Physics-Informed Neural Networks for Structural Dynamic Response Analysis: Theoretical and Numerical Verification

Accurate and rapid prediction of structural responses to seismic loading is critical for ensuring structural safety. Recently, there has been active research focusing on the application of deep learning techniques, including Physics-Informed Neural Networks (PINNs) and Long Short-Term Memory (LSTM) networks, to predict the dynamic behavior of structures. While these methods have shown promise, each comes with distinct limitations. PINNs offer physical consistency but struggle with capturing long-term temporal dependencies in nonlinear systems, while LSTMs excel in learning sequential data but lack physical interpretability. To address these complementary limitations, this study proposes a hybrid LSTM-PINN model, combining the temporal learning ability of LSTMs with the physics-based constraints of PINNs. This hybrid approach allows the model to capture both nonlinear, time-dependent behaviors and maintain physical consistency. The proposed model is evaluated on both single-degree-of-freedom (SDOF) and multi-degree-of-freedom (MDOF) structural systems subjected to the El-Centro ground motion. For validation, the 1940 El-Centro NS earthquake record was used, and the ground acceleration data were normalized and discretized for numerical simulation. The proposed LSTM-PINN is trained under the same conditions as the conventional PINN models (e.g., same optimizer, learning rate, and loss structure), but with fewer training epochs, to evaluate learning efficiency. Prediction accuracy is quantitatively assessed using mean error and mean squared error (MSE) for displacement, velocity, and acceleration, and results are compared with PINN-only models (PINN-1, PINN-2). The results show that LSTM-PINN consistently achieves the most stable and precise predictions across the entire time domain. Notably, it outperforms the baseline PINNs even with fewer training epochs. Specifically, it achieved up to 50% lower MSE with only 10,000 epochs, compared to the PINN's 50,000 epochs, demonstrating improved generalization through temporal sequence learning. This study empirically validates the potential of physics-guided time-series AI models for dynamic structural response prediction. The proposed approach is expected to contribute to future applications such as real-time response estimation, structural health monitoring, and seismic performance evaluation.

The tank front platform in the Liquefied Natural Gas (LNG) terminal concentrates numerous natural gas pipelines and valves. Once gas leakage occurs, there is a significant explosion risk. This paper investigates the analysis methods for explosion overpressure and structural dynamic response of the tank front platform under blast loading. Firstly, an explosion shock wave analysis model was established based on the Arbitrary Lagrangian-Eulerian (ALE) algorithm and fluid-structure coupling theory, analyzing the propagation trends of explosion pressure with time. Subsequently, the blast loading was applied to the platform structure through fluid-structure coupling for structural dynamic response analysis, determining stress distribution and its time-dependent variation trend. The results show that the explosion overpressure exhibits distinct transient characteristics after detonation, requiring sufficiently small computational time steps to capture the maximum pressure. The steel structures near the explosion source experience extremely high stresses, necessitating reinforcement in the design of steel structures in leakage and explosion risk zones. Even after the peak explosion pressure subsides, steel structure stresses continue to fluctuate at relatively high levels, which also requires special attention.

In recent years, successful applications of deep learning (DL) have inspired scientists to explore the possibilities of applying DL approaches to modeling scientific problems. Existing studies have revealed that to bake the physics into the DL makes a good supplement to the traditional numerical methods (e.g., finite element, finite volume method) which primarily rely on partial differential equations (PDEs). While DL models are ordinarily trained in a purely data-driven manner, integrating physics into them for simulating scientific problems has several benefits such as (i) physics constraints could regularize the over-parameterized model and hence mitigate the overfitting issue commonly seen in DL; (ii) physics information could also effectively reduce the amount of data needed for training the model; (iii) the resultant physics-informed DL models feature better interpretability and generalizability compared with the conventional black-box model. Furthermore, the powerful expressiveness of the deep network, guaranteed by the universal approximation theorem, makes it a suitable approximator for the solution to a physical system. In this dissertation, we develop two different DL architectures (or approaches), one being continuous scheme-based while the other discrete scheme-based, that leverage physics knowledge for modeling scientific problems. Through comprehensive numerical experiments, we demonstrate the proposed models can be used in solving general PDEs, establishing predictive data-driven models for dynamical systems, identifying the parameters in governing PDEs or even discovering the entire governing PDEs of dynamical systems from scarce and noisy measurements. The continuous model roots on the physics-informed neural network (PINN) which uses a fully connected neural network (FCNN) to approximate the physical fields of a system globally. This model is mesh-free as the residual of the physics (e.g., PDEs, initial/boundary values) is evaluated on a set of collocation points within the physical domain. Several applications including the forward simulations, data-driven simulations and solving inverse problems are presented to exemplify the advantages of PINN over traditional numerical methods. However, the original PINN suffers from inaccurate initial/boundary values due to the weak enforcement of the initial/boundary conditions (I/BCs). To overcome this issue, we propose an improved PINN model by utilizing multiple deep neural networks (DNNs) to construct the solution. Through a DNN pre-trained to represent the initial/boundary values, the approximated solution would obey the given I/BCs forcibly. With several numerical examples, we show that the improved PINN is characterized with much better accuracy on the I/BCs. Though the PINN shows great promise in data-driven modeling and solving inverse problems, some inherent limitations of PINN still exist, such as (i) the solution might lacks fine-scale details due to the global approximation of FCNN; (ii) high computational expense caused by the FCNN it roots on; (iii) incapability to incorporate existing PDE terms (e.g., $\Delta u$) into the network architecture. To overcome these drawbacks, this dissertation also proposes a discrete model - Physics-encoded Recurrent Convolutional Neural Network (PeRCNN) which recurrently updates the solution (or state variable) for time marching. Specifically, it utilizes convolutional neural network (CNN) to capture the spatial patterns of the solution while the recurrent network mimics the forward Euler scheme (or Runge-Kutta scheme) in numerical methods. PeRCNN is a mesh-based and discrete model due to the discretization in time and spatial dimension. The local connectivity of CNN makes PeRCNN more computationally efficient. In addition, the coercive encoding mechanism of physics in PeRCNN, fundamentally different from the PINN relying on soft penalty, ensures the network to rigorously obey given physics. The proposed PeRCNN is successfully applied to solving general PDEs, the data-driven modeling of dynamical systems and the data-driven discovery of governing PDEs from scarce and noisy measurements. Comparisons with the state-of-the-art DL models demonstrate that the proposed PeRCNN possesses excellent computational efficiency, accuracy and generalizability. --Author's abstract

