A new probabilistic distribution matching PATC formulation using polynomial chaos expansion

In the existing probabilistic hierarchical optimization approaches, such as probabilistic analytical target cascading (PATC), all the stochastic interrelated responses are characterized only by the first two statistical moments. However, due to the high nonlinear relation between the inputs and outputs, the interrelated responses are not necessarily normally distributed. The existing approaches, therefore, may not accurately quantify the probabilistic characteristics of the interrelated responses, and would further prevent achieving the real optimal solution. To overcome this deficiency, a novel PATC approach, named PATC-PCE is developed. By employing the polynomial chaos expansion (PCE) technique, the entire distribution of interrelated response can be characterized by a PCE coefficients vector, and then matched and propagated in the hierarchy. Comparative studies show that PATC-PCE outperforms PATC in terms of yielding more accurate optimal solutions and fewer design cycles when the interrelated response are random non-normal quantities, while at a sacrifice of extra function evaluations.     

基于风振性能的高层建筑抗风设计优化

提出了一个基于风振性能的结构优化设计方法来解决高层建筑在各种风灾条件下的风致性能设计问题。借签抗震性能设计方法,划分了高层建筑的三水准抗风设防标准。各水准下的风灾烈度及其概率要素可以通过对气象部门的极端风速记录开展统计极值分析来进行。基于高层建筑风致动力响应的频域内风振模态分析,给出了各种风致加速度指标的实用表达式。通过数学建模对风致性能的结构优化设计问题作出了精确描述,用结构构件尺寸等设计变量显式的表达出各种风振性能设计指标。发展了基于优化准则法的数值算法来求解高层建筑在多种风致性能设计约束条件下的结构最优设计。最后,通过一个40层住宅楼的工程实例验证了文中提出的多级抗风性能自动化结构设计方法的有效性和实用性。     

Topology optimization of buildings subjected to stochastic wind loads

Traditionally, the main structural system of tall buildings is designed iteratively to resist extreme wind loads, which provides safe, but typically suboptimal building systems. Topology optimization provides a general approach to obtain optimal material layout to carry the required load within specified design constraints. The wind loading on a structure is typically modeled as an equivalent static load or as a spatio-temporal stochastic field. While a few models for stochastic wind excitation are available, these approaches are focused on obtaining samples for time history analyses. In this study the stochastic wind excitation is modeled as a filtered vector white noise; the state space representation of the filter is obtained by solving a regularized optimization problem from known stationary wind power spectral densities. An augmented state space representation is formed by combining the equation of motion for the structure with the excitation filter. The stationary covariances of the structural responses of interest are then obtained by solving the associated Lyapunov equation. Dynamic condensation of the equations of motion is employed to increase the efficiency of the proposed approach. Objective functions for the optimization scheme are defined in terms of the stationary covariance of the response; various objectives are considered corresponding to typical design considerations. An equivalent smooth formulation is employed to solve the non-smooth min–max problem. A gradient-based method is used to update the design variables, while the sensitivities are computed from the solution of an adjoint Lyapunov equation. The proposed topology optimization scheme is illustrated for a tall building subjected to along-wind and across-wind loads. The results presented herein demonstrate the efficacy of the proposed approach for efficient topology optimization of buildings subjected to stochastic wind excitation.     

Shape optimal design of elastic planar frames with non-linear response

This paper describes an approach to shape optimal design of elastic planar frames with non-linear response. The foundation of the proposed approach forms an appropriate strategy of shape representation of the structure, based on the design element technique. A frame structure is treated as to be assembled from several frame design elements, which in turn may consist of several appropriately joined beam finite elements. The shape of each frame design element is defined by convenient functions involving Bezier blending polynomials. The original formulation of the beam finite element, proposed by Saje, is modified in order to fit nicely into the context of the frame design element technique. The formulation of the shape optimal design problem in a form of a problem of non-linear mathematical programming and its solution by employing gradient-based methods of mathematical programming are discussed briefly. The theory is illustrated with two numerical examples.     

Multi‐response optimization in industrial experiments using Taguchi's quality loss function and principal component analysis

Many industrial experiments based on Taguchi's parameter design (PD) methodology deal with the optimization of a single performance quality characteristic. Studies have shown that the optimal factor settings for one performance characteristic are not necessarily compatible with those of other performance characteristics. Multi‐response problems have received very little attention among industrial engineers and Taguchi practitioners. Many Taguchi practitioners have employed engineering judgement for determining the final optimal condition when several responses are to be optimized. However, this approach always brings some level of uncertainty to the decision‐making process and is very subjective in nature. In order to rectify this problem, the author proposes an alternative approach using a powerful multivariate statistical method called principal component analysis (PCA). The paper also presents a case study in order to demonstrate the potential of this approach. Copyright © 2000 John Wiley & Sons, Ltd.     

