Robust Design Optimization Research Articles

Multi-objective reliability-based robust design optimization of uncertain PMS involving parametric uncertainty and correlation

Multi-objective reliability-based robust design optimization of uncertain PMS involving parametric uncertainty and correlation

Improving the potential of fifth-generation district heating and cooling networks through robust design and operational optimization under future energy market and demand uncertainties

Network investments and projected energy prices greatly impact the financial viability and environmental impact of fifth-generation district heating and cooling (5GDHC) networks. Compared to the previous generation, the upfront investment costs in 5GDHC networks are relatively high due to the decentralized energy demand and supply of the participating prosumers. The transition to 5GDHC is also hindered by uncertain energy markets and the fluctuating energy demands of the prosumers. It makes investors hesitant to commit, even for promising business cases. In this work, a framework is presented to assess the economic and environmental performance of a 5GDHC business case, based on a multi-objective robust optimization algorithm (MO-RDO), to identify the optimal trade-offs between environmental and economic designs. The proposed methodology consists of four steps. In the first two steps, a specific business case is modeled, and techno-economic and environmental performances are analyzed against uncertain energy markets. In the third step, the techno-economic and environmental indicator responses are used to develop a data-driven machine-learning model. This model offers better computational efficiency than a high-fidelity nonlinear model and assesses the most influential parameters driving economic and environmental performance via a global sensitivity analysis. This aids in refining the multi-objective robust design optimization (MO-RDO) framework by pinpointing key design variables and objectives amid data uncertainties. Additionally, the formulated MO-RDO provides a mix of environmentally and economically optimal network designs and operational parameters and highlights the trade-offs between them. The designs and trade-offs are evaluated using financial metrics like net present value (NPV), return on investment, and discounted payback period. These metrics are calculated over the project life, considering the initial investment and net savings. For the considered case, the environmental design has a 10% higher investment cost compared to the economically optimal solution but reduces total emissions (TE) by 13% and improves operational cost savings, which is crucial for better financial indicators. The proposed framework aligns with the EU transition plan to incorporate waste energy sources and decarbonization paths, estimating only a 3% increase in NPV as a risk for the environmentally optimal solution on a financial scale.

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.

Design optimization of continuous fiber composites with thermo-mechanical coupling and load uncertainties

This paper introduces a theoretical framework for the design optimization of continuous fiber composites reinforced with continuous fiber trajectories subject to thermo-mechanical coupling and load uncertainties. Different uniform temperature variations are applied in the structure to investigate the influence of ambient temperature change on the structural performance. To consider the external load uncertainties, a robust design optimization model is proposed where the loads are modeled as hybrid variables, namely magnitudes as random variables and directions as interval variables, with the robust objective determined through a hybrid orthogonal polynomial expansion method. Furthermore, we use a level-set function to represent the structural boundary, with its evolution driven by shape derivatives calculated based on uncertainty analysis. The continuous fiber paths are subsequently determined by the level-set isoline extracted from the structural boundary, which in turn influences the structural mechanical performance due to the material anisotropy of composites. The continuity of continuous fiber and the equal space between adjacent trajectories largely ensure the additive manufacturability of the composites. Three numerical examples are presented to demonstrate the effectiveness of the developed framework. The results show that the ambient temperature variations and load uncertainties largely impact the optimized topology and fiber infill patterns of composites, thus are important to be considered in the design stage. Moreover, the optimized structure can have a 5-fold stiffness per unit mass compared with the initial design thus largely increasing the material efficiency in carrying external uncertain loads.

