Deterministic Design Optimization Research Articles

Multidisciplinary design optimization of an autonomous underwater vehicles based on random uncertainty

Herein, discipline decomposition was performed for an autonomous underwater vehicle (AUV) based on multidisciplinary design optimization (MDO), and its design parameters were determined. Parameterized analysis of the hull-form and structure disciplines was performed, and an approximate model of these disciplines was established with a high fitting accuracy. An analysis model of propulsion, energy, and system disciplines was developed based on the formula method. Based on collaborative optimization as well as discipline and system level models, a deterministic MDO framework for AUVs was established. Subsequently, an optimized solution was obtained. By considering the random uncertainty of design variables, an MDO framework for the uncertainty of AUVs was established and optimized solutions were obtained. The distribution of solutions obtained from uncertainty optimization was more concentrated and the distribution range was smaller than that obtained using deterministic optimization. Robustness analysis was performed on the initial scheme, typical deterministic optimization schemes, and typical uncertainty optimization schemes. Results showed that fluctuations in design variables may lead to constraints that exceed boundary conditions in the deterministic optimization design scheme. Using uncertainty and objective function optimization, the robustness of the overall scheme of AUVs was improved.

Surrogate based design space exploration and exploitation for an efficient airfoil optimization under uncertainties using transition models

The pursuit of sustainable, zero-emission air travel is heavily dependent on the creation of energy-efficient aircraft. Key strategies for achieving this sustainability in aviation include reducing fuel consumption through low-drag designs harnessing laminar flow. However, designing aircraft with laminar flow characteristics is complex due to their sensitivity to environmental and operational factors. This study tackles the challenge of developing energy-efficient aircraft by using computational fluid dynamics models and sophisticated optimization techniques that account for uncertainty. Our approach demonstrates the effectiveness of surrogate-based optimization and uncertainty quantification in optimizing airfoil drag for a natural laminar airfoil (NLF) design. We use surrogate models, trained with data from detailed airfoil simulations, which include a boundary layer code coupled with a linear stability method and a newly developed transition transport model. Transition location predicted using transition models facilitate an accurate drag prediction used in the optimization process. The accuracy of these surrogate models is enhanced through active sampling strategies. Our robust optimization method considers uncertainties in environmental and operational conditions, offering a deeper insight into their effects on crucial design parameters. Unlike traditional deterministic aerodynamic design optimization, our findings highlight the efficacy and precision of uncertainty-based optimization in achieving robust NLF airfoil designs over large (exploration mode) and small (exploitation mode) design spaces. Investigating design space parameterization based on the size of design variables reveals significant differences in optimal airfoil configurations. The optimized designs we propose favor delayed transition, in contrast to deterministic designs which often result in significant loss of laminarity when facing uncertainties. This study represents a significant advancement in aerospace engineering, providing a practical and effective methodology for creating energy-efficient airfoil designs. The application of these advanced optimization and uncertainty quantification techniques shows great potential for the wider field of aerospace engineering, paving the way for more resilient and robust aircraft designs.

Novel study on active reliable PID controller design based on probability density evolution method and interval-oriented sequential optimization strategy

Vibration active control is a significant problem in practical engineering, but the influence of hybrid uncertainties in practical engineering cannot be ignored. For the uncertainties with adequate data, probability theory can be used while for the uncertainties with insufficient data, probability theory is limited and interval quantification can be used. Taking the widely used proportional-integral-differential (PID) control as an example, a novel study on controller design based on probability density evolution method and interval-oriented sequential optimization strategy considering interval and random uncertainties is proposed. Firstly, under a certain sample point of interval uncertainties, probability density evolution equation is utilized to calculate the probability density function of uncertain response considering only random uncertainties. In this way the probabilistic reliability corresponding to the sample point can be obtained. Then the hybrid reliability is defined as the average probability within the interval uncertainty and the adaptive dichotomy method is utilized to choose proper sample points of interval uncertainties, in which the reliability curve and the interval uncertainty axis is regarded as the judgement of convergence. Next the hybrid reliability is introduced to the optimization as a constraint and the PID gains reliable design optimization is established to balance the energy-consuming and safety. Finally, the sequential optimization strategy is adopted to improve the efficiency of optimization. In every iteration step of the optimization, the constraint of deterministic optimal design is adjusted according to the static reliability at the moment before the passage or reaching the maximum value. Three numerical examples are presented to demonstrate the accuracy and practicability of the proposed framework.

6σ Robust Lightweight Design of the GIL/GIS Bus Based on Co‐Simulation of Insulation, Current Flow, and Mechanical Stress

GIL/GIS bus is widely used in the field of power transmission and distribution, and the demand for lightweight is becoming more and more prominent. To reduce the weight of the bus and ensure the safety and reliability of the structure, a new 6σ robust design method of the GIL/GIS bus based on electric‐thermal‐mechanical co‐simulation is proposed. Taking a 550 kV/5000A bus as the initial structure, considering the uncertainty of product size and performance, a 6σ robust lightweight model of bus insulation electric field strength, current temperature rise, and mechanical stress multi‐performance co‐simulation is established. The finite element numerical calculation method is used to calculate and analyze the performance parameters of the bus. The Kriging surrogate model is used to form the response surface, and the optimal bus structure is obtained by genetic algorithm. The results show that after the robustness optimization, the performance of the bus meets the design requirements, the weight per meter decreased from 65.11 to 39.12 kg, and the lightweight effect is remarkable. Compared with the traditional deterministic optimization design, the reliability of the structure is improved. The new method of the GIL/GIS bus lightweight design proposed in this study has high feasibility and application prospects, which can provide new ideas for the development of gas‐insulated electrical equipment. © 2024 Institute of Electrical Engineers of Japan and Wiley Periodicals LLC.

