#include "fesa/elements/hex8_element.h"

#include "fesa/math/vector3.h"

#include <cmath>
#include <memory>
#include <set>
#include <stdexcept>
#include <utility>

#include "../math/dense_blas_internal.h"
#include "hex8_geometry.h"
#include "hex8_linear_tangent_cache.h"

namespace fesa {
namespace {
using hex8_internal::Tensor;
using hex8_internal::Triple;
/// @brief Checks every tensor component before recovery or inverse use.
bool Finite(const Tensor& value) {
  for (const auto& row : value)
    for (double entry : row)
      if (!std::isfinite(entry)) return false;
  return true;
}
/// @brief Packs actual tensor shear, never engineering shear, in schema order.
std::array<double, 6> Pack(const Tensor& value) {
  return {value[0][0], value[1][1], value[2][2],
          value[0][1], value[0][2], value[1][2]};
}
/// @brief Computes a Euclidean contraction in deterministic component order.
double Dot(const Triple& a, const Triple& b) {
  return a[0] * b[0] + a[1] * b[1] + a[2] * b[2];
}
/// @brief Forms a scaled 24-component outer product through dense MKL.
Matrix Outer(const std::array<Triple, 8>& values, double coefficient) {
  Matrix left{24U, 1U};
  Matrix right{1U, 24U};
  for (std::size_t a = 0; a < 8; ++a)
    for (std::size_t i = 0; i < 3; ++i) {
      left(3U * a + i, 0) = coefficient * values[a][i];
      right(0, 3U * a + i) = values[a][i];
    }
  return left.Multiply(right);
}
/// @brief Uses MKL entry products and the original ordered Gram/stress sums.
Matrix GradientProduct(const std::array<Triple, 8>& gradients,
                       const Tensor* stress = nullptr) {
  const std::size_t terms = stress == nullptr ? 3U : 9U;
  Vector left{64U * terms};
  Vector right{64U * terms};
  Vector final_factor{64U * terms};
  for (std::size_t a = 0; a < 8; ++a)
    for (std::size_t b = 0; b < 8; ++b)
      for (std::size_t j = 0; j < 3; ++j)
        for (std::size_t l = 0; l < (stress == nullptr ? 1U : 3U); ++l) {
          const auto entry = (8U * a + b) * terms +
                             (stress == nullptr ? j : 3U * j + l);
          left[entry] = gradients[a][j];
          right[entry] = stress == nullptr ? gradients[b][j] : (*stress)[l][j];
          final_factor[entry] = gradients[b][l];
        }
  auto products = dense_blas_internal::MultiplyDiagonal(left, right);
  if (stress != nullptr)
    products = dense_blas_internal::MultiplyDiagonal(products, final_factor);
  Matrix result{8U, 8U};
  for (std::size_t a = 0; a < 8; ++a)
    for (std::size_t b = 0; b < 8; ++b) {
      const auto first = (8U * a + b) * terms;
      if (stress == nullptr) {
        result(a, b) = (products[first] + products[first + 1U]) +
                       products[first + 2U];
      } else {
        for (std::size_t term = 0; term < terms; ++term)
          result(a, b) += products[first + term];
      }
    }
  return result;
}
/// @brief Retains product and cancellation errors in fixed-order contractions.
void AccumulateProduct(double left, double right, double& sum,
                       double& correction) {
  const double product = left * right;
  const double next = sum + product;
  correction += std::fma(left, right, -product) +
                (std::abs(sum) >= std::abs(product)
                     ? (sum - next) + product
                     : (product - next) + sum);
  sum = next;
}
/// @brief Recovers low bits lost when subtracting unequal nodal translations.
double DifferenceCorrection(double left, double right, double difference) {
  const double right_virtual = left - difference;
  const double left_virtual = difference + right_virtual;
  return (left - left_virtual) + (right_virtual - right);
}
/// @brief Gives the finite-control failure without accepting a geometry guess.
