Explicit Time-domain Research Articles

E-PINN: A fast physics-informed neural network based on explicit time-domain method for dynamic response prediction of nonlinear structures

The physics-informed neural networks (PINNs), serving as the surrogate models, have emerged in the dynamic response prediction of nonlinear systems. However, in the conventional PINN model, the global differential equation of motion of the nonlinear system is directly integrated into the loss function, which requires the network to capture the intricate global dynamic evolution mechanism of the system and thus poses significant challenges for network learning. In this study, inspired by the explicit time-domain method (ETDM), a novel PINN based on ETDM, called E-PINN, is proposed to address the challenges arising from the network learning of the existing PINNs. A lightweight long short-term memory (LSTM) module is trained to learn the nonlinear evolution mechanism of restoring forces, while a single convolutional layer (SCL) module is utilized to reflect the linear evolution mechanism of the primary structure under the combined action of the external excitations and the nonlinear restoring forces. The weights of the SCL module can be directly retrieved from the discrete convolutional formulation of ETDM, and only the parameters in the LSTM module need to be optimized through a loss function solely based on the nonlinear restoring forces, thereby enabling easy network learning of the proposed E-PINN by decoupling the linear and nonlinear evolution mechanisms embedded in the nonlinear system. Three numerical examples are investigated by the present approach, and the results demonstrate the superior training efficiency and prediction accuracy of E-PINN in dynamic response prediction of nonlinear systems.

Dynamic response of a pipe conveying pulsating fluid under random vibration

This paper studies the dynamic response of a pipe conveying pulsating fluid under random vibration. Its nonlinear equation of motion is first developed using the method of Newton. The method of Galerkin is then used to discretize the equation of motion into a system of ordinary differential equations. Afterwards, the statistical moments and mean peak values of the displacement and velocity responses of the pipe are obtained using an efficient Monte Carlo simulation (MCS) method based on the explicit time-domain method (ETDM). Furthermore, the displacement-sensitive and velocity-sensitive locations of the pipe are obtained for different boundary conditions. The impacts of the fluid, random vibration, and structural parameters on the mean peak value of the dynamic response of the pipe are finally analyzed. The obtained results provide a reference for better understanding the dynamic characteristics of pulsating fluid transport pipes under random vibration and for accurately designing pipe structures.

Stochastic optimization of levitation control system for maglev vehicles subjected to random guideway irregularity

The dynamic performance of maglev vehicle-bridge coupled systems subjected to random guideway irregularity, especially the levitation stability of the running vehicle subsystem, is a critical issue affecting the safe operation and the comfort riding of maglev railways. This study is devoted to stochastic optimization of the levitation control system considering the coupled vibration of maglev vehicle and bridge under the random guideway irregularity. To alleviate the computational burden in the stochastic optimization process, the explicit expressions of the critical responses of the maglev vehicle-bridge coupled systems and the sensitivities of critical responses with respect to the feedback gains of the proportional-integral-derivative (PID) levitation controller are respectively constructed in terms of the guideway irregularity by virtue of the dimension-reduced formulation capability of the explicit time-domain method (ETDM), enabling stochastic dynamic analysis and stochastic sensitivity analysis to be executed at high efficiency. The stochastic optimization problem is formulated as the minimization of the standard deviation of the air gap variation with the constraint on the standard deviation of the current variation, and the optimal values of the PID gains are searched by the gradient-based method of moving asymptotes (MMA), in which the statistical moments and moment sensitivities of the critical responses required in the optimization process are obtained by ETDM. A numerical example with a 2-degree-of-freedom maglev vehicle moving on a multi-span simply-supported bridge is used to demonstrate the efficacy of ETDM for the stochastic analysis of maglev vehicle-bridge coupled systems under guideway irregularity. An engineering example involving a 3-car maglev train traversing a multi-span simply-supported bridge is further presented to show the effectiveness of the ETDM-based approach for the stochastic optimization of large-scale engineering problems. The levitation stability of the maglev train under the action of random guideway irregularity is considerably improved with the use of optimal feedback gains.

