GaussianQuadratureRule.cpp

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*/

#include <cmath>

#include "GaussianQuadratureRule.hpp"
#include "Exception.hpp"
#include "UblasCustomFunctions.hpp"

template<unsigned ELEMENT_DIM>
const ChastePoint<ELEMENT_DIM>& GaussianQuadratureRule<ELEMENT_DIM>::rGetQuadPoint(unsigned index) const
{
    assert(index < mNumQuadPoints);
    return mPoints[index];
}

template<unsigned ELEMENT_DIM>
double GaussianQuadratureRule<ELEMENT_DIM>::GetWeight(unsigned index) const
{
    assert(index < mNumQuadPoints);
    return mWeights[index];
}

template<unsigned ELEMENT_DIM>
unsigned GaussianQuadratureRule<ELEMENT_DIM>::GetNumQuadPoints() const
{
    return mNumQuadPoints;
}

template<>
GaussianQuadratureRule<0>::GaussianQuadratureRule(unsigned quadratureOrder)
{
    mNumQuadPoints = 1;
    mWeights.push_back(1);
    mPoints.push_back(ChastePoint<0>());
}

template<>
GaussianQuadratureRule<1>::GaussianQuadratureRule(unsigned quadratureOrder)
{
    switch (quadratureOrder)
    {
        case 0:
        case 1: // 1d, 1st order
                // 1 point rule
            mWeights.push_back(1);
            mPoints.push_back(ChastePoint<1>(0.5));
            break;
        case 2:
        case 3: // 1d, 3rd order
                // 2 point rule
            mWeights.push_back(0.5);
            mWeights.push_back(0.5);
            {
                double sqrt_one_third = sqrt(1.0/3.0);
                mPoints.push_back(ChastePoint<1>((-sqrt_one_third+1.0)/2.0));
                mPoints.push_back(ChastePoint<1>((sqrt_one_third+1.0)/2.0));
            }
            break;
        case 4:
        case 5: // 1d, 5th order
                // 3 point rule
            mWeights.push_back(5.0/18.0);
            mWeights.push_back(4.0/9.0);
            mWeights.push_back(5.0/18.0);

            {
                double sqrt_three_fifths = sqrt(3.0/5.0);
                mPoints.push_back(ChastePoint<1>((-sqrt_three_fifths+1.0)/2.0));
                mPoints.push_back(ChastePoint<1>(0.5));
                mPoints.push_back(ChastePoint<1>((sqrt_three_fifths+1.0)/2.0));
            }
            break;
         default:
            EXCEPTION("Gauss quadrature order not supported.");
    }
    assert(mPoints.size() == mWeights.size());
    mNumQuadPoints = mPoints.size();
}

template<>
GaussianQuadratureRule<2>::GaussianQuadratureRule(unsigned quadratureOrder)
{
    double one_third = 1.0/3.0;
    double one_sixth = 1.0/6.0;
    double two_thirds = 2.0/3.0;
    switch (quadratureOrder)
    {
        case 0: // 2d, 0th order
        case 1: // 2d, 1st order
                // 1 point rule
            mWeights.push_back(0.5);
            mPoints.push_back(ChastePoint<2>(one_third, one_third));
            break;

        case 2: // 2d, 2nd order
                // 3 point rule
            mWeights.push_back(one_sixth);
            mWeights.push_back(one_sixth);
            mWeights.push_back(one_sixth);

            mPoints.push_back(ChastePoint<2>(two_thirds, one_sixth));
            mPoints.push_back(ChastePoint<2>(one_sixth,  one_sixth));
            mPoints.push_back(ChastePoint<2>(one_sixth,  two_thirds));
            break;

        case 3: // 2d, 3rd order - derived by hand and using a Macsyma script to solve the cubic
                //                60*x^3  - 60*x^2  + 15*x - 1;
                // 6 point rule
            {
                double w = 1.0/12.0;
                mWeights.push_back(w);
                mWeights.push_back(w);
                mWeights.push_back(w);
                mWeights.push_back(w);
                mWeights.push_back(w);
                mWeights.push_back(w);

                double inverse_tan = atan(0.75);
                double cos_third = cos(inverse_tan/3.0);
                double sin_third = sin(inverse_tan/3.0);
                // a = 0.23193336461755
                double a =   sin_third/(2*sqrt(3.0)) - cos_third/6.0 + 1.0/3.0;
                // b = 0.10903901046622
                double b = - sin_third/(2*sqrt(3.0)) - cos_third/6.0 + 1.0/3.0;
                // c = 0.659028
                double c = cos_third/3.0 + 1.0/3.0;