Simultaneous Optimization of Robust Parameter and Tolerance Design Based on Generalized Linear Models

Robust parameter design (RPD) and tolerance design (TD) are two important stages in design process for quality improvement. Simultaneous optimization of RPD and TD is well established on the basis of linear models with constant variance assumption. However, little attention has been paid to RPD and TD with non‐constant variance of residuals or non‐normal responses. In order to obtain further quality improvement and cost reduction, a hybrid approach for simultaneous optimization of RPD and TD with non‐constant variance or non‐normal responses is proposed from generalized linear models (GLMs). First, the mathematical relationship among the process mean, process variance and control factors, noise factors and tolerances is derived from a dual‐response approach based on GLMs, and the quality loss function integrating with tolerance is developed. Second, the total cost model for RPD‐TD concurrent optimization based on GLMs is proposed to determine the best control factors settings and the optimal tolerance values synchronously, which is solved by genetic algorithm in detail. Finally, the proposed approach is applied into an example of electronic circuit design with non‐constant variance, and the results show that the proposed approach performs better on quality improvement and cost reduction. Copyright © 2012 John Wiley & Sons, Ltd.     

A review of robust design methods for multiple responses

Problems in engineering design often involve determining design variable settings to optimize individual product performance for multiple criteria, which are often in conflict. We review mathematically rigorous techniques from the statistical literature for finding a vector x of design variable settings, which produces an optimal compromise solution among a group of prioritized response variables. The best compromise solution is typically gained by optimizing an objective function, which incorporates the prioritized demands of multiple responses. Since most multi-response objective functions are constructed by combining the functions used to optimize univariate responses, a review of the prominent univariate approaches is presented first. A multivariate approach from the engineering literature called the compromise decision support problem (cDSP) is also reviewed. Finally, a table comparing the relative merits of the different multivariate approaches summarizes the article in a concise and user-friendly fashion.     

Stochastic sensitivity analysis of energy-dissipating structures with nonlinear viscous dampers by efficient equivalent linearization technique based on explicit time-domain method

Nonlinear fluid viscous dampers have been widely used in energy-dissipating structures due to their stable and high dissipation capacity and low maintenance cost. However, the literature on stochastic optimization of nonlinear viscous dampers under nonstationary excitations is limited. This paper is devoted to the stochastic response and sensitivity analysis of large-scale energy-dissipating structures equipped with nonlinear viscous dampers subjected to nonstationary seismic excitations. The analysis procedure is developed in the frame of the equivalent linearization method (ELM) in conjunction with the explicit time-domain method (ETDM). The equivalent linear system and the corresponding statistical moments of responses at a specific time instant are first obtained through a series of stochastic response analyses of the linearized systems. Then the sensitivities of the statistical moments of responses are determined via a series of stochastic sensitivity analyses of the equivalent linear system at the corresponding time instant. The above two iterative procedures are facilitated at high efficiency using ETDM with explicit formulations of the statistical moments of responses and the sensitivities of the statistical moments. This process is repeated for different time instants, and the time histories of the statistical moments and their sensitivities can be obtained. The stochastic response and sensitivity results are further utilized to conduct the stochastic optimal parametric design of the nonlinear viscous dampers. A one-storey building model equipped with a nonlinear viscous damper is analyzed to demonstrate the accuracy of the proposed method, and a suspension bridge with a main span of 1200 m equipped with 4 nonlinear viscous dampers is further investigated to illustrate the feasibility of the proposed method for stochastic optimal design of large-scale structures.     

A Novel Approach to Simultaneous Robust Design of Product Parameters and Tolerances Using Quality Loss and Multivariate ANOVA Concepts

Design and development of high quality products are of utmost importance to any production plant. Product design consists of parameter design and tolerance design, which affect the product performances and the manufacturing costs, respectively. Most products involve more than one quality feature. Design and development of such products raise multi‐response surface problems in which it is necessary to determine the optimal values of parameters and the tolerances for all responses simultaneously. In this research, an approach for simultaneous robust parameter and tolerance design is proposed to deal with multi‐response problems. The proposed method employs quality loss concept and one‐way multivariate analysis of variance. Two simulation studies are performed to validate the applicability of the proposed method. Research findings show that the proposed method performs better in quality improvement as well as in cost reduction than the existing methods. The variances of the responses are also lower than those of the other methods, that is, the proposed method results in a more robust approach to product design. Copyright © 2016 John Wiley & Sons, Ltd.     

