fix(elements): compensate cancelled Hex8 linear force products

src/fesa/elements/hex8_element.cpp

@@ -243,7 +243,22 @@ Result<Hex8Response> Hex8Element::Evaluate(const Vector& u,
     for (std::size_t a = 0; a < 8; ++a)
       for (std::size_t i = 0; i < 3; ++i)
         relative[3U * a + i] = u[3U * a + i] - u[i];
-    response.internal_force = stiffness_.Multiply(relative);
+    // 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) {
+        const double product = stiffness_(i, j) * relative[j];
+        const double next = sum + product;
+        correction += std::fma(stiffness_(i, j), relative[j], -product) +
+                      (std::abs(sum) >= std::abs(product)
+                           ? (sum - next) + product
+                           : (product - next) + sum);
+        sum = next;
+      }
+      response.internal_force[i] = sum + correction;
+    }
     response.strain_energy = 0.5 * relative.Dot(response.internal_force);
   }
   for (std::size_t g = 0; g < 8; ++g) {