Sensitivity analysis and shape optimization of axisymmetric structures

An efficient method is developed for sensitivity analysis in shape optimization of axisymmetric structures. The technique of isoparametric mapping is used to generate the finite element mesh from a small set of master elements and master nodes. Co-ordinates of selected master nodes are used as design variables. Shape function values of master elements at derived finite element nodes obtained during the isoparametric mapping process are utilized to calculate the gradients of weight and response of the structures with respect to the design variables. Analytic formulations of the gradients are developed for sensitivity analysis of axisymmetric structures. An optimization procedure using a sequential linear programming method is applied to effectively utilize the calculated gradients. Numerical examples of optimum design of disks subject to thermo-mechanical loadings are presented.     

Stochastic design sensitivity in structural dynamics

A method to analyse stochastic design sensitivity for problems of structural dynamics is presented. A combination of the adjoint variable approach and the second order perturbation method is used in the finite element context. An alternative form of the constraint functional that holds for all times is introduced to consider the time response of dynamic sensitivity. The terminal problem of the adjoint system is solved using equivalent homogeneous equations exicited by initial velocities. The numerical procedures are shown to be much more efficient when based on the fold superposition technique: the generalized co-ordinates are normalized and the correlated random variables are transformed to uncorrelated variables, whereas the secularities are eliminated by the fast Fourier transform of complex valued sequences. Numerical algorithms have been worked out and proved to be accurate and efficient; they can be readily adapted to fit into the existing finite element codes whose element derivative matrices can be explicitly generated. A number of numerical results for the deterministic and stochastic sensitivity analysis of beams and shells illustrates the paper.     

Robust optimal design using a second-order tolerance model

In this paper we present a general, quantitative method for developing designs that are robust to variation in design variables and parameters. Variation is defined in terms of tolerances which bracket the expected deviation of uncertain quantities about nominal values. We specifically address the case where input variations are assumed to be random variables that are normally distributed. The method incorporates a second-order tolerance model as part of a nonlinear optimization process. The second-order tolerance model makes it possible to estimate the skewness of function distributions, which are modeled with a three-parameter gamma distribution. We apply the method to determine robust designs for 11 test cases that span a variety of problems; robustness is verified with Monte Carlo simulation. The method enables a designer to understand and account for the effects of tolerances, making it possible to build robustness into an engineering design.     

Compromise design of stochastic dynamical systems: A reliability-based approach

This paper presents a procedure for obtaining compromise designs of structural systems under stochastic excitation. In particular, an effective strategy for determining specific Pareto optimal solutions is implemented. The design goals are defined in terms of deterministic performance functions and/or performance functions involving reliability measures. The associated reliability problems are characterized by means of a large number of uncertain parameters (hundreds or thousands). The designs are obtained by formulating a compromise programming problem which is solved by a first-order interior point algorithm. The sensitivity information required by the proposed solution strategy is estimated by an approach that combines an advanced simulation technique with local approximations of some of the quantities associated with structural performance. An efficient Pareto sensitivity analysis with respect to the design variables becomes possible with the proposed formulation. Such information is used for decision making and tradeoff analysis. Numerical validations show that only a moderate number of stochastic analyses (reliability estimations) has to be performed in order to find compromise designs. Two example problems are presented to illustrate the effectiveness of the proposed approach.     

一座现有拱桥面内失稳的可靠度随机有限元分析

用基于一次可靠度方法的可靠度随机有限元对一座现有的钢筋混凝土拱桥面内稳定性进行剩余可靠度计算,并对影响稳定性可靠度的主要参数进行了灵敏度分析。以随机变量和随机场表示现状荷载(汽车荷载、人群荷载、桥面二期恒载和结构自重)及结构参数(主拱圈弹性模量)。用作者提出的基于线性回归的随机场离散方法离散上述随机场,以有限元稳定分析的解作为复杂结构的隐式功能函数。上述功能以及失稳特征值对基本变量的梯度计算均已包含在作者开发的可靠度随机有限元程序RESFEP中。分析给出此桥稳定性可靠度值。灵敏度分析表明:在各随机变量中,拱肋弹性模量随机场离散变量对拱桥可靠度指标影响最大,汽车荷载随机场离散变量次之;在各随机变量的均值和标准差中,拱肋弹性模量随机场离散变量均值和标准差对拱桥可靠度影响最大,汽车荷载随机场离散变量的均值和标准差次之。     

