Deterministic Design Optimization Research Articles

Reliability-based layout optimization of concentrically braced in 3D steel frames

For an economical and safe design of building systems with concentrically braced frames, the layout of the braces should be done by considering all limitations. The current study proposes an approach for a reliability-based optimal design of concentric bracing layouts in 3D steel frames. The objective function was considered to be a reduction in the total weight. The layout of the braces was considered as the variable of the optimization problem. The deterministic design constraints of limitations of strength, drift and torsional irregularities were considered according to AISC 360-16 and ASCE 7-16. The distance between the center of mass and the center of rigidity was defined as the probability constraint. The probabilistic variables were the intensity of the distributed loads and the cross-sectional area of the braces and columns. The reliability index was evaluated using the Monte Carlo simulation method. The reliability-based design optimization (RBDO) problem was solved for 4-, 5- and 9-story steel space frames using the EVPS algorithm. To investigate the effect of the probabilistic constraint on the optimal design, numerical examples were solved without applying this constraint and the solutions to the RBDO and deterministic design optimization problems were compared.

Robust method for the identification of dynamical anisotropic flexible bearing parameters using multi-objective optimization and structural modification technique

In this paper, a deterministic multi-objective design optimization problem is addressed in order to identify the dynamical joint parameters such as mass, stiffness, damping and cross-coupling parameters. The proposed formulation uses directly the measured frequency transfer functions of the constrained system (with supports) and the mode of the unconstrained system (without supports) to predict the joint dynamic parameters. Since measurement noise often leads to erroneous identifications, the deterministic multi-objective optimization (DMOO) is used to eliminate the redistribution and amplification of noise due to the inversion of the frequency transfer function matrices. The proposed optimization design gives satisfactory results when using noise-free data as well as under realistic noise levels. A parametric study was also performed in order to assert the sensitivity of dynamical joints parameters against the level of noise. The results show also that the constrained non-dominated sorting genetic algorithm (C-NSGA-II) was capable to generate well-distributed reliable Pareto solutions.

열교환기 냉각팬의 베어링 하우징 신뢰성 기반 최적설계

Purpose: This study aimed to reduce the weight of the bearing housing for heat exchanger cooling fans and satisfy reliability of the design constraints at 95% through reliability-based design optimization (RBDO).BRMethods: The design of experiments method was used in conjunction with an orthogonal array to establish the design variables of the bearing housing, and finite element analysis was performed to analyze the sensitivity trends of the performance factors with changes in the magnitudes of the design variables. To minimize epistemic uncertainty arising from the surrogate model generated using the response surface method (RSM), the optimal values obtained from deterministic design optimization process were sequentially added to update the surrogate models. Subsequently, deterministic and reliability-based optimization was performed using the updated RSM.BRResults: The optimal RBDO-based solution was obtained by considering variations in dimensional tolerance. Consequently, the weight of the bearing housing was reduced by 52.48% compared to that of the initial design, and the reliability of the design constraint was satisfied at 95%.BRConclusion: The conservative optimum design of a bearing housing was obtained through RBDO by considering safety factors to avoid overdesign. This design procedure can be applied to various mechanical parts to avoid overdesign.

Reliability-Based Robust Design Optimization for Maximizing the Output Torque of Brushless Direct Current (BLDC) Motors Considering Manufacturing Uncertainty

In recent years, the deterministic design optimization method has been widely used to improve the output performance of brushless direct current (BLDC) motors. However, it does not contribute to reducing the failure rate and performance variation of products because it cannot determine the manufacturing uncertainty. In this study, we proposed reliability-based robust design optimization to improve the output torque of a BLDC motor while reducing the failure rate and performance variation. We calculated the output torque and vibration response of the BLDC motor using the electromagnetic–structural coupled analysis. We selected the tooth thickness, slot opening width, slot radius, slot depth, tooth width, magnet thickness, and magnet length as the design variables related to the shape of the stator and rotor that affect the output torque. We considered the distribution of design variables with manufacturing tolerances. We performed a reliability analysis of the BLDC motor considering the distribution of design variables with manufacturing tolerances. Using the reliability analysis results, we performed reliability-based robust design optimization (RBRDO) to maximize the output torque; consequently, the output torque increased by 8.8% compared to the initial BLDC motor, the standard deviation in output performance decreased by 46.9% with improved robustness, and the failure rate decreased by 99.2% with enhanced reliability. The proposed reliability-based robust design optimization is considered to be useful in the actual product design field because it can evaluate both the reliability and robustness of the product and improve its performance in the design stage.

