feat(hex8): preserve full high-low state in dynamic element responses
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KOKO\Mimi
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#include "fesa/elements/hex8_element.h"
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#include <gtest/gtest.h>
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#include <cmath>
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#include "fesa/materials/isotropic_linear_elastic_material.h"
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#include "../../../src/fesa/math/high_low_arithmetic.h"
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#include "../../../src/fesa/elements/hex8_geometry.h"
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namespace {
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using fesa::math_internal::HighLow;
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using fesa::math_internal::Subtract;
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fesa::Hex8Element UnitBrick(double offset = 0, double nu = 0) {
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fesa::ElementDofLayout layout;
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layout.node_indices = {0, 1, 2, 3, 4, 5, 6, 7};
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layout.components_per_node = {fesa::DofComponent::kUx,
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fesa::DofComponent::kUy,
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fesa::DofComponent::kUz};
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fesa::Hex8Element::Coordinates coordinates{{{0, 0, 0}, {1, 0, 0},
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{1, 1, 0}, {0, 1, 0},
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{0, 0, 1}, {1, 0, 1},
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{1, 1, 1}, {0, 1, 1}}};
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for (auto& x : coordinates) x[0] += offset;
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fesa::IsotropicLinearElasticMaterial material("test", 12, nu, {});
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return fesa::Hex8Element::Create(0, layout, coordinates, material, 2).Value();
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}
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TEST(Hex8HighLow, LowOnlyAffineStrainSurvivesUniformHighTranslation) {
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auto brick = UnitBrick();
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fesa::Vector high(24), low(24);
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const double strain = std::ldexp(1.0, -60);
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for (std::size_t a = 0; a < 8; ++a) high[3 * a] = 1;
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for (auto a : {1, 2, 5, 6}) low[3 * a] = strain;
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const auto response = brick.EvaluateLinearHighLow(high, low);
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ASSERT_TRUE(response.HasValue());
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for (std::size_t a = 0; a < 8; ++a) {
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const bool positive = a == 1 || a == 2 || a == 5 || a == 6;
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EXPECT_NEAR(response.Value().high.internal_force[3 * a],
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(positive ? 3 : -3) * strain, 1e-13 * strain);
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}
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EXPECT_NEAR(response.Value().high.strain_energy, 6 * strain * strain,
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1e-13 * strain * strain);
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for (const auto& row : response.Value().high.points)
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EXPECT_NEAR(row.strain[0], strain, 1e-13 * strain);
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}
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TEST(Hex8HighLow, NonlinearLowGradientProducesSecondOrderForce) {
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auto brick = UnitBrick();
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fesa::Vector high(24), low(24);
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const double strain = std::ldexp(1.0, -60);
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for (std::size_t a = 0; a < 8; ++a) high[3 * a] = 1;
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for (auto a : {1, 2, 5, 6}) low[3 * a] = strain;
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const auto linear = brick.EvaluateLinearHighLow(high, low);
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const auto nonlinear = brick.EvaluateNonlinearHighLow(high, low);
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ASSERT_TRUE(linear.HasValue());
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ASSERT_TRUE(nonlinear.HasValue());
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const auto& l = linear.Value();
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const auto& n = nonlinear.Value();
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const auto increment = Subtract(
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HighLow{n.high.internal_force[3], n.internal_force_low[3]},
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HighLow{l.high.internal_force[3], l.internal_force_low[3]});
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EXPECT_NEAR(increment.high, 4.5 * strain * strain,
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1e-12 * strain * strain);
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}
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TEST(Hex8HighLow, RejectsInvalidLowShapeAndOverlappingComponents) {
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auto brick = UnitBrick();
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fesa::Vector high(24), low(24);
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EXPECT_FALSE(brick.EvaluateLinearHighLow(high, fesa::Vector(23)).HasValue());
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high[0] = 1;
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low[0] = 1;
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EXPECT_FALSE(brick.EvaluateNonlinearHighLow(high, low).HasValue());
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}
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TEST(Hex8HighLow, LinearAndNonlinearZeroStateHaveSameFullHessian) {
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auto brick = UnitBrick();
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fesa::Vector zero(24);
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const auto linear = brick.EvaluateLinearHighLow(zero, zero);
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const auto nonlinear = brick.EvaluateNonlinearHighLow(zero, zero);
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ASSERT_TRUE(linear.HasValue());
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ASSERT_TRUE(nonlinear.HasValue());
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for (std::size_t i = 0; i < 24; ++i)
