An efficient reliability-based optimization scheme for uncertain linear systems subject to general Gaussian excitation

A very efficient methodology to carry out reliability-based optimization of linear systems with random structural parameters and random excitation is presented. The reliability-based optimization problem is formulated as the minimization of an objective function for a specified reliability. The probability that design conditions are satisfied within a given time interval is used as a measure of the system reliability. Approximation concepts are used to construct high quality approximations of dynamic responses in terms of the design variables and uncertain structural parameters during the design process. The approximations are combined with an efficient simulation technique to generate explicit approximations of the reliability measures with respect to the design variables. In particular, an efficient importance sampling technique is used to estimate the failure probabilities. The number of dynamic analyses as well as reliability estimations required during the optimization process are reduced dramatically. Several example problems are presented to illustrate the effectiveness and feasibility of the suggested approach.     

A computational procedure for response statistics-based optimization of stochastic non-linear FE-models

An approach for an efficient solution of response statistics-based optimization problems of non-linear FE systems under stochastic loading is presented. A sequential approximate optimization approach, where approximate stochastic analyses are used during portions of the optimization process, is implemented in the proposed formulation. In this approach, analytical approximations of the performance functions in terms of the design variables are considered during the optimization process. The analytical approximations are constructed by combining a mixed linearization approach with a stochastic response sensitivity analysis. The state of the system is defined in terms of the statistical second-moment characteristics of the structural response. The stochastic loading and the response of the system are represented by an orthogonal series expansion of the corresponding covariance matrices. In particular, a truncated Karhunen-Loève (K-L) expansion is applied. The system of non-linear equations is replaced by a statistical equivalent linear system. The evaluation of the K-L vectors is carried out by an efficient procedure that combines local linearization, modal analysis and static response of higher structural modes. An illustrative example is presented that shows the efficiency of the proposed methodology: it considers a building finite element model enforced with non-linear hysteretic devices and subject to a stochastic ground acceleration. Two types of problems are considered: a minimum structural weight design problem and an optimal non-linear device design problem.     

Structural optimization of linear dynamical systems under stochastic excitation: a moving reliability database approach

A method to carry out structural synthesis of deterministic linear dynamical systems under stochastic excitation is introduced. The structural optimization problem is written as a nonlinear mathematical programming problem with reliability constraints. Probability that design conditions are satisfied within a given time period is used as a measure of system reliability. The solution of the original optimization problem is replaced by the solution of a sequence of approximate sub-optimization problems. An explicit approximation of the system reliability in terms of the design variables is constructed in each sub-optimization problem. The approximations are locally adjusted to a reliability database, which is obtained by an efficient importance sampling technique. Each approximate optimization problem is solved in an efficient manner due to the availability of the system reliability in explicit form. Numerical examples are presented to illustrate the performance and efficiency of the proposed methodology.     

基于控制参数化的注塑工业过程最优反馈控制方法

本文针对一类典型的注塑工业过程系统, 研究了注塑填充过程中产生的熔体流动速度最优跟踪控制问题, 提出了一种基于控制参数化的计算最优反馈控制器设计方法以实现注塑过程中熔融聚合物流动前沿位移的最优跟 踪控制, 进而达到改善注塑零件性能的高效生产目标. 首先, 面向注塑工艺复杂生产过程建立了动态过程系统数学 模型, 提出了注塑机内部熔融聚合物流动前沿位置的动态最优跟踪控制问题; 其次, 设计了一种多级反馈控制律, 通 过控制参数化方法将控制反馈核进行了参数化表示, 将控制器设计问题转化为一序列最优参数决策问题; 然后, 通 过状态灵敏度方程分析方法, 求解出了目标函数及约束条件关于决策变量参数梯度信息的显式表达式, 并基于所提 供的梯度信息结合序列二次规划算法进行了高效优化迭代求解; 最后, 通过实验仿真验证了本文所提出的最优反 馈控制器设计方法的可行性和有效性.     

