V_GaussIntegration.cpp

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/*=========================================================================

  Module:    $RCSfile: V_GaussIntegration.cpp,v $

  Copyright (c) 2006 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.

=========================================================================*/

/*
 *
 * GaussIntegration.cpp performs Gauss Integrations
 *
 * This file is part of VERDICT
 *
 */

#define VERDICT_EXPORTS

#include "moab/verdict.h"
#include "V_GaussIntegration.hpp"

#include <cmath>

double verdictSqrt2 = sqrt( (double)2.0 );

int numberGaussPoints;
int numberNodes;
int numberDims;
double gaussPointY[maxNumberGaussPoints];
double gaussWeight[maxNumberGaussPoints];
double shapeFunction[maxTotalNumberGaussPoints][maxNumberNodes];
double dndy1GaussPts[maxTotalNumberGaussPoints][maxNumberNodes];
double dndy2GaussPts[maxTotalNumberGaussPoints][maxNumberNodes];
double dndy3GaussPts[maxTotalNumberGaussPoints][maxNumberNodes];
double totalGaussWeight[maxTotalNumberGaussPoints];
int totalNumberGaussPts;
double y1Area[maxNumberGaussPointsTri];
double y2Area[maxNumberGaussPointsTri];
double y1Volume[maxNumberGaussPointsTet];
double y2Volume[maxNumberGaussPointsTet];
double y3Volume[maxNumberGaussPointsTet];
double y4Volume[maxNumberGaussPointsTet];

void GaussIntegration::initialize( int n, int m, int dim, int tri )
{
    numberGaussPoints = n;
    numberNodes       = m;
    numberDims        = dim;

    if( tri == 1 )
    // triangular element
    {
        if( numberDims == 2 )
            totalNumberGaussPts = numberGaussPoints;
        else if( numberDims == 3 )
            totalNumberGaussPts = numberGaussPoints;
    }
    else if( tri == 0 )
    {
        if( numberDims == 2 )
            totalNumberGaussPts = numberGaussPoints * numberGaussPoints;
        else if( numberDims == 3 )
            totalNumberGaussPts = numberGaussPoints * numberGaussPoints * numberGaussPoints;
    }
}

void GaussIntegration::get_shape_func( double shape_function[],
                                       double dndy1_at_gauss_pts[],
                                       double dndy2_at_gauss_pts[],
                                       double gauss_weight[] )
{
    int i, j;
    for( i = 0; i < totalNumberGaussPts; i++ )
    {
        for( j = 0; j < numberNodes; j++ )
        {
            shape_function[i * maxNumberNodes + j]     = shapeFunction[i][j];
            dndy1_at_gauss_pts[i * maxNumberNodes + j] = dndy1GaussPts[i][j];
            dndy2_at_gauss_pts[i * maxNumberNodes + j] = dndy2GaussPts[i][j];
        }
    }

    for( i = 0; i < totalNumberGaussPts; i++ )
        gauss_weight[i] = totalGaussWeight[i];
}

void GaussIntegration::get_shape_func( double shape_function[],
                                       double dndy1_at_gauss_pts[],
                                       double dndy2_at_gauss_pts[],
                                       double dndy3_at_gauss_pts[],
                                       double gauss_weight[] )
{
    int i, j;
    for( i = 0; i < totalNumberGaussPts; i++ )
    {
        for( j = 0; j < numberNodes; j++ )
        {
            shape_function[i * maxNumberNodes + j]     = shapeFunction[i][j];
            dndy1_at_gauss_pts[i * maxNumberNodes + j] = dndy1GaussPts[i][j];
            dndy2_at_gauss_pts[i * maxNumberNodes + j] = dndy2GaussPts[i][j];
            dndy3_at_gauss_pts[i * maxNumberNodes + j] = dndy3GaussPts[i][j];
        }
    }

    for( i = 0; i < totalNumberGaussPts; i++ )
        gauss_weight[i] = totalGaussWeight[i];
}

void GaussIntegration::get_gauss_pts_and_weight()
{

    switch( numberGaussPoints )
    {
        case 1:
            gaussPointY[0] = 0.0;
            gaussWeight[0] = 2.0;
            break;
        case 2:
            gaussPointY[0] = -0.577350269189626;
            gaussPointY[1] = 0.577350269189626;
            gaussWeight[0] = 1.0;
            gaussWeight[1] = 1.0;
            break;
        case 3:
            gaussPointY[0] = -0.774596669241483;
            gaussPointY[1] = 0.0;
            gaussPointY[2] = 0.774596669241483;
            gaussWeight[0] = 0.555555555555555;
            gaussWeight[1] = 0.888888888888889;
            gaussWeight[2] = 0.555555555555555;
            break;
    }
}

void GaussIntegration::calculate_shape_function_2d_quad()
{
    int ife = 0, i, j;
    double y1, y2;
    get_gauss_pts_and_weight();

