Optimal design of composite shells based on minimum weight and maximum feasibility robustness

A robust design optimization (RDO) approach for minimum weight and safe shell composite structures with minimal variability into design constraints under uncertainties is proposed. A new concept of feasibility robustness associated to the variability of design constraints is considered. So, the feasibility robustness is defined through the determinant of variance–covariance matrix of constraint functions introducing in this way the joint effects of the uncertainty propagations on structural response. A new framework considering aleatory uncertainty into RDO of composite structures is proposed. So, three classes of variables and parameters are identified: deterministic design variables, random design variables and random parameters. The bi-objective optimization search is performed using on a new approach based on two levels of dominance denoted by Co-Dominance-based Genetic Algorithm (CoDGA). The use of evolutionary concepts together sensitivity analysis based on adjoint variable method is a new proposal. The examples with different sources of uncertainty show that the Pareto front definition depends on random design variables and/or random parameters considered in RDO. Furthermore, the importance to control the uncertainties on the feasibility of constraints is demonstrated. CoDGA approach is a powerfully tool to help designers to make decision establishing the priorities between performance and robustness.     

Variation-aware clock network buffer sizing using robust multi-objective optimization

Many engineering optimization problems include unavoidable uncertainties in parameters or variables. Ignoring such uncertainties when solving the optimization problems may lead to inferior solutions that may even violate problem constraints. Another challenge in most engineering optimization problems is having different conflicting objectives that cannot be minimized simultaneously. Finding a balanced trade-off between these objectives is a complex and time-consuming task. In this paper, an optimization framework is proposed to address both of these challenges. First, we exploit a self-calibrating multi-objective framework to achieve a balanced trade-off between the conflicting objectives. Then, we develop the robust counterpart of the uncertainty-aware self-calibrating multi-objective optimization framework. The significance of this framework is that it does not need any manual tuning by the designer. We also develop a mathematical demonstration of the objective scale invariance property of the proposed framework. The engineering problem considered in this paper to illustrate the effectiveness of the proposed framework is a popular sizing problem in digital integrated circuit design. However, the proposed framework can be applied to any uncertain multi-objective optimization problem that can be formulated in the geometric programming format. We propose to consider variations in the sizes of circuit elements during the optimization process by employing ellipsoidal uncertainty model. For validation, several industrial clock networks are sized by the proposed framework. The results show a significant reduction in one objective (power, on average 38 %) as well as significant increase in the robustness of solutions to the variations. This is achieved with no significant degradation in the other objective (timing metrics of the circuit) or reduction in its standard deviation which demonstrates a more robust solution.     

Robust design of tuned mass dampers installed on multi-degree-of-freedom structures subjected to seismic action

This study deals with robust optimum design of tuned mass dampers installed on multi-degree-of-freedom systems subjected to stochastic seismic actions, assuming the structural and seismic model parameters to be uncertain. A new global performance index for evaluating the efficiency of protection systems is proposed, as an alternative to commonly used local performance indices such as the maximum interstorey drift. The latter can be considered a good estimator of seismic damage, but it does not measure the whole structural integrity. The direct perturbation method based on first order approximation is adopted to evaluate the effects of uncertainties on the response. The robust design is formulated as a multi-objective optimization problem, in which both the mean and the standard deviation of the performance index are simultaneously minimized. A comparison of the effectiveness and robustness of tuned mass dampers designed using local or global performance indices is carried out, considering different levels of uncertainty.     

