Explicit Time-domain Research Articles

Solution Analysis and Novel Admissibility Conditions of SFOSs: The 1 < α < 2 Case

This article considers the regularity, nonimpulsiveness, stability, as well as admissibility of singular fractional-order systems (SFOSs) with the fractional-order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha \in (1,2)$ </tex-math></inline-formula> . First, the existence and uniqueness of time-domain solutions of the systems are analyzed by using the Kronecker equivalent standard form, and then the necessary and sufficient condition for the regularity is proposed. Second, the explicit time-domain solutions of the systems are presented, which are the basis of the necessary and sufficient conditions for the nonimpulsiveness and stability. Third, two novel necessary and sufficient conditions for the admissibility of the systems are proposed in terms of the nonstrict linear matrix inequalities (LMIs) and strict LMIs, respectively. Finally, two numerical examples about the SFOSs are provided to illustrate the validity of the obtained conclusions.

Stochastic heat transfer analysis and reliability assessment under non-stationary random thermal load using the explicit time-domain method

Efficient and accurate reliability assessment is of utmost importance for a safe and economical design of engineering equipment involving stochastic heat transfer. However, the stochastic analysis of large-scale heat transfer problems under non-stationary random thermal loads is still a challenging task due to the heavy computational burden. This paper extends the explicit time-domain method (ETDM), an efficient method initially developed for random structural vibration, to stochastic heat transfer problems. In ETDM, the underlying physical law driving the uncertainty propagation is first converted to a discrete and algebraic form through deterministic impulse analyses, resulting in the explicit expressions of concerned responses in the time domain. Using the explicit expressions, the statistics of the random system responses can be efficiently evaluated. With the deterministic impulse analyses serving as a bridge, the physical mechanism is treated separately from the stochastic evolution mechanism. Thus, ETDM can be deemed a general method for stochastic problems governed by various physical laws. Its capability is demonstrated by solving large-scale heat transfer problems subjected to non-stationary random thermal loads. The efficiency and accuracy of the method are highlighted by the reliability assessment of a CPU heat sink discretized by a very fine finite element mesh.

Runge–Kutta HIE-FDTD formulations for analyzing optical transmission through periodic array of magnetically-biased graphene micro-patch structures

In this paper, simple and efficient hybrid implicit–explicit finite difference time domain (HIE-FDTD) formulations are presented for analyzing optical transmission through periodic array of magnetically-biased graphene structures. In this respect, the Runge–Kutta (RK) time differencing scheme is extended for incorporating the gaphene’s anisotropic conductivity into the HIE-FDTD scheme. The temporal discretization of the presented formulations, which are named as RK-HIE-FDTD, is not confined by the graphene’s nano thickness and this will allow using larger time-step than that of the classical explicit-FDTD counterpart. Compared with the previous magnetized HIE-FDTD approaches, the presented formulations not only involve less memory indexing overhead and fewer arithmetic operations, but also found to be numerically stable at the conventional HIE-FDTD stability limit. The accuracy and stability of the presented RK-HIE-FDTD formulations is verified by first investigating the transmission coefficient for a plane-wave normally incident on an infinite magnetized graphene layer and very good agreement with the analytical solution is observed. Furthermore, a numerical test conducted on a periodic array of magnetically-biased graphene micro-patches deposited on top of SiO2 glass substrate shows that the number and the intensity of the generated resonance frequencies are greatly affected by the graphene’s chemical potential as well as by the magnetic field bias intensity.

Stability study of the modified HIE-FDTD method for modeling graphene as a surface boundary condition

An impressive approach to model graphene in the finite-difference time-domain (FDTD) method is using a surface boundary condition (SBC). In our previous work, a method for modeling graphene as an SBC in the hybrid implicit–explicit finite-difference time-domain (HIE-FDTD) was extracted. In this paper, the impact of applying an SBC on the stability of the HIE-FDTD method is numerically analyzed using an amplification matrix method. We show that the application of the SBC does not degrade the stability of the original HIE-FDTD algorithm.

