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 <unsigned ELEMENT_DIM>
GaussianQuadratureRule<ELEMENT_DIM>::GaussianQuadratureRule([[maybe_unused]] unsigned quadratureOrder)
{
    if constexpr (ELEMENT_DIM == 0)
    {
        mNumQuadPoints = 1;
        mWeights.push_back(1);
        mPoints.push_back(ChastePoint<0>());
    }
    else if constexpr (ELEMENT_DIM == 1)
    {
        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();
    }
    else if constexpr (ELEMENT_DIM == 2)
    {
        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();
    }
    else if constexpr (ELEMENT_DIM == 3)
    {
        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();
    }
    else
    {
        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>;