Robust global optimization of plasmonic filter based on Q factor analysis using graded index

Abstract To reduce the impact of uncertainties in various manufacturing processes while maintaining satisfactory performance level, we propose a robust global optimization approach for plasmonic filters based on extraordinary optical transmission by use of Q factor analysis with reasonable computational cost. The gradient index is employed as a metric for robustness, and the multiobjective genetic algorithm is selected as a global optimization tool. The figures of merit and gradient index are chosen as two objective functions with the design variables being the slit width, slit height, and period of the slit array, respectively. The finite element method is employed to investigate a variety of optical properties rigorously. The numerical optimization results demonstrate that the proposed approach provides improved effectiveness and efficiency in designing plasmonic filters under fabrication uncertainties by using Q factor analysis. The method proposed in this study enables systematic robust optimization while accounting for manufacturing uncertainties, which would be infeasible under typical time constraints. Specifically, within the proposed optimization framework, Q factor analysis can be employed due to a 20.60-times reduction in computational time. To the best of our knowledge, this represents the first robust optimization design for enhanced filter performance based on Q factor analysis in EOT, necessitating comprehensive spectrum calculations alongside fabrication tolerance considerations.

A comprehensive validation of GRA with ant colony optimization for robust optimal design: A case study on the forming process

Sheet metal forming design involves numerous factors and controlling all these parameters to prevent cracks is a major challenge to maintain the quality. Therefore, the careful selection of process parameters is critical for ensuring product longevity. Several researchers employed different techniques for optimal selection of process parameters but consistency among the techniques and validation of results in real time scenarios will play a pivotal role in determining the suitable parameters. Therefore, the current work presents a validation method of Grey Relational Analysis (GRA) results with Ant Colony Optimization (ACO) for the robust optimum design with an illustrative example of forming process. Al 6061 – T6 alloy was chosen as a material for deep drawing process and L27 Orthogonal Array was planned to conduct experiments with four input parameters such as blank thickness, Die and Blank Temperature, Die Speed, Lubrication at three levels each. Punch force and Thickness variation were considered as the output parameters. GRA Technique was implemented to find the best optimal combination and regression equation was developed to predict the nature of objective function. These results were given as an input to ACO, a nature inspired algorithm to search for the optimal solution with in the design space. Sensitivity analysis has also been performed to highlight the effectiveness of the proposed solution in terms of solution quality and computational efficiency. The ACO method, GRA technique and the experimental results are almost similar with a margin of acceptable deviation. The results indicated the robustness of the projected approach in achieving the quality of solution.

Robust design optimization of Critical Quality Indicators (CQIs) of medical-graded polycaprolactone (PCL) in bioplotting

Polycaprolactone (PCL), either in its pure grade or as a polymeric matrix for bio-composites, plays a key role in the biomedical and bioengineering industries. It is also considered a multifunctional and versatile polymer for bioprinting and bioplotting purposes, especially in tissue engineering. Herein, an undiscovered yet valuable aspect of PCL extrusion-based bioprinting, such as the predictability of Critical Quality Indicators (CQIs), is investigated in depth. With the aid of the robust L25 orthogonal matrix design, the six most generic and device-independent control factors proved their impact on quality metrics such as global porosity, dimensional conformity, and surface roughness, determined with the aid of highly evolved Nondestructive Testing (NDT) and algorithms. To this end, 25 experimental runs were set, and 125 specimens were fabricated using an industrial-scale bio-plotter and medical-graded polycaprolactone. Various infill densities (ID), layer thicknesses (LT), raster deposition angles (RDA), printing speeds (PS), nozzle temperatures (NT), and bed temperatures (BT) were applied. CQIs were determined using optical profilometry and microscopy, and micro-computed tomography. Quadratic predictive equations were compiled and verified using two additional, well-chosen experimental runs. These generally applicable predictive models carry a massive amount of research and industrial merit, as they ensure visibility in bioprinting with PCL.

Design, analysis, and optimization of PMSM for ultra-deep water pile hammer application

