Uncertain Structural Responses Research Articles

A GRU-based ensemble learning method for time-variant uncertain structural response analysis

Owing to the influence of manufacturing and assembly errors, material performance degradation, external loads and unpredictability of the environment during service, structural response analysis should consider the time-invariant uncertainties and time-variant uncertainties simultaneously. In this paper, a mixed uncertainty model with random variable and stochastic process is adopted to handle this issue. A time-variant uncertain structural response analysis method is proposed based on recurrent neural network using gated recurrent units (GRU) combined with ensemble learning. In the proposed method, by performing Latin hypercube sampling (LHS) of random variables, multiple GRU networks can be trained to estimate the time-variant system response under fixed random variables. During the process of training GRU models, an active learning strategy is developed and applied to improve model accuracy and reduce training samples. On this basis, a set of augmented data is generated using the trained GRU models. Then the mapping relationship between random variables and structural responses through the Gaussian process (GP) regression is built accordingly. Eventually, the global surrogate model of time-variant uncertain structural response can be obtained by integrating the GRU networks and the GP models. Two numerical examples are used to demonstrate the effectiveness and accuracy of the proposed method. The results indicate that the proposed method can effectively calculate the expectation and standard deviation of the system response under the mixed uncertainty model with random variables and stochastic processes. In addition, it has higher computational efficiency under the premise of ensuring the calculation accuracy.

Novel solution framework for inverse problem considering interval uncertainty

AbstractThis article proposes a novel solution framework for the inverse problem considering interval uncertainty in structural or systemic responses, which provides an efficient tool for the uncertain inverse problems of nonlinear structures or systems. Interval is used to model and characterize the uncertainty, and the bounds of uncertain structural responses are only required. In each iterative step, the approximate deterministic inverse problem is constructed according to the interval analysis results, and then the identified intervals of inputs are updated by solving the approximate deterministic inverse problem. Therefore, the interval inverse problem is decoupled into a series of interval analyses and deterministic inverse problems that are alternately solved, which dramatically promotes the computational efficiency of the interval inverse problem. Besides, an iterative mechanism is proposed to ensure the convergence of the whole procedure. Finally, two numerical examples and an engineering application are investigated to demonstrate the accuracy and efficiency of the proposed method.

A two-step interval structural damage identification approach based on model updating and set-membership technique

External conditions unavoidably cause deterioration to structures in service, and the degeneration in material property may accumulate and lead to huge accidents. Meanwhile, practical engineering includes a variety of uncertainties from responses, which can negatively influence the damage identification result. In this study, a two-step interval structural damage identification approach is proposed with consideration of response uncertainty. Deterministic damage identification strategy by which damage index can be solved is firstly introduced. Then, damage locations are identified as the first step based on model updating. For the second step, uncertain structural dynamic responses are viewed as interval variables. Then, the bounds of the damage extents of the identified elements are estimated using Chebyshev polynomial approximation and interval mathematics. For a tighter solution, set-membership technique is used to obtain the fusion interval damage index. Two numerical verifications and an experimental verification show the use of the presented two-step identification method. In addition, the Taylor expansion and interval combination methods are also analyzed for comparable purpose.

Applications of multifidelity reduced order modeling to single and multiphysics nonlinear structural problems

The present investigation focuses on the response of uncertain structures that are excited well beyond their linear regime so that significant nonlinear geometric effects take place. Such situations may occur due to the application of large mechanical loads but also, and especially, when the structure is part of a multiphysics problem, e.g., coupled to aerodynamics and/or heating. It is demonstrated here that the computational burden associated with the prediction of the statistics of the response can be dramatically reduced by combining nonlinear reduced order modeling and multifidelity Monte Carlo (MFMC) simulation. The MFMC approach combines the simulations of several different models of different fidelity to accurately estimates the desired statistics. This approach is well suited to reduced order models where different fidelities are easily obtained with different number of basis functions. Three such nonlinear applications are considered the first of which is a purely structural problem. The second one is a multiphysics problem, a panel undergoing a simulated high speed trajectory with aerodynamic-structural-thermal coupling. The third application is also multiphysics and focuses on the limit cycle oscillation behavior of a wing past flutter due to structural nonlinearity. In all of these applications, the MFMC performed very well leading to accurate predictions of the statistics of the response at a reduced/much reduced computational cost.

