Efficient strategies for reliability-based design optimization of variable stiffness composite structures

This study investigates efficient design optimization frameworks for composite structures with uncertainties related to material properties and loading. The integration of two decoupled reliability-based design optimization methodologies with a decoupled discrete material optimization is proposed to determine material and fiber orientation for three-dimensional composite structures. First, a deterministic and decoupled discrete material optimization is used for baseline comparison. The objective is to minimize the cost of composite structures with the design variables comprising of the piecewise patch orientations and material properties of the fiber reinforced composites. The reliability-based design optimization includes a hybrid method, and also the sequential optimization and reliability assessment method. In the sequential optimization and reliability assessment method, the inverse reliability analysis is evaluated using a stochastic response surface method and a first order reliability approach. Comparing the methods based on the optimal material and fiber orientations, the uncertainties in loads and material properties lead to different optimal layouts compared to the deterministic solutions. The numerical results also reveal that the hybrid method applied in reliability based designs results in negligible additional computational cost.     

Low cycle fatigue of single crystal silicon thin films

Silicon will continue to be the critical structure material for micromechanical components for years to come so that reliability must be a key concern. Consequently, in order to ensure reliability design approaches must account for fatigue behavior. This work is aimed at studying the fatigue of single crystal silicon (SCS) thin films by a specially designed method. The films were tested using cantilever microbeam deflection with respect to the effect of loading conditions. To conduct a fatigue test under cyclic strain would be more realistic because many MEMS applications, such as micro-accelerometers and micro-filters, consist of beams vibrating in the same mode. A micro-force testing machine with a micro-probe and specially designed fixtures is used to contact and load the beams under the cyclic strain. Before the fatigue test, the failure strain _f of beams in the flexural test is achieved as the testing criterion. In fatigue testing, various percentages of failure strain _f, two times of the strain amplitude, are adopted. First of all, fatigue strain–life cycle (S/N) curve is achieved. Further, the curve of fatigue force detected on the SCS versus fatigue life is measured. SEM is also employed to observe the fracture modes of films under fatigue. Based on the SEM observation and force/life curve, the failure mechanism of the fatigued SCS films under the cyclic strain is proposed. This provides a viable method to evaluate the reliability of the SCS.     

Stochastic inverse method to identify parameter random fields in a structure

The parameters in a structure such as geometric and material properties are generally uncertain due to manufacturing tolerance, wear, fatigue and material irregularity. Such parameters are random fields because the uncertain properties vary along the spatial domain of a structure. Since the parameter uncertainties in a structure result in the uncertainty of the structural dynamic behavior, they need to be identified accurately for structural analysis or design. In order to identify the random fields of geometric parameters, the parameters can be measured directly using a 3-dimensional coordinate measuring machine. However, it is often very expensive to measure them directly. It is even impossible to directly measure some parameters such as density and Young’s modulus. For that case, the parameter random fields should be identified from measurable response data samples. In this paper, a stochastic inverse method to identify parameter random fields in a structure using modal data is proposed. The proposed method consists of the following three steps: (i) obtaining realizations of the parameter random field from modal data samples by solving an optimization problem, (ii) obtaining the deterministic terms in the Karhunen-Loève expansion by solving an eigenvalue problem and (iii) estimating the distributions of random variables in the Karhunen-Loève expansion using a maximum likelihood estimation method with kernel density.     

Fatigue constrained topology optimization

We present a contribution to a relatively unexplored application of topology optimization: structural topology optimization with fatigue constraints. A probability based high-cycle fatigue analysis is combined with principal stress calculations in order to find the topology with minimum mass that can withstand prescribed variable-amplitude loading conditions for a specific life time. This allows us to generate optimal conceptual designs of structural components where fatigue life is the dimensioning factor. We describe the fatigue analysis and present ideas that make it possible to separate the fatigue analysis from the topology optimization. The number of constraints is kept low as they are applied to stress clusters, which are created such that they give adequate representations of the local stresses. Optimized designs constrained by fatigue and static stresses are shown and a comparison is also made between stress constraints based on the von Mises criterion and the highest tensile principal stresses. The paper is written with focus on structural parts in the avionic industry, but the method applies to any load carrying structure, made of linear elastic isotropic material, subjected to repeated loading conditions.     

