Robust Design Optimization Approach Research Articles

ROBUST DESIGN OPTIMIZATION OF A COMPRESSOR ROTOR USING RECURSIVE COKRIGING BASED MULTI-FIDELITY UNCERTAINTY QUANTIFICATION AND MULTI-FIDELITY OPTIMIZATION

Abstract This work focuses on the application of multi-fidelity methods for the robust design optimization of engine components. The robust design optimization approach yields geometric designs with high efficiency and low sensitivity to uncertainties from manufacturing and wear. The uncertainty quantification techniques required to evaluate the robustness are computationally expensive, which limits their use in robust optimization. Multi-fidelity methods offer a promising solution to reduce the computational cost while maintaining the accuracy in both uncertainty quantification and optimization. A Kriging and a multi-fidelity recursive Cokriging framework are developed and applied to a test function. Furthermore, a multi-fidelity super efficient global optimization algorithm is developed. The optimizer is surrogate based and handles constraints. The developed methods are then applied to a compressor test case of a high pressure compressor blade row with 9 uncertainty and 24 design parameters of the geometry. The 2.5 % quantile of the stage efficiency is used as a robustness measure and is optimized. Design bounds and performance constraints are applied. Various uncertainty quantification techniques are analyzed. A multi-fidelity uncertainty quantification approach is developed that combines simplified coarse-grid low-fidelity with high-fidelity results to reduce the computational cost while maintaining a high accuracy. Uncertainty quantification techniques of three fidelity levels are developed and used for the multi-fidelity approach in the design space. The robust design optimization is performed and the optimal designs obtained from the multi-fidelity approach show superior performance compared to existing robust design optima in the literature.

A Heuristic Robust Design Optimization Approach for Batch Production Permanent Magnet Actuator Consider Various Uncertainties and Life-Cycle Quality Loss

A Heuristic Robust Design Optimization Approach for Batch Production Permanent Magnet Actuator Consider Various Uncertainties and Life-Cycle Quality Loss

Fabrication uncertainty guided design optimization of a photonic crystal cavity by using Gaussian processes

We present a fabrication uncertainty aware and robust design optimization approach that can be used to obtain robust design estimates for nonlinear, nonconvex, and expensive model functions. It is founded on Gaussian processes and a Monte Carlo sampling procedure, and assumes knowledge about the uncertainties associated with a manufacturing process. The approach itself is iterative. First, a large parameter domain is sampled in a coarse fashion. This coarse sampling is used primarily to determine smaller candidate regions to investigate in a second, more refined sampling pass. This finer step is used to obtain an estimate of the expected performance of the found design parameter under the assumed manufacturing uncertainties. We apply the presented approach to the robust optimization of the Purcell enhancement of a photonic crystal nanobeam cavity. We obtain a predicted robust Purcell enhancement of F¯P≈3.6. For comparison we also perform an optimization without robustness. We find that an unrobust optimum of FP≈256.5 dwindles to only F¯P≈0.7 when fabrication uncertainties are taken into account. We thus demonstrate that the presented approach is able to find designs of significantly higher performance than those obtained with conventional optimization.

Surrogate-Based Calculation Method for Robust Design Optimization Considering the Fatigue Probability for Variable Service Loads of eBike Drive Units

This paper proposes a robust design-optimization approach for eBike drive units that incorporates the highly variable driver-dependent load collectives and system conditions into a fatigue calculation. In an initial step, the relevant influences and loads on the investigated system are examined and reviewed in relation to the current normative requirements. From a methodical viewpoint, this paper presents a surrogate-based simulation-based approach to assess reliability across the entire geometry according to a probabilistic fatigue calculation. The probabilistic evaluation considers the several measured load collectives of different drivers and driving scenarios to enable a robust and type-oriented bike design. In addition to methods of fatigue calculation, this approach also includes common methods of order reduction and reliability-based design optimization. To avoid additional uncertainties in the calculation, this approach considers a complex critical-plane-based multiaxial-fatigue calculation to correctly evaluate the multiaxial and non-proportional stress state across the whole geometry. A data-based surrogate model that supports the fatigue calculation by predicting the load across the given uncertainties is the key to the efficient assessment of the service life of the eBike. Lastly, the identified uncertainties in the design of eBike drive units are investigated and evaluated by this method.