    switch( numberNodes )
    {
        case 4:
            for( i = 0; i < numberGaussPoints; i++ )
            {
                for( j = 0; j < numberGaussPoints; j++ )
                {
                    y1                    = gaussPointY[i];
                    y2                    = gaussPointY[j];
                    shapeFunction[ife][0] = 0.25 * ( 1 - y1 ) * ( 1 - y2 );
                    shapeFunction[ife][1] = 0.25 * ( 1 + y1 ) * ( 1 - y2 );
                    shapeFunction[ife][2] = 0.25 * ( 1 + y1 ) * ( 1 + y2 );
                    shapeFunction[ife][3] = 0.25 * ( 1 - y1 ) * ( 1 + y2 );

                    dndy1GaussPts[ife][0] = -0.25 * ( 1 - y2 );
                    dndy1GaussPts[ife][1] = 0.25 * ( 1 - y2 );
                    dndy1GaussPts[ife][2] = 0.25 * ( 1 + y2 );
                    dndy1GaussPts[ife][3] = -0.25 * ( 1 + y2 );

                    dndy2GaussPts[ife][0] = -0.25 * ( 1 - y1 );
                    dndy2GaussPts[ife][1] = -0.25 * ( 1 + y1 );
                    dndy2GaussPts[ife][2] = 0.25 * ( 1 + y1 );
                    dndy2GaussPts[ife][3] = 0.25 * ( 1 - y1 );

                    totalGaussWeight[ife] = gaussWeight[i] * gaussWeight[j];
                    ife++;
                }
            }
            break;
        case 8:
            for( i = 0; i < numberGaussPoints; i++ )
            {
                for( j = 0; j < numberGaussPoints; j++ )
                {
                    y1                    = gaussPointY[i];
                    y2                    = gaussPointY[j];
                    shapeFunction[ife][0] = 0.25 * ( 1 - y1 ) * ( 1 - y2 ) * ( -y1 - y2 - 1 );
                    shapeFunction[ife][1] = 0.25 * ( 1 + y1 ) * ( 1 - y2 ) * ( y1 - y2 - 1 );
                    shapeFunction[ife][2] = 0.25 * ( 1 + y1 ) * ( 1 + y2 ) * ( y1 + y2 - 1 );
                    shapeFunction[ife][3] = 0.25 * ( 1 - y1 ) * ( 1 + y2 ) * ( -y1 + y2 - 1 );
                    shapeFunction[ife][4] = 0.5 * ( 1 - y1 * y1 ) * ( 1 - y2 );
                    shapeFunction[ife][5] = 0.5 * ( 1 - y2 * y2 ) * ( 1 + y1 );
                    shapeFunction[ife][6] = 0.5 * ( 1 - y1 * y1 ) * ( 1 + y2 );
                    shapeFunction[ife][7] = 0.5 * ( 1 - y2 * y2 ) * ( 1 - y1 );

                    dndy1GaussPts[ife][0] = 0.25 * ( 1 - y2 ) * ( 2.0 * y1 + y2 );
                    dndy1GaussPts[ife][1] = 0.25 * ( 1 - y2 ) * ( 2.0 * y1 - y2 );
                    dndy1GaussPts[ife][2] = 0.25 * ( 1 + y2 ) * ( 2.0 * y1 + y2 );
                    dndy1GaussPts[ife][3] = 0.25 * ( 1 + y2 ) * ( 2.0 * y1 - y2 );

                    dndy1GaussPts[ife][4] = -y1 * ( 1 - y2 );
                    dndy1GaussPts[ife][5] = 0.5 * ( 1 - y2 * y2 );
                    dndy1GaussPts[ife][6] = -y1 * ( 1 + y2 );
                    dndy1GaussPts[ife][7] = -0.5 * ( 1 - y2 * y2 );

                    dndy2GaussPts[ife][0] = 0.25 * ( 1 - y1 ) * ( 2.0 * y2 + y1 );
                    dndy2GaussPts[ife][1] = 0.25 * ( 1 + y1 ) * ( 2.0 * y2 - y1 );
                    dndy2GaussPts[ife][2] = 0.25 * ( 1 + y1 ) * ( 2.0 * y2 + y1 );
                    dndy2GaussPts[ife][3] = 0.25 * ( 1 - y1 ) * ( 2.0 * y2 - y1 );

                    dndy2GaussPts[ife][4] = -0.5 * ( 1 - y1 * y1 );
                    dndy2GaussPts[ife][5] = -y2 * ( 1 + y1 );
                    dndy2GaussPts[ife][6] = 0.5 * ( 1 - y1 * y1 );
                    dndy2GaussPts[ife][7] = -y2 * ( 1 - y1 );

                    totalGaussWeight[ife] = gaussWeight[i] * gaussWeight[j];
                    ife++;
                }
            }
            break;
    }
}

void GaussIntegration::calculate_shape_function_3d_hex()
{
    int ife = 0, i, j, k, node_id;
    double y1, y2, y3, sign_node_y1, sign_node_y2, sign_node_y3;
    double y1_term, y2_term, y3_term, y123_temp;

    get_gauss_pts_and_weight();

    switch( numberNodes )
    {
        case 8:
            for( i = 0; i < numberGaussPoints; i++ )
            {
                for( j = 0; j < numberGaussPoints; j++ )
                {
                    for( k = 0; k < numberGaussPoints; k++ )
                    {
                        y1 = gaussPointY[i];
                        y2 = gaussPointY[j];
                        y3 = gaussPointY[k];

