A unified stochastic framework for robust topology optimization of continuum and truss-like structures

In this article, a unified framework is introduced for robust structural topology optimization for 2D and 3D continuum and truss problems. The uncertain material parameters are modelled using a spatially correlated random field which is discretized using the Karhunen–Loève expansion. The spectral stochastic finite element method is used, with a polynomial chaos expansion to propagate uncertainties in the material characteristics to the response quantities. In continuum structures, either 2D or 3D random fields are modelled across the structural domain, while representation of the material uncertainties in linear truss elements is achieved by expanding 1D random fields along the length of the elements. Several examples demonstrate the method on both 2D and 3D continuum and truss structures, showing that this common framework provides an interesting insight into robustness versus optimality for the test problems considered.     

Simulation of local material properties based on moving-window GMC

When analyzing the behavior of composite materials under various loading conditions, the assumption is generally made that the behavior due to randomness in the material can be represented by a homogenized, or effective, set of material properties. This assumption may be valid when considering displacement, average strain, or even average stress of structures much larger than the inclusion size. The approach is less valid, however, when considering either behavior of structures of size at the scale of the inclusions or local stress of structures in general. In this paper, Monte Carlo simulation is used to assess the effects of microstructural randomness on the local stress response of composite materials. In order to achieve these stochastic simulations, the mean, variance and spectral density functions describing the randomly varying elastic properties are required as input. These are obtained here by using a technique known as moving-window generalized method of cells (moving-window GMC). This method characterizes a digitized composite material microstructure by developing fields of local effective material properties. Once these fields are generated, it is straightforward to obtain estimates of the associated probabilistic parameters required for simulation. Based on the simulated property fields, a series of local stress fields, associated with the random material sample under uniaxial tension, is calculated using finite element analysis. An estimation of the variability in the local stress response for the given random composite is obtained from consideration of these simulations.     

Influence of uncertainties on the response and reliability of self-adaptive composite rotors

Advanced composite structures are becoming increasingly popular because of their high specific strength and stiffness, as well as ability to provide improved performance through passive morphing via intrinsic bend–twist deformation coupling. Self-adaptive composite structures tend to be more susceptible to geometric, material, and loading uncertainties because of their complex configuration, manufacturing process, and dependence on fluid–structure interaction (FSI) response. The objective of this work is to quantify the effects of material, geometric, and loading uncertainties on the response of self-adaptive composite propellers and overall system reliability. A fully-coupled, 3-D boundary element method–finite element method is used to compute the dynamic FSI response. Variability in propeller performance is estimated by considering variations in operating conditions, as well as blade geometry and stiffness. Modeling uncertainties are considered by employing various mechanistic-based failure initiation models. Random variations in material strengths are implemented and an estimate of the structural reliability is determined. The results indicate that adaptive composite structures that depend on FSI are more sensitive to natural, random variations than equivalent rigid, isotropic structures. Therefore, it is necessary to quantify the effects of material, geometric, and loading uncertainties on the responses, safe operating envelopes, and reliability of self-adaptive composite structures.     

Free vibration and reliability of composite cantilevers featuring uncertain properties

The problem of free vibration and reliability of cantilever composite beams featuring structural uncertainties is analyzed. The random structural uncertainties involve material properties, thickness and fiber orientation of the individual constituent laminae. Such uncertainties undoubtedly affect the achievable performance as well as their structural reliabilities. In order to investigate the effects of random structural uncertainties on free vibration problem, a stochastic eigenvalue problem of self-adjoint systems is formulated to provide first and second moments of eigenvalues, i.e., their mean and variance. In this context, a stochastic finite element method based on the mean-centered-second-moment method and first-order perturbation technique are employed during the probabilistic discretization of uncertain distributed-parameter structural systems.Sensitivity and reliability analyses for the uncertain beam when subjected to an external oscillatory load are performed. In addition, in order to mitigate the detrimental effects of uncertainties and so, to render the structure more robust to such effects, the structural tailoring technique is implemented and its beneficial effects are revealed.     

Level-set topology optimization for robust design of structures under hybrid uncertainties

This paper will develop a new robust topology optimization (RTO) method based on level sets for structures subject to hybrid uncertainties, with a more efficient Karhunen-Loève hyperbolic Polynomial Chaos–Chebyshev Interval method to conduct the hybrid uncertain analysis. The loadings and material properties are considered hybrid uncertainties in structures. The parameters with sufficient information are regarded as random fields, while the parameters without sufficient information are treated as intervals. The Karhunen-Loève expansion is applied to discretize random fields into a finite number of random variables, and then, the original hybrid uncertainty analysis is transformed into a new process with random and interval parameters, to which the hyperbolic Polynomial Chaos–Chebyshev Interval is employed for the uncertainty analysis. RTO is formulated to minimize a weighted sum of the mean and standard variance of the structural objective function under the worst-case scenario. Several numerical examples are employed to demonstrate the effectiveness of the proposed RTO, and Monte Carlo simulation is used to validate the numerical accuracy of our proposed method.     

