A Thesis entitled An Efficient Method for Reliability-based Design Optimization when the Design Variables are Random By Zhong Ren Submitted to the Graduate Faculty as partial fulfillment of the requirements for The Master of Science Degree in Industrial Engineering Dr. Efstratios Nikolaidis, Committee Chair Dr. Lesley Berhan, Committee Member Dr. Mehdi Pourazady , Committee Member Dr.
Matthew Franchetti, Committee Member Dr. Komuniecki, Dean College of Graduate Studies The University of Toledo August 2013 Copyright 2013, Zhong Ren This document is copyrighted material. Under copyright law, no parts of this document may be reproduced without the expressed permission of the author. An Abstract of An Efficient Method for Reliability-based Design Optimization when the Design Variables are Random By Zhong Ren Submitted to the Graduate Faculty as partial fulfillment of the requirements for The Master of Science in Industrial Engineering The University of Toledo August 2013 In many design problems, designers typically utilize finite element models to predict the behavior and assess the safety of a system.
It is challenging to perform probabilistic analysis, and design a reliable system, because repeated finite element analyses of large models are required, and these models must be coupled with an optimizer, which is often prohibitively expensive. This thesis presents a methodology for probabilistic analysis and reliability based design optimization (RBDO) to overcome the above challenge. RBDO incorporates probabilistic reanalysis (PRRA) into the optimization process so that the optimum design has a great chance of staying in the feasible design space despite the inevitable variability in the design variables/parameters. PRRA calculates very efficiently the system iii reliability for many probability distributions of the design variables by performing a single Monte Carlo simulation.
Another part of work integrates PRRA with two alternative methods to create a new design tool that can perform reliability based optimization efficiently. The first is Trust Region methodology and the second is a Global-Local methodology. These two methods are demonstrated and compared on a ten-bar truss structure. iv Acknowledgements I would like to express my sincere gratitude to Dr.
Efstratios Nikolaidis for his invaluable guidance and support throughout my graduate study and my Master research work at The University of Toledo. Most importantly I would like to thank him for his patience and understanding, as well as all the mentorship that he has given me through the years. I would like to thank Dr. Matthew Franchetti, Dr.
Lesley Berhan and Dr. Mehdi Pourazady for serving on my thesis defense committee. I would like to thank the MIME department at the University of Toledo for the financial support during my graduate study. I would like to thank my family and friends for encouraging me to pursue my master’s degree and supporting me throughout.
Without them everything I have done would not have been possible. v Contents Abstract iii Acknowledgements v Contents vi List of Tables viii List of Figures ix List of Abbreviations xi List of Symbols xii 1.1 Reliability-Based Design Optimization(RBDO) .4 Outline of Thesis .1 Probabilistic Design Overview .3 Trust Region Method .1 Overview of Trust Region Approach in RBDO .4 Global-local Method .1 Description of a Ten-bar Truss Model .2 Problem Definition and Objective Function .2 Trust Region Optimization Design .2 Results and Discussion .3 Global-Local Optimization Design .2 Results and Discussion .4 Discussion and Lessons Learnt. Conclusion 64 References 66 vii List of Tables 3.1 Cross sectional areas and weight of the optimal designs in studies [7], [26], and [27] .2 Properties of initial design in [7].3 The initial design and boundary for the 1^th iteration trust region .4 Optimization design results of Trust Region method .5 Initial design and bounds in global design .6 Initial designs and boundaries in local design.7 Global level design results.8 Local level design results .9 Final optimum design of the two design techniques.10 Time for simulation with 10,000 replications. 61 viii List of Figures 2-1 Demonstration of Trust Region method concept.
21 2-2 Flowchart of Trust Region design optimization. 24 2-3 Schematic representation of the Global-Local method. 28 3-1 Geometry and loads for the ten-bar truss example. 31 3-2 PDF of strengths of members 1 to 8, and 10 (red dotted curve), and 9 (solid blue curve).
34 3-3 PDF and CDF of applied loads 𝑃1 and 𝑃2. 35 3-4 The boundary of 𝐴𝑟𝑒𝑎𝑖. 38 3-5 Sample values of 𝐴𝑟𝑒𝑎1 and 𝐴𝑟𝑒𝑎2 , and the initial trust region represented by the red rectangle. 39 3-6 Illustration of the trust region half-width ∆.
41 3-7 Moving trail of iterations. 42 3-8 Failure probability and mass of optimum designs in ten iterations: First iteration results are depicted by the rightmost point and tenth iteration results by the leftmost point. 45 3-9 Global level optimization design results. 57 3-10 Local level optimization design results.
58 3-11 Optimum design results of the two methods. 60 ix 3-12 Moving trend of the optimum point in the Trust Region method. 63 x List of Abbreviations CDF……………. Cumulative Distribution Function DPC …………….Design Potential Concept ECARD………….
Exact-Capacity-Approximate-Response-Distribution ESORA…………. Efficient Sequential Optimization and Reliability Assessment FORM……………First-Order Reliability Method MC………………Monte Carlo MPP……………. Most Probable Point MV……………… Mean Value OCA …………….Optimality Criteria Algorithm PDF……………. Probability Density Function PRRA……………Probabilistic Reanalysis RBDO………….