Robust design – A concept for imperfection insensitive composite structures

Robust design is a philosophy that aims to ensure that a structure will be tolerant to unknown variations and imperfections. This is an important consideration as highly optimised critical structures are required to survive unexpected loading and operating conditions. In some ways, robust design appears to be similar to damage tolerant design but its application to aerospace structural design is neither well established nor understood. In order to demonstrate the differences between the two concepts, a stiffened composite panel has been analysed for damage tolerance and robustness properties. Damage tolerance has been studied experimentally with the panel subjected to impact damage. The effect of laminate stacking sequence on the robustness of the panel has been assessed using finite element analysis and a Robust Index applied to quantify the robustness. The differences between designs are discussed together with the possible future directions for robust design applied to aerospace composite structures.     

High-Fidelity Computational Optimization for 3-D Flexible Wings: Part II—Effect of Random Geometric Uncertainty on Design

The effect of geometric uncertainty due to statistically independent, random, normally distributed shape parameters is demonstrated in the computational design of a 3-D flexible wing. A first-order second-moment statistical approximation method is used to propagate the assumed input uncertainty through coupled Euler CFD aerodynamic/finite element structural codes for both analysis and sensitivity analysis. First-order sensitivity derivatives obtained by automatic differentiation are used in the input uncertainty propagation. These propagated uncertainties are then used to perform a robust design of a simple 3-D flexible wing at supercritical flow conditions. The effect of the random input uncertainties is shown by comparison with conventional deterministic design results. Sample results are shown for wing planform, airfoil section, and structural sizing variables.     

A contribution to the SFE-based reliability assessment of nonlinear structures under dynamic loading

This paper presents a procedure which allows for a stochastic finite element (SFE)-based reliability analysis of large nonlinear structures under dynamic loading involving both structural and loading randomness with relatively little computational effort when compared to traditional Monte Carlo methods. The analysis is based on the identification of important random variables by means of a transformation of the vector of original random variables to the uncorrelated space and subsequent sensitivity analyses. Only few nonlinear computations using the most important identified random variables are performed to determine points on the limit state surface. Subsequently, the response surface method (RSM) is employed to estimate the reliability of the structure.     

Design Optimization of Composite Cylindrical Shells under Uncertainty

Four different approaches for the design of axially compressed cylindrical shells are presented, namely (1) the knockdown factor (KDF) concept, (2) the single perturbation load approach, (3) a probabilistic design procedure and (4) the convex anti-optimization approach. The different design approaches take the imperfection sensitivity and the scatter of input parameters into account differently. In this paper, the design of a composite cylinder is optimized considering the ply angles as design variables. The KDF concept provides a very conservative design load and in addition an imperfection sensitive design, whereas the other approaches lead to a significantly less conservative design load and to a less imperfection sensitive design configuration. The ways in which imperfection sensitivity is treated by the different approaches and how these influence the optimal design configuration is discussed.     

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.     

Collocation-based stochastic finite element analysis for random field problems

A stochastic response surface method (SRSM) which has been previously proposed for problems dealing only with random variables is extended in this paper for problems in which physical properties exhibit spatial random variation and may be modeled as random fields. The formalism of the extended SRSM is similar to the spectral stochastic finite element method (SSFEM) in the sense that both of them utilize Karhunen–Loeve (K–L) expansion to represent the input, and polynomial chaos expansion to represent the output. However, the coefficients in the polynomial chaos expansion are calculated using a probabilistic collocation approach in SRSM. This strategy helps us to decouple the finite element and stochastic computations, and the finite element code can be treated as a black box, as in the case of a commercial code. The collocation-based SRSM approach is compared in this paper with an existing analytical SSFEM approach, which uses a Galerkin-based weighted residual formulation, and with a black-box SSFEM approach, which uses Latin Hypercube sampling for the design of experiments. Numerical examples are used to illustrate the features of the extended SRSM and to compare its efficiency and accuracy with the existing analytical and black-box versions of SSFEM.     