Development of a deviation package method for low-cost robust optimization in compressor blade design

Manufacture variations can greatly increase the performance variability of compressor blades. Current robust design optimization methods have a critical role in reducing the adverse impact of the variations, but can be affected by errors if the assumptions of the deviation models and distribution parameters are inaccurate. A new approach for robust design optimization without the employment of the deviation models is proposed. The deviation package method and the interval estimation method are exploited in this new approach. Simultaneously, a stratified strategy is used to reduce the computational cost and assure the optimization accuracy. The test case employed for this study is a typical transonic compressor blade profile, which resembles most of the manufacture features of modern compressor blades. A set of 96 newly manufactured blades was measured using a coordinate measurement machine to obtain the manufacture variations and produce a deviation package. The optimization results show that the scatter of the aerodynamic performance for the optimal robust design is 20% less than the baseline value. By comparing the optimization results obtained from the deviation package method with those obtained from widely-used methods employing the deviation model, the efficiency and accuracy of the deviation package method are demonstrated. Finally, the physical mechanisms that control the robustness of different designs were further investigated, and some statistical laws of robust design were extracted.

Memeplex-based memetic algorithm for the multi-objective optimal design of composite structures

Hybrid construction is proposed to decrease costs in lightweight structures using fiber-reinforced plastics (FRP) composite materials. Minimum weight, minimum strain energy (stiffness), and minimum energy variability of the structural system response are the objectives of the robust design optimization approach applied to composite shell structures with stiffeners. The trade-off depends on given stress, displacement, and buckling constraints imposed on composite structures considering non-linear geometric behavior. The design variables are ply angles and ply thicknesses of shell laminates, the cross-section dimensions of stiffeners, and the variables related to the selection of materials and their distribution at the laminate level and subsequently of the laminates along the structure. A Multi-Objective Memetic Algorithm (MOMA) applies multiple learning procedures exploring the synergy of different cultural transmission rules. An enlarged virtual population with a dual age-dominance nature captures and updates the Pareto curve. The concept of memeplex controls the meme selection and their propagation along the hybrid genetic and cultural evolution path. The success of memetic learning procedures is analyzed by measuring each meme's relative and absolute success events according to different approaches, depending on the reference time used in evolutionary history. Results show that MOMA is promising in multi-objective optimization of composite hybrid structures.

An efficient and non-intrusive approach for robust design optimization with the first-order second-moment method

A modified robust design optimization approach is presented, which uses the first-order second-moment method to compute the mean value and the standard deviation for arbitrary objective functions. Existing approaches compute the gradient of the variance using the adjoint method, direct differentiation or finite differences, respectively. These approaches either access to the FE-code and/or have high computational cost. In this paper, a new approach for the computation of the gradient of the variance is provided. It can be easily implemented as a non-intrusive method, which behaves similar to finite differences with the cost of only one additional objective evaluation, independent of the number of variables. Here, a step-size has to be chosen carefully and therefore, a procedure to determine a problem-independent step-size is provided. As an alternative, the approach can be implemented as an analytic method with the same cost like the adjoint method, but providing wider applicability (e.g. eigenvalue problems). The provided approach is derived, analyzed and applied to several benchmark examples.

Semi-intrusive approach for stiffness and strength topology optimization under uncertainty

A semi-intrusive approach for robust design optimization is presented. The stochastic moments of the objective function and constraints are estimated using a Taylor series-based approach, which requires derivatives with respect to design variables, random variables as well as mixed derivatives. The required derivatives with respect to design variables are determined using the intrusive adjoint method available in commercial software. The partial derivatives with respect to random parameters as well as the mixed second derivatives are approximated non-intrusively using finite differences. The presented approach provides a semi-intrusive procedure for robust design optimization at reasonable computational cost while allowing an arbitrary choice of random parameters. The approach is implemented as an add-on for commercial software. The method and its limitations are demonstrated by academic test cases and industrial applications.

