KOKO\Mimi
1140 lines
46 KiB
C++
1140 lines
46 KiB
C++
#include "fesa/elements/euler_beam_3d.h"
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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 <cstddef>
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#include <limits>
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#include <stdexcept>
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#include <string>
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#include <utility>
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#include <vector>
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#include "fesa/math/vector3.h"
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#include "fesa/model/model_types.h"
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namespace fesa {
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namespace {
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constexpr std::size_t kElementDofCount = 12U;
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constexpr double kMatrixTolerance = 1.0e-12;
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constexpr double kRigidTolerance = 1.0e-10;
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constexpr double kAnalyticalTolerance = 1.0e-9;
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Node MakeNode(std::array<double, 3> coordinates, std::size_t line) {
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return {{"Beam-1", static_cast<std::int64_t>(line), std::to_string(line)},
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coordinates,
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{"beam-test.inp", line}};
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}
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LinearElasticMaterial MakeMaterial(double youngs_modulus = 210.0e9,
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double poisson_ratio = 0.3) {
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return {"Steel", youngs_modulus, poisson_ratio, {"beam-test.inp", 20U}};
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}
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GeneralBeamSection MakeSection(
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std::array<double, 3> first_axis = {0.0, 1.0, 0.0},
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std::vector<std::array<double, 2>> section_points = {}) {
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return {"Section-1",
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0.012,
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2.5e-5,
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0.0,
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4.0e-5,
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1.5e-5,
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first_axis,
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std::move(section_points),
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{"beam-test.inp", 30U}};
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}
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EulerBeam3D RequireBeam(const Node& first_node, const Node& second_node,
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const GeneralBeamSection& section,
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const LinearElasticMaterial& material) {
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auto result = EulerBeam3D::Create(first_node, second_node, section, material);
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if (!result.HasValue()) {
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throw std::runtime_error{"Expected a valid EulerBeam3D fixture."};
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}
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return std::move(result.Value());
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}
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EulerBeam3D AlignedBeam(
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double length, const GeneralBeamSection& section = MakeSection(),
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const LinearElasticMaterial& material = MakeMaterial()) {
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return RequireBeam(MakeNode({0.0, 0.0, 0.0}, 1U),
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MakeNode({length, 0.0, 0.0}, 2U), section, material);
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}
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double MaximumAbsoluteEntry(const Matrix& matrix) {
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double maximum = 0.0;
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for (std::size_t row = 0; row < matrix.Rows(); ++row) {
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for (std::size_t column = 0; column < matrix.Columns(); ++column) {
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maximum = (std::max)(maximum, std::abs(matrix(row, column)));
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}
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}
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return maximum;
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}
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bool MatrixIsFinite(const Matrix& matrix) {
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for (std::size_t row = 0; row < matrix.Rows(); ++row) {
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for (std::size_t column = 0; column < matrix.Columns(); ++column) {
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if (!std::isfinite(matrix(row, column))) {
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return false;
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}
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}
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}
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return true;
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}
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double NormalizedMatrixError(const Matrix& actual, const Matrix& expected) {
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if (actual.Rows() != expected.Rows() ||
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actual.Columns() != expected.Columns()) {
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throw std::invalid_argument{"Matrix comparison requires equal shapes."};
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}
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double maximum_difference = 0.0;
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for (std::size_t row = 0; row < actual.Rows(); ++row) {
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for (std::size_t column = 0; column < actual.Columns(); ++column) {
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maximum_difference =
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(std::max)(maximum_difference,
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std::abs(actual(row, column) - expected(row, column)));
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}
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}
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const double scale =
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(std::max)(1.0, (std::max)(MaximumAbsoluteEntry(actual),
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MaximumAbsoluteEntry(expected)));
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return maximum_difference / scale;
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}
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double VectorNorm(const Vector& vector) {
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double sum = 0.0;
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for (std::size_t index = 0; index < vector.Size(); ++index) {
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sum += vector[index] * vector[index];
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}
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return std::sqrt(sum);
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}
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double QuadraticEnergy(const Matrix& matrix, const Vector& vector) {
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const Vector product = matrix.Multiply(vector);
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double value = 0.0;
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for (std::size_t index = 0; index < vector.Size(); ++index) {
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value += vector[index] * product[index];
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}
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return value;
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}
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void ExpectScaledNear(double actual, double expected,
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double relative_tolerance) {
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const double scale = (std::max)(1.0, std::abs(expected));
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EXPECT_LE(std::abs(actual - expected), relative_tolerance * scale);
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}
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void ExpectRelativeNear(double actual, double expected,
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double relative_tolerance) {
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ASSERT_NE(expected, 0.0);
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EXPECT_LE(std::abs(actual - expected) / std::abs(expected),
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relative_tolerance);
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}
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Matrix ExpectedClosedStiffness(double length, const GeneralBeamSection& section,
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const LinearElasticMaterial& material) {
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Matrix expected{kElementDofCount, kElementDofCount};
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const double shear_modulus =
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material.youngs_modulus / (2.0 * (1.0 + material.poisson_ratio));
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const auto add_block = [&expected](const std::vector<std::size_t>& indices,
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const std::vector<double>& values) {
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const std::size_t width = indices.size();
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for (std::size_t row = 0; row < width; ++row) {
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for (std::size_t column = 0; column < width; ++column) {
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expected(indices[row], indices[column]) = values[row * width + column];
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}
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}
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};
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const double axial = material.youngs_modulus * section.area / length;
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add_block({0U, 6U}, {axial, -axial, -axial, axial});
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const double torsion = shear_modulus * section.torsional_constant / length;
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add_block({3U, 9U}, {torsion, -torsion, -torsion, torsion});
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const auto bending_block = [length](double flexural_rigidity,
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double rotation_sign) {
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const double v = 12.0 * flexural_rigidity / (length * length * length);
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const double c =
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rotation_sign * 6.0 * flexural_rigidity / (length * length);
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const double d = 4.0 * flexural_rigidity / length;
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const double e = 2.0 * flexural_rigidity / length;
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return std::vector<double>{v, c, -v, c, c, d, -c, e,
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-v, -c, v, -c, c, e, -c, d};
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};
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add_block({1U, 5U, 7U, 11U},
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bending_block(material.youngs_modulus * section.i22, 1.0));
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add_block({2U, 4U, 8U, 10U},
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bending_block(material.youngs_modulus * section.i11, -1.0));
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return expected;
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}
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std::array<double, kElementDofCount> SymmetricEigenvalues(Matrix matrix) {
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for (std::size_t iteration = 0; iteration < 100U * kElementDofCount;
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++iteration) {
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std::size_t p = 0U;
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std::size_t q = 1U;
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double maximum_off_diagonal = 0.0;
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for (std::size_t row = 0; row < kElementDofCount; ++row) {
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for (std::size_t column = row + 1U; column < kElementDofCount; ++column) {
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const double candidate = std::abs(matrix(row, column));
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if (candidate > maximum_off_diagonal) {
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maximum_off_diagonal = candidate;
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p = row;
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q = column;
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}
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}
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}
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if (maximum_off_diagonal <=
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1.0e-14 * (std::max)(1.0, MaximumAbsoluteEntry(matrix))) {
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break;
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}
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const double app = matrix(p, p);
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const double aqq = matrix(q, q);
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const double apq = matrix(p, q);
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const double angle = 0.5 * std::atan2(2.0 * apq, aqq - app);
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const double cosine = std::cos(angle);
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const double sine = std::sin(angle);
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for (std::size_t index = 0; index < kElementDofCount; ++index) {
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if (index == p || index == q) {
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continue;
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}
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const double aip = matrix(index, p);
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const double aiq = matrix(index, q);
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matrix(index, p) = cosine * aip - sine * aiq;
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matrix(p, index) = matrix(index, p);
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matrix(index, q) = sine * aip + cosine * aiq;
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matrix(q, index) = matrix(index, q);
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}
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matrix(p, p) =
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cosine * cosine * app - 2.0 * sine * cosine * apq + sine * sine * aqq;
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matrix(q, q) =
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sine * sine * app + 2.0 * sine * cosine * apq + cosine * cosine * aqq;
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matrix(p, q) = 0.0;
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matrix(q, p) = 0.0;
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}
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std::array<double, kElementDofCount> eigenvalues{};
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for (std::size_t index = 0; index < kElementDofCount; ++index) {
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eigenvalues[index] = matrix(index, index);
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}
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return eigenvalues;
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}
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std::size_t SymmetricRank(const Matrix& matrix, double relative_tolerance) {
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const auto eigenvalues = SymmetricEigenvalues(matrix);
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double maximum = 0.0;
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for (const double value : eigenvalues) {
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maximum = (std::max)(maximum, std::abs(value));
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}
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return static_cast<std::size_t>(
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std::count_if(eigenvalues.begin(), eigenvalues.end(),
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[maximum, relative_tolerance](double value) {
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return std::abs(value) > relative_tolerance * maximum;
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}));
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}
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Matrix TestOnlyOnePointStiffness(double length,
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const GeneralBeamSection& section,
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const LinearElasticMaterial& material) {
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// At xi=0 the two bending curvature rows retain only the nodal rotations.