Integrated layout design of multi‐component system

A new integrated layout optimization method is proposed here for the design of multi‐component systems. By introducing movable components into the design domain, the components layout and the supporting structural topology are optimized simultaneously. The developed design procedure mainly consists of three parts: (i) Introduction of non‐overlap constraints between components. The finite circle method (FCM) is used to avoid the components overlaps and also overlaps between components and the design domain boundaries. (ii) Layout optimization of the components and supporting structure. Locations and orientations of the components are assumed as geometrical design variables for the optimal placement while topology design variables of the supporting structure are defined by the density points. Meanwhile, embedded meshing techniques are developed to take into account the finite element mesh change caused by the component movements. (iii) Consistent material interpolation scheme between element stiffness and inertial load. The commonly used solid isotropic material with penalization model is improved to avoid the singularity of localized deformation in the presence of design dependent loading when the element stiffness and the involved inertial load are weakened by the element material removal. Finally, to validate the proposed design procedure, a variety of multi‐component system layout design problems are tested and solved on account of inertia loads and gravity center position constraint. Solutions are compared with traditional topology designs without component. Copyright © 2008 John Wiley & Sons, Ltd.     

工程弹塑性断裂随机分析的多维插值方法

为解决用确定性有限元或已有无量纲断裂参数表值时进行弹塑性断裂随机分析的问题，本文研究了利用数值插值方法进行随机参数计算的可行性。以单边裂纹板为例，根据确定性有限元计算的无量纲断裂参数表值，通过多维插值计算获得断裂参数及其对基本随机变量变化率，应用“弹塑性断裂随机分析的工程方法”进行弹塑性断裂随机分析。算例表明在敏感区用插值方法计算的结果小于用随机有限元方法计算的结果。     

Bi-Directional Evolutionary Topology Optimization Using Element Replaceable Method

In the present paper, design problems of maximizing the structural stiffness or natural frequency are considered subject to the material volume constraint. A new element replaceable method (ERPM) is proposed for evolutionary topology optimization of structures. Compared with existing versions of evolutionary structural optimization methods, contributions are twofold. On the one hand, a new automatic element deletion/growth procedure is established. The deletion of a finite element means that a solid element is replaced with an orthotropic cellular microstructure (OCM) element. The growth of an element means that an OCM element is replaced with a solid element of full materials. In fact, both operations are interchangeable depending upon how the value of element sensitivity is with respect to the objective function. The OCM design strategy is beneficial in preventing artificial modes for dynamic problems. Besides, the iteration validity is greatly improved with the introduction of a check position (CP) technique. On the other hand, a new checkerboard control algorithm is proposed to work together with the above procedure. After the identification of local checkerboards and detailed structures over the entire design domain, the algorithm will fill or delete elements depending upon the prescribed threshold of sensitivity values. Numerical results show that the ERPM is efficient and a clear and valuable material pattern can be achieved for both static and dynamic problems.     

Design sensitivity analysis of non-linear structural systems part I: Theory

A unified approach is presented for design sensitivity analysis of non-linear structural systems that include truss, beam, plane elastic solid and plate components. Both geometric and material non-linearities are treated. Sizing design variables, such as thickness and cross-sectional areas of components of individual members and built-up structures, are considered. A distributed parameter structural design sensitivity analysis approach is used that retains the continuum elasticity formulation throughout the derivation of design sensitivity analysis results. Using this approach and an adjoint variable method, expressions for design sensitivity in terms of design variations are derived in the continuous setting which can be evaluated numerically using analysis results of finite element analysis. Both total Lagrangian and updated Lagrangian formulations in non-linear analysis of solid mechanics are used for design sensitivity analysis. Numerical implementation of design sensitivity analysis results using existing finite element code will be presented in Part II of this paper.     

