Robust Design Optimization Problem Research Articles

Data-driven aerodynamic shape design with distributionally robust optimization approaches

We formulate and solve data-driven aerodynamic shape design problems with distributionally robust optimization (DRO) approaches. DRO aims to minimize the worst-case expected performance in a set of distributions that is informed by observed data with uncertainties. Building on the findings of the work Gotoh, et al. (2018), we study the connections between a class of DRO and robust design optimization, which is classically based on the mean–variance (standard deviation) optimization formulation pioneered by Taguchi. Our results provide a new perspective to the understanding and formulation of robust design optimization problems. It enables data-driven and statistically principled approaches to quantify the trade-offs between robustness and performance, in contrast to the classical robust design formulation that captures uncertainty only qualitatively. Our preliminary computational experiments on aerodynamic shape optimization in transonic turbulent flow show promising design results.

Robust design optimization of a multi-body system with aleatory and epistemic uncertainty

There are several issues in robust design optimization of the multi-body system with aleatory and epistemic uncertainty, such as the difficulty in the multi-source uncertainty characterization and low efficiency in dynamic modeling, analysis and the corresponding solving strategy for its robust design optimization. To address these problems, this paper firstly constructs a unified analysis framework for aleatory and epistemic uncertainty for the multi-body system based on probability theory and evidence theory. Taking a multi-body system of a shipborne artillery model as an example, the transfer matrix method is next employed to rapidly calculate the dynamic response of the muzzle. The results are compared with those by the finite element method to verify the accuracy and efficiency of the transfer matrix method. A decision model of the robust design optimization with both aleatory and epistemic uncertainty is proposed. Based on the polynomial chaos expansion-Kriging model, subset simulation optimization method and a weighted single-objective solving strategy, the multi-objective robust design optimization problem involving aleatory and epistemic uncertainty is then transformed into a single-objective optimization one. Finally, results of both the multi-objective and weighted single-objective robust design optimization methods for the shipborne artillery model are compared to demonstrate the performance of the proposed method.

Active-learning Kriging-assisted robust design optimization of tuned mass dampers: Vibration mitigation of a steel-arch footbridge

Many typical footbridges are designed as slender structures to highlight their aesthetic characteristic. As these bridges can be highly susceptible to pedestrian-induced excitations, tuned mass dampers have been adopted as energy absorbers in many applications to mitigate this issue. The design of tuned mass dampers can be formulated as a robust design optimization problem considering various sources of uncertainties related to the bridge state and environment. To alleviate the computational burden of analyzing the statistical moments, surrogate model-based approaches have been introduced to build cheaper-to-evaluate substitutes, and the analytical expression of the robustness has been discussed to facilitate the optimization procedure. Motivated by this, the present paper derives an analytical expression for the robustness index, considering both deterministic and uncertain variables as inputs of the ordinary Kriging surrogate model. An active-learning framework is proposed to streamline the design optimization of tuned mass dampers for the vibration mitigation of slender footbridges. Two numerical examples are firstly investigated to validate the performance of the proposed method; and then, a practical application of a steel-arch footbridge is carefully studied to demonstrate the engineering feasibility and superiority of the proposed method. The results exhibit that the proposed method can efficiently and accurately facilitate the robust design of vibration mitigation devices for practical engineering projects, and showcases the potential to be used in other engineering problems.

Adaptive decoupled robust design optimization

Robust design optimization (RDO) is a valuable technique in the design of engineering structures as it can provide an optimum design solution that is relatively insensitive to input uncertainties. However, the nested double-loop estimation process required in RDO often results in significant computational costs. To address this issue, we propose an adaptive decoupled RDO method based on the Kriging surrogate model. This method transforms the nested double-loop estimation process into a traditional deterministic optimization procedure, thus reducing computational costs. Furthermore, a novel estimation expression for the performance standard deviation that can simultaneously reflect the uncertainties in both the prediction and the performance mean is established. The closed-form expressions of the performance mean and performance standard deviation under different design parameters are deduced, which are further implemented to the uncertainty propagation during the design optimization. Moreover, an adaptive framework is introduced to improve the computational accuracy of uncertainty propagation as well as optimization procedure to guarantee the estimation accuracy of RDO problems. Several numerical examples along with engineering cases are introduced to illustrate the effectiveness of the established adaptive decoupled adaptive RDO method, and the results demonstrate that the proposed method can effectively optimize the design of structures while reducing computational costs.