Global Reliability Sensitivity of the Structure Under Fuzzy Uncertainty and Its Estimations

Global reliability sensitivity (GRS) index under fuzzy uncertainty is defined as the mean of the absolute difference between unconditional failure credibility and conditional failure credibility, which is important to quantify the effect of fuzzy inputs on failure credibility of the structure. The solution of this index requires nested analysis of conditional failure credibility and fuzzy mean, which introduces a large amount of computation. To efficiently estimate the GRS under fuzzy uncertainty, a method combining fuzzy simulation (FS) with fuzzy first-order and second moment (FS–FFOSM) method as well as a fuzzy sequential optimization and reliability assessment (FSORA) method are proposed. Compared with the direct double-loop FS method, in which both the conditional failure credibility and the fuzzy mean are estimated by FS, the proposed FS–FFOSM uses a more efficient FFOSM method to estimate the conditional failure credibility in the inner loop, which improves the computational efficiency of estimating GRS under fuzzy uncertainty. To further improve the computational efficiency, this paper also transforms the estimation of GRS into the solution of the optimization model under the constraint of failure credibility and establishes the FSORA method to solve the optimization model. The FSORA method uses a series of cycles of deterministic optimization and inverse design point analysis corresponding to the required fuzzy reliability index by GRS to reduce the iterations, and then the computational efficiency is further improved. Several examples illustrate the computational efficiency and accuracy of the proposed methods.

Reliability-based optimal control design for seismic-excited structures: A hybrid IS-MTLBO pseudo-double loop method

This study addressed a reliability-based optimal control design (RBOCD) approach for seismic-excited structures. For this aim, a hybrid importance sampling-modified teaching-learning based optimization (IS-MTLBO) pseudo-double loop method is developed. Uncertainty sources such as structural model, stochastic model of the ground acceleration, structural response measurements in sensors, and designed/applied forces in actuators are considered in this study. Numerical studies are carried out on a 10-story building subjected to earthquake excited. A deterministic optimal control design (DOCD) approach based on the MTLBO algorithm is firstly conducted for the structure. Then, a sensitivity analysis using the Borgonovo sensitivity method is carried out to identify and measure the effectiveness of the mentioned uncertainty sources on the seismic responses. Finally, the IS-MTLBO pseudo-double loop approach as an RBOCD approach is implemented for the studied structure. The average reductions of the seismic responses in terms of maximum displacement, acceleration, and drift of floors are obtained as 18.95%, 27.7%, and 24.65% for the RBOCD approach, while, the corresponding reduction are 7.30%, 19.45%, and 14.05% for the DOCD approach. It is shown that although the designed controller based on the DOCD approach satisfies the limitation considered for the maximum floor drift in the nominal model, it is not able to satisfy it in the presence of uncertainties. In other words, the RBOCD approach represents a robust control design against uncertainties, while the DOCD approach gives an uncertainty-sensitive control scheme. Considering four well-known earthquakes, the failure probabilities of the structure controlled by both DOCD and RBOCD approaches are compared. The results indicate the failure probability of the structure controlled by the RBOCD approach is much less than those given using the DOCD approach.

