Probability Distribution Of Response Research Articles

Dimension reduction for constructing high-dimensional response distributions by accounting for unimportant and important variables

Probability distributions of responses have been widely used in structural analysis and design because of their complete statistical information. In practice, the dimensionality of input variables could easily reach hundreds or thousands, making it computationally expensive to obtain accurate distributions. In this paper, a generalized most probable point (MPP) method is developed to effectively build the response distributions of high-dimensional problems. First, a global index based on one-iteration MPPs is presented for dimension reduction, which is to divide the input variables into important and unimportant variables. After fixing the unimportant variables at their one-iteration MPP components, the MPP components of the important variables are obtained by performing the inverse first-order reliability method (FORM) in the reduced space. Predictive models of the all MPP components are then established to quickly estimate the MPPs of other cumulative distribution function (CDF) values. To accurately calculate CDF points of limit state functions with different shapes, a comprehensive uncertainty analysis method that accommodates the contributions of the important and unimportant variables is proposed by multiple combinations of FORM, second-order reliability method, and second-order saddlepoint approximation. Finally, the response distributions are generated based on Gaussian mixture distribution and all CDF points. The effectiveness of the proposed method is verified by a mathematical example and two engineering cases.

QMU Analysis of Flexoelectric Timoshenko Beam by Evidence Theory

In recent years, with the rapid development of nanotechnology, a new type of electromechanical coupling effect similar to the piezoelectric effect, the flexoelectric effect, has gradually come into the public’s view. The flexoelectric beam that is the main structural unit of the flexoelectric signal output has broad application prospects in the next generation of micro- and nanoelectromechanical systems. Therefore, the investigation of flexoelectric materials and structures has important scientific and engineering application significances for the design of flexoelectric devices. In this paper, a model of flexoelectric Timoshenko beam is established, the deflection, rotation angle, and dynamic electrical signal output of the forced vibration are taken as the system response, and the density ρ , shear correction factor κ , and frequency ratio λ are selected as the key performance parameters of the system. The combination of available data and engineers’ experience suggests that there are random and cognitive uncertainties in the parameters. Therefore, the probability distribution of the system performance response is expressed by the likelihood function and belief function through the quantification of margins and uncertainties (QMUs) analysis methodology under the framework of evidence theory, and the system reliability or performance evaluation is measured by the calculated confidence factors. These results provide a theoretical basis for accurate analysis of flexoelectric components and provide guidance for the design of flexoelectric components with excellent performance.

A multi-region active learning Kriging method for response distribution construction of highly nonlinear problems

Probability distributions of structural responses have been widely used in many engineering applications and their accuracy could significantly affect the performance and credibility of these applications. To obtain accurate distributions, existing methods often need massive calculations of the original response function, especially for highly nonlinear problems. To alleviate the computational burden, this paper proposes a multi-region active learning Kriging method to construct response distributions of highly nonlinear problems. First, a low-precision CDF curve is built based on the one-iteration MPP search and MPP prediction. Multiple regions and response values are then identified to obtain the limit state surfaces for Kriging modeling. To determine the best training points, a multi-region learning strategy including rough and precise selections is developed based on the U learning function and a hybrid index of the reward-based probability and nonlinearity. A two-level stopping criterion is further provided to achieve fast convergence and high accuracy. Finally, the response distributions are constructed using the obtained Kriging model. The effectiveness of the proposed method is verified by four examples.

Direct Probability Integral Method for Seismic Performance Assessment of Earth Dam Subjected to Stochastic Mainshock–Aftershock Sequences