Multi‐Objective Optimization Models for the Design of Water Distribution Networks by Exploring Scenario‐Based Approaches

AbstractDemand in a water distribution network (WDN) is an aleatory variable, owing to the unpredictable behaviors of water users. Therefore, it is one of the main reasons for uncertainty in the design of this infrastructure. The increasing number of water demand data sets offers opportunities to improve the traditional deterministic design approaches of WDNs by combining statistical and optimization methods. Robust optimization (RO) takes demand uncertainty into account by studying solutions that perform well under any possible demand scenario, that is, any possible realizations of this variable in the lifetime of a WDN. The right choice of scenario is therefore essential to ensure the reliability of the designed network. This paper presents a statistical methodology for generating scenarios to be used to solve a robust design optimization problem. It involves three steps: (a) descriptive analytics of historical data to derive the marginal distributions of peak hour demand in each node of the WDN, (b) generation of a very large number of snapshots by stratified sampling from the correlated marginal distributions of nodal peak demand, (c) generation of the peak demand scenarios by reducing the number of snapshots. Two heuristic techniques are proposed to reduce the number of snapshots, and for each of them, two different numbers of scenarios are derived. Two multi‐objective RO models are solved: the first model includes cost minimization and a mean‐variance Generalized Resilience and Failure index maximization objectives, and the second one additionally considers the minimization of the maximum undelivered demand, formalized using a regret function.

Multiple parameter optimization methodology by integrating a game theory principle into priority-based decision making

Mathematical model development for multivariate decision-making problems (i.e., multiple parameter and response optimization problems) has received significant attention and has been investigated by many researchers and practitioners. Although many different optimization models and methods, including priority and weight allocation problems, have been proposed in the literature, there is significant room for improvement. The majority of existing optimization methods may have difficulties in determining reliable priorities and weights for practical industrial situations. In addition, because they often focus on tradeoffs only among multiple output responses, these methods may not incorporate the multiple priority and/or weight effects directly from input factors to output responses (i.e., quality characteristics). To address these problems, the primary objective of this research is to propose a new multi-objective optimization approach by integrating a game theory principle (i.e., the Stackelberg leadership game) into a robust parameter design model to determine the optimal factor settings. First, a multi-response robust design optimization (RDO) problem was formulated using a mean squared error model and response surface methodology. A Stackelberg leadership game is then integrated into this RDO problem, where multiple responses play the role of game participants. Second, the integrated RDO model using the Stackelberg game-based multi-response (SGMR) formulation approach is analyzed by decomposition into various sequential optimization models in terms of different leader–follower relationships. Finally, non-dominated solutions are obtained from the proposed model by evaluating the overall quality loss (OQL). A pharmaceutical numerical example was performed to demonstrate the proposed integrated model and to examine whether this model can determine the most efficient solutions when the relative importance among multiple responses is unidentified. In addition, a comparative study associated with the conventional multi-objective optimization methods and the newly proposed RDO model is presented, and the proposed RDO model provided significantly better solutions than the conventional methods in the relevant comparative study.

Swarm intelligence hybridized with genetic search in multi-objective design optimization under constrained-Pareto dominance

The inclusion of uncertainty in structural design optimization led to more complex design optimization formulations, where uncertainty quantification has significant influence on solution methods and computing times. Designs maintaining steady levels of performance, under uncertainty, are called robust and the combination of both robustness and performance optimality leads to a methodology called Robust Design Optimization (RDO). In this work a new approach to the RDO of angle-ply composite laminate structures is proposed. The key concept of this methodology is the hybridization between the principles of Particle Swarm Optimization and Genetic Algorithms, through the evolution of multiple populations based on local and global constrain-dominance. The RDO problem is here defined as the bi-objective minimization of the structural weight (optimality) and the determinant of the variance–covariance matrix (robustness) of the system’s response functionals, subject to stress and displacement constraints. A numerical structural design problem, representing a composite laminate engine hood shell, shows the capabilities of the proposed approach. Results display the good convergence properties of the proposed hybridizations both in the search and objective spaces.

