10節点四面体要素(TET10)
Theory and Physics
TET10 — The Main Character of 3D Automatic Meshing
Professor, you said "don't use" TET4, but is it okay to actively use TET10?
TET10 is the practical standard for 3D structural analysis. It can generate automatic meshes for almost any shape and has high accuracy. It's fair to say that the majority of modern 3D FEM is performed with TET10.
Structure of TET10
How is TET10 different from TET4?
In addition to the 4 vertices of TET4, it has 6 intermediate nodes at the midpoints of each edge, making it a 10-node tetrahedron. Each node has 3 degrees of freedom, totaling 30 degrees of freedom.
The shape functions are quadratic polynomials:
Because the shape functions are quadratic, displacement varies quadratically within the element. Strain varies linearly.
Correct. This is the decisive difference from TET4 (constant strain). It can represent stress gradients due to bending within a single element.
Theoretical Background of Accuracy
Why are quadratic elements more accurate?
It's easier to understand by thinking about Taylor expansion. When approximating the true displacement field with a polynomial:
- TET4 (linear element): constant term and up to linear term → error in quadratic and higher terms
- TET10 (quadratic element): constant term, linear term, and up to quadratic term → error in cubic and higher terms
Rate of convergence (error reduction rate when mesh is refined):
| Quantity | TET4 | TET10 |
|---|---|---|
| Displacement Error | $O(h^2)$ | $O(h^3)$ |
| Stress Error | $O(h)$ | $O(h^2)$ |
| Convergence Rate | Slow | Fast |
For TET10, stress error decreases as $h^2$... if the mesh is halved, the error becomes 1/4. For TET4, halving only reduces it to 1/2.
That's why TET10 provides sufficient accuracy even with coarse meshes. For the same number of DOFs, TET10 is far more accurate than TET4.
Numerical Integration
What about numerical integration for TET10?
Since TET10 shape functions are quadratic, the B matrix is linear. $B^T D B$ becomes a quadratic polynomial, and to integrate this accurately, 4-point Gauss integration is required.
What about numerical integration for TET10?
Since TET10 shape functions are quadratic, the B matrix is linear. $B^T D B$ becomes a quadratic polynomial, and to integrate this accurately, 4-point Gauss integration is required.
| Integration Scheme | Number of Points | Accuracy | Application |
|---|---|---|---|
| 4-point (Full Integration) | 4 | Integrates quadratic polynomials exactly | Standard |
| 1-point (Reduced Integration) | 1 | Reduced accuracy | Special applications |
Is reduced integration ever used with TET10?
Almost never. Unlike TET4, volume locking usually does not occur with TET10's full integration (4-point). Using reduced integration only reduces accuracy with few benefits.
However, Abaqus's C3D10M (Modified) uses a special integration technique. It uses reduced integration with hourglass control, improving stability in contact problems. The distinction between standard C3D10 and C3D10M will be explained in detail later.
Curved Surface Approximation Accuracy
Can TET10 accurately represent curved surfaces?
Because the mid-edge nodes are on the edges, edges can be curved. By snapping the mid-edge nodes to CAD surfaces, curved surfaces like cylinders or spheres can be approximated much more accurately than with TET4.
For example, modeling the perimeter of a circular hole with TET4 approximates the circle with a polygon. With TET10, each edge becomes a curve, following the circle. This directly affects the accuracy of stress concentration evaluation.
What happens if the mid-edge nodes are not snapped to the CAD surface?
It becomes a "straight-edged TET10" with mid-edge nodes placed linearly, and the curved surface approximation accuracy drops to a level similar to TET4. Many preprocessors automatically snap to CAD surfaces, but you should check the settings.
Summary
Let me organize the theory of TET10.
Key points:
- 10 nodes, quadratic shape functions, linear strain variation within element — A qualitative improvement over TET4
- Fast convergence — Stress converges as $O(h^2)$. Twice the order of TET4
- Easy automatic meshing — Compatible with almost any 3D shape
- Accurate representation of curved surfaces — Quadratic approximation of surfaces by snapping mid-edge nodes to CAD
- Practical standard for 3D FEM — "When in doubt, use TET10"
So TET10 is the most well-balanced element in practice, achieving both "ease of automatic meshing" and "sufficient accuracy".
Exactly. It's not perfect (HEX20 is more efficient), but in terms of total cost including mesh generation effort, TET10 is the most superior.
Derivation of TET10 Quadratic Shape Functions
The 10-node tetrahedral element (TET10) has 4 vertices and 6 edge midpoints, totaling 10 nodes, and possesses complete quadratic polynomial (10-term) shape functions. Using volume coordinates (L1, L2, L3, L4, where L1+L2+L3+L4=1), the vertex shape functions are Li(2Li-1), and the edge midpoint functions are 4LiLj. This formulation was systematized by Zienkiewicz et al. at the University of Wales, Swansea in the late 1960s and became widely popular due to its good compatibility with automatic mesh generation (Delaunay triangulation) for arbitrary-shaped parts.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, i.e., "mass × acceleration". Haven't you experienced being thrown forward when slamming on the brakes? That "feeling of being carried away" 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, which is the assumption that "forces are applied slowly enough that 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 pull a spring, you feel a "force trying to return it", right? That's Hooke's law $F=kx$, and it's the essence of the stiffness term. So here's a question — an iron rod and a rubber band, which stretches more when pulled with the same force? Obviously the rubber. 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 $f_b$ (e.g., gravity) 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 contents" (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 it becomes "compression" — 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 away. That's because the vibration energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle — they intentionally absorb vibration 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 important.
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 stress-strain relationship is linear
- Isotropic material (unless otherwise specified): 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 inconsistency 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
C3D10 vs. C3D10M
Abaqus has both C3D10 and C3D10M. What's the difference?
This is one of the most important distinctions in practice.
| Characteristic | C3D10 | C3D10M |
|---|---|---|
| Integration | 4-point full integration | Modified reduced integration (hourglass control) |
| Contact Stability | Slightly unstable | Stable |
| Volume Locking | Caution for $\nu > 0.45$ | Countermeasures implemented |
| Computational Cost | Low | Slightly higher |
| Recommended Scenario | No contact, linear elastic | With contact, incompressible materials, general use |
If C3D10M is more versatile, should I always use C3D10M?
Abaqus's manual generally recommends C3D10M. C3D10's weakness is that it tends to produce a checkerboard pattern (stress oscillation) on contact surfaces. C3D10M is an improved version that solves this problem.
However, C3D10M has one hourglass mode, so in very rare cases hourglass deformation can become problematic. C3D10 may produce "cleaner" results in some situations.
Element Names by Solver
| Solver | Standard TET10 | Improved TET10 | Recommendation |
|---|---|---|---|
| Abaqus | C3D10 | C3D10M, C3D10MH | C3D10M |
| Nastran | CTETRA(10) | — | CTETRA(10) |
| Ansys | SOLID187 | — | SOLID187 |
| LS-DYNA | ELFORM=17 | — | ELFORM=17 |
Don't Nastran and Ansys have improved versions?
Nastran's CTETRA(10) and Ansys's SOLID187 each have built-in stabilization techniques, incorporating countermeasures equivalent to C3D10M. They are not separately named but are internally improved.
Mesh Generation
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