Towards error bounds of the failure probability of elastic structures using reduced basis models

Structural reliability methods aim at computing the probability of failure of systems with respect to prescribed limit state functions. A common practice to evaluate these limit state functions is using Monte Carlo simulations. The main drawback of this approach is the computational cost, because it requires computing a large number of deterministic finite element solutions. Surrogate models, which are built from a limited number of runs of the original model, have been developed, as substitute of the original model, to reduce the computational cost. However, these surrogate models, while decreasing drastically the computational cost, may fail in computing an accurate failure probability. In this paper, we focus on the control of the error introduced by a reduced basis surrogate model on the computation of the failure probability obtained by a Monte Carlo simulation. We propose a technique to determine bounds of this failure probability, as well as a strategy of enrichment of the reduced basis, based on limiting the bounds of the error of the failure probability for a multi‐material elastic structure. Copyright © 2017 John Wiley & Sons, Ltd.     

Parametric uncertainty quantification using proper generalized decomposition applied to neutron diffusion

In this paper, a proper generalized decomposition (PGD) approach is employed for uncertainty quantification purposes. The neutron diffusion equation with external sources, a diffusion-reaction problem, is used as the parametric model. The uncertainty parameters include the zone-wise constant material diffusion and reaction coefficients as well as the source strengths, yielding a large uncertain space in highly heterogeneous geometries. The PGD solution, parameterized in all uncertain variables, can then be used to compute mean, variance, and more generally probability distributions of various quantities of interest. In addition to parameterized properties, parameterized geometrical variations of three-dimensional models are also considered in this paper. To achieve and analyze a parametric PGD solution, algorithms are developed to decompose the model's parametric space and semianalytically integrate solutions for evaluating statistical moments. Varying dimensional problems are evaluated to showcase PGD's ability to solve high-dimensional problems and analyze its convergence.     

Error bounds for the reliability index in finite element reliability analysis

This work presents an extension of the goal‐oriented error estimation techniques to the reliability analysis of a linear elastic structure. We use a first‐order reliability method in conjunction with a finite element analysis (FEA) to compute the failure probability of the structure. In such a situation the output of interest that is computed from the FEA is the reliability index β. The accuracy of this output, and thus of the reliability analysis, depends, in particular, on the accuracy of the FEA. In this paper, upper and lower bounds of the reliability index are proposed, as well as simple bounds of the failure probability. An application to linear fracture mechanics is presented. Copyright © 2011 John Wiley & Sons, Ltd.     

Experimental Characterisation and Numerical Modelling of Hexagonal Honeycomb Cellular Solids under Multi‐axial Loading

Abstract: Cellular solids are becoming increasingly popular for sandwich core and energy‐absorbing applications in many automotive and other transportation structures. This paper investigates experimentally and numerically the strength and post‐failure energy absorption of a popular hexagonal aluminium honeycomb material under multi‐axial loading conditions. For the experimental work, an improved Arcan test apparatus is used so that interaction of multi‐axial compression and shear loading on failure and crushing may be studied; optical measuring methods are used to extract deformation data. In addition, experimental work to characterise the material with pre‐deformation in the in‐plane directions has also been conducted. This experimental work provides input for computational modelling of the material and two alternative modelling approaches have been investigated. First, a three‐dimensional anisotropic, elastic–plastic model, with coupling of loading components is used to represent the material at the macro‐level and, second, a meso‐modelling approach using a fine shell representation of the thin‐walled honeycomb cellular structure is applied. For practical analysis of large‐scale structures, the former approach is computationally efficient and can reasonably treat the most important failure and crush characteristics of the material. However, for more accurate analysis, particularly in the case of complex non‐proportional loading, the meso‐shell model may provide a more realistic solution.     