const char* CertificateMessage(hex8_internal::Certificate certificate) {
  switch (certificate) {
    case hex8_internal::Certificate::kNonpositive:
      return "Whole-cell Jacobian has a nonpositive witness.";
    case hex8_internal::Certificate::kUncertain:
      return "Whole-cell Jacobian positivity is unresolved at depth 10 or cell "
             "budget 65536.";
    case hex8_internal::Certificate::kNonfinite:
      return "Whole-cell Jacobian interval arithmetic is nonfinite.";
    case hex8_internal::Certificate::kPositive:
      break;
  }
  return "Invalid Jacobian certificate.";
}
}  // namespace

Hex8Element::Hex8Element(EntityIndex index, ElementDofLayout layout,
                         Coordinates coordinates, double kappa, double mu,
                         std::optional<double> density, SourceLocation location)
    : element_index_{index},
      layout_{std::move(layout)},
      coordinates_{coordinates},
      kappa_{kappa},
      mu_{mu},
      density_{density},
      location_{std::move(location)},
      paired_linear_tangent_{
          std::make_shared<hex8_internal::LinearTangentCache>()} {}

Status Hex8Element::Failure(const char* message) const {
  return Status::Failure(FailureCategory::kModel,
                         {{Severity::kError, "invalid-hex8", location_, "C3D8",
                           layout_.source_id.instance_name + ":" +
                               layout_.source_id.source_label_text,
                           message}});
}

Result<Hex8Element> Hex8Element::Create(EntityIndex index,
                                        ElementDofLayout layout,
                                        Coordinates coordinates,
                                        const LinearElasticMaterial& material,
                                        std::optional<double> density,
                                        SourceLocation location) {
  const double e = material.YoungsModulus();
  const double nu = material.PoissonsRatio();
  Hex8Element candidate{index, std::move(layout), coordinates,        0.0,
                        0.0,   density,           std::move(location)};
  const std::vector<DofComponent> components{
      DofComponent::kUx, DofComponent::kUy, DofComponent::kUz};
  const std::set<EntityIndex> nodes(candidate.layout_.node_indices.begin(),
                                    candidate.layout_.node_indices.end());
  if (candidate.layout_.node_indices.size() != 8U || nodes.size() != 8U ||
      candidate.layout_.components_per_node != components)
    return Result<Hex8Element>::Failure(candidate.Failure(
        "Hex8 requires eight distinct nodes with exactly UX, UY, UZ."));
  if (!std::isfinite(e) || e <= 0.0 || !std::isfinite(nu) || nu <= -1.0 ||
      nu >= 0.5)
    return Result<Hex8Element>::Failure(
        candidate.Failure("Hex8 requires finite E>0 and -1<nu<0.5."));
  candidate.mu_ = e / (2.0 * (1.0 + nu));
  candidate.kappa_ = e / (3.0 * (1.0 - 2.0 * nu));
  if (!std::isfinite(candidate.mu_) || candidate.mu_ <= 0.0 ||
      !std::isfinite(candidate.kappa_) || candidate.kappa_ <= 0.0)
    return Result<Hex8Element>::Failure(
        candidate.Failure("Hex8 shear and direct bulk moduli must remain "
                          "finite and representably positive."));
  if (density && (!std::isfinite(*density) || *density <= 0.0))
    return Result<Hex8Element>::Failure(candidate.Failure(
        "Hex8 reference density must be finite and positive."));
  const auto certificate = hex8_internal::Certify(coordinates);
  if (certificate != hex8_internal::Certificate::kPositive)
    return Result<Hex8Element>::Failure(
        candidate.Failure(CertificateMessage(certificate)));
  const double gauss = 1.0 / std::sqrt(3.0);
  for (std::size_t g = 0; g < 8; ++g) {
    auto& point = candidate.points_[g];
    point.natural = {g % 2U == 0U ? -gauss : gauss,
                     (g / 2U) % 2U == 0U ? -gauss : gauss,
                     g < 4U ? -gauss : gauss};
    std::array<Triple, 8> derivatives{};
    hex8_internal::Shape(point.natural, point.shape, derivatives);
    const auto jacobian = hex8_internal::Jacobian(coordinates, derivatives);
    point.weight = hex8_internal::Determinant(jacobian);
    if (!Finite(jacobian) || !std::isfinite(point.weight) ||
        point.weight <= 0.0)
      return Result<Hex8Element>::Failure(candidate.Failure(
          "Reference quadrature Jacobian must be finite and positive."));
    const Vector3 c0{jacobian[0][0], jacobian[1][0], jacobian[2][0]};
    const Vector3 c1{jacobian[0][1], jacobian[1][1], jacobian[2][1]};
    const Vector3 c2{jacobian[0][2], jacobian[1][2], jacobian[2][2]};
    if (!c0.Cross(c1).IsFinite() || !c1.Cross(c2).IsFinite() ||
        !c2.Cross(c0).IsFinite())
      return Result<Hex8Element>::Failure(
          candidate.Failure("Reference Jacobian inverse is nonfinite."));
    Matrix coefficients{3U, 3U};
    Matrix natural_gradients{3U, 8U};
    for (std::size_t i = 0; i < 3U; ++i) {
      for (std::size_t d = 0; d < 3U; ++d)
        coefficients(i, d) = jacobian[i][d];
      for (std::size_t a = 0; a < 8U; ++a)
        natural_gradients(i, a) = derivatives[a][i];
    }
    const auto factor = dense_blas_internal::DenseFactor::Lu(coefficients);
    if (!factor.HasValue())
      return Result<Hex8Element>::Failure(
          candidate.Failure("Reference Jacobian inverse is nonfinite."));
    // The original geometry gate required every reciprocal action to be
    // finite, including entries not exposed by the smaller shape-gradient RHS.
