feat(hex8): use reference-weighted mean-volume energy

include/fesa/elements/hex8_element.h

@@ -30,7 +30,7 @@ struct Hex8HighLowResponse {
   double strain_energy_low{0.0};
 };
 
-/// @brief Implements full-integration C3D8 linear and TL StVK mechanics.
+/// @brief Implements C3D8 full deviatoric and mean-volume Green energy.
 /// @note Reference geometry is owned and immutable; no trial is committed.
 class Hex8Element final : public Element {
  public:
@@ -53,14 +53,14 @@ class Hex8Element final : public Element {
 
   /// @brief Returns f=K0*u and infinitesimal tensor strain at eight points.
   Result<Hex8Response> EvaluateLinear(const Vector& displacement) const;
-  /// @brief Returns one certified StVK trial and its analytic energy Hessian.
+  /// @brief Returns one certified mean-Green TL trial and its energy Hessian.
   Result<Hex8Response> EvaluateNonlinear(const Vector& displacement) const;
   /// @brief Evaluates linear energy/force/Hessian on the complete paired state.
   /// @note Uses the same reference-relative gradient operator as the paired TL
   /// path. Inputs must have 24 finite normalized high+low translations.
   Result<Hex8HighLowResponse> EvaluateLinearHighLow(
       const Vector& high, const Vector& low) const;
-  /// @brief Evaluates certified TL StVK mechanics without collapsing low state.
+  /// @brief Evaluates mean-Green TL mechanics without collapsing low state.
   /// @note Failure preserves immutable reference data and all caller state.
   Result<Hex8HighLowResponse> EvaluateNonlinearHighLow(
       const Vector& high, const Vector& low) const;
@@ -81,7 +81,7 @@ class Hex8Element final : public Element {
   };
   /// @brief Constructs an owned candidate before reference validation.
   Hex8Element(EntityIndex element_index, ElementDofLayout layout,
-              Coordinates coordinates, double lambda, double mu,
+              Coordinates coordinates, double kappa, double mu,
               std::optional<double> density, SourceLocation location);
   /// @brief Evaluates without mutating the cached reference or previous values.
   Result<Hex8Response> Evaluate(const Vector& displacement,
@@ -95,7 +95,7 @@ class Hex8Element final : public Element {
   EntityIndex element_index_;
   ElementDofLayout layout_;
   Coordinates coordinates_;
-  double lambda_;
+  double kappa_;
   double mu_;
   std::optional<double> density_;
   SourceLocation location_;

src/fesa/elements/hex8_element.cpp

@@ -61,12 +61,12 @@ const char* CertificateMessage(hex8_internal::Certificate certificate) {
 }  // namespace
 