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for (std::size_t j = 0; j < 24; ++j) {
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EXPECT_EQ(linear.Value().high.tangent(i, j),
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nonlinear.Value().high.tangent(i, j));
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EXPECT_EQ(linear.Value().tangent_low(i, j),
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nonlinear.Value().tangent_low(i, j));
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}
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}
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TEST(Hex8HighLow, CoupledBricksTransmitLowStrainAcrossSharedFace) {
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const auto left = UnitBrick(), right = UnitBrick(1);
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fesa::Vector high(24), left_low(24), right_low(24);
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const double strain = std::ldexp(1.0, -60);
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for (std::size_t a = 0; a < 8; ++a) {
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high[3 * a] = 1;
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const bool positive = a == 1 || a == 2 || a == 5 || a == 6;
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left_low[3 * a] = positive ? strain : 0;
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right_low[3 * a] = positive ? 2 * strain : strain;
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}
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const auto l = left.EvaluateLinearHighLow(high, left_low);
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const auto r = right.EvaluateLinearHighLow(high, right_low);
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ASSERT_TRUE(l.HasValue());
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ASSERT_TRUE(r.HasValue());
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const std::array<std::size_t, 4> face_left{1, 2, 5, 6};
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const std::array<std::size_t, 4> face_right{0, 3, 4, 7};
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for (std::size_t a = 0; a < 4; ++a) {
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const auto i = 3 * face_left[a], j = 3 * face_right[a];
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const auto assembled = fesa::math_internal::Add(
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{l.Value().high.internal_force[i], l.Value().internal_force_low[i]},
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{r.Value().high.internal_force[j], r.Value().internal_force_low[j]});
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EXPECT_NEAR(assembled.high, 0, 1e-13 * strain);
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}
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EXPECT_NEAR(l.Value().high.strain_energy + r.Value().high.strain_energy,
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12 * strain * strain, 1e-13 * strain * strain);
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}
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TEST(Hex8HighLow, NodeZeroSignedDirectionMatchesForceAndEnergyDerivatives) {
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using namespace fesa::math_internal;
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auto brick = UnitBrick();
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fesa::Vector high(24), low(24);
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for (std::size_t a = 0; a < 8; ++a) {
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high[3 * a] = 1;
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low[3 * a] = (a == 1 || a == 2 || a == 5 || a == 6)
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? std::ldexp(1.0, -60) : 0;
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}
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const auto baseline = brick.EvaluateNonlinearHighLow(high, low);
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ASSERT_TRUE(baseline.HasValue());
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const auto& base = baseline.Value();
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for (int exponent : {-65, -70, -75}) {
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const double step = std::ldexp(1.0, exponent);
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auto plus_high = high, minus_high = high;
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auto plus_low = low, minus_low = low;
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const auto plus = Add({high[0], low[0]}, HighLow{step});
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const auto minus = Subtract({high[0], low[0]}, HighLow{step});
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plus_high[0] = plus.high; plus_low[0] = plus.low;
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minus_high[0] = minus.high; minus_low[0] = minus.low;
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const auto positive = brick.EvaluateNonlinearHighLow(plus_high, plus_low);
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const auto negative = brick.EvaluateNonlinearHighLow(minus_high, minus_low);
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ASSERT_TRUE(positive.HasValue());
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ASSERT_TRUE(negative.HasValue());
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for (std::size_t i = 0; i < 24; ++i) {
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const auto derivative = Divide(Subtract(
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{positive.Value().high.internal_force[i], positive.Value().internal_force_low[i]},
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{negative.Value().high.internal_force[i], negative.Value().internal_force_low[i]}),
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HighLow{2 * step});
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const auto error = Subtract(derivative,
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{base.high.tangent(i, 0), base.tangent_low(i, 0)});
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EXPECT_LE(std::abs(error.high), 1e-25);
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}
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const auto gradient = Divide(Subtract(
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{positive.Value().high.strain_energy, positive.Value().strain_energy_low},
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{negative.Value().high.strain_energy, negative.Value().strain_energy_low}),
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HighLow{2 * step});
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const auto energy_error = Subtract(gradient,
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{base.high.internal_force[0], base.internal_force_low[0]});
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// Centered energy differences retain the StVK cubic term: O(step^2).
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// For this unit E=12 brick its directional coefficient is bounded by 32.
