HHT-α Method (Hilber-Hughes-Taylor)
HHT-α Method (Hilber-Hughes-Taylor): Theoretical Foundations
What is the HHT-α Method?
Professor, is the HHT-α method an improved version of the Newmark method?
Yes. It's an improved version by Hilber, Hughes, Taylor (1977) that added a numerical damping parameter $\alpha$ to the Newmark method. It's the default time integration method in Abaqus and Ansys.
Problems with the Newmark Method
The Newmark method ($\beta=1/4, \gamma=1/2$) is second-order accurate and unconditionally stable, but has zero numerical dissipation. High-frequency numerical noise, once introduced, never disappears. High-frequency noise is prone to occur with contact impacts or sudden load changes.
It's problematic that high-frequency noise doesn't disappear.
The HHT-α method solves this problem. It preserves low-frequency accuracy while selectively damping only high frequencies.
HHT-α Method Algorithm
Modified equation of motion:
Parameter relationships:
So it matches the Newmark method when $\alpha = 0$, right?
The range for $\alpha$ is $-1/3 \leq \alpha \leq 0$. $\alpha = 0$: No damping (Newmark method). $\alpha = -0.05$: Gentle high-frequency damping. $\alpha = -1/3$: Maximum high-frequency damping (but accuracy degrades).
Practical recommendation: Around $\alpha = -0.05$. This effectively damps high-frequency noise while maintaining second-order accuracy.
Abaqus
```
*DYNAMIC, ALPHA=-0.05 $ HHT-α α value
0.001, 1.0
```
Abaqus default is equivalent to $\alpha = -0.05$ (APPLICATION=MODERATE DISSIPATION).
Ansys
```
TINTP, , , , , 0.05 $ γ = 1/2 + 0.05 → equivalent α
```
In Ansys, set Newmark parameters with the TINTP command. γ > 1/2 gives numerical damping.
Nastran
```
PARAM, NDAMP, 0.01 $ Numerical damping parameter
```
Summary
Key points:
- Newmark method + numerical damping $\alpha$ — Selective damping of high-frequency noise
- $\alpha = -0.05$ is recommended — Suppresses high frequencies while maintaining 2nd-order accuracy
- Default in Abaqus/Ansys — Often used without realizing it
- Degenerates to Newmark method at $\alpha = 0$ — No numerical damping
- HHT-α is effective when high-frequency noise appears from contact or sudden load changes
So many engineers are "using the HHT-α method without knowing it."
The default for *DYNAMIC in Abaqus is the HHT-α method. Appropriate numerical damping is applied even without changing settings. However, understanding the $\alpha$ value allows you to adjust it when noise appears.
HHT-α: A Numerical Damping Scheme Born in 1977
The HHT-α algorithm, published by Hilber, Hughes, and Taylor in 1977, extends Newmark-β to selectively damp only high-frequency components. The α parameter ranges from −1/3 ≤ α ≤ 0, where α = 0 matches the Newmark method, and α ≈ −0.1 can suppress high-frequency noise while maintaining second-order accuracy and unconditional stability. Abaqus's *DYNAMIC procedure adopts α = −0.05 as default, making it a practical standard for seismic response calculations in building structures.
Computational Methods for the HHT-α Method (Hilber-Hughes-Taylor)
Numerical Characteristics of HHT-α Method
Please explain the numerical characteristics of the HHT-α method in detail.
| $\alpha$ | Numerical Damping | Accuracy | Application |
|---|---|---|---|
| 0 | None | 2nd order | Pure Newmark method |
| -0.05 | Gentle | Nearly 2nd order | Recommended (Standard) |
| -0.1 | Moderate | Slightly degraded | Problems with much noise |
| -0.33 | Maximum | Close to 1st order | Special applications only |
So accuracy drops if you make $\alpha$ larger (more negative) than -0.1.
Numerical damping also affects low frequencies. Larger $|\alpha|$ damps low-frequency response more. $\alpha = -0.05$ strikes a good balance of "damping high frequencies while hardly affecting low frequencies".
Relationship with Generalized-α Method
Chung-Hulbert's (1993) Generalized-α method further generalizes the HHT-α method. It allows independent control of low-frequency accuracy and high-frequency damping. Abaqus's APPLICATION=MODERATE DISSIPATION is based on the Generalized-α method.
Summary
Analysis Accuracy Changes with α Selection
Setting HHT-α's α between −0.05 and −0.10 results in a numerical damping ratio ξnum of a few percent to about 10% for the highest mode. Making α too small (e.g., α = −0.3) damps even physical low-order modes, so for structural engineering, it's desirable to have it affect modes with natural periods below about 0.01s. In MSC Nastran SOL 109 direct transient response, HHT-α can be set with the DTI,DIRECTT,ALPHA card.
HHT-α Method (Hilber-Hughes-Taylor) in Practice
HHT-α Method in Practice
In practice, situations where you "consciously use the HHT-α method" are limited. It's fine to rely on the solver's default for most cases.
When to Adjust $\alpha$
| Situation | $\alpha$ Adjustment |
|---|---|
| High-frequency noise present in response | Strengthen to $\alpha = -0.1$ |
| Low-frequency accuracy is critical (flutter, etc.) | Revert to $\alpha = 0$ (Newmark method) |
| Spike noise from contact impact | $\alpha = -0.05 \sim -0.1$ |
| No issues with default | No change needed |