6節点三角形要素(TRIA6)
Theory and Physics
LST Element — The Superior Version of CST
Does the 6-node triangular element (TRIA6, LST) solve the problems of CST?
Yes. The LST (Linear Strain Triangle) is a quadratic triangular element where strain varies linearly within the element. It completely eliminates the constant strain problem of CST.
Shape Function
TRIA6 has three vertex nodes and three mid-side nodes. The shape functions are quadratic polynomials of the area coordinates $L_1, L_2, L_3$:
TRIA6 has three vertex nodes and three mid-side nodes. The shape functions are quadratic polynomials of the area coordinates $L_1, L_2, L_3$:
Vertex nodes:
Mid-side nodes:
It's the 2D version of TET10, right?
Exactly. TRIA6 = 2D version of TET10, CST = 2D version of TET4. It's the same relationship.
Accuracy
- Displacement: $O(h^3)$ (CST is $O(h^2)$)
- Stress: $O(h^2)$ (CST is $O(h)$)
Same convergence rate as Q8, right?
Yes. Since TRIA6 and Q8 are elements of the same order (quadratic), they have the same convergence rate. However, Q8 has slightly higher accuracy per element (8-term vs. 6-term polynomial).
TRIA6 vs. Q8
| Characteristic | TRIA6 | Q8 |
|---|---|---|
| Automatic Meshing | Easy | Somewhat Difficult |
| Accuracy per DOF | 70-80% of Q8 | Baseline |
| Curved Edge Handling | Has mid-side nodes | Has mid-side nodes |
| Sensitivity to Mesh Quality | More robust than Q8 | Sensitive to distortion |
| Adaptive Meshing | Easy | Somewhat Difficult |
TRIA6 is inferior to Q8 in accuracy efficiency but superior in ease of mesh generation. It's the same pattern as TET10 vs. HEX20, right?
Perfect understanding. The pattern "triangles/tetrahedra are easy for automatic meshing, quadrilaterals/hexahedra have higher accuracy" is common in both 2D and 3D.
Numerical Integration
| Integration | Points | Accuracy |
|---|---|---|
| 3-point Gauss (Full Integration) | 3 | Integrates quadratic polynomials exactly |
| 1-point Gauss | 1 | Insufficient accuracy (not used) |
| 7-point Gauss | 7 | Used for higher-order integration |
Usually, 3 points are sufficient, right?
Yes. 3-point full integration is standard for TRIA6. Shear locking does not occur (because it's a quadratic element). Volume locking is also usually not a problem. It's a very stable element.
Summary
Let me organize the theory of TRIA6.
Key Points:
- Quadratic Triangular Element (LST) — Superior version of CST. Strain varies linearly.
- 2D Version of TET10 — Inherits the advantage of automatic meshing.
- Same Convergence Rate as Q8 — However, Q8 has higher DOF efficiency.
- Robust to Mesh Quality — More stable than Q8 even with distorted shapes.
- 3-point Gauss Integration is Sufficient — No locking.
If CST is the "do not use" element, then TRIA6 is the "use with confidence" element, right?
Exactly. TRIA6 is the most reliable triangular element for 2D automatic mesh analysis.
TRIA6's Complete Quadratic Polynomial and Accuracy
The 6-node triangular element (TRIA6) has three vertex nodes and three mid-side nodes, using a complete quadratic polynomial (6 terms) for its shape functions. In area coordinates, the vertex shape functions are Li(2Li-1), and the mid-side ones are 4LiLj. At three times the computational cost of TRIA3 (linear), the stress error convergence rate for bending problems improves from h-squared to h-cubed. Records indicate that Browning, Abrahamson, and others first practically applied quadratic triangular elements to aircraft wing stress analysis in the late 1960s.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, meaning "mass × acceleration". Have you ever experienced being thrown forward when slamming on the brakes? That "being carried away" feeling 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, assuming "forces are applied slowly enough that acceleration is negligible". 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", right? That's Hooke's law $F=kx$, the essence of the stiffness term. Now a question — an iron bar and a rubber band, which stretches more under 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 force $f_b$ (e.g., gravity) and surface force $f_s$ (pressure, contact force, 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 pitfall here: getting the load direction wrong. Intending "tension" but ending up with "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. 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 keep swaying 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 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
TRIA6 Implementation
Please tell me the implementation considerations for TRIA6.
TRIA6 is more complex than CST but can be implemented standardly using numerical integration (3-point Gauss).
Element Names by Solver
| Variation | Nastran | Abaqus | Ansys |
|---|---|---|---|
| Standard | CTRIA6 | CPS6 (Plane Stress) | PLANE183 (degenerated) |
| Plane Strain | CTRIA6 + PLPLANE | CPE6 | PLANE183(degenerated,PE) |
| Axisymmetric | CTRIAX6 | CAX6 | PLANE183(degenerated,Axi) |
In Ansys, is TRIA6 not a dedicated element but a degenerated form of PLANE183?
Ansys's PLANE183 is an 8-node quadrilateral but is used degenerated into a 6-node triangle by placing mid-side nodes on three edges. However, internally, the correct TRIA6 formulation is used, not a degenerated CST.
Handling of Mid-Side Nodes
Similar to Q8/TET10, CAD curve snapping for mid-side nodes is important for TRIA6. It improves the approximation accuracy of curved boundaries.
Points to note:
- Mid-side nodes should be within 25% to 75% of the edge length.
- The Jacobian must be positive.
- Use finer mesh in areas with high curvature.
TRIA6 Mesh Quality
| Metric | Ideal Value | Acceptable Range |
|---|---|---|
| Aspect Ratio | 1.0 | < 5.0 |
| Minimum Angle | 60° | > 20° |
| Jacobian | Positive | Must be positive |
Are the quality requirements more lenient than for CST?
TRIA6 is more robust to element shape distortion than CST. The quadratic shape functions partially absorb distortion. However, accuracy degrades with extreme distortion like minimum angles below 10°.
Summary
Let me organize the numerical methods for TRIA6.
Key Points:
- 3-point Gauss Integration is Standard — Full integration, no locking.
- CAD Snapping for Mid-Side Nodes — Same caution as Q8, TET10.
- Robust to Shape Distortion — More stable than CST.
- Dramatic Accuracy Improvement from CST→TRIA6 Conversion — Effect of linear→quadratic.
3-Point Gauss Integration for TRIA6
TRIA6 typically uses 3-point or 7-point Gauss integration. The integration points for the 3-point scheme are placed at equal distances from the triangle's centroid (e.g., L1=L2=1/6, L3=2/3) and can integrate polynomials up to 5th order exactly. The 7-point scheme is a high-precision version proposed by Strang-Fix (1973), capable of fully integrating polynomials up to 7th order. Since the 3-point integration points become superconvergence points for stress sampling, the Zienkiewicz-Zhu method, which extrapolates stresses from these points to obtain nodal stresses, is widely used.
Linear Elements (1st Order Elements)
Linear interpolation between nodes. Computational cost is low, but stress accuracy is low. Beware of shear locking (reduced integration...
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