With the development of global deepwater oil and gas and offshore wind power resources, there is an increasing demand for ultra-deep water pile hammers (UDPH). In this paper, a permanent magnet synchronous motor (PMSM) with oil fill is proposed for UDPH on the basis of underwater 2500 m depth and 450 kJ impact energy requirements. First, the rated parameters of the PMSM are calculated based on the power required by hydraulic pumps, and then, the method of combining empirical formulas and path algorithms is used to complete the electromagnetic scheme design. Second, finite element analysis of the electromagnetic field is carried out by Maxwell under both no-load and load conditions to verify the rationality of the PMSM. Furthermore, aiming at improving efficiency and reducing cogging, parameter optimization is carried out by a robust parameter optimization design method, and the optimal parameter scheme of the PMSM is obtained. In addition, loss analysis and temperature field coupling calculations of the optimized PMSM are conducted to verify that the temperature of this motor can meet the requirements under different working conditions. Finally, a comparative analysis is conducted between the proposed PMSM and the three-phase asynchronous motor used in the underwater 2,500 m-450 kJ UDPH. The results indicate that the new PMSM introduced in this paper demonstrates significant advantages in terms of weight, volume, and efficiency; moreover, the new PMSM is reasonable and more suitable for UDPH. Therefore, the PMSM proposed in this paper is reasonable and can provide a theoretical basis for improving the traditional UDPH.

A lever-enhanced tuned inerter damper for controlling vibrations due to rotary unbalance and bounded parametric uncertainty: Classic and robust equal-peak optimization

This study proposes a lever-enhanced tuned inerter damper (LTID) for controlling the excessive vibrations induced by rotary unbalance within the primary system, whose stiffness is further considered a bounded uncertain parameter. Classic and robust equal-peak optimal design of the proposed LTID are carried out according to the methodologies stemming from the fixed-points phenomenon. Closed-form solutions to the H∞ optimization problem are analytically derived, whose accuracy is validated by comparing with either the long-established results in the literature or exact solutions numerically obtained in this study. Numerical results clearly suggest that in contrast to the classic one, robust equal-peak optimization can always produce the same and minimized vibration amplitude at the leftmost and rightmost peaks, regardless of the stiffness uncertainty magnitude. Compared to the classic LTID, the improvement made by the robust LTID in terms of minimizing and equalizing worst-case vibration amplitudes becomes more evident as the stiffness uncertainty magnitude increases. Meanwhile, the effectiveness of LTID is controlled by the amount of inertance and the lever amplification ratio simultaneously. Finally, the proposed LTID is exempt from the need for a large oscillating mass and is appropriate for lightweight applications of vibration control, as also inferred from the fact that the inerter device could generate an inertance far greater than its physical mass and the amplification ratio of a lever could be easily adjusted by changing the location of its fulcrum.

A novel bi-level decision-making approach based on game theory for mechanism design

This study addresses the bi-level multi-objective optimization problems raised in reliability-based robust design optimization (RBRDO) of engineering applications through establishing a state-of-the-art game theoretic scenario. A novel bi-level decentralized decision-making approach is proposed using the synergy of RBRDO, game theory, Monte Carlo simulation (MCS), and genetic programming (GP). The application of the proposed approach is elaborated in a case study of robust synthesis of high-speed path-generating four-bar mechanisms. The four performance criteria, namely, accuracy ( TE), robustness ([Formula: see text] TE and [Formula: see text]2 TE), reliability ( f G) at the upper level, and quality of motion ( TA) at the lower level are assigned to four players so that each of whom is in charge of one objective criterion. The peak input driving torque ( T S) is associated with the upper-level problem. The GP meta-model is used to capture the Stackelberg protocol that is, constructing the follower’s rational reaction set (RRS) and the Nash bargaining function is hired to model the cooperative behavior. The obtained results show a considerable enhancement in reliability and robust behavior of mechanism, while the deterministic criteria of accuracy and quality of motion are preserved.

Robust retrofits for rural house envelope considering construction quality and occupant behavior uncertainties: A MOO-integrated Taguchi method

