[2] * RSM * (1.0 - 2.0 * (rst[0] + rst[1]) + 3.0 * RS);
dhdr[10] = TP * TM * (4.0 * rst[0] - 1.0 + 3.0 * rst[1] - 6.0 * RS - 3.0 * SS);
dhds[10] = TP * TM * (3.0 * rst[0] - 6.0 * RS - 3.0 * RR);
dhdt[10] = -2.0 * rst[2] * (rst[0] - 2.0 * (RSM * rst[0] + RS) + 3.0 * RSM * RS);
dhdr[11] = TP * TM * (3.0 * rst[1] - 6.0 * RS - 3.0 * SS);
dhds[11] = TP * TM * (4.0 * rst[1] - 1.0 + 3.0 * rst[0] - 6.0 * RS - 3.0 * RR);
dhdt[11] = -2.0 * rst[2] * (rst[1] - 2.0 * (RSM * rst[1] + RS) + 3.0 * RSM * RS);
dhdr[16] = -0.5 * 27.0 * rst[2] * TM * (rst[1] - 2.0 * RS - SS);
dhds[16] = -0.5 * 27.0 * rst[2] * TM * (rst[0] - RR - 2.0 * RS);
dhdt[16] = -0.5 * 27.0 * T2M * RSM * RS;
dhdr[17] = 0.5 * 27.0 * rst[2] * TP * (rst[1] - 2.0 * RS - SS);
dhds[17] = 0.5 * 27.0 * rst[2] * TP * (rst[0] - RR - 2.0 * RS);
dhdt[17] = 0.5 * 27.0 * T2P * RSM * RS;
dhdr[20] = TP * TM * (4.0 - 8.0 * rst[0] - 16.0 * rst[1] + 24.0 * RS + 12.0 * SS);
dhds[20] = TP * TM * (-16.0 * rst[0] + 12.0 * RR + 24.0 * RS);
dhdt[20] = -2.0 * rst[2] * RSM * (4.0 * rst[0] - 12.0 * RS);
dhdr[18] = TP * TM * (-8.0 * rst[1] + 24.0 * RS + 12.0 * SS);
dhds[18] = TP * TM * (-8.0 * rst[0] + 12.0 * RR + 24.0 * RS);
dhdt[18] = -2.0 * rst[2] * (4.0 * RS - 12.0 * RSM * RS);
dhdr[19] = TP * TM * (-16.0 * rst[1] + 24.0 * RS + 12.0 * SS);
dhds[19] = TP * TM * (4.0 - 16.0 * rst[0] - 8.0 * rst[1] + 12.0 * RR + 24.0 * RS);
dhdt[19] = -2.0 * rst[2] * RSM * (4.0 * rst[1] - 12.0 * RS);
dhdr[15] = 27.0 * TM * TP * (rst[1] - 2.0 * RS - SS);
dhds[15] = 27.0 * TM * TP * (rst[0] - RR - 2.0 * RS);
dhdt[15] = -2.0 * 27.0 * rst[2] * RSM * RS;
}
VERDICT_HOST_DEVICE double wedge_equiangle_skew(int /*num_nodes*/, const double coordinates[][3])
{
double tri1[3][3];
double tri2[3][3];
double quad1[4][3];
double quad2[4][3];
double quad3[4][3];
make_wedge_faces(coordinates, tri1, tri2, quad1, quad2, quad3);
double tri1_skew = tri_equiangle_skew(3, tri1);
double tri2_skew = tri_equiangle_skew(3, tri2);
double quad1_skew = quad_equiangle_skew(4, quad1);
double quad2_skew = quad_equiangle_skew(4, quad2);
double quad3_skew = quad_equiangle_skew(4, quad3);
double max_skew = tri1_skew;
max_skew = max_skew > tri2_skew ? max_skew : tri2_skew;
max_skew = max_skew > quad1_skew ? max_skew : quad1_skew;
max_skew = max_skew > quad2_skew ? max_skew : quad2_skew;
max_skew = max_skew > quad3_skew ? max_skew : quad3_skew;
return max_skew;
}
/*!
calculate the volume of a wedge
this is done by dividing the wedge into 11 tets
and summing the volume of each tet
*/
VERDICT_HOST_DEVICE double wedge_volume(int /*num_nodes*/, const double coordinates[][3])
{
// We need to divide the wedge into 11 tets.