Robust Design Optimization of Car-Door Structures with Spatially Varied Material Uncertainties

This paper presents an effective approach for robust design optimization of car-door structures with spatially varied material properties. This spatially varied material property causes structural response quantities; for example, the natural frequency and the lateral stiffness coefficient become random variables. In this regard, the Karhunen-Loève expansion is first used to represent the elastic modulus and the mass density random fields as a series of random variables. Then, a stochastic finite-element model is formulated for uncertainty quantification of the car-door structure. Combined with a polynomial-based response surface model to mimic the true performance indicator, this allows one to efficiently evaluate probability constraints for the robust design optimization of the uncertain car-door structure. In numerical simulations, design variables of the uncertain car-door structure are defined as thickness values of the tailor rolled blank structure at various regions, whereas multiple design objectives are formulated via the structural weight, the first-order natural frequency, and the lateral stiffness coefficient. Results have shown that the mean value of performance indicators can be generally improved, whereas the response variance is further minimized to archive the robust design objective. The probability-based constraint is significant to relate the Pareto optimum set to the targeted structural safety level. The proposed approach is simple, suggesting an attractive tool for the robust design optimization of car-door structures with spatially varied material uncertainties.

Robust optimization of an organic Rankine cycle for geothermal application

A robust design optimization (RDO) methodology for Organic Rankine Cycles (ORC) is presented, allowing to ensure an improved, stable performance over a large range of operating conditions. In contrast with classical ORC design methods, whereby all modeling hypotheses and operating conditions are considered as perfectly known, i.e. deterministic, the RDO approach allows to account for the manifold sources of uncertainty affecting the system. For geothermal ORC, the latter are related on one hand with the ill-known properties of the geothermal source and, on the other, with intrinsically random parameters, such as the condensation temperature. The proposed RDO approach selects values of the design parameters that maximize the expected (average) performance while minimizing its variance under uncertain nominal operating conditions. The optimal design delivered by the proposed strategy outperforms the one derived from the standard deterministic approach: specifically, the expected power output is increased by 1.5%, while its standard deviation is reduced by 8.5% and the surface of the heat exchangers by 34%. • Geothermal Organic Rankine cycle performance is subject to many uncertainties. • A robust optimization strategy is devised for cycle design under uncertainty. • The design maximizes average performance while minimizing the variance. • Nested Kriging surrogates are used to efficiently speed-up the design loop. • Robust optimum has +1.5% power output and smaller variance wrt standard design.

Robust Design Optimization of Permanent Magnet Linear Synchronous Motor Based on Quantified Constraint Satisfaction Problem

This study focuses on the robust design optimization of permanent magnet linear synchronous motors (PMLSM) considering the effect of uncertainties (noise factors) such as manufacturing errors, installation errors and working wears. A robust design optimization approach based on quantified constraint satisfaction problem (QCSP) is proposed. It establishes a robust PMLSM thrust ripple model considering noise factors by QCSP, and obtains the optimal design parameters that have lower thrust ripple and smaller thrust ripple fluctuation within their error range by branch-and-prune algorithm. First, an analytical model of thrust ripple for PMLSM is established. Second, the robust QCSP model of thrust ripple is established by quantifier concept based on analytical model. It defines the design parameters and noise variables as existential and universal quantifiers, respectively, defines the variables of QCSP model as quantifier parameters that are composed of existential and universal quantifiers, then QCSP model are established by quantifier parameters. Third, the algorithm of branch-and-prune based on interval arithmetic is proposed to optimize the QCSP model. This approach is proven to be robust and superior compared with traditional deterministic approach and another robust approach. Finally, Finite element analysis and motor prototype experiments confirm the feasibility and validity of the proposed method.

An effective approach for robust design optimization of wind turbine airfoils with random aerodynamic variables

The robust design optimization of an airfoil needs to continuously realize the probability-based aerodynamic simulation for various combinations of geometry and wind climate parameters. The simulation time is lengthy when a full aerodynamic model is embedded for the numerical iteration. To this end, a second-order polynomial-based response surface model is first presented to relate the airfoil performance indicator with geometry and random aerodynamic variables. This allows to quickly evaluate the response moments and optimization constraints. Then, the robust design optimization is formulated to simultaneously maximize the mean aerodynamic performance and minimize the variance of design results due to the variation of geometry and aerodynamic parameters. The robust design optimization based on the NACA63418 and the DU93-W-210 airfoils with random Mach and Reynolds numbers is presented to demonstrate potential applications of this proposed model. Results have shown that the mean-value aerodynamic indicator is generally improved, whereas the variance is minimized to archive the robust design objective. The proposed approach is simple and accurate, suggesting an attractive tool for robust design optimization of airfoils with random aerodynamic variables.