Due to the underdeveloped socio-economic environment, the uncertainties in construction quality and occupant behavior are more prominent for rural house than urban residential buildings. If these uncertainties are not considered, there are higher risks of inaccurate performance prediction, unrealistic retrofit decisions and causing serious performance gap. To address this challenge, this paper proposes a robust envelope retrofitting workflow that combine deterministic multi objective optimization (MOO) and Taguchi method. This workflow stands out for its utilizing mature deterministic MOO to fast approximate Alternative schemes, and utilizing Taguchi method to handle uncertainties, reducing some key disadvantages of traditional robust design optimization (RDO), such as heavy calculating burden, overly conservative solution and difficulty to predefine proper robust objectives or performance target value. The proposed workflow consists of four main steps, formulating Alternative schemes by deterministic MOO, screening highly sensitive uncertain parameters by uncertainty and global sensitivity analysis, formulating Typical uncertain scenarios by orthogonal experiments (OE) and calculating the comprehensive robustness indicators. Results show that by employing the proposed workflow, the selected robust optimal solution not only proves to be robust but also leads to preferable performance enhancements. Compared with the reference mode, the energy efficiency has been improved by 33.2% to 49.4%, the discomfort hours have been reduced by 0.4% to 26.5%, and the robustness has been enhanced by 36.1%.

Robust Design Optimization for a Hairpin Winding Permanent Magnet Motor Considering Permanent Magnet Inconsistency and Manufacturing Uncertainty

Robust Design Optimization for a Hairpin Winding Permanent Magnet Motor Considering Permanent Magnet Inconsistency and Manufacturing Uncertainty

Multi-Objective Robust Design Optimization for Crashworthiness Enhancement of Hybrid 2D Triaxially Braided Composite Tube Using Evolutionary Algorithms.

An innovative optimal design framework is developed aiming at enhancing the crashworthiness while ensuring the lightweight design of a hybrid two-dimensional triaxial braided composite (2DTBC) tube, drawing insights from the mesostructure of the composite material. To achieve these goals, we first compile the essential mechanical properties of the 2DTBC using a concentric cylinder model (CCM) and an analytical laminate model. Subsequently, a kriging surrogate model to elucidate the intricate relationship between design variables and macroscopic crashworthiness is developed and validated. Finally, employing multi-objective evolutionary optimization, we identify Pareto optimal solutions, highlighting that reducing the total fiber volume and increasing the glass fiber content in the total fiber volume are crucial for optimal crashworthiness and the lightweight design of the hybrid 2DTBC tube. By integrating advanced predictive modeling techniques with multi-objective evolutionary optimization, the proposed approach not only sheds light on the fundamental principles governing the crashworthiness of hybrid 2DTBC but also provides valuable insights for the design of robust and lightweight composite structures.

Multi-objective robust optimization design framework for low-pollution emission burners

The optimal design of low-pollution emission burners plays an important role in controlling pollutant emissions of industrial equipment, and is crucial for the sustainable development of the national economy and environmental protection. However, many uncertain factors challenge the optimal design of low-pollution emission burners. The Latin hypercube sampling (LHS) method was used to obtain sampling data representing the distribution of the uncertain variable. The training dataset was obtained using the turbulent combustion coupling model. A high-precision sparse polynomial chaos expansion (PCE) model was constructed by the degree-adaptive scheme and least angle regression (LAR) algorithm. Furthermore, the Legendre polynomial is introduced to establish a continuous robust optimization model. The model is carried out by the non-dominated sorting genetic algorithm II (NSGA-II). The results show that the excess air coefficient of 1.227 is optimal. Compared with the excess air coefficient of 1.20 under the discrete robust optimization, the optimal coefficient can further reduce pollutant emissions and bring strong robustness to the ethylene cracking furnace. It has also been proven that the continuous robust optimization scheme improves the optimization granularity. Compared with discrete robust optimization, this method reduces the number of samples by 66.7 %.

Reliability-Based Robust Design Optimization with Fourth-Moment Method for Ball Bearing Wear

Ball bearings operating at low speeds and under heavy loads are susceptible to wear failure, leading to significant economic losses. The existing reliability-based robust design optimization method of the fourth-moment method has high accuracy and does not need to determine the random distribution of the input variables, but it is not possible to apply it to ball bearing wear due to the complexity of the bearing wear state function that cannot be characterized as an explicit form. To address this issue, this paper proposes a novel design method for ball bearing wear. Firstly, a surrogate model is constructed using the Kriging model method to establish a relationship between the bearing design parameters and the mechanical response. Subsequently, a wear reliability model is developed on the basis of the fourth-moment method, and reliability sensitivity analysis is conducted. Finally, the ball bearing wear reliability-based robust design optimization is accomplished through the use of a genetic algorithm. The results of the case calculations demonstrate that the proposed method effectively calculates the ball bearing wear reliability and analyzes the impact of design parameter randomness on reliability. Furthermore, optimizing the design parameters reduces the sensitivity of wear reliability to parameter randomness.

