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. Solver defaults and settings depend on the product and application.
Problems with the Newmark Method
The Newmark method ($\beta=1/4, \gamma=1/2$) is second-order accurate and unconditionally stable for linear problems, but has zero numerical dissipation. High-frequency numerical noise may persist when physical damping is small. 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 prioritizes high-frequency numerical damping while limiting low-frequency effects.
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 numerical damping (Newmark method). $\alpha = -0.05$: Gentle high-frequency damping. $\alpha = -1/3$: Maximum high-frequency damping (strong damping; check time-step error).
Practical recommendation: Around $\alpha = -0.05$. This effectively damps high-frequency noise while maintaining second-order accuracy.
Abaqus
*DYNAMIC, ALPHA=-0.05
0.001, 1.0This is a setting fragment for an analysis step; a model, boundary conditions, loads and step definition are also required. Abaqus documentation: *DYNAMIC
Abaqus/Standard generally selects TRANSIENT FIDELITY (HHT-TF, default α = −0.05) without contact, and MODERATE DISSIPATION (HHT-MD) with contact. Quasi-static applications use a different integrator.
Ansys
TRNOPT,FULL,,,,,HHT ! Select HHT integration
TINTP,0.05 ! GAMMA amplitude decay factor (first argument)In Ansys Mechanical APDL, TRNOPT selects the integrator and the first TINTP argument GAMMA sets an amplitude decay factor; this is not the same quantity as Newmark γ.
Nastran
Nastran settings and parameter meanings depend on the product, version and solution procedure; consult the applicable solver documentation.
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
- Available in Abaqus/Ansys — Select and check the integration settings
- 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."
Abaqus/Standard uses α=−0.05 by default for HHT-TF; contact problems default to HHT-MD and quasi-static applications use BWE. Verify the selected scheme and time step.
HHT-α: A Numerical Damping Scheme Born in 1977
HHT-α extends Newmark integration to reduce high-frequency numerical oscillations. At α=0 it recovers the average-acceleration method. Damping depends on both the time step and frequency.
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 | 2nd order | Recommended (Standard) |
| -0.1 | Moderate | 2nd order | Problems with much noise |
| -0.33 | Near maximum | 2nd order | Special applications only |
With the β and γ relationships above, each α in the table is second order. Convergence order is distinct from amplitude and phase error at a finite time step; check time-step convergence for the problem. Sandia: HHT integration theory
So stronger numerical damping can change the response error while the scheme remains second order.
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/Standard APPLICATION=MODERATE DISSIPATION selects HHT-MD.
Summary
Analysis Accuracy Changes with α Selection
Nastran settings and parameter meanings depend on the product, version and solution procedure; consult the applicable solver documentation.
HHT-α Method (Hilber-Hughes-Taylor) in Practice
HHT-α Method in Practice
In practice, check the selected scheme and time step and compare how changes affect the response of interest.
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 |
Practical Checklist
References: Abaqus/Standard and Ansys Mechanical APDL official documentation： Abaqus *DYNAMIC · Ansys TINTP · Ansys TRNOPT