Studying the impact of mainshock–aftershock sequences on dam reliability is crucial for effective disaster prevention measures. With this purpose in mind, a new method for stochastic dynamic response analyses and reliability assessments of dams during seismic sequences has been proposed. Firstly, a simulation method of stochastic seismic sequences is described, considering the dependence between mainshock and aftershock based on Copula function. Then, a novel practical framework for stochastic dynamic analysis is established, combined with the improved point selection strategy and the direct probability integration method (DPIM). The DPIM is employed on a nonlinear system with one degree of freedom and compared with Monte Carlo simulation (MCS). The findings reveal that the method boasts exceptional precision and efficiency. Finally, the seismic performance of a practical dam was evaluated based on the above method, which not only accurately estimates the response probability distribution and dynamic reliability of the dam, but also greatly reduces the required calculations. Furthermore, the impact of aftershocks on dam seismic performance is initially evaluated through a probability approach in this research. It is found that seismic sequences will significantly increase the probability of earth dam failure compared with sequences of only mainshocks. In addition, the influence of aftershocks on reliability will further increase when the limit state is more stringent. Specifically, the novel analysis method proposed in this paper provides more abundant and objective evaluation indices, providing a dynamic reliability assessment for dams that is more effective than traditional evaluation methods.

Prediction of Long-Term Offshore Structural Responses Based on Nonlinear Wave Modeling

Wind-generated random wave loads are the dominant loads to consider for maintaining the reliability of fixed offshore structures. Based on probabilistic techniques, the inherent randomness of the wave loading can be used to predict extreme offshore structural response, which is Gaussian in nature. However, researchers have found that the hydrodynamic component and structural dynamics substantially impact the frequency spectrum, leading to a non-Gaussian stochastic offshore structural response. A finite-memory non-linear system (FMNSNL) has been proven to be an efficient approach to evaluate the non-Gaussian stochastic offshore structural response compared to the conventional method, Monte Carlo time simulation. However, the analysis has been conducted based on short-term distribution only. The most satisfactory analysis is based on long-term distribution. Hence, further investigation in this paper will evaluate the long-term probability distribution of extreme offshore structural responses. As a result, a 100-year extreme offshore structural response prediction achieves 80% to 96% accuracy compared to the Monte Carlo time simulation. The probability distribution has been evaluated using the Gumbel distribution function throughout this investigation. Still, there is a little deviation at the tail end of the distribution between the simulated response values and the fitted line. A different distribution function, such as the Generalised Extreme Value (GEV) distribution, is advised for future work.

Stationary response analysis for a linear system under non-Gaussian random excitation by the equivalent non-Gaussian excitation method and the Hermite moment model

An approximate analytical method is developed to obtain the stationary response probability density and mean upcrossing rate for a linear system subjected to non-Gaussian random excitation. The random excitation is a stationary stochastic process prescribed by an arbitrary first-order probability density and a class of power spectral density with a bandwidth parameter, and is described by an Itô stochastic differential equation. In the proposed method, the equivalent non-Gaussian excitation method is first utilized to derive a closed set of moment equations for the system response. The exact solutions of response moments up to the fourth order are calculated from the moment equations. Next, using the response moments, the Hermite moment model is built for the stationary displacement response, and its probability density function is obtained through the model. Finally, the mean upcrossing rate of the displacement is determined from the response probability distribution by making two approximations that the displacement and velocity of the system are independent and the velocity is Gaussian. The proposed method can be used for any excitation probability density and bandwidth parameter value as long as the skewness and kurtosis of the response are within the applicable range of the Hermite moment model. In illustrative examples, the proposed method is applied to linear systems under random excitations with two-types of non-Gaussian probability densities. The effectiveness of the method is demonstrated by comparing the analytical results with the pertinent Monte Carlo simulation results while widely varying the bandwidth ratio between the excitation spectral density and the frequency response function of the system. Moreover, the effect of excitation non-Gaussianity on the mean upcrossing rate is discussed. It is shown that for small bandwidth ratio, the crossing rate differs considerably depending on the excitation probability density. The difference due to the excitation distribution decreases monotonically as the bandwidth ratio increases, and the crossing rate becomes almost the same as that in the case of Gaussian excitation when the bandwidth ratio is large.

