8節点六面体要素(HEX8)
Theory and Physics
Characteristics of HEX8 Elements
Professor, how does HEX8 compare to TET10?
HEX8 (8-node hexahedral element) is the basic element for structured meshes. If TET10 is the star of automatic meshing, HEX8 is the star of manual meshing (mapped meshing).
Shape Functions
The shape function for HEX8 is expressed in natural coordinates $(\xi, \eta, \zeta)$ as a trilinear function:
$$ N_i = \frac{1}{8}(1 + \xi_i \xi)(1 + \eta_i \eta)(1 + \zeta_i \zeta) $$
The shape function for HEX8 is expressed in natural coordinates $(\xi, \eta, \zeta)$ as a trilinear function:
Here, $(\xi_i, \eta_i, \zeta_i)$ are the natural coordinates of node $i$ (combinations of $\pm 1$).
Trilinear means it's a first-order polynomial in each direction, right? So it's a first-order element like TET4?
This is an important difference. TET4 uses a complete first-order polynomial (4 terms: $1, x, y, z$), but HEX8 uses a trilinear function (8 terms: $1, \xi, \eta, \zeta, \xi\eta, \eta\zeta, \zeta\xi, \xi\eta\zeta$). That means it includes cross terms.
What's the difference when cross terms are present?
TET4 is a constant strain element, but HEX8 can partially represent linear strain. Thanks especially to the cross term $\xi\eta$, it can (imperfectly) represent bending deformation. This is something TET4 cannot do.
Advantages and Disadvantages of HEX8
| Characteristic | Advantages | Disadvantages |
|---|---|---|
| DOF Efficiency | Achieves comparable accuracy with fewer DOFs than TET10 | — |
| Mesh Generation | — | Requires manual (mapped) meshing |
| Bending Accuracy | Much better than TET4 | Shear locking with full integration |
| Incompressible Materials | Can be addressed with reduced integration | Volumetric locking with full integration |
| Contact Surfaces | Stable | — |
Is shear locking the same problem as with Q4 (2D quadrilateral)?
Exactly the same. When HEX8 is used with full integration (2×2×2 = 8 Gauss points), parasitic shear strain occurs during bending deformation, underestimating displacement. The standard approach to avoid shear locking is to use reduced integration (1×1×1 = 1 point).
Reduced Integration and Hourglass Modes
With 1-point integration, hourglass modes appear, right?
Yes. Reduced integration for HEX8 has 12 hourglass modes (zero-energy modes). Even if the element deforms in a zigzag, hourglass shape, the stress remains zero.
Countermeasures include hourglass control:
- Viscous Hourglass Control — For dynamic analysis. Suppresses using artificial viscosity.
- Stiffness Hourglass Control — For static analysis. Suppresses using artificial stiffness.
- Enhanced Assumed Strain (EAS) — Abaqus's C3D8I. Adds internal degrees of freedom to eliminate hourglassing.
Does the "I" in C3D8I stand for "Incompatible modes"?
Yes. C3D8I is an incompatible mode element that adds 13 internal degrees of freedom. It's an excellent element that solves both shear locking and hourglassing. It's more stable than reduced integration elements (C3D8R) and more accurate than full integration elements (C3D8).
When to Use HEX8
Why use HEX8 when TET10 exists?
There are three reasons:
1. DOF Efficiency — Requires 1/2 to 1/5 the number of DOFs for the same accuracy as TET10.
2. Contact Stability — Contact surfaces are more stable than with TET10.
3. Large Deformation Analysis — HEX elements are less prone to distortion under large deformation (TET elements collapse easily).
TET10 collapses easily under large deformation?
Tetrahedra have lower shape flexibility, so they are more prone to element degeneration (negative Jacobian) under large deformation. Hexahedra have more shape tolerance. HEX8 is preferred for large deformation problems like forging and metal forming.
Summary
Let me organize the theory of HEX8.
Key points:
- Trilinear Shape Functions — More accurate than TET4 (includes cross terms).
- Shear Locking with Full Integration — Counter with reduced integration (C3D8R) or EAS (C3D8I).
- Hourglass Modes with Reduced Integration — Hourglass control is required.
- C3D8I (Incompatible Modes) offers the best balance — No locking or hourglassing.
- Requires Manual Meshing — Automatic meshing for HEX8 is difficult.
- More advantageous than TET10 for Large Deformation and contact — Also has higher DOF efficiency.
So TET10 and HEX8 represent a trade-off between "convenience of automatic meshing" vs. "accuracy, efficiency, and stability".
Exactly. Choose based on project requirements (shape complexity, accuracy needs, computational budget). Engineers who can use both are the strongest.