多级性态隔震体系试验研究和结构动力响应分析

铅芯橡胶支座力学性能受竖向压应力和水平剪应变影响较大,且在极罕遇地震下传统隔震结构易发生位移过大而引发隔震沟碰撞风险。该文提出一种新型高性能多级性态隔震支座,可在单个支座中兼顾竖向高承载特性和稳定的水平滞回特性,所提出支座具有与结构多水准抗震性能相匹配的多级刚度特性。制作多级性态原型支座并进行力学性能试验,得到多级性态支座不同工况下的水平滞回性能,基于试验结果提出了多级性态支座的力学滞回模型。对某高层框架剪力墙结构进行多级性态隔震设计并与原设计铅芯橡胶支座（lead rubber bearing,LRB）隔震方案对比分析,结果表明多级性态隔震系统具有理想的水平隔震效果,且可有效控制支座的竖向拉应力,在极罕遇地震下有效控制隔震层水平位移,实现隔震结构具有多级的抗震性能。     

An approach to optimal design of structures with non-linear response

A commonly adopted procedure to solve a structural optimum design problem is to improve the design iteratively by solving a sequence of simpler problems, where each of these is obtained as an approximation of the original problem. Considering structures with linear response, this procedure works very well and can be implemented quite straightforwardly. However, considering structures with non-linear response, the situation is more sophisticated since structural stability properties have to be taken into account. The paper presents an approach to optimal design of statically loaded structures with non-linear response. It is proposed to replace the original problem with a sequence of similar problems from which corresponding approximate problems can easily be obtained. These similar problems additionally contain an estimation of the critical load constraint while other constraints are imposed at some properly chosen load level, which is lower or at most equal to the critical load level. By this approach, the structural stability requirements are taken into account in an effective manner. The efficiency is illustrated with several examples involving kinematic as well as material non-linear effects.     

随机波浪作用下海洋平台响应分析与结构优化设计

根据随机响应分析的确定性算法,提出了随机波浪作用下海洋平台结构响应灵敏度分析方法。采用虚拟激励算法,将随机响应灵敏度分析问题转化为稳态简谐响应分析,进一步求出结构随机响应的均值、方差、协方差等统计量的灵敏度。这一过程可直接利用响应分析的程序模块,避免计算振型导数的困难,具有较高的计算效率和精度。与差分法计算结果的比较验证了该方法的效率和精度。在此基础上,研究了随机波浪作用下海洋平台结构尺寸与形状优化方法。本文方法在有限元分析与优化设计软件系统JIFEX中实现,数值算例验证了算法的有效性。     

A Prediction Region‐based Approach to Model Uncertainty for Multi‐response Optimization

Multi‐response optimization methods rely on empirical process models based on the estimates of model parameters that relate response variables to a set of design variables. However, in determining the optimal conditions for the design variables, model uncertainty is typically neglected, resulting in an unstable optimal solution. This paper proposes a new optimization strategy that takes model uncertainty into account via the prediction region for multiple responses. To avoid obtaining an overly conservative design, the location and dispersion performances are constructed based on the best‐case strategy and the worst‐case strategy of expected loss. We reveal that the traditional loss function and the minimax/maximin strategy are both special cases of the proposed approach. An example is illustrated to present the procedure and the effectiveness of the proposed loss function. The results show that the proposed approach can give reasonable results when both the location and dispersion performances are important issues. Copyright © 2015 John Wiley & Sons, Ltd.     

A review of robust design methods for multiple responses

Problems in engineering design often involve determining design variable settings to optimize individual product performance for multiple criteria, which are often in conflict. We review mathematically rigorous techniques from the statistical literature for finding a vector x of design variable settings, which produces an optimal compromise solution among a group of prioritized response variables. The best compromise solution is typically gained by optimizing an objective function, which incorporates the prioritized demands of multiple responses. Since most multi-response objective functions are constructed by combining the functions used to optimize univariate responses, a review of the prominent univariate approaches is presented first. A multivariate approach from the engineering literature called the compromise Decision Support Problem is also reviewed. Finally a table comparing the relative merits of the different multivariate approaches summarizes the article in a concise and user-friendly fashion.     

