feat(elements): implement certified full-integration Hex8 kernel
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KOKO\Mimi
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#include "fesa/elements/hex8_element.h"
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#include <cmath>
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#include <set>
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#include <utility>
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#include "hex8_geometry.h"
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namespace fesa {
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namespace {
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using hex8_internal::Tensor;
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using hex8_internal::Triple;
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/// @brief Checks every tensor component before recovery or inverse use.
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bool Finite(const Tensor& value) {
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for (const auto& row : value)
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for (double entry : row)
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if (!std::isfinite(entry)) return false;
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return true;
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}
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/// @brief Packs actual tensor shear, never engineering shear, in schema order.
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std::array<double, 6> Pack(const Tensor& value) {
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return {value[0][0], value[1][1], value[2][2],
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value[0][1], value[0][2], value[1][2]};
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}
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/// @brief Computes a Euclidean contraction in deterministic component order.
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double Dot(const Triple& a, const Triple& b) {
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return a[0] * b[0] + a[1] * b[1] + a[2] * b[2];
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}
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/// @brief Gives the finite-control failure without accepting a geometry guess.
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const char* CertificateMessage(hex8_internal::Certificate certificate) {
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switch (certificate) {
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case hex8_internal::Certificate::kNonpositive:
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return "Whole-cell Jacobian has a nonpositive witness.";
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case hex8_internal::Certificate::kUncertain:
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return "Whole-cell Jacobian positivity is unresolved at depth 10 or cell "
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"budget 65536.";
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case hex8_internal::Certificate::kNonfinite:
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return "Whole-cell Jacobian interval arithmetic is nonfinite.";
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case hex8_internal::Certificate::kPositive:
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break;
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}
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return "Invalid Jacobian certificate.";
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}
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} // namespace
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Hex8Element::Hex8Element(EntityIndex index, ElementDofLayout layout,
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Coordinates coordinates, double lambda, double mu,
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std::optional<double> density, SourceLocation location)
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: element_index_{index},
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layout_{std::move(layout)},
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coordinates_{coordinates},
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lambda_{lambda},
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mu_{mu},
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density_{density},
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location_{std::move(location)} {}
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Status Hex8Element::Failure(const char* message) const {
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return Status::Failure(FailureCategory::kModel,
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{{Severity::kError, "invalid-hex8", location_, "C3D8",
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layout_.source_id.instance_name + ":" +
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layout_.source_id.source_label_text,
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message}});
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}
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Result<Hex8Element> Hex8Element::Create(EntityIndex index,
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ElementDofLayout layout,
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Coordinates coordinates,
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const LinearElasticMaterial& material,
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std::optional<double> density,
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SourceLocation location) {
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const double e = material.YoungsModulus();
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const double nu = material.PoissonsRatio();
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Hex8Element candidate{index, std::move(layout), coordinates, 0.0,
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0.0, density, std::move(location)};
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const std::vector<DofComponent> components{
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DofComponent::kUx, DofComponent::kUy, DofComponent::kUz};
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const std::set<EntityIndex> nodes(candidate.layout_.node_indices.begin(),
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candidate.layout_.node_indices.end());
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if (candidate.layout_.node_indices.size() != 8U || nodes.size() != 8U ||
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candidate.layout_.components_per_node != components)
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return Result<Hex8Element>::Failure(candidate.Failure(
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"Hex8 requires eight distinct nodes with exactly UX, UY, UZ."));
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if (!std::isfinite(e) || e <= 0.0 || !std::isfinite(nu) || nu <= -1.0 ||
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nu >= 0.5)