 Hex8Element::Hex8Element(EntityIndex index, ElementDofLayout layout,
-                         Coordinates coordinates, double lambda, double mu,
+                         Coordinates coordinates, double kappa, double mu,
                          std::optional<double> density, SourceLocation location)
     : element_index_{index},
       layout_{std::move(layout)},
       coordinates_{coordinates},
-      lambda_{lambda},
+      kappa_{kappa},
       mu_{mu},
       density_{density},
       location_{std::move(location)} {}
@@ -102,23 +102,12 @@ Result<Hex8Element> Hex8Element::Create(EntityIndex index,
     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.lambda_ = e * nu / ((1.0 + nu) * (1.0 - 2.0 * nu));
+  candidate.kappa_ = e / (3.0 * (1.0 - 2.0 * nu));
   if (!std::isfinite(candidate.mu_) || candidate.mu_ <= 0.0 ||
-      !std::isfinite(candidate.lambda_) ||
+      !std::isfinite(candidate.kappa_) || candidate.kappa_ <= 0.0)
-      !std::isfinite(candidate.lambda_ + 2.0 * candidate.mu_))
     return Result<Hex8Element>::Failure(
-        candidate.Failure("Hex8 Lame moduli must remain finite with "
+        candidate.Failure("Hex8 shear and direct bulk moduli must remain "
-                          "representable positive shear modulus."));
+                          "finite and representably positive."));
-  // Validate the rounded moduli actually used by the kernel, not merely the
-  // positive exact-real E/nu formula: cancellation near nu=-1 can erase bulk.
-  const double bulk = candidate.lambda_ + (2.0 * candidate.mu_) / 3.0;
-  const double volumetric_stress =
-      3.0 * candidate.lambda_ + 2.0 * candidate.mu_;
-  if (!std::isfinite(bulk) || bulk <= 0.0 ||
-      !std::isfinite(volumetric_stress) || volumetric_stress <= 0.0)
-    return Result<Hex8Element>::Failure(
-        candidate.Failure("Hex8 rounded Lame moduli require finite "
-                          "representable positive bulk stiffness."));
   if (density && (!std::isfinite(*density) || *density <= 0.0))
     return Result<Hex8Element>::Failure(candidate.Failure(
         "Hex8 reference density must be finite and positive."));
@@ -156,20 +145,43 @@ Result<Hex8Element> Hex8Element::Create(EntityIndex index,
           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;
+  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];
     for (std::size_t a = 0; a < 8; ++a)
       for (std::size_t b = 0; b < 8; ++b) {
-        const auto& ga = point.gradients[a];
+        const auto& ga = gradients[a];
-        const auto& gb = point.gradients[b];
+        const auto& gb = gradients[b];
         const double dot = Dot(ga, gb);
         for (std::size_t i = 0; i < 3; ++i)
           for (std::size_t k = 0; k < 3; ++k) {
             candidate.stiffness_(3U * a + i, 3U * b + k) +=
-                point.weight * (candidate.lambda_ * ga[i] * gb[k] +
+                point.weight * (-(2.0 / 3.0) * candidate.mu_ * ga[i] * gb[k] +
                                 candidate.mu_ * ga[k] * gb[i] +
                                 (i == k ? candidate.mu_ * dot : 0.0));
           }
       }
   }
+  if (!std::isfinite(volume) || volume <= 0.0)
+    return Result<Hex8Element>::Failure(
+        candidate.Failure("Hex8 reference volume is nonfinite."));
+  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)
+          candidate.stiffness_(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)
       if (!std::isfinite(candidate.stiffness_(i, j)))
@@ -278,48 +290,33 @@ Result<Hex8Response> Hex8Element::Evaluate(const Vector& u,
   if (!nonlinear) {
     response.material_tangent = stiffness_;
     response.tangent = stiffness_;
-    // K0 annihilates uniform translation. Use the same relative field as
-    // strain recovery to avoid cancellation of a large rigid displacement.
-    // Retain low product bits and cancelled sum terms in bending-dominated
-    // rows; the stored stiffness and its tangent are unchanged.
-    for (std::size_t i = 0; i < 24; ++i) {
-      double sum = 0.0;
-      double correction = 0.0;
-      for (std::size_t j = 0; j < 24; ++j) {
-        AccumulateProduct(stiffness_(i, j), relative[j], sum, correction);
-        AccumulateProduct(stiffness_(i, j), relative_correction[j], sum,
-                          correction);
-      }
-      response.internal_force[i] = sum + correction;
-    }
-    double energy = 0.0;
-    double energy_correction = 0.0;
-    for (std::size_t i = 0; i < 24; ++i) {
-      AccumulateProduct(relative[i], response.internal_force[i], energy,
-                        energy_correction);
-      AccumulateProduct(relative_correction[i], response.internal_force[i],
-                        energy, energy_correction);
-    }
-    response.strain_energy = 0.5 * (energy + energy_correction);
   }
+  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 = 0; a < 8; ++a)
+    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],
+                            point.gradients[a][j], gradient[i][j],
-                            gradient[i][j], gradient_correction[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 = gradient;
+    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)
@@ -330,51 +327,76 @@ Result<Hex8Response> Hex8Element::Evaluate(const Vector& u,
           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{};
+  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{};
+    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)
+      for (std::size_t j = 0; j < 3; ++j) {
-        s[i][j] = 2.0 * mu_ * strain[i][j] + (i == j ? lambda_ * trace : 0.0);
+        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."));
-      Tensor p{};
-      Tensor fft{};
-      double energy = 0.5 * lambda_ * trace * trace;
-      for (std::size_t i = 0; i < 3; ++i)
-        for (std::size_t j = 0; j < 3; ++j) {
-          energy += mu_ * strain[i][j] * strain[i][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 += energy * point.weight;
       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) {
+    }
-        const auto& ga = point.gradients[a];
+    for (std::size_t a = 0; a < 8; ++a) {
-        for (std::size_t i = 0; i < 3; ++i) {
+      for (std::size_t i = 0; i < 3; ++i)
-          double contraction = 0.0;
+        integrated_fg[a][i] += point.weight * Dot(f[i], gradients[g][a]);
-          double contraction_correction = 0.0;
+    }
-          for (std::size_t j = 0; j < 3; ++j)
+    for (std::size_t a = 0; a < 8; ++a) {
-            AccumulateProduct(p[i][j], ga[j], contraction,
+      const auto& ga = gradients[g][a];
-                              contraction_correction);
+      for (std::size_t i = 0; i < 3; ++i) {
-          AccumulateProduct(contraction, point.weight,
+        double contraction = 0.0;
-                            response.internal_force[3U * a + i],
+        double contraction_correction = 0.0;
-                            force_correction[3U * a + i]);
+        for (std::size_t j = 0; j < 3; ++j)
-          AccumulateProduct(contraction_correction, point.weight,
+          AccumulateProduct(p[i][j], ga[j], contraction,
-                            response.internal_force[3U * a + i],
+                            contraction_correction);
-                            force_correction[3U * a + i]);
+        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 = point.gradients[b];
+          const auto& gb = gradients[g][b];
           double geometric = 0.0;
           for (std::size_t j = 0; j < 3; ++j)
             for (std::size_t l = 0; l < 3; ++l)
@@ -384,26 +406,33 @@ Result<Hex8Response> Hex8Element::Evaluate(const Vector& u,
               const auto row = 3U * a + i;
               const auto column = 3U * b + k;
               response.material_tangent(row, column) +=
-                  point.weight * (lambda_ * Dot(f[i], ga) * Dot(f[k], gb) +
+                  point.weight *
-                                  mu_ * Dot(f[i], gb) * Dot(f[k], ga) +
+                  (-(2.0 / 3.0) * mu_ * Dot(f[i], ga) * Dot(f[k], gb) +
-                                  mu_ * fft[i][k] * Dot(ga, gb));
+                   mu_ * Dot(f[i], gb) * Dot(f[k], ga) +
+                   mu_ * fft[i][k] * Dot(ga, gb));
               if (i == k)
                 response.geometric_tangent(row, column) +=
                     geometric * point.weight;
             }
         }
-      }
     }
-    if (!Finite(f) || !Finite(strain) || !Finite(s) || !Finite(stress))
+    if (!Finite(f) || !Finite(strain) || !Finite(s) || !Finite(p) ||
+        !Finite(fft) || !Finite(stress))
       return Result<Hex8Response>::Failure(
           Failure("Hex8 trial tensor is nonfinite."));
     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)
-    for (std::size_t i = 0; i < 24; ++i)
+    for (std::size_t a = 0; a < 8; ++a)
-      response.internal_force[i] += force_correction[i];
+      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)
+            response.material_tangent(3 * a + i, 3 * b + k) +=
+                (kappa_ / volume) * integrated_fg[a][i] * integrated_fg[b][k];
   if (!std::isfinite(response.strain_energy))
     return Result<Hex8Response>::Failure(
         Failure("Hex8 trial energy is nonfinite."));