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// This deliberately under-integrated negative control is independent of
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// production.
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Matrix b{4U, kElementDofCount};
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b(0U, 0U) = -1.0 / length;
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b(0U, 6U) = 1.0 / length;
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b(1U, 3U) = -1.0 / length;
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b(1U, 9U) = 1.0 / length;
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b(2U, 4U) = -1.0 / length;
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b(2U, 10U) = 1.0 / length;
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b(3U, 5U) = -1.0 / length;
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b(3U, 11U) = 1.0 / length;
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const double shear_modulus =
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material.youngs_modulus / (2.0 * (1.0 + material.poisson_ratio));
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const std::array<double, 4> diagonal = {
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material.youngs_modulus * section.area,
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shear_modulus * section.torsional_constant,
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material.youngs_modulus * section.i11,
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material.youngs_modulus * section.i22};
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Matrix stiffness{kElementDofCount, kElementDofCount};
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for (std::size_t row = 0; row < kElementDofCount; ++row) {
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for (std::size_t column = 0; column < kElementDofCount; ++column) {
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for (std::size_t component = 0; component < diagonal.size();
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++component) {
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stiffness(row, column) += b(component, row) * diagonal[component] *
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b(component, column) * length;
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}
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}
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}
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return stiffness;
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}
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Vector SolveFixedFirstNode(const Matrix& stiffness,
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const std::array<double, 6>& free_end_load) {
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std::array<std::array<double, 7>, 6> augmented{};
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for (std::size_t row = 0; row < 6U; ++row) {
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for (std::size_t column = 0; column < 6U; ++column) {
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augmented[row][column] = stiffness(row + 6U, column + 6U);
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}
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augmented[row][6U] = free_end_load[row];
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}
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for (std::size_t pivot = 0; pivot < 6U; ++pivot) {
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std::size_t pivot_row = pivot;
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for (std::size_t row = pivot + 1U; row < 6U; ++row) {
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if (std::abs(augmented[row][pivot]) >
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std::abs(augmented[pivot_row][pivot])) {
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pivot_row = row;
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}
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}
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if (std::abs(augmented[pivot_row][pivot]) <=
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std::numeric_limits<double>::min()) {
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throw std::runtime_error{"Cantilever fixture is singular."};
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}
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std::swap(augmented[pivot], augmented[pivot_row]);
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const double pivot_value = augmented[pivot][pivot];
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for (std::size_t column = pivot; column < 7U; ++column) {
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augmented[pivot][column] /= pivot_value;
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}
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for (std::size_t row = 0; row < 6U; ++row) {
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if (row == pivot) {
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continue;
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}
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const double factor = augmented[row][pivot];
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for (std::size_t column = pivot; column < 7U; ++column) {
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augmented[row][column] -= factor * augmented[pivot][column];
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}
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}
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}
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Vector displacement{kElementDofCount};
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for (std::size_t component = 0; component < 6U; ++component) {
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displacement[component + 6U] = augmented[component][6U];
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}
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return displacement;
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}
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Vector SolveDenseSystem(Matrix matrix, Vector right_hand_side) {
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if (matrix.Rows() != matrix.Columns() ||
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matrix.Rows() != right_hand_side.Size()) {
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throw std::invalid_argument{"Dense test solve requires a square system."};
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}
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for (std::size_t pivot = 0; pivot < matrix.Rows(); ++pivot) {
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std::size_t pivot_row = pivot;
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for (std::size_t row = pivot + 1U; row < matrix.Rows(); ++row) {
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if (std::abs(matrix(row, pivot)) > std::abs(matrix(pivot_row, pivot))) {
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pivot_row = row;
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}
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}
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if (!(std::abs(matrix(pivot_row, pivot)) > 0.0) ||
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!std::isfinite(matrix(pivot_row, pivot))) {
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throw std::runtime_error{"Uniform-load test fixture is singular."};
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}
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for (std::size_t column = pivot; column < matrix.Columns(); ++column) {
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std::swap(matrix(pivot, column), matrix(pivot_row, column));
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}
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std::swap(right_hand_side[pivot], right_hand_side[pivot_row]);
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const double pivot_value = matrix(pivot, pivot);
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for (std::size_t column = pivot; column < matrix.Columns(); ++column) {
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matrix(pivot, column) /= pivot_value;
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}
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right_hand_side[pivot] /= pivot_value;
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for (std::size_t row = 0; row < matrix.Rows(); ++row) {
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if (row == pivot) {
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continue;
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}
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const double factor = matrix(row, pivot);
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for (std::size_t column = pivot; column < matrix.Columns(); ++column) {
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matrix(row, column) -= factor * matrix(pivot, column);
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}
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right_hand_side[row] -= factor * right_hand_side[pivot];
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}
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}
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return right_hand_side;
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}
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Vector SolveUniformTransverseCantilever(std::size_t element_count,
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double length, double line_load,
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const GeneralBeamSection& section,
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const LinearElasticMaterial& material) {
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const double element_length = length / static_cast<double>(element_count);
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const std::size_t system_size = 2U * (element_count + 1U);
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Matrix assembled_stiffness{system_size, system_size};
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Vector assembled_load{system_size};
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const std::array<std::size_t, 4> bending_dofs = {1U, 5U, 7U, 11U};
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// Test-only direct assembly keeps this evidence at the formulation boundary:
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// two [v,rz] DOFs per node, with no Domain, parser, or DLOAD path.