Aerodynamic Performance Uncertainty Analysis and Optimization of a Conventional Axisymmetric Vehicle Based on Parallel Polynomial Chaos Expansions

In this study, the aerodynamic uncertainty analysis and optimization of a conventional axisymmetric vehicle with an aerodynamic configuration were investigated. The prediction precision of the typical aerodynamic performance estimating methods, namely, engineering estimation and numerical simulation, was compared using the wind tunnel test data of the vehicle. Then, using a modified missile data compendium (DATCOM) software, a high-efficiency and high-precision method was developed, which was applied to analyze and characterize the aerodynamic parameters of the axisymmetric vehicle. To enhance the robustness and reliability of aerodynamic performance, an uncertainty-based design optimization (UDO) framework was established. The design space was scaled by parameter sensitivity analysis, and improved computational efficiency was achieved by developing parallel polynomial chaos expansions (PCEs). The optimized results show that the modified method exhibits high accuracy in predicting aerodynamic performance. For the same constraints, the results of the deterministic design optimization (DDO) showed that compared with the initial scheme, the probability of the controllability-to-stability ratio satisfying the constraint decreased from 98.8% to 72.4%, and this value increased to 99.9% in the case of UDO. Compared with the results of the initial scheme and DDO, UDO achieved a considerable reduction in mean values and standard deviation of aerodynamic performances, which can ensure a higher probability of constraints meeting the design requirements, thereby, realizing a reliable and robust design.

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.

Influences of usage intensity and weather on optimal building energy system design with multiple storage options

In non-residential buildings, many factors affect thermal and electrical loads. Some of these factors are not known during the planning process but are essential for a proper energy system design.Conducting a simulation- and optimization-based sensitivity analysis, this study investigates the effects of usage intensity and weather uncertainties on building energy system sizing. We developed a toolchain that comprehensively covers the stages, from determining user-related electricity demand to thermal demand calculation and energy conversion system design, up to deterministic design optimization of storage systems. By means of design of experiments, we systematically applied the toolchain to an all-electric non-residential building as a use case.We find that weather has the strongest correlation with optimal storage sizes and the size of energy conversion systems. They are also affected by the usage intensity of certain building zones. In addition, the storage sizes influence each other in the complex building energy system of our use case. Depending on the scenario and objective, annuities deviate 6%, annual emissions deviate up to 10% from the base scenario.We conclude that normative design methods do not suffice for optimal building energy system design and offer a viable method that can be incorporated into existing planning processes.

Optimization Design for the Planetary Gear Train of an Electric Vehicle under Uncertainties

The planetary gear train is often used as the main device for decelerating and increasing the torque of the drive motor of electric vehicles. Considering the lightweight requirement and existing uncertainty in structural design, a multi-objective uncertainty optimization design (MUOD) framework is developed for the planetary gear train of the electric vehicle in this study. The volume and transmission efficiency of the planetary gear train are taken into consideration as optimization objectives. The manufacturing size, material, and load input of the planetary gear train are considered as uncertainties. An approximate direct decoupling model, based on subinterval Taylor expansion, is applied to evaluate the propagation of uncertainties. To improve the convergence ability of the multi-objective evolutionary algorithm, the improved non-dominated sorting genetic algorithm II (NSGA-II) is designed by using chaotic and adaptive strategies. The improved NSGA-II has better convergence efficiency than classical NSGA-II and multi-objective particle swarm optimization (MOPSO). In addition, the multi-criteria decision making (MCDM) method is applied to choose the most satisfactory solution in Pareto sets from the multi-objective evolutionary algorithm. Compared with the multi-objective deterministic optimization design (MDOD), the proposed MUOD framework has better reliability than MDOD under different uncertainty cases. This MUOD method enables further guidance pertaining to the uncertainty optimization design of transportation equipment, containing gear reduction mechanisms, in order to reduce the failure risk.