A Telegrapher’s equations-based implicit finite difference solution for fault location with current transformer saturation

This paper introduces a fault location method based on time analysis, utilizing synchronized data collected from both ends of the faulted line. The data is obtained either from Common Format for Transient Data Exchange (COMTRADE) files or Sampled Measured Value (SMV) of the IEC-61850 protocol. The method proposed in this study employs the implicit finite difference solution to the Telegrapher’s equations. In certain fault scenarios, such as when one or both transformers experience saturation, a narrow time window of data can be utilized to accurately pinpoint the fault location. The implicit solution method is unconditionally stable and therefore decouples the choice of the line discretization step from the sampling time interval, which is commonly required to maintain numerical stability of the explicit strategies for fault location. Furthermore, the numerical stability of the implicit process allows for devising a systematic search procedure that improves the accuracy of the calculated fault location, regardless of the transmission line length. By conducting numerical comparisons with an explicit time-domain fault locator and recent frequency-domain algorithms, the superiority of the proposed approach becomes evident. The results highlight the significant improvements achieved through the utilization of this method. Specifically, the results demonstrate that Telegrapher’s Equations implicit finite difference solution maintains accuracy under current transformer saturation, as it can locate the fault using data from a brief interval where the current transformer is not saturated.

Small failure probability analysis of stochastic structures based on a new hybrid approach

The small failure probability problem of stochastic structures is investigated by using two types of surrogate models and the subset simulation method in conjunction with parallel computation. To achieve high computational efficiency, the explicit expression of dynamic responses of stochastic structures is first derived in the form based on the explicit time-domain method. Then, the small failure probability analysis of stochastic structures is efficiently carried out through the Monte Carlo simulation method utilizing explicit expressions. To avoid the repeated calculation for the coefficient matrices or vectors of the explicit expression of stochastic structures, two types of surrogate models, e.g., the backpropagation neural network model and the Kriging model, are introduced to obtain these matrices or vectors for each parameter sample of the stochastic structures. The computational cost is further reduced by using the subset simulation method to generate conditional samples which follow the rule of Metropolis-Hastings. Furthermore, in virtue of the independence of the surrogate models for each time instant and the independence of dynamic analysis for each sample, parallel computation is embedded in the proposed approach, which can fully exploit the characteristics of the proposed approach and further improve the computational efficiency of dynamic reliability analysis. Numerical examples are given to illustrate the validity of the proposed hybrid approach.

A high-order explicit time-domain FEM using 15-node tetrahedral elements for room acoustics modeling: Basic performance

Time-domain room acoustic modeling using mass-lumped higher-order tetrahedral finite elements with an explicit time-marching scheme is highly attractive because of its excellent geometrical flexibility with unstructured meshing and applicability into massively parallel computing. However, the standard mass-lumped tetrahedral elements yield an unstable scheme, and only implicit time-marching schemes are available. This study proposes a novel wave-based room acoustics solver based on a high-order explicit time-domain FEM. The present solver uses mass-lumped 15-node tetrahedral elements for spatial discretization and a dissipation-free two-stage partitioned Runge-Kutta time integration for time discretization. The 15-node tetrahedral elements had been developed to apply mass-lumping methods to 10-node tetrahedral elements, which are the most common finite element in engineering applications, but its applicability in room acoustics problems is unknown. Higher accuracy and efficiency of the proposed method over the standard method using 10-node tetrahedral elements is presented through an eigenvalue analysis of a long duct model and acoustic simulations in a small room. Additionally, frequency-dependent sound absorbing boundary is implemented by revising an auxiliary differential equation method.