// This is a better solution than 3 tets or 3 hexes because
// if the wedge is twisted then the 3 quads will be twisted.
// This presents a problem when you have multiple wedges next to
// each other. A hex or tet representation of a wedge may vary
// from one wedge to another. This means that if wedge A splits
// a quad one way, wedge B may split the matching quad the other direction.
// this will produce an error in the total volume calculation across
// multiple wedges. Placing a center point on each quad and dividing the
// wedge into 11 tets avoids this problem because each wedge will
// split the quads the same way. This eliminates error in the total
// volume calculation across multiple wedges.
double center_coords[3][3];
// calculate the center of the quads
center_coords[0][0] =
(coordinates[0][0] + coordinates[1][0] + coordinates[3][0] + coordinates[4][0]) / 4;
center_coords[0][1] =
(coordinates[0][1] + coordinates[1][1] + coordinates[3][1] + coordinates[4][1]) / 4;
center_coords[0][2] =
(coordinates[0][2] + coordinates[1][2] + coordinates[3][2] + coordinates[4][2]) / 4;
center_coords[1][0] =
(coordinates[1][0] + coordinates[2][0] + coordinates[4][0] + coordinates[5][0]) / 4;
center_coords[1][1] =
(coordinates[1][1] + coordinates[2][1] + coordinates[4][1] + coordinates[5][1]) / 4;
center_coords[1][2] =
(coordinates[1][2] + coordinates[2][2] + coordinates[4][2] + coordinates[5][2]) / 4;
center_coords[2][0] =
(coordinates[2][0] + coordinates[0][0] + coordinates[3][0] + coordinates[5][0]) / 4;
center_coords[2][1] =
(coordinates[2][1] + coordinates[0][1] + coordinates[3][1] + coordinates[5][1]) / 4;
center_coords[2][2] =
(coordinates[2][2] + coordinates[0][2] + coordinates[3][2] + coordinates[5][2]) / 4;
// create the tets.
double tet_coords[11][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[3][0];
tet_coords[0][1][1] = coordinates[3][1];
tet_coords[0][1][2] = coordinates[3][2];
tet_coords[0][2][0] = center_coords[0][0];
tet_coords[0][2][1] = center_coords[0][1];
tet_coords[0][2][2] = center_coords[0][2];
tet_coords[0][3][0] = center_coords[2][0];
tet_coords[0][3][1] = center_coords[2][1];
tet_coords[0][3][2] = center_coords[2][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[4][0];
tet_coords[1][1][1] = coordinates[4][1];
tet_coords[1][1][2] = coordinates[4][2];
tet_coords[1][2][0] = center_coords[1][0];
tet_coords[1][2][1] = center_coords[1][1];
tet_coords[1][2][2] = center_coords[1][2];
tet_coords[1][3][0] = center_coords[0][0];
tet_coords[1][3][1] = center_coords[0][1];
tet_coords[1][3][2] = center_coords[0][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[5][0];
tet_coords[2][1][1] = coordinates[5][1];
tet_coords[2][1][2] = coordinates[5][2];
tet_coords[2][2][0] = center_coords[2][0];
tet_coords[2][2][1] = center_coords[2][1];
tet_coords[2][2][2] = center_coords[2][2];
tet_coords[2][3][0] = center_coords[1][0];
tet_coords[2][3][1] = center_coords[1][1];
tet_coords[2][3][2] = center_coords[1][2];
tet_coords[3][0][0] = center_coords[0][0];