    Matrix canonical{3U, 1U};
    for (std::size_t axis = 0; axis < 3U; ++axis) {
      if (axis != 0U) canonical(axis - 1U, 0U) = 0.0;
      canonical(axis, 0U) = 1.0;
      if (!factor.Value().Solve(canonical, true).HasValue())
        return Result<Hex8Element>::Failure(
            candidate.Failure("Reference Jacobian inverse is nonfinite."));
    }
    // J has Cartesian rows and natural columns: J^T grad_x N = grad_nat N.
    const auto gradients = factor.Value().Solve(natural_gradients, true);
    if (!gradients.HasValue())
      return Result<Hex8Element>::Failure(candidate.Failure(
          "Reference gradient or point coordinate is nonfinite."));
    point.position = coordinates[0];
    for (std::size_t a = 0; a < 8; ++a)
      for (std::size_t i = 0; i < 3; ++i) {
        point.position[i] +=
            point.shape[a] * (coordinates[a][i] - coordinates[0][i]);
        point.gradients[a][i] = gradients.Value()(i, a);
        if (!std::isfinite(point.gradients[a][i]) ||
            !std::isfinite(point.position[i]))
          return Result<Hex8Element>::Failure(candidate.Failure(
              "Reference gradient or point coordinate is nonfinite."));
      }
  }
  // Zero Hessian of the same reference-relative mean-volume energy.
  std::array<Triple, 8> integrated_gradients{};
  double volume = 0.0;
  bool backend_inputs_failed = false;
  for (const auto& point : candidate.points_) {
    auto gradients = point.gradients;
    gradients[0] = {};
    for (std::size_t a = 1; a < 8; ++a)
      for (std::size_t i = 0; i < 3; ++i) gradients[0][i] -= gradients[a][i];
    volume += point.weight;
    for (std::size_t a = 0; a < 8; ++a)
      for (std::size_t i = 0; i < 3; ++i)
        integrated_gradients[a][i] += point.weight * gradients[a][i];
    Matrix deviatoric{0U, 0U};
    Matrix shear{0U, 0U};
    Matrix gram{0U, 0U};
    bool point_products_failed = false;
    try {
      deviatoric = Outer(gradients, -(2.0 / 3.0) * candidate.mu_);
      shear = Outer(gradients, candidate.mu_);
      gram = GradientProduct(gradients);
    } catch (const std::invalid_argument&) {
      backend_inputs_failed = true;
      point_products_failed = true;
    }
    for (std::size_t a = 0; a < 8; ++a)
      for (std::size_t b = 0; b < 8; ++b) {
        for (std::size_t i = 0; i < 3; ++i)
          for (std::size_t k = 0; k < 3; ++k) {
            if (point_products_failed) {
              // Diagnostic replay only: this candidate can never succeed.