Efficient Robust Design Optimization of a Stacker–Reclaimer Structure Under Uncertainty

A stacker–reclaimer structure (SRS) is a massive structure used for bulk material exploration. Performance of SRS is sensitive to the effect of uncertainty which may lead to catastrophic failure consequences. Thus, in this paper a comparatively new robust design optimization (RDO) approach for design of SRS is explored. The involved parameter of SRS e.g., material loading, incrustation, normal digging, etc., may not have well-defined probability density functions and can be expressed as uncertain but bounded (UBB) type parameters. Hence, RDO is explored for probabilistic as well as UBB cases. Solution of such RDO problem in full simulation approach would require extensive computational time. Hence, response surface method (RSM) based metamodeling approach has been adopted here to alleviate computational burden. Also, as conventional least squares method (LSM) based RSM may be a source of error, this study adopts a comparatively new moving LSM (MLSM) based adaptive RSM in RDO. The RDO results depict that UBB type uncertainty is more critical than the probabilistic case. The proposed MLSM based RDO approach yields reasonably accurate design solutions in a computationally efficient way. The proposed MLSM based RDO approach yields design solutions which ensures safe performance of SRS even in the presence of uncertainty.

Efficient Robust Design Optimization of Optical Systems

To accelerate the virtual product development of using optical simulation software, the Robust Design Optimization approach is very promising. Optical designs can be explored thoroughly by means of sensitivity analysis. This includes the identification of relevant input parameters and the modelling of inputs vs. outputs to understand their dependencies and interactions. Furthermore, the intelligent definition of objective functions for an efficient subsequent optimization is of high importance for multi-objective optimization tasks. To find the best trade-off between two or more merit functions, a Pareto optimization is the best choice. As a result, not only one design, but a front of best designs is obtained and the most appropriate design can be selected by the decision maker. Additionally, the best trade-off between output variation of the robustness (tolerance) and optimization targets can be found to secure the manufacturability of the optical design by several advanced approaches. The benefit of this Robust Design Optimization approach will be demonstrated.

A New Approach for Robust Design Optimization Based on the Concepts of Fuzzy Logic and Preference Function

In this study, an efficient methodology is proposed for robust design optimization by using preference function and fuzzy logic concepts. In this method, the experience of experts is used as an important source of information during the design optimization process. The case study in this research is wing design optimization of Boeing 747. Optimization problem has two objective functions (wing weight and wing drag) so that they are transformed into new forms of objective functions based on fuzzy preference functions. Design constraints include transformation of fuel tank volume and lift coefficient into new constraints based on fuzzy preference function. The considered uncertainties are cruise velocity and altitude, which Monte Carlo simulation method is used for modeling them. The non-dominated sorting genetic algorithm is used as the optimization algorithm that can generate set of solutions as Pareto frontier. Ultimate distance concept is used for selecting the best solution among Pareto frontier. The results of the probabilistic analysis show that the obtained configuration is less sensitive to uncertainties.

Response surface approach to robust design of assembly cells through simulation

PurposeThis paper aims to present a multi-response robust design (RD) optimization approach for U-shaped assembly cells (ACs) with multi-functional walking-workers by using operational design (OD) factors in a simulation setting. The proposed methodology incorporated the design factors related to the operation of ACs into an RD framework. Utilization of OD factors provided a practical design approach for ACs addressing system robustness without modifying the cell structure.Design/methodology/approachTaguchi’s design philosophy and response surface meta-models have been combined for robust simulation optimization (SO). Multiple performance measures have been considered for the study and concurrently optimized by using a multi-response optimization (MRO) approach. Simulation setting provided flexibility in experimental design selection and facilitated experiments by avoiding cost and time constraints in real-world experiments.FindingsThe present approach is illustrated through RD of an AC for performance measures: average throughput time, average WIP inventory and cycle time. Findings are in line with expectations that a significant reduction in performance variability is attainable by trading-off optimality for robustness. Reductions in expected performance (optimality) values are negligible in comparison to reductions in performance variability (robustness).Practical implicationsACs designed for robustness are more likely to meet design objectives once they are implemented, preventing changes or roll-backs. Successful implementations serve as examples to shop-floor personnel alleviating issues such as operator/supervisor resistance and scepticism, encouraging participation and facilitating teamwork.Originality/valueACs include many activities related to cell operation which can be used for performance optimization. The proposed framework is a realistic design approach using OD factors and considering system stochasticity in terms of noise factors for RD optimization through simulation. To the best of the authors’ knowledge, it is the first time a multi-response RD optimization approach for U-shaped manual ACs with multi-functional walking-workers using factors related to AC operation is proposed.

