8-Node Hexahedral Element (HEX8)
8-Node Hexahedral Element (HEX8): Theoretical Foundations
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:
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.
Computational Methods for 8-Node Hexahedral Element (HEX8)
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 | — |
I see. So different solvers have different naming conventions and formulations.
Summary
To summarize the computational methods for HEX8:
- Full Integration (C3D8) — Accurate but suffers from shear locking. Use for plane strain problems only.
- Reduced Integration (C3D8R) — Good for explicit dynamics. Requires hourglass control.
- Selective Reduced Integration — Best for static metal forming problems. Eliminates both locking and hourglassing.
- Incompatible Modes (C3D8I) — Best for general-purpose static analysis. No locking or hourglassing.