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

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