Stochastic optimization using a sparse grid collocation scheme

In computational sciences, optimization problems are frequently encountered in solving inverse problems for computing system parameters based on data measurements at specific sensor locations, or to perform design of system parameters. This task becomes increasingly complicated in the presence of uncertainties in boundary conditions or material properties. The task of computing the optimal probability density function (PDF) of parameters based on measurements of physical fields of interest in the form of a PDF, is posed as a stochastic optimization problem. This stochastic optimization problem is solved by dividing it into two problems—an auxiliary optimization problem to construct stochastic space representations from the PDF of measurement data, and a stochastic optimization problem to compute the PDF of problem parameters. The auxiliary optimization problem is solved using a downhill simplex method, whilst a gradient based approach is employed for solving the stochastic optimization problem. The gradients required for stochastic optimization are defined, using appropriate stochastic sensitivity problems. A computationally efficient sparse grid collocation scheme is utilized to compute the solution of these stochastic sensitivity problems. The implementation discussed, requires minimum intrusion into existing deterministic solvers, and it is thus applicable to a variety of problems. Numerical examples involving stochastic inverse heat conduction problems, contamination source identification problems and large deformation robust design problems are discussed.     

Sensitivity importance‐based robust optimization of structures with incomplete probabilistic information

Robust design optimization (RDO) is usually performed by minimizing the nominal value of a performance function and its dispersion considering equal importance to each individual gradient of the performance function. However, it is well known that all gradients are not equally important. An efficient sensitivity importance‐based RDO technique is proposed in the present study for optimum design of structures characterized by bounded uncertain input parameters. The basic idea of the proposed RDO formulation is to improve the robustness of a performance function by using a new gradient index that utilizes the importance factors proportional to the importance of the gradients of the performance function. The same concept is also extended to the constraints. To enhance the robustness of the constraints, the constraint functions are also modified by using the importance factor proportional to the importance of the associated gradient of the constraint. Because all the variables are not equally important to capture the presence of uncertainty, an improved robust solution is obtained by the proposed approach compared with the conventional RDO approach. The present formulation is illustrated with the help of three informative examples. The results are compared with the conventional RDO results to study the effectiveness of the proposed RDO approach. Copyright © 2012 John Wiley & Sons, Ltd.     

Distributionally robust optimization for engineering design under uncertainty

This paper addresses the challenge of design optimization under uncertainty when the designer only has limited data to characterize uncertain variables. We demonstrate that the error incurred when estimating a probability distribution from limited data affects the out-of-sample performance (ie, performance under the true distribution) of optimized designs. We demonstrate how this can be mitigated by reformulating the engineering design problem as a distributionally robust optimization (DRO) problem. We present computationally efficient algorithms for solving the resulting DRO problem. The performance of the DRO approach is explored in a practical setting by applying it to an acoustic horn design problem. The DRO approach is compared against traditional approaches to optimization under uncertainty, namely, sample-average approximation and multiobjective optimization incorporating a risk reduction objective. In contrast with the multiobjective approach, the proposed DRO approach does not use an explicit risk reduction objective but rather specifies a so-called ambiguity set of possible distributions and optimizes against the worst-case distribution in this set. Our results show that the DRO designs, in some cases, significantly outperform those designs found using the sample-average or the multiobjective approach.     

Gearbox design for uncertain load requirements using active robust optimization

Design and optimization of gear transmissions have been intensively studied, but surprisingly the robustness of the resulting optimal design to uncertain loads has never been considered. Active Robust (AR) optimization is a methodology to design products that attain robustness to uncertain or changing environmental conditions through adaptation. In this study the AR methodology is utilized to optimize the number of transmissions, as well as their gearing ratios, for an uncertain load demand. The problem is formulated as a bi-objective optimization problem where the objectives are to satisfy the load demand in the most energy efficient manner and to minimize production cost. The results show that this approach can find a set of robust designs, revealing a trade-off between energy efficiency and production cost. This can serve as a useful decision-making tool for the gearbox design process, as well as for other applications.     