A Hybrid MPI/OpenMP Parallelization Scheme Based on Nested FDTD for Parametric Decay Instability

Parametric decay instability (PDI) generated in milliseconds is an important physical phenomenon in ionospheric heating. Usually, numerical simulations are used to study PDI mechanisms. They can intuitively investigate the generation and development process of PDI, which is necessary in experimental studies. When simulating the PDI phenomenon through the explicit finite-difference time-domain (FDTD), the spatial scale spans from kilometers to centimeters, and the time scale needs to meet the Courant–Friedrichs–Lewy condition. Simulating the PDI phenomenon is time-consuming and difficult due to the high spatial resolution and strict restriction on the discrete time step. Although a nested mesh technique can boost the computational efficiency, the application of a parallel strategy is imperative to further improve it. In this study, we present a hybrid Message Passing Interface (MPI)/OpenMP parallelization scheme to solve the above-mentioned problems. This scheme can achieve an adaptive calculation and automatic allocation of MPI tasks and OpenMP threads, proving its flexibility and portability. Under the EISCAT background parameters, the PDI phenomenon was simulated. The results of the wave mode conversion and intense localized turbulence were identical to those of the serial program. Furthermore, a new simulation example and the effect of the cavity depth on electrostatic waves and negative ion density cavity were investigated. By utilizing the proposed parallelization scheme, the simulation time can be reduced from 70 h for the serial program to 3.6 h.

Explicit predictor–corrector method for nonlinear acoustic waves excited by a moving wave emitting boundary

We present an explicit finite difference time domain method to solve the lossless Westervelt equation for a moving wave emitting boundary in one dimension and in spherical symmetry. The approach is based on a coordinate transformation between a moving physical domain and a fixed computational domain. This allows to simulate the combined effects of wave profile distortion due to the constitutive nonlinearity of the medium and the nonlinear Doppler modulation of a pressure wave due to the acceleration of the wave emitting boundary. A predictor–corrector method is employed to enhance the numerical stability of the method in the presence of shocks and grid motion. It is demonstrated that the method can accurately predict the Doppler shift of nonlinear wave distortion and the amplitude modulation caused by an oscillating motion of the wave emitting boundary. The novelty of the presented methodology lies in its capability to reflect the Doppler shift of the rate of nonlinear wave profile distortion and shock attenuation for finite amplitude acoustic waves emitted from an accelerating boundary.

Nonstationary random vibration analysis of fractionally-damped systems by numerical explicit time-domain method

Random vibration analysis of fractionally-damped systems has received considerable attentions over the past two decades. In this paper, an efficient explicit time-domain approach is developed to determine the response statistics of multi-degrees-of-freedom linear systems endowed with fractional derivative terms subjected to nonstationary random excitations. The explicit time-domain expression of the state vector is constructed for the fractionally-damped system, and on this basis, the statistical moments of responses are directly formulated with the moment operation rule or efficiently calculated with the Monte-Carlo simulation. Dimension-reduced statistical analysis can be easily conducted just focusing on certain responses of interest with the use of the explicit expressions of dynamic responses. The present approach can account for nonstationary random excitations with arbitrary evolutionary power spectrum forms, even of the non-separable kind. A 20-degrees-of-freedom shear-type fractionally-damped system subjected to fully nonstationary random excitation is analysed to illustrate the accuracy and efficiency of the proposed method.

Ground vibration induced by high speed trains on an embankment with pile-board foundation: Modelling and validation with in situ tests

To investigate the train-induced ground vibration, an explicit time-domain, three dimensional (3D) finite element (FE) model is developed. The train, track, embankment, pile-board structure and nearby ground soils are all fully coupled in this model. The complex geometries involving the track components and pile-board structure are all modelled in detail, which makes the simulation of wave propagation more realistic from the train to the ground. The model is validated with in situ tests data collected in the Beijing-Shanghai high speed railway line. Good agreements have been achieved between the numerical results and experimental results both in time domain and frequency domain. The proposed model is thus capable of reproducing the dynamic ground response induced by a typical high speed train. Soil responses induced by different number of vehicles are compared. With more vehicles, the spectral peaks of soil responses are more prominent at the integral multiples of the vehicle passing frequency. Too few vehicles will not bring about such phenomenon, thus sufficient number of vehicles should be included in a train to properly model train-induced ground vibration. With the proposed model, the influence of the pile-board foundation on the ground vibration is investigated. It is found that the pile-board foundation can significantly attenuate the low frequency ground vibration. The attenuation of the ground vibration as a function of distance from the track is simulated and the influential factors to the local vibration amplification are investigated. It is found that soil Young’s modulus and soil impedance contrast are the two main factors influential to the local vibration amplification. The softer the natural soil, the larger the amplification. The larger soil impedance contrast makes the amplification more obvious. The soil stratification and geometric discontinuity at ground surface are not the main cause of the local vibration amplification in this work.