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EXPECT_LE(std::abs(energy_error.high), 32 * step * step + 1e-44);
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}
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}
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TEST(Hex8HighLow, GeometryCertificateIncludesLowOnlyCurrentCellWidth) {
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fesa::Hex8Element::Coordinates coordinates{{{0, 0, 0}, {1, 0, 0},
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{1, 1, 0}, {0, 1, 0},
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{0, 0, 1}, {1, 0, 1},
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{1, 1, 1}, {0, 1, 1}}};
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fesa::Vector high(24), positive_low(24), negative_low(24);
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for (std::size_t a = 0; a < 8; ++a) {
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high[3 * a] = -coordinates[a][0];
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positive_low[3 * a] = std::ldexp(coordinates[a][0], -60);
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negative_low[3 * a] = -positive_low[3 * a];
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}
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using fesa::hex8_internal::Certificate;
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EXPECT_EQ(fesa::hex8_internal::Certify(coordinates, &high, &positive_low),
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Certificate::kPositive);
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EXPECT_EQ(fesa::hex8_internal::Certify(coordinates, &high, &negative_low),
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Certificate::kNonpositive);
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EXPECT_NE(fesa::hex8_internal::Certify(coordinates, &high),
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Certificate::kPositive);
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}
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TEST(Hex8HighLow, SignedZeroMaterialCoefficientHasIdenticalResponse) {
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auto positive = UnitBrick(0, 0.0), negative = UnitBrick(0, -0.0);
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fesa::Vector high(24), low(24);
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high[3] = 0.01;
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for (bool nonlinear : {false, true}) {
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const auto p = nonlinear ? positive.EvaluateNonlinearHighLow(high, low)
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: positive.EvaluateLinearHighLow(high, low);
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const auto n = nonlinear ? negative.EvaluateNonlinearHighLow(high, low)
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: negative.EvaluateLinearHighLow(high, low);
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ASSERT_TRUE(p.HasValue());
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ASSERT_TRUE(n.HasValue());
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EXPECT_EQ(p.Value().high.strain_energy, n.Value().high.strain_energy);
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EXPECT_EQ(p.Value().strain_energy_low, n.Value().strain_energy_low);
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for (std::size_t i = 0; i < 24; ++i) {
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EXPECT_EQ(p.Value().high.internal_force[i], n.Value().high.internal_force[i]);
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EXPECT_EQ(p.Value().internal_force_low[i], n.Value().internal_force_low[i]);
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for (std::size_t j = 0; j < 24; ++j) {
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EXPECT_EQ(p.Value().high.tangent(i,j), n.Value().high.tangent(i,j));
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EXPECT_EQ(p.Value().tangent_low(i,j), n.Value().tangent_low(i,j));
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}
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}
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}
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}
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TEST(Hex8HighLow, FiniteRotationRetainsTinyShearForceCovariance) {
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using namespace fesa::math_internal;
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auto brick = UnitBrick();
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fesa::Hex8Element::Coordinates x{{{0, 0, 0}, {1, 0, 0},
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{1, 1, 0}, {0, 1, 0},
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{0, 0, 1}, {1, 0, 1},
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{1, 1, 1}, {0, 1, 1}}};
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fesa::Vector zero(24), rotation(24), shear(24), rotated_shear(24);
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const double gamma = std::ldexp(1.0, -60);
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for (std::size_t a = 0; a < 8; ++a) {
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// Q is an exact quarter turn; Q*(gamma*y,0,0)=(0,gamma*y,0).
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rotation[3*a] = HighLow{-x[a][1] - x[a][0]}.high;
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rotation[3*a+1] = x[a][0] - x[a][1];
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shear[3*a] = gamma*x[a][1];
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rotated_shear[3*a+1] = gamma*x[a][1];
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}
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fesa::Vector turned_high(24), turned_low(24);
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for (std::size_t j = 0; j < 24; ++j) {
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const auto full = Add(HighLow{rotation[j]}, HighLow{rotated_shear[j]});
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turned_high[j] = full.high; turned_low[j] = full.low;
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}
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const auto origin = brick.EvaluateNonlinearHighLow(zero, zero);
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const auto deformed = brick.EvaluateNonlinearHighLow(shear, zero);
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const auto turned = brick.EvaluateNonlinearHighLow(rotation, zero);
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const auto turned_deformed = brick.EvaluateNonlinearHighLow(turned_high, turned_low);
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ASSERT_TRUE(origin.HasValue()); ASSERT_TRUE(deformed.HasValue());
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ASSERT_TRUE(turned.HasValue()); ASSERT_TRUE(turned_deformed.HasValue());
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// The fixed F64 reference gradients leave coefficient-level rotation error.
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// Subtract the same rigid baseline; do not project that force to zero.
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for (std::size_t a = 0; a < 8; ++a) {
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std::array<HighLow, 3> baseline{}, rotated{};
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for (std::size_t i = 0; i < 3; ++i) {
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const auto j = 3*a+i;
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baseline[i] = Subtract(
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{deformed.Value().high.internal_force[j], deformed.Value().internal_force_low[j]},
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{origin.Value().high.internal_force[j], origin.Value().internal_force_low[j]});
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rotated[i] = Subtract(
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{turned_deformed.Value().high.internal_force[j], turned_deformed.Value().internal_force_low[j]},
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{turned.Value().high.internal_force[j], turned.Value().internal_force_low[j]});
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}
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EXPECT_LE(std::abs(Add(rotated[0], baseline[1]).high), 1e-13*gamma);
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EXPECT_LE(std::abs(Subtract(rotated[1], baseline[0]).high), 1e-13*gamma);
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EXPECT_LE(std::abs(Subtract(rotated[2], baseline[2]).high), 1e-13*gamma);
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}
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}
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} // namespace
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