tet_coords[3][0][1] = center_coords[0][1];
tet_coords[3][0][2] = center_coords[0][2];
tet_coords[3][1][0] = center_coords[2][0];
tet_coords[3][1][1] = center_coords[2][1];
tet_coords[3][1][2] = center_coords[2][2];
tet_coords[3][2][0] = center_coords[1][0];
tet_coords[3][2][1] = center_coords[1][1];
tet_coords[3][2][2] = center_coords[1][2];
tet_coords[3][3][0] = coordinates[0][0];
tet_coords[3][3][1] = coordinates[0][1];
tet_coords[3][3][2] = coordinates[0][2];
tet_coords[4][0][0] = coordinates[1][0];
tet_coords[4][0][1] = coordinates[1][1];
tet_coords[4][0][2] = coordinates[1][2];
tet_coords[4][1][0] = center_coords[0][0];
tet_coords[4][1][1] = center_coords[0][1];
tet_coords[4][1][2] = center_coords[0][2];
tet_coords[4][2][0] = center_coords[1][0];
tet_coords[4][2][1] = center_coords[1][1];
tet_coords[4][2][2] = center_coords[1][2];
tet_coords[4][3][0] = coordinates[0][0];
tet_coords[4][3][1] = coordinates[0][1];
tet_coords[4][3][2] = coordinates[0][2];
tet_coords[5][0][0] = coordinates[2][0];
tet_coords[5][0][1] = coordinates[2][1];
tet_coords[5][0][2] = coordinates[2][2];
tet_coords[5][1][0] = coordinates[1][0];
tet_coords[5][1][1] = coordinates[1][1];
tet_coords[5][1][2] = coordinates[1][2];
tet_coords[5][2][0] = center_coords[1][0];
tet_coords[5][2][1] = center_coords[1][1];
tet_coords[5][2][2] = center_coords[1][2];
tet_coords[5][3][0] = coordinates[0][0];
tet_coords[5][3][1] = coordinates[0][1];
tet_coords[5][3][2] = coordinates[0][2];
tet_coords[6][0][0] = coordinates[2][0];
tet_coords[6][0][1] = coordinates[2][1];
tet_coords[6][0][2] = coordinates[2][2];
tet_coords[6][1][0] = center_coords[1][0];
tet_coords[6][1][1] = center_coords[1][1];
tet_coords[6][1][2] = center_coords[1][2];
tet_coords[6][2][0] = center_coords[2][0];
tet_coords[6][2][1] = center_coords[2][1];
tet_coords[6][2][2] = center_coords[2][2];
tet_coords[6][3][0] = coordinates[0][0];
tet_coords[6][3][1] = coordinates[0][1];
tet_coords[6][3][2] = coordinates[0][2];
tet_coords[7][0][0] = center_coords[0][0];
tet_coords[7][0][1] = center_coords[0][1];
tet_coords[7][0][2] = center_coords[0][2];
tet_coords[7][1][0] = center_coords[1][0];
tet_coords[7][1][1] = center_coords[1][1];
tet_coords[7][1][2] = center_coords[1][2];
tet_coords[7][2][0] = center_coords[2][0];
tet_coords[7][2][1] = center_coords[2][1];
tet_coords[7][2][2] = center_coords[2][2];
tet_coords[7][3][0] = coordinates[3][0];
tet_coords[7][3][1] = coordinates[3][1];
tet_coords[7][3][2] = coordinates[3][2];
tet_coords[8][0][0] = coordinates[5][0];
tet_coords[8][0][1] = coordinates[5][1];
tet_coords[8][0][2] = coordinates[5][2];
tet_coords[8][1][0] = center_coords[2][0];
tet_coords[8][1][1] = center_coords[2][1];
tet_coords[8][1][2] = center_coords[2][2];
tet_coords[8][2][0] = center_coords[1][0];
tet_coords[8][2][1] = center_coords[1][1];
tet_coords[8][2][2] = center_coords[1][2];
tet_coords[8][3][0] = coordinates[3][0];
tet_coords[8][3][1] = coordinates[3][1];
tet_coords[8][3][2] = coordinates[3][2];
