feat: implement consistent-mass HHT shell dynamics
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KOKO\Mimi
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#include <gtest/gtest.h>
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#include <algorithm>
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#include <array>
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#include <cmath>
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#include <complex>
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#include <cstddef>
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#include <iostream>
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#include <stdexcept>
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#include "fesa/analysis/hht_time_integration.h"
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namespace {
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struct Oscillator {
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fesa::Vector q{1U};
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fesa::Vector v{1U};
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fesa::Vector a{1U};
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fesa::Vector auxiliary{1U};
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};
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void Advance(Oscillator* state, const double mass, const double stiffness,
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const double force, const double h,
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const fesa::HhtParameters& parameters) {
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const auto predictor =
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fesa::PredictHht(state->q, state->v, state->auxiliary, h, parameters);
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if (!predictor.HasValue())
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throw std::runtime_error{"Invalid oscillator predictor."};
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fesa::Vector q{1U};
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q[0] =
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(mass * predictor.Value().coordinates[0] +
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parameters.beta * h * h *
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((1.0 + parameters.alpha) * force -
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mass * parameters.alpha * state->a[0])) /
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(mass + (1.0 + parameters.alpha) * stiffness * parameters.beta * h * h);
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const auto rates =
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fesa::EvaluateHhtRates(q, predictor.Value(), state->a, h, parameters);
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if (!rates.HasValue()) throw std::runtime_error{"Invalid oscillator rates."};
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state->q = q;
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state->v = rates.Value().velocity;
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state->a = rates.Value().acceleration;
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state->auxiliary = rates.Value().algorithmic_acceleration;
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}
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using Matrix3 = std::array<std::array<double, 3>, 3>;
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Matrix3 Amplification(const double frequency_step, const double alpha) {
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// Independent dimensionless recurrence in (q,h*v,h^2*A), with m=1.
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const double beta = (1.0 - alpha) * (1.0 - alpha) / 4.0;
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const double gamma = 0.5 - alpha;
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const double stiffness = frequency_step * frequency_step;
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Matrix3 matrix{};
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for (std::size_t column = 0; column < 3; ++column) {
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const double q = column == 0U ? 1.0 : 0.0;
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const double velocity = column == 1U ? 1.0 : 0.0;
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const double auxiliary = column == 2U ? 1.0 : 0.0;
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const double predictor = q + velocity + (0.5 - beta) * auxiliary;
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const double next_auxiliary =
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(-(1.0 + alpha) * stiffness * predictor + alpha * stiffness * q) /
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(1.0 + (1.0 + alpha) * stiffness * beta);
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matrix[0][column] = predictor + beta * next_auxiliary;
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matrix[1][column] =
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velocity + (1.0 - gamma) * auxiliary + gamma * next_auxiliary;
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matrix[2][column] = next_auxiliary;
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}
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return matrix;
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}
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std::array<std::complex<double>, 3> Eigenvalues(const Matrix3& matrix) {
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const double trace = matrix[0][0] + matrix[1][1] + matrix[2][2];
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const double second =
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matrix[0][0] * matrix[1][1] + matrix[0][0] * matrix[2][2] +
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matrix[1][1] * matrix[2][2] - matrix[0][1] * matrix[1][0] -
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matrix[0][2] * matrix[2][0] - matrix[1][2] * matrix[2][1];
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const double determinant =
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matrix[0][0] *
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(matrix[1][1] * matrix[2][2] - matrix[1][2] * matrix[2][1]) -
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matrix[0][1] *
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(matrix[1][0] * matrix[2][2] - matrix[1][2] * matrix[2][0]) +
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matrix[0][2] *
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(matrix[1][0] * matrix[2][1] - matrix[1][1] * matrix[2][0]);
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if (determinant == 0.0) {
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const auto discriminant =
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std::sqrt(std::complex<double>{trace * trace - 4.0 * second, 0.0});
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return {std::complex<double>{0.0, 0.0}, 0.5 * (trace + discriminant),
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0.5 * (trace - discriminant)};
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}
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std::array<std::complex<double>, 3> roots{
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std::complex<double>{1.1, 0.0}, {-0.6, 0.9}, {-0.6, -0.9}};
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bool converged = false;
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for (std::size_t iteration = 0; iteration < 500U; ++iteration) {
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const auto previous = roots;
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double change = 0.0;
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for (std::size_t i = 0; i < 3; ++i) {
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const auto value =
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((previous[i] - trace) * previous[i] + second) * previous[i] -
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determinant;
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const auto denominator = (previous[i] - previous[(i + 1U) % 3U]) *
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(previous[i] - previous[(i + 2U) % 3U]);
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roots[i] = previous[i] - value / denominator;
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change = std::max(change, std::abs(roots[i] - previous[i]));
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}
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if (change <= 1.0e-13) {
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converged = true;
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break;
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}
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}
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EXPECT_TRUE(converged);
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for (const auto root : roots) {
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const double numerator =
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std::abs(((root - trace) * root + second) * root - determinant);
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const double scale =
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std::pow(std::abs(root), 3) + std::abs(trace) * std::norm(root) +