Electrical Response Estimation of Vibratory Energy Harvesters via Hilbert Transform Based Stochastic Averaging

Abstract Converting mechanical vibrations into electrical power with vibratory energy harvesters can ensure the portability, efficiency, and sustainability of electronic devices and batteries. Vibratory energy harvesters are typically modeled as nonlinear oscillators subject to random excitation, and their design requires a complete characterization of their probabilistic responses. However, simulation techniques such as Monte Carlo are computationally prohibitive when the accurate estimation of the response probability distribution is needed. Alternatively, approximate methods such as stochastic averaging can estimate the probabilistic response of such systems at a reduced computational cost. In this paper, the Hilbert transform based stochastic averaging is used to model the output voltage amplitude as a Markovian stochastic process with dynamics governed by a stochastic differential equation with nonlinear drift and diffusion terms. Moreover, the voltage amplitude dependent damping and stiffness terms are determined via an appropriate equivalent linearization, and the stationary probability distribution of the output voltage amplitude is obtained analytically by solving the corresponding Fokker–Plank equation. Two examples are used to demonstrate the accuracy of the obtained analytical probability distributions via comparisons with Monte Carlo simulation data.

Probability Distribution of Transient Response of an Electric Circuit

In this work, we explore the transient response of an electric circuit using probabilistic analysis when the precise value of the characteristic parameter of a circuit element is not known and is instead constrained to a range of values. To accomplish this, a combination of probability theory and traditional circuit analysis techniques is employed. In this approach, the circuit element’s parameter is treated as a random variable, and probability methods are used to derive the probability distribution of the voltage or current across it. To demonstrate this approach, we apply it to a resistor capacitor (RC) circuit, where the capacitance parameter of the capacitor is regarded as a random variable. We derive probability distributions for the transient voltage across the capacitor and the voltage at the output terminals to illustrate the method’s effectiveness.

Selection of hazard-consistent ground motions for risk-based analyses of structures

In seismic risk analysis, probabilistic seismic demand analysis (PSDA) is required to determine the probability distribution of structural seismic responses. One of the key steps in PSDA is to select a suite of ground motions that are consistent with seismic hazards at a target site. Current methods for selecting hazard-consistent ground motions only achieve consistency at one or several predefined periods and choose several discrete intensity levels to consider uncertainty of ground motions. To avoid drawbacks of the current methods, this study proposes a novel method for selecting hazard-consistent ground motions. In this method, a scenario earthquake set for ground motion selection is firstly obtained from seismic hazard disaggregation to a hazard level corresponding to a very small value of spectral acceleration at a specific period. Then, a suite of random target response spectra that are consistent with target site seismic hazards are simulated based on the scenario earthquake set. Finally, one suite of hazard-consistent ground motions are selected from a ground motion database. Through a numerical example, this study concludes, a suite of selected ground motions using the proposed method presents very good consistency with the target site seismic hazards. Since hazard-consistent ground motions selected using the proposed method do not depend on the building information, they can be used to accurately perform PSDA of any building and accurately predict structural seismic responses.

Dynamic reliability analysis of structures under nonstationary stochastic excitations using tail-modified extreme value distribution

In practical engineering, it is usually necessary to compute the small first-passage probability of dynamic systems exposed to nonstationary stochastic excitations. However, it is still a challenging problem to efficiently estimate the super-small failure probability using an approximate extreme value distribution (EVD) based on a small number of simulated samples. In this study, a novel EVD model is proposed to describe the probability distribution of the structural extreme response induced by nonstationary stochastic excitations. This EVD model consists of two parts: the main body is modeled by a truncated-shifted generalized lognormal distribution, and a monotonic exponential model is proposed to fit the tail region. Besides that, a criterion for determining the breakpoint between the main body and tail region is proposed, which ensures the non-negativity and normativity of the proposed EVD model and allows the EVD to be reconstructed accurately and efficiently, particularly in its tail region. In this regard, the corresponding first passage probabilities under different thresholds can be straightly evaluated. The precision and completeness of the proposed EVD model are investigated using two numerical examples that consider linear and nonlinear frame structures subjected to nonstationary ground motion accelerations. The results show that the proposed model is in excellent agreement with the results of Monte Carlo simulation, even when the failure probability is very small.