Physics-informed deep learning for structural vibration identification and its application on a benchmark structure

Structural vibration identification is an important task in civil engineering that is based on processing measured data from structural monitoring. However, predicting the response at unsensed locations based on limited sensor data can be challenging. Deep learning (DL) methods have shown promise in vibration data feature extraction and generation, but they struggle to capture the underlying physics laws and dynamic equations that govern vibration identification. This paper presents a novel framework called physics-informed deep learning (PIDL) that combines deep generative networks with structural dynamics knowledge to address these challenges. The PIDL framework consists of a data-driven convolutional neural network for structural excitation identification and a physics-informed variational autoencoder for explicit time-domain (ETD) vibration analysis with the generated unit impulse response (UIR) signal of the measured structure. The proposed framework is evaluated on a benchmark structure for structural health monitoring, demonstrating its effectiveness in extracting physics-related dynamics features and accurately identifying excitation signals and latent physics parameters across different damage patterns. Additionally, the incorporation of an ETD method-aided convolution function in the loss function aligns the generated UIR signals with the dynamic properties of the measured structure. Compared with conventional DL-based vibration analysis methods, the PIDL framework offers improved accuracy and reliability by integrating structural dynamics knowledge. This study contributes to the advancement of structural vibration identification and showcases the potential of the PIDL framework in civil structure monitoring applications. This article is part of the theme issue 'Physics-informed machine learning and its structural integrity applications (Part 2)'.

Reliability-based topology optimization of fractionally-damped structures under nonstationary random excitation

This contribution proposes an effective reliability-based topology optimization (RBTO) framework for the layout of viscoelastic dampers of energy-dissipating structures under nonstationary seismic excitation, in which the constitutive behavior of viscoelastic damper is described by the novel fractional derivative models. To formulate the RBTO problem, the installation number of fractional viscoelastic dampers is minimized under the constraints of dynamic reliability, and the design variables are chosen as the damper parameters and the existence variables of the potential fractional viscoelastic dampers. The explicit responses are first established for the fractionally-damped structure, and the explicit response sensitivities with respect to the design variables are further derived through the adjoint variable method (AVM). On this basis, an explicit time-domain method (ETDM) is developed for explicit formulation of the statistical moments of responses and the moment sensitivities of the fractionally-damped structure, which are further employed to analytically derive the first-passage dynamic reliabilities and the reliability sensitivities of the structure based on the classical level-crossing process theory. Finally, the RBTO problem for the layout of fractional viscoelastic dampers is solved with the gradient-based method of moving asymptotes (MMA), in which the existence variables of fractional viscoelastic dampers can be driven to binary solutions by the solid isotropic material with penalization (SIMP) technique. An engineering application involving a building frame structure installed with fractional viscoelastic dampers is presented to validate the feasibility of the proposed ETDM-based RBTO framework.

The Analysis and Calculation of Power Angle Transient Characteristics in VSG Control Using Parameter-Perturbation-Based Averaging Method

With the increasing penetration of power converters in modern power system, virtual synchronous generator (VSG) control is considered to be a promising method due to its grid friendliness. However, the lack of clear and accurate description of its power angle dynamics poses challenges to the stability analysis and converter parameters design. In this paper, Krylov-Bogoliubov (K-B) averaging method, an effective parameter-perturbation-based analytical calculation method for dissipative process, is first presented to characterize the dynamic trajectory of VSG control. Moreover, the solvable problem is transformed from typical simple harmonic motion (SHM) into damped oscillation, which redistributes the weights of solvable and perturbed problems. On top of that, an unperturbed-result-based (URB) averaging method is further proposed to obtain the explicit time-domain expression of power angle dynamics, which may provide good potential for transient stability analysis and grid-connected converter design. Finally, simulations and control-hardware-in-loop (CHIL) experiments have been both demonstrated to verify the effectiveness of proposed method.

