Transient analysis

Time-marching response \(\mathbf{M}\, \ddot{\mathbf{U}} + \mathbf{C}\, \dot{\mathbf{U}} + \mathbf{K}\, \mathbf{U} = \mathbf{F}(t)\), driven by the upcoming Model.transient_solve() API.

Note

The transient solver path is partial as of the current release. Modal-superposition transient on a pre-computed eigen basis is shipped via the harmonic / transient roadmap (TA-14 / TA-16). Direct time-stepping on the full DOF system (Newmark-β / generalised-α) is on the roadmap.

Approach: Newmark-β family

For time step \(\Delta t\) and Newmark parameters \((\beta, \gamma)\), the implicit one-step update solves

\[\tilde{\mathbf{K}}\, \mathbf{U}_{n + 1} = \tilde{\mathbf{F}}_{n + 1},\]

with

\[\tilde{\mathbf{K}} = \mathbf{K} + \frac{\gamma}{\beta\, \Delta t}\, \mathbf{C} + \frac{1}{\beta\, \Delta t^{2}}\, \mathbf{M}.\]

Standard parameter choices:

• \((\beta, \gamma) = (1/4, 1/2)\) — the classical trapezoidal / average-acceleration rule. Unconditionally stable, second-order accurate, no numerical damping. Default.

• \((\beta, \gamma) = (1/6, 1/2)\) — linear- acceleration rule. Conditionally stable (\(\Delta t \le 0.55\, T_n\)); often preferred for short simulations where the highest-frequency content is benign.

The factor of \(\tilde{\mathbf{K}}\) is shared across time steps for a constant-\(\Delta t\) solve — the linear backend (see Linear-solver backends) factors it once and re-uses the factor for every step.

Modal superposition (TA-14 path)

For damped systems where a modal basis already exists (Model.modal_solve()), the modal-superposition path projects the full system onto the first \(r\) modes:

\[\mathbf{U}(t) = \boldsymbol{\Phi}_{r}\, \boldsymbol{\eta}(t), \qquad \ddot{\eta}_{i} + 2 \zeta_{i}\, \omega_{i}\, \dot{\eta}_{i} + \omega_{i}^{2}\, \eta_{i} = \boldsymbol{\phi}_{i}^{\!\top}\, \mathbf{F}(t).\]

The \(r\) decoupled modal SDOFs are integrated cheaply (closed-form for harmonic, RK / Newmark-β per mode for arbitrary forcing). Reconstruct the full DOF response via \(\mathbf{U}(t) = \boldsymbol{\Phi}_{r}\, \boldsymbol{\eta}(t)\) for plotting / stress recovery.

Damping

Rayleigh damping \(\mathbf{C} = \alpha\, \mathbf{M} + \beta\, \mathbf{K}\) is the practitioner default; the \((\alpha, \beta)\) constants are chosen to hit a target damping ratio at two reference frequencies. Modal- superposition supports per-mode damping ratios \(\zeta_i\) directly (no Rayleigh fit needed).

Public API (forthcoming)

• Model.transient_solve() — direct time-stepping (TA-16 roadmap).

• HarmonicResult / TransientResult — modal-superposition response containers (TA-14 shipping).

References

• Newmark, N. M. (1959) “A method of computation for structural dynamics,” J. Eng. Mech. 85 (3), 67–94.

• Hilber, H. M., Hughes, T. J. R. and Taylor, R. L. (1977) “Improved numerical dissipation for time integration algorithms in structural dynamics,” EESD 5 (3), 283–292 (HHT-α / generalised-α family).

• Chung, J. and Hulbert, G. M. (1993) “A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-α method,” J. Appl. Mech. 60 (2), 371–375.

• Bathe, K.-J. (2014) Finite Element Procedures, 2nd ed., §9 (direct time integration), §12 (mode superposition).

• Cook, R. D., Malkus, D. S., Plesha, M. E., Witt, R. J. (2002) Concepts and Applications of Finite Element Analysis, 4th ed., Wiley, §11 (modal analysis), §13 (transient response).

• Hughes, T. J. R. (2000) The Finite Element Method — Linear Static and Dynamic Finite Element Analysis, Dover, §9 (algorithms for hyperbolic and parabolic problems).