A response band-based method for time-dependent reliability-based robust design optimization

In this paper, a response band-based method for time-dependent reliability-based robust design optimization is proposed. The proposed method provides a novel alternative framework, consist of a two-step transformation stage and a solving stage, to solve the time-dependent reliability-based robust design optimization problem. The original time-dependent reliability-based robust design optimization problem is transformed into an equivalent deterministic robust design optimization problem in the transformation stage, and the equivalent problem is settled in the solving stage. In the transformation stage, the dynamic modal decomposition technique and the kriging technique are combined to overcome the problem that there is no standard for both time division and observation sampling in the commonly used transformation methods. In the solving stage, an approach for constructing the response band of the objective function is presented, which significantly reduces the computational consumption of the variation evaluation of the objective function. Five cases are employed to verify the effectiveness of the proposed method.

RELIABILITY-BASED ROBUST DESIGN OPTIMIZATION FOR BUILDING-CONNECTING INERTIAL-MASS DAMPER

In this paper, we propose an accurate and efficient reliability evaluation method for robust design optimization using a probabilistic approach. First, we verify the validity of this method using two benchmark functions. Then, we apply this method to the robust design optimization problem of building-connecting damper parameters and verify the effect of robust design. We consider the variation of the natural period and structural damping ratio of buildings, and directly evaluate the criteria satisfaction rate (CSR) using the response value calculated by time-history response analysis.

Reliability‐based robust design optimization for building‐connecting inertial mass damper

AbstractThis paper proposes an accurate and efficient reliability evaluation method for robust design optimization using a probabilistic approach. First, the validity of this method is verified using two benchmark functions. This method is then applied to the robust design optimization problem of building‐connecting damper parameters, and the effect of robust design is verified. The variation in the natural period and structural damping ratio of buildings is considered, and the criteria satisfaction rate (CSR) are directly evaluated using the response value calculated by time history response analysis.

Performance analysis of a robust design optimization of a solenoid with different sensitivity metrics

Optimization is an essential part of designing electrical machines and devices. Considering the uncertainties and tolerances from the beginning of the design process can lead to a more robust design and significantly reduce the number of waste products during the manufacturing process. However, the computational demand for these robust design optimization problems is amazingly high. This paper examines how the computational demand and the robustness of the results depend on the applied sensitivity metric. The multi-objective TEAM 35 benchmark problem has been used as the basis of the comparison. This optimization task aims to create a homogeneous magnetic field in a predefined coil region, which is insensitive to the positioning errors of the coil turns. The original definition of the problem uses a simple worst-case sensitivity metric calculated from the extreme values of the optimized turn parameters in the given tolerance range. This sensitivity metric has been replaced by Plackett–Burman, Box–Behnken, and Central Composite Design-based metrics. It was shown in this simple geometry that there is a significant difference between the resulting sensitivities. Nevertheless, Plackett–Burman and Central Composite Design provided the more accurate and consistent estimate of the sensitivity of the examined layouts, with a higher but still reasonable computation demand than the worst-case metric.

A Sequential Sampling Approach for Multi-Fidelity Surrogate Modeling-Based Robust Design Optimization

Abstract Multi-fidelity surrogate modeling has been extensively used in engineering design to achieve a balance between computational efficiency and prediction accuracy. Sequential sampling strategies have been investigated to improve the computational efficiency of surrogate-assisted design optimization. The existing sequential sampling approaches, however, are dedicated to either deterministic multi-fidelity design optimization or robust design under uncertainty using single-fidelity models. This paper proposes a sequential sampling method for robust design optimization based on multi-fidelity modeling. The proposed method considers both design variable uncertainty and interpolation uncertainty during the sequential sampling. An extended upper confidence boundary (EUCB) function is developed to determine both the sampling locations and the fidelity levels of the sequential samples. In the EUCB function, the cost ratio between high- and low-fidelity models and the sampling density are considered. Moreover, the EUCB function is extended to handle constrained robust design optimization problems by combining the probability of feasibility. The performance of the proposed approach is verified using two analytical examples and an engineering case. Results show that the proposed sequential approach is more efficient than the one-shot sampling method for robust design optimization problems.

