Nonlinear Time-History Response Analysis

Updated 23 December 2025

Nonlinear Time-History Response Analysis is a simulation technique that models seismic effects on structures by incorporating material hysteresis and geometric nonlinearities.
It integrates advanced damping formulations and time-stepping schemes to accurately predict dynamic responses and energy dissipation in performance-based earthquake engineering.
Recent advances include surrogate and operator-based models that accelerate computations by orders of magnitude while maintaining high accuracy in risk and resilience evaluations.

Nonlinear Time-History Response Analysis (NLTHA) is the computational process of estimating the detailed, time-dependent behavior of structural systems subjected to highly variable, typically seismic, dynamic loads while explicitly accounting for material and geometric nonlinearities as well as other sources of inelastic response. NLTHA forms the core computational engine of modern performance-based seismic engineering, enables inelastic demand assessment under site-specific ground motions, and underpins next-generation surrogate modeling paradigms for structural risk and resilience evaluation.

1. Theoretical Foundations and Governing Equations

In matrix-vector notation, the governing equations for NLTHA of an $n$ -degree-of-freedom (DOF) system are: $\mathbf{M}\,\ddot{\mathbf{u}}(t) \;+\; \mathbf{C}\,\dot{\mathbf{u}}(t) \;+\; \mathbf{K}(t)\,\mathbf{u}(t) \;+\; \mathbf{f}_{\mathrm{nl}}\bigl(\mathbf{u}(t)\bigr) \;=\; -\mathbf{M}\mathbf{r}\,\ddot{u}_g(t)$ where

$\mathbf{u}(t)\in\mathbb{R}^n$ : displacement vector,
$\mathbf{M}$ : lumped mass matrix (typically diagonal),
$\mathbf{C}$ : viscous damping matrix,
$\mathbf{K}(t)$ : (generally time-dependent) tangent or secant stiffness matrix accounting for material softening, stiffness degradation, or geometric nonlinearity,
$\mathbf{f}_{\mathrm{nl}}$ : vector of nonlinear restoring (hysteretic/inelastic) forces,
$\mathbf{r}$ : influence vector for base excitation,
$\ddot{u}_g(t)$ : prescribed ground acceleration input.

Material nonlinearity is regularly modeled with phenomenological constitutive laws such as Bouc–Wen or bilinear/hardening rules, and geometric nonlinearity is incorporated via displacement-dependent $K(\mathbf{u})$ terms. For multi-system simulations (e.g., frames coupled with rocking walls), additional kinematic constraints and rotary/axial degrees of freedom augment the system and require careful enforcement of interface equilibrium (Aghagholizadeh et al., 2019).

For general MDOF systems, time integration employs schemes such as Newmark- $\beta$ , Hilber–Hughes–Taylor $\alpha$ -method (to introduce controllable numerical dissipation), or explicit, symplectic algorithms as dictated by stability and energy preservation needs (Pan et al., 2021).

2. Damping Models and Spurious Dissipation Effects

Damping, necessary to represent energy dissipation not directly accounted for by hysteresis or frictional models, is typically introduced as an added viscous term. Classical formulations—Rayleigh ( ${C}=a_0M+a_1K$ ), generalized Caughey series, or Wilson–Penzien modal matrices—can produce unintended or "spurious" damping forces under nonlinear evolution:

Type I (massless DOF, S-type) spurious damping: Arises when damping matrices have nonzero subblocks in massless DOFs, generating nonphysical forces after yielding.
Type II (modal-parameter drift, P-type) spurious damping: Manifests as drift in effective damping ratios $\xi_m(t)$ due to nonlinear changes in modal properties (e.g., frequency shifts), if damping coefficients are held fixed.

The explicit conditions for these effects are formalized as: $\xi_m^{\mathrm{Rayleigh}}(t) = \frac{1}{2}\biggl(\frac{a_0}{\omega_m(t)} + a_1\,\omega_m(t)\biggr)$ wherein reduction of $\omega_m$ during inelastic action can artificially inflate $\xi_m$ . Avoiding such effects requires careful assembly of $C$ (e.g., using negative-power Caughey series, elemental Wilson–Penzien construction, or time-updating of damping parameters), and remains an area of active methodological development (Jehel et al., 2017).

3. Advanced Modeling: Soil-Structure Interaction and Coupled Systems

NLTHA frameworks increasingly demand integration of frequency-dependent soil–foundation interaction, using substructure approaches that decouple the superstructure from a generalized impedance boundary. For nonlinear superstructures on frequency-dependent soil, the Hybrid Time-Frequency Domain (HTFD) method decomposes the impedance into:

a lumped (reference) part, modeled as conventional mass-dashpot-spring,
a regular (frequency-dependent) component, imposed as a correction ("pseudo-force") via convolution with the interface velocity history.

The time-stepping implementation in OpenSees leverages partitioned solution windows, with the convolutional correction updated using fast Fourier transforms in each window. Convergence and stability are controlled by compatible selection of reference damping terms and window sizes (Ghahari et al., 2023).

For complex systems such as frame–rocking wall coupling, nonlinear equations for each subsystem (e.g., Bouc–Wen oscillator + stepping rocking wall with tendon) are fully coupled at the acceleration and force level. Time integration enforces both equilibrium and kinematic compatibility, and modal properties are validated against push-over calibration and full NLTHA time-history consistency (Aghagholizadeh et al., 2019).