src/fesa/elements/hex8_high_low.cpp

@@ -86,7 +86,10 @@ Result<Hex8HighLowResponse> Hex8Element::EvaluateHighLow(
   Hex8HighLowResponse response;
   std::array<HighLow, 24> force{};
   std::array<std::array<HighLow, 24>, 24> material{}, geometric{};
-  HighLow energy;
+  HighLow energy, volume, trace_integral;
+  std::array<Tensor, 8> deformations{}, strains{};
+  std::array<std::array<Triple, 8>, 8> point_gradients{};
+  std::array<Triple, 8> integrated_fg{};
   for (std::size_t g = 0; g < 8; ++g) {
     const auto& point = points_[g];
     std::array<Triple, 8> gradients{};
@@ -104,7 +107,7 @@ Result<Hex8HighLowResponse> Hex8Element::EvaluateHighLow(
     for (std::size_t j = 0; j < 3; ++j)
       gradients[0][j] = math_internal::Negate(gradient_sum[j]);
     Tensor f{};
-    Tensor strain{}, s{}, p{}, fft{};
+    Tensor strain{};
     for (std::size_t i = 0; i < 3; ++i)
       for (std::size_t j = 0; j < 3; ++j) {
         f[i][j] = nonlinear ? h[i][j] : HighLow{};
@@ -114,13 +117,48 @@ Result<Hex8HighLowResponse> Hex8Element::EvaluateHighLow(
           for (std::size_t k = 0; k < 3; ++k)
             strain[i][j] += HighLow{0.5} * h[k][i] * h[k][j];
       }
+    if (!Finite(h) || !Finite(f) || !Finite(strain))
+      return Result<Hex8HighLowResponse>::Failure(Failure(
+          "Hex8 high+low strain arithmetic is outside its supported range."));
+    point_gradients[g] = gradients;
+    deformations[g] = f;
+    strains[g] = strain;
+    volume += HighLow{point.weight};
+    trace_integral +=
+        HighLow{point.weight} * (strain[0][0] + strain[1][1] + strain[2][2]);
+  }
+  if (!IsNormalized(volume) || math_internal::Compare(volume, HighLow{}) != 1 ||
+      !IsNormalized(trace_integral))
+    return Result<Hex8HighLowResponse>::Failure(
+        Failure("Hex8 high+low reference volume or trace is unsupported."));
+  const HighLow mean_trace = trace_integral / volume;
+  if (!IsNormalized(mean_trace))
+    return Result<Hex8HighLowResponse>::Failure(
+        Failure("Hex8 high+low weighted mean is unsupported."));
+  const HighLow twice_mu = HighLow{2} * HighLow{mu_};
+  const HighLow deviatoric_coefficient =
+      math_internal::Negate(twice_mu) / HighLow{3};
+  const HighLow bulk_coefficient = HighLow{kappa_} / volume;
+  if (!IsNormalized(twice_mu) || !IsNormalized(deviatoric_coefficient) ||
+      !IsNormalized(bulk_coefficient))
+    return Result<Hex8HighLowResponse>::Failure(
+        Failure("Hex8 high+low constitutive coefficients are unsupported."));
+  energy = HighLow{0.5} * HighLow{kappa_} * volume * mean_trace * mean_trace;
+  for (std::size_t g = 0; g < 8; ++g) {
+    const auto& point = points_[g];
+    const auto& gradients = point_gradients[g];
+    const auto& f = deformations[g];
+    const auto& strain = strains[g];
     const HighLow trace = strain[0][0] + strain[1][1] + strain[2][2];
-    HighLow density_energy = HighLow{0.5} * HighLow{lambda_} * trace * trace;
+    Tensor s{}, p{}, fft{};
+    HighLow density_energy;
     for (std::size_t i = 0; i < 3; ++i)
       for (std::size_t j = 0; j < 3; ++j) {
-        s[i][j] = HighLow{2} * HighLow{mu_} * strain[i][j];
+        const HighLow dev =
-        if (i == j) s[i][j] += HighLow{lambda_} * trace;
+            strain[i][j] - (i == j ? trace / HighLow{3} : HighLow{});
-        density_energy += HighLow{mu_} * strain[i][j] * strain[i][j];
+        s[i][j] = twice_mu * dev;
+        if (i == j) s[i][j] += HighLow{kappa_} * mean_trace;
+        density_energy += HighLow{mu_} * dev * dev;
       }
     energy += density_energy * HighLow{point.weight};
     for (std::size_t i = 0; i < 3; ++i)
@@ -144,8 +182,8 @@ Result<Hex8HighLowResponse> Hex8Element::EvaluateHighLow(
           stress[i][j] = stress[i][j] / det;
         }
     }
-    if (!Finite(h) || !Finite(f) || !Finite(strain) || !Finite(s) ||
+    if (!Finite(f) || !Finite(strain) || !Finite(s) || !Finite(p) ||
-        !Finite(p) || !Finite(fft) || !Finite(stress))
+        !Finite(fft) || !Finite(stress))
       return Result<Hex8HighLowResponse>::Failure(Failure(
           "Hex8 high+low tensor arithmetic is outside its supported range."));
     response.high.points[g] = {element_index_, static_cast<int>(g + 1U),
@@ -160,6 +198,9 @@ Result<Hex8HighLowResponse> Hex8Element::EvaluateHighLow(
     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[a]);
+    for (std::size_t a = 0; a < 8; ++a)
+      for (std::size_t i = 0; i < 3; ++i)
+        integrated_fg[a][i] += fg[a][i] * HighLow{point.weight};
     for (std::size_t a = 0; a < 8; ++a)
       for (std::size_t b = 0; b < 8; ++b) {
         const auto dot = Dot(gradients[a], gradients[b]);
@@ -172,7 +213,7 @@ Result<Hex8HighLowResponse> Hex8Element::EvaluateHighLow(
           for (std::size_t k = 0; k < 3; ++k) {
             const auto row = 3U * a + i;
             const auto column = 3U * b + k;
-            const auto mat = HighLow{lambda_} * fg[a][i] * fg[b][k] +
+            const auto mat = deviatoric_coefficient * fg[a][i] * fg[b][k] +
                              HighLow{mu_} * fg[b][i] * fg[a][k] +
                              HighLow{mu_} * fft[i][k] * dot;
             material[row][column] += mat * HighLow{point.weight};
@@ -181,6 +222,13 @@ Result<Hex8HighLowResponse> Hex8Element::EvaluateHighLow(
           }
       }
   }
+  if (nonlinear || !paired_linear_tangent_ready_)
+    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)
+            material[3 * a + i][3 * b + k] +=
+                bulk_coefficient * integrated_fg[a][i] * integrated_fg[b][k];
   if (!IsNormalized(energy))
     return Result<Hex8HighLowResponse>::Failure(
         Failure("Hex8 high+low energy is unsupported or nonfinite."));

tests/unit/analysis/solid_frequency_procedure_test.cpp

@@ -99,8 +99,11 @@ std::vector<double> DenseOracle(const std::vector<std::size_t>& free) {
       for (std::size_t d = 0; d < 3; ++d)
         gradient_dot += GradientIntegral(a, b, d, d);
       k[row * n + col] =
-          48.0 * (GradientIntegral(a, b, i, j) + GradientIntegral(a, b, j, i) +
+          48.0 *
-                  (i == j ? gradient_dot : 0.0));
+              (-(2.0 / 3.0) * GradientIntegral(a, b, i, j) +
+               GradientIntegral(a, b, j, i) + (i == j ? gradient_dot : 0.0)) +
+          (80.0 / 48.0) * (48.0 * kSigns[a][i] / (8.0 * (i + 1))) *
+              (48.0 * kSigns[b][j] / (8.0 * (j + 1)));
       if (i == j) {
         m[row * n + col] = 2.0 * 48.0 / 64.0;
         for (std::size_t d = 0; d < 3; ++d)