A surrogate based multi-fidelity approach for robust design optimization

• A surrogate based multi-fidelity framework for RDO is proposed. • The first approach is highly efficient and requires very few actual simulations. • Second approach yields highly accurate result from slightly increased simulation. Robust design optimization (RDO) is a field of optimization in which certain measure of robustness is sought against uncertainty. Unlike conventional optimization, the number of function evaluations in RDO is significantly more which often renders it time consuming and computationally cumbersome. This paper presents two new methods for solving the RDO problems. The proposed methods couple differential evolution algorithm (DEA) with polynomial correlated function expansion (PCFE). While DEA is utilized for solving the optimization problem, PCFE is utilized for calculating the statistical moments. Three examples have been presented to illustrate the performance of the proposed approaches. Results obtained indicate that the proposed approaches provide accurate and computationally efficient estimates of the RDO problems. Moreover, the proposed approaches outperforms popular RDO techniques such as tensor product quadrature, Taylor’s series and Kriging. Finally, the proposed approaches have been utilized for robust hydroelectric flow optimization, demonstrating its capability in solving large scale problems.

Cost efficient robust global supply chain system design under uncertainty

Brutal competitions in today’s global supply chain have forced business enterprises to focus attention on their supply chains. These issues have guided this research to introduce a supply chain service level measure and develop a novel robust design optimization (RDO) approach to assure service level rate requirements and minimize total service level costs. A supply chain system design must maintain its efficiency over time while coping with the uncertainty in production and transportation. Unexpected delay in the product flow can render the supply chain system inefficient with respect to service level costs. Analytical models for the determination of the risk involved in a supply chain system design are difficult to derive when uncertainty is involved in the supply chain. In this paper, the service level rate measure approach is detailed first. Then, the methodology for the robust design optimization-based approach is provided. Two case studies are performed to demonstrate the effectiveness of the proposed service level measure and developed robust design optimization model. Results from the two case studies have shown that by implementing the developed robust design optimization approach, a system is able to achieve a 90 % service level rate while minimizing performance cost and uncertainty impact.

Reliability based multi-objective robust design optimization of steel moment resisting frame considering spatial variability of connection parameters

The focus of this paper is on achieving robustness in the seismic design against uncertainties in the design process. In this regard, we present a reliability based robust design optimization approach to achieve designs that are not only safe and cost efficient, but also robust against uncertainties. The aim here is to adjust easy-to-control design parameters to reduce the variability of the seismic response to hard-to-control noise parameters while also considering design safety and budget. The proposed approach is demonstrated with a steel moment resisting frame design, considering uncertainties and spatial variability in connection parameters. Here, the sizes of the steel sections constitute the design parameters, while the uncertain and spatially variable parameters of the Ibarra–Krawinkler connection model are treated as noise parameters. The design objectives include the collapse prevention reliability conditional on a maximum earthquake intensity level, as well as the initial cost of the structure. To reduce computation demands in calculating the seismic response variation, design parameters that have a negligible effect on seismic response are first eliminated through sensitivity analysis. Furthermore, the seismic response variation is evaluated parametrically to investigate the effect of connection parameter spatial variability. Finally, the authors demonstrate the use of proposed robust design optimization approach to obtain a Pareto Front, a collection of optimal designs that are optimized for both reliability and cost.

Robust Aerodynamic Design Optimization via Variable Fidelity Surrogate Model

In this paper, a robust design optimization framework is proposed with a variable fidelity Kriging model. By the use of the variable fidelity Kriging model approach, an accurate surrogate model can be constructed efficiently by the absolute values of a high-fidelity function as well as the trends obtained by low-fidelity function values. The high- and low-fidelity levels can be defined by utilizing different physical models, computational meshes and so on. The robustness of a candidate design is efficiently evaluated by a Monte Carlo simulation which is executed on the variable fidelity Kriging model. The efficiencies of robust design optimization approaches are investigated in a 2D airfoil drag minimization problem. In this problem, free-stream Mach number as well as target lift coefficient are supposed as uncertain parameters. The mean and standard deviation of drag coefficient are simultaneously minimized to obtain non-dominated robust optimal designs. The developed robust design optimization approach via the variable fidelity Kriging model is shown to be useful for efficient search of robust airfoil designs.