A Parallel Dissipation-Free and Dispersion-Optimized Explicit Time-Domain FEM for Large-Scale Room Acoustics Simulation

Wave-based acoustics simulation methods such as finite element method (FEM) are reliable computer simulation tools for predicting acoustics in architectural spaces. Nevertheless, their application to practical room acoustics design is difficult because of their high computational costs. Therefore, we propose herein a parallel wave-based acoustics simulation method using dissipation-free and dispersion-optimized explicit time-domain FEM (TD-FEM) for simulating room acoustics at large-scale scenes. It can model sound absorbers with locally reacting frequency-dependent impedance boundary conditions (BCs). The method can use domain decomposition method (DDM)-based parallel computing to compute acoustics in large rooms at kilohertz frequencies. After validation studies of the proposed method via impedance tube and small cubic room problems including frequency-dependent impedance BCs of two porous type sound absorbers and a Helmholtz type sound absorber, the efficiency of the method against two implicit TD-FEMs was assessed. Faster computations and equivalent accuracy were achieved. Finally, acoustics simulation of an auditorium of 2271 m3 presenting a problem size of about 150,000,000 degrees of freedom demonstrated the practicality of the DDM-based parallel solver. Using 512 CPU cores on a parallel computer system, the proposed parallel solver can compute impulse responses with 3 s time length, including frequency components up to 3 kHz within 9000 s.

Random analysis of coupled vehicle–bridge systems with local nonlinearities based on explicit time-domain method

Based on the explicit time-domain method, an efficient analysis algorithm is developed for the random vibration analysis of the coupled vehicle–bridge system with local nonlinear components. In this work, the coupled vehicle–bridge system is divided into two subsystems, including the linear bridge subsystem and the nonlinear vehicle subsystem. Firstly, using the equivalent linearization technique, the equivalent linearized subsystem is constructed for the vehicle subsystem with the hysteretic suspension spring at a given time instant. Then, the explicit expressions of the responses of the linearized vehicle subsystem and the linear bridge subsystem are constructed corresponding to contact forces, respectively. Further, the explicit expression of the contact forces is derived in consideration of the compatibility condition. Lastly, the dimension-reduction vibration analysis for the equivalent linearized coupled vehicle–bridge system can be carried out based on the explicit time-domain method. The numerical example about a coupled vehicle–bridge system under the random irregular excitation is investigated, and the results indicate that the proposed approach is of feasibility.

Balanced Energy/Time Optimal Defibrillation

Abstract The synthesis of a cardiac defibrillating pulse is cast as a standard optimal feedback control problem that minimizes a weighted measure of consumed energy and elapsed time to reach a nominal defibrillated state. The solution is developed for a general first-order system that includes the widely used parallel resistor/capacitor circuit and energy source as a special case. The novel optimal pulse comprises an exponentially ascending and a rectangular component; it is agile and energy conscious; and it therefore outperforms the waveforms developed thus far that minimize energy or time expenditures alone. Explicit time-domain expressions are derived which may be used for comparing against other commonly studied defibrillating functions. The analytic formulas and computer simulations may be useful to implement performance improvements in defibrillating devices.

Elastodynamic wave propagation modelling in geological structures considering fully-adaptive explicit time-marching procedures

In this paper, two fully-adaptive explicit time-marching procedures are discussed for the time-domain solution of wave propagation in elastic media. In these procedures, the time integration parameters of the methods are adaptively locally evaluated, following the properties of the discretized model and the values of the calculated responses, engendering more accurate solution procedures. In addition, automated domain decomposition and sub-cycling procedures are also performed, providing more efficient analyses. In this case, a domain decomposition procedure divides the domain of the model into different time-marching subdomains according to the properties of the discretized problem and the stability limit of the adopted time-marching technique, so that a different time-step value is applied to each subdomain, leading to more efficient (and yet stable) explicit time-domain computations. As it is indicated in this work, the adoption of multi-time-steps/sub-cycling splitting procedures further improves the accuracy of the discussed adaptive time-marching formulations. At the end of the paper, benchmark analyses are performed to illustrate the effectiveness of the proposed techniques, followed by synthetic case analyses, with degrees of complexity equivalent to that of real geophysical applications in the OIL & GAS industry, demonstrating the robustness of the proposed explicit approaches.