tests/unit/analysis/solid_hht_dynamic_procedure_test.cpp

@@ -573,7 +573,7 @@ TEST(SolidHhtDynamics,
   auto result = RunDynamics(OneFree());
   ASSERT_TRUE(result.HasValue()) << solid_hht_test::Message(result.GetStatus());
   const auto d = solid_hht_test::Brick();
-  constexpr double lambda = 4.8;
+  constexpr double bulk = 8.0;
   constexpr double mu = 4.8;
   const double roundoff = 512 * std::numeric_limits<double>::epsilon();
   const auto& frames = result.Value().state.AcceptedFrames();
@@ -592,17 +592,20 @@ TEST(SolidHhtDynamics,
         double stiffness;
         double stiffness_term_scale;
         if (c == 0) {
-          stiffness = (lambda + 2 * mu) * (-sign[0]) * product[1] * product[2] +
+          stiffness = (4 * mu / 3) * (-sign[0]) * product[1] * product[2] +
                       mu * (-sign[1]) * product[0] * product[2] +
                       mu * (-sign[2]) * product[0] * product[1];
-          stiffness_term_scale = (lambda + 2 * mu) * product[1] * product[2] +
+          stiffness_term_scale = (4 * mu / 3) * product[1] * product[2] +
                                  mu * product[0] * product[2] +
                                  mu * product[0] * product[1];
         } else {
           const unsigned other = c == 1 ? 2 : 1;
-          stiffness = -.25 * (lambda * sign[c] + mu * sign[0]) * product[other];
+          stiffness =
-          stiffness_term_scale = .25 * (lambda + mu) * product[other];
+              -.25 * ((-2 * mu / 3) * sign[c] + mu * sign[0]) * product[other];
+          stiffness_term_scale = .25 * (2 * mu / 3 + mu) * product[other];
         }
+        stiffness -= bulk * sign[c] / 16;
+        stiffness_term_scale += bulk / 16;
         const double inertia = c == 0 ? scalar_mass * f.acceleration[0] : 0;
         const double internal = stiffness * f.displacement[0];
         const double applied = b == 0 && c == 0 ? .01 : 0;

tests/unit/analysis/solid_hht_temporal_verification_test.cpp

@@ -24,7 +24,7 @@ AffineMotion Motion(double time, bool nonlinear) {
   const double dq = 2 * std::sin(2 * time);
   const double ddq = 4 * std::cos(2 * time);
   AffineMotion m;
-  constexpr double lambda = 4.8;
+  constexpr double bulk = 8;
   constexpr double mu = 4.8;
   if (!nonlinear) {
     const M3 d{{{.015, .008, -.004}, {.006, -.01, .003}, {.005, .004, .012}}};
@@ -35,8 +35,8 @@ AffineMotion Motion(double time, bool nonlinear) {
         m.b[i][j] = q * d[i][j];
         m.velocity[i][j] = dq * d[i][j];
         m.acceleration[i][j] = ddq * d[i][j];
-        m.piola[i][j] =
+        m.piola[i][j] = (i == j ? (bulk * trace - 2 * mu * trace / 3) : 0) +
-            (i == j ? lambda * trace : 0) + mu * q * (d[i][j] + d[j][i]);
+                        mu * q * (d[i][j] + d[j][i]);
       }
   } else {
     const double s = .004 * q, ds = .004 * dq, dds = .004 * ddq;
@@ -46,7 +46,7 @@ AffineMotion Motion(double time, bool nonlinear) {
     const M3 jr{{{-sn, -c, 0}, {c, -sn, 0}, {0, 0, 0}}};
     const M3 jjr{{{-c, sn, 0}, {-sn, -c, 0}, {0, 0, 0}}};
     const double eg = .5 * ((1 + s) * (1 + s) - 1);
-    const double stress = (3 * lambda + 2 * mu) * eg;
+    const double stress = 3 * bulk * eg;
     for (unsigned i = 0; i < 3; ++i)
       for (unsigned j = 0; j < 3; ++j) {
         m.b[i][j] = (1 + s) * r[i][j] - (i == j ? 1 : 0);
@@ -506,7 +506,7 @@ TEST(SolidHhtTemporal,
 
 TEST(SolidHhtTemporal, IndependentThreeDofBrickOscillatorHasSecondOrderUVA) {
   // Node0's three translations are free. Exact polynomial integrals yield
-  // M=(rho/27)I, Kii=(lambda+4mu)/9, Kij=(lambda+mu)/12.
+  // M=(rho/27)I, Kii=41/18, Kij=19/30 for mu=4.8 and direct kappa=8.
   std::array<std::array<double, 3>, 3> errors{};
   for (unsigned level = 0; level < 3; ++level) {
     const double h = .02 / static_cast<double>(1U << level);
@@ -529,8 +529,8 @@ TEST(SolidHhtTemporal, IndependentThreeDofBrickOscillatorHasSecondOrderUVA) {
         << solid_hht_test::Message(result.GetStatus());
     const auto& f = result.Value().state.CommittedState();
     const double m = 2.0 / 27;
-    const double diagonal = (4.8 + 4 * 4.8) / 9;
+    const double diagonal = 41.0 / 18;
-    const double off = (4.8 + 4.8) / 12;
+    const double off = 19.0 / 30;
     const std::array<double, 2> k{diagonal + 2 * off, diagonal - off};
     for (unsigned c = 0; c < 3; ++c) {
       std::array<double, 3> truth{};
@@ -563,7 +563,7 @@ TEST(SolidHhtTemporal, IndependentThreeDofBrickOscillatorHasSecondOrderUVA) {
 TEST(SolidHhtTemporal, IndependentScalarBrickOscillatorHasSecondOrderUVA) {
   std::array<std::array<double, 3>, 3> errors{};
   const double m = 2.0 / 27;
-  const double k = (4.8 + 4 * 4.8) / 9;
+  const double k = 41.0 / 18;
   const double w = std::sqrt(k / m);
   for (unsigned level = 0; level < 3; ++level) {
     const double h = .02 / static_cast<double>(1U << level);
@@ -608,46 +608,95 @@ TEST(SolidHhtTemporal, IndependentScalarBrickOscillatorHasSecondOrderUVA) {
   }
 }
 