Approximate structural reanalysis based on series expansion

In most optimal design procedures the analysis of the structure must be repeated many times. This operation, which involves much computational effort, is one of the main difficulties in applying optimization methods to large systems. This study deals with approximate reanalysis methods based on series expansion. Both design variables and inverse variables formulations are presented. It is shown that a Taylor series expansion of the nodal displacements or the redundant forces is equivalent to a series obtained from a simple iteration procedure. The series coefficients can readily be computed, providing efficient and high-degree polynomial approximations.To further improve the quality of the approximations, a modified nonpolynomial series is proposed. To reduce the amount of calculations, the possibility of reanalysis along a given line in the variables space is demonstrated. All the proposed procedures require a single exact analysis to obtain an explicit behaviour model along a line.The relationship between the various methods is discussed and numerical examples demonstrate applications. The results obtained are encouraging and indicate that the proposed methods provide efficient and high quality approximations for the structural behavior. This may lead to a wider use of optimization methods in the design of large structural systems.     

A preference aggregation rule approach for structural optimization

In this paper, preference aggregation rules are used to define overall design evaluation measures in optimal design problems. A methodology for the efficient solution of the corresponding design optimization problems is presented. Each design criterion as well as the constraints imposed on the design variables and problem parameters are characterized by preference functions. The nondifferentiable nature of the optimization problems which arise in this formulation is coped with using a first-order algorithm combined with approximation concepts. High-quality approximations for the system response functions are constructed using the concepts of intermediate response quantities and intermediate variables. These approximations are used to replace the original problem by a sequence of approximate problems. Example problems are presented to study the performance of the proposed optimization technique as well as the methodology based on approximation concepts.     

Least-weight design of frameworks under multiple dynamic loads

A computer-based structural design methodology is presented for the least-weight design of planar frameworks subjected to multiple dynamic loads. The method makes use of dynamic finite element analysis, sensitivity analysis, first-order Taylor series approximations and the identification of response extrema to convert the time-parametric design problem into an explicit non-parametric form, which is solved iteratively using a dual optimization routine. The focus of the study is not to develop a general-purpose design capability but, rather, to examine the computational aspects of accounting for multiple dynamic loads in the design process. Two example truss designs under multiple regular sinusoidal wave loadings and multiple irregular earthquake loadings are presented.     

Robust design optimization by polynomial dimensional decomposition

This paper introduces four new methods for robust design optimization (RDO) of complex engineering systems. The methods involve polynomial dimensional decomposition (PDD) of a high-dimensional stochastic response for statistical moment analysis, a novel integration of PDD and score functions for calculating the second-moment sensitivities with respect to the design variables, and standard gradient-based optimization algorithms. New closed-form formulae are presented for the design sensitivities that are simultaneously determined along with the moments. The methods depend on how statistical moment and sensitivity analyses are dovetailed with an optimization algorithm, encompassing direct, single-step, sequential, and multi-point single-step design processes. Numerical results indicate that the proposed methods provide accurate and computationally efficient optimal solutions of RDO problems, including an industrial-scale lever arm design.     

Stochastic Subset Optimization for reliability optimization and sensitivity analysis in system design

Design problems that involve the system reliability as the objective function are discussed. In order to appropriately address the challenges of such applications when complex system models are involved, stochastic simulation is selected to evaluate the probability of failure. An innovative algorithm, called Stochastic Subset Optimization (SSO), is discussed for performing the reliability optimization as well as an efficient sensitivity analysis. The basic principle in SSO is the formulation of an augmented problem where the design variables are artificially considered as uncertain. Stochastic simulation techniques are implemented in order to simulate samples of these variables that lead to system failure. The information that these samples provide is then exploited in an iterative approach in SSO to identify a smaller subset of the design space that consists of near-optimal design variables and also that has high plausibility of containing the optimal design. At the same time, a sensitivity analysis for the influence of both the design variables and the uncertain model parameters is established.     

Adaptive gradient-enhanced kriging model for variable-stiffness composite panels using Isogeometric analysis

A vibration isolation system is designed using novel hybrid optimization techniques, where locations of machines, locations of isolators and layout of supporting structure are all taken as design variables. Instead of conventional parametric optimization model, the 0-1 programming model is established to optimize the locations of machines and isolators so that the time-consuming remeshing procedure and the complicated sensitivity analysis with respect to position parameters can be circumvented. The 0-1 sequence for position design variables is treated as binary bits so as to reduce the actual number of design variables to a great extent. This way the 0-1 programming can be solved in a quite efficient manner using a special version of genetic algorithm(GA) that has been published by the authors. The layout of supporting structure is optimized using SIMP based topology optimization method, where the fictitious elemental densities are taken as design variables ranging from 0 to 1. Influence of different design variables is firstly investigated by numerical examples. Then a hybrid multilevel optimization method is proposed and implemented to simultaneously take all design variables into account.     