                        for( node_id = 0; node_id < numberNodes; node_id++ )
                        {
                            get_signs_for_node_local_coord_hex( node_id, sign_node_y1, sign_node_y2, sign_node_y3 );

                            y1_term = 1 + sign_node_y1 * y1;
                            y2_term = 1 + sign_node_y2 * y2;
                            y3_term = 1 + sign_node_y3 * y3;

                            shapeFunction[ife][node_id] = 0.125 * y1_term * y2_term * y3_term;
                            dndy1GaussPts[ife][node_id] = 0.125 * sign_node_y1 * y2_term * y3_term;
                            dndy2GaussPts[ife][node_id] = 0.125 * sign_node_y2 * y1_term * y3_term;
                            dndy3GaussPts[ife][node_id] = 0.125 * sign_node_y3 * y1_term * y2_term;
                        }
                        totalGaussWeight[ife] = gaussWeight[i] * gaussWeight[j] * gaussWeight[k];
                        ife++;
                    }
                }
            }
            break;
        case 20:
            for( i = 0; i < numberGaussPoints; i++ )
            {
                for( j = 0; j < numberGaussPoints; j++ )
                {
                    for( k = 0; k < numberGaussPoints; k++ )
                    {
                        y1 = gaussPointY[i];
                        y2 = gaussPointY[j];
                        y3 = gaussPointY[k];

                        for( node_id = 0; node_id < numberNodes; node_id++ )
                        {
                            get_signs_for_node_local_coord_hex( node_id, sign_node_y1, sign_node_y2, sign_node_y3 );

                            y1_term   = 1 + sign_node_y1 * y1;
                            y2_term   = 1 + sign_node_y2 * y2;
                            y3_term   = 1 + sign_node_y3 * y3;
                            y123_temp = sign_node_y1 * y1 + sign_node_y2 * y2 + sign_node_y3 * y3 - 2.;

                            switch( node_id )
                            {
                                case 0:
                                case 1:
                                case 2:
                                case 3:
                                case 4:
                                case 5:
                                case 6:
                                case 7: {
                                    shapeFunction[ife][node_id] = 0.125 * y1_term * y2_term * y3_term * y123_temp;
                                    dndy1GaussPts[ife][node_id] = 0.125 * sign_node_y1 * y123_temp * y2_term * y3_term +
                                                                  0.125 * y1_term * y2_term * y3_term * sign_node_y1;
                                    dndy2GaussPts[ife][node_id] = 0.125 * sign_node_y2 * y1_term * y3_term * y123_temp +
                                                                  0.125 * y1_term * y2_term * y3_term * sign_node_y2;
                                    dndy3GaussPts[ife][node_id] = 0.125 * sign_node_y3 * y1_term * y2_term * y123_temp +
                                                                  0.125 * y1_term * y2_term * y3_term * sign_node_y3;
                                    break;
                                }
                                case 8:
                                case 10:
                                case 16:
                                case 18: {
                                    shapeFunction[ife][node_id] = 0.25 * ( 1 - y1 * y1 ) * y2_term * y3_term;
                                    dndy1GaussPts[ife][node_id] = -0.5 * y1 * y2_term * y3_term;
                                    dndy2GaussPts[ife][node_id] = 0.25 * ( 1 - y1 * y1 ) * sign_node_y2 * y3_term;
                                    dndy3GaussPts[ife][node_id] = 0.25 * ( 1 - y1 * y1 ) * y2_term * sign_node_y3;
                                    break;
                                }
                                case 9:
                                case 11:
                                case 17:
                                case 19: {
                                    shapeFunction[ife][node_id] = 0.25 * ( 1 - y2 * y2 ) * y1_term * y3_term;
                                    dndy1GaussPts[ife][node_id] = 0.25 * ( 1 - y2 * y2 ) * sign_node_y1 * y3_term;
                                    dndy2GaussPts[ife][node_id] = -0.5 * y2 * y1_term * y3_term;
                                    dndy3GaussPts[ife][node_id] = 0.25 * ( 1 - y2 * y2 ) * y1_term * sign_node_y3;
                                    break;
                                }
                                case 12:
                                case 13:
                                case 14:
                                case 15: {
                                    shapeFunction[ife][node_id] = 0.25 * ( 1 - y3 * y3 ) * y1_term * y2_term;
                                    dndy1GaussPts[ife][node_id] = 0.25 * ( 1 - y3 * y3 ) * sign_node_y1 * y2_term;
                                    dndy2GaussPts[ife][node_id] = 0.25 * ( 1 - y3 * y3 ) * y1_term * sign_node_y2;
                                    dndy3GaussPts[ife][node_id] = -0.5 * y3 * y1_term * y2_term;
                                    break;
                                }
                            }
                        }
                        totalGaussWeight[ife] = gaussWeight[i] * gaussWeight[j] * gaussWeight[k];
                        ife++;
                    }
                }
            }
            break;
    }
}

void GaussIntegration::calculate_derivative_at_nodes( double dndy1_at_nodes[][maxNumberNodes],
                                                      double dndy2_at_nodes[][maxNumberNodes] )
{
    double y1 = 0., y2 = 0.;
    int i;
    for( i = 0; i < numberNodes; i++ )
    {
        switch( i )
        {
            case 0:
                y1 = -1.;
                y2 = -1.;
                break;
            case 1:
                y1 = 1.;
                y2 = -1.;
                break;
            case 2:
                y1 = 1.;
                y2 = 1.;
                break;
            case 3:
                y1 = -1.;
                y2 = 1.;
                break;