A Numerical Fatigue Damage Model for Life Scatter of MEMS Devices

This paper presents a fatigue damage model to estimate fatigue lives of microelectromechanical systems (MEMS) devices and account for the effects of topological randomness of material microstructure. For this purpose, the damage mechanics modeling approach is incorporated into a new Voronoi finite-element model (VFEM). The VFEM developed for this investigation is able to consider both intergranular crack initiation (debonding) and propagation stages. The model relates the fatigue life to a damage parameter "D" which is a measure of the gradual material degradation under cyclic loading. The fatigue damage model is then used to investigate the effects of microstructure randomness on the fatigue of MEMS. In this paper, three different types of randomness are considered: (1) randomness in the microstructure due to random shapes and sizes of the material grains; (2) the randomness in the material properties considering a normally (Gaussian) distributed elastic modulus; and (3) the randomness in the material properties considering a normally distributed resistance stress, which is the experimentally determined material property controlling the ability of a material to resist the damage accumulation. Thirty-one numerical models of MEMS specimens are considered under cyclic axial and bending loading conditions. It is observed that the stress-life results obtained are in good agreement with the experimental study. The effects of material inhomogeneity and internal voids are numerically investigated.     

A Phenomenological Discrete Brittle Damage-Mechanics Model for Fatigue of MEMS Devices With Application to LIGA Ni

Fatigue initiation and failure of various microelectromechanical systems (MEMS) is of significant importance as they gain widespread acceptance in sensors and electronics. This paper presents an approach for utilizing available experimental fatigue data to evaluate the fatigue lives of MEMS components. The approach is based on a phenomenological discrete material representation in which a domain is represented by a collection of rigid elements that interacts via springs along their boundaries. The principles of continuum damage mechanics are used to degrade the spring stiffnesses as brittle damage occurs when the domain is subjected to fatigue loading. The model utilizes experimental stress–life data for LIGA Ni to identify the material properties used in the model. The proposed model captures the statistical distribution of material properties and geometrical randomness of the microstructure commonly observed in a wide variety of MEMS. Consequently, simulations that account for the variability in fatigue life can be readily performed. The model is applied to a dog-bone-shaped specimen to evaluate the influence of material heterogeneity and material flaws on fatigue crack initiation life and scatter. The ability of the model to predict the fatigue life of different types of MEMS devices and loading conditions is also demonstrated by simulating the fatigue stress–life behavior of a MEMS resonator support beam. $hfill$[2008-0087]      

Buckling analysis of imperfect frames using a stochastic finite element method

A stochastic finite element method is developed for the buckling analysis of frames with random initial imperfections, uncertain sectional and material properties. The random geometrical imperfections of the frames are described by member initial crookednesses which are modeled as given initial displacement functions with amplitudes treated as random variables. The effects of the random initial geometric imperfections are formulated as a set of equivalent random nodal coordinates in the finite element discretization of the members. The mean-centered second-order perturbation technique is used to formulate the stochastic finite element method for the buckling analysis of the imperfect frames. Use of the present method is illustrated by several examples of buckling analysis of random frames. Results derived from the Monte Carlo method are also obtained for comparison.     

基于nCode Design-Life的某车架疲劳可靠性分析

针对某型半挂牵引车车架局部出现裂纹或者断裂现象,通过有限元理论和疲劳分析理论相结合进行了车架的有限元分析和疲劳可靠性分析。首先对该车架进行了建模和有限元静态特性分析,然后定义了特定的时间载荷序列数据和材料参数,选用了S-N疲劳设计和静态疲劳分析方法,利用疲劳分析软件nCode Design-Life对该车架进行疲劳可靠性分析,得出该车架的疲劳结果云图和各节点的疲劳寿命,并从相关云图中确定出车架容易破坏的位置和其寿命之后再进行样车实验验证。     

Random eigenvalue problems for large systems

The inherent uncertainties in geometry, material properties, etc. of engineering structures can be represented by stochastic models, where the parameters are described by probabilistic laws. Results from any analysis based on stochastic models inherit probabilistic information as well, which can be used e.g. for reliability analysis. Particularly in linear dynamics of structures the calculation and analysis of random eigenvalues and eigenvectors is crucial. A quite versatile, however computationally intensive way to analyze such systems is direct Monte Carlo simulation. In this paper procedures are shown, which allow a significant reduction of computational efforts of the simulation using a subspace iteration scheme with “optimally” selected start-vectors. As the subspace iteration procedure, although quite accurate, requires a factorization of the stiffness matrix, as an alternative, a procedure based on component mode synthesis is suggested.     