Stochastic nonlocal damage analysis by a machine learning approach

A machine learning aided stochastic nonlocal damage analysis framework is proposed for quasi-brittle materials. The uncertain system parameters, including the material properties and loading actions, have been incorporated and analysed within a unified safety assessment framework against various working conditions. A three-dimensional integral-type nonlocal damage model through finite element method (FEM) has been adopted. For the purpose of investigating the probabilistic damage analysis problems, a freshly established machine learning approach, namely the capped-extended-support vector regression method (C-X-SVR), is proposed to eliminate the influences of random outliers in the first step, then establish the relationship between the uncertain systemic inputs and structural responses. Such that the training robustness and computational adaptability of the proposed regression model can be reinforced. Moreover, the proposed approach is competent of efficiently predicting the statistical information (i.e., means, standard deviations, probability density functions and cumulative density functions) of structural behaviours under continuous information update of the uncertain working condition from mercurial environment. One real-life experimental validation and two numerical investigations are implemented to further verify the effectiveness and efficiency of the uncertainty quantification framework against probabilistic damage analysis.

Application of Recurrent Neural Networks in the numerical analysis of reinforced concrete structures considering polymorphic uncertainty

AbstractThe realistic modelling of structures is fundamental for their numerical simulation and is mainly characterized by the physical approach and the consideration of the available data by an adequate uncertainty description. The key idea in the contribution at hand is the consideration of polymorphic uncertainty in the numerical structural analysis and in the mechanical modelling for reinforced concrete structures, which are characterized by a combination of heterogeneous concrete and different types of reinforcement (e.g. steel bars or woven carbon fibres). Typically, the reinforcement is determined by another length scale compared to the overall structural size. The formulation and development of a computational homogenization approach, considering the different homogeneous and heterogeneous characteristics of a macroscopic structure, are essential for a realistic numerical computation. In recent years, focal point of research was on structural analysis considering uncertain material or geometry parameters. Probabilistic approaches are dominating the uncertainty consideration currently, although they are associated with specific disadvantages and limitations. In this contribution, a generalized uncertainty model is utilized in order to take variability, impression as well as inaccuracy, vagueness and incompleteness into account. This allows a separated evaluation of the influence for each uncertainty source on the results. Therefore, polymorphic uncertainty models are applied and developed by combining aleatoric and epistemic uncertainty descriptions, resulting e.g. in the formulation of the uncertainty model “fuzzy probability based randomness”. The information of the different length scales is considered to be uncertain, e.g. the geometry or the material properties of a representative volume element (RVE) at the meso scale. Subsequently, the uncertainty of a macro structure is derived from uncertain results on the meso structure. In the contribution, a parametrized RVE for concrete structures, including cement phase as well as aggregates, is presented. Various material parameters are considered as uncertain, which results in uncertain effective quantities. Assuming, that an uncertain quantity is substitutable by a combination of a representative measure and multiple uncertainty characterizing measures, an approximation of both type of quantities by a recurrent neural network is carried out. The surrogate models are utilized as constitutive description in a numerical structural analysis on macro level, providing uncertain structural responses.

Unified uncertainty analysis under probabilistic, evidence, fuzzy and interval uncertainties

The uncertainty analysis of structures generally involves uncertain parameters of different types. In order to derive predictions regarding uncertain structural responses, it is crucial to represent the uncertainty appropriately according to the underlying information available. This paper presents a unified framework for uncertainty analysis under probabilistic, evidence, fuzzy and interval uncertainties, by which the quantities with sufficient data, sparse data, and subjective information can be simultaneously considered. A Taylor expansion-based unified uncertainty analysis (T-UUA) method is first proposed for small uncertainty problems. By temporarily neglecting the evidence, fuzzy and interval variables, the probability-evidence-interval-fuzzy model is degraded into a random problem, in which the expectations and variances of responses can be obtained as functions in terms of evidence, interval and fuzzy uncertainties. Then, through dealing with the evidence variables, the previous expectations and variances are further expressed as a summation of functions in terms of fuzzy and interval variables with basic probability assignments (BPAs). The fuzziness is then discretized by using α-cut technique and thus the expectations and variances are further expressed as functions of only intervals. Afterwards, by reconsidering the interval uncertainties, the bounds of the expectations and variances are computed via combining Taylor expansion with interval arithmetic. In addition, a dimensional reduction (DR)/efficient global optimization (EGO)-based unified uncertainty analysis (DR/EGO-UUA) method is also presented to solve the large uncertainty problems. The framework of DR/EGO-UUA is similar as T-UUA. However, in DR/EGO-UUA, the second moments of responses are computed by DR integrations, and their upper and lower bounds are calculated by the EGO. Finally, three numerical examples are investigated to demonstrate the effectiveness of the proposed methods.