Sequential approximate robust design optimization using radial basis function network

This paper proposes a sequential approximate robust design optimization (SARDO) with the radial basis function (RBF) network. In RDO, the mean and the standard deviation of objective should be minimized simultaneously. Therefore, the RDO is generally formulated as bi-objective design optimization. Our goal is to find a robust optimal solution with a small number of function evaluations, not identifying a set of Pareto-optimal solution using Multi-Objective Evolutionary Algorithms. The weighted sum is often used to find a robust optimal solution. In contrast, the weighted lp norm method is used in this paper. Through illustrative examples, some validations of the weighted lp norm method to the RDO are clarified. Next, SARDO with the RBF network is discussed. In general, the standard deviation of functions is obtained by using the finite difference method. Thus, in order to obtain the standard deviation of functions, the finite difference method is directly applied to the response surface. High accuracy of the finite difference method will leads to highly accurate robust optimal solution. In order to avoid the inaccurate numerical calculation, the standard deviation is expressed by only the Gaussian kernel. As the result, it is expected that a highly accurate robust optimal solution can be found with a small number of function evaluations. Through numerical examples, the validity of the proposed approach is examined. Finally, the variable blank holder force trajectory for reducing springback is examined.     

An efficient ensemble of radial basis functions method based on quadratic programming

Radial basis function (RBF) surrogate models have been widely applied in engineering design optimization problems to approximate computationally expensive simulations. Ensemble of radial basis functions (ERBF) using the weighted sum of stand-alone RBFs improves the approximation performance. To achieve a good trade-off between the accuracy and efficiency of the modelling process, this article presents a novel efficient ERBF method to determine the weights through solving a quadratic programming subproblem, denoted ERBF-QP. Several numerical benchmark functions are utilized to test the performance of the proposed ERBF-QP method. The results show that ERBF-QP can significantly improve the modelling efficiency compared with several existing ERBF methods. Moreover, ERBF-QP also provides satisfactory performance in terms of approximation accuracy. Finally, the ERBF-QP method is applied to a satellite multidisciplinary design optimization problem to illustrate its practicality and effectiveness for real-world engineering applications.     

Interval-Based Uncertain Multi-Objective Optimization Design of Vehicle Crashworthiness

In this paper, an uncertain multi-objective optimization method is suggested to deal with crashworthiness design problem of vehicle, in which the uncertainties of the parameters are described by intervals. Considering both lightweight and safety performance, structural weight and peak acceleration are selected as objectives. The occupant distance is treated as constraint. Based on interval number programming method, the uncertain optimization problem is transformed into a deterministic optimization problem. The approximation models are constructed for objective functions and constraint based on Latin Hypercube Design (LHD). Thus, the interval number programming method is combined with the approximation model to solve the uncertain optimization problem of vehicle crashworthiness efficiently. The present method is applied to two practical full frontal impact (FFI) problems.     

An accurate penalty-based approach for reliability-based design optimization

A typical reliability-based design optimization (RBDO) problem is usually formulated as a stochastic optimization model where the performance of a system is optimized with the reliability requirements being satisfied. Most existing RBDO methods divide the problem into two sub-problems: one relates to reliability analysis, the other relates to optimization. Traditional approaches nest the two sub-problems with the reliability analysis as the inner loop and the optimization as the outer loop. Such nested approaches face the challenge of prohibitive computational expense that drives recent research focusing on decoupling the two loops or even fundamentally transforming the two-loop structure into one deterministic optimization problem. While promising, the potential issue in these computationally efficient approaches is the lowered accuracy. In this paper, a new decoupled approach, which performs the two loops sequentially, is proposed. First, a deterministic optimization problem is solved to locate the means of the uncertain design variables. After the mean values are determined, the reliability analysis is performed. A new deterministic optimization problem is then restructured with a penalty added to each limit-state function to improve the solution iteratively. Most existing research on decoupled approaches linearizes the limit-state functions or introduces the penalty into the limit-state functions, which may suffer the approximation error. In this research, the penalty term is introduced to change the right hand side (RHS) value of the deterministic constraints. Without linearizing or transforming the formulations of limit-state function, this penalty-based approach effectively improves the accuracy of RBDO. Comparison experiments are conducted to illustrate how the proposed method obtains improved solutions with acceptable computational cost when compared to other RBDO approaches collected from literature.     