A Bi-stage Multi-objective Reliability-based Design Optimization Using Surrogate Model for Reusable Thrust Chambers

To reduce the computational burden of Multi-Objective Reliability-Based Design Optimization (MORBDO) and promote its application in complex practical systems, a bi-stage MORBDO procedure using surrogate models is proposed in this paper. The novel process consists of two stages: in the first stage, a narrowed solution space named as MORBDO design space is obtained by a pre-multi-objective deterministic design optimization in the whole original design space; In the second stage, the MORBDO design space is extended to form an augmented space by considering uncertainties, and then the MORBDO is executed with the double-loop strategy over the augmented space. Besides, the multiple response surface (MRS) is adopted in the bi-stage MORBDO procedure to replace the time-consuming finite element analysis. The proposed method is validated by conducting MORBDO on a lab-scale reusable LOX/H2 thrust chamber aiming at maximum residual strength and cyclic life. Compared with the traditional MORBDO implemented in the whole original design space, the bi-stage MORBDO can significantly reduce the computational burden of the surrogate model by 49.33% and only one-quarter of the optimization population is required in NSGA II to search for the optimal solution. Finally, a sensitivity analysis is performed to quantify the importance ranking of design variables of thrust chambers. The results indicate that the geometry of cooling channels near the throat plays the most important role in the thrust chamber's residual strength and cyclic life. The proposed method is easy to be conducted and holds great potentials in the MORBDO of other complex practical systems.

Optimization design of uncertain parameters for improving the stability of photovoltaic system

The uncertainty of photovoltaic module material parameters have a great impact on the overall output performance of photovoltaic cells. Ignoring the uncertainty of these parameters results in low reliability in photovoltaic cell applications. In this study, the deterministic and robust optimization design of photovoltaic cells was demonstrated by considering the randomness of the actual working environment of photovoltaic cells and the interval uncertainty of material properties of photovoltaic components, and by using the theoretical efficiency of photovoltaic cells as constraint. The corresponding parameter values were obtained after optimization, and the optimization results are analyzed with Monte Carlo simulations. Making parameters after deterministic optimization obey the normal distribution, and the parameter value obtained after optimization is the mean value, then compared with the robust optimization. It can be concluded that compared with the traditional deterministic design, the mean value of the constraint conditions after robust optimization is far removed from the upper limit, the standard deviation is smaller, and the failure probability is far less than that after setting variables through deterministic optimization. Based on these findings, the influence of the uncertain factors can be reduced, the output is more reliable, and the performance optimization of photovoltaic cells can be provided guidance.

Reliability-Based Robust Design Optimization of Lithium-Ion Battery Cells for Maximizing the Energy Density by Increasing Reliability and Robustness

Lithium-ion batteries (LIBs) are increasingly employed in electric vehicles (EVs) owing to their advantages, such as low weight, and high energy and power densities. However, the uncertainty encountered in the manufacturing of LIB cells increases the failure rate and causes cell-to-cell variations, thereby degrading the battery capacity and lifetime. In this study, the reliability and robustness of LIB cells were improved using the design of experiments (DOE), and the reliability-based robust design optimization (RBRDO) approaches. First, design factors sensitive to the energy density and power density were selected as design variables through sensitivity analysis using the DOE. RBRDO was performed to maximize the energy density while reducing the failure rate and cell-to-cell variations. To verify the superiority of the reliability and robustness offered by RBRDO, the obtained results were compared with those from conventional deterministic design optimization (DDO), and reliability-based design optimization (RBDO). RBRDO increased the mean of the energy density by 33.5% compared to the initial value and reduced the failure rate by 98.9%, due to improved reliability, compared to DDO. Moreover, RBRDO reduced the standard deviation in the energy density (i.e., cell-to-cell variations) by 30.0% due to the improved robustness compared to RBDO.

Reliability-Based Design Optimization for Reducing the Performance Failure and Maximizing the Specific Energy of Lithium-Ion Batteries Considering Manufacturing Uncertainty of Porous Electrodes