              const auto& ga = gradients[a];
              const auto& gb = gradients[b];
              candidate.stiffness_(3U * a + i, 3U * b + k) +=
                  point.weight *
                  (-(2.0 / 3.0) * candidate.mu_ * ga[i] * gb[k] +
                   candidate.mu_ * ga[k] * gb[i] +
                   (i == k ? candidate.mu_ * Dot(ga, gb) : 0.0));
            } else {
              candidate.stiffness_(3U * a + i, 3U * b + k) +=
                  point.weight * (deviatoric(3U * a + i, 3U * b + k) +
                                  shear(3U * a + k, 3U * b + i) +
                                  (i == k ? candidate.mu_ * gram(a, b) : 0.0));
            }
          }
      }
  }
  if (!std::isfinite(volume) || volume <= 0.0)
    return Result<Hex8Element>::Failure(
        candidate.Failure("Hex8 reference volume is nonfinite."));
  Matrix bulk{0U, 0U};
  try {
    bulk = Outer(integrated_gradients, candidate.kappa_ / volume);
  } catch (const std::invalid_argument&) {
    backend_inputs_failed = true;
    bulk = Matrix{24U, 24U};
    for (std::size_t a = 0; a < 8; ++a)
      for (std::size_t b = 0; b < 8; ++b)
        for (std::size_t i = 0; i < 3; ++i)
          for (std::size_t k = 0; k < 3; ++k)
            bulk(3U * a + i, 3U * b + k) =
                (candidate.kappa_ / volume) * integrated_gradients[a][i] *
                integrated_gradients[b][k];
  }
  for (std::size_t i = 0; i < 24; ++i)
    for (std::size_t j = 0; j < 24; ++j)
      candidate.stiffness_(i, j) += bulk(i, j);
  for (std::size_t i = 0; i < 24; ++i)
    for (std::size_t j = 0; j < 24; ++j)
      if (!std::isfinite(candidate.stiffness_(i, j)))
        return Result<Hex8Element>::Failure(
            candidate.Failure("Hex8 reference stiffness is nonfinite."));
  for (std::size_t i = 0; i < 24; ++i)
    if (candidate.stiffness_(i, i) <= 0.0)
      return Result<Hex8Element>::Failure(
          candidate.Failure("Hex8 reference stiffness has an unrepresentable "
                            "positive diagonal."));
  if (backend_inputs_failed)
    return Result<Hex8Element>::Failure(candidate.Failure(
        "Hex8 dense MKL operands failed finite validation."));
  return Result<Hex8Element>::Success(std::move(candidate));
}

const ElementDofLayout& Hex8Element::DofLayout() const noexcept {
  return layout_;
}
Result<ElementStiffnessContribution> Hex8Element::ComputeStiffness() const {
  return Result<ElementStiffnessContribution>::Success({layout_, stiffness_});
}
std::array<double, 8> Hex8Element::ReferenceWeights() const noexcept {
  std::array<double, 8> weights{};
  for (std::size_t g = 0; g < weights.size(); ++g)
    weights[g] = points_[g].weight;
  return weights;
}

Result<ElementStiffnessContribution> Hex8Element::ComputeMass() const {
  if (!density_)
    return Result<ElementStiffnessContribution>::Failure(
        Failure("Hex8 mass requires reference density."));
  Matrix consistent{24U, 24U};
  for (const auto& point : points_) {
    Matrix left{8U, 1U};
    Matrix right{1U, 8U};
    for (std::size_t a = 0; a < 8; ++a) {
      left(a, 0) = *density_ * point.shape[a];
      right(0, a) = point.shape[a];
    }
    Matrix products{0U, 0U};
    try {
      products = left.Multiply(right);
    } catch (const std::invalid_argument&) {
      return Result<ElementStiffnessContribution>::Failure(
          Failure("Hex8 reference mass is nonfinite."));
    }
    for (std::size_t a = 0; a < 8; ++a)
      for (std::size_t b = 0; b < 8; ++b) {
        const double value = products(a, b) * point.weight;
        if (!std::isfinite(value))
          return Result<ElementStiffnessContribution>::Failure(
              Failure("Hex8 reference mass is nonfinite."));
        for (std::size_t i = 0; i < 3; ++i)
          consistent(3U * a + i, 3U * b + i) += value;
      }
  }
  Matrix mass{24U, 24U};
  for (std::size_t i = 0; i < 24; ++i)
    for (std::size_t j = 0; j < 24; ++j) {
      if (!std::isfinite(consistent(i, j)))