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for (std::size_t element = 0; element < element_count; ++element) {
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const EulerBeam3D beam = AlignedBeam(element_length, section, material);
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const Matrix element_stiffness = beam.LocalStiffness();
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const Vector element_load =
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beam.LocalEquivalentLoad({0.0, line_load, 0.0, 0.0});
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const std::array<std::size_t, 4> assembled_dofs = {
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2U * element, 2U * element + 1U, 2U * (element + 1U),
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2U * (element + 1U) + 1U};
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for (std::size_t row = 0; row < bending_dofs.size(); ++row) {
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assembled_load[assembled_dofs[row]] += element_load[bending_dofs[row]];
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for (std::size_t column = 0; column < bending_dofs.size(); ++column) {
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assembled_stiffness(assembled_dofs[row], assembled_dofs[column]) +=
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element_stiffness(bending_dofs[row], bending_dofs[column]);
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}
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}
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}
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const std::size_t free_size = system_size - 2U;
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Matrix free_stiffness{free_size, free_size};
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Vector free_load{free_size};
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for (std::size_t row = 0; row < free_size; ++row) {
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free_load[row] = assembled_load[row + 2U];
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for (std::size_t column = 0; column < free_size; ++column) {
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free_stiffness(row, column) = assembled_stiffness(row + 2U, column + 2U);
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}
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}
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const Vector free_displacement =
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SolveDenseSystem(std::move(free_stiffness), std::move(free_load));
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Vector nodal_displacement{system_size};
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for (std::size_t dof = 0; dof < free_size; ++dof) {
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nodal_displacement[dof + 2U] = free_displacement[dof];
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}
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return nodal_displacement;
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}
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double UniformLoadInteriorDisplacementError(const Vector& nodal_displacement,
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std::size_t element_count,
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double length, double line_load,
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double flexural_rigidity) {
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const double element_length = length / static_cast<double>(element_count);
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const std::array<double, 5> gauss_points = {
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-0.9061798459386640, -0.5384693101056831, 0.0, 0.5384693101056831,
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0.9061798459386640};
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const std::array<double, 5> gauss_weights = {
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0.2369268850561891, 0.4786286704993665, 0.5688888888888889,
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0.4786286704993665, 0.2369268850561891};
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double squared_error = 0.0;
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double squared_reference = 0.0;
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// Five-point integration is independent of production and exactly integrates
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// the squared error between cubic Hermite interpolation and the quartic beam
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// solution.
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for (std::size_t element = 0; element < element_count; ++element) {
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for (std::size_t point = 0; point < gauss_points.size(); ++point) {
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const double r = 0.5 * (1.0 + gauss_points[point]);
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const double r_squared = r * r;
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const double r_cubed = r_squared * r;
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const double h1 = 1.0 - 3.0 * r_squared + 2.0 * r_cubed;
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const double h2 = element_length * (r - 2.0 * r_squared + r_cubed);
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const double h3 = 3.0 * r_squared - 2.0 * r_cubed;
|
|
const double h4 = element_length * (-r_squared + r_cubed);
|
|
const double interpolated =
|
|
h1 * nodal_displacement[2U * element] +
|
|
h2 * nodal_displacement[2U * element + 1U] +
|
|
h3 * nodal_displacement[2U * (element + 1U)] +
|
|
h4 * nodal_displacement[2U * (element + 1U) + 1U];
|
|