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std::abs(second) * std::abs(root) + std::abs(determinant);
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EXPECT_LE(scale > 0.0 ? numerator / scale : numerator, 1.0e-12);
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}
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return roots;
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}
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TEST(HhtEnergyVerification, C07AlphaZeroConservesLinearOscillatorEnergy) {
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const auto parameters = fesa::MakeHhtParameters(0.0);
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ASSERT_TRUE(parameters.HasValue());
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Oscillator state;
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state.q[0] = 1.0;
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state.a[0] = -4.0;
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state.auxiliary = state.a;
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double maximum = 0.0;
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for (std::size_t step = 0; step < 640U; ++step) {
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Advance(&state, 1.0, 4.0, 0.0, 1.0 / 64.0, parameters.Value());
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const double energy =
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0.5 * state.v[0] * state.v[0] + 2.0 * state.q[0] * state.q[0];
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maximum = std::max(maximum, std::abs(energy - 2.0) / 2.0);
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}
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std::cout << "C07 alpha0 max energy defect=" << maximum << '\n';
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EXPECT_LE(maximum, 1.0e-10);
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}
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TEST(HhtEnergyVerification, C07DefaultAlphaReportsEnergyAndPhaseChanges) {
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const auto parameters = fesa::MakeHhtParameters(-0.05);
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ASSERT_TRUE(parameters.HasValue());
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Oscillator state;
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state.q[0] = 1.0;
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state.a[0] = -4.0;
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state.auxiliary = state.a;
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double maximum = 0.0;
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for (std::size_t step = 0; step < 640U; ++step) {
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Advance(&state, 1.0, 4.0, 0.0, 1.0 / 64.0, parameters.Value());
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const double energy =
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0.5 * state.v[0] * state.v[0] + 2.0 * state.q[0] * state.q[0];
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maximum = std::max(maximum, std::abs(energy - 2.0) / 2.0);
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}
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const double energy =
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0.5 * state.v[0] * state.v[0] + 2.0 * state.q[0] * state.q[0];
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std::cout << "C07 default alpha max energy defect=" << maximum
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<< " signed endpoint=" << (energy - 2.0) / 2.0
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<< " displacement error=" << state.q[0] - std::cos(20.0)
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<< " phase error="
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<< std::remainder(std::atan2(-state.v[0] / 2.0, state.q[0]) - 20.0,
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2.0 * std::acos(-1.0))
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<< '\n';
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EXPECT_TRUE(std::isfinite(energy));
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EXPECT_LT(energy, 2.0);
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}
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TEST(HhtEnergyVerification,
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C07IndependentAmplificationHasThePrescribedSpectrum) {
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for (const double alpha : {0.0, -0.05}) {
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double previous_limit_error = 1.0;
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for (const double frequency_step : {0.01, 0.1, 1.0, 10.0, 100.0, 1000.0}) {
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auto roots = Eigenvalues(Amplification(frequency_step, alpha));
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std::sort(roots.begin(), roots.end(), [](const auto& a, const auto& b) {
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return std::abs(a) < std::abs(b);
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});
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const double radius = std::abs(roots.back());
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std::cout << "C07 alpha=" << alpha << " omega*h=" << frequency_step
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<< " spectral radius=" << radius << '\n';
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if (alpha == 0.0) {
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EXPECT_NEAR(std::abs(roots[1]), 1.0, 1.0e-10);
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EXPECT_NEAR(std::abs(roots[2]), 1.0, 1.0e-10);
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} else {
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EXPECT_LE(radius, 1.0 + 1.0e-10);
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const double limit = (1.0 + alpha) / (1.0 - alpha);
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if (frequency_step >= 100.0) {
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const double error = std::abs(radius - limit);
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EXPECT_LT(error, previous_limit_error);
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previous_limit_error = error;
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std::cout << "C07 high-frequency limit=" << limit
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<< " error=" << error << '\n';
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}
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}
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}
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}
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}
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TEST(HhtEnergyVerification,
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C07SupportReactionUsesPhysicalAccelerationAndCoupling) {
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const auto parameters = fesa::MakeHhtParameters(-0.05);
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ASSERT_TRUE(parameters.HasValue());
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Oscillator state;
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state.a[0] = 1.0;
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state.auxiliary = state.a;
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double mutation_difference = 0.0;
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for (std::size_t step = 1; step <= 64U; ++step) {
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Advance(&state, 2.0, 8.0, 2.0, 1.0 / 64.0, parameters.Value());
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const std::array<double, 2> displacement{state.q[0], 0.0};
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const std::array<double, 2> acceleration{state.a[0], 0.0};
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const std::array<std::array<double, 2>, 2> mass{{{2.0, 0.5}, {0.5, 1.0}}};
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const double reaction = mass[1][0] * acceleration[0] +
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mass[1][1] * acceleration[1] +
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3.0 * displacement[1];
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EXPECT_EQ(reaction, 0.5 * state.a[0]);
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mutation_difference = std::max(
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mutation_difference, std::abs(reaction - 0.5 * state.auxiliary[0]));
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const double momentum_rate = 2.5 * state.a[0];
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const double balance = 2.0 + reaction - 8.0 * state.q[0] - momentum_rate;
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const double scale = 2.0 + std::abs(reaction) + std::abs(8.0 * state.q[0]) +
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std::abs(momentum_rate);
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EXPECT_LE(std::abs(balance) / scale, 1.0e-10);
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}
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EXPECT_GT(mutation_difference, 1.0e-6);
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}
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} // namespace
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