4. Computational Acceleration via Surrogate and Operator-Based Models

Classical NLTHA is computationally intensive due to fine timestep requirements and repeated solution of nonlinear equations. Machine learning surrogates, notably neural operators (Fourier neural operator—FNO, DeepONet, hybrid DeepFNOnet) and tailored deep neural networks, now accelerate NLTHA by 1–4 orders of magnitude:

Deep operator surrogates: Learn the ground motion $\to$ response functional mapping directly, enabling mesh-free, physically consistent predictions across unseen ground motions and system variations (Goswami et al., 16 Feb 2025, Kim et al., 12 Jun 2025).
Composite physics–neural frameworks: Use a simplified physics simulator (linearized, reduced-order, or coarse-stepped) as a preprocessor, with FNO corrected for hysteresis/path-dependence, and optional postprocessing via linear regression for uncertainty quantification (Kim et al., 12 Jun 2025).
Transfer learning surrogates: Pretrain on inexpensive SDOF dynamic systems and transfer to high-fidelity MDOF systems (e.g., 20-story steel moment frame) using a small set of high-fidelity runs, yielding accurate hazard-consistent predictions with only $\sim$ 20 training samples (Ishikawa et al., 16 Dec 2025).
Adaptive neural networks: Employ dynamic widening and deepening of the network architecture based on validation loss trends, achieving L $^2$ -norm errors $\sim$ 1–5% and up to 43% computational savings in parametric spectrum generation (Pan et al., 2021).

These surrogates are validated for structures ranging from shear frames and bridges to high-rise moment frames, maintaining accuracy in peak/displacement and RMSE norms under broadband or spectrum-matched ground motions.

5. Integration with Performance-Based Earthquake Engineering (PBEE) and Parametric Studies

Contemporary PBEE workflows rely on NLTHA to extract engineering demand parameters (EDPs) under suites of ground motions, for subsequent derivation of fragility and exceedance curves. NLTHA enables direct computation of floor accelerations, interstory drifts, and member force demands needed for loss modeling:

Surrogate acceleration enables tens of thousands of time-histories (as required in Monte Carlo PBEE pipelines) to be feasibly analyzed, with accuracy in exceedance probability curves for peak floor acceleration (PFA) and interstory drift ratio (IDR) maintained to within 3–6% of direct simulation (Ishikawa et al., 16 Dec 2025).
Operator and ANN-based surrogates support spectrum-based design, e.g., rapid parameter sweeps for rocking amplitude vs. system period, enabling design iteration and uncertainty quantification in real time (Pan et al., 2021).
Physics-guided, modular surrogates may be embedded into end-to-end workflows that incorporate stochastic ground motion generators, fragility estimators, and record selection algorithms (Kim et al., 12 Jun 2025, Ishikawa et al., 16 Dec 2025).

6. Limitations, Model Trade-Offs, and Research Directions

Despite dramatic efficiency improvements, fundamental limitations remain:

Surrogate models can under-predict tail/extreme events, particularly those outside the training distribution envelope; active learning and Bayesian neural operators are under consideration.
Epistemic uncertainty in material and geometric parameters is not inherently captured; recent work proposes including parameter fields as additional inputs or outputs in operator surrogates (Kim et al., 12 Jun 2025).
Damping model selection continues to pose accuracy challenges, with no universal damping construction avoiding both Type I and Type II spurious forces—elemental, discrepancy-calibrated, or dynamically updated schemes are advocated (Jehel et al., 2017).
NLTHA incorporating both nonlinear superstructure and nonlinear soil remains computationally burdensome; as of now, HTFD implementations leverage linearized impedance only (Ghahari et al., 2023).
Extrapolation to severe collapse or post-yield behavior may not be guaranteed without explicit training data in that regime; structural equilibrium constraints can be regularized within loss functions for improved physical fidelity (Ishikawa et al., 16 Dec 2025).

7. Practical Implementation and Benchmarking

Open-source finite element platforms (e.g., OpenSees, MATLAB) provide ready-to-use modules for NLTHA, including advanced soil–structure interaction via HTFD and parameterized nonlinear elements. For surrogate-based analysis, standard ANN architectures (FCNNs, operator networks) are implemented in Python and MATLAB, with interfacing wrappers for input/output time-history mapping.

Comparative benchmarking consistently demonstrates surrogate models achieve:

RMS displacement error: $\sim$ 0.001–0.01 m depending on system,
Relative L $^2$ : $\sim$ 0.16–0.65 (C-PhysFNO vs. standard FNO),
Surrogate vs. direct simulation speed-up: $10^2$ – $10^6\times$ ,
Hazard-consistent curve accuracy: correlation $r \approx 0.95$ (Ishikawa et al., 16 Dec 2025, Kim et al., 12 Jun 2025, Goswami et al., 16 Feb 2025).

Correct implementation of soil, damping, and hysteretic material models as specified in each referenced study is essential for fidelity to benchmark results, and pipeline integration for PBEE or parametric studies demands systematic validation against direct simulation (Aghagholizadeh et al., 2019, Ghahari et al., 2023, Pan et al., 2021).

References:

(Jehel et al., 2017, Aghagholizadeh et al., 2019, Pan et al., 2021, Ghahari et al., 2023, Goswami et al., 16 Feb 2025, Kim et al., 12 Jun 2025, Ishikawa et al., 16 Dec 2025).