Influence of uncertainties on the response and reliability of self-adaptive composite rotors

Advanced composite structures are becoming increasingly popular because of their high specific strength and stiffness, as well as ability to provide improved performance through passive morphing via intrinsic bend–twist deformation coupling. Self-adaptive composite structures tend to be more susceptible to geometric, material, and loading uncertainties because of their complex configuration, manufacturing process, and dependence on fluid–structure interaction (FSI) response. The objective of this work is to quantify the effects of material, geometric, and loading uncertainties on the response of self-adaptive composite propellers and overall system reliability. A fully-coupled, 3-D boundary element method–finite element method is used to compute the dynamic FSI response. Variability in propeller performance is estimated by considering variations in operating conditions, as well as blade geometry and stiffness. Modeling uncertainties are considered by employing various mechanistic-based failure initiation models. Random variations in material strengths are implemented and an estimate of the structural reliability is determined. The results indicate that adaptive composite structures that depend on FSI are more sensitive to natural, random variations than equivalent rigid, isotropic structures. Therefore, it is necessary to quantify the effects of material, geometric, and loading uncertainties on the responses, safe operating envelopes, and reliability of self-adaptive composite structures.     

Model order reduction based on proper generalized decomposition for the propagation of uncertainties in structural dynamics

A priori model reduction methods based on separated representations are introduced for the prediction of the low frequency response of uncertain structures within a parametric stochastic framework. The proper generalized decomposition method is used to construct a quasi‐optimal separated representation of the random solution at some frequency samples. At each frequency, an accurate representation of the solution is obtained on reduced bases of spatial functions and stochastic functions. An extraction of the deterministic bases allows for the generation of a global reduced basis yielding a reduced order model of the uncertain structure, which appears to be accurate on the whole frequency band under study and for all values of input random parameters. This strategy can be seen as an alternative to traditional constructions of reduced order models in structural dynamics in the presence of parametric uncertainties. This reduced order model can then be used for further analyses such as the computation of the response at unresolved frequencies or the computation of more accurate stochastic approximations at some frequencies of interest. Because the dynamic response is highly nonlinear with respect to the input random parameters, a second level of separation of variables is introduced for the representation of functions of multiple random parameters, thus allowing the introduction of very fine approximations in each parametric dimension even when dealing with high parametric dimension. Copyright © 2011 John Wiley & Sons, Ltd.     

Optimal design of composite shells based on minimum weight and maximum feasibility robustness

A robust design optimization (RDO) approach for minimum weight and safe shell composite structures with minimal variability into design constraints under uncertainties is proposed. A new concept of feasibility robustness associated to the variability of design constraints is considered. So, the feasibility robustness is defined through the determinant of variance–covariance matrix of constraint functions introducing in this way the joint effects of the uncertainty propagations on structural response. A new framework considering aleatory uncertainty into RDO of composite structures is proposed. So, three classes of variables and parameters are identified: deterministic design variables, random design variables and random parameters. The bi-objective optimization search is performed using on a new approach based on two levels of dominance denoted by Co-Dominance-based Genetic Algorithm (CoDGA). The use of evolutionary concepts together sensitivity analysis based on adjoint variable method is a new proposal. The examples with different sources of uncertainty show that the Pareto front definition depends on random design variables and/or random parameters considered in RDO. Furthermore, the importance to control the uncertainties on the feasibility of constraints is demonstrated. CoDGA approach is a powerfully tool to help designers to make decision establishing the priorities between performance and robustness.     

Interval spectral stochastic finite element analysis of structures with aggregation of random field and bounded parameters

This paper presents the study on non‐deterministic problems of structures with a mixture of random field and interval material properties under uncertain‐but‐bounded forces. Probabilistic framework is extended to handle the mixed uncertainties from structural parameters and loads by incorporating interval algorithms into spectral stochastic finite element method. Random interval formulations are developed based on K–L expansion and polynomial chaos accommodating the random field Young's modulus, interval Poisson's ratios and bounded applied forces. Numerical characteristics including mean value and standard deviation of the interval random structural responses are consequently obtained as intervals rather than deterministic values. The randomised low‐discrepancy sequences initialized particles and high‐order nonlinear inertia weight with multi‐dimensional parameters are employed to determine the change ranges of statistical moments of the random interval structural responses. The bounded probability density and cumulative distribution of the interval random response are then visualised. The feasibility, efficiency and usefulness of the proposed interval spectral stochastic finite element method are illustrated by three numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Spectral stochastic analysis of structures with uncertain parameters