EESD special issue: AI and data‐driven methods in earthquake engineering – (Part 1)

EESD special issue: AI and data‐driven methods in earthquake engineering – (Part 1)

Nonparametric probabilistic seismic demand model and fragility analysis of subway stations using deep learning techniques

A novel and computationally efficient method for developing a nonparametric probabilistic seismic demand model (PSDM) is proposed to conduct the fragility analysis of subway stations accurately and efficiently. The probability density evolution method (PDEM) is used to calculate the evolutionary probability density function of demand measure (DM) without resort to any assumptions of the distribution pattern of DM. To reduce the computational cost of a large amount of nonlinear time history analyses (NLTHAs) in the PDEM, the one-dimensional convolutional neural network (1D-CNN) is used as a surrogate model to predict the time history of structural seismic responses in a data-driven fashion. The proposed nonparametric PSDM is adopted to conduct the fragility analysis of a two-story and three-span subway station, and the results are compared with those from two existing parametric PSDMs, i.e., two-parameter lognormal distribution model and probabilistic neural network (PNN)-based PSDM. The results show that the PDEM-based PSDM has the best performance in describing the probability distribution of seismic responses of underground structures. Different from the fragility curves, the time-dependent fragility surface of the subway station shows how the exceedance probability of damage state changes over time. It can be used to estimate the escape time and thus the number of casualties in an earthquake, which are important indexes when conducting the resilience-based seismic evaluation.

Surrogate modeling of structural seismic response using probabilistic learning on manifolds

AbstractNonlinear response history analysis (NLRHA) is generally considered to be a reliable and robust method to assess the seismic performance of buildings under strong ground motions. While NLRHA is fairly straightforward to evaluate individual structures for a select set of ground motions at a specific building site, it becomes less practical for performing large numbers of analyses to evaluate either (1) multiple models of alternative design realizations with a site‐specific set of ground motions, or (2) individual archetype building models at multiple sites with multiple sets of ground motions. In this regard, surrogate models offer an alternative to running repeated NLRHAs for variable design realizations or ground motions. In this paper, a recently developed surrogate modeling technique, called probabilistic learning on manifolds (PLoM), is presented to estimate structural seismic response. Essentially, the PLoM method provides an efficient stochastic model to develop mappings between random variables, which can then be used to efficiently estimate the structural responses for systems with variations in design/modeling parameters or ground motion characteristics. The PLoM algorithm is introduced and then used in two case studies of 12‐story buildings for estimating probability distributions of structural responses. The first example focuses on the mapping between variable design parameters of a multidegree‐of‐freedom analysis model and its peak story drift and acceleration responses. The second example applies the PLoM technique to estimate structural responses for variations in site‐specific ground motion characteristics. In both examples, training data sets are generated for orthogonal input parameter grids, and test data sets are developed for input parameters with prescribed statistical distributions. Validation studies are performed to examine the accuracy and efficiency of the PLoM models. Overall, both examples show good agreement between the PLoM model estimates and verification data sets. Moreover, in contrast to other common surrogate modeling techniques, the PLoM model is able to preserve correlation structure between peak responses. Parametric studies are conducted to understand the influence of different PLoM tuning parameters on its prediction accuracy.

Probability Distributions of Asphalt Pavement Responses and Performance under Random Moving Loads and Pavement Temperature

Asphalt pavements are damaged by traffic load repetitions. Conventionally, the allowed number of load repetitions until pavement failure is calculated based on empirical transfer functions from deterministic pavement mechanical responses to performance. However, the mechanical responses and damage to the pavement are uncertain under a random realistic traffic load and pavement temperature. Therefore, the non-deterministic problem—that is, the probability distributions of asphalt pavement responses and performance under random moving loads and pavement temperatures—was investigated in this study. Random factors include the load pressure, vehicle wandering, speed, and temperature inside the asphalt layer. A combination of the response surface and first-order reliability methodologies was recommended to calculate the probability of mechanical responses at any point within the pavement, for reasons of computational efficiency. The accuracy of this method was verified by a Monte-Carlo simulation. Then, the effects of the mean values and standard deviations of the random factors on the probability distributions of the mechanical responses were discussed. Finally, probability distributions of pavement performance (i.e., probability density distributions of cumulative damage for fatigue failure and rutting after repeated random loads) were calculated using transfer functions and the probability distributions of the mechanical responses; thereby, the failure probability of the pavement after a given number of load repetitions was obtained. The results show that the previous deterministic analysis could not fully reflect the random characteristics of pavement mechanical responses under realistic random moving loads, and the mean values and standard deviations of the random factors have significant effects on the probability distributions of mechanical responses and performance. The failure probability of the pavement after a given number of load repetitions can be used as a guide to reliability-based pavement design. This study on the probability distributions of asphalt pavement responses and performance exhibits the potential to understand pavement behavior and could be beneficial as a complement during reliability-based pavement design.