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return Result<Hex8Element>::Failure(
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candidate.Failure("Hex8 requires finite E>0 and -1<nu<0.5."));
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candidate.mu_ = e / (2.0 * (1.0 + nu));
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candidate.lambda_ = e * nu / ((1.0 + nu) * (1.0 - 2.0 * nu));
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if (!std::isfinite(candidate.mu_) || candidate.mu_ <= 0.0 ||
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!std::isfinite(candidate.lambda_) ||
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!std::isfinite(candidate.lambda_ + 2.0 * candidate.mu_))
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return Result<Hex8Element>::Failure(
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candidate.Failure("Hex8 Lame moduli must remain finite with "
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"representable positive shear modulus."));
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if (density && (!std::isfinite(*density) || *density <= 0.0))
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return Result<Hex8Element>::Failure(candidate.Failure(
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"Hex8 reference density must be finite and positive."));
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const auto certificate = hex8_internal::Certify(coordinates);
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if (certificate != hex8_internal::Certificate::kPositive)
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return Result<Hex8Element>::Failure(
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candidate.Failure(CertificateMessage(certificate)));
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const double gauss = 1.0 / std::sqrt(3.0);
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for (std::size_t g = 0; g < 8; ++g) {
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auto& point = candidate.points_[g];
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point.natural = {g % 2U == 0U ? -gauss : gauss,
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(g / 2U) % 2U == 0U ? -gauss : gauss,
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g < 4U ? -gauss : gauss};
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std::array<Triple, 8> derivatives{};
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hex8_internal::Shape(point.natural, point.shape, derivatives);
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const auto jacobian = hex8_internal::Jacobian(coordinates, derivatives);
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point.weight = hex8_internal::Determinant(jacobian);
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if (!Finite(jacobian) || !std::isfinite(point.weight) ||
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point.weight <= 0.0)
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return Result<Hex8Element>::Failure(candidate.Failure(
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"Reference quadrature Jacobian must be finite and positive."));
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const auto inverse = hex8_internal::Inverse(jacobian, point.weight);
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if (!Finite(inverse))
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return Result<Hex8Element>::Failure(
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candidate.Failure("Reference Jacobian inverse is nonfinite."));
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point.position = coordinates[0];
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for (std::size_t a = 0; a < 8; ++a)
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for (std::size_t i = 0; i < 3; ++i) {
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point.position[i] +=
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point.shape[a] * (coordinates[a][i] - coordinates[0][i]);
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for (std::size_t d = 0; d < 3; ++d)
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point.gradients[a][i] += inverse[d][i] * derivatives[a][d];
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if (!std::isfinite(point.gradients[a][i]) ||
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!std::isfinite(point.position[i]))
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return Result<Hex8Element>::Failure(candidate.Failure(
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"Reference gradient or point coordinate is nonfinite."));
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}
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for (std::size_t a = 0; a < 8; ++a)
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for (std::size_t b = 0; b < 8; ++b) {
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const auto& ga = point.gradients[a];
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const auto& gb = point.gradients[b];
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const double dot = Dot(ga, gb);
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for (std::size_t i = 0; i < 3; ++i)
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for (std::size_t k = 0; k < 3; ++k) {
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candidate.stiffness_(3U * a + i, 3U * b + k) +=
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point.weight * (candidate.lambda_ * ga[i] * gb[k] +
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candidate.mu_ * ga[k] * gb[i] +
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(i == k ? candidate.mu_ * dot : 0.0));
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}
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}
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}
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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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if (!std::isfinite(candidate.stiffness_(i, j)))
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return Result<Hex8Element>::Failure(
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candidate.Failure("Hex8 reference stiffness is nonfinite."));
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for (std::size_t i = 0; i < 24; ++i)
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if (candidate.stiffness_(i, i) <= 0.0)
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return Result<Hex8Element>::Failure(
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candidate.Failure("Hex8 reference stiffness has an unrepresentable "
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"positive diagonal."));
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return Result<Hex8Element>::Success(std::move(candidate));
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}
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const ElementDofLayout& Hex8Element::DofLayout() const noexcept {
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return layout_;
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}
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Result<ElementStiffnessContribution> Hex8Element::ComputeStiffness() const {