-TEST(SolidHhtTemporal,
+namespace {
-     BendingMeshRefinementQuantifiesFullIntegrationShearLocking) {
+// Compatible nodal pure-bending field; no equilibrium solve is claimed.
-  constexpr double thickness = .125;
+std::array<double, 3> BendingEnergy(unsigned nx, unsigned ny) {
-  constexpr double curvature = .01;
+  constexpr double thickness = .125, curvature = .01, bulk = 8;
-  constexpr double lambda = 4.8;
+  std::array<double, 3> result{};
-  constexpr double mu = 4.8;
+  auto d = solid_hht_test::Brick();
-  const double continuum = .5 * (lambda + 2 * mu) * curvature * curvature *
+  for (unsigned ex = 0; ex < nx; ++ex)
-                           thickness * thickness * thickness / 12;
+    for (unsigned ey = 0; ey < ny; ++ey) {
-  double previous_error = 0;
+      const auto index = ex * ny + ey;
-  for (unsigned nx : {1U, 2U, 4U, 8U}) {
-    double numerical = 0;
-    for (unsigned e = 0; e < nx; ++e) {
-      auto d = solid_hht_test::Brick();
       fesa::Hex8Element::Coordinates x{};
       fesa::ElementDofLayout layout{
-          {"I", e + 1, std::to_string(e + 1)},
+          {"I", index + 1, std::to_string(index + 1)},
           {0, 1, 2, 3, 4, 5, 6, 7},
           {fesa::DofComponent::kUx, fesa::DofComponent::kUy,
            fesa::DofComponent::kUz}};
       fesa::Vector u(24);
       for (unsigned a = 0; a < 8; ++a) {
-        x[a] = {static_cast<double>(e) / nx + d.nodes[a].coordinates[0] / nx,
+        x[a] = {(ex + d.nodes[a].coordinates[0]) / nx,
-                thickness * (d.nodes[a].coordinates[1] - .5),
+                thickness * ((ey + d.nodes[a].coordinates[1]) / ny - .5),
                 d.nodes[a].coordinates[2]};
         u[3 * a] = -curvature * x[a][0] * x[a][1];
         u[3 * a + 1] = .5 * curvature * x[a][0] * x[a][0];
       }
-      auto element = fesa::Hex8Element::Create(e, layout, x, d.materials[0], 2);
+      auto element =
-      ASSERT_TRUE(element.HasValue());
+          fesa::Hex8Element::Create(index, layout, x, d.materials[0], 2);
+      EXPECT_TRUE(element.HasValue());
+      if (!element.HasValue()) return {};
       auto response = element.Value().EvaluateLinear(u);
-      ASSERT_TRUE(response.HasValue());
+      EXPECT_TRUE(response.HasValue());
-      numerical += response.Value().strain_energy;
+      if (!response.HasValue()) return {};
+      const auto weights = element.Value().ReferenceWeights();
+      double volume = 0, trace_integral = 0;
+      for (unsigned g = 0; g < 8; ++g) {
+        volume += weights[g];
+        const auto& strain = response.Value().points[g].strain;
+        trace_integral += weights[g] * (strain[0] + strain[1] + strain[2]);
+      }
+      const double volumetric =
+          .5 * bulk * trace_integral * trace_integral / volume;
+      result[0] += response.Value().strain_energy;
+      result[1] += response.Value().strain_energy - volumetric;
+      result[2] += volumetric;
     }
-    const double error = numerical / continuum - 1;
+  return result;
-    const double exact_parasitic =
+}
-        mu / (lambda + 2 * mu) / std::pow(nx * thickness, 2);
+}  // namespace
-    EXPECT_NEAR(error, exact_parasitic, 2e-11);
+
-    if (previous_error != 0) EXPECT_NEAR(previous_error / error, 4, 1e-10);
+TEST(SolidHhtTemporal, FixedThicknessBendingRetainsDiscreteBulkBias) {
-    std::cout << "Bending nx=" << nx
+  constexpr double thickness = .125, curvature = .01, mu = 4.8, bulk = 8;
-              << " energy/continuum=" << numerical / continuum << '\n';
+  constexpr double axial = bulk + 4 * mu / 3;
-    previous_error = error;
+  const double continuum =
+      axial * curvature * curvature * std::pow(thickness, 3) / 24;
+  const double limit = 4 * mu / (3 * axial);
+  double previous_parasitic = 0;
+  for (unsigned nx : {1U, 2U, 4U, 8U}) {
+    const auto energy = BendingEnergy(nx, 1);
+    const double parasitic = mu / (axial * std::pow(nx * thickness, 2));
+    EXPECT_NEAR(energy[0] / continuum, limit + parasitic, 2e-11);
+    EXPECT_NEAR(energy[2] / continuum, 0, 2e-13);
+    if (previous_parasitic != 0)
+      EXPECT_NEAR(previous_parasitic / parasitic, 4, 1e-13);
+    previous_parasitic = parasitic;
+    std::cout << "Fixed thickness nx=" << nx
+              << " U/Ucont=" << energy[0] / continuum
+              << " limiting ratio=" << limit << " shear=" << parasitic << '\n';
+  }
+}
+
+TEST(SolidHhtTemporal,
+     BendingMeshRefinementQuantifiesFullIntegrationShearLocking) {
+  constexpr double thickness = .125, curvature = .01, mu = 4.8, bulk = 8;
+  constexpr double axial = bulk + 4 * mu / 3;
+  const double continuum =
+      axial * curvature * curvature * std::pow(thickness, 3) / 24;
+  double previous_error = 0;
+  for (unsigned n : {1U, 2U, 4U, 8U}) {
+    const auto energy = BendingEnergy(n, n);
+    const double shear = mu / (axial * std::pow(n * thickness, 2));
+    const double missing_bulk = bulk / (axial * n * n);
+    const double error = energy[0] / continuum - 1;
+    EXPECT_NEAR(error, shear - missing_bulk, 2e-11);
+    // Independent moments separate competing errors, so cancellation cannot
+    // make the continuum-convergence claim pass accidentally.
+    EXPECT_NEAR(energy[1] / continuum, 4 * mu / (3 * axial) + shear, 2e-11);
+    EXPECT_NEAR(energy[2] / continuum, bulk / axial - missing_bulk, 2e-11);
+    if (previous_error != 0) EXPECT_NEAR(previous_error / error, 4, 1e-10);
+    previous_error = error;
+    std::cout << "Bending nx=ny=" << n << " U/Ucont=" << energy[0] / continuum
+              << " positive shear=" << shear
+              << " negative bulk=" << missing_bulk << '\n';
   }
 }