In the long-term service, the wharf structure can be damaged by ship impact, wave load, and even earthquake, which will affect the safe production and smooth operation of the port. Based on the theory of structural dynamic response analysis and wavelet packet analysis principle, this paper established the damage identification index of the wharf frame structure. Combining with the finite element method and the dynamic response theory of the wharf frame structure, it set up a finite element analysis model of the dynamic response of the wharf frame structure under the action of multiple loads, with the impact load of the ship as the dynamic load under the non-damaged state and the different damaged states. In addition, the characteristic response point location was drawn up. Furthermore, the transient dynamic analysis and damage index analysis of the frame structure in the non-damaged and damaged state were conducted respectively. In addition, the model test and numerical simulation results were combined to compare and analyze the identification of damage indicators, so as to verify the identification effect of the established damage identification indicators on the structural damage, which lays a foundation for the next step of structural damage identification.

Dynamic response and damage analysis of masonry structures and masonry infilled RC frames to blast ground motion

ABSTRACTThis study presents a novel neural network architecture called spectral integrated neural networks (SINNs), which combines physics‐informed neural networks (PINNs) with time‐spectral integration techniques to efficiently solve two‐ and three‐dimensional dynamic piezoelectric problems. To avoid the numerical instability associated with time‐differential operators, the coupled system of mechanical and electrical equilibrium equations is reformulated into a weak time‐integral form. The temporal derivatives of displacement and voltage fields, treated as the primary unknown physical quantities, can be approximated utilizing fully connected neural networks (FCNNs). The displacements and electric potential are subsequently recovered through time‐spectral integration of their respective derivatives. A physical‐informed loss function is formulated by the weak time‐integral type of the governing equations and boundary conditions, with the initial conditions embedded within the integral expressions. The proposed SINNs demonstrate superior stability and accuracy, even under large time steps conditions. Numerical verification is accomplished through three representative test cases of the method, and a comparison analysis is presented between the results obtained by the SINNs and those from the PINNs.

A new method, to parallel the dominating computing work of structural dynamic response analysis, is presented in this paper. Then, based on “ShenWei I” supercomputer, MSC.NASTRAN and PATRAN, a parallel and integrated system is implemented. The system tightly couples advantages of the FEA code and high performance of the supercomputer. The system can improve the structural scale and velocity of dynamic response analysis greatly. Finally two examples are presented to demonstrate the validity and efficiency of the system.

Dynamic Response and Failure Mechanism Analysis of a Kiewitt-Type Cable Dome Structure Under Impact Loads

Three-dimensional dynamic response analysis of a single-ring structure in a prefabricated subway station

Piping system response induced by thermal-hydraulic transient: A review of recent analysis methods and experiments

Discretization-based methods like the finite element method have proven to be effective for solving the Helmholtz equation in computational acoustics. However, it is very challenging to incorporate measured data into the model or infer model input parameters based on observed response data. Machine learning approaches have shown promising potential in data-driven modeling. In practical applications, purely supervised approaches suffer from poor generalization and physical interpretability. Physics-informed neural networks (PINNs) incorporate prior knowledge of the underlying partial differential equation by including the residual into the loss function of an artificial neural network. Training the neural network minimizes the residual of both the differential equation and the boundary conditions and learns a solution that satisfies the corresponding boundary value problem. In this contribution, PINNs are applied to solve the Helmholtz equation within a two-dimensional acoustic duct and mixed boundary conditions are considered. The results show that PINNs are able to solve the Helmholtz equation very accurately and provide a promising data-driven method for physics-based surrogate modeling.