                // midside nodes if there is any

            case 4:
                y1 = 0.;
                y2 = -1.;
                break;

            case 5:
                y1 = 1.;
                y2 = 0.;
                break;

            case 6:
                y1 = 0.;
                y2 = 1.;
                break;

            case 7:
                y1 = -1.;
                y2 = 0.;
                break;
        }

        switch( numberNodes )
        {
            case 4:
                // dn_i/dy1 evaluated at node i
                dndy1_at_nodes[i][0] = -0.25 * ( 1 - y2 );
                dndy1_at_nodes[i][1] = 0.25 * ( 1 - y2 );
                dndy1_at_nodes[i][2] = 0.25 * ( 1 + y2 );
                dndy1_at_nodes[i][3] = -0.25 * ( 1 + y2 );

                // dn_i/dy2 evaluated at node i
                dndy2_at_nodes[i][0] = -0.25 * ( 1 - y1 );
                dndy2_at_nodes[i][1] = -0.25 * ( 1 + y1 );
                dndy2_at_nodes[i][2] = 0.25 * ( 1 + y1 );
                dndy2_at_nodes[i][3] = 0.25 * ( 1 - y1 );
                break;

            case 8:

                dndy1_at_nodes[i][0] = 0.25 * ( 1 - y2 ) * ( 2.0 * y1 + y2 );
                dndy1_at_nodes[i][1] = 0.25 * ( 1 - y2 ) * ( 2.0 * y1 - y2 );
                dndy1_at_nodes[i][2] = 0.25 * ( 1 + y2 ) * ( 2.0 * y1 + y2 );
                dndy1_at_nodes[i][3] = 0.25 * ( 1 + y2 ) * ( 2.0 * y1 - y2 );

                dndy1_at_nodes[i][4] = -y1 * ( 1 - y2 );
                dndy1_at_nodes[i][5] = 0.5 * ( 1 - y2 * y2 );
                dndy1_at_nodes[i][6] = -y1 * ( 1 + y2 );
                dndy1_at_nodes[i][7] = -0.5 * ( 1 - y2 * y2 );

                dndy2_at_nodes[i][0] = 0.25 * ( 1 - y1 ) * ( 2.0 * y2 + y1 );
                dndy2_at_nodes[i][1] = 0.25 * ( 1 + y1 ) * ( 2.0 * y2 - y1 );
                dndy2_at_nodes[i][2] = 0.25 * ( 1 + y1 ) * ( 2.0 * y2 + y1 );
                dndy2_at_nodes[i][3] = 0.25 * ( 1 - y1 ) * ( 2.0 * y2 - y1 );

                dndy2_at_nodes[i][4] = -0.5 * ( 1 - y1 * y1 );
                dndy2_at_nodes[i][5] = -y2 * ( 1 + y1 );
                dndy2_at_nodes[i][6] = 0.5 * ( 1 - y1 * y1 );
                dndy2_at_nodes[i][7] = -y2 * ( 1 - y1 );
                break;
        }
    }
}

void GaussIntegration::calculate_derivative_at_nodes_3d( double dndy1_at_nodes[][maxNumberNodes],
                                                         double dndy2_at_nodes[][maxNumberNodes],
                                                         double dndy3_at_nodes[][maxNumberNodes] )
{
    double y1, y2, y3, sign_node_y1, sign_node_y2, sign_node_y3;
    double y1_term, y2_term, y3_term, y123_temp;
    int node_id, node_id_2;
    for( node_id = 0; node_id < numberNodes; node_id++ )
    {
        get_signs_for_node_local_coord_hex( node_id, y1, y2, y3 );

        switch( numberNodes )
        {
            case 8:
                for( node_id_2 = 0; node_id_2 < numberNodes; node_id_2++ )
                {
                    get_signs_for_node_local_coord_hex( node_id_2, sign_node_y1, sign_node_y2, sign_node_y3 );
                    y1_term = 1 + sign_node_y1 * y1;
                    y2_term = 1 + sign_node_y2 * y2;
                    y3_term = 1 + sign_node_y3 * y3;

                    dndy1_at_nodes[node_id][node_id_2] = 0.125 * sign_node_y1 * y2_term * y3_term;

                    dndy2_at_nodes[node_id][node_id_2] = 0.125 * sign_node_y2 * y1_term * y3_term;

                    dndy3_at_nodes[node_id][node_id_2] = 0.125 * sign_node_y3 * y1_term * y2_term;
                }
                break;
            case 20:
                for( node_id_2 = 0; node_id_2 < numberNodes; node_id_2++ )
                {
                    get_signs_for_node_local_coord_hex( node_id_2, sign_node_y1, sign_node_y2, sign_node_y3 );