Optimization Approaches for Durable Reinforced Concrete Structures considering Interval and Stochastic Parameter Uncertainty

AbstractThe durability of reinforced concrete structures is dominated by steel reinforcement corrosion and uncertain service loads, which have to be considered in the lifetime oriented structural design. The transport of corrosive substances into the structure is considerably influenced by load induced cracking. The crack width therefore is a major controlling factor for the lifetime of reinforced concrete structures. To improve the design for durability, finite element models in combination with optimization approaches for polymorphic uncertain data are presented. Here, the crack width at the reinforcement layer is used as the optimization objective to be minimized. The structural reliability is treated as a constraint of the optimization task in terms of the accepted failure probability. The concrete covers of the reinforcement layers are chosen as interval design parameters, with midpoints to be optimized and a given radius to take construction imprecision into account. The structural loading, the Young's modulus of concrete and the corresponding tensile strength are considered as stochastic a priori parameters within the optimization, which is solved by a particle swarm optimization approach in combination with an artificial neural network surrogate model. The polymorphic uncertain structural response is computed by a combination of Monte Carlo simulations and optimization‐based interval analysis to consider the stochastic and interval parameters within the optimization task, respectively. An application example demonstrates the performance of the proposed approach.

Enhanced uncertain structural analysis with time- and spatial-dependent (functional) fuzzy results

Abstract Uncertain structural analysis is the computation of uncertain structural response for uncertain input quantities. In terms of structural analysis, the result quantities (e.g. displacements, stresses, damages, …) are time- and spatial-dependent (τ and θ respectively), such that x u ↦ z u ( τ , θ ) needs to be solved. The usual procedure is to a priori select a small amount of Quantities of Interest (QoI), e.g. for a specific time and spatial location. For this small amount, the uncertainty analysis is performed, but these results do not represent the structural behaviour; they are just an excerpt based on a priori knowledge (in general not available). The consideration of time- and spatial-dependent results yields a high amount of uncertain result quantities. The main goal of this contribution is the formulation of time and spatial dependency based on the continuous space, which allows the general application to structural analysis. Furthermore, the computation of a large amount of fuzzy result quantities is addressed. Therefore, a fuzzy analysis on the basis of a specific fuzzy sampling is introduced. The third aspect discussed is the enhanced fuzzy structural analysis, an approach to compute, visualise and evaluate functional fuzzy results. The necessity and usefulness of these procedures, in terms of identifying failure modes and weak points, is discussed by two examples.

Eigenfrequency and deflection analysis of layered structure using uncertain elastic properties – a fuzzy finite element approach

This is the first-time a higher-order fuzzy finite element model has been proposed and implemented to compute the laminated structural responses (modal frequency and static deflection) numerically including the uncertain elastic properties. The composite structure is modelled via a higher-order mid-plane theory which accounts the parabolic variation of the shear stress in conjunction with the fuzzified properties. The fuzzy arithmetic steps are implemented for the composite property evaluation via the triangular membership functions (α-cut method) and solved numerically via finite element techniques. The variational technique and the classical Hamilton's principle are utilized to derive the necessary structural governing equation of the laminated structure including the fuzzified properties. Subsequently, the frequency and the static deflection values are obtained computationally via an efficient home-made computer code (MATLAB environment) with the help of current higher-order finite element model. Further, different kinds of numerical examples are solved using the current higher-order fuzzified model to demonstrate the convergence and comparison behaviour. The differences between the available deterministic (3D-FEM and exact solution) and the uncertain (fuzzified) structural responses indicate the necessity of the current model including the accuracy level. Lastly, the influence of the individual and the combined effect of the material (fuzzified elastic property) and the geometrical parameters on the final structural responses are computed and inferred in details.

Uncertainty analysis for structures with hybrid random and interval parameters using mathematical programming approach

A novel computational method, namely the unified perturbation mathematical programming (UPMP) approach, for hybrid uncertainty analysis of engineering structures is proposed in this paper. The presented study considers a mixture of random and interval system parameters which are frequently encountered in engineering applications. Within the UPMP approach, matrix perturbation theory is adopted in combination with the mathematical programming approach. The proposed computational method provides a non-simulative hybrid uncertainty analysis framework, which is competent to offer the extreme bounds of the statistical characteristics (i.e., mean and variance) of any concerned structural responses in computationally tractable fashion. In order to thoroughly explore various intricate aspects of the engineering system involving hybrid uncertainties, systematic numerical experiments have also been conducted. Diverse statistical analyses are implemented to identify the bounded probability profile of the uncertain structural responses. Both academic and practical engineering structures are investigated to justify the applicability, accuracy and efficiency of the proposed UPMP approach.