        return Result<ElementStiffnessContribution>::Failure(
            Failure("Hex8 accumulated reference mass is nonfinite."));
      mass(i, i) += consistent(i, j);
    }
  for (std::size_t i = 0; i < 24; ++i)
    if (!std::isfinite(mass(i, i)) || mass(i, i) <= 0.0)
      return Result<ElementStiffnessContribution>::Failure(Failure(
          "Hex8 reference mass has an unrepresentable positive diagonal."));
  return Result<ElementStiffnessContribution>::Success(
      {layout_, std::move(mass)});
}
Result<ElementResultBundle> Hex8Element::Recover(const Vector& u) const {
  auto response = EvaluateLinear(u);
  if (!response.HasValue())
    return Result<ElementResultBundle>::Failure(response.GetStatus());
  SolidElementResultRows rows;
  rows.rows.assign(response.Value().points.begin(),
                   response.Value().points.end());
  rows.physical_strain_energy = response.Value().strain_energy;
  return Result<ElementResultBundle>::Success(
      {layout_.source_id, std::move(rows)});
}
Result<Hex8Response> Hex8Element::EvaluateLinear(const Vector& u) const {
  return Evaluate(u, false);
}
Result<Hex8Response> Hex8Element::EvaluateNonlinear(const Vector& u) const {
  return Evaluate(u, true);
}
Result<Hex8HighLowResponse> Hex8Element::EvaluateLinearHighLow(
    const Vector& high, const Vector& low) const {
  return EvaluateHighLow(high, low, false);
}
Result<Hex8HighLowResponse> Hex8Element::EvaluateNonlinearHighLow(
    const Vector& high, const Vector& low) const {
  return EvaluateHighLow(high, low, true);
}
Result<Hex8Response> Hex8Element::Evaluate(const Vector& u,
                                           bool nonlinear) const {
  if (u.Size() != 24U)
    return Result<Hex8Response>::Failure(
        Failure("Hex8 trial requires 24 translations."));
  const auto certificate = hex8_internal::Certify(coordinates_, &u);
  if (certificate != hex8_internal::Certificate::kPositive)
    return Result<Hex8Response>::Failure(
        Failure(CertificateMessage(certificate)));
  Hex8Response response;
  Vector relative{24U};
  Vector relative_correction{24U};
  for (std::size_t a = 0; a < 8; ++a)
    for (std::size_t i = 0; i < 3; ++i) {
      const auto row = 3U * a + i;
      relative[row] = u[row] - u[i];
      relative_correction[row] =
          DifferenceCorrection(u[row], u[i], relative[row]);
    }
  if (!nonlinear) {
    response.material_tangent = stiffness_;
    response.tangent = stiffness_;
  }
  std::array<Tensor, 8> deformations{}, strains{};
  std::array<std::array<Triple, 8>, 8> gradients{};
  double volume = 0.0, trace_integral = 0.0;
  Vector force_correction{24U};
  for (std::size_t g = 0; g < 8; ++g) {
    const auto& point = points_[g];
    gradients[g] = point.gradients;
    Triple sum{};
    for (std::size_t a = 1; a < 8; ++a)
      for (std::size_t j = 0; j < 3; ++j) sum[j] += gradients[g][a][j];
    for (std::size_t j = 0; j < 3; ++j) gradients[g][0][j] = -sum[j];
    Tensor gradient{};
    Tensor gradient_correction{};
    for (std::size_t a = 1; a < 8; ++a)
      for (std::size_t i = 0; i < 3; ++i)
        for (std::size_t j = 0; j < 3; ++j) {
          AccumulateProduct(relative[3U * a + i], point.gradients[a][j],
                            gradient[i][j], gradient_correction[i][j]);
          AccumulateProduct(relative_correction[3U * a + i],
                            point.gradients[a][j], gradient[i][j],
                            gradient_correction[i][j]);
        }
    for (std::size_t i = 0; i < 3; ++i)
      for (std::size_t j = 0; j < 3; ++j)
        gradient[i][j] += gradient_correction[i][j];
    Tensor f = nonlinear ? gradient : Tensor{};
    for (std::size_t i = 0; i < 3; ++i) f[i][i] += 1.0;
    Tensor strain{};
    for (std::size_t i = 0; i < 3; ++i)
      for (std::size_t j = 0; j < 3; ++j) {
        strain[i][j] = 0.5 * (gradient[i][j] + gradient[j][i]);
        // E = sym(H) + H^T*H/2 avoids subtracting I at tiny strain.