不确定性转子系统的随机有限元建模及响应分析

随机特性和随机载荷会引起转子系统动力响应的不确定性,是转子动力学分析中的重要影响因素.本文基于Timosheke梁理论,把转轴的材料和几何随机特性表示为一维随机场函数,推导出随机转轴有限元列式,建立转子系统随机动力学模型,并给出随机载荷作用下随机转子系统动力响应统计量的分析方法.分别对线性和非线性涡轮泵转子系统进行了随机动力响应分析,并同Monte Carlo仿真结果进行对比,结果表明所建立的随机有限元动力学模型和给出的随机响应分析方法是合理可行的,可以有效应用于实际转子系统随机动力学分析和设计中.     

航天器电子元器件疲劳寿命分析

为研究空间环境效应对航天器电子设备元器件疲劳寿命的影响,应用Miner累积损伤理论及三带宽技术,根据表贴元器件经历的试验、发射和在轨运行等实际环境进行振动及热疲劳寿命分析．给出振动及热循环疲劳寿命的计算方法,并以某BGA（Ball Grid Array）器件为例进行分析．该方法可为长寿命、高可靠的航天器电子设备抗力学及热设计提供分析途径．     

Continuum topology optimization with non-probabilistic reliability constraints based on multi-ellipsoid convex model

Using a quantified measure for non-probab ilistic reliability based on the multi-ellipsoid convex model, the topology optimization of continuum structures in presence of uncertain-but-bounded parameters is investigated. The problem is formulated as a double-loop optimization one. The inner loop handles evaluation of the non-probabilistic reliability index, and the outer loop treats the optimum material distribution using the results from the inner loop for checking feasibility of the reliability constraints. For circumventing the numerical difficulties arising from its nested nature, the topology optimization problem with reliability constraints is reformulated into an equivalent one with constraints on the concerned performance. In this context, the adjoint variable schemes for sensitivity analysis with respect to uncertain variables as well as design variables are discussed. The structural optimization problem is then solved by a gradient-based algorithm using the obtained sensitivity. In the present formulation, the uncertain-but bounded uncertain variations of material properties, geometrical dimensions and loading conditions can be realistically accounted for. Numerical investigations illustrate the applicability and the validity of the present problem statement as well as the proposed numerical techniques. The computational results also reveal that non-probabilistic reliability-based topology optimization may yield more reasonable material layouts than conventional deterministic approaches. The proposed method can be regarded as an attractive supplement to the stochastic reliability-based topology optimization.     

Stochastic elastic–plastic finite elements

A computational framework has been developed for simulations of the behavior of solids and structures made of stochastic elastic–plastic materials. Uncertain elastic–plastic material properties are modeled as random fields, which appear as the coefficient term in the governing partial differential equation of mechanics. A spectral stochastic elastic–plastic finite element method with Fokker–Planck–Kolmogorov equation based probabilistic constitutive integrator is proposed for solution of this non-linear (elastic–plastic) partial differential equation with stochastic coefficient. To this end, the random field material properties are discretized, in both spatial and stochastic dimension, into finite numbers of independent basic random variables, using Karhunen–Loève expansion. Those random variables are then propagated through the elastic–plastic constitutive rate equation using Fokker–Planck–Kolmogorov equation approach, to obtain the evolutionary material properties, as the material plastifies. The unknown displacement (solution) random field is then assembled - using polynomial chaos - as a function of known basic random variables and unknown deterministic coefficients, which are obtained by minimizing the error of finite representation, by Galerkin technique.     

Simulation-based time-dependent reliability analysis for composite hydrokinetic turbine blades

The reliability of blades is vital to the system reliability of a hydrokinetic turbine. A time-dependent reliability analysis methodology is developed for river-based composite hydrokinetic turbine blades. Coupled with the blade element momentum theory, finite element analysis is used to establish the responses (limit-state functions) for the failure indicator of the Tsai–Hill failure criterion and blade deflections. The stochastic polynomial chaos expansion method is adopted to approximate the limit-state functions. The uncertainties considered include those in river flow velocity and composite material properties. The probabilities of failure for the two failure modes are calculated by means of time-dependent reliability analysis with joint upcrossing rates. A design example for the Missouri river is studied, and the probabilities of failure are obtained for a given period of operation time.     