Structural time-dependent reliability assessment of the vibration active control system with unknown-but-bounded uncertainties

The active control system for structural vibration is extremely sensitive to the parametric uncertainty so that more and more concerns of its reliability estimation have been given recently. In view of the insufficiency of the uncertainty information in practical engineering, a non-probabilistic time-dependent reliability method that combines the active vibration control theory with interval analysis is proposed in this paper to effectively estimate the dynamic safety of the controlled structures, in which circumstances the unknown-but-bounded uncertainties in structural parameters are considered. The uncertain structural responses based on the closed-loop control are firstly analyzed and embodied by the interval process model. By virtue of the first-passage theory, an integral procedure of non-probabilistic time-dependent reliability analysis of the active control system for structural vibration is then conducted. Two engineering examples and one experimental application are eventually presented to demonstrate the validity and applicability of the methodology developed.

Inverse system method for dynamic loads identification via noisy measured dynamic responses

Purpose – The purpose of this paper is to focus on the influences of the uncertain dynamic responses on the reconstruction of loads. Design/methodology/approach – Based on the assumption of unknown-but-bounded (UBB) noise, a time-domain approach to estimate the uncertain time-dependent external loads is presented by combining the inverse system method in modern control theory and interval analysis in interval mathematics. Inspired by the concept of set membership identification in control theory, an interval analysis model of external loads time history, which is indeed a region or feasible set containing all possible loads being consistent with the bounded structural acceleration responses is established and further solved by two interval algorithms. Findings – Unlike traditional loads identification methods which only give a point estimation, an interval estimation of external loads time history, which is a region containing all the possible loads being consistent with the uncertain structural responses, is determined. The correlation characteristics among the responses of acceleration, velocity, and displacement are also discussed in consideration of the UBB uncertainty. Originality/value – For one hand, the solution of the inverse problem in original system is transformed to the solution of the direct problem in inverse system; for another, the authors deal with the uncertainty by use of interval analysis method, and the identified interval process, which contains any possible external loads time history being consistent with the bounded structural responses can be approximately obtained.

Time-variant random interval response of concrete-filled steel tubular composite curved structures

Abstract Comprehensive investigation into uncertain time-dependent structural responses of concrete-filled steel tubular composite curved structures is presented. Diverse uncertainties including variational material properties, inevitable viscoelastic effects, as well as random environmental influences are addressed simultaneously. A new hybrid probabilistic and interval computational method is proposed for robustly assessing both time-dependent serviceability and strength limit of concrete-filled steel tubular arch structures involving both random and interval variables. The proposed approach offers a unified analysis framework, which adequately delivers the time-dependent bounds of statistical characteristics (i.e., lower and upper bounds of means and standard deviations) of structural behaviours. In this non-simulative computational scheme, general formulations on the time-dependent bounds of means and standard deviations are explicitly derived. Applicability, accuracy and efficiency of the proposed method are revealed by practically motivated examples. In addition, time-dependent hybrid probabilistic interval features of structural responses of concrete-filled steel tubular composite arch structures are systematically studied.

Static response analysis of structures with interval parameters using the second-order Taylor series expansion and the DCA for QB

In this paper, based on the second-order Taylor series expansion and the difference of convex functions algorithm for quadratic problems with box constraints (the DCA for QB), a new method is proposed to solve the static response problem of structures with fairly large uncertainties in interval parameters. Although current methods are effective for solving the static response problem of structures with interval parameters with small uncertainties, these methods may fail to estimate the region of the static response of uncertain structures if the uncertainties in the parameters are fairly large. To resolve this problem, first, the general expression of the static response of structures in terms of structural parameters is derived based on the second-order Taylor series expansion. Then the problem of determining the bounds of the static response of uncertain structures is transformed into a series of quadratic problems with box constraints. These quadratic problems with box constraints can be solved using the DCA approach effectively. The numerical examples are given to illustrate the accuracy and the efficiency of the proposed method when comparing with other existing methods. Although the first-order parameter perturbation method (FM) is effective for solving the static response problem of structures with interval parameters with small uncertainties, the method may fail to estimate the region of the static response of uncertain structures if the uncertainties in the parameters are fairly large. To resolve this problem, a new approach (the proposed method:PM) is presented based on the second-order Taylor series expansion and the DCA for QB(the difference of convex functions algorithm for quadratic problems with box constraints).