tet_coords[9][0][0] = coordinates[4][0];
tet_coords[9][0][1] = coordinates[4][1];
tet_coords[9][0][2] = coordinates[4][2];
tet_coords[9][1][0] = coordinates[5][0];
tet_coords[9][1][1] = coordinates[5][1];
tet_coords[9][1][2] = coordinates[5][2];
tet_coords[9][2][0] = center_coords[1][0];
tet_coords[9][2][1] = center_coords[1][1];
tet_coords[9][2][2] = center_coords[1][2];
tet_coords[9][3][0] = coordinates[3][0];
tet_coords[9][3][1] = coordinates[3][1];
tet_coords[9][3][2] = coordinates[3][2];
tet_coords[10][0][0] = coordinates[4][0];
tet_coords[10][0][1] = coordinates[4][1];
tet_coords[10][0][2] = coordinates[4][2];
tet_coords[10][1][0] = center_coords[1][0];
tet_coords[10][1][1] = center_coords[1][1];
tet_coords[10][1][2] = center_coords[1][2];
tet_coords[10][2][0] = center_coords[0][0];
tet_coords[10][2][1] = center_coords[0][1];
tet_coords[10][2][2] = center_coords[0][2];
tet_coords[10][3][0] = coordinates[3][0];
tet_coords[10][3][1] = coordinates[3][1];
tet_coords[10][3][2] = coordinates[3][2];
double volume = 0.0;
for (int t = 0; t < 11; t++)
{
volume += tet_volume(4, tet_coords[t]);
}
return (double)volume;
}
/* Edge ratio
The edge ratio quality metric is the ratio of the longest to shortest edge of
a wedge.
q = L_max / L_min
Dimension : 1
Acceptable range : --
Normal range : [1,DBL_MAX]
Full range : [1,DBL_MAX]
q for right, unit wedge : 1
Reference : -
*/
VERDICT_HOST_DEVICE double wedge_edge_ratio(int /*num_nodes*/, const double coordinates[][3])
{
VerdictVector a, b, c, d, e, f, g, h, i;
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[4][0] - coordinates[3][0], coordinates[4][1] - coordinates[3][1],
coordinates[4][2] - coordinates[3][2]);
e.set(coordinates[5][0] - coordinates[4][0], coordinates[5][1] - coordinates[4][1],
coordinates[5][2] - coordinates[4][2]);
f.set(coordinates[3][0] - coordinates[5][0], coordinates[3][1] - coordinates[5][1],
coordinates[3][2] - coordinates[5][2]);
g.set(coordinates[3][0] - coordinates[0][0], coordinates[3][1] - coordinates[0][1],
coordinates[3][2] - coordinates[0][2]);
h.set(coordinates[4][0] - coordinates[1][0], coordinates[4][1] - coordinates[1][1],
coordinates[4][2] - coordinates[1][2]);
i.set(coordinates[5][0] - coordinates[2][0], coordinates[5][1] - coordinates[2][1],
coordinates[5][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 g2 = g.length_squared();
double h2 = h.length_squared();
double i2 = i.length_squared();
double max = a2, min = a2;
if (max <= b2)
{
max = b2;
}
if (b2 <= min)
{
min = b2;
}
if (max <= c2)
{
max = c2;
}
if (c2 <= min)
{
min = c2;
}
if (max <= d2)
{
max = d2;
}
if (d2 <= min)
{
min = d2;
}
if (max <= e2)
{
max = e2;
}
if (e2 <= min)
{
min = e2;
}
if (max <= f2)
{
max = f2;
}
if (f2 <= min)
{
min = f2;
}
if (max <= g2)
{
max = g2;
}
if (g2 <= min)
{
min = g2;
}
if (max <= h2)
{
max = h2;
}
if (h2 <= min)
{
min = h2;
}
if (max <= i2)
{
max = i2;
}
if (i2 <= min)
{
min = i2;
}
double edge_ratio = sqrt(max / min);
if (isnan(edge_ratio))
{
return VERDICT_DBL_MAX;
}
if (edge_ratio < 1.)