Removing the Courant-Friedrichs-Lewy stability criterion of the explicit time-domain very high degree spectral-element method with eigenvalue perturbation

The explicit time-domain spectral-element method (SEM) for synthesizing seismograms has gained tremendous credibility within the seismological community at all scales. Although the recent introduction of nonperiodic homogenization has addressed the spatial meshing difficulty of the mechanical discontinuities, the Courant-Friedrichs-Lewy (CFL) stability criterion strictly constrains the maximum time step, which still puts a great burden on the numerical simulation. In the explicit time-domain SEM, the source of instability of using a time step beyond the stability criterion is that some unstable eigenvalues of the updated matrix are larger than what can be accurately simulated. We have succeeded in removing the CFL stability condition in explicit time-domain SEM by combining the forward time dispersion-transform method, the eigenvalue perturbation, and the inverse time dispersion-transform method. Our theoretical analyses and numerical experiments in the homogeneous, moderate, and strong heterogeneous models show that this combination can accurately simulate waveforms with time steps several times the size of the CFL limit even toward the Nyquist limit especially for the efficient very high degree SEM, which abundantly saves iteration times without suffering from time-dispersion error. It demonstrates a potential application prospect in some situations such as full-waveform inversion that require multiple numerical simulations for the same model.

An explicit time-domain FEM for acoustic simulation in rooms with frequency-dependent impedance boundary: Comparison of performance in 2D simulation with frequency-domain FEM

Accurate boundary modelings that address the frequency-dependent sound absorption characteristics of various sound absorbers are crucial for wave-based room acoustic simulation. In time-domain simulations, however, a computationally demanding convolution appears in frequency-dependent impedance boundary conditions. The present paper proposes a room acoustic solver with a fourth-order accurate explicit TD-FEM, incorporating a frequency-dependent absorbing boundary condition efficiently using a recursive convolution method, namely the auxiliary differential equation (ADE) method. Its performance against the fourth-order accurate frequency-domain FEM is examined via 2D real-scale room acoustic problems, solving a sound propagation in an office room up to 4.5 kHz. Firstly, we describe briefly the formulation of the proposed room acoustics solver based on the explicit TD-FEM. Then, the discretization error property of the proposed method is evaluated via an impedance tube problem, including a frequency-dependent impedance boundary of porous sound absorber. Finally, the accuracy and efficiency of the proposed method are demonstrated with the comparison of frequency-domain FEM solver, which uses a sparse direct solver for the solution of the linear system at each frequency. Results showed the proposed method can perform an acoustic simulation with significantly low computational costs compared to the frequency-domain solver while keeping an acceptable level of accuracy.

Stochastic optimization of uncertain viscous dampers for energy-dissipation structures under random seismic excitations

Energy-dissipation devices have been widely used for seismic mitigation of civil structures. Stochastic optimization of energy-dissipation devices has emerged as an appealing means to achieve optimal seismic performance of structures in uncertain environment. This paper proposes an effective optimization framework for nonlinear viscous dampers of large-scale structures in consideration of both random seismic excitations and uncertain damper parameters. A dimension-reduced explicit time-domain method (ETDM) is employed for nonlinear time-history analysis of seismic responses and response sensitivities with respect to the nominal parameters of uncertain viscous dampers. To obtain the statistical moments of critical responses and the corresponding moment sensitivities, the high-efficient ETDM is then used in conjunction with the Monte-Carlo simulation (MCS) with sufficient samples of seismic excitation and damper parameters. The stochastic response and sensitivity results are further utilized for stochastic optimization of the nominal values of damper parameters with the method of moving asymptotes (MMA). The present optimization framework represents a hybrid approach combining the existing ETDM, MCS and MMA. A large-scale suspension bridge installed with nonlinear viscous dampers with uncertain parameters is analyzed to demonstrate the feasibility of the proposed optimization framework, and the influence of the uncertainties of damper parameters on stochastic optimization is also investigated.

Sensitivity analysis in seismic reliability of an urban self-anchored suspension bridge

Seismic resistance and disaster prevention for self-anchored suspension bridge (SASB) are important, but the seismic reliability of this complex structure is time-consuming or low-accuracy based on existing methods, let alone the relative sensitivity analysis. A sampling-based hybrid methodology of sensitivity analysis in seismic reliability, combining subset simulation (SuS), explicit time domain method (ETDM), BP neural network (BPNN) and Pearson’s Linear Correlation Coefficient (PLCC), is proposed herein. The separated treatment of low and high variabilities of structural and earthquake parameters based on ETDM with BPNN accelerates the computing efficiency. With the application to the SASB, sensitivities of structural parameters to seismic reliability are obtained, where three largest ones are ambient temperature, Young’s modulus and density of girder. The failure probability of this bridge is mostly smaller than 1.0 × 10-3, but it takes only about 10 ~ 15 min to obtain single failure probability with a deterministic parameter vector based on the proposed method, which is in lieu of hundreds of days based on Monte Carlo simulation. Hence, it is verified a sound approach with great accuracy and efficiency used in engineering structures.