We present a probabilistic analysis of a structure with uncertain parameters subject to arbitrary stochastic excitations in a frequency domain. The problem of stochastic dynamic analysis of a linear system in a frequency domain is formulated by taking into consideration the uncertainty of structural parameters. The solution is based on the idea of a random frequency response vector for stationary input excitation and a transient random frequency response vector for nonstationary one which are used in the context of spectral analysis in order to determine the influence of structural uncertainty on the random response of structure. The numerical spectral analysis of the building structure under wind and earthquake excitation is provided to demonstrate the described algorithms in the context of computer implementation.     

On the dynamics of uncertain coupled structures through a wave based method in mid- and high-frequency ranges

This paper focuses on guided wave propagation in elastic random structures. A numerical tool, referred to as the ’stochastic wave finite element method’ (SWFE) describing uncertain spectral parameters in periodic structures is presented. This approach represents an extension of the wave finite element for homogenous randomness media. The statistics of the kinematic diffusion matrix for two semi-infinite waveguides connected through an uncertain coupling element is offered. The diffusion relationships presented evaluate the statistics of reflection and transmission coefficients for semi-infinite connected waveguides subject to structural and geometrical variabilities on a coupling element. Finally, the effects of the uncertainties on kinematic and energetic parameters are investigated for two finite coupled structures based on the stochastic spectral approach. Numerical experiments show the effectiveness of the proposed formulation to predict the dynamics of periodic systems in mid- and high-frequency ranges with low CPU consumption.     

The use of Monte Carlo techniques in statistical finite element methods for the determination of the structural behaviour of composite materials structural components

The main purpose of this paper is to develop an alternative approach to the classical deterministic design to account for uncertainties encountered during design, construction and lifetime of structures. This approach is based on the use of statistical tools in material characterisation and structural design by means of the finite element method combined with Monte Carlo techniques. In the first instance, the mechanical behaviour of different materials, including composite materials, is characterised by means of stochastic tools. A procedure based on the combination of various methods for estimating distribution parameters has been set up to ensure correct estimation. The second part of the paper focuses on the finite element modelling of structures combined with Monte Carlo simulation to deal with the stochastic aspects of the input parameters (material properties, structure geometry and loading conditions) and determine the probability distribution characterising the structural response.     

复合材料层合板非线性系统随机振动的样条有限元分析

工程结构中的复合材料的几何参数往往具有随机性质,如何研究随机参数非线性系统的随机响应及统计特性,对结构的可靠性设计和优化设计有着非常重要的意义。应用摄动法、随机中心差分法和线化和校正法,建立了复合材料非线性系统的振动方程和计算模型,采用样条有限元法研究了复合材料层合板具有随机参数的非线性系统在确定性荷载下的随机响应,数值算例说明了本算法的正确性。     

Probabilistic analysis of concrete cracking using neural networks and random fields

In this paper an algorithm for the probabilistic analysis of concrete structures is proposed which considers material uncertainties and failure due to cracking. The fluctuations of the material parameters are modeled by means of random fields and the cracking process is represented by a discrete approach using a coupled meshless and finite element discretization. In order to analyze the complex behavior of these nonlinear systems with low numerical costs a neural network approximation of the performance functions is realized. As neural network input parameters the important random variables of the random field in the uncorrelated Gaussian space are used and the output values are the interesting response quantities such as deformation and load capacities. The neural network approximation is based on a stochastic training which uses wide spanned Latin hypercube sampling to generate the training samples. This ensures a high quality approximation over the whole domain investigated, even in regions with very small probability.     

Energy-based collapse assessment of concrete structures subjected to random damage evolutions