ABSTRACTLong-duration ground motions may be down-sampled to speed up the computational process. However, using ground motions with large time step (Δt) would inevitably bring in numerical errors. The influence of Δt on the site effect and structural response analyses was quantitatively examined in this study. The results show that the nonlinear site response method is more sensitive to a change of Δt than the equivalent-linear method. For the structural analysis, the impact of Δt is highly dependent on the magnitude of damage parameters. Thus, using input motions with Δt as 0.005 s is recommended for structures subjected to strong shakings.

Numerical analysis of dynamic responses of laminated glass window subjected to gas explosions

A well-validated Finite Element (FE) model of a jacket structure (OC4 jacket) with pile foundation together with a Computational Fluid Dynamic (CFD) model (OpenFOAM-based) set-up for extreme waves on the structure are applied to perform systematic comparative analyses of the dynamic response of the structure to extreme wave loads with and without scour effects. The pile foundation model is considered by implementing a non-linear soil model to the foundation using load-deflection (p-y) approach to capture the non-linear behaviour of the soil for large pile foundation displacements. The results show that, the scour protection improves the foundation bearing capacity and consequently, decreases the failure probability in terms of both serviceability and ultimate limit states. Moreover, it enhances the dynamic response of the structure under extreme load events caused by breaking waves. Accordingly, an innovative solution for scour protection of offshore-fixed bottom structures has been introduced by “Biodegradable Soft Rock” (sand filled containers produced from bio-degradable geotextile product) which is the newest development in this section.

As composite materials have been used for marine structures such as boats, ships, offshore structures, etc., the effect of FSI should be understood.In particular, FSI effect is expected to be significant for polymer composites because the water density is very comparable to the composites' densities.For example, the density of a carbon composite is approximately 50% greater than the water density.Sandwich composites consisting of very low densities of core materials have lower densities than that of water.As a result, the hydrodynamic mass associated with FSI would be very critical to composite structures under water.The growing use of composites in ship masts, superstructures, deck grates, piping, ducting, rudders, propellers, stacks, and various submarine structures requires extensive modeling and testing to help designers, builders and operators better understand composite response [Mouritz, et al., 2001].These materials are subjected to a wide spectrum of loads during manufacturing and service life.Dynamic loadings, in particular, impact type event, represent a serious design concern for use of composite.Composite structures are more susceptible to impact damage than similar metallic structures which are more ductile in nature and can absorb typically large amounts of energy without failure.Furthermore, the damage in composites from impact can go undetected even when the mechanical properties may be drastically reduced from an impact.For these reasons, numerous experimental and analytical studies have been conducted to study the dynamic response of composites subjected to impact loading [Abrate, 1994;Aslan, et al., 2003;Kwon & Wojcik, 1998].According to the review of past works, most of the research effort has been focused on low velocity impact damage, specifically, the damage predictions, and the evaluation and prediction of residual properties of damaged laminates.All of the research completed thus far has focused on damage in composites under impact loading in dry environments to support development of composites in aircraft structures.As far as dynamic response of structures under water is concerned, a great deal amount of analytical and experimental studies have been conducted on the effect of fluid force on the natural frequencies, damping ratios and mode shapes of vibrating structures in contact with fluid.This is commonly known as the Fluid Structure Interaction (FSI) problem.FSI investigations have supported many problems in submarine signaling, offshore oil structure stability, and ship structure vibrations.Through these studies, many numerical and analytical methods have been developed in order to predict the added mass and the www.intechopen.com Advances in Composites Materials -Ecodesign and Analysis 338resulting changes in natural frequency of a structure in contact with fluid.It has been determined and widely proven that the effect of fluid surrounding a structure decreases the natural frequency of a structure due to the increase in total kinetic energy of the vibrating structure and fluid from the addition of kinetic energy of the fluid.This effect can be interpreted as an added mass to the vibrating structure in the analysis of the dynamic response.Essentially as the structure vibrates, its mass is increased by the mass of the vibrating fluid with which it is in contact, consequently decreasing its natural frequency.Studies of fluid structure interaction and the added mass effect, also known as virtual mass effect, hydrodynamic mass, and hydroelastic vibration of structures, started with Lamb [Lamb, 1921] who calculated the first bending mode of a submerged circular plate.In response to a problem of submarine signaling, Lamb investigated the vibrations of a thin elastic circular plate in contact with water.In his investigation he discovered that the natural frequencies for structures in contact with fluid are lower than the frequencies in air, based on the assumption that the modes shapes are virtually the same in water as in a vacuum.The resonant frequency was determined using Rayleigh's method.Lamb's theortical results were verified experimentally [Powell & Roberts, 1923].Much later, more research was conducted for measuing and calculating natural frequencies of free vibartions of beams and plates under water [Lindholm, et al., 1965; Fu & price, 1979;Kwak, 1996].Another branch of FSI studies is underwater explosion.Both experimental and numerical studies have been conducted for metallic structures [