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return Result<ElementStiffnessContribution>::Success({layout_, stiffness_});
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}
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Result<ElementStiffnessContribution> Hex8Element::ComputeMass() const {
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if (!density_)
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return Result<ElementStiffnessContribution>::Failure(
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Failure("Hex8 mass requires reference density."));
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Matrix mass{24U, 24U};
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for (const auto& point : points_)
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for (std::size_t a = 0; a < 8; ++a)
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for (std::size_t b = 0; b < 8; ++b) {
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const double value =
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*density_ * point.shape[a] * point.shape[b] * point.weight;
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if (!std::isfinite(value))
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return Result<ElementStiffnessContribution>::Failure(
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Failure("Hex8 reference mass is nonfinite."));
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for (std::size_t i = 0; i < 3; ++i)
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mass(3U * a + i, 3U * b + i) += value;
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}
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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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if (!std::isfinite(mass(i, j)))
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return Result<ElementStiffnessContribution>::Failure(
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Failure("Hex8 accumulated reference mass is nonfinite."));
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for (std::size_t i = 0; i < 24; ++i)
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if (mass(i, i) <= 0.0)
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return Result<ElementStiffnessContribution>::Failure(Failure(
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"Hex8 reference mass has an unrepresentable positive diagonal."));
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return Result<ElementStiffnessContribution>::Success(
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{layout_, std::move(mass)});
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}
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Result<ElementResultBundle> Hex8Element::Recover(const Vector& u) const {
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auto response = EvaluateLinear(u);
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if (!response.HasValue())
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return Result<ElementResultBundle>::Failure(response.GetStatus());
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SolidElementResultRows rows;
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rows.rows.assign(response.Value().points.begin(),
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response.Value().points.end());
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rows.physical_strain_energy = response.Value().strain_energy;
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return Result<ElementResultBundle>::Success(
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{layout_.source_id, std::move(rows)});
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}
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Result<Hex8Response> Hex8Element::EvaluateLinear(const Vector& u) const {
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return Evaluate(u, false);
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}
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Result<Hex8Response> Hex8Element::EvaluateNonlinear(const Vector& u) const {
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return Evaluate(u, true);
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}
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Result<Hex8Response> Hex8Element::Evaluate(const Vector& u,
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bool nonlinear) const {
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if (u.Size() != 24U)
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return Result<Hex8Response>::Failure(
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Failure("Hex8 trial requires 24 translations."));
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const auto certificate = hex8_internal::Certify(coordinates_, &u);
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if (certificate != hex8_internal::Certificate::kPositive)
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return Result<Hex8Response>::Failure(
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Failure(CertificateMessage(certificate)));
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Hex8Response response;
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if (!nonlinear) {
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response.material_tangent = stiffness_;
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response.tangent = stiffness_;
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response.internal_force = stiffness_.Multiply(u);
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response.strain_energy = 0.5 * u.Dot(response.internal_force);
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}
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for (std::size_t g = 0; g < 8; ++g) {
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const auto& point = points_[g];
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Tensor gradient{};
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for (std::size_t a = 0; a < 8; ++a)
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for (std::size_t i = 0; i < 3; ++i)
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for (std::size_t j = 0; j < 3; ++j)
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gradient[i][j] += (u[3U * a + i] - u[i]) * point.gradients[a][j];
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Tensor f = gradient;
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for (std::size_t i = 0; i < 3; ++i) f[i][i] += 1.0;
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Tensor strain{};
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for (std::size_t i = 0; i < 3; ++i)
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for (std::size_t j = 0; j < 3; ++j) {
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if (nonlinear) {
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for (std::size_t k = 0; k < 3; ++k)
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strain[i][j] += 0.5 * f[k][i] * f[k][j];
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if (i == j) strain[i][j] -= 0.5;
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} else