The assessment of structural capacity against collapse is conducive to the optimal design of new structures as well as checking the safety of existing structures. However, the evaluation cannot be typically carried out by means of destructive tests on prototype or reduced scale structures. In this regard, the numerical models that adequately represent the prototype structures can be alternatively used. Specifically, both the nonlinearities and randomness as well as their coupling effect of materials need to be represented in a unified manner in structural analysis. The present paper aims at providing an effective approach to incorporate the stochastic nature of damage constitutive relationships in collapse analysis and assessment of concrete structures subjected to earthquake ground motions. Within the framework of stochastic damage mechanics, the spatial variability of concrete is represented by a two-scale stationary random fields. The concept of covariance constraint is introduced to bridge the two-scale random fields such that the scale-of-fluctuation of the random material property is satisfied at both scales. Random damage evolution induced structural collapse analysis is achieved via the nonlinear stochastic finite element method. To address the randomness propagation across scales, the probability density evolution method is employed. By exerting the absorbing boundary condition associated with an energy-based collapse criterion on the generalized probability density evolution equation, the anti-collapse reliability of concrete structures can be evaluated with fair accuracy and efficiency. Numerical investigation regarding an actual high-rise reinforced concrete frame-shear wall structure indicates that the random damage evolution of concrete dramatically affects the structural nonlinear behaviors and even leads to entirely different collapse modes. The proposed method provides a systematic treatment of both uncertainties and nonlinearities in collapse assessment of complex concrete structures.     

Truss layout design under nonprobabilistic reliability-based topology optimization framework with interval uncertainties

A nonprobabilistic reliability-based topology optimization (NRBTO) method for truss structures with interval uncertainties (or unknown-but-bounded uncertainties) is proposed in this paper. The cross-sectional areas of levers are defined as design variables, while the material properties and external loads are regard as interval parameters. A modified perturbation method is applied to calculate structural response bounds, which are the prerequisite to obtain structural reliability. A deviation distance between the current limit state plane and the objective limit state plane, of which the expression is explicit, is defined as the nonprobabilistic reliability index, which serves as a constraint function in the optimization model. Compared with the deterministic topology optimization problem, the proposed NRBTO formulation is still a single-loop optimization problem, as the reliability index is explicit. The sensitivity results are obtained from an analytical approach as well as a direct difference method. Eventually, the NRBTO problem is solved by a sequential quadratic programming method. Two numerical examples are used to testify the validity and effectiveness of the proposed method. The results show significant effects of uncertainties to the topology configuration of truss structures.     

Quantification of uncertainty modelling in stochastic analysis of FRP composites

The extensive use of FRP composite materials in a wide range of industries, and their inherent variability, has prompted many researchers to assess their performance from a probabilistic perspective. This paper attempts to quantify the uncertainty in FRP composites and to summarise the different stochastic modelling approaches suggested in the literature. Researchers have considered uncertainties starting at a constituent (fibre/matrix) level, at the ply level or at a coupon or component level. The constituent based approach could be further classified as a random variable based stochastic computational mechanics approach (whose usage is comparatively limited due to complex test data requirements and possible uncertainty propagation errors) and the more widely used morphology based random composite modelling which has been recommended for exploring local damage and failure characteristics. The ply level analysis using either stiffness/strength or fracture mechanics based models is suggested when the ply characteristics influence the composite properties significantly, or as a way to check the propagation of uncertainties across length scales. On the other hand, a coupon or component level based uncertainty modelling is suggested when global response characteristics govern the design objectives. Though relatively unexplored, appropriate cross-fertilisation between these approaches in a multi-scale modelling framework seems to be a promising avenue for stochastic analysis of composite structures. It is hoped that this review paper could facilitate and strengthen this process.     

随机有限元-最大熵法

本文提出一种用于结构可靠性分析的随机有限元-最大熵法。它是利用随机有限元法计算结构响应量的前几阶矩,然后利用最大熵法拟会响应量的概率分布,据此算出结构的失效概率。此法具有精度较高、计算量较小的优点。     

钢筋砼结构非线性随机地震响应分析

考虑地震作用和结构参数的随机性,建立了钢筋砼结构药非线性随机动力学模型。文中导出了随机结构动力分析的非线性随机有限元法的增量列式,并据此对多层钢筋砼结构进行了弹塑性随机地震响应分析。计算结果与该建筑物的实际震害作了对比,效果良好。还讨论了动力模型中随机变量对响应量的影响。     

Non-Gaussian simulation of local material properties based on a moving-window technique

In this paper, a moving-window micromechanics technique, Monte Carlo simulation, and finite element analysis are used to assess the effects of microstructural randomness on the local stress response of composite materials. The randomly varying elastic properties are characterized in terms of a field of local effective elastic constitutive matrices using a moving-window technique based on a finite element model of a given digitized composite material microstructure. Once the fields are generated, estimates of the random properties are obtained for use as input to a simulation algorithm that was developed to retain spectral, correlation, and non-Gaussian probabilistic characteristics. Rapidly generated Monte Carlo simulations of the constitutive matrix fields are used in a finite element analysis to create a series of local stress fields associated with the random material sample under uniaxial tension. This series allows estimation of the statistical variability in the local stress response for the random composite. The identification of localized extreme stress deviations from those of the aggregate or effective properties approach highlight the importance of modeling the stochastic variability of the microstructure.