Nonstationary random vibration analysis of structures with uncertain parameters based on a polynomial dimensional decomposition

AbstractBased on polynomial dimensional decomposition (PDD), a novel method for analysing nonstationary random vibration of structures with uncertain parameters is proposed in this paper. The uncertain structural parameters are assumed to be independent random variables, with certain types of probability distributions; thus, the nonstationary random vibration responses of the structures become stochastic functions of these uncertain parameters. A dimensional decomposition and Fourier expansion were applied to calculate the statistical characteristics of the nonstationary random vibration responses of the structures. To calculate the conditional power spectral density (CPSD) and conditional standard deviation (CSD) of the nonstationary random vibration responses, CPSD and CSD were explicitly expressed using expansion coefficients and polynomial basis. Subsequently, the probability distributions of the structural responses can be effectively calculated in terms of these explicit expressions. Dimension‐reduction integration and Gauss numerical integration were applied to calculate expansion coefficients. The validity of the proposed method was verified by numerical examples, including a single‐degree‐of‐freedom system and a multi‐degree‐of‐freedom system (reinforced concrete shear wall). The results indicate that the proposed method is as accurate as Monte Carlo simulation (MCS), whereas the proposed method requires considerably smaller number of nonstationary random vibration analyses than MCS. Moreover, the nonstationary random vibration analysis of a long‐span cable‐stayed bridge demonstrates that the proposed method is feasible for quantifying the uncertainties of the nonstationary random vibration responses of complex engineering structures with uncertain parameters.

Approximate analytical methods for stationary response probability density of a SDOF system with a nonlinear spring under non-Gaussian random excitation

Approximate analytical methods for finding the stationary response probability density of a SDOF system with a wide class of nonlinear springs under non-Gaussian random excitation are proposed. The non-Gaussian excitation is a stationary stochastic process prescribed by an arbitrary probability density with finite variance and a power spectral density with a bandwidth parameter. In order to obtain the response probability distribution of the system, two approximate methods are developed by theoretically extending the Monte Carlo simulation observations on the properties of the response distribution obtained in the previous study. One of the methods is applied when the bandwidth of the excitation power spectrum is narrower than that of the primary resonance peak of the system. In this case, the system becomes quasi-static, and its equation of motion reduces to the equilibrium equation for the stiffness and excitation terms. This equilibrium equation is used to derive the approximate solution of the response distribution. The only requirement of this method is that the nonlinear spring of the system is continuous and monotonically increasing. The other method is proposed for the case of the wide excitation bandwidth compared to the system bandwidth. In this method, the exact solution of the response distribution under Gaussian white noise is utilized. The value of the white noise spectral density parameter in the solution is determined by the equivalent linearization. The second method is applicable to systems with arbitrary integrable nonlinear springs. In numerical examples, we consider three nonlinear systems whose spring characteristics are quite different. Besides, two types of non-Gaussian excitation probability densities with differences in shape are used. It is demonstrated that the proposed methods can reproduce the unique shape of the response probability density caused by the spring nonlinearity and the excitation non-Gaussianity. From the results, the range of the bandwidth ratio between the excitation and the system where the response distribution is accurately obtainable is also examined for each method.

Analysis on Influence of Residents’ Response Probability Distribution on Load Aggregation Effect

Residential loads are essential resources, and they can participate in grid scheduling through aggregation, with great potential. However, inherent uncertainties challenge the realization of accurate load aggregation (LA), leading to the failure to fully play a role in grid operation. In this context, this paper analyzes the influence of the probability of residents participating in LA on the actual aggregation effect. Firstly, the LA problem is modeled, and the optimization objective is equivalently transformed from minimizing the mismatch between the actual load regulation and the target to minimizing the sum of variances of the overall response probabilities. Response probabilities of residents are then simulated based on the β distribution model, and various typical distributions can be generated by modifying only two hyperparameters. In order to evaluate the actual aggregation effect, a multi-armed bandit model that can be used for LA is adopted, and an evaluation framework is designed. The simulation results show that for different probability distributions, the smaller the sum of variances of all residents’ response probabilities are, the more accurate aggregation can be achieved.