{
return 1.;
}
return (double)fmin(edge_ratio, VERDICT_DBL_MAX);
}
VERDICT_HOST_DEVICE static void aspects(int num_nodes, const double coordinates[][3], double& aspect1, double& aspect2,
double& aspect3, double& aspect4, double& aspect5, double& aspect6)
{
if (num_nodes < 6)
{
aspect1 = 0;
aspect2 = 0;
aspect3 = 0;
aspect4 = 0;
aspect5 = 0;
aspect6 = 0;
return;
}
double mini_tris[4][3];
int i = 0;
// Take first tetrahedron
for (i = 0; i < 3; i++)
{
mini_tris[0][i] = coordinates[0][i];
}
for (i = 0; i < 3; i++)
{
mini_tris[1][i] = coordinates[1][i];
}
for (i = 0; i < 3; i++)
{
mini_tris[2][i] = coordinates[2][i];
}
for (i = 0; i < 3; i++)
{
mini_tris[3][i] = coordinates[3][i];
}
aspect1 = tet_aspect_frobenius(4, mini_tris);
// Take second tet
for (i = 0; i < 3; i++)
{
mini_tris[0][i] = coordinates[1][i];
}
for (i = 0; i < 3; i++)
{
mini_tris[1][i] = coordinates[2][i];
}
for (i = 0; i < 3; i++)
{
mini_tris[2][i] = coordinates[0][i];
}
for (i = 0; i < 3; i++)
{
mini_tris[3][i] = coordinates[4][i];
}
aspect2 = tet_aspect_frobenius(4, mini_tris);
// 3rd tet
for (i = 0; i < 3; i++)
{
mini_tris[0][i] = coordinates[2][i];
}
for (i = 0; i < 3; i++)
{
mini_tris[1][i] = coordinates[0][i];
}
for (i = 0; i < 3; i++)
{
mini_tris[2][i] = coordinates[1][i];
}
for (i = 0; i < 3; i++)
{
mini_tris[3][i] = coordinates[5][i];
}
aspect3 = tet_aspect_frobenius(4, mini_tris);
// 4th tet
for (i = 0; i < 3; i++)
{
mini_tris[0][i] = coordinates[3][i];
}
for (i = 0; i < 3; i++)
{
mini_tris[1][i] = coordinates[5][i];
}
for (i = 0; i < 3; i++)
{
mini_tris[2][i] = coordinates[4][i];
}
for (i = 0; i < 3; i++)
{
mini_tris[3][i] = coordinates[0][i];
}
aspect4 = tet_aspect_frobenius(4, mini_tris);
// 5th tet
for (i = 0; i < 3; i++)
{
mini_tris[0][i] = coordinates[4][i];
}
for (i = 0; i < 3; i++)
{
mini_tris[1][i] = coordinates[3][i];
}
for (i = 0; i < 3; i++)
{
mini_tris[2][i] = coordinates[5][i];
}
for (i = 0; i < 3; i++)
{
mini_tris[3][i] = coordinates[1][i];
}
aspect5 = tet_aspect_frobenius(4, mini_tris);
// 6th tet
for (i = 0; i < 3; i++)
{
mini_tris[0][i] = coordinates[5][i];
}
for (i = 0; i < 3; i++)
{
mini_tris[1][i] = coordinates[4][i];
}
for (i = 0; i < 3; i++)
{
mini_tris[2][i] = coordinates[3][i];
}
for (i = 0; i < 3; i++)
{
mini_tris[3][i] = coordinates[2][i];
}
aspect6 = tet_aspect_frobenius(4, mini_tris);
}
/* For wedges, there is not a unique definition of the aspect Frobenius. Rather,
* this metric uses the aspect Frobenius defined for tetrahedral (see section
* 6.4) and is comparable in methodology to the maximum aspect Frobenius defined
* for hexahedra (see section 7.7). This value is normalized for a unit wedge.
q = max(F_0123, F_1204, F_2015, F_3540, F_4351, F_5432)
This is also known as the wedge condition number.