tests/unit/elements/hex8_element_test.cpp

@@ -11,6 +11,8 @@
 #include <string>
 #include <vector>
 
+#include "../../../src/fesa/math/high_low_arithmetic.h"
+
 namespace {
 using Coordinates = fesa::Hex8Element::Coordinates;
 constexpr int kSigns[8][3] = {{-1, -1, -1}, {1, -1, -1}, {1, 1, -1},
@@ -64,11 +66,14 @@ TEST(Hex8Element, ExactBrickStiffnessAndConsistentMass) {
         scalar *= kSigns[a][d] == kSigns[b][d] ? 2.0 : 1.0;
       for (std::size_t i = 0; i < 3; ++i)
         for (std::size_t j = 0; j < 3; ++j) {
-          double exact = 48.0 * GradientProduct(a, b, i, j) +
+          double exact = -(2.0 / 3.0) * 48.0 * GradientProduct(a, b, i, j) +
                          48.0 * GradientProduct(a, b, j, i);
           if (i == j)
             for (std::size_t d = 0; d < 3; ++d)
               exact += 48.0 * GradientProduct(a, b, d, d);
+          // Mean bulk term is the product of two independent first moments.
+          exact += (80.0 / 48.0) * (48.0 * kSigns[a][i] / (8.0 * (i + 1))) *
+                   (48.0 * kSigns[b][j] / (8.0 * (j + 1)));
           EXPECT_NEAR(k(3 * a + i, 3 * b + j), exact, 2e-12);
           EXPECT_NEAR(m(3 * a + i, 3 * b + j), i == j ? scalar : 0.0, 2e-14);
         }
@@ -542,8 +547,9 @@ TEST(Hex8Element, EnergyForceAndForceTangentDirectionalStepSweep) {
   ASSERT_TRUE(trial.HasValue());
   const double derivative = direction.Dot(trial.Value().internal_force);
   const auto tangent_direction = trial.Value().tangent.Multiply(direction);
-  std::array<double, 6> energy_error{}, force_error{};
+  std::array<double, 8> energy_error{}, force_error{};
-  const std::array<double, 6> steps{1e-2, 1e-3, 1e-4, 1e-5, 1e-6, 1e-7};
+  const std::array<double, 8> steps{1e-2, 1e-3, 1e-4, 1e-5,
+                                    1e-6, 1e-7, 1e-8, 1e-9};
   for (std::size_t s = 0; s < steps.size(); ++s) {
     auto plus = u, minus = u;
     plus.Axpy(steps[s], direction);
@@ -751,14 +757,14 @@ TEST(Hex8Element, RejectsUnrepresentableShearAndMassInsteadOfReturningZeros) {
   ASSERT_TRUE(element.HasValue());
   EXPECT_FALSE(element.Value().ComputeMass().HasValue());
 }
-TEST(Hex8Element, RejectsRoundedLossOfPositiveVolumetricModulus) {
+TEST(Hex8Element, DirectBulkSurvivesNearNegativePoissonBound) {
-  // Binary64 Lame arithmetic at this valid nu yields lambda+2mu/3 == 0
-  // even though exact E/[3(1-2nu)] is approximately1/9, which is positive.
   const double nu = std::nextafter(-1.0, 0.0);
   auto element = Make(Brick(), 1.0, nu);
-  ASSERT_FALSE(element.HasValue());
+  ASSERT_TRUE(element.HasValue());
-  EXPECT_NE(element.GetStatus().Diagnostics()[0].message.find("bulk"),
+  // Direct kappa is positive even where a lambda-based difference loses it.
-            std::string::npos);
+  EXPECT_GT(1.0 / (3.0 * (1.0 - 2.0 * nu)), 0.0);
+  fesa::Vector zero(24);
+  EXPECT_TRUE(element.Value().EvaluateLinear(zero).HasValue());
 }
 TEST(Hex8Element, OrdinaryNegativePoissonRatioRetainsPositiveVolumetricEnergy) {
   auto element = Make(Brick(), 120.0, -.5);
@@ -838,9 +844,11 @@ Triple Cross(const Triple& a, const Triple& b) {
           a[0] * b[1] - a[1] * b[0]};
 }
 std::array<double, 2> WarpedQuadratureEnergies(const fesa::Vector& u,
-                                               unsigned count) {
+                                               unsigned count,
+                                               bool nonlinear = false) {
   const auto rule = LegendreRule(count);
   std::array<double, 2> result{};
+  double volume = 0.0, trace_integral = 0.0;
   for (const auto& xi : rule)
     for (const auto& eta : rule)
       for (const auto& zeta : rule) {
@@ -877,19 +885,30 @@ std::array<double, 2> WarpedQuadratureEnergies(const fesa::Vector& u,
           for (std::size_t j = 0; j < 3; ++j)
             for (std::size_t d = 0; d < 3; ++d)
               gradient[i][j] += natural_gradient[i][d] * reciprocal[d][j];
+        Tensor strains{};
+        for (std::size_t i = 0; i < 3; ++i)
+          for (std::size_t j = 0; j < 3; ++j) {
+            strains[i][j] = .5 * (gradient[i][j] + gradient[j][i]);
+            if (nonlinear)
+              for (std::size_t k = 0; k < 3; ++k)
+                strains[i][j] += .5 * gradient[k][i] * gradient[k][j];
+          }
         double trace = 0.0, strain_squared = 0.0, velocity_squared = 0.0;
         for (std::size_t i = 0; i < 3; ++i) {
-          trace += gradient[i][i];
+          trace += strains[i][i];
           velocity_squared += velocity[i] * velocity[i];
           for (std::size_t j = 0; j < 3; ++j) {
-            const double strain = .5 * (gradient[i][j] + gradient[j][i]);
+            const double strain = strains[i][j];
             strain_squared += strain * strain;
           }
         }
         const double weight = xi[1] * eta[1] * zeta[1] * det;
-        result[0] += weight * (24.0 * trace * trace + 48.0 * strain_squared);
+        result[0] += weight * 48.0 * (strain_squared - trace * trace / 3.0);
+        volume += weight;
+        trace_integral += weight * trace;
         result[1] += weight * velocity_squared;  // .5*rho*v^2 with rho=2.
       }
+  result[0] += 40.0 * trace_integral * trace_integral / volume;
   return result;
 }
 TEST(Hex8Element, WarpedNonaffineQuadratureDifferenceIsQuantifiedAndRefines) {
@@ -922,4 +941,199 @@ TEST(Hex8Element, WarpedNonaffineQuadratureDifferenceIsQuantifiedAndRefines) {
             << " K3-vs-K7=" << std::abs(q3[0] - q7[0]) / q7[0]
             << " K5-vs-K7=" << std::abs(q5[0] - q7[0]) / q7[0] << '\n';
 }
+Coordinates MeanVolumeUnitBrick() {
+  Coordinates x{};
+  for (std::size_t a = 0; a < 8; ++a)
+    for (std::size_t i = 0; i < 3; ++i) x[a][i] = .5 * (kSigns[a][i] + 1);
+  return x;
+}
+
+TEST(Hex8Element,
+     MeanVolumeNonaffinePolynomialEnergyForceHessianAndActualStrain) {
+  const auto x = MeanVolumeUnitBrick();
+  const auto element = Make(x, 12.0, .25);
+  ASSERT_TRUE(element.HasValue());
+  constexpr double mu = 4.8, bulk = 8.0, c = .125;
+  // Exact monomial integrals of u_x=c*x*y over the unit cube. The other
+  // translations vanish; this direction varies trace among all eight points.
+  constexpr double a = 7 * mu / 18 + bulk / 8;
+  constexpr double b = 5 * mu / 18 + bulk / 6;
+  // Fixed two-point quadrature has Q[x^4]=7/36, not the continuum 1/5.
+  constexpr double d = 11 * mu / 108 + bulk / 18;
+  fesa::Vector u(24);
+  for (std::size_t n = 0; n < 8; ++n) u[3 * n] = c * x[n][0] * x[n][1];
+  for (bool nonlinear : {false, true}) {
+    const auto response = nonlinear ? element.Value().EvaluateNonlinear(u)
+                                    : element.Value().EvaluateLinear(u);
+    ASSERT_TRUE(response.HasValue());
+    const auto& r = response.Value();
+    const double energy =
+        a * c * c + (nonlinear ? b * c * c * c + d * c * c * c * c : 0);
+    const double force =
+        2 * a * c + (nonlinear ? 3 * b * c * c + 4 * d * c * c * c : 0);
+    const double hessian = 2 * a + (nonlinear ? 6 * b * c + 12 * d * c * c : 0);
+    EXPECT_NEAR(r.strain_energy, energy, 2e-14);
+    EXPECT_NEAR(r.internal_force[6] + r.internal_force[18], force, 2e-13);
+    EXPECT_NEAR(r.tangent(6, 6) + r.tangent(6, 18) + r.tangent(18, 6) +
+                    r.tangent(18, 18),
+                hessian, 2e-12);
+    const double mean = c / 2 + (nonlinear ? c * c / 3 : 0);
+    for (const auto& point : r.points) {