Establishing the Pharmacokinetics of Genetic Vaccines is Essential for Maximising their Safety and Efficacy.

In a typical course of drug development, thorough pharmacokinetic (PK) studies are essential for the determination of drug biodistribution, dosage and efficacy without toxicity. For vaccines, however, unless a new formulation component is used, most regulatory agencies rule out the need for studying the biodistribution of the vaccine antigenic material per se, and only dose-immunogenicity studies are performed. This is because traditional vaccines are meant to directly induce immunogenicity by locally recruiting immunocytes that will carry on with the pursuing immunogenic processes. Thus, the clinical outcome from traditional vaccines is determined mainly by an immunological response phase. Yet, the case is significantly different for the emergent genetic vaccines (vectorised DNA or mRNA vaccines), where the clinical outcome is dependent on a combination of two major response phases: a pharmacological phase that involves biodistribution, assimilation, gene translation and epitope(s) presentation, followed by an immunological phase, which is similar to that of traditional vaccines. From a mathematical perspective, processes involved in drug administration are typically subject to inter- and intra-patient statistical distributions like most physiological processes. Therefore, the clinical outcome after administering genetic vaccines obeys a statistical probability distribution combined of the sum of two major response probability distributions, pharmacological and immunological. This implies that the variance coefficient of the summed response probability distributions has a larger value than the variance of each underlying distribution. In other words, due to the multi-phased mode of action of genetic vaccines, their clinical outcome has more variability than that of traditional vaccines. This observation points toward the necessity for regulating genetic vaccines in a similar manner to bio-therapeutics to ensure better efficacy and safety. A structural PK model is provided to predict the sources of variability, biodistribution and dose optimisation.

New non-intrusive stochastic finite element method for plate structures

Currently, several intrusive stochastic finite element methods (SFEMs) such as perturbation method and spectral SFEM are widely applied for stochastic response analysis of continuous structures. However, the intrusive SFEMs need to modify conventional finite element formulations to establish the stochastic stiffness matrix, and cannot calculate the probability density function of structural response and the reliability straightforwardly. This paper proposes a new non-intrusive SFEM for efficiently computing stochastic responses and reliabilities of plates in a unified way. Firstly, the direct probability integral method (DPIM) is developed to obtain the probability density function of stochastic response by solving probability density integral equation (PDIE). Secondly, the non-intrusive SFEM based on DPIM decouples the deterministic finite element analysis and PDIE to calculate the stochastic responses and reliabilities of uncertain plate structures, and the discretization and quantification of random fields of elastic modulus and thickness are implemented through Karhunen-Loève expansion. Finally, several examples of uncertain Kirchhoff and Mindlin plates demonstrate the efficiency and versatility of the proposed non-intrusive method by comparing with the results from Monte Carlo simulation and literature. The effects of correlation length, mean and variability of random field on the probability distribution of responses and the reliabilities of plates are revealed. For Gaussian random thickness, the linearly elastic plate yields non-Gaussian distributed responses. Increasing the correlation length and variability of random field reduces the reliabilities of plates.

On High-Dimensional Time-Variant Reliability Analysis with the Maximum Entropy Principle

The structural reliability analysis suffers from the curse of dimensionality if the associated limit state function involves a large number of inputs. This study develops a reliability analysis method that deals with high-dimensional inputs over time. The probability distribution of the structural response is reconstructed by the maximum entropy principle which is achieved by solving an optimization problem derived from the concept of relative entropy. The optimization problem is transformed into a convex one with respect to the orders of fractional moments and the Lagrange multipliers. Additionally, considering the associated computational issues, it is reformulated with side constraints on the parameters of the maximum entropy distribution. Then, a global optimization procedure is performed. The proposed method is successfully applied to the reliability analysis of a linear and a nonlinear structural system, which involves a large number of inputs deriving from the discretization of the input random processes.