Stochastic evolution mechanism in random thermoelastic vibration captured using the explicit time-domain method

Thermoelastic vibration under random excitations is a frequently encountered challenge in modern engineering. The stochastic evolution mechanism involved needs to be quantified using novel and efficient procedures. This paper extends the explicit time-domain method (ETDM), which is a powerful tool originally developed for structural random vibration, to the stochastic analysis of coupled thermoelastic problems. To efficiently solve the stochastic evolution equations, the explicit time-domain expressions of critical responses are first constructed via a deterministic impulse response analysis. Then, using the explicit expressions the statistics of the random system responses can be efficiently computed. Thus, in ETDM the physical mechanism and the stochastic evolution mechanism are treated separately. This feature makes the method particularly suitable for stochastic multiphysics problems like thermoelastic vibration, where the physical mechanisms involved are relatively complex. Its capability is shown by considering coupled displacement and temperature field in large scale problems with non-stationary random inputs. Numerical examples are presented to demonstrate the efficiency and accuracy of the method with emphasis on structural reliability assessments.

A two-step framework for stochastic dynamic analysis of uncertain vehicle-bridge system subjected to random track irregularity

This paper presents a two-step probabilistic framework for stochastic dynamic analysis of the coupled vehicle-bridge system (VBS) involving both uncertain parameters and random track irregularity. Firstly, an explicit time-domain method is extended to evaluate the conditional response statistics of the VBS subjected to random irregularity at specific parameter points, and an efficient spectral representation-based formulation is proposed to calculate the excitation correlation matrix required in the analysis. Then, the total response statistics are further evaluated by Gaussian integration of the conditional ones based on the total probability theorem, and an efficient dimension-adaptive algorithm is proposed to collocate the required quadrature points. In the numerical examples, the proposed method is used to study a typical VBS model with both deterministic or random parameters; several other existing methods are employed to illustrate the advantages of the proposed method. The results indicate that the proposed framework is an accurate and efficient method for stochastic dynamic analysis of the vehicle-bridge interaction problem.

An Efficient Monte-Carlo Simulation for the Dynamic Reliability Analysis of Jacket Platforms Subjected to Random Wave Loads

The growing demand for the application of jacket platforms in deep water requires more attention on the assessment of structural reliability. This paper is devoted to the dynamic reliability analysis of jacket platforms subjected to random wave loads with the Monte-Carlo simulation (MCS), in which a sample size of the order of magnitude of 104 to 105 for repeated time–history analyses is required for small failure probability problems, and a duration time up to three hours needs to be considered in the time–history analyses for a specific sea condition. To tackle the difficulty involved in the MCS, the explicit time-domain method (ETDM) is used for the required time–history analyses of jacket platforms, in which truncated explicit expressions of critical responses with regards to the contributing loading terms are first established and then used for numerous repeated sample analyses. The use of ETDM greatly enhances the computational efficiency of MCS, making it feasible for the dynamic reliability analysis of jacket platforms under random wave loads. A jacket platform with 11,688 degrees of freedom was analyzed for the evaluation of dynamic reliability under a given sea condition, indicating the accuracy and efficiency of the present approach and its feasibility to practical structures.

Efficient non-stationary random vibration analysis of vehicle-bridge system based on an improved explicit time-domain method

Based on the recently developed explicit time-domain method (ETDM), this paper presents a new efficient computational framework for non-stationary random vibration analysis of the vehicle-bridge system (VBS) subjected to random track irregularity excitation. In this framework, the time-varying VBS equation is discretized using the time-stepping integration method, leading to an explicit expression for the stochastic system response in terms of the track irregularity excitation. Based on this expression, the response moments of the VBS can be easily evaluated in a direct way; the response probability density can also be obtained by using the Gaussian characteristic of the irregularity excitation. To further improve the computational efficiency, an efficient spectral representation-based formulation is proposed to calculate the excitation covariance matrix required in the analysis, which avoids tedious Fourier integrals required in the traditional ETDM. In the numerical application, the accuracy and efficiency of the proposed method are validated by the solutions obtained using the Monte Carlo and pseudo-excitation methods; the effects of several parameters on the stochastic responses of the VBS are investigated.