Reliability-based design optimization for problems with interval distribution parameters

Traditional reliability-based design optimization (RBDO) generally describes uncertain variables using random distributions, while some crucial distribution parameters in practical engineering problems can only be given intervals rather than precise values due to the limited information. Then, an important probability-interval hybrid reliability problem emerged. For uncertain problems in which interval variables are included in probability distribution functions of the random parameters, this paper establishes a hybrid reliability optimization design model and the corresponding efficient decoupling algorithm, which aims to provide an effective computational tool for reliability design of many complex structures. The reliability of an inner constraint is an interval since the interval distribution parameters are involved; this paper thus establishes the probability constraint using the lower bound of the reliability degree which ensures a safety design of the structure. An approximate reliability analysis method is given to avoid the time-consuming multivariable optimization of the inner hybrid reliability analysis. By using an incremental shifting vector (ISV) technique, the nested optimization problem involved in RBDO is converted into an efficient sequential iterative process of the deterministic design optimization and the hybrid reliability analysis. Three numerical examples are presented to verify the proposed method, which include one simple problem with explicit expression and two complex practical applications.     

基于混沌多项式的指令鲁棒优化及在飞行控制中的应用

本文提出一种新的方法对随机系统进行运动预测和控制指令设计, 该方法可以充分利用已知信息设计控 制指令以提高闭环随机系统的鲁棒性. 首先采用混沌多项式对随机信息进行数学表述, 并利用Galerkin投影法将随 机变量的混沌多项式引入常微分方程中. 然后, 将随机变量的均值和方差考虑至优化问题的成本函数中, 并利用伪 谱法对控制指令进行鲁棒优化. 最后, 将该方法应用于飞行器的动力学预测以及控制指令设计. 仿真结果表明, 该 方法能够预测飞行器飞行过程中不确定性的演化, 其精度与蒙特卡罗方法相当, 并且计算效率更高. 此外, 获得的 控制指令对存在不确定参数或初始条件的随机系统具有强鲁棒性.     

A hybrid surrogate and pattern search optimization method and application to microelectronics

A hybrid method for robust and efficient optimization process is developed by integrating a new response surface method and pattern search algorithm. The method is based on: (1) multipoint approximations of the objective and constraint functions, (2) a multiquadric radial basis function (RBF) for the zeroth-order function approximation and a new RBF plus polynomial-based moving least-squares approximation for the first-order enhanced function approximation, and (3) a pattern search algorithm to impose a descent condition and applied adaptive subregion management strategy. Several numerical examples are presented to illustrate accuracy and computational efficiency of the proposed method for both function approximation and design optimization. To demonstrate the effectiveness of the proposed hybrid method, it is applied to obtain optimum designs of a microelectronic packaging system. A two-stage optimization approach is proposed for the design optimization. The material properties of microelectronic packaging system and the shape parameters of solder ball are selected as design variables. Through design optimization, significant improvements of durability performances are obtained using the proposed hybrid optimization method.     

Sensitivity analysis in shape optimization of continuum structures

This paper describes an efficient method for sensitivity analysis in shape optimum design. One feature is the use of limited number of master nodes to characterize the surfaces of a set of isoparametric finite elements, and the adoption of their coordinates as design variables of the shape optimization. Another is the derivation of analytical formulations of the gradients of both the stiffness terms and the load vectors with respect to the design variables. A finite element analysis code is adapted to the purposes of the method and numerical examples are performed and comparisons made with sensitivity analysis based on forward finite differences.     