Sensitivity analysis and robust optimization to high-dimensional uncertainties of compressors with active subspace method

The performance of compressors often deviates significantly from the design value due to the substantial geometric deviations arising from manufacturing errors and in-service degradation. Robust design optimization (RDO) plays a crucial role in ensuring the high performance and reliability of compressors. The study proposes an active subspace-based dimensionality-reduction RDO method for high-dimensional uncertainties of compressors. By employing the active subspace (AS) method, a dimensionality-reduction artificial neural network (ANN) is raised to solve the "dimension disaster" problem for high-dimensional uncertainty quantification (UQ) analysis. Additionally, to decrease the computation cost of the original AS method, a data-driven method is utilized to calculate the gradient of the quantity of interest (QoI) within the AS dimensionality-reduction framework. Furthermore, Shapley method is applied to study the sensitivity of geometric variables and their variation direction on the mean and variance of the performance, and the robustness improvement method by only modifying key geometric variables based on Shapley results is discussed. The RDO of high dimensional geometrical uncertainties of a transonic compressor stage is studied by using this method. The results demonstrate a reduction from the original 24-dimensional uncertainties to a 4-dimensional space, achieved through this method. Consequently, the required sample size is reduced by 75 % while maintaining nearly unchanged model accuracy. Furthermore, the findings reveal that after RDO, the mean efficiency increases by 1.04 %, with a 42 % reduction in variance. Compared to RDO, which requires simultaneous optimization of all 24 geometric variables, the robustness improvement based on Shapley only necessitates modifying 4 key geometric variables, resulting in a substantial increase in both the mean and variance of efficiency.

Robust design optimization using a non-intrusive second-order approximation of stochastic moments

This paper presents a new formulation of the second-order fourth-moment method (sometimes referred to as second-order perturbation method or second-order method of moments). The method allows to efficiently predict the stochastic moments of a response function and is therefore often used within robust design optimization. The new approach allows a non-intrusive implementation at the same cost as existing, highly intrusive formulations. Therefore, the new approach can be applied to any objective function without significant implementation effort. It is based on a few finite difference steps into special directions and hence is dependent on the corresponding step sizes. An automatic step size procedure is supplied beside a detailed convergence analysis. The advantages of the new formulation are demonstrated by robust design optimizations of a 2D and a 3D example using the geometrically nonlinear finite element method.

Surrogate-based robust design optimization by using Chebyshev-transformed orthogonal grid

Surrogate-based robust design optimization by using Chebyshev-transformed orthogonal grid

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.

Analytical robust design optimization for hybrid design variables: An active-learning methodology based on polynomial chaos Kriging

In robust design optimization, statistical moments of performance are widely adopted in formulating robustness metrics. To address the high computational costs stemming from the many-query nature of such optimizations with respect to robustness metrics, analytical formulas of the statistical moments have been developed based on surrogate models. However, existing methods consider random variables as the sole model input, which excludes, from the application scope, problems that also involve deterministic design variables. To remedy this issue, this paper proposes a new Polynomial Chaos Kriging-based methodology for efficient and accurate analytical robust design optimization. The analytical solutions for the statistical moments of performance are developed considering that the Polynomial Chaos Kriging model is established in the augmented space of the deterministic design and random variables. This is achieved by systematically decoupling associations with deterministic input from random input, providing effective solutions even when the orthonormality of the basis function is not applicable in the augmented space. This work also presents an active-learning framework enabling seamless implementation of various numerical optimization methods. Several numerical examples and a practical application illustrate the performance and superiority of the proposed method.