+      const double px = point.reference_coordinates[0],
+                   py = point.reference_coordinates[1];
+      const double t = 1 + c * py;
+      const double exx = c * py + (nonlinear ? .5 * c * c * py * py : 0);
+      const double eyy = nonlinear ? .5 * c * c * px * px : 0;
+      const double exy = .5 * c * px + (nonlinear ? .5 * c * c * px * py : 0);
+      const double trace = exx + eyy;
+      const double sxx = 2 * mu * (exx - trace / 3) + bulk * mean;
+      const double syy = 2 * mu * (eyy - trace / 3) + bulk * mean;
+      const double szz = -2 * mu * trace / 3 + bulk * mean;
+      const double sxy = 2 * mu * exy;
+      EXPECT_NEAR(point.strain[0], exx, 2e-15);
+      EXPECT_NEAR(point.strain[1], eyy, 2e-15);
+      EXPECT_NEAR(point.strain[3], exy, 2e-15);
+      EXPECT_NEAR(
+          point.cauchy_stress[0],
+          nonlinear
+              ? (t * t * sxx + 2 * t * c * px * sxy + c * c * px * px * syy) / t
+              : sxx,
+          2e-13);
+      EXPECT_NEAR(point.cauchy_stress[1], nonlinear ? syy / t : syy, 2e-13);
+      EXPECT_NEAR(point.cauchy_stress[2], nonlinear ? szz / t : szz, 2e-13);
+      EXPECT_NEAR(point.cauchy_stress[3],
+                  nonlinear ? (t * sxy + c * px * syy) / t : sxy, 2e-13);
+    }
+  }
+  const auto k = element.Value().ComputeStiffness().Value().values;
+  EXPECT_NEAR(k(0, 0), 41.0 / 18, 2e-14);
+  EXPECT_NEAR(k(0, 1), 19.0 / 30, 2e-14);
+}
+
+TEST(Hex8Element, MeanVolumeWarpedStressUsesReferenceWeightedTrace) {
+  auto element = Make(Warped());
+  ASSERT_TRUE(element.HasValue());
+  fesa::Vector u(24);
+  u[1] = 1;  // Independent derivative coefficients separate the two means by
+             // >1e-8.
+  const auto response = element.Value().EvaluateLinear(u);
+  ASSERT_TRUE(response.HasValue());
+  const auto weights = element.Value().ReferenceWeights();
+  double volume = 0, integral = 0, arithmetic = 0;
+  for (std::size_t g = 0; g < 8; ++g) {
+    const auto& e = response.Value().points[g].strain;
+    const double trace = e[0] + e[1] + e[2];
+    volume += weights[g];
+    integral += weights[g] * trace;
+    arithmetic += trace / 8;
+  }
+  const double mean = integral / volume;
+  ASSERT_GT(std::abs(mean - arithmetic), 1e-8);
+  for (const auto& point : response.Value().points) {
+    const auto& s = point.cauchy_stress;
+    EXPECT_NEAR((s[0] + s[1] + s[2]) / 3, 80 * mean, 2e-13);
+  }
+  EXPECT_NEAR(response.Value().strain_energy, WarpedQuadratureEnergies(u, 2)[0],
+              2e-12);
+}
+
+TEST(Hex8Element,
+     MeanVolumeDirectBulkSupportsBoundsWithoutLambdaOrThreeBulkGuard) {
+  EXPECT_TRUE(
+      Make(MeanVolumeUnitBrick(), 12, std::nextafter(-1.0, 0.0)).HasValue());
+  EXPECT_TRUE(
+      Make(MeanVolumeUnitBrick(), 1, std::nextafter(.5, 0.0)).HasValue());
+  const auto huge = Make(MeanVolumeUnitBrick(), 1e308, .4);
+  ASSERT_TRUE(huge.HasValue());
+  const double direct_bulk = 1e308 / (3 * (1 - 2 * .4));
+  ASSERT_TRUE(std::isfinite(direct_bulk));
+  ASSERT_FALSE(std::isfinite(3 * direct_bulk));
+  fesa::Vector u(24);
+  constexpr double shear = 1e-150;
+  const auto x = MeanVolumeUnitBrick();
+  for (unsigned n = 0; n < 8; ++n) u[3 * n] = shear * x[n][1];
+  const auto response = huge.Value().EvaluateLinear(u);
+  ASSERT_TRUE(response.HasValue());
+  const double expected = .5 * (1e308 / (2 * (1 + .4))) * shear * shear;
+  EXPECT_NEAR(response.Value().strain_energy, expected, 1e-13 * expected);
+  EXPECT_FALSE(Make(MeanVolumeUnitBrick(), 1e308, .499).HasValue());
+  EXPECT_FALSE(Make(MeanVolumeUnitBrick(),
+                    std::numeric_limits<double>::denorm_min(), -.5)
+                   .HasValue());
+}
+TEST(Hex8Element,
+     MeanVolumeWarpedIndependentEnergyHasCoupledNodeZeroDerivatives) {
+  auto element = Make(Warped());
+  ASSERT_TRUE(element.HasValue());
+  fesa::Vector u(24), direction(24), zero(24);
+  for (std::size_t i = 0; i < 24; ++i) u[i] = .03 * std::sin(.7 * i);
+  direction[0] = 1;
+  direction[8] = -.375;
+  direction[22] = .25;
+  constexpr double h = .01;
+  for (bool nonlinear : {false, true}) {
+    const auto response = nonlinear ? element.Value().EvaluateNonlinear(u)
+                                    : element.Value().EvaluateLinear(u);
+    const auto paired = nonlinear
+                            ? element.Value().EvaluateNonlinearHighLow(u, zero)
+                            : element.Value().EvaluateLinearHighLow(u, zero);
+    ASSERT_TRUE(response.HasValue());
+    ASSERT_TRUE(paired.HasValue());
+    const double energy = WarpedQuadratureEnergies(u, 2, nonlinear)[0];
+    EXPECT_NEAR(response.Value().strain_energy, energy, 2e-12);
+    EXPECT_NEAR(paired.Value().high.strain_energy, energy, 2e-12);
+    std::array<double, 2> first{}, second{};
+    for (unsigned level = 0; level < 2; ++level) {
+      const double step = h / (1U << level);
+      auto plus = u, minus = u;
+      plus.Axpy(step, direction);
+      minus.Axpy(-step, direction);
+      const double ep = WarpedQuadratureEnergies(plus, 2, nonlinear)[0];
+      const double em = WarpedQuadratureEnergies(minus, 2, nonlinear)[0];
+      first[level] = (ep - em) / (2 * step);
+      second[level] = (ep - 2 * energy + em) / (step * step);
+    }
+    // This energy is quartic in nodal translations; Richardson removes the
+    // exact centered-difference h² term, independent of either kernel path.
+    const double df = (4 * first[1] - first[0]) / 3;
+    const double ddf = (4 * second[1] - second[0]) / 3;
+    EXPECT_NEAR(direction.Dot(response.Value().internal_force), df, 2e-10);
+    EXPECT_NEAR(direction.Dot(paired.Value().high.internal_force), df, 2e-10);
+    EXPECT_NEAR(direction.Dot(response.Value().tangent.Multiply(direction)),
+                ddf, 2e-8);
+    EXPECT_NEAR(direction.Dot(paired.Value().high.tangent.Multiply(direction)),
+                ddf, 2e-8);
+  }
+}
+
+TEST(Hex8Element, MeanVolumeWarpedLowOnlyStateRetainsWeightedCubicEnergy) {
+  using namespace fesa::math_internal;
+  auto element = Make(Warped());
+  ASSERT_TRUE(element.HasValue());
+  const double c = std::ldexp(1.0, -60);
+  fesa::Vector high(24), low(24), zero(24), basis(24), minus(24);
+  for (unsigned a = 0; a < 8; ++a)
+    for (unsigned i = 0; i < 3; ++i) high[3 * a + i] = 1;
+  low[1] = c;
+  basis[1] = 1;
+  minus[1] = -1;
+  const auto linear = element.Value().EvaluateLinearHighLow(high, low);
+  const auto nonlinear = element.Value().EvaluateNonlinearHighLow(high, low);
+  const auto high_only = element.Value().EvaluateNonlinearHighLow(high, zero);
+  ASSERT_TRUE(linear.HasValue());
+  ASSERT_TRUE(nonlinear.HasValue());
+  ASSERT_TRUE(high_only.HasValue());
+  EXPECT_EQ(high_only.Value().high.strain_energy, 0);
+  const double quadratic = WarpedQuadratureEnergies(basis, 2)[0];
+  const double cubic = .5 * (WarpedQuadratureEnergies(basis, 2, true)[0] -
+                             WarpedQuadratureEnergies(minus, 2, true)[0]);
+  EXPECT_NEAR(linear.Value().high.strain_energy, quadratic * c * c,
+              1e-12 * c * c);
+  const auto actual_cubic = Subtract(
+      {nonlinear.Value().high.strain_energy,
+       nonlinear.Value().strain_energy_low},
+      {linear.Value().high.strain_energy, linear.Value().strain_energy_low});
+  EXPECT_NEAR(actual_cubic.high, cubic * c * c * c, 1e-11 * c * c * c);
+  EXPECT_NE(actual_cubic.high, 0);
+}
 }  // namespace