Response and reliability analysis of nonlinear uncertain dynamical structures by the probability density evolution method

The paper deals with the response and reliability analysis of hysteretic or geometric nonlinear uncertain dynamical systems of arbitrary dimensionality driven by stochastic processes. The approach is based on the probability density evolution method proposed by Li and Chen (Stochastic dynamics of structures, 1st edn. Wiley, London, 2009; Probab Eng Mech 20(1):33–44, 2005), which circumvents the dimensional curse of traditional methods for the determination of non-stationary probability densities based on Markov process assumptions and the numerical solution of the related Fokker–Planck and Kolmogorov–Feller equations. The main obstacle of the method is that a multi-dimensional convolution integral needs to be carried out over the sample space of a set of basic random variables, for which reason the number of these need to be relatively low. In order to handle this problem an approach is suggested, which reduces the number of basic random variables to merely a single one. Correspondingly, the response and reliability problems reduce to the solution of one-dimensional quadratures.

Reliability sensitivity estimation of nonlinear structural systems under stochastic excitation: A simulation-based approach

Abstract This work explores the feasibility of using advanced simulation-based methods for reliability sensitivity analysis of nonlinear structural systems subject to stochastic excitation. Reliability is defined in terms of a first excursion probability which measures the chances that uncertain structural responses exceed in magnitude prescribed threshold levels within a specified time interval. The proposed reliability sensitivity analysis consists of applying a simple post-processing step associated with an advanced sampling-based reliability analysis. It does not require any additional system response evaluations. In addition, the method provides information about the whole trend of the reliability sensitivity estimates in terms of the threshold levels. The approach is validated on two nonlinear structural models under stochastic ground excitation.

Generalized variability response functions for two-dimensional elasticity problems

The ability to determine probabilistic characteristics of response quantities in structural mechanics (e.g. displacements, stresses) as well as effective material properties is restricted due to lack of information on the probabilistic characteristics of the uncertain system parameters. The concept of the variability response function (VRF) has been proposed as a means to systematically capture the effect of the stochastic spectral characteristics of uncertain system parameters modeled by homogeneous random fields on the uncertain structural response. The key property of the VRF in its classical sense is its independence from the marginal probability distribution function (PDF) and the spectral density function (SDF) of the uncertain system parameters (it depends only on the deterministic structural configuration and boundary conditions). Proofs have been provided for the existence of VRFs for linear and some nonlinear statically determinate beams. For statically indeterminate structures, the Monte Carlo based generalized variability response function (GVRF) methodology has been proposed recently as a generalization of the VRF concept to indeterminate linear and some nonlinear beams. The methodology computes GVRFs, which are analogous to VRFs for statically determinate structures, and evaluates their dependence (or lack thereof) on the PDF and SDF of the random field, thereby providing an estimate of the accuracy of the GVRF. In this paper, the GVRF methodology is extended to problems involving two-dimensional, linear continua whose stochasticity is characterized by statistically homogeneous random fields. After detailing the GVRF methodology for two-dimensional random fields, two numerical examples are provided: GVRFs are computed for the displacement response and for the effective compliance of linear plane stress systems.

Evidence-theory-based structural static and dynamic response analysis under epistemic uncertainties

Evidence theory has a strong ability to deal with epistemic uncertainty, based on which the imprecise parameters with limited information can be conveniently treated. In this paper, a numerical method is developed to compute the linear elastic static and dynamic responses of structures with epistemic uncertainty based on evidence theory. Inspired by the moment concept in probability theory, the Raw Moments, Central Moments and Mixed Central Moments are proposed to describe the distribution characteristics of evidence variables, and the corresponding moments of functions with evidence variables are also defined. By integrating the moment concept and finite element method, a linear elastic static and dynamic response analysis technique is formulated to compute the moments of uncertain structural responses. To reduce the computational cost, the interval analysis technique is adopted to obtain the approximate response bounds for each focal element. Three numerical examples are investigated to demonstrate the effectiveness of the present method.

Optimized and Robust Design of Tires Based on Numerical Simulation

ABSTRACT: This publication presents a numerical approach for analyzing tires based on multiobjective optimization, with particular consideration of uncertainties. Within the optimization, which uses evolutionary algorithms, the evaluation of a three-dimensional, finite element tire model at steady-state rolling is performed. To obtain a reliable and high-quality design, data uncertainty caused, e.g., by variation in production conditions of the tire components, as well as incomplete information concerning loading, have to be considered. Among several design goals, this study looked at durability as an example. An improvement is achieved by the consideration of two objective functions: one focusing on reducing wear, and the other on providing resistance to fatigue. In addition, the proposed optimization measures robustness implicitly. A tire model is regarded as robust when large variations of the uncertain influencing factors mentioned, e.g., loading or material properties, lead to only minor variations in uncertain structural responses, e.g., strains, stresses, or contact pressures. To improve the numerical efficiency of the proposed design approach, a response-surface approximation, based on artificial neural networks, is applied.