                    y1_term   = 1 + sign_node_y1 * y1;
                    y2_term   = 1 + sign_node_y2 * y2;
                    y3_term   = 1 + sign_node_y3 * y3;
                    y123_temp = sign_node_y1 * y1 + sign_node_y2 * y2 + sign_node_y3 * y3 - 2.;
                    switch( node_id_2 )
                    {
                        case 0:
                        case 1:
                        case 2:
                        case 3:
                        case 4:
                        case 5:
                        case 6:
                        case 7: {
                            dndy1_at_nodes[node_id][node_id_2] = 0.125 * sign_node_y1 * y2_term * y3_term * y123_temp +
                                                                 0.125 * y1_term * y2_term * y3_term * sign_node_y1;
                            dndy2_at_nodes[node_id][node_id_2] = 0.125 * sign_node_y2 * y1_term * y3_term * y123_temp +
                                                                 0.125 * y1_term * y2_term * y3_term * sign_node_y2;
                            dndy3_at_nodes[node_id][node_id_2] = 0.125 * sign_node_y3 * y1_term * y2_term * y123_temp +
                                                                 0.125 * y1_term * y2_term * y3_term * sign_node_y3;
                            break;
                        }
                        case 8:
                        case 10:
                        case 16:
                        case 18: {
                            dndy1_at_nodes[node_id][node_id_2] = -0.5 * y1 * y2_term * y3_term;
                            dndy2_at_nodes[node_id][node_id_2] = 0.25 * ( 1 - y1 * y1 ) * sign_node_y2 * y3_term;
                            dndy3_at_nodes[node_id][node_id_2] = 0.25 * ( 1 - y1 * y1 ) * y2_term * sign_node_y3;
                            break;
                        }
                        case 9:
                        case 11:
                        case 17:
                        case 19: {
                            dndy1_at_nodes[node_id][node_id_2] = 0.25 * ( 1 - y2 * y2 ) * sign_node_y1 * y3_term;
                            dndy2_at_nodes[node_id][node_id_2] = -0.5 * y2 * y1_term * y3_term;
                            dndy3_at_nodes[node_id][node_id_2] = 0.25 * ( 1 - y2 * y2 ) * y1_term * sign_node_y3;
                            break;
                        }
                        case 12:
                        case 13:
                        case 14:
                        case 15: {
                            dndy1_at_nodes[node_id][node_id_2] = 0.25 * ( 1 - y3 * y3 ) * sign_node_y1 * y2_term;
                            dndy2_at_nodes[node_id][node_id_2] = 0.25 * ( 1 - y3 * y3 ) * y1_term * sign_node_y2;
                            dndy3_at_nodes[node_id][node_id_2] = -0.5 * y3 * y1_term * y2_term;
                            break;
                        }
                    }
                }
                break;
        }
    }
}

void GaussIntegration::get_signs_for_node_local_coord_hex( int node_id,
                                                           double& sign_node_y1,
                                                           double& sign_node_y2,
                                                           double& sign_node_y3 )
{
    switch( node_id )
    {
        case 0:
            sign_node_y1 = -1.;
            sign_node_y2 = -1.;
            sign_node_y3 = -1.;
            break;
        case 1:
            sign_node_y1 = 1.;
            sign_node_y2 = -1.;
            sign_node_y3 = -1.;
            break;
        case 2:
            sign_node_y1 = 1.;
            sign_node_y2 = 1.;
            sign_node_y3 = -1.;
            break;
        case 3:
            sign_node_y1 = -1.;
            sign_node_y2 = 1.;
            sign_node_y3 = -1.;
            break;
        case 4:
            sign_node_y1 = -1.;
            sign_node_y2 = -1.;
            sign_node_y3 = 1.;
            break;
        case 5:
            sign_node_y1 = 1.;
            sign_node_y2 = -1.;
            sign_node_y3 = 1.;
            break;
        case 6:
            sign_node_y1 = 1.;
            sign_node_y2 = 1.;
            sign_node_y3 = 1.;
            break;
        case 7:
            sign_node_y1 = -1.;
            sign_node_y2 = 1.;
            sign_node_y3 = 1.;
            break;
        case 8:
            sign_node_y1 = 0;
            sign_node_y2 = -1.;
            sign_node_y3 = -1.;
            break;
        case 9:
            sign_node_y1 = 1.;
            sign_node_y2 = 0;
            sign_node_y3 = -1.;
            break;
        case 10:
            sign_node_y1 = 0;
            sign_node_y2 = 1.;
            sign_node_y3 = -1.;
            break;
        case 11:
            sign_node_y1 = -1.;
            sign_node_y2 = 0.;
            sign_node_y3 = -1.;
            break;
        case 12:
            sign_node_y1 = -1.;
            sign_node_y2 = -1.;
            sign_node_y3 = 0.;
            break;
        case 13:
            sign_node_y1 = 1.;
            sign_node_y2 = -1.;
            sign_node_y3 = 0.;
            break;
        case 14:
            sign_node_y1 = 1.;
            sign_node_y2 = 1.;
            sign_node_y3 = 0.;
            break;
        case 15:
            sign_node_y1 = -1.;
            sign_node_y2 = 1.;
            sign_node_y3 = 0.;
            break;
        case 16:
            sign_node_y1 = 0;
            sign_node_y2 = -1.;
            sign_node_y3 = 1.;
            break;
        case 17:
            sign_node_y1 = 1.;
            sign_node_y2 = 0;
            sign_node_y3 = 1.;
            break;
        case 18:
            sign_node_y1 = 0;
            sign_node_y2 = 1.;
            sign_node_y3 = 1.;
            break;
        case 19:
            sign_node_y1 = -1.;
            sign_node_y2 = 0.;
            sign_node_y3 = 1.;
            break;
    }
}