平流层飞艇充气柔性膜结构的区间不确定优化

根据随体坐标描述法,以变形后的构型作为参考基准,研究构建准确的充气膜结构应变/位移几何关系。利用虚功原理建立平衡方程,同时使用更新的拉格朗日格式(UL格式)保证膜结构分析求解的精度和稳定性。在充气柔性膜结构的分析和优化中引入不确定性参数,通过区间数学理论对其进行刻画。基于区间不确定性优化的模型和算法,对充气膜结构进行区间可靠性优化,为提高膜结构在复杂工况下的可靠性,降低失效概率提供借鉴和参考。     

A coupled mechanical‐charge/dipole molecular dynamics finite element method,with multi‐scale applications to the design of graphene nano‐devices

A new Molecular Dynamics Finite Element Method (MDFEM) with a coupled mechanical‐charge/dipole formulation is proposed. The equilibrium equations of Molecular Dynamics (MD) are embedded exactly within the computationally more favourable Finite Element Method (FEM). This MDFEM can readily implement any force field because the constitutive relations are explicitly uncoupled from the corresponding geometric element topologies. This formal uncoupling allows to differentiate between chemical‐constitutive, geometric and mixed‐mode instabilities. Different force fields, including bond‐order reactive and polarisable fluctuating charge–dipole potentials, are implemented exactly in both explicit and implicit dynamic commercial finite element code. The implicit formulation allows for larger length and time scales and more varied eigenvalue‐based solution strategies. The proposed multi‐physics and multi‐scale compatible MDFEM is shown to be equivalent to MD, as demonstrated by examples of fracture in carbon nanotubes (CNT), and electric charge distribution in graphene, but at a considerably reduced computational cost. The proposed MDFEM is shown to scale linearly, with concurrent continuum FEM multi‐scale couplings allowing for further computational savings. Moreover, novel conformational analyses of pillared graphene structures (PGS) are produced. The proposed model finds potential applications in the parametric topology and numerical design studies of nano‐structures for desired electro‐mechanical properties (e.g. stiffness, toughness and electric field induced vibrational/electron‐emission properties). Copyright © 2014 John Wiley & Sons, Ltd.     

An embedded statistical method for coupling molecular dynamics and finite element analyses

The coupling of molecular dynamics (MD) simulations with finite element methods (FEM) yields computationally efficient models that link fundamental material processes at the atomistic level with continuum field responses at higher length scales. The theoretical challenge involves developing a seamless connection along an interface between two inherently different simulation frameworks. Various specialized methods have been developed to solve particular classes of problems. Many of these methods link the kinematics of individual MD atoms with finite element (FE) nodes at their common interface, necessarily requiring that the FE mesh be refined to atomic resolution. Some of these coupling approaches also require simulations to be carried out at 0 K and restrict modelling to two‐dimensional material domains due to difficulties in simulating full three‐dimensional material processes. In the present work, a new approach to MD–FEM coupling is developed based on a restatement of the standard boundary value problem used to define a coupled domain. The method replaces a direct linkage of individual MD atoms and FE nodes with a statistical averaging of atomistic displacements in local atomic volumes associated with each FE node in an interface region. The FEM and MD computational systems are effectively independent and communicate only through an iterative update of their boundary conditions. Thus, the method lends itself for use with any FEM or MD code. With the use of statistical averages of the atomistic quantities to couple the two computational schemes, the developed approach is referred to as an embedded statistical coupling method (ESCM). ESCM provides an enhanced coupling methodology that is inherently applicable to three‐dimensional domains, avoids discretization of the continuum model to atomic scale resolution, and permits finite temperature states to be applied. Published in 2009 by John Wiley & Sons, Ltd.     

Optimization of a pipe end upsetting process

A method is proposed for the optimization, by finite element analysis, of design variables of sheet metal forming processes. The method is useful when the non-controllable process parameters (e.g. the coefficient of friction or the material properties) can be modelled as random variables, introducing a degree of uncertainty into any process solution. The method is suited for problems with large FEM computational times and small process window. The problem is formulated as the minimization of a cost function, subject to a reliability constraint. The cost function is indirectly optimized through a “metamodel”, built by “Kriging” interpolation. The reliability, i.e. the failure probability, is assessed by a binary logistic regression analysis of the simulation results. The method is applied to the u-channel forming and springback problem presented in Numisheet 1993, modified by handling the blankholder force as a time-dependent variable.     