A probability of improvement-based multi-fidelity robust optimization approach for aerospace products design

Uncertainty is inevitable in aerospace products design optimization, which may lead to the undesired performance of aerospace products and even infeasible designs. Robust design optimization (RDO) aims to obtain a solution with a desirable performance mean and is insensitive to uncertainty. However, it is computationally intensive during the RDO process, which is unaffordable for aerospace product design optimization. Multi-fidelity (MF) surrogate modeling is a widely used strategy to alleviate the heavy computational burden in RDO problems. The existing MF surrogate modeling-based RDO approaches, however, generate the samples in a one-shot way, which requires sufficient samples distributed in the entire design space. In this work, a probability of improvement-based adaptive sampling approach is proposed for multi-fidelity robust design optimization. The multi-level hierarchical Kriging (MHK) model is used for multi-fidelity modeling, which allows the proposed approach to deal with data of multiple fidelities. Both design variable uncertainty and interpolation uncertainty are considered within the approach. An extended PI (EPI) function is developed to simultaneously select the design location and fidelity level of the updated sample. To deal with RDO problems with constraints, the proposed EPI function is extended by combining with the probability of feasibility functions. The proposed approach is demonstrated using four numerical examples and an engineering example of robust design optimization for a micro-aerial vehicle fuselage.

Robust design optimization by spline dimensional decomposition

This article highlights new spline-empowered computational methods for solving robust design optimization (RDO) problems in complex mechanical systems. The methods are predicated on a spline dimensional decomposition (SDD) of a high-dimensional, discontinuous, or nonsmooth stochastic response for statistical moment analysis, a novel fusion of SDD 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 derived for the design sensitivities that are concurrently determined along with the moments. The methods depend on how the statistical moment and sensitivity analyses are engaged with an optimization algorithm, engendering direct and multi-point single-step design processes. Numerical results reveal that the proposed methods deliver accurate and computationally efficient optimal solutions to RDO problems, including an industrial-scale shape design of a robotic gripper jaw.

Design optimization-under-uncertainty of a forward swept wing unmanned aerial vehicle using SAMURAI

In this paper the design optimization-under-uncertainty of a forward swept wing (FSW) blended wing body (BWB) unmanned aerial vehicle (UAV) is examined. Conventional BWBs are often tailless, which leads to a backward swept wing to ensure longitudinal static stability. This in turn can induce flow separation at the tip, leading to a loss of lift, controllability and the appearance of a nose-up pitching moment. A possible solution to this problem is a conceptual redesign by introducing a forward swept wing, which is inherently free of tip-stall, but needs a careful design in order to be controllable. However, fixed wing UAVs are often produced by means of direct injection expanded foam moulding, which is characterized by not negligible production tolerances. This lead to a reliability-based robust design optimization problem, for which a novel framework is employed: SAMURAI. Firstly, the method accounts for computational cost by means of surrogate modelling, an analytical treatment of the problem and an asynchronous updating scheme that balances design space exploration and objective exploitation. Secondly, the method treats the problem as a multi-objective problem, which leads to a Pareto front of robust and reliable designs. The result is a novel series of UAV designs that are inherently free of tip stall, perform robustly and meet the stability requirements with the target reliability obtained with a computationally feasible budget.

Robust geometric parameter optimization of a crossed beveloid gear pair with approximate line contact

For a crossed beveloid gear pair with approximate line contact, the macroscopic geometric parameter is a dominant factor that affects meshing characteristics. However, geometric parameter design and reliability under misalignment are of great difficulty. To deal with these issues, an enhanced design method for a beveloid gear with approximate line contact is firstly proposed. Based on this, a robust optimization model of geometric parameters for crossed beveloid gear pairs with approximate line contact is developed. Three evaluation items are considered, namely the maximum contact pressure, contact trace offset distance and contact ratio. To reduce the calculation time, the influence coefficient method is applied to calculate the contact pressure. Considering the characteristics of installation deviation uncertainty and distribution, the mean maximum contact pressure, contact trace offset distance and ratio are considered as objective functions, formulated as a robust design optimization problem. Finally, robust optimization of a crossed beveloid gear pair based on a fast elitist nondominated sorting genetic algorithm (NSGA-II) is applied to solve the model to obtain the Pareto front of the macroscopic geometric parameters. A numerical example is provided to demonstrate the effectiveness of the proposed method.