void GaussIntegration::get_tri_rule_pts_and_weight()
{
    // get triangular rule integration points and weight

    switch( numberGaussPoints )
    {
        case 6:
            y1Area[0] = 0.09157621;
            y2Area[0] = 0.09157621;

            y1Area[1] = 0.09157621;
            y2Area[1] = 0.8168476;

            y1Area[2] = 0.8168476;
            y2Area[2] = 0.09157621;

            y1Area[3] = 0.4459485;
            y2Area[3] = 0.4459485;

            y1Area[4] = 0.4459485;
            y2Area[4] = 0.1081030;

            y1Area[5] = 0.1081030;
            y2Area[5] = 0.4459485;

            int i;
            for( i = 0; i < 3; i++ )
            {
                totalGaussWeight[i] = 0.06348067;
            }
            for( i = 3; i < 6; i++ )
            {
                totalGaussWeight[i] = 0.1289694;
            }
            break;
    }
}

void GaussIntegration::calculate_shape_function_2d_tri()
{
    int ife;
    double y1, y2, y3;
    get_tri_rule_pts_and_weight();

    for( ife = 0; ife < totalNumberGaussPts; ife++ )
    {
        y1 = y1Area[ife];
        y2 = y2Area[ife];
        y3 = 1.0 - y1 - y2;

        shapeFunction[ife][0] = y1 * ( 2. * y1 - 1. );
        shapeFunction[ife][1] = y2 * ( 2. * y2 - 1. );
        shapeFunction[ife][2] = y3 * ( 2. * y3 - 1. );

        shapeFunction[ife][3] = 4. * y1 * y2;
        shapeFunction[ife][4] = 4. * y2 * y3;
        shapeFunction[ife][5] = 4. * y1 * y3;

        dndy1GaussPts[ife][0] = 4 * y1 - 1.;
        dndy1GaussPts[ife][1] = 0;
        dndy1GaussPts[ife][2] = 1 - 4. * y3;

        dndy1GaussPts[ife][3] = 4. * y2;
        dndy1GaussPts[ife][4] = -4. * y2;
        dndy1GaussPts[ife][5] = 4. * ( 1 - 2 * y1 - y2 );

        dndy2GaussPts[ife][0] = 0.0;
        dndy2GaussPts[ife][1] = 4. * y2 - 1.;
        dndy2GaussPts[ife][2] = 1 - 4. * y3;

        dndy2GaussPts[ife][3] = 4. * y1;
        dndy2GaussPts[ife][4] = 4. * ( 1 - y1 - 2. * y2 );
        dndy2GaussPts[ife][5] = -4. * y1;
    }
}

void GaussIntegration::calculate_derivative_at_nodes_2d_tri( double dndy1_at_nodes[][maxNumberNodes],
                                                             double dndy2_at_nodes[][maxNumberNodes] )
{
    double y1 = 0., y2 = 0., y3;
    int i;
    for( i = 0; i < numberNodes; i++ )
    {
        switch( i )
        {
            case 0:
                y1 = 1.;
                y2 = 0.;
                break;
            case 1:
                y1 = 0.;
                y2 = 1.;
                break;
            case 2:
                y1 = 0.;
                y2 = 0.;
                break;
            case 3:
                y1 = 0.5;
                y2 = 0.5;
                break;
            case 4:
                y1 = 0.;
                y2 = 0.5;
                break;
            case 5:
                y1 = 0.5;
                y2 = 0.0;
                break;
        }

        y3 = 1. - y1 - y2;

        dndy1_at_nodes[i][0] = 4 * y1 - 1.;
        dndy1_at_nodes[i][1] = 0;
        dndy1_at_nodes[i][2] = 1 - 4. * y3;

        dndy1_at_nodes[i][3] = 4. * y2;
        dndy1_at_nodes[i][4] = -4. * y2;
        dndy1_at_nodes[i][5] = 4. * ( 1 - 2 * y1 - y2 );

        dndy2_at_nodes[i][0] = 0.0;
        dndy2_at_nodes[i][1] = 4. * y2 - 1.;
        dndy2_at_nodes[i][2] = 1 - 4. * y3;

        dndy2_at_nodes[i][3] = 4. * y1;
        dndy2_at_nodes[i][4] = 4. * ( 1 - y1 - 2. * y2 );
        dndy2_at_nodes[i][5] = -4. * y1;
    }
}
void GaussIntegration::get_tet_rule_pts_and_weight()
{
    // get tetrahedron rule integration points and weight