Input-response space-filling designs

Traditional space-filling designs are a convenient way to explore throughout an input space of flexible dimension and have design points close to any region where future predictions might be of interest. In some applications, there may be a model connecting the input factors to the response(s), which provides an opportunity to consider the spacing not only in the input space but also in the response space. In this paper, we present an approach for leveraging current understanding of the relationship between inputs and responses to generate designs that allow the experimenter to flexibly balance the spacing in these two regions to find an appropriate design for the experimental goals. Applications where good spacing of the observed response values include calibration problems where the goal is to demonstrate the adequacy of the model across the range of the responses, sensitivity studies where the outputs from a submodel may be used as inputs for subsequent models, and inverse problems where the outputs of a process will be used in the inverse prediction for the unknown inputs. We use the multi-objective optimization method of Pareto fronts to generate multiple non-dominated designs with different emphases on the input and response space-filling criteria from which the experimenter can choose. The methods are illustrated through several examples and a chemical engineering case study.     

声场-结构耦合系统灵敏度分析及优化设计研究

给出了低频声－结构耦合系统的有限元方程，并在此基础上提出声一结构耦合系统的包含尺寸和形状设计变量的优化设计模型，建立基于灵敏度分析求解的优化设计方法，重点推导了耦合系统的特征频率和声压级响应关于设计变量的灵敏度方程。在JIFEX软件中实现上述理论和算法，并通过灵敏度比较和优化设计的数值算例，进一步说明该研究方法对声结构耦合系统的工程设计具有实用意义。     

Stochastic analysis and robust optimization for a deck lid inner panel stamping

FE-simulation and optimization are widely used in the stamping process to improve design quality and shorten development cycle. However, the current simulation and optimization may lead to non-robust results due to not considering the variation of material and process parameters. In this study, a novel stochastic analysis and robust optimization approach is proposed to improve the stamping robustness, where the uncertainties are involved to reflect manufacturing reality. A meta-model based stochastic analysis method is developed, where FE-simulation, uniform design and response surface methodology (RSM) are used to construct meta-model, based on which Monte-Carlo simulation is performed to predict the influence of input parameters variation on the final product quality. By applying the stochastic analysis, uniform design and RSM, the mean and the standard deviation (SD) of product quality are calculated as functions of the controllable process parameters. The robust optimization model composed of mean and SD is constructed and solved, the result of which is compared with the deterministic one to show its advantages. It is demonstrated that the product quality variations are reduced significantly, and quality targets (reject rate) are achieved under the robust optimal solution. The developed approach offers rapid and reliable results for engineers to deal with potential stamping problems during the early phase of product and tooling design, saving more time and resources.     

Stochastic Finite element analysis of the free vibration of functionally graded material plates

The superior properties of functionally graded materials (FGM) are usually accompanied by randomness in their properties due to difficulties in tailoring the gradients during manufacturing processes. Using the stochastic finite element method (SFEM) proved to be a powerful tool in studying the sensitivity of the static response of FGM plates to uncertainties in their material properties. This tool is yet to be used in studying free vibration of FGM plates. The aim of this work is to use both a First Order Reliability Method (FORM) and the Second Order Reliability Method (SORM), combined with a nine-noded isoparametric Lagrangian element based on the third order shear deformation theory to investigate sensitivity of the fundamental frequency of FGM plates to material uncertainties. These include the effect of uncertainties on both the metal and ceramic constituents. The basic random variables include ceramic and metal Young’s modulus and Poisson’s ratio, their densities and ceramic volume fraction. The developed code utilizes MATLAB capabilities to derive the derivatives of the stiffness and mass matrices symbolically with a considerable reduction in calculation time. Calculating the eigenvectors at the mean values of the variables proves to be a reasonable simplification which significantly increases solution speed. The stochastic finite element code is validated using available data in the literature, in addition to comparisons with results of the well-established Monte Carlo simulation technique with importance sampling. Results show that SORM is an excellent rapid tool in the stochastic analysis of free vibration of FGM plates, when compared to the slower Monte Carlo simulation techniques.