Non-stationary non-Gaussian random vibration analysis of Duffing systems based on explicit time-domain method

Non-stationary non-Gaussian random vibration problems of structures are challenging and drawing increasing attention. In the present study, firstly, an explicit time-domain method (ETDM) is proposed to determine the higher-order response statistics of linear systems subjected to non-stationary non-Gaussian random excitations, in which the first four orders of cumulants of dynamic responses are directly formulated through the cumulant operation rule based on the explicit expressions of responses. Secondly, an equivalent linearization – explicit time-domain method (EL-ETDM) is further developed to solve the non-stationary non-Gaussian random vibration problems of Duffing systems, in which the equivalent linear system is derived discarding the assumption of Gaussian response, and the corresponding higher-order cumulant analyses of the linearized system are accomplished by the efficient ETDM. The present approach can account for non-Gaussian random excitations with arbitrary forms, and two specific applications to the Poisson white noise and the square form of Gaussian random process are investigated. Four numerical examples are presented to demonstrate the effectiveness of the proposed methods.

Modeling impedance boundary conditions and acoustic barriers using the immersed boundary method: The one-dimensional case

Immersed boundary methods are heavily used in computational fluid dynamics, as an alternative to volumetric meshing, when a problem contains irregular geometric features. In wave-based architectural and room acoustics, the dynamics are simplified, but boundary conditions and acoustic barriers are usually described in terms of frequency-dependent impedance and transmittance functions. In this article, a formulation of the immersed boundary method is developed in the informative special case of one-dimensional linear acoustics. It relies on dual driving terms applied to the conservation of mass and momentum equations separately and is directly tunable against boundary impedances and barrier transmittances. It is shown how the driving terms may be combined to model either an impermeable frequency-dependent boundary condition or a barrier with a given transmittance. An explicit time-domain numerical method of finite-difference time-domain type is presented, and it is shown how the immersed boundary condition may be included, at minimal additional computational cost. Special attention is paid to the discrete approximation of the Dirac delta function, necessary in immersed boundary methods, as well as the discretisation strategy for frequency-dependent boundary and barrier conditions. Numerical results are presented. A complete derivation of numerical stability conditions for this immersed boundary method appears in an appendix.

Binaural Auralization of Room Acoustics with a Highly Scalable Wave-Based Acoustics Simulation

This paper presents a proposal of an efficient binaural room-acoustics auralization method, an essential goal of room-acoustics modeling. The method uses a massively parallel wave-based room-acoustics solver based on a dispersion-optimized explicit time-domain finite element method (TD-FEM). The binaural room-acoustics auralization uses a hybrid technique of first-order Ambisonics (FOA) and head-related transfer functions. Ambisonics encoding uses room impulse responses computed by a parallel wave-based room-acoustics solver that can model sound absorbers with complex-valued surface impedance. Details are given of the novel procedure for computing expansion coefficients of spherical harmonics composing the FOA signal. This report is the first presenting a parallel wave-based solver able to simulate room impulse responses with practical computational times using an HPC cloud environment. A meeting room problem and a classroom problem are used, respectively, having 35 million degrees of freedom (DOF) and 100 million DOF, to test the parallel performance of up to 6144 CPU cores. Then, the potential of the proposed binaural room-acoustics auralization method is demonstrated via an auditorium acoustics simulation of up to 5 kHz having 750,000,000 DOFs. Room-acoustics auralization is performed with two acoustics treatment scenarios and room-acoustics evaluations that use an FOA signal, binaural room impulse response, and four room acoustical parameters. The auditorium acoustics simulation showed that the proposed method enables binaural room-acoustics auralization within 13,000 s using 6144 cores.

Investigating the stability of an explicit ADE-FDTD scheme for modeling graphene: Avoiding erroneous conclusions