An approach for robust PDE-constrained optimization with application to shape optimization of electrical engines and of dynamic elastic structures under uncertainty

We present a robust optimization framework that is applicable to general nonlinear programs (NLP) with uncertain parameters. We focus on design problems with partial differential equations (PDE), which involve high computational cost. Our framework addresses the uncertainty with a deterministic worst-case approach. Since the resulting min–max problem is computationally intractable, we propose an approximate robust formulation that employs quadratic models of the involved functions that can be handled efficiently with standard NLP solvers. We outline numerical methods to build the quadratic models, compute their derivatives, and deal with high-dimensional uncertainties. We apply the presented approach to the parametrized shape optimization of systems that are governed by different kinds of PDE and present numerical results.     

Multi-objective design optimization of electrostatically actuated microbeam resonators with and without parameter uncertainty

Electrostatically actuated microbeam resonators are widely used components in microelectromechanical systems for sensing and signal filtering purposes. Due to the uncertainties resulting from manufacturing processes, material properties, and modeling assumptions, microbeam resonators may exhibit significant variations in their performance compared to nominal designs. There has been limited research on the performance prediction and the design optimization of such microsystems while accounting for relevant uncertainties. In this study, such uncertainties are considered in terms of the variability of parameters that define the dimensions, the material properties, and the operating conditions of the device. In addition, uncertainties with respect to a two-dimensional model of a microbeam resonator subject to electrostatic actuation are considered. A finite element model consisting of both the microbeam and the substrate is developed. The actuation forces are predicted by a reduced order electrostatic model, which accounts for the electromechanical interaction. A computationally efficient procedure is presented for simulating the steady-state dynamic response under electrostatic forces. The probabilistic performance of the microresonator is investigated using Monte Carlo simulation. A genetic algorithm is used to optimize the stochastic behavior of the microbeam resonator. The design is posed as combinatorial multi-objective optimization problem. Two design criteria describing the filter performance in terms of the shape of the frequency–response curve are simultaneously considered. The numerical results demonstrate the effectiveness of this procedure for the multi-objective optimization design of microbeam resonators and the importance of considering parameter uncertainty in the design of these devices.     

Reliability based optimum design of Tuned Mass Damper in seismic vibration control of structures with bounded uncertain parameters

A reliability based optimization of Tuned Mass Damper (TMD) parameters in seismic vibration control under bounded uncertain system parameters is presented. The study on TMD with random parameters in a probabilistic framework is noteworthy. But, it cannot be applied when the necessary information about parameters uncertainties is limited. In such cases, the interval method is a viable alternative. Applying matrix perturbation theory through a first order Taylor series expansion about the mean values of the uncertain parameters’ conservative dynamic response bounds are obtained assuming a small degree of parameter uncertainty. The first-passage probability of failure of the system is taken as the performance objective. Using the interval extension of the performance objective, the vibration control problem under bounded uncertainties is transformed to the appropriate deterministic optimization problems yielding the lower and upper bound solutions. A numerical study is performed to elucidate the effect of parameters’ uncertainties on the TMD parameters’ optimization and the safety of the structure.     

Bi-directional evolutionary level set method for topology optimization

A bi-directional evolutionary level set method for solving topology optimization problems is presented in this article. The proposed method has three main advantages over the standard level set method. First, new holes can be automatically generated in the design domain during the optimization process. Second, the dependency of the obtained optimized configurations upon the initial configurations is eliminated. Optimized configurations can be obtained even being started from a minimum possible initial guess. Third, the method can be easily implemented and is computationally more efficient. The validity of the proposed method is tested on the mean compliance minimization problem and the compliant mechanisms topology optimization problem.     