    double a, b;
    switch( numberGaussPoints )
    {
        case 1:
            // 1 integration point formula, degree of precision 1
            y1Volume[0]         = 0.25;
            y2Volume[0]         = 0.25;
            y3Volume[0]         = 0.25;
            y4Volume[0]         = 0.25;
            totalGaussWeight[0] = 1.;
            break;
        case 4:
            // 4 integration points formula, degree of precision 2
            a = 0.58541020;
            b = 0.13819660;

            y1Volume[0] = a;
            y2Volume[0] = b;
            y3Volume[0] = b;
            y4Volume[0] = b;

            y1Volume[1] = b;
            y2Volume[1] = a;
            y3Volume[1] = b;
            y4Volume[1] = b;

            y1Volume[2] = b;
            y2Volume[2] = b;
            y3Volume[2] = a;
            y4Volume[2] = b;

            y1Volume[3] = b;
            y2Volume[3] = b;
            y3Volume[3] = b;
            y4Volume[3] = a;

            int i;
            for( i = 0; i < 4; i++ )
            {
                totalGaussWeight[i] = 0.25;
            }
            break;
    }
}

void GaussIntegration::calculate_shape_function_3d_tet()
{
    int ife;
    double y1, y2, y3, y4;
    get_tet_rule_pts_and_weight();

    switch( numberNodes )
    {
        case 10:  // 10 nodes quadratic tet
        {
            for( ife = 0; ife < totalNumberGaussPts; ife++ )
            {
                // y1,y2,y3,y4 are the volume coordinates
                y1 = y1Volume[ife];
                y2 = y2Volume[ife];
                y3 = y3Volume[ife];
                y4 = y4Volume[ife];

                // shape function is the same as in ABAQUS
                // it is different from that in all the FEA book
                // in which node is the first node
                // here at node 1 y4=1
                shapeFunction[ife][0] = y4 * ( 2. * y4 - 1. );
                shapeFunction[ife][1] = y1 * ( 2. * y1 - 1. );
                shapeFunction[ife][2] = y2 * ( 2. * y2 - 1. );
                shapeFunction[ife][3] = y3 * ( 2. * y3 - 1. );

                shapeFunction[ife][4] = 4. * y1 * y4;
                shapeFunction[ife][5] = 4. * y1 * y2;
                shapeFunction[ife][6] = 4. * y2 * y4;
                shapeFunction[ife][7] = 4. * y3 * y4;
                shapeFunction[ife][8] = 4. * y1 * y3;
                shapeFunction[ife][9] = 4. * y2 * y3;

                dndy1GaussPts[ife][0] = 1 - 4 * y4;
                dndy1GaussPts[ife][1] = 4 * y1 - 1.;
                dndy1GaussPts[ife][2] = 0;
                dndy1GaussPts[ife][3] = 0;

                dndy1GaussPts[ife][4] = 4. * ( y4 - y1 );
                dndy1GaussPts[ife][5] = 4. * y2;
                dndy1GaussPts[ife][6] = -4. * y2;
                dndy1GaussPts[ife][7] = -4. * y3;
                dndy1GaussPts[ife][8] = 4. * y3;
                dndy1GaussPts[ife][9] = 0;

                dndy2GaussPts[ife][0] = 1 - 4 * y4;
                dndy2GaussPts[ife][1] = 0;
                dndy2GaussPts[ife][2] = 4. * y2 - 1.;
                dndy2GaussPts[ife][3] = 0;

                dndy2GaussPts[ife][4] = -4. * y1;
                dndy2GaussPts[ife][5] = 4. * y1;
                dndy2GaussPts[ife][6] = 4. * ( y4 - y2 );
                dndy2GaussPts[ife][7] = -4. * y3;
                dndy2GaussPts[ife][8] = 0.;
                dndy2GaussPts[ife][9] = 4. * y3;

                dndy3GaussPts[ife][0] = 1 - 4 * y4;
                dndy3GaussPts[ife][1] = 0;
                dndy3GaussPts[ife][2] = 0;
                dndy3GaussPts[ife][3] = 4. * y3 - 1.;

                dndy3GaussPts[ife][4] = -4. * y1;
                dndy3GaussPts[ife][5] = 0;
                dndy3GaussPts[ife][6] = -4. * y2;
                dndy3GaussPts[ife][7] = 4. * ( y4 - y3 );
                dndy3GaussPts[ife][8] = 4. * y1;
                dndy3GaussPts[ife][9] = 4. * y2;
            }
            break;
        }
        case 4:  // four node linear tet for debug purpose
        {
            for( ife = 0; ife < totalNumberGaussPts; ife++ )
            {
                y1 = y1Volume[ife];
                y2 = y2Volume[ife];
                y3 = y3Volume[ife];
                y4 = y4Volume[ife];

                shapeFunction[ife][0] = y4;
                shapeFunction[ife][1] = y1;
                shapeFunction[ife][2] = y2;
                shapeFunction[ife][3] = y3;

                dndy1GaussPts[ife][0] = -1.;
                dndy1GaussPts[ife][1] = 1;
                dndy1GaussPts[ife][2] = 0;
                dndy1GaussPts[ife][3] = 0;