In recent years, an explicit auxiliary differential equation finite difference time domain (ADE-FDTD) scheme has been frequently used in graphene simulations. In this respect, a differencing scheme in which the magnetic field and the current density are collocated in time was developed and it has been recently stated that this explicit ADE-FDTD implementation is “conditionally stable with the maximum time-step size bounded by the common Courant–Friedrichs–Lewy (CFL) limit”. In this communication, the stability of this explicit ADE-FDTD scheme is revisited and it is shown that the conventional CFL constraint is not retained, and a time-step less than the CFL limit must be used in all domain grids to insure the stability in the whole system. Although this time-step stringent can be of little effect for some graphene applications, and in order to avoid drawing erroneous conclusions, it is shown that when the given ADE-FDTD scheme is used for simulating graphene structures that include noble and other plasmonic materials the derived time-step stringent can be of significant effect. To retain the CFL stability constraint, alternative formulation based on the Runge–Kutta (RK) time-differencing scheme is also presented in this communication. Contrary to the other existing CFL-stable ADE-FDTD approaches, the presented RK-FDTD method is equally applicable to both magnetized and unmagnetized graphene structures without incurring additional computational cost. Furthermore, the RK-FDTD scheme is extended for modeling the complex-frequency-shifted perfectly matched layer (CFS-PML) mesh truncating technique. Finally, the stability and accuracy of the explicit ADE/RK-FDTD schemes are demonstrated through numerical examples.

A Response Spectrum Method for Base-Isolated Structures with Equivalent Base Excitations

Base isolation is an effective seismic design strategy that has received considerable attention in recent decades. This research is dedicated to developing a response spectrum method (RSM) for analysis of base-isolated structures under seismic excitation. A novel analytical model consisting of a linear superstructure under equivalent base excitations is proposed in the present study. To determine the response spectra corresponding to the base excitations, termed as the base response spectra, a sufficient number of samples of base excitations comprising base responses need to be obtained through time-history analysis of the nonlinear base-isolated structure in conjunction with the Monte Carlo simulation (MCS). To improve the efficiency for each sample analysis, the recently proposed explicit time-domain method (ETDM) is utilized for fast analysis of the base acceleration and velocity responses with a dimension-reduced iteration scheme. Once the base response spectra are obtained, the mean peak values (MPVs) of structural responses of the linear superstructure can be conveniently evaluated through the complete quadratic combination (CQC) rule. A 4-story nonlinear base-isolated frame structure was analyzed to validate the effectiveness of the proposed analytical model and the feasibility of the present approach for seismic response analysis of nonlinear base-isolated structures.

Model updating and damage detection of jacket type platform using explicit and exact time domain sensitivity equation

In this paper, a new sensitivity equation is formulated for structural parameter estimation based on the time-history response. There are three methods for calculating the structural response variation with respect to changes in structural parameters in the time domain: Finite Difference Method (FDM), Direct Differentiation Method (DDM), and Adjoint variable Method (AVM). These methods are gradient-based, and they cannot compute the actual sensitivity. The proposed sensitivity equation explicitly calculates the sensitivity (closed-form sensitivity) in an exact manner and provides a linear relationship between the time history response and the variation of structural parameters. The efficiency and accuracy of the proposed method are examined for the finite element model updating of a jacket-type platform. Moreover, the effectiveness of the proposed method is evaluated in the presence of measurement and environmental errors. The performance of the proposed sensitivity equation is compared with the mentioned direct differentiation method.

Dynamic reliability analysis of stochastic structures under non-stationary random excitations based on an explicit time-domain method

For the analysis of the dynamic reliability of stochastic structures subjected to non-stationary random excitations, an effective hybrid approach is proposed. In this approach, the explicit time-domain method is employed to obtain the dynamic responses of each deterministic structure. Dynamic responses for the explicit time-domain method can be expressed as a series of products of deterministic coefficient matrices and random load vectors, which can significantly enhance computational efficiency. For the issue of stochastic structures, the double-hidden-layer backpropagation neural network is introduced as a surrogate model for reconstructing the coefficient matrices to avoid repeated construction of the coefficient matrices for each sample with different structural parameters using the extremely time-intensive regular method. For more effective training of the backpropagation neural network, the Latin hypercube sampling technique is introduced to generate representative samples of random structural parameters. In consideration of the intrinsic benefit of the explicit time-domain method, parallel computation is incorporated into the method. Finally, the explicit time-domain method is used to perform the Monte Carlo simulation in conjunction with the parallel computing technique to resolve the problem of small failure probabilities. To demonstrate the efficacy of the proposed method, numerical examples are presented.