        if (nonlinear)
          for (std::size_t k = 0; k < 3; ++k)
            strain[i][j] += 0.5 * gradient[k][i] * gradient[k][j];
      }
    deformations[g] = f;
    strains[g] = strain;
    volume += point.weight;
    trace_integral +=
        point.weight * (strain[0][0] + strain[1][1] + strain[2][2]);
    if (!Finite(gradient) || !Finite(f) || !Finite(strain))
      return Result<Hex8Response>::Failure(
          Failure("Hex8 trial tensor is nonfinite."));
  }
  const double mean_trace = trace_integral / volume;
  if (!std::isfinite(volume) || volume <= 0 || !std::isfinite(mean_trace))
    return Result<Hex8Response>::Failure(
        Failure("Hex8 mean volume is nonfinite."));
  response.strain_energy = (0.5 * kappa_ * mean_trace) * (volume * mean_trace);
  std::array<Triple, 8> integrated_fg{};
  bool backend_inputs_failed = false;
  for (std::size_t g = 0; g < 8; ++g) {
    const auto& point = points_[g];
    const auto& f = deformations[g];
    const auto& strain = strains[g];
    const double trace = strain[0][0] + strain[1][1] + strain[2][2];
    Tensor s{}, p{}, fft{};
    double density_energy = 0.0;
    for (std::size_t i = 0; i < 3; ++i)
      for (std::size_t j = 0; j < 3; ++j) {
        const double dev = strain[i][j] - (i == j ? trace / 3.0 : 0.0);
        s[i][j] = 2.0 * mu_ * dev + (i == j ? kappa_ * mean_trace : 0.0);
        density_energy += mu_ * dev * dev;
      }
    for (std::size_t i = 0; i < 3; ++i)
      for (std::size_t j = 0; j < 3; ++j) {
        for (std::size_t k = 0; k < 3; ++k) {
          p[i][j] += f[i][k] * s[k][j];
          fft[i][j] += f[i][k] * f[j][k];
        }
      }
    response.strain_energy += density_energy * point.weight;
    Tensor stress = s;
    if (nonlinear) {
      const double det_f = hex8_internal::Determinant(f);
      if (!std::isfinite(det_f) || det_f <= 0.0)
        return Result<Hex8Response>::Failure(Failure(
            "Hex8 trial deformation determinant must be finite and positive."));
      stress = {};
      for (std::size_t i = 0; i < 3; ++i)
        for (std::size_t j = 0; j < 3; ++j)
          for (std::size_t k = 0; k < 3; ++k)
            stress[i][j] += p[i][k] * f[j][k] / det_f;
    }
    for (std::size_t a = 0; a < 8; ++a) {
      for (std::size_t i = 0; i < 3; ++i)
        integrated_fg[a][i] += point.weight * Dot(f[i], gradients[g][a]);
    }
    Matrix deviatoric{0U, 0U};
    Matrix shear{0U, 0U};
    Matrix gram{0U, 0U};
    Matrix geometric{0U, 0U};
    bool point_products_failed = false;
    if (!Finite(f) || !Finite(strain) || !Finite(s) || !Finite(p) ||
        !Finite(fft) || !Finite(stress))
      return Result<Hex8Response>::Failure(
          Failure("Hex8 trial tensor is nonfinite."));
    if (nonlinear) {
      std::array<Triple, 8> fg{};
      for (std::size_t a = 0; a < 8; ++a)
        for (std::size_t i = 0; i < 3; ++i)
          fg[a][i] = Dot(f[i], gradients[g][a]);
      try {
        deviatoric = Outer(fg, -(2.0 / 3.0) * mu_);
        shear = Outer(fg, mu_);
        gram = GradientProduct(gradients[g]);
        geometric = GradientProduct(gradients[g], &s);
      } catch (const std::invalid_argument&) {
        backend_inputs_failed = true;
        point_products_failed = true;
      }
    }
    for (std::size_t a = 0; a < 8; ++a) {
      const auto& ga = gradients[g][a];
      for (std::size_t i = 0; i < 3; ++i) {
        double contraction = 0.0;
        double contraction_correction = 0.0;
        for (std::size_t j = 0; j < 3; ++j)
          AccumulateProduct(p[i][j], ga[j], contraction,
                            contraction_correction);
        AccumulateProduct(contraction, point.weight,
                          response.internal_force[3U * a + i],