Dimension : 1
Acceptable Range :
Normal Range :
Full Range :
q for right, unit wedge : 1
Reference : Adapted from section 7.7
Verdict Function : wedge_max_aspect_frobenius or wedge_condition
*/
VERDICT_HOST_DEVICE double wedge_max_aspect_frobenius(int num_nodes, const double coordinates[][3])
{
double aspect1, aspect2, aspect3, aspect4, aspect5, aspect6;
aspects(num_nodes, coordinates, aspect1, aspect2, aspect3, aspect4, aspect5, aspect6);
double max_aspect = fmax( aspect1, fmax(aspect2, fmax(aspect3, fmax(aspect4, fmax(aspect5, aspect6 )))));
if (max_aspect >= VERDICT_DBL_MAX)
{
return VERDICT_DBL_MAX;
}
max_aspect /= 1.16477;
return fmax(max_aspect, 1.);
}
/*
For wedges, there is not a unique definition of the aspect Frobenius. Rather,
this metric uses the aspect Frobenius defined for tetrahedra (see section
6.4) and is comparable in methodology to the mean aspect Frobenius defined
for hexahedra (see section 7.8). This value is normalized for a unit wedge.
q = 1/6 * (F_0123 + F_1204 + F+2015 + F_3540 + F_4351 + F_5432)
Dimension : 1
Acceptable Range :
Normal Range :
Full Range :
q for right, unit wedge : 1
Reference : Adapted from section 7.8
Verdict Function : wedge_mean_aspect_frobenius
*/
VERDICT_HOST_DEVICE double wedge_mean_aspect_frobenius(int num_nodes, const double coordinates[][3])
{
double aspect1, aspect2, aspect3, aspect4, aspect5, aspect6;
aspects(num_nodes, coordinates, aspect1, aspect2, aspect3, aspect4, aspect5, aspect6);
double mean_aspect = (aspect1 + aspect2 + aspect3 + aspect4 + aspect5 + aspect6);
if (mean_aspect >= VERDICT_DBL_MAX)
{
return VERDICT_DBL_MAX;
}
mean_aspect /= (6. * 1.16477);
return fmax(mean_aspect, 1.);
}
/* This is the minimum determinant of the Jacobian matrix evaluated at each
* corner of the element.
q = min[((L_2 X L_0) * L_3)_k]
where ((L_2 X L_0) * L_3)_k is the determinant of the Jacobian of the
tetrahedron defined at the kth corner node, and L_2, L_0 and L_3 are the edges
defined according to the standard for tetrahedral elements.
Dimension : L^3
Acceptable Range : [0,DBL_MAX]
Normal Range : [0,DBL_MAX]
Full Range : [-DBL_MAX,DBL_MAX]
q for right, unit wedge : sqrt(3)/2
Reference : Adapted from section 6.10
Verdict Function : wedge_jacobian
*/
VERDICT_HOST_DEVICE double wedge_jacobian(int num_nodes, const double coordinates[][3])
{
if (num_nodes == 21)
{
double dhdr[21];
double dhds[21];
double dhdt[21];
double min_determinant = VERDICT_DBL_MAX;
for (int i = 0; i < 15; i++)
{
WEDGE21_gradients_of_the_shape_functions_for_RST(
WEDGE21_node_local_coord(i), dhdr, dhds, dhdt);
double jacobian[3][3] = { { 0, 0, 0 }, { 0, 0, 0 }, { 0, 0, 0 } };
for (int j = 0; j < 21; j++)
{
jacobian[0][0] += coordinates[j][0] * dhdr[j];
jacobian[0][1] += coordinates[j][0] * dhds[j];
jacobian[0][2] += coordinates[j][0] * dhdt[j];
jacobian[1][0] += coordinates[j][1] * dhdr[j];
jacobian[1][1] += coordinates[j][1] * dhds[j];
jacobian[1][2] += coordinates[j][1] * dhdt[j];
jacobian[2][0] += coordinates[j][2] * dhdr[j];
jacobian[2][1] += coordinates[j][2] * dhds[j];
jacobian[2][2] += coordinates[j][2] * dhdt[j];
}
double det =
(VerdictVector(jacobian[0]) * VerdictVector(jacobian[1])) % VerdictVector(jacobian[2]);
min_determinant = fmin(det, min_determinant);
}
return min_determinant;
}
else
{
double min_jacobian = 0, current_jacobian = 0;