Component and system reliability-based topology optimization using a single-loop method

We perform reliability-based topology optimization by combining reliability analysis and material distribution topology design methods to design linear elastic structures subject to random inputs, such as random loadings. Both component reliability and system reliability are considered. In component reliability, we satisfy numerous probabilistic constraints which quantify the failure of different events. In system reliability, we satisfy a single probabilistic constraint which encompasses the component events. We adopt the first-order reliability method to approximate the component reliabilities and the inclusion-exclusion rule to approximate the system reliability. To solve the probabilistic optimization problem, we use a variant of the single loop method, which eliminates the need for an inner reliability analysis loop. The proposed method is amenable to implementation with existing deterministic topology optimization software, and hence suitable for practical applications. Designs obtained from component and system reliability-based topology optimization are compared to those obtained from traditional deterministic topology optimization and validated via Monte Carlo simulation.     

桥梁疲劳监测传感器设计

设计了一种桥梁疲劳监测传感器,它通过电阻不可逆变化的疲劳载荷响应机理拾取结构损伤信号。所设计的独特应变倍增器可将微弱的交变应变信号进行机械放大,使敏感元件在桥梁高周疲劳载荷作用下仍工作在敏感区。解决了疲劳敏感元件存在的较高动应变响应门槛值与结构较低动应变损伤间的矛盾,采用弓形弹性元件与硅橡胶预应力组合结构,得到了可靠性好、刚度低、放大率高及使用方便的理想结构,从而满足了桥梁结构长寿命、低应力工作特点的寿命检测要求。静载和疲劳试验说明:所设计的传感器具有很高的疲劳灵敏度、长期稳定性,完全适用于桥梁结构的高周疲劳检测。     

Geometrically non-linear behavior of structural systems with random material property: An asymptotic spectral stochastic approach

An asymptotic spectral stochastic approach is presented for computing the statistics of the equilibrium path in the post-bifurcation regime for structural systems with random material properties. The approach combines numerical implementation of Koiter’s asymptotic theory with a stochastic Galerkin scheme and collocation in stochastic space to quantify uncertainties in the parametric representation of the load–displacement relationship, specifically in the form of uncertain post-buckling slope, post-buckling curvature, and a family of stochastic displacement fields. Using the proposed method, post-buckling response statistics for two plane frames are obtained and shown to be in close agreement with those obtained from Monte Carlo simulation, provided a fine enough spectral representation is used to model the variability in the random dimension.     

Spectral techniques for stochastic finite elements

Summary A formulation for the stochastic finite element method is presented which is a natural extension of the deterministic finite element method. Discretization of the random dimension is achieved via two spectral expansions. One of them is used to represent the coefficients of the differential, equation which model the random material properties, the other is used to represent the random solution process. The method relies on viewing the random aspect of the problem as an added dimension, and on treating random variables and processes as functions defined over that dimension. The versatility of the method is demonstrated by discussing, as well, some non-traditional problems of stochastic mechanics.     

The stochastic finite-element method

A generic stochastic finite-element method for modeling structures is proposed as a means to analyze and design structures in a probabilistic framework. Stochastic differential and difference equation theory is applied in structures discretized with the finite-element methodology.Transient structural loads, idealized as stochastic processes, are incorporated into finite-element dynamic models with uncertain parameters. An estimate of the probability of failure based on known and established procedures in second-moment reliability analysis can be made with the aid of a transformation to gaussian space of the random variables that define structural reliability.The stochastic finite-element method will facilitate the use of probabilistic mathematical structural models for structural code development or design of important structures. It will also permit better estimation of structural reliability, which, when combined with risk analysis, could lead to improved decision-making processes.     

多晶硅双端固支梁高周循环下疲劳特性的分析

多晶硅固支梁是MEMS器件中较常见的可动部件,通过静电激励的方式对其进行疲劳振动加载;所用结构为面外运动结构,为了测试样品的加速疲劳特性,通过在固支梁面内引入缺陷的方式来增大应力水平值;器件在经历了1.72×1011次循环之后,微梁的谐振频率、振动幅度发生了较大偏移,其谐振频率的偏移量达到14.531 kHz,器件性能发生了严重的退化.研究结果表明,利用谐振频率的改变来表征材料性能的退化是一种准确、可行的方法,同时本文进一步分析指出,器件上引入凹槽缺陷的方法确实可起到加速疲劳的作用;可利用此方法制作不同应力水平幅度的结构进行振动载荷疲劳加载实验,从而得到固支梁结构疲劳加速因子.