                mPoints.push_back(ChastePoint<2>(a, b));
                mPoints.push_back(ChastePoint<2>(a, c));
                mPoints.push_back(ChastePoint<2>(b, a));
                mPoints.push_back(ChastePoint<2>(b, c));
                mPoints.push_back(ChastePoint<2>(c, a));
                mPoints.push_back(ChastePoint<2>(c, b));
            }
            break;
        default:
            EXCEPTION("Gauss quadrature order not supported.");
    }
    assert(mPoints.size() == mWeights.size());
    mNumQuadPoints = mPoints.size();
}

template<>
GaussianQuadratureRule<3>::GaussianQuadratureRule(unsigned quadratureOrder)
{
    switch (quadratureOrder)
    {
        case 0: // 3d, 0th order
        case 1: // 3d, 1st order
                // 1 point rule
            mWeights.push_back(1.0/6.0);
            mPoints.push_back(ChastePoint<3>(0.25, 0.25, 0.25));
            break;

        case 2: //2nd order
                // 4 point rule
            {
                double sqrt_fifth = 1.0/sqrt(5.0);
                double a = (1.0 + 3.0*sqrt_fifth)/4.0; //0.585410196624969;
                double b = (1.0 - sqrt_fifth)/4.0; //0.138196601125011;
                double w = 1.0/24.0;

                mWeights.push_back(w);
                mWeights.push_back(w);
                mWeights.push_back(w);
                mWeights.push_back(w);

                mPoints.push_back(ChastePoint<3>(a,b,b));
                mPoints.push_back(ChastePoint<3>(b,a,b));
                mPoints.push_back(ChastePoint<3>(b,b,a));
                mPoints.push_back(ChastePoint<3>(b,b,b));
            }
            break;

        case 3: // 3d, 3rd order
            // 8 point rule
            /* The main options were
             *  5-point rule.  Commonly published rule has four symmetric points and
             *                 a negative weight in the centre.  We would like to avoid
             *                 negative weight (certainly for interpolation.
             *  8-point rule.  Uses two sets of symmetric points (as 4 point rule with a,b and then with c,d).
             *                 This one is hard to derive a closed form solution to.
             */
            {
                double root_seventeen = sqrt(17.0);
                double root_term = sqrt(1022.0-134.0*root_seventeen);
                double b = (55.0 - 3.0*root_seventeen + root_term)/196; //b = 0.328055
                double d = (55.0 - 3.0*root_seventeen - root_term)/196; //d = 0.106952

                double a = 1.0 - 3.0*b; // a = 0.0158359
                double c = 1.0 - 3.0*d; // c = 0.679143

                // w1 = 0.023088 (= 0.138528/6)
                double w1 = (20.0*d*d - 10.0*d + 1.0)/(240.0*(2.0*d*d - d - 2.0*b*b + b)); // w1 = 0.0362942
                double w2 = 1.0/24.0 - w1; // w2 = 0.0185787 (=0.111472/6)

                mWeights.push_back(w1);
                mWeights.push_back(w1);
                mWeights.push_back(w1);
                mWeights.push_back(w1);

                mWeights.push_back(w2);
                mWeights.push_back(w2);
                mWeights.push_back(w2);
                mWeights.push_back(w2);

                mPoints.push_back(ChastePoint<3>(a, b, b));
                mPoints.push_back(ChastePoint<3>(b, a, b));
                mPoints.push_back(ChastePoint<3>(b, b, a));
                mPoints.push_back(ChastePoint<3>(b, b, b));

                mPoints.push_back(ChastePoint<3>(c, d, d));
                mPoints.push_back(ChastePoint<3>(d, c, d));
                mPoints.push_back(ChastePoint<3>(d, d, c));
                mPoints.push_back(ChastePoint<3>(d, d, d));

            }
break;

        default:
            EXCEPTION("Gauss quadrature order not supported.");
    }
    assert(mPoints.size() == mWeights.size());
    mNumQuadPoints = mPoints.size();
}

template<unsigned ELEMENT_DIM>
GaussianQuadratureRule<ELEMENT_DIM>::GaussianQuadratureRule(unsigned quadratureOrder)
{
    EXCEPTION("Gauss quadrature rule not available for this dimension.");
}

// Explicit instantiation

template class GaussianQuadratureRule<0>;
template class GaussianQuadratureRule<1>;
template class GaussianQuadratureRule<2>;
template class GaussianQuadratureRule<3>;
template class GaussianQuadratureRule<4>;