const double x = element_length * (static_cast<double>(element) + r);
|
|
const double analytical =
|
|
line_load * x * x *
|
|
(6.0 * length * length - 4.0 * length * x + x * x) /
|
|
(24.0 * flexural_rigidity);
|
|
const double weight = 0.5 * element_length * gauss_weights[point];
|
|
const double difference = interpolated - analytical;
|
|
squared_error += weight * difference * difference;
|
|
squared_reference += weight * analytical * analytical;
|
|
}
|
|
}
|
|
return std::sqrt(squared_error / squared_reference);
|
|
}
|
|
|
|
Matrix TransformationFromKnownRows(
|
|
const std::array<std::array<double, 3>, 3>& rotation) {
|
|
Matrix transformation{kElementDofCount, kElementDofCount};
|
|
for (std::size_t block = 0; block < 4U; ++block) {
|
|
for (std::size_t row = 0; row < 3U; ++row) {
|
|
for (std::size_t column = 0; column < 3U; ++column) {
|
|
transformation(block * 3U + row, block * 3U + column) =
|
|
rotation[row][column];
|
|
}
|
|
}
|
|
}
|
|
return transformation;
|
|
}
|
|
|
|
Vector TransposeMultiply(const Matrix& matrix, const Vector& vector) {
|
|
if (matrix.Rows() != vector.Size()) {
|
|
throw std::invalid_argument{"Transpose multiply dimension mismatch."};
|
|
}
|
|
Vector result{matrix.Columns()};
|
|
for (std::size_t column = 0; column < matrix.Columns(); ++column) {
|
|
for (std::size_t row = 0; row < matrix.Rows(); ++row) {
|
|
result[column] += matrix(row, column) * vector[row];
|
|
}
|
|
}
|
|
return result;
|
|
}
|
|
|
|
double Determinant(const std::array<std::array<double, 3>, 3>& matrix) {
|
|
return matrix[0][0] *
|
|
(matrix[1][1] * matrix[2][2] - matrix[1][2] * matrix[2][1]) -
|
|
matrix[0][1] *
|
|
(matrix[1][0] * matrix[2][2] - matrix[1][2] * matrix[2][0]) +
|
|
matrix[0][2] *
|
|
(matrix[1][0] * matrix[2][1] - matrix[1][1] * matrix[2][0]);
|
|
}
|
|
|
|
TEST(EulerBeam3D, HermiteAndBMatrixMatchReviewedSigns) {
|
|
const double length = 2.5;
|
|
const auto section = MakeSection();
|
|
const auto material = MakeMaterial();
|
|
const auto beam = AlignedBeam(length, section, material);
|
|
|
|
const double axial_strain = 0.012;
|
|
const double twist = -0.021;
|
|
const std::array<double, 4> v = {1.2, -0.4, 0.3, -0.07};
|
|
const std::array<double, 4> w = {-0.8, 0.6, -0.2, 0.05};
|
|
const auto value = [](const std::array<double, 4>& coefficients, double x) {
|
|
return coefficients[0] + coefficients[1] * x + coefficients[2] * x * x +
|
|
coefficients[3] * x * x * x;
|
|
};
|
|
const auto slope = [](const std::array<double, 4>& coefficients, double x) {
|
|
return coefficients[1] + 2.0 * coefficients[2] * x +
|
|
3.0 * coefficients[3] * x * x;
|
|
};
|
|
const auto curvature = [](const std::array<double, 4>& coefficients,
|
|
double x) {
|
|
return 2.0 * coefficients[2] + 6.0 * coefficients[3] * x;
|
|
};
|
|
|
|
Vector displacement{kElementDofCount};
|
|
displacement[0U] = 0.2;
|
|
displacement[1U] = value(v, 0.0);
|
|
displacement[2U] = value(w, 0.0);
|
|
displacement[3U] = -0.1;
|
|
displacement[4U] = -slope(w, 0.0);
|
|
displacement[5U] = slope(v, 0.0);
|
|
displacement[6U] = displacement[0U] + axial_strain * length;
|
|
displacement[7U] = value(v, length);
|
|
displacement[8U] = value(w, length);
|
|
displacement[9U] = displacement[3U] + twist * length;
|
|
displacement[10U] = -slope(w, length);
|
|
displacement[11U] = slope(v, length);
|
|
|
|
const BeamRecovery recovery = beam.Recover(displacement);
|
|
const double inverse_sqrt_three = 1.0 / std::sqrt(3.0);
|
|
const std::array<double, 2> gauss_xi = {-inverse_sqrt_three,
|
|
inverse_sqrt_three};
|
|
for (std::size_t point = 0; point < gauss_xi.size(); ++point) {
|
|
const double x = 0.5 * length * (1.0 + gauss_xi[point]);
|
|
EXPECT_NEAR(recovery.gauss_generalized_strains[point][0U], axial_strain,
|
|
1.0e-14);
|
|
EXPECT_NEAR(recovery.gauss_generalized_strains[point][1U], twist, 1.0e-14);
|
|
EXPECT_NEAR(recovery.gauss_generalized_strains[point][2U], -curvature(w, x),
|
|
1.0e-13);
|
|
EXPECT_NEAR(recovery.gauss_generalized_strains[point][3U], curvature(v, x),
|
|
1.0e-13);
|
|
}
|
|
|
|
EXPECT_NEAR(recovery.endpoint_section_resultants[0U][2U],
|
|
material.youngs_modulus * section.i11 * -curvature(w, 0.0),
|
|
1.0e-5);
|
|
EXPECT_NEAR(recovery.endpoint_section_resultants[1U][2U],
|
|
material.youngs_modulus * section.i11 * -curvature(w, length),
|
|
1.0e-5);
|
|
EXPECT_NEAR(recovery.endpoint_section_resultants[0U][3U],
|
|
material.youngs_modulus * section.i22 * curvature(v, 0.0),
|
|
1.0e-5);
|
|
EXPECT_NEAR(recovery.endpoint_section_resultants[1U][3U],
|
|
material.youngs_modulus * section.i22 * curvature(v, length),
|
|
1.0e-5);
|
|
}
|
|
|
|
TEST(EulerBeam3D, TwoPointGaussMatchesClosedStiffness) {
|
|
const double length = 3.7;
|
|
auto section = MakeSection();
|
|
section.area = 0.019;
|
|
section.i11 = 3.1e-5;
|
|
section.i22 = 7.4e-5;
|
|
section.torsional_constant = 2.2e-5;
|
|
const auto material = MakeMaterial(73.0e9, 0.27);
|
|
const auto beam = AlignedBeam(length, section, material);
|
|
|
|
const Matrix actual = beam.LocalStiffness();
|
|
const Matrix closed = ExpectedClosedStiffness(length, section, material);
|
|
EXPECT_LE(NormalizedMatrixError(actual, closed), kMatrixTolerance);
|
|
|
|
Matrix transpose{actual.Rows(), actual.Columns()};
|
|
for (std::size_t row = 0; row < actual.Rows(); ++row) {
|
|
for (std::size_t column = 0; column < actual.Columns(); ++column) {
|
|
transpose(row, column) = actual(column, row);
|
|
}
|
|
}
|
|
EXPECT_LE(NormalizedMatrixError(actual, transpose), kMatrixTolerance);
|
|
}
|
|
|
|
TEST(EulerBeam3D, HasSixRigidModesRankSixAndPositiveDeformationEnergy) {
|
|
const double length = 2.0;
|
|
auto section = MakeSection();
|
|
section.area = 1.4;
|
|
section.i11 = 0.8;
|
|
section.i22 = 1.1;
|
|
section.torsional_constant = 0.6;
|
|
const auto material = MakeMaterial(5.0, 0.25);
|
|
const Matrix stiffness =
|
|
AlignedBeam(length, section, material).LocalStiffness();