Enhanced high‐dimensional model representation for reliability analysis

This paper presents a new and alternative computational tool for predicting failure probability of structural/mechanical systems subject to random loads, material properties, and geometry based on high‐dimensional model representation (HDMR) generated from low‐order function components. HDMR is a general set of quantitative model assessment and analysis tools for capturing the high‐dimensional relationships between sets of input and output model variables. It is a very efficient formulation of the system response, if higher‐order variable correlations are weak, allowing the physical model to be captured by the lower‐order terms and facilitating lower‐dimensional approximation of the original high‐dimensional implicit limit state/performance function. When first‐order HDMR approximation of the original high‐dimensional implicit limit state/performance function is not adequate to provide the desired accuracy to the predicted failure probability, this paper presents an enhanced HDMR (eHDMR) method to represent the higher‐order terms of HDMR expansion by expressions similar to the lower‐order ones with monomial multipliers. The accuracy of the HDMR expansion can be significantly improved using preconditioning with a minimal number of additional input–output samples without directly invoking the determination of second‐ and higher‐order terms. The mathematical foundation of eHDMR is presented along with its applicability to approximate the original high‐dimensional implicit limit state/performance function for subsequent reliability analysis, given that conventional methods for reliability analysis are computationally demanding when applied in conjunction with complex finite element models. This study aims to assess how accurately and efficiently the eHDMR approximation technique can capture complex model output uncertainty. The limit state/performance function surrogate is constructed using moving least‐squares interpolation formula by component functions of eHDMR expansion. Once the approximate form of implicit response function is defined, the failure probability can be obtained by statistical simulation. Results of five numerical examples involving elementary mathematical functions and structural/solid‐mechanics problems indicate that the failure probability obtained using the eHDMR approximation method for implicit limit state/performance function, provides significant accuracy when compared with the conventional Monte Carlo method, while requiring fewer original model simulations. Copyright © 2008 John Wiley & Sons, Ltd.     

Behaviour of cellular materials under impact loading

The paper describes experimental and computational testing of regular open‐cell cellular structures behaviour under impact loading. Open‐cell cellular specimens made of aluminium alloy and polymer were experimentally tested under quasi‐static and dynamic compressive loading in order to evaluate the failure conditions and the strain rate sensitivity. Additionally, specimens with viscous fillers have been tested to determine the increase of the energy absorption due to filler effects. The tests have shown that brittle behaviour of the cellular structure due to sudden rupture of intercellular walls observed in quasi‐static and dynamic tests is reduced by introduction of viscous filler, while at the same time the energy absorption is increased. The influence of fluid filler on open‐cell cellular material behaviour under impact loading was further investigated with parametric computational simulations, where fully coupled interaction between the base material and the pore filler was considered. The explicit nonlinear finite element code LS‐DYNA was used for this purpose. Different failure criteria were evaluated to simulate the collapsing of intercellular walls and the failure mechanism of cellular structures in general. The new computational models and presented procedures enable determination of the optimal geometric and material parameters of cellular materials with viscous fillers for individual application demands. For example, the cellular structure stiffness and impact energy absorption through controlled deformation can be easily adapted to improve the efficiency of crash absorbers.     

Reliability–sensitivity analysis using dimension reduction methods and saddlepoint approximations

Reliability–sensitivity, which is considered as an essential component in engineering design under uncertainty, is often of critical importance toward understanding the physical systems underlying failure and modifying the design to mitigate and manage risk. This paper presents a new computational tool for predicting reliability (failure probability) and reliability–sensitivity of mechanical or structural systems subject to random uncertainties in loads, material properties, and geometry. The dimension reduction method is applied to compute response moments and their sensitivities with respect to the distribution parameters (e.g., shape and scale parameters, mean, and standard deviation) of basic random variables. Saddlepoint approximations with truncated cumulant generating functions are employed to estimate failure probability, probability density functions, and cumulative distribution functions. The rigorous analytic derivation of the parameter sensitivities of the failure probability with respect to the distribution parameters of basic random variables is derived. Results of six numerical examples involving hypothetical mathematical functions and solid mechanics problems indicate that the proposed approach provides accurate, convergent, and computationally efficient estimates of the failure probability and reliability–sensitivity. Copyright © 2012 John Wiley & Sons, Ltd.     