An Asymptotic-Homogenization Explicit Time-Domain Method for Random Multiscale Vibration Analysis of Porous Material Structures

Porous material structures have been widely used in civil engineering, mechanical engineering, aerospace engineering and other fields due to their high specific strength and specific stiffness. The stochastic response analysis of porous material structures under random excitations deserves more attention. The multiscale differential governing equations of porous material structures are derived based on the multiscale asymptotic homogenization method (AHM), and the macroscale and microscale explicit time-domain expressions of structural responses are further established. On this basis, the statistical moments of dynamic responses of porous material structures under non-stationary random excitations can be achieved with the explicit time-domain method (ETDM). The purposed method combines the advantages of AHM for high-efficient explicit formulation of macroscale and microscale dynamic responses of porous material structures and the advantages of ETDM for fast analysis of non-stationary random vibration problems. A numerical example is presented to validate the computational accuracy and efficiency of the present approach for non-stationary random vibration analysis of porous material structures.

An HIE‐FDTD algorithm for dispersive media with modified Lorentz model

AbstractIn this letter, a hybrid implicit–explicit finite difference time domain (HIE‐FDTD) method is developed to simulate general dispersive media with modified Lorentz model. The dispersive media is described through modified Lorentz model and is incorporated into the HIE‐FDTD method by two different auxiliary differential equations (ADEs). One ADE establishes the relationship between the electric displacement and electric field intensity and the other one establishes the connection between the electric current density and electric field intensity. Numerical examples are conducted and it can be seen from the numerical results that the proposed algorithm can simulate dispersive media correctly and also improve computational efficiency compared with the FDTD method and one unconditionally stable FDTD method.

Spectral properties of Fluid Structure Interaction pressure/stress waves in liquid filled pipes

We hereby develop a theoretical framework for analyzing Fluid Structure Interaction (FSI) waves propagation occurring in liquid filled pipes to manage a large family set of boundary conditions (e.g. junctions coupling effects). A self-adjoint operator theory framework leads to the analytical derivation of a transcendental equations for operator’s spectrum. The latter provides the system’s natural resonant frequencies as well as permit to find the discrete mode orthogonal basis decomposition. This theoretical framework also permits to demonstrate that the spectrum is uniquely composed into simple eigenvalues enabling explicit time-domain solutions from inverse-Laplace transform. The analysis is directly conducted in the time-domain but the obtained spectrum also applies to Fourier transformed frequency analysis. The obtained analytical solutions are successfully confronted with numerical simulation obtained using the Method of characteristic (MOC) for the same four equations (FSI) model on the very same configurations. The spectrum sensitivity matrix is also explicitly evaluated.

An explicit time-domain method for non-stationary random analysis of nonlinear frame structures with plastic hinges

In this study, a novel approach for random vibration analysis of nonlinear frame structures under seismic random excitations is developed. The explicit time-domain method is improved in this approach by integrating the plastic hinge model, which can simulate the nonlinear behaviors caused by material property changes. Specifically, the hysteretic system’s equation of motion is constructed using auxiliary differential equations that govern the plastic rotational displacements and their corresponding hysteretic displacements. Additionally, by introducing the concept of equivalent excitations, an explicit iteration scheme for solving the equation of the hysteretic system is developed, in which the auxiliary differential equations are solved under the assumption that the plastic rotational velocity changes linearly with time between two adjacent time instants. Finally, by combining the Monte Carlo simulation method and the proposed explicit time-domain method, the non-stationary random responses of nonlinear frame structures can be obtained. As illustrated by numerical examples, the proposed method achieves satisfactory solution accuracy and efficiency when applied to nonlinear frame structures with plastic hinges. Moreover, the proposed iterative method resolves equations involving displacements describing the frame’s global state, plastic rotational displacements, and corresponding hysteretic parameters, introducing a novel concept for solving problems with nonlinear coupled variables of multiple types.