tests/unit/elements/hex8_high_low_test.cpp

@@ -295,4 +295,40 @@ TEST(Hex8HighLow, FiniteRotationRetainsTinyShearForceCovariance) {
     EXPECT_LE(std::abs(Subtract(rotated[2], baseline[2]).high), 1e-13*gamma);
   }
 }
+TEST(Hex8HighLow,
+     MeanVolumeLowOnlyNonuniformTraceRetainsMeanAndCrossPointHessian) {
+  auto brick = UnitBrick(0, .25);
+  fesa::Vector high(24), low(24), zero(24);
+  const double c = std::ldexp(1.0, -60);
+  for (std::size_t n = 0; n < 8; ++n) high[3 * n] = 1;
+  low[6] = c;
+  low[18] = c;
+  const double coefficient = 7 * 4.8 / 18 + 8.0 / 8;
+  const auto response = brick.EvaluateLinearHighLow(high, low);
+  const auto high_only = brick.EvaluateLinearHighLow(high, zero);
+  ASSERT_TRUE(response.HasValue());
+  ASSERT_TRUE(high_only.HasValue());
+  EXPECT_EQ(high_only.Value().high.strain_energy, 0);
+  EXPECT_NEAR(response.Value().high.strain_energy, coefficient * c * c,
+              1e-13 * c * c);
+  EXPECT_NEAR(response.Value().high.internal_force[6] +
+                  response.Value().high.internal_force[18],
+              2 * coefficient * c, 1e-13 * c);
+  for (const auto& point : response.Value().high.points) {
+    const auto& s = point.cauchy_stress;
+    EXPECT_NEAR((s[0] + s[1] + s[2]) / 3, 4 * c, 1e-13 * c);
+    EXPECT_NEAR(point.strain[0], c * point.reference_coordinates[1], 1e-13 * c);
+  }
+  const auto& k = response.Value().high.tangent;
+  EXPECT_NEAR(k(0, 0), 41.0 / 18, 2e-14);
+  EXPECT_NEAR(k(0, 1), 19.0 / 30, 2e-14);
+  const auto nonlinear = brick.EvaluateNonlinearHighLow(high, low);
+  ASSERT_TRUE(nonlinear.HasValue());
+  const auto cubic_energy = Subtract({nonlinear.Value().high.strain_energy,
+                                      nonlinear.Value().strain_energy_low},
+                                     {response.Value().high.strain_energy,
+                                      response.Value().strain_energy_low});
+  EXPECT_NEAR(cubic_energy.high, (5 * 4.8 / 18 + 8.0 / 6) * c * c * c,
+              1e-12 * c * c * c);
+}
 }  // namespace