                          force_correction[3U * a + i]);
        AccumulateProduct(contraction_correction, point.weight,
                          response.internal_force[3U * a + i],
                          force_correction[3U * a + i]);
      }
      if (nonlinear)
        for (std::size_t b = 0; b < 8; ++b) {
          const auto& gb = gradients[g][b];
          double diagnostic_geometric = 0.0;
          if (point_products_failed)
            for (std::size_t j = 0; j < 3; ++j)
              for (std::size_t l = 0; l < 3; ++l)
                diagnostic_geometric += ga[j] * s[l][j] * gb[l];
          for (std::size_t i = 0; i < 3; ++i)
            for (std::size_t k = 0; k < 3; ++k) {
              const auto row = 3U * a + i;
              const auto column = 3U * b + k;
              if (point_products_failed) {
                // Reproduce original diagnostic priority, never acceptance.
                response.material_tangent(row, column) +=
                    point.weight *
                    (-(2.0 / 3.0) * mu_ * Dot(f[i], ga) * Dot(f[k], gb) +
                     mu_ * Dot(f[i], gb) * Dot(f[k], ga) +
                     mu_ * fft[i][k] * Dot(ga, gb));
              } else {
                response.material_tangent(row, column) +=
                    point.weight *
                    (deviatoric(row, column) +
                     shear(3U * b + i, 3U * a + k) +
                     mu_ * fft[i][k] * gram(a, b));
              }
              if (i == k)
                response.geometric_tangent(row, column) +=
                    (point_products_failed ? diagnostic_geometric
                                           : geometric(a, b)) * point.weight;
            }
        }
    }
    response.points[g] = {element_index_, static_cast<int>(g + 1U),
                          point.natural,  point.position,
                          Pack(stress),   Pack(strain)};
  }
  for (std::size_t i = 0; i < 24; ++i)
    response.internal_force[i] += force_correction[i];
  if (nonlinear) {
    Matrix bulk{0U, 0U};
    try {
      bulk = Outer(integrated_fg, kappa_ / volume);
    } catch (const std::invalid_argument&) {
      backend_inputs_failed = true;
      bulk = Matrix{24U, 24U};
      for (std::size_t a = 0; a < 8; ++a)
        for (std::size_t b = 0; b < 8; ++b)
          for (std::size_t i = 0; i < 3; ++i)
            for (std::size_t k = 0; k < 3; ++k)
              bulk(3U * a + i, 3U * b + k) =
                  (kappa_ / volume) * integrated_fg[a][i] * integrated_fg[b][k];
    }
    for (std::size_t i = 0; i < 24; ++i)
      for (std::size_t j = 0; j < 24; ++j)
        response.material_tangent(i, j) += bulk(i, j);
  }
  if (!std::isfinite(response.strain_energy))
    return Result<Hex8Response>::Failure(
        Failure("Hex8 trial energy is nonfinite."));
  for (std::size_t i = 0; i < 24; ++i) {
    if (!std::isfinite(response.internal_force[i]))
      return Result<Hex8Response>::Failure(
          Failure("Hex8 trial force is nonfinite."));
    for (std::size_t j = 0; j < 24; ++j) {
      if (nonlinear)
        response.tangent(i, j) =
            response.material_tangent(i, j) + response.geometric_tangent(i, j);
      if (!std::isfinite(response.tangent(i, j)) ||
          !std::isfinite(response.material_tangent(i, j)) ||
          !std::isfinite(response.geometric_tangent(i, j)))
        return Result<Hex8Response>::Failure(
            Failure("Hex8 trial tangent is nonfinite."));
    }
  }
  if (backend_inputs_failed)
    return Result<Hex8Response>::Failure(
        Failure("Hex8 dense MKL operands failed finite validation."));
  return Result<Hex8Response>::Success(std::move(response));
}
}  // namespace fesa