Reliability analysis of universal joint of a compliant platform

The paper describes a methodology for computation of fatigue reliability of universal joint in an articulated offshore tower. Failure criteria were formulated using the conventional Palmgren‐Miner rule (S‐N curve approach) and the fracture mechanics (F‐M) principle. The dynamic analysis of double hinged articulated tower under wind and waves is carried out in time domain. The response histories of hinge shear stresses are employed for the reliability analysis. Advanced first‐order reliability method and Monte Carlo simulation method were used to estimate the reliability. Various parametric studies were carried out, which yield important information for the reliability based design. The S‐N curve approach yields a significantly conservative estimate of probability of failure when compared to the F‐M approach.     

Hazard maps and global probability as a way to transfer standard fracture results to reliable design of real components

Design and lifetime prediction of structural and mechanical components require the assessment of the global probability of failure to be determined from stress and strain distributions obtained by FEM, as well as calculation of hazard maps in order to facilitate redesign and recognition of critical parts to be inspected regularly. The so-called generalized probabilistic local approach (GPLA), developed by the authors, allows the primary failure cumulative distribution function (PFCDF) owning to a certain failure type to be determined for a given material from experimental data and used subsequently for probabilistic design. The approach ensures a realistic safety margin provided that the failure criterion represented by an adequate generalized parameter (GP) and the corresponding failure criterion is properly recognized as a reference variable to be considered in the failure assessment. The way in which the results of such a reliability analysis are interpreted encompasses a variety of concepts under which failure can be understood and may be classified as global probability of failure and hazard maps, the former providing the conclusive failure probability for definitive design, and the latter representing, presumably, a risk of local failure that facilitates the possible component redesign but without providing the global probability of failure. In order to promote the implementation of the methodology proposed, an application is exemplary presented for the particular case of experimental results of glass plates. A finite element subroutine is developed for calculation of hazard maps and global probabilities of failure.     

Reliability analysis of structures based on a probability‐uncertainty hybrid model

The traditional reliability analysis method based on probabilistic method requires probability distributions of all the uncertain parameters. However, in practical applications, the distributions of some parameters may not be precisely known due to the lack of sufficient sample data. The probabilistic theory cannot directly measure the reliability of structures with epistemic uncertainty, ie, subjective randomness and fuzziness. Hence, a hybrid reliability analysis (HRA) problem will be caused when the aleatory and epistemic uncertainties coexist in a structure. In this paper, by combining the probability theory and the uncertainty theory into a chance theory, a probability‐uncertainty hybrid model is established, and a new quantification method based on the uncertain random variables for the structural reliability is presented in order to simultaneously satisfy the duality of random variables and the subadditivity of uncertain variables; then, a reliability index is explored based on the chance expected value and variance. Besides, the formulas of the chance theory‐based reliability and reliability index are derived to uniformly assess the reliability of structures under the hybrid aleatory and epistemic uncertainties. The numerical experiments illustrate the validity of the proposed method, and the results of the proposed method can provide a more accurate assessment of the structural system under the mixed uncertainties than the ones obtained separately from the probability theory and the uncertainty theory.     

Reliability analysis for multidisciplinary systems with the mixture of epistemic and aleatory uncertainties

Epistemic and aleatory uncertain variables always exist in multidisciplinary system simultaneously and can be modeled by probability and evidence theories, respectively. The propagation of uncertainty through coupled subsystem and the strong nonlinearity of the multidisciplinary system make the reliability analysis difficult and computational cost expensive. In this paper, a novel reliability analysis procedure is proposed for multidisciplinary system with epistemic and aleatory uncertain variables. First, the probability density function of the aleatory variables is assumed piecewise uniform distribution based on Bayes method, and approximate most probability point is solved by equivalent normalization method. Then, important sampling method is used to calculate failure probability and its variance and variation coefficient. The effectiveness of the procedure is demonstrated by two numerical examples. Copyright © 2013 John Wiley & Sons, Ltd.     