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strain[i][j] = 0.5 * (gradient[i][j] + gradient[j][i]);
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}
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const double trace = strain[0][0] + strain[1][1] + strain[2][2];
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Tensor s{};
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for (std::size_t i = 0; i < 3; ++i)
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for (std::size_t j = 0; j < 3; ++j)
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s[i][j] = 2.0 * mu_ * strain[i][j] + (i == j ? lambda_ * trace : 0.0);
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Tensor stress = s;
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if (nonlinear) {
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const double det_f = hex8_internal::Determinant(f);
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if (!std::isfinite(det_f) || det_f <= 0.0)
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return Result<Hex8Response>::Failure(Failure(
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"Hex8 trial deformation determinant must be finite and positive."));
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Tensor p{};
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Tensor fft{};
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double energy = 0.5 * lambda_ * trace * trace;
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for (std::size_t i = 0; i < 3; ++i)
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for (std::size_t j = 0; j < 3; ++j) {
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energy += mu_ * strain[i][j] * strain[i][j];
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for (std::size_t k = 0; k < 3; ++k) {
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p[i][j] += f[i][k] * s[k][j];
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fft[i][j] += f[i][k] * f[j][k];
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}
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}
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response.strain_energy += energy * point.weight;
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stress = {};
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for (std::size_t i = 0; i < 3; ++i)
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for (std::size_t j = 0; j < 3; ++j)
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for (std::size_t k = 0; k < 3; ++k)
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stress[i][j] += p[i][k] * f[j][k] / det_f;
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for (std::size_t a = 0; a < 8; ++a) {
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const auto& ga = point.gradients[a];
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for (std::size_t i = 0; i < 3; ++i)
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response.internal_force[3U * a + i] += Dot(p[i], ga) * point.weight;
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for (std::size_t b = 0; b < 8; ++b) {
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const auto& gb = point.gradients[b];
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double geometric = 0.0;
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for (std::size_t j = 0; j < 3; ++j)
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for (std::size_t l = 0; l < 3; ++l)
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geometric += ga[j] * s[l][j] * gb[l];
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for (std::size_t i = 0; i < 3; ++i)
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for (std::size_t k = 0; k < 3; ++k) {
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const auto row = 3U * a + i;
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const auto column = 3U * b + k;
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response.material_tangent(row, column) +=
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point.weight * (lambda_ * Dot(f[i], ga) * Dot(f[k], gb) +
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mu_ * Dot(f[i], gb) * Dot(f[k], ga) +
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mu_ * fft[i][k] * Dot(ga, gb));
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if (i == k)
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response.geometric_tangent(row, column) +=
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geometric * point.weight;
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}
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}
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}
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}
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if (!Finite(f) || !Finite(strain) || !Finite(s) || !Finite(stress))
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return Result<Hex8Response>::Failure(
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Failure("Hex8 trial tensor is nonfinite."));
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response.points[g] = {element_index_, static_cast<int>(g + 1U),
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point.natural, point.position,
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Pack(stress), Pack(strain)};
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}
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if (!std::isfinite(response.strain_energy))
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return Result<Hex8Response>::Failure(
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Failure("Hex8 trial energy is nonfinite."));
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for (std::size_t i = 0; i < 24; ++i) {
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if (!std::isfinite(response.internal_force[i]))
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return Result<Hex8Response>::Failure(
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Failure("Hex8 trial force is nonfinite."));
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for (std::size_t j = 0; j < 24; ++j) {
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if (nonlinear)
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response.tangent(i, j) =
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response.material_tangent(i, j) + response.geometric_tangent(i, j);
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if (!std::isfinite(response.tangent(i, j)) ||
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!std::isfinite(response.material_tangent(i, j)) ||
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!std::isfinite(response.geometric_tangent(i, j)))
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return Result<Hex8Response>::Failure(
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Failure("Hex8 trial tangent is nonfinite."));
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}
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}
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return Result<Hex8Response>::Success(std::move(response));
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}
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} // namespace fesa
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