Reliability-based design optimization RSM …………….Response Surface Method SORA ……………Sequential Optimization and Reliability Assessment SORM……………Second-Order Reliability Methods SQP………………Sequential Quadratic Programming xi List of Symbols 𝝁𝑿 ……… .the vector of the mean values of the random design variables X 𝝁𝑿𝐿 ……….lower bounds of the mean values 𝝁𝑿𝑈 ……….upper bounds of the mean values 𝜌 ……………the material density 𝐴𝑟𝑒𝑎𝑖 𝑘 …….the optimum design in the 𝑘 𝑡ℎ iteration 𝐸( ) ……….the expected value operator 𝑓𝑿,𝑷 𝒙, 𝒑 ……true probability density function 𝑠 𝑓𝑿,𝑷 𝒙, 𝒑 …….importance density function 𝐼(𝒙, 𝒑)………the system failure indicator function 𝐿𝑖 ………….the bar length 𝑷 ………….vector of random parameters 𝑃𝐹𝑠𝑦𝑠 ……….the allowable value of failure probability 𝑝𝑓𝑎𝑙𝑙 𝑛𝑓 ……………number of failures 𝑡𝑛−1,1−𝛼 ….Student’s t-distribution with 𝑛 − 1 degrees of freedom at the point that 2 𝛼 has probability lower than 1 − 2 𝑊𝐴𝑟𝑒𝑎 ……….weight of the truss ∆……………. trust region half-width xii Chapter 1 Introduction 1.1 Problem statement In the globally competitive marketplace, many industries have endeavored to improve the quality of the design procedures and manufacturing processes for their products. Manufacturers should be able to predict the reliability of a product as it evolves during the development process, so as to design the product so that it has a prescribed performance. Early in the design phase, it is necessary to make decisions based on incomplete information.
In order to quantify the performance of a system under uncertainty, a method is needed to determine its reliability. An efficient computational tool for estimating the system reliability of a given design is required in order to optimize the design, while maintaining a prescribed reliability. A designer faces many challenges when applying reliability analysis and design to real-world problems. Finding the structural reliability of a given design requires repeated analyses for many values of the random variables.
This is computationally expensive when using large finite element models. The computational cost increases dramatically 1 when calculating the probability of failure of different designs during the search for an optimum reliability-based design. Reliability-based design seeks to satisfy given requirements on the probability of failure, instead of the safety factor. This approach enables designers to consider the uncertainties involved in the design, construction, and service life of structures [1].
Reliability-based design optimization (RBDO) is a procedure that optimizes attributes of a design such as the performance, cost and weight while maintaining an acceptable reliability level. RBDO incorporates probabilistic analysis into the optimization process to ensure that the optimum design is likely to stay in the feasible region of the design space despite the variability in the design variables. One drawback of RBDO is its high computational cost in large-scale problems. A designer can reduce the cost by using Probabilistic Reanalysis (PRRA).
This method calculates very efficiently the system reliability for many different probability distributions of the design variables by performing a single Monte Carlo simulation. The objective of this thesis is to develop efficient computational analysis and design tools that use PRRA for RBDO problems with many design variables with low dispersion. This thesis presents a methodology for simulation-based RBDO using a trust region approach and global-local design models in conjunction with PRRA. The methodology performs well for problems with many random variables.
A key advantage of the method is that it performs only one Monte Carlo (MC) simulation in each trust region. A 2 Global-local design methodology performs reliability analysis at two levels. At the global level, the overall sample region of a design is determined, while at the local level the refined local regions are explored to find the optimum using simulation.2 Background There are many studies and approaches on reducing the computational cost of probabilistic design. The background and literature survey for this study is divided into two parts.
The first part covers the reliability-based design optimization, while the second part covers the probabilistic reanalysis.1 Reliability-based design optimization (RBDO) Many researchers have proposed methods to improve the effectiveness of reliability-based design optimization (RBDO) methods. Two challenging tasks in RBDO are to construct probabilistic models of the random variables from limited data, and to increase the computational efficiency of this method. Here, we review studies that address the second issue. Deterministic design optimization is traditionally used for most structural design problems.
The uncertainties in the variables are accounted for by safety factors in deterministic design [2]. Qu and Haftka(2004) stated: “Freudenthal (1962) showed that reliability can be expressed in terms of the probability distribution function of the safety factor.” This factor is the ratio of the design values of the strength (capacity) and the applied load (demand). Elishakoff [3] investigated the relationship between the safety 3 factor and reliability, and showed that, in some cases, the safety factor can be expressed explicitly in terms of the reliability. The standard safety factor is defined with respect to the response obtained with the mean values of the random variables.
Thus, a safety factor of 1.5 implies that with the mean values of the random variables we have a 50% margin between the response (e. the applied stress) and the capacity (e. stress at failure). An alternative definition uses design values of the random variables, which are conservative estimates of the values of these variables.
However, the value of the safety factor is not sufficient to specify the reliability of a design. To address this issue, Birger [3] introduced a factor, called the probabilistic sufficiency factor, that correlates better than the traditional safety factor with the target reliability [5]. Approaches based on the most probable point (MPP) concept can be used to estimate reliability. The seminal work by Hasofer and Lind in 1974 sparked the development of these approaches.
The MPP (also called design point) is defined as that point that is most likely to satisfy the limit-state function in the space of the random variables. In the standardized (reduced) normal space, the MPP is the point on the limit-state that has the minimum distance to the origin. A simple closed-form expression exists for approximation of the system reliability as a function of the distance of the MPP from the origin. This distance is called safety index.
First-order and second-order approximations exist for reliability assessment. These are called first-order reliability method (FORM) and second-order reliability methods (SORM) [6]. 4 Estimation of the probability of failure can be challenging when there are multiple failure points. These are local minima of the distance of a point of the limit state surface to the origin.
Therefore, an analyst should understand the physics of the system very well in order to use effectively FORM and SORM. In addition, the analyst must validate the reliability estimates by using Monte Carlo simulation. Kumar et al.