VerdictVector vec1, vec2, vec3;
// Node 0
vec1.set(coordinates[1][0] - coordinates[0][0], coordinates[1][1] - coordinates[0][1],
coordinates[1][2] - coordinates[0][2]);
vec2.set(coordinates[3][0] - coordinates[0][0], coordinates[3][1] - coordinates[0][1],
coordinates[3][2] - coordinates[0][2]);
vec3.set(coordinates[2][0] - coordinates[0][0], coordinates[2][1] - coordinates[0][1],
coordinates[2][2] - coordinates[0][2]);
current_jacobian = vec2 % (vec1 * vec3);
min_jacobian = current_jacobian;
// node 1
vec1.set(coordinates[2][0] - coordinates[1][0], coordinates[2][1] - coordinates[1][1],
coordinates[2][2] - coordinates[1][2]);
vec2.set(coordinates[4][0] - coordinates[1][0], coordinates[4][1] - coordinates[1][1],
coordinates[4][2] - coordinates[1][2]);
vec3.set(coordinates[0][0] - coordinates[1][0], coordinates[0][1] - coordinates[1][1],
coordinates[0][2] - coordinates[1][2]);
current_jacobian = vec2 % (vec1 * vec3);
min_jacobian = fmin(current_jacobian, min_jacobian);
// node 2
vec1.set(coordinates[0][0] - coordinates[2][0], coordinates[0][1] - coordinates[2][1],
coordinates[0][2] - coordinates[2][2]);
vec2.set(coordinates[5][0] - coordinates[2][0], coordinates[5][1] - coordinates[2][1],
coordinates[5][2] - coordinates[2][2]);
vec3.set(coordinates[1][0] - coordinates[2][0], coordinates[1][1] - coordinates[2][1],
coordinates[1][2] - coordinates[2][2]);
current_jacobian = vec2 % (vec1 * vec3);
min_jacobian = fmin(current_jacobian, min_jacobian);
// node 3
vec1.set(coordinates[0][0] - coordinates[3][0], coordinates[0][1] - coordinates[3][1],
coordinates[0][2] - coordinates[3][2]);
vec2.set(coordinates[4][0] - coordinates[3][0], coordinates[4][1] - coordinates[3][1],
coordinates[4][2] - coordinates[3][2]);
vec3.set(coordinates[5][0] - coordinates[3][0], coordinates[5][1] - coordinates[3][1],
coordinates[5][2] - coordinates[3][2]);
current_jacobian = vec2 % (vec1 * vec3);
min_jacobian = fmin(current_jacobian, min_jacobian);
// node 4
vec1.set(coordinates[1][0] - coordinates[4][0], coordinates[1][1] - coordinates[4][1],
coordinates[1][2] - coordinates[4][2]);
vec2.set(coordinates[5][0] - coordinates[4][0], coordinates[5][1] - coordinates[4][1],
coordinates[5][2] - coordinates[4][2]);
vec3.set(coordinates[3][0] - coordinates[4][0], coordinates[3][1] - coordinates[4][1],
coordinates[3][2] - coordinates[4][2]);
current_jacobian = vec2 % (vec1 * vec3);
min_jacobian = fmin(current_jacobian, min_jacobian);
// node 5
vec1.set(coordinates[3][0] - coordinates[5][0], coordinates[3][1] - coordinates[5][1],
coordinates[3][2] - coordinates[5][2]);
vec2.set(coordinates[4][0] - coordinates[5][0], coordinates[4][1] - coordinates[5][1],
coordinates[4][2] - coordinates[5][2]);
vec3.set(coordinates[2][0] - coordinates[5][0], coordinates[2][1] - coordinates[5][1],
coordinates[2][2] - coordinates[5][2]);
current_jacobian = vec2 % (vec1 * vec3);
min_jacobian = fmin(current_jacobian, min_jacobian);
if (min_jacobian > 0)
{
return (double)fmin(min_jacobian, VERDICT_DBL_MAX);
}
return (double)fmax(min_jacobian, -VERDICT_DBL_MAX);
}
}
/* distortion is a measure of how well a particular wedge element maps to a
* 'master' wedge with vertices:
P0 - (0, 0, 0)
P1 - (1, 0, 0)
P2 - (1/2, sqrt(3)/2, 0)
P3 - (0, 0, 1)
P4 - (1, 0, 1)
P2 - (1/2, sqrt(3)/2, 1)
and volume (V_m).