Bi-objective cyclic scheduling in a robotic cell with processing time windows and non-Euclidean travel times

This paper addresses bi-objective cyclic scheduling in a robotic cell with processing time windows. In particular, we consider a more general non-Euclidean travel time metric where robot’s travel times are not required to satisfy the well-known triangular inequality. We develop a tight bi-objective mixed integer programming (MIP) model with valid inequalities for the cyclic robotic cell scheduling problem with processing time windows and non-Euclidean travel times. The objective is to minimise the cycle time and the total robot travel distance simultaneously. We propose an iterative ε-constraint method to solve the bi-objective MIP model, which can find the complete Pareto front. Computational results both on benchmark instances and randomly generated instances indicate that the proposed approach is efficient in solving the cyclic robotic cell scheduling problems.     

Polymorphic uncertain nonlinear programming approach for maximizing the capacity of V-belt driving

In this paper, a polymorphic uncertain nonlinear programming (PUNP) approach is developed to formulate the problem of maximizing the capacity in a system of V-belt driving with uncertainties. The constructed optimization model is found to consist of a nonlinear objective function and some nonlinear constraints with some parameters which are of uncertain nature. These uncertain parameters are interval parameters, random interval parameters, fuzzy parameters or fuzzy interval parameters. To find a robust solution of the problem, a deterministic equivalent formulation (DEF) is established for the polymorphic uncertain nonlinear programming model. For a given satisfaction level, this DEF turns out to be a nonlinear programming involving only interval parameters. A solution method, called a sampling based interactive method, is developed such that a robust solution of the original model with polymorphic uncertainties is obtained by using standard smooth optimization techniques. The proposed method is applied into a real-world design of V-belt driving, and the results indicate that both the PUNP approach and the developed algorithm are useful to the optimization problem with polymorphic uncertainty.     

Comparison of robust,reliability-based and non-probabilistic topology optimization under uncertain loads and stress constraints

It is nowadays widely acknowledged that optimal structural design should be robust with respect to the uncertainties in loads and material parameters. However, there are several alternatives to consider such uncertainties in structural optimization problems. This paper presents a comprehensive comparison between the results of three different approaches to topology optimization under uncertain loading, considering stress constraints: (1) the robust formulation, which requires only the mean and standard deviation of stresses at each element; (2) the reliability-based formulation, which imposes a reliability constraint on computed stresses; (3) the non-probabilistic formulation, which considers a worst-case scenario for the stresses caused by uncertain loads. The information required by each method, regarding the uncertain loads, and the uncertainty propagation approach used in each case is quite different. The robust formulation requires only mean and standard deviation of uncertain loads; stresses are computed via a first-order perturbation approach. The reliability-based formulation requires full probability distributions of random loads, reliability constraints are computed via a first-order performance measure approach. The non-probabilistic formulation is applicable for bounded uncertain loads; only lower and upper bounds are used, and worst-case stresses are computed via a nested optimization with anti-optimization. The three approaches are quite different in the handling of uncertainties; however, the basic topology optimization framework is the same: the traditional density approach is employed for material parameterization, while the augmented Lagrangian method is employed to solve the resulting problem, in order to handle the large number of stress constraints. Results are computed for two reference problems: similarities and differences between optimized topologies obtained with the three formulations are exploited and discussed.     

Performance-based design and seismic reliability analysis using designed experiments and neural networks

Seismic design involves many uncertainties that arise from the earthquake motions, structural geometries, material properties, and analytical models. Taking into account all major uncertainties, reliability analysis is applied to estimate probability of failure in each of a set of performance requirements. The probability estimation is best conducted through Monte Carlo simulations with variance reduction techniques. However, this may involve many performance function evaluations, each requiring a non-linear dynamic analysis, which may be very computationally demanding. In order to improve computational efficiency, this paper explores Design of Computer Experiments and Neural Networks for representation of structural behavior. The neural networks are directly employed for reliability assessment and design optimization. Performance-based seismic design is formulated as an optimization problem, with design parameters optimally calculated. Two case studies are presented to demonstrate efficiency and applicability of the methodology: a bridge bent with or without seismic isolation and a steel pipe pile foundation.