Formulation of First-Order Hexahedral Elements
The 8-node hexahedral element, along with tetrahedral elements, was proposed in the 1956 paper "Stiffness and Deflection Analysis of Complex Structures" by Turner, Clough, Martin, and Topp in the Journal of Aeronautical Sciences, making it a historical element. Due to the simplicity of linear interpolation, it remains the mainstay of CAE practice today, accounting for over 80% of all elements in automotive crash analysis.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, meaning "mass × acceleration". Haven't you experienced your body being thrown forward during sudden braking? That "feeling of being pulled" is precisely the inertial force. Heavier objects are harder to set in motion and harder to stop once moving. Buildings shake during earthquakes because the ground moves suddenly while the building's mass "gets left behind". In static analysis, this term is set to zero, based on the assumption that "forces are applied slowly, so acceleration can be ignored". It absolutely cannot be omitted for impact loads or vibration problems.
- Stiffness Term (Elastic Restoring Force): $Ku$ or $\nabla \cdot \sigma$. When you stretch a spring, you feel a "force trying to return it", right? That's Hooke's law $F=kx$, the essence of the stiffness term. Now a question — an iron rod and a rubber band, which stretches more under the same force? Obviously the rubber band. This "resistance to stretching" is the Young's modulus $E$, which determines stiffness. A common misconception: "high stiffness ≠ strong". Stiffness is "resistance to deformation", strength is "resistance to failure" — they are different concepts.
- External Force Term (Load Term): Body forces $f_b$ (gravity, etc.) and surface forces $f_s$ (pressure, contact forces, etc.). Think of it this way — the weight of a truck on a bridge is a "force acting on the entire volume" (body force), while the force of the tires pushing on the road surface is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A common mistake here: getting the load direction wrong. Intending "tension" but ending up with "compression" — it sounds like a joke, but it actually happens when coordinate systems are rotated in 3D space.
- Damping Term: Rayleigh damping $C\dot{u} = (\alpha M + \beta K)\dot{u}$. Try plucking a guitar string. Does the sound continue forever? No, it gradually fades. That's because vibrational energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle — intentionally absorbing vibrational energy to improve ride comfort. What if damping were zero? Buildings would continue shaking forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is crucial.
Assumptions and Applicability Limits
- Continuum Assumption: Treats material as a continuous medium, ignoring microscopic heterogeneity.
- Small Deformation Assumption (for linear analysis): Deformation is sufficiently small compared to initial dimensions, and the stress-strain relationship is linear.
- Isotropic Material (unless specified otherwise): Material properties are independent of direction (anisotropic materials require separate tensor definitions).
- Quasi-Static Assumption (for static analysis): Ignores inertial and damping forces, considering only the balance between external and internal forces.
- Non-Applicable Cases: Large deformation/large rotation problems require geometric nonlinearity. Nonlinear material behavior like plasticity or creep requires constitutive law extensions.
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify loads and elastic modulus to MPa/N system. |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Be careful of unit system inconsistencies when comparing with yield stress. |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note the distinction between engineering strain and logarithmic strain (for large deformation). |
| Elastic Modulus $E$ | Pa | Steel: ~210 GPa, Aluminum: ~70 GPa. Note temperature dependence. |
| Density $\rho$ | kg/m³ | In mm system: tonne/mm³ (= 10⁻⁹ tonne/mm³ for steel). |
| Force $F$ | N (Newton) | Unify as N in mm system, N in m system. |
Numerical Methods and Implementation
Comparison of HEX8 Integration Schemes
Could you explain the differences in HEX8 integration schemes in more detail?
| Integration | Gauss Points | Shear Locking | Hourglass | Applications |
|---|---|---|---|---|
| Full Integration (2×2×2) | 8 | Present | None | Plane strain problems |
| Reduced Integration (1×1×1) | 1 | None | Present (12 modes) | Explicit impact analysis |
| Selective Reduced Integration | 8/1 mixed | None | None | Static analysis (some solvers) |
| Incompatible Modes (EAS) | 8 | None | None | Static analysis (recommended) |
What is selective reduced integration?
It evaluates the volumetric (dilatational) component with reduced integration (1 point) and the deviatoric (shear/bending) component with full integration (8 points). Also called the B-bar method. It avoids volumetric locking while preventing hourglassing.
Abaqus doesn't have an explicit B-bar HEX8, but its C3D8RH hybrid element has a similar effect. LS-DYNA's ELFORM=2 (selective reduced integration) is the standard for metal forming analysis.
Element Names by Solver
| Variation | Abaqus | Nastran | Ansys | LS-DYNA |
|---|---|---|---|---|
| Full Integration | C3D8 | CHEXA(8) | SOLID185(full) | ELFORM=2(sel.) |
| Reduced Integration | C3D8R | — | SOLID185(red.) | ELFORM=1 |
| Incompatible Modes | C3D8I | — | SOLID185(EAS) | — |
| Hybrid | C3D8H, C3D8RH | — | u-P formulation | — |
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