Uncertainty quantification in LIB manufacturing has received interest in order to improve the reliability of LIB. The uncertainty generated during the manufacturing causes variations in the performance of LIBs, thereby increasing capacity degradation and leading to failure. In this study, a reliability-based design optimization (RBDO) of LIBs is conducted to reduce performance failure while maximizing the specific energy. The design variables with uncertainty are the thickness, porosity, and particle size of the anode and cathode. The specific energy is defined as the objective function in the optimization design problem. To maintain the specific power in the initial design of the LIB, it is defined as the constraint function. Reliability is evaluated as the probability that the battery satisfies the performance of the required design. The results indicate that the design optimized through RBDO increases the specific energy by 42.4% in comparison with that of the initial design while reducing the failure rate to 1.53%. Unlike the conventional deterministic design optimization method (DDO), which exhibits 55.09% reliability, the proposed RBDO method ensures 98.47% reliability. It is shown that the proposed RBDO approach is an effective design method to reduce the failure rate while maximizing the specific energy.

Multi-objective robust optimization of a solar power tower plant under uncertainty

The optimal design of a molten salt solar power tower (SPT) plant is sensitive to the variations of uncertainties, such as solar radiation, which result in dispersion of the model output. To mitigate the impacts of uncertainties on the thermo-economic performance of SPT plant, this study develops an uncertainty-based multi-objective robust optimization design method for the case of a SPT plant in Sevilla with the expected value (i.e. the average energy cost) and the standard deviation (i.e. the dispersion of the model output) of the levelized cost of energy (LCOE) as the objectives. The Monte Carlo (MC) simulation and simulated annealing (SA) algorithm are combined to solve the robust optimization problem. The results of Pareto frontier indicate that a trade-off is needed through decision-making. The final optimal solution is determined with expectation of LCOE of 23.09 c/kWhe and standard deviation of LCOE of 1.25 c/kWhe. Compared with the deterministic optimal design, the standard deviation of LCOE of the multi-objective robust optimum is reduced by 17.22 %, which turns out to be less sensitive to the uncertainties. Moreover, the Sobol’ global sensitivity analysis results show that the direct solar radiation, heliostat field cost and receiver cost are the most sensitive to LCOE.

Robust design optimization considering inlet flow angle variations of a turbine cascade

The stochastic variation of inlet flow is one common uncertainty in turbomachines, the performance impact of which requires to be quantified and taken into account in the design optimization of turbomachinery blades. The paper introduces the robust aerodynamic design optimization improving the mean performance and the robustness of a turbine cascade at off-design condition. The consistent-refinement of the grids used for solving the Euler equations is checked and the employed numerical scheme is verified and validated. Performance impact of the inlet flow angle variations is statistically evaluated by using a non-intrusive polynomial chaos with an adaptive sparse grid sampling method. The camber of the turbine blade is redesigned with the enforced blade thickness distribution to minimize the statistical mean and standard deviation (Stdv) of total pressure loss coefficient. The results of both the deterministic and robust aerodynamic design optimization are presented and compared in detail. The effects of camber redesign on reducing the statistical mean and Stdv of total pressure loss coefficient are discussed. Moreover, the statistical flow solutions are illustrated to demonstrate the influence mechanisms of flow variations in the blade passage on the improvements of aerodynamic robustness.

Aerodynamic analysis and design optimization of a centrifugal compressor impeller considering realistic manufacturing uncertainties

Centrifugal compressor impeller blades inevitably suffer from manufacturing uncertainties. Such manufacturing uncertainties result in geometric deviations of blade profiles, and have been increasingly recognized to be detrimental to compressor performance. However, relevant studies on the uncertainty analysis and design optimization of centrifugal impeller blades are quite rare. The effect of realistic manufacturing uncertainties on impeller aerodynamic performance and flow field still remains unclear. Here we present a comprehensive investigation on deterministic analysis, uncertainty analysis and design optimization of a centrifugal compressor impeller with considerations of three-dimensional realistic manufacturing uncertainties. Through statistical analysis of blade profile measurements of machined centrifugal impellers, both the blade camber and thickness errors are confirmed to obey Gaussian distributions. The deterministic analysis results show that the positive thickness errors near the main blade leading edge are mainly responsible for impeller performance degradations because of aggravated flow separations especially at large flow rate. A bi-fidelity metamodel, namely sparse grid stochastic collocation-support vector regression (SGSC-SVR), is used to balance the efficiency and accuracy of the uncertainty analysis. The mean impeller performance is found to become more deteriorated as the mass flow rate increases while the uncertain band is smaller at the design point than at off design points. Finally, a sequential infilling sampling-based optimization method is employed to solve the design optimization. It is demonstrated that reduced blade angles are able to compensate the adverse effect of positive blade thickness errors and thus enhance the impeller performance robustness against manufacturing uncertainties. The present study lays a theoretical foundation for the further uncertainty quantification and robust design of advanced centrifugal compressors.