Quantile-based sequential optimization and reliability assessment method under random and interval hybrid uncertainty

Under random and interval hybrid uncertainties, solving hybrid reliability based design optimization (HRBDO) can acquire an optimal balance between structural performance and reliability. Since solving HRBDO includes a triple nested framework involving minimum analysis of performance function (PF), failure probability constraint analysis and design parameter optimization, the computational complexity of HRBDO is high, especially for dealing with complex structures. Therefore, a quantile-based sequential optimization and reliability assessment method (QSORA) is proposed for reducing the computational complexity of HRBDO. In the proposed QSORA for HRBDO, failure probability constraint is firstly transformed into minimum PF (MPF) quantile one corresponding to target failure probability. Then, approximating the difference between PF and its target quantile at current iteration by that at previous one, the failure probability constraint analysis is decoupled from the design parameter optimization. Moreover, by approximating the minimum point of the PF with respect to the interval input in the current iteration by that in the previous one, the minimum analysis of PF is separated from the design parameter optimization. By the separation of minimum analysis and failure probability constraint analysis from the design parameter optimization in the proposed QSORA, the triple nested framework of HRBDO is decoupled sequentially as the deterministic design optimization, the minimum analysis of the PF and the target MPF quantile estimation, and this way of reconstructing the HRBDO from the triple nested framework to three single-loop frameworks can significantly enhance the efficiency of solving HRBDO. Furthermore, the MPF quantile at the current design parameter is estimated by stochastic collocation based statistical moment method, in which the stochastic collocation method is employed to efficiently estimate the MPF moment to approximate the probability density function of MPF. The efficiency and accuracy of the QSORA are validated by four numerical and engineering examples finally.

Comparative analysis of deterministic and reliability-based structural optimization methods

Optimization is the act of obtaining the best possible result under established conditions. Usually, the optimization of a structural design is done considering the structure's dimensions, the materials' properties, and the loads as deterministic values. This way, the optimization process can lead to a more economical design without guaranteeing that this structure is safe. In practice, there are always uncertainties about the final dimensions of the built structure, material properties, and loads. Then, the need arises to use design optimization techniques based on reliability to guarantee a project that is both economical and safe. This objective is achieved by including uncertainties in the optimization process. This article evaluates the parameters that determine the global minimum of the optimization methods DDO (Deterministic Design Optimization) and RBDO (Reliability-Based Design Optimization). This work aims to compare the structural optimization methods of DDO and RBDO through an example. The results are obtained through the codes of the methods implemented in the Python language and show that when comparing the two optimization methods, the presence of uncertainties alters the optimal solution.

Reliability-based design optimization of the spiral water jacket for motorized spindle

Spiral water jackets are essential refrigeration units in motorized spindles. The design of spiral water jackets directly affects the machining accuracy and reliability of motorized spindles. In this regard, a reliability-based design optimization (RBDO) method is presented to obtain the optimal configuration of the spiral water jacket. The proposed RBDO algorithm aims to minimize the volume of the spiral water jacket and power loss of the cooling equipment while ensuring machining reliability. The geometric correlation and machining reliability are selected as constraints. The Kriging surrogate model is established to approximate the time-consuming finite element model for improving computational efficiency. An efficient metaheuristic algorithm is introduced to perform optimal exploration for the proposed RBDO problem. The risk speed of the motorized spindle is investigated by the experiment and finite element analysis. Subsequently, the effects of various cooling configurations on thermal error are investigated thoroughly. Eventually, the proposed RBDO is conducted on a motorized spindle. Through probabilistic evaluation, machining reliability is guaranteed by the proposed RBDO compared to deterministic design optimization. This paper provides a new design paradigm for cooling configurations in motorized spindles from the perspective of machining reliability.

A discrete adjoint framework coupled with adaptive PCE for robust aerodynamic optimization of turbomachinery under flow uncertainty

Flow uncertainty is commonly encountered in turbomachinery. To mitigate the negative effects caused by the flow uncertainty, a framework coupled with adaptive polynomial chaos expansion (PCE) and discrete adjoint method for robust aerodynamic design optimization (RADO) of turbomachinery blades is developed. The optimization framework relies on gradient algorithms, which effectively avoids “dimensional curse” problem. Adaptive PCE method improves the efficiency of uncertainty quantification and only once PCE model need to be constructed in per optimization step, making the framework highly efficient. The PCE matrix and chain rule are employed to determine the gradient of the RADO objective function. This framework is employed for optimizing turbine blade, considering both inviscid and viscous flow. Monte Carlo simulation and finite difference PCE are used to validate the reliability of uncertainty quantification and gradient propagation methods. A comparative analysis of the optimization results obtained by both RADO and deterministic aerodynamic design optimization (DADO) is demonstrated. The higher robustness brought by RADO indicates the practicality of this framework in robust optimization of turbomachinery blades.

EVTOL Tilt-Wing Aircraft Design under Uncertainty Using a Multidisciplinary Possibilistic Approach

Recent development in Electric Vertical Take-off and Landing (eVTOL) aircraft makes it a popular design approach for urban air mobility (UAM). When designing these configurations, due to the uncertainty present in semi-empirical estimations, often used for aerodynamic characteristics during the conceptual design phase, results can only be trusted to approximately 80% accuracy