Stochastic Subset Optimization for optimal reliability problems

Reliability-based design of a system often requires the minimization of the probability of system failure over the admissible space for the design variables. For complex systems this probability can rarely be evaluated analytically and so it is often calculated using stochastic simulation techniques, which involve an unavoidable estimation error and significant computational cost. These features make efficient reliability-based optimal design a challenging task. A new method called Stochastic Subset Optimization (SSO) is proposed here for iteratively identifying sub-regions for the optimal design variables within the original design space. An augmented reliability problem is formulated where the design variables are artificially considered as uncertain and Markov Chain Monte Carlo techniques are implemented in order to simulate samples of them that lead to system failure. In each iteration, a set with high likelihood of containing the optimal design parameters is identified using a single reliability analysis. Statistical properties for the identification and stopping criteria for the iterative approach are discussed. For problems that are characterized by small sensitivity around the optimal design choice, a combination of SSO with other optimization algorithms is proposed for enhanced overall efficiency.     

Design and multi-objective optimization for a broad self-amplified 2-DOF monolithic mechanism

This paper proposes a structural design and multi-objective optimization of a two-degree-of-freedom (DOF) monolithic mechanism. The mechanism is designed based on compliant mechanism with flexure hinge and is compact in size (126 mm by 107 mm). Unlike traditional one-lever mechanisms, a new double-lever mechanism is developed to increase the working travel amplification ratio of the monolithic mechanism. The ideal amplification ratio, the working travel, the statics and the dynamics of the mechanism are taken into consideration. The effects of design variables on the output responses such as the displacement and first natural frequency are investigated via finite-element analysis based on response surface methodology. The fuzzy-logic-based Taguchi method is then used to simultaneously optimize the displacement and the first natural frequency. Experimental validations are conducted to verify the optimal results, which are compared to those of the original design. On using a finite-element method, the validation results indicated that the displacement and frequency are enhanced by up to 12.47% and 33.27%, respectively, over those of the original design. The experiment results are in a good agreement with the simulations. It also revealed that the developed fuzzy-logic-based Taguchi method is an effectively systematic reasoning approach for optimizing the multiple quality characteristics of compliant mechanisms. It was noted that the working travel/displacement of the double-lever mechanism is much larger than that of the traditional one-lever mechanism. It leads to the conclusion that the proposed mechanism has good performances for manipulations and positioning systems.     

Statistical modelling of the variation in advanced process technologies using a multi-level partitioned response

The advances in semiconductor processing technologies have led to the need for a detailed understanding and stringent control of the variations in device performance. Statistical techniques provide methods, such as response surface modelling (RSM), to measure, characterise and model the variations, thus enabling an understanding and identification of the impact of these on both yield and performance of the devices and circuits built from advanced process technologies. The construction of response surface (RS) models, however, has been restricted to only a few variables, due to the number of TCAD simulations and hence the statistical analyses required for fitting sufficiently accurate models. The problem of modelling a large number of manufacturing process parameters is addressed by partitioning the parameters and subsequently building multi-level RS models which can analyse and predict the process variability. This approach greatly reduces (by approximately two to three orders of magnitude) the large number of TCAD simulations necessary to fit the RS models. The application of multi-level partitioned RSM is demonstrated on a 65 nm CMOS technology. With the device dimensions shrinking and the impact of manufacturing process variations becoming dominant on the device performance, the proposed approach plays a vital role in design for manufacturability. The variability information obtained from these models is important not only to control and optimise the process variation but also to quantify its effects on device and circuits designs.     

The Taguchi's Performance Statistic to Optimize Theophylline Beads Production in a High-Speed Granulator

An original process using a simple procedure is developed to produce theophylline active pellets. In order to improve this process, an optimization approach is applied. But rather than only trying to bring the process to the target optimal values, attempt is made to find operating conditions leading also to stable and non-sensitive pellets characteristics. In this purpose, the classic experimental design approach and response surface methodology are completed by using Taguchi's philosophy.     

The Taguchi's Performance Statistic to Optimize Theophylline Beads Production in a High-Speed Granulator

An original process using a simple procedure is developed to produce theophylline active pellets. In order to improve this process, an optimization approach is applied. But rather than only trying to bring the process to the target optimal values, attempt is made to find operating conditions leading also to stable and non-sensitive pellets characteristics. In this purpose, the classic experimental design approach and response surface methodology are completed by using Taguchi's philosophy.