A New Adaptive Response Surface Model for Reliability Analysis of 2.5D C/SiC Composite Turbine Blade

In order to calculate the failure probability of complex structures such as a 2.5D/SiC composites turbine blade and improve the structure safety, a new adaptive model of Response Surface (RS) analysis has been developed in this paper, which can improve the computational efficiency of structural failure problem while ensure the accuracy. The Gaussian Process Regression (GPR) theory was used to establish the RS and reconstruct the performance function of structure. And, an Adaptive Latin hypercube Sampling (ALHS) strategy was adopted in the process of establishing and correcting the RS. Finally the Direct Simulation Monte Carlo(DSMC)was utilized to calculate the failure probability of the performance function replacing the complex structure. Two numerical examples were calculated to validate the accuracy and computational efficiency of the proposed method. Additionally the finite element stress analysis results of 2.5D C/SiC composite turbine blade were used to structural reliability analysis by the proposed method. The approach in this paper provides a new way to evaluate the risk of the complex structures.     

Reliability analysis of structures using neural network method

In order to predict the failure probability of a complicated structure, the structural responses usually need to be estimated by a numerical procedure, such as finite element method. To reduce the computational effort required for reliability analysis, response surface method could be used. However the conventional response surface method is still time consuming especially when the number of random variables is large. In this paper, an artificial neural network (ANN)-based response surface method is proposed. In this method, the relation between the random variables (input) and structural responses is established using ANN models. ANN model is then connected to a reliability method, such as first order and second moment (FORM), or Monte Carlo simulation method (MCS), to predict the failure probability. The proposed method is applied to four examples to validate its accuracy and efficiency. The obtained results show that the ANN-based response surface method is more efficient and accurate than the conventional response surface method.     

Inverse Sub‐Structuring Method for Multi‐Coordinate Coupled Product Transport System

A new high‐accuracy transfer function is selected, and an inverse sub‐structuring method is developed for the analysis of the dynamic characteristics of a three‐sub‐structure coupled product transport system. The closed‐form analytical solution to inverse sub‐structuring analysis of multi‐coordinate coupled multi‐ sub‐structure product transport system is derived. The proposed method is validated by a lumped mass spring damper model; the predicted frequency response functions (FRFs) of sub‐structures and the coupling stiffness, in addition to the most concerned system‐level FRF, are compared with the direct computations, showing exact agreement. Then, FRF tests of a physical prototype of the multi‐coordinate coupled product transport system with three sub‐structures are performed to further check the accuracy of the suggested method. The method developed offers an approach to predict the unknown sub‐structure‐level FRFs and coupling stiffness purely from system‐level FRFs. The suggested method may help obtain the main controlling factors and contributions from the various structure‐borne paths for the product transport system, which may certainly facilitate the cushioning packaging design. Copyright © 2013 John Wiley & Sons, Ltd.     

Stochastic multiscale analysis in hydrodynamic lubrication

A stochastic multiscale analysis framework is developed for hydrodynamic lubrication problems with random surface roughness. The approach is based on a multi‐resolution computational strategy wherein the deterministic solution of the multiscale problem for each random surface realization is achieved through a coarse‐scale analysis with a local upscaling that is achieved through homogenization theory. The stochastic nature of this solution because of the underlying randomness is then characterized through local and global quantities of interest, accompanied by a detailed discussion regarding suitable choices of the numerical parameters in order to achieve a desired stochastic predictive capability while ensuring numerical efficiency. Finally, models of the stochastic interface response are constructed, and their performance is demonstrated for representative problem settings. Overall, the developed approach offers a computational framework, which can essentially predict the significant influence of interface heterogeneity in the absence of a strict scale separation. Copyright © 2017 John Wiley & Sons, Ltd.