Reliability-based design optimization for RV reducer with experimental constraint

Due to the limited joint position space and the consideration of reducing the moment of inertia and vibration for industrial robot, the design optimization for rotate vector (RV) reducer is becoming a new and urgent problem in industry. Currently, the existing researches focus on deterministic design optimization, which may cause unreliable designs without considering uncertainties. Therefore, the study focuses on the implementation of reliability-based design optimization (RBDO) to the RV reducer. The aim is to make the RV reducer smaller in size while ensuring a higher reliability. Firstly, a modified advanced mean value (MAMV) method is proposed to improve efficiency and robustness of the advanced mean value method, which encounters inefficiency and numerical instability for the concave or highly nonlinear performance measure functions. Secondly, a mathematical model of RBDO for the RV reducer is established. Thirdly, the proposed MAMV method is integrated into double-loop method to optimize the established mathematical model of RBDO with different target reliability. The results show that the proposed MAMV method is efficient compared with other methods. In addition, the volume of the RV reducer is correspondingly reduced by 9.44%, 7.89%, and 5.66% compared with that before optimization when the target reliabilities are 99.38%, 99.87%, and 99.98%.

A PDF-based performance shift approach for reliability-based design optimization

A probability density function (PDF)-based performance shift approach (PPSA) is proposed for sequential reliability-based design optimization (RBDO) in this paper, which provides an effective tool for complex engineering design with consideration of uncertainties associated with design variables and simulation models. It converts the double-loop probabilistic optimization into a serial of cycles of reliability assessment and deterministic optimization through a performance shift strategy. The approach further proceeds by calculating the performance shift scalar and then solving a deterministic design optimization at each cycle, and eventually converges to the optimum design solution. The novelty of the proposed approach lies in two main aspects. First, the shift scalar is deducted for each probabilistic constraint in the response space, rather than in the design variable space, through which the violated constraints are moved towards the reliable region and the design solutions are improved progressively. Second, reliability analysis methods that calculate the moments or probability density functions, rather than most probable points or reliability indexes, are able to be integrated in the sequential RBDO framework, through which we could enhance the performance of existing sequential RBDO methods, especially the computational efficiency. Three numerical examples are investigated to demonstrate the effectiveness of the proposed approach, and the proposed approach is further utilized to perform crashworthiness design of a vehicle side impact problem.

An efficient reliability-based design optimization study for PCM-based heat-sink used for cooling electronic devices

Nowadays, development of new technologies requires efficient mechatronic components which needs more sophisticated cooling systems. Furthermore, phase change material (PCM) based heat sinks presents an appropriate technique for electronic devices. Then, the aim of this work is to propose an efficient methodology that determines an optimal design of such a PCM-based cooling system. Despite of the satisfaction of the optimal solution founded by the deterministic design optimization (DDO), the reliability level is not controlled. For this reason, reliability-based design optimization (RBDO) studies were performed. In fact, RBDO approaches aim to find the best compromise between safety and cost by taking into consideration uncertainties for the studied model. Therefore, several methods are studied, such as the optimum safety factor approach and the hybrid method. Yet, the application of these methods is limited only to linear materials. Hence, this study presents an extension in the case of nonlinear materials. In addition, to overcome problems of the classical hybrid approach, a Robust Hybrid Method (RHM) is proposed. A numerical application is used to study the different DDO and RBDO methods. Then, the efficiency of the RHM method for PCM-based heat sinks is verified.