Mapping method for sensitivity analysis of composite material property

Composite properties are dependent on the microstructure of materials, which is depicted with a base cell. The parameters for representing the microstructure should include the shape parameters of the base cell and those used to describe the distribution of materials in the base cell. The goal of material design optimization is to find appropriate values of these parameters to make the materials have specific properties. Design optimization needs the sensitivity information of the material properties with respect to the shape parameter of the base cell and the material distribution parameters. Moreover, sensitivity calculation is often expensive. Thus, it is very important to develop an efficient sensitivity analysis method. In this paper, a mapping method is proposed for predicting the material properties and computing their sensitivities with respect to the shape parameters of the base cell. Through mapping transformation, solutions to the micro-scale homogenization problem defined on the domain of a base cell can be obtained by solving a homogenization problem defined on an initial given domain. The composite properties and their sensitivities with respect to the shape parameters of the base cell are explicitly expressed in terms of the properties and their sensitivities of a virtual material with respect to the distribution parameters. This virtual material has an initially given base cell domain. Thus re-meshing for discretizing the problem is avoided and computing cost savings are realized. Numerical examples show that the proposed method is accurate and efficient in both the prediction of material properties and sensitivity calculation.     

Mechanism-based emulation of dynamic simulation models: Concept and application in hydrology

Many model-based investigation techniques, such as sensitivity analysis, optimization, and statistical inference, require a large number of model evaluations to be performed at different input and/or parameter values. This limits the application of these techniques to models that can be implemented in computationally efficient computer codes. Emulators, by providing efficient interpolation between outputs of deterministic simulation models, can considerably extend the field of applicability of such computationally demanding techniques. So far, the dominant techniques for developing emulators have been priors in the form of Gaussian stochastic processes (GASP) that were conditioned with a design data set of inputs and corresponding model outputs. In the context of dynamic models, this approach has two essential disadvantages: (i) these emulators do not consider our knowledge of the structure of the model, and (ii) they run into numerical difficulties if there are a large number of closely spaced input points as is often the case in the time dimension of dynamic models. To address both of these problems, a new concept of developing emulators for dynamic models is proposed. This concept is based on a prior that combines a simplified linear state space model of the temporal evolution of the dynamic model with Gaussian stochastic processes for the innovation terms as functions of model parameters and/or inputs. These innovation terms are intended to correct the error of the linear model at each output step. Conditioning this prior to the design data set is done by Kalman smoothing. This leads to an efficient emulator that, due to the consideration of our knowledge about dominant mechanisms built into the simulation model, can be expected to outperform purely statistical emulators at least in cases in which the design data set is small. The feasibility and potential difficulties of the proposed approach are demonstrated by the application to a simple hydrological model.     

Discrete variable optimization of plate structures using dual methods

This study presents an efficient method for optimum design of plate and shell structures, when the design variables are continuous or discrete. Both sizing and shape design variables are considered. First the structural responses, such as element forces, are approximated in terms of some intermediate variables. By substituting these approximate relations into the original design problem, an explicit nonlinear approximate design task with high quality approximation is achieved. This problem with continuous variables can be solved very efficiently by means of numerical optimization techniques, the results of which are then used for discrete variable optimization. Now, the approximate problem is converted into a sequence of second level approximation problems of separable form, each of which is solved by a dual strategy with discrete design variables. The approach is efficient in terms of the number of required structural analyses, as well as the overall computational cost of optimization. Examples are offered and compared with other methods to demonstrate the features of the proposed method.     

Topology optimization of piezoelectric smart structures for minimum energy consumption under active control

This paper investigates topology optimization of the electrode coverage over piezoelectric patches attached to a thin-shell structure to reduce the energy consumption of active vibration control under harmonic excitations. The constant gain velocity feedback control method is employed, and the structural frequency response under control is analyzed with the finite element method. In the mathematical formulation of the proposed topology optimization model, the total energy consumption of the control system is taken as the objective function, and a constraint of the maximum allowable dynamic compliance is considered. The pseudo-densities indicating the distribution of surface electrode coverage over the piezoelectric layers are chosen as the design variables, and a penalized model is employed to relate the active damping effect and these design variables. The sensitivity analysis scheme of the control energy consumption with respect to the design variables is derived with the adjoint-variable method. Numerical examples demonstrate that the proposed optimization model is able to generate optimal topologies of electrode coverage over the piezoelectric layers, which can effectively reduce the energy consumption of the control system. Also, numerical comparisons with a minimum-volume optimization model show the advantage of the proposed method with respect to energy consumption. The proposed method may provide useful guidance to the layout optimization of piezoelectric smart structures where the energy supply is limited, such as miniature vibration control systems.