|
|
|
|
std::array<Vector, 6> rigid_modes = {
|
|
Vector{kElementDofCount}, Vector{kElementDofCount},
|
|
Vector{kElementDofCount}, Vector{kElementDofCount},
|
|
Vector{kElementDofCount}, Vector{kElementDofCount}};
|
|
rigid_modes[0U][0U] = rigid_modes[0U][6U] = 1.0;
|
|
rigid_modes[1U][1U] = rigid_modes[1U][7U] = 1.0;
|
|
rigid_modes[2U][2U] = rigid_modes[2U][8U] = 1.0;
|
|
rigid_modes[3U][3U] = rigid_modes[3U][9U] = 1.0;
|
|
rigid_modes[4U][4U] = rigid_modes[4U][10U] = 1.0;
|
|
rigid_modes[4U][8U] = -length;
|
|
rigid_modes[5U][5U] = rigid_modes[5U][11U] = 1.0;
|
|
rigid_modes[5U][7U] = length;
|
|
|
|
const double stiffness_scale =
|
|
(std::max)(1.0, MaximumAbsoluteEntry(stiffness));
|
|
for (const Vector& mode : rigid_modes) {
|
|
const double normalizedResidual =
|
|
VectorNorm(stiffness.Multiply(mode)) /
|
|
(stiffness_scale * (std::max)(1.0, VectorNorm(mode)));
|
|
EXPECT_LE(normalizedResidual, kRigidTolerance);
|
|
}
|
|
|
|
const std::array<double, kElementDofCount> q = {
|
|
1.0, 1.0, 1.0, length, length, length,
|
|
1.0, 1.0, 1.0, length, length, length};
|
|
Matrix scaled{kElementDofCount, kElementDofCount};
|
|
for (std::size_t row = 0; row < kElementDofCount; ++row) {
|
|
for (std::size_t column = 0; column < kElementDofCount; ++column) {
|
|
scaled(row, column) = stiffness(row, column) / (q[row] * q[column]);
|
|
}
|
|
}
|
|
const auto eigenvalues = SymmetricEigenvalues(scaled);
|
|
double maximum_singular_value = 0.0;
|
|
for (const double value : eigenvalues) {
|
|
maximum_singular_value =
|
|
(std::max)(maximum_singular_value, std::abs(value));
|
|
}
|
|
const auto positive_count = std::count_if(
|
|
eigenvalues.begin(), eigenvalues.end(),
|
|
[maximum_singular_value](double value) {
|
|
return std::abs(value) > kRigidTolerance * maximum_singular_value;
|
|
});
|
|
EXPECT_EQ(positive_count, 6);
|
|
for (const double value : eigenvalues) {
|
|
EXPECT_GE(value, -kRigidTolerance * maximum_singular_value);
|
|
}
|
|
|
|
for (std::size_t component = 0; component < 6U; ++component) {
|
|
Vector deformation{kElementDofCount};
|
|
deformation[6U + component] = 1.0;
|
|
EXPECT_GT(QuadraticEnergy(stiffness, deformation), 0.0);
|
|
}
|
|
}
|
|
|
|
TEST(EulerBeam3D, RotatedTransformPreservesWorkAndEnergy) {
|
|
const double inverse_sqrt_two = 1.0 / std::sqrt(2.0);
|
|
const std::array<std::array<double, 3>, 3> rotation = {
|
|
{{{2.0 / 3.0, 2.0 / 3.0, 1.0 / 3.0}},
|
|
{{-inverse_sqrt_two, inverse_sqrt_two, 0.0}},
|
|
{{-inverse_sqrt_two / 3.0, -inverse_sqrt_two / 3.0,
|
|
4.0 * inverse_sqrt_two / 3.0}}}};
|
|
const Matrix transformation = TransformationFromKnownRows(rotation);
|
|
|
|
for (std::size_t row = 0; row < 3U; ++row) {
|
|
for (std::size_t column = 0; column < 3U; ++column) {
|
|
double dot = 0.0;
|
|
for (std::size_t component = 0; component < 3U; ++component) {
|
|
dot += rotation[row][component] * rotation[column][component];
|
|
}
|
|
EXPECT_NEAR(dot, row == column ? 1.0 : 0.0, 1.0e-14);
|
|
}
|
|
}
|
|
EXPECT_NEAR(Determinant(rotation), 1.0, 1.0e-14);
|
|
|
|
const auto section = MakeSection({-2.0, 2.0, 0.0});
|
|
const auto material = MakeMaterial();
|
|
const auto beam =
|
|
RequireBeam(MakeNode({1.0, -2.0, 0.5}, 1U), MakeNode({3.0, 0.0, 1.5}, 2U),
|
|
section, material);
|
|
const Matrix local = beam.LocalStiffness();
|
|
const Matrix global = beam.GlobalStiffness();
|
|
|
|
Matrix expected_global{kElementDofCount, kElementDofCount};
|
|
const Matrix local_times_transform = local.Multiply(transformation);
|
|
for (std::size_t row = 0; row < kElementDofCount; ++row) {
|
|
for (std::size_t column = 0; column < kElementDofCount; ++column) {
|
|
for (std::size_t inner = 0; inner < kElementDofCount; ++inner) {
|
|
expected_global(row, column) +=
|
|
transformation(inner, row) * local_times_transform(inner, column);
|
|
}
|
|
}
|
|
}
|
|
EXPECT_LE(NormalizedMatrixError(global, expected_global), kMatrixTolerance);
|
|
|
|
Vector local_displacement{kElementDofCount};
|
|
for (std::size_t index = 0; index < local_displacement.Size(); ++index) {
|
|
local_displacement[index] = 0.01 * static_cast<double>(index + 1U) - 0.04;
|
|
}
|
|
const Vector global_displacement =
|
|
TransposeMultiply(transformation, local_displacement);
|
|
const Vector local_force = local.Multiply(local_displacement);
|
|
const Vector global_force = global.Multiply(global_displacement);
|
|
const Vector expected_global_force =
|
|
TransposeMultiply(transformation, local_force);
|
|
for (std::size_t index = 0; index < kElementDofCount; ++index) {
|
|
ExpectScaledNear(global_force[index], expected_global_force[index],
|
|
kMatrixTolerance);
|
|
}
|
|
ExpectScaledNear(QuadraticEnergy(global, global_displacement),
|
|
QuadraticEnergy(local, local_displacement),
|
|
kMatrixTolerance);
|
|
|
|
Vector global_variation{kElementDofCount};
|
|
for (std::size_t index = 0; index < global_variation.Size(); ++index) {
|
|
global_variation[index] = 0.03 - 0.002 * static_cast<double>(index);
|
|
}
|
|
const Vector local_variation = transformation.Multiply(global_variation);
|
|
ExpectScaledNear(global_variation.Dot(global_force),
|
|
local_variation.Dot(local_force), kMatrixTolerance);
|
|
|
|
const BeamRecovery recovery = beam.Recover(global_displacement);
|
|
EXPECT_NEAR(recovery.gauss_generalized_strains[0U][0U],
|
|
(local_displacement[6U] - local_displacement[0U]) / 3.0, 1.0e-14);
|
|
}
|
|
|
|
TEST(EulerBeam3D, PreservesExactRotatedResultsAcrossVector3Migration) {
|
|
const auto section = MakeSection({-2.0, 2.0, 0.0});
|
|
const auto material = MakeMaterial();
|
|
const Node first_node = MakeNode({1.0, -2.0, 0.5}, 1U);
|
|
const Node second_node = MakeNode({3.0, 0.0, 1.5}, 2U);
|
|
const Vector3 delta =
|
|
Vector3(second_node.coordinates) - Vector3(first_node.coordinates);
|
|
EXPECT_DOUBLE_EQ(delta.Norm(), 3.0);
|
|
|
|
const auto beam = RequireBeam(first_node, second_node, section, material);