                dndy2GaussPts[ife][0] = -1.;
                dndy2GaussPts[ife][1] = 0;
                dndy2GaussPts[ife][2] = 1;
                dndy2GaussPts[ife][3] = 0;

                dndy3GaussPts[ife][0] = -1.;
                dndy3GaussPts[ife][1] = 0;
                dndy3GaussPts[ife][2] = 0;
                dndy3GaussPts[ife][3] = 1;
            }
            break;
        }
    }
}

void GaussIntegration::calculate_derivative_at_nodes_3d_tet( double dndy1_at_nodes[][maxNumberNodes],
                                                             double dndy2_at_nodes[][maxNumberNodes],
                                                             double dndy3_at_nodes[][maxNumberNodes] )
{
    double y1, y2, y3, y4;
    int i;

    switch( numberNodes )
    {
        case 10: {
            for( i = 0; i < numberNodes; i++ )
            {
                get_node_local_coord_tet( i, y1, y2, y3, y4 );

                dndy1_at_nodes[i][0] = 1 - 4 * y4;
                dndy1_at_nodes[i][1] = 4 * y1 - 1.;
                dndy1_at_nodes[i][2] = 0;
                dndy1_at_nodes[i][3] = 0;

                dndy1_at_nodes[i][4] = 4. * ( y4 - y1 );
                dndy1_at_nodes[i][5] = 4. * y2;
                dndy1_at_nodes[i][6] = -4. * y2;
                dndy1_at_nodes[i][7] = -4. * y3;
                dndy1_at_nodes[i][8] = 4. * y3;
                dndy1_at_nodes[i][9] = 0;

                dndy2_at_nodes[i][0] = 1 - 4 * y4;
                dndy2_at_nodes[i][1] = 0;
                dndy2_at_nodes[i][2] = 4. * y2 - 1.;
                dndy2_at_nodes[i][3] = 0;
                dndy2_at_nodes[i][4] = -4. * y1;
                dndy2_at_nodes[i][5] = 4. * y1;
                dndy2_at_nodes[i][6] = 4. * ( y4 - y2 );
                dndy2_at_nodes[i][7] = -4. * y3;
                dndy2_at_nodes[i][8] = 0.;
                dndy2_at_nodes[i][9] = 4. * y3;

                dndy3_at_nodes[i][0] = 1 - 4 * y4;
                dndy3_at_nodes[i][1] = 0;
                dndy3_at_nodes[i][2] = 0;
                dndy3_at_nodes[i][3] = 4. * y3 - 1.;

                dndy3_at_nodes[i][4] = -4. * y1;
                dndy3_at_nodes[i][5] = 0;
                dndy3_at_nodes[i][6] = -4. * y2;
                dndy3_at_nodes[i][7] = 4. * ( y4 - y3 );
                dndy3_at_nodes[i][8] = 4. * y1;
                dndy3_at_nodes[i][9] = 4. * y2;
            }
            break;
        }
        case 4: {
            for( i = 0; i < numberNodes; i++ )
            {
                get_node_local_coord_tet( i, y1, y2, y3, y4 );
                dndy1_at_nodes[i][0] = -1.;
                dndy1_at_nodes[i][1] = 1;
                dndy1_at_nodes[i][2] = 0;
                dndy1_at_nodes[i][3] = 0;

                dndy2_at_nodes[i][0] = -1.;
                dndy2_at_nodes[i][1] = 0;
                dndy2_at_nodes[i][2] = 1;
                dndy2_at_nodes[i][3] = 0;

                dndy3_at_nodes[i][0] = -1.;
                dndy3_at_nodes[i][1] = 0;
                dndy3_at_nodes[i][2] = 0;
                dndy3_at_nodes[i][3] = 1;
            }
            break;
        }
    }
}

void GaussIntegration::get_node_local_coord_tet( int node_id, double& y1, double& y2, double& y3, double& y4 )
{
    switch( node_id )
    {
        case 0:
            y1 = 0.;
            y2 = 0.;
            y3 = 0.;
            y4 = 1.;
            break;
        case 1:
            y1 = 1.;
            y2 = 0.;
            y3 = 0.;
            y4 = 0.;
            break;
        case 2:
            y1 = 0.;
            y2 = 1.;
            y3 = 0.;
            y4 = 0.;
            break;
        case 3:
            y1 = 0.;
            y2 = 0.;
            y3 = 1.;
            y4 = 0.;
            break;
        case 4:
            y1 = 0.5;
            y2 = 0.;
            y3 = 0.;
            y4 = 0.5;
            break;
        case 5:
            y1 = 0.5;
            y2 = 0.5;
            y3 = 0.;
            y4 = 0.;
            break;
        case 6:
            y1 = 0.;
            y2 = 0.5;
            y3 = 0.;
            y4 = 0.5;
            break;
        case 7:
            y1 = 0.;
            y2 = 0.0;
            y3 = 0.5;
            y4 = 0.5;
            break;
        case 8:
            y1 = 0.5;
            y2 = 0.;
            y3 = 0.5;
            y4 = 0.0;
            break;
        case 9:
            y1 = 0.;
            y2 = 0.5;
            y3 = 0.5;
            y4 = 0.;
            break;
    }
}