The behavior of the map is measured by sampling the determinant of the Jacobian
at the vertices (k). Thus the distortion is given by:
q = ( min_k { det(J_k)} * V_m ) / V
Dimension : 1
Acceptable Range : [0.5,1]
Normal Range : [0,1]
Full Range : [-DBL_MAX,DBL_MAX]
q for right, unit wedge : 1
Reference : Adapted from section 7.3
Verdict Function : wedge_distortion
*/
VERDICT_HOST_DEVICE double wedge_distortion(int num_nodes, const double coordinates[][3])
{
double jacobian = wedge_jacobian(num_nodes, coordinates);
double master_volume = 0.433013;
double current_volume = wedge_volume(num_nodes, coordinates);
double distortion = VERDICT_DBL_MAX;
if (fabs(current_volume) > 0.0)
distortion = jacobian * master_volume / current_volume / 0.866025;
if (isnan(distortion))
{
return VERDICT_DBL_MAX;
}
if (distortion >= VERDICT_DBL_MAX)
{
return VERDICT_DBL_MAX;
}
if (distortion <= -VERDICT_DBL_MAX)
{
return -VERDICT_DBL_MAX;
}
return distortion;
}
/*
The stretch of a wedge element is here defined to be the maximum value of the
stretch (S) of the three quadrilateral faces (see section 5.21):
q = max[S_1043, S_1254, S_2035]
Dimension : 1
Acceptable Range :
Normal Range :
Full Range : [0,DBL_MAX]
q for right, unit wedge : 1
Reference : Adapted from section 5.21
Verdict Function : wedge_max_stretch
*/
VERDICT_HOST_DEVICE double wedge_max_stretch(int /*num_nodes*/, const double coordinates[][3])
{
// This function finds the stretch of the 3 quadrilateral faces and returns the maximum value
double stretch = 42, quad_face[4][3], stretch1 = 42, stretch2 = 42, stretch3 = 42;
int i = 0;
// first face
for (i = 0; i < 3; i++)
{
quad_face[0][i] = coordinates[0][i];
}
for (i = 0; i < 3; i++)
{
quad_face[1][i] = coordinates[1][i];
}
for (i = 0; i < 3; i++)
{
quad_face[2][i] = coordinates[4][i];
}
for (i = 0; i < 3; i++)
{
quad_face[3][i] = coordinates[3][i];
}
stretch1 = quad_stretch(4, quad_face);
// second face
for (i = 0; i < 3; i++)
{
quad_face[0][i] = coordinates[1][i];
}
for (i = 0; i < 3; i++)
{
quad_face[1][i] = coordinates[2][i];
}
for (i = 0; i < 3; i++)
{
quad_face[2][i] = coordinates[5][i];
}
for (i = 0; i < 3; i++)
{
quad_face[3][i] = coordinates[4][i];
}
stretch2 = quad_stretch(4, quad_face);
// third face
for (i = 0; i < 3; i++)
{
quad_face[0][i] = coordinates[2][i];
}
for (i = 0; i < 3; i++)
{
quad_face[1][i] = coordinates[0][i];
}
for (i = 0; i < 3; i++)
{
quad_face[2][i] = coordinates[3][i];
}
for (i = 0; i < 3; i++)
{
quad_face[3][i] = coordinates[5][i];
}
stretch3 = quad_stretch(4, quad_face);
stretch = fmax( stretch1, fmax(stretch2, stretch3 ));
if (stretch > 0)
{
return (double)fmin(stretch, VERDICT_DBL_MAX);
}
return (double)fmax(stretch, -VERDICT_DBL_MAX);
}
/*
This is the minimum determinant of the Jacobian matrix evaluated at each
corner of the element, divided by the corresponding edge lengths and
normalized to the unit wedge:
q = min( 2 / sqrt(3) * ((L_2 X L_0) * L_3)_k / sqrt(mag(L_2) * mag(L_0) * mag(L_3)))
where ((L_2 X L_0) * L_3)_k is the determinant of the Jacobian of the
tetrahedron defined at the kth corner node, and L_2, L_0 and L_3 are the