|
|
const Matrix global = beam.GlobalStiffness();
|
|
Vector displacement{kElementDofCount};
|
|
for (std::size_t index = 0U; index < displacement.Size(); ++index) {
|
|
displacement[index] = 0.01 * static_cast<double>(index + 1U) - 0.04;
|
|
}
|
|
const BeamRecovery recovery = beam.Recover(displacement);
|
|
|
|
EXPECT_DOUBLE_EQ(global(0U, 0U), 0x1.65f135da12f68p+28);
|
|
EXPECT_DOUBLE_EQ(global(0U, 1U), 0x1.6261c084bda12p+28);
|
|
EXPECT_DOUBLE_EQ(global(0U, 2U), 0x1.630ca684bda13p+27);
|
|
EXPECT_DOUBLE_EQ(global(4U, 4U), 0x1.068e359b59b58p+22);
|
|
EXPECT_DOUBLE_EQ(global(5U, 11U), 0x1.2d14a7ee7ee7bp+22);
|
|
EXPECT_DOUBLE_EQ(recovery.gauss_generalized_strains[0U][0U],
|
|
0x1.1111111111110p-5);
|
|
EXPECT_DOUBLE_EQ(recovery.gauss_generalized_strains[0U][1U],
|
|
0x1.1111111111111p-5);
|
|
EXPECT_DOUBLE_EQ(recovery.gauss_generalized_strains[0U][2U],
|
|
-0x1.382425a7d2473p-6);
|
|
EXPECT_DOUBLE_EQ(recovery.gauss_generalized_strains[0U][3U],
|
|
-0x1.a9389137b051dp-6);
|
|
EXPECT_DOUBLE_EQ(recovery.endpoint_section_resultants[1U][2U],
|
|
0x1.525c94a87359fp+17);
|
|
}
|
|
|
|
TEST(EulerBeam3D, ConstantLineLoadMatchesAllSignedComponents) {
|
|
const double length = 4.0;
|
|
const ConstantLocalLineLoad load{2.5, -3.0, 5.5, -7.0};
|
|
const Vector equivalent = AlignedBeam(length).LocalEquivalentLoad(load);
|
|
const std::array<double, kElementDofCount> expected = {
|
|
5.0, -6.0, 11.0, -14.0, -22.0 / 3.0, -4.0,
|
|
5.0, -6.0, 11.0, -14.0, 22.0 / 3.0, 4.0};
|
|
ASSERT_EQ(equivalent.Size(), expected.size());
|
|
for (std::size_t index = 0; index < expected.size(); ++index) {
|
|
ExpectScaledNear(equivalent[index], expected[index], kMatrixTolerance);
|
|
}
|
|
|
|
auto convergence_section = MakeSection();
|
|
convergence_section.area = 1.0;
|
|
convergence_section.i11 = 1.0;
|
|
convergence_section.i22 = 1.0;
|
|
convergence_section.torsional_constant = 1.0;
|
|
const auto convergence_material = MakeMaterial(5.0, 0.25);
|
|
const double transverse_load = -3.0;
|
|
const std::array<std::size_t, 3> element_counts = {1U, 2U, 4U};
|
|
std::array<double, 3> relative_errors{};
|
|
for (std::size_t mesh = 0; mesh < element_counts.size(); ++mesh) {
|
|
const double element_length =
|
|
length / static_cast<double>(element_counts[mesh]);
|
|
const Vector element_load =
|
|
AlignedBeam(element_length, convergence_section, convergence_material)
|
|
.LocalEquivalentLoad({0.0, transverse_load, 0.0, 0.0});
|
|
ExpectRelativeNear(element_load[1U], transverse_load * element_length / 2.0,
|
|
kMatrixTolerance);
|
|
ExpectRelativeNear(element_load[5U],
|
|
transverse_load * element_length * element_length / 12.0,
|
|
kMatrixTolerance);
|
|
ExpectRelativeNear(element_load[7U], transverse_load * element_length / 2.0,
|
|
kMatrixTolerance);
|
|
ExpectRelativeNear(
|
|
element_load[11U],
|
|
-transverse_load * element_length * element_length / 12.0,
|
|
kMatrixTolerance);
|
|
|
|
const Vector nodal_displacement = SolveUniformTransverseCantilever(
|
|
element_counts[mesh], length, transverse_load, convergence_section,
|
|
convergence_material);
|
|
relative_errors[mesh] = UniformLoadInteriorDisplacementError(
|
|
nodal_displacement, element_counts[mesh], length, transverse_load,
|
|
convergence_material.youngs_modulus * convergence_section.i22);
|
|
}
|
|
EXPECT_GT(relative_errors[0U], relative_errors[1U]);
|
|
EXPECT_GT(relative_errors[1U], relative_errors[2U]);
|
|
EXPECT_NEAR(
|
|
std::log(relative_errors[0U] / relative_errors[1U]) / std::log(2.0), 4.0,
|
|
1.0e-8);
|
|
EXPECT_NEAR(
|
|
std::log(relative_errors[1U] / relative_errors[2U]) / std::log(2.0), 4.0,
|
|
1.0e-8);
|
|
}
|
|
|
|
TEST(EulerBeam3D, AnalyticalAxialTorsionAndTwoPlaneBendingRecover) {
|
|
const double length = 3.0;
|
|
const auto section = MakeSection();
|
|
const auto material = MakeMaterial();
|
|
const auto beam = AlignedBeam(length, section, material);
|
|
const Matrix stiffness = beam.LocalStiffness();
|
|
const double shear_modulus =
|
|
material.youngs_modulus / (2.0 * (1.0 + material.poisson_ratio));
|
|
|
|
const double axial_force = 1250.0;
|
|
const Vector axial =
|
|
SolveFixedFirstNode(stiffness, {axial_force, 0.0, 0.0, 0.0, 0.0, 0.0});
|
|
ExpectRelativeNear(
|
|
axial[6U],
|
|
axial_force * length / (material.youngs_modulus * section.area),
|
|
kAnalyticalTolerance);
|
|
const BeamRecovery axial_recovery = beam.Recover(axial);
|
|
ExpectRelativeNear(axial_recovery.equilibrium_end_actions[0U][0U],
|
|
-axial_force, kAnalyticalTolerance);
|
|
ExpectRelativeNear(axial_recovery.equilibrium_end_actions[1U][0U],
|
|
axial_force, kAnalyticalTolerance);
|
|
ExpectRelativeNear(axial_recovery.endpoint_section_resultants[0U][0U],
|
|
axial_force, kAnalyticalTolerance);
|
|
ExpectRelativeNear(axial_recovery.endpoint_section_resultants[1U][0U],
|
|
axial_force, kAnalyticalTolerance);
|
|
|
|
const double torque = -870.0;
|
|
const Vector torsion =
|
|
SolveFixedFirstNode(stiffness, {0.0, 0.0, 0.0, torque, 0.0, 0.0});
|
|
ExpectRelativeNear(
|
|
torsion[9U],
|
|
torque * length / (shear_modulus * section.torsional_constant),
|
|
kAnalyticalTolerance);
|
|
const BeamRecovery torsion_recovery = beam.Recover(torsion);
|
|
ExpectRelativeNear(torsion_recovery.equilibrium_end_actions[0U][3U], -torque,
|
|
kAnalyticalTolerance);
|
|
ExpectRelativeNear(torsion_recovery.equilibrium_end_actions[1U][3U], torque,
|
|
kAnalyticalTolerance);
|
|
ExpectRelativeNear(torsion_recovery.endpoint_section_resultants[0U][1U],
|
|
torque, kAnalyticalTolerance);
|
|
|
|
const double local_yforce = 640.0;
|
|
const Vector local_y =
|
|
SolveFixedFirstNode(stiffness, {0.0, local_yforce, 0.0, 0.0, 0.0, 0.0});
|
|
ExpectRelativeNear(local_y[7U],
|
|
local_yforce * length * length * length /
|
|
(3.0 * material.youngs_modulus * section.i22),
|
|
kAnalyticalTolerance);
|
|
ExpectRelativeNear(local_y[11U],
|
|
local_yforce * length * length /
|
|