ERGO: A New Robust Design Optimization Technique Combining Multi-Objective Bayesian Optimization With Analytical Uncertainty Quantification

Abstract In this work, robust design optimization (RDO) is treated, motivated by the increasing desire to account for variability in the design phase. The problem is formulated in a multi-objective setting with the objective of simultaneously minimizing the mean of the objective and its variance due to variability of design variables and/or parameters. This allows the designer to choose its robustness level without the need to repeat the optimization as typically encountered when formulated as a single objective. To account for the computational cost that is often encountered in RDO problems, the problem is fitted in a Bayesian optimization framework. The use of surrogate modeling techniques to efficiently solve problems under uncertainty has effectively found its way in the optimization community leading to surrogate-assisted optimization-under-uncertainty schemes. The Gaussian processes, the surrogates on which Bayesian optimization builds, are often considered cheap-to-sample black-boxes and are sampled to obtain the desired quantities of interest. However, since the analytical formulation of these surrogates is known, an analytical treatment of the problem is available. To obtain the quantities of interest without sampling an analytical uncertainty, propagation through the surrogate is presented. The multi-objective Bayesian optimization framework and the analytical uncertainty quantification are linked together through the formulation of the robust expected improvement, obtaining the novel efficient robust global optimization scheme. The method is tested on a series of test cases to examine its behavior for varying difficulties and validated on an aerodynamic test function which proves the effectiveness of the novel scheme.

Local Latin hypercube refinement for multi-objective design uncertainty optimization

Optimizing the reliability and the robustness of a design is important but often unaffordable due to high sample requirements. Surrogate models based on statistical and machine learning methods are used to increase the sample efficiency. However, for higher dimensional or multi-modal systems, surrogate models may also require a large amount of samples to achieve good results. We propose a sequential sampling strategy for the surrogate based solution of multi-objective reliability based robust design optimization problems. Proposed local Latin hypercube refinement (LoLHR) strategy is model-agnostic and can be combined with any surrogate model because there is no free lunch but possibly a budget one. The proposed method is compared to stationary sampling as well as other proposed strategies from the literature. Gaussian process and support vector regression are both used as surrogate models. Empirical evidence is presented, showing that LoLHR achieves on average better results compared to other surrogate based strategies on the tested examples.

Bayesian optimization for robust design of steel frames with joint and individual probabilistic constraints

This work proposes a Bayesian optimization (BO) method for solving multi-objective robust design optimization (RDO) problems of steel frames under aleatory uncertainty in external loads and material properties. Joint and individual probabilistic constrained RDO problems are formulated to consider two different ways the frame reaches its collapse state. Each problem involves three conflicting objective functions, namely, the total mass of the frame, the mean and variance of the maximum inter-story drift. Since the uncertain objective and probabilistic constraint functions of both problems are implicit within a finite element analysis program and the computation of the probabilistic constraints is an NP-hard problem, BO is used to guide the optimization process toward better solutions after it completes an iteration and offers a set of near Pareto-optimal solutions when it terminates. Specifically, Bayesian regression models called Gaussian processes (GPs) serve as surrogates for the structural responses. Two acquisition functions are then developed for the two RDO problems and a maximization problem of these functions is formulated as a mixed-integer nonlinear programming (MINLP) problem. A new random search coupled with simulated annealing is devised to solve the MINLP problem, thereby locating the most promising point in the input variable space at which the current solutions maximize their chance to be improved and the GP models are refined before the BO starts a new iteration. A test problem and two design examples show that exact or good Pareto-optimal solutions to the RDO problems can be found by the proposed method with 20 iterations.

Projection-based robust optimization with symbolic computation

Robust counterpart reformulation is a common technique used to deal with data uncertainty in robust optimization (RO) problems. The derivation of the robust counterpart formulation using the duality theory is nontrivial, especially for complex uncertainty sets. To reduce the dependence on robust counterparts, a novel method is proposed in this article for RO problems. Based on the feasible space projection, the proposed method can locate robust solutions without formulating the robust counterparts. RO model can be reformulated as a semi-algebraic system and a modified cylindrical algebraic decomposition method is applied to project the high-dimensional feasible space on the low-dimensional space of the objective function and uncertainty parameters. By solving the maximization problem and using the max-max decision criterion, the final robust solution can be selected. The case studies, involving problems of nonlinear programming (NLP) and robust design optimization problems, show that the proposed method can obtain the robust solution effectively.