(2.0 * material.youngs_modulus * section.i22),
|
|
kAnalyticalTolerance);
|
|
const BeamRecovery local_yrecovery = beam.Recover(local_y);
|
|
ExpectRelativeNear(local_yrecovery.equilibrium_end_actions[0U][1U],
|
|
-local_yforce, kAnalyticalTolerance);
|
|
ExpectRelativeNear(local_yrecovery.equilibrium_end_actions[1U][1U],
|
|
local_yforce, kAnalyticalTolerance);
|
|
ExpectRelativeNear(local_yrecovery.equilibrium_end_actions[0U][5U],
|
|
-local_yforce * length, kAnalyticalTolerance);
|
|
ExpectRelativeNear(local_yrecovery.endpoint_section_resultants[0U][3U],
|
|
local_yforce * length, kAnalyticalTolerance);
|
|
EXPECT_NEAR(local_yrecovery.endpoint_section_resultants[1U][3U], 0.0, 1.0e-8);
|
|
|
|
const double local_zforce = -510.0;
|
|
const Vector local_z =
|
|
SolveFixedFirstNode(stiffness, {0.0, 0.0, local_zforce, 0.0, 0.0, 0.0});
|
|
ExpectRelativeNear(local_z[8U],
|
|
local_zforce * length * length * length /
|
|
(3.0 * material.youngs_modulus * section.i11),
|
|
kAnalyticalTolerance);
|
|
ExpectRelativeNear(local_z[10U],
|
|
-local_zforce * length * length /
|
|
(2.0 * material.youngs_modulus * section.i11),
|
|
kAnalyticalTolerance);
|
|
const BeamRecovery local_zrecovery = beam.Recover(local_z);
|
|
ExpectRelativeNear(local_zrecovery.equilibrium_end_actions[0U][2U],
|
|
-local_zforce, kAnalyticalTolerance);
|
|
ExpectRelativeNear(local_zrecovery.equilibrium_end_actions[1U][2U],
|
|
local_zforce, kAnalyticalTolerance);
|
|
ExpectRelativeNear(local_zrecovery.equilibrium_end_actions[0U][4U],
|
|
local_zforce * length, kAnalyticalTolerance);
|
|
ExpectRelativeNear(local_zrecovery.endpoint_section_resultants[0U][2U],
|
|
-local_zforce * length, kAnalyticalTolerance);
|
|
EXPECT_NEAR(local_zrecovery.endpoint_section_resultants[1U][2U], 0.0, 1.0e-8);
|
|
}
|
|
|
|
TEST(EulerBeam3D, RejectsInvalidGeometryAndProperties) {
|
|
const Node origin = MakeNode({0.0, 0.0, 0.0}, 1U);
|
|
const Node unit_x = MakeNode({1.0, 0.0, 0.0}, 2U);
|
|
const auto valid_section = MakeSection();
|
|
const auto valid_material = MakeMaterial();
|
|
|
|
const auto expect_failure = [](const Result<EulerBeam3D>& result,
|
|
const std::string& code) {
|
|
if (result.HasValue()) {
|
|
const Matrix stiffness = result.Value().LocalStiffness();
|
|
ADD_FAILURE() << "Invalid fixture was accepted; local stiffness finite="
|
|
<< MatrixIsFinite(stiffness) << ", maximum absolute entry="
|
|
<< MaximumAbsoluteEntry(stiffness);
|
|
return;
|
|
}
|
|
EXPECT_EQ(result.GetStatus().Category(), FailureCategory::kModel);
|
|
ASSERT_EQ(result.GetStatus().Diagnostics().size(), 1U);
|
|
EXPECT_EQ(result.GetStatus().Diagnostics()[0U].code, code);
|
|
};
|
|
|
|
expect_failure(
|
|
EulerBeam3D::Create(origin, origin, valid_section, valid_material),
|
|
"invalid-beam-length");
|
|
expect_failure(EulerBeam3D::Create(origin, MakeNode({1.0e-12, 0.0, 0.0}, 2U),
|
|
valid_section, valid_material),
|
|
"invalid-beam-length");
|
|
|
|
const double coordinate = 1048576.0;
|
|
const Node scaled_first = MakeNode({coordinate, 0.0, 0.0}, 1U);
|
|
const Node below_threshold =
|
|
MakeNode({coordinate + coordinate * 0.5e-12, 0.0, 0.0}, 2U);
|
|
const Node above_threshold =
|
|
MakeNode({coordinate + coordinate * 2.0e-12, 0.0, 0.0}, 2U);
|
|
expect_failure(EulerBeam3D::Create(scaled_first, below_threshold,
|
|
valid_section, valid_material),
|
|
"invalid-beam-length");
|
|
EXPECT_TRUE(EulerBeam3D::Create(scaled_first, above_threshold, valid_section,
|
|
valid_material)
|
|
.HasValue());
|
|
|
|
auto parallel_guide = valid_section;
|
|
parallel_guide.first_axis = {1.0, 0.0, 0.0};
|
|
expect_failure(
|
|
EulerBeam3D::Create(origin, unit_x, parallel_guide, valid_material),
|
|
"invalid-beam-guide-vector");
|
|
auto guide_at_threshold = valid_section;
|
|
guide_at_threshold.first_axis = {1.0, 1.0e-12, 0.0};
|
|
expect_failure(
|
|
EulerBeam3D::Create(origin, unit_x, guide_at_threshold, valid_material),
|
|
"invalid-beam-guide-vector");
|
|
auto guide_above_threshold = valid_section;
|
|
guide_above_threshold.first_axis = {1.0, 2.0e-12, 0.0};
|
|
EXPECT_TRUE(
|
|
EulerBeam3D::Create(origin, unit_x, guide_above_threshold, valid_material)
|
|
.HasValue());
|
|
|
|
auto invalid_material = valid_material;
|
|
invalid_material.youngs_modulus = 0.0;
|
|
expect_failure(
|
|
EulerBeam3D::Create(origin, unit_x, valid_section, invalid_material),
|
|
"invalid-beam-property");
|
|
invalid_material = valid_material;
|
|
invalid_material.poisson_ratio = -2.0;
|
|
expect_failure(
|
|
EulerBeam3D::Create(origin, unit_x, valid_section, invalid_material),
|
|
"invalid-beam-property");
|
|
|
|
for (std::size_t property = 0; property < 4U; ++property) {
|
|
auto invalid_section = valid_section;
|
|
double* properties[] = {&invalid_section.area, &invalid_section.i11,
|
|
&invalid_section.i22,
|
|
&invalid_section.torsional_constant};
|
|
*properties[property] = 0.0;
|
|
expect_failure(
|
|
EulerBeam3D::Create(origin, unit_x, invalid_section, valid_material),
|
|
"invalid-beam-property");
|
|
}
|
|
|
|
auto overflow_material =
|
|
MakeMaterial(std::numeric_limits<double>::max() / 4.0, 0.25);
|
|
auto overflow_section = valid_section;
|
|
overflow_section.area = 8.0;
|
|
expect_failure(
|
|
EulerBeam3D::Create(origin, unit_x, overflow_section, overflow_material),
|
|
"invalid-beam-property");
|
|
|
|
auto underflow_section = valid_section;
|
|
underflow_section.area = std::numeric_limits<double>::denorm_min();
|
|
underflow_section.i11 = std::numeric_limits<double>::denorm_min();
|
|
underflow_section.i22 = std::numeric_limits<double>::denorm_min();
|
|
underflow_section.torsional_constant =
|
|
std::numeric_limits<double>::denorm_min();
|
|
expect_failure(EulerBeam3D::Create(origin, unit_x, underflow_section,
|
|
MakeMaterial(0.5, 0.25)),
|
|
"invalid-beam-property");
|
|
|
|
auto length_scaled_section = valid_section;
|
|
length_scaled_section.area = 1.0;
|
|
length_scaled_section.i11 = 1.0;
|
|
length_scaled_section.i22 = 1.0;
|
|
length_scaled_section.torsional_constant = 1.0;
|
|
expect_failure(
|
|
EulerBeam3D::Create(origin, MakeNode({1.0e103, 0.0, 0.0}, 2U),
|
|
length_scaled_section, MakeMaterial(1.0, 0.25)),
|
|
"invalid-beam-property");
|
|
|
|
auto coupled_section = valid_section;
|
|
coupled_section.i12 = 1.0e-9;
|
|
expect_failure(
|
|
EulerBeam3D::Create(origin, unit_x, coupled_section, valid_material),
|
|
"unsupported-coupled-section");
|
|
}
|
|
|
|
TEST(EulerBeam3D, RecoversSectionPointAndDefaultCentroidS11) {
|
|
const double length = 2.0;
|
|
const double epsilon = 0.01;
|
|
const double kappa_y = 0.02;
|
|
const double kappa_z = -0.03;
|
|
const auto material = MakeMaterial();
|
|
auto section = MakeSection({0.0, 1.0, 0.0}, {{0.25, -0.5}, {-0.4, 0.3}});
|
|
const auto beam = AlignedBeam(length, section, material);
|
|
|
|
Vector displacement{kElementDofCount};
|
|
displacement[6U] = epsilon * length;
|
|
displacement[7U] = 0.5 * kappa_z * length * length;
|
|
displacement[8U] = -0.5 * kappa_y * length * length;
|
|
displacement[10U] = kappa_y * length;
|
|
displacement[11U] = kappa_z * length;
|
|
|
|
const BeamRecovery recovery = beam.Recover(displacement);
|
|
ASSERT_EQ(recovery.stress_points.size(), 4U);
|
|
for (std::size_t gauss_point = 0; gauss_point < 2U; ++gauss_point) {
|
|
for (std::size_t point = 0; point < section.section_points.size();
|
|
++point) {
|
|
const BeamStressPoint& stress =
|
|
recovery.stress_points[gauss_point * section.section_points.size() +
|
|
point];
|
|
const double x1 = section.section_points[point][0U];
|
|
const double x2 = section.section_points[point][1U];
|
|
EXPECT_EQ(stress.gauss_point, static_cast<int>(gauss_point + 1U));
|
|
EXPECT_EQ(stress.section_point, point + 1U);
|
|
EXPECT_DOUBLE_EQ(stress.x1, x1);
|
|
EXPECT_DOUBLE_EQ(stress.x2, x2);
|
|
ExpectScaledNear(
|
|
stress.s11,
|
|
material.youngs_modulus * (epsilon + x2 * kappa_y - x1 * kappa_z),
|
|
kMatrixTolerance);
|
|
}
|
|
}
|
|
|
|
const auto default_beam = AlignedBeam(length, MakeSection(), material);
|
|
const BeamRecovery default_recovery = default_beam.Recover(displacement);
|
|
ASSERT_EQ(default_recovery.stress_points.size(), 2U);
|
|
for (std::size_t gauss_point = 0; gauss_point < 2U; ++gauss_point) {
|
|
const BeamStressPoint& stress = default_recovery.stress_points[gauss_point];
|
|
EXPECT_EQ(stress.gauss_point, static_cast<int>(gauss_point + 1U));
|
|
EXPECT_EQ(stress.section_point, 0U);
|
|
EXPECT_DOUBLE_EQ(stress.x1, 0.0);
|
|
EXPECT_DOUBLE_EQ(stress.x2, 0.0);
|
|
EXPECT_EQ(stress.source, "fesa-default");
|
|
ExpectScaledNear(stress.s11, material.youngs_modulus * epsilon,
|
|
kMatrixTolerance);
|
|
}
|
|
}
|
|
|
|
TEST(EulerBeam3D, ReproducesConstantStrainTwistAndCurvaturePatches) {
|
|
const double length = 2.8;
|
|
const double epsilon = -0.014;
|
|
const double twist = 0.023;
|
|
const double kappa_y = -0.031;
|
|
const double kappa_z = 0.047;
|
|
const auto section = MakeSection();
|
|
const auto material = MakeMaterial();
|
|
const double shear_modulus =
|
|
material.youngs_modulus / (2.0 * (1.0 + material.poisson_ratio));
|
|
const auto beam = AlignedBeam(length, section, material);
|
|
|
|
Vector displacement{kElementDofCount};
|
|
displacement[6U] = epsilon * length;
|
|
displacement[7U] = 0.5 * kappa_z * length * length;
|
|
displacement[8U] = -0.5 * kappa_y * length * length;
|
|
displacement[9U] = twist * length;
|
|
displacement[10U] = kappa_y * length;
|
|
displacement[11U] = kappa_z * length;
|
|
|
|
const std::array<double, 4> expected_strain = {epsilon, twist, kappa_y,
|
|
kappa_z};
|
|
const std::array<double, 4> expected_resultant = {
|
|
material.youngs_modulus * section.area * epsilon,
|
|
shear_modulus * section.torsional_constant * twist,
|
|
material.youngs_modulus * section.i11 * kappa_y,
|
|
material.youngs_modulus * section.i22 * kappa_z};
|
|
const BeamRecovery recovery = beam.Recover(displacement);
|
|
for (std::size_t point = 0; point < 2U; ++point) {
|
|
for (std::size_t component = 0; component < 4U; ++component) {
|
|
ExpectScaledNear(recovery.gauss_generalized_strains[point][component],
|
|
expected_strain[component], kMatrixTolerance);
|
|
ExpectScaledNear(recovery.gauss_generalized_resultants[point][component],
|
|
expected_resultant[component], kMatrixTolerance);
|
|
ExpectScaledNear(recovery.endpoint_section_resultants[point][component],
|
|
expected_resultant[component], kMatrixTolerance);
|
|
}
|
|
}
|
|
|
|
const std::array<std::size_t, 4> end_action_components = {0U, 3U, 4U, 5U};
|
|
for (std::size_t component = 0; component < expected_resultant.size();
|
|
++component) {
|
|
ExpectScaledNear(
|
|
recovery.equilibrium_end_actions[0U][end_action_components[component]],
|
|
-expected_resultant[component], kMatrixTolerance);
|
|
ExpectScaledNear(
|
|
recovery.equilibrium_end_actions[1U][end_action_components[component]],
|
|
expected_resultant[component], kMatrixTolerance);
|
|
}
|
|
EXPECT_NEAR(recovery.equilibrium_end_actions[0U][1U], 0.0, 1.0e-8);
|
|
EXPECT_NEAR(recovery.equilibrium_end_actions[0U][2U], 0.0, 1.0e-8);
|
|
EXPECT_NEAR(recovery.equilibrium_end_actions[1U][1U], 0.0, 1.0e-8);
|
|
EXPECT_NEAR(recovery.equilibrium_end_actions[1U][2U], 0.0, 1.0e-8);
|
|
}
|
|
|
|
TEST(EulerBeam3D, OnePointNegativeControlHasRankFour) {
|
|
const double length = 3.7;
|
|
auto section = MakeSection();
|
|
section.area = 1.0;
|
|
section.i11 = 0.7;
|
|
section.i22 = 1.2;
|
|
section.torsional_constant = 0.9;
|
|
const auto material = MakeMaterial(4.0, 0.25);
|
|
|
|
const Matrix one_point = TestOnlyOnePointStiffness(length, section, material);
|
|
const Matrix production =
|
|
AlignedBeam(length, section, material).LocalStiffness();
|
|
EXPECT_EQ(SymmetricRank(one_point, kRigidTolerance), 4U);
|
|
EXPECT_EQ(SymmetricRank(production, kRigidTolerance), 6U);
|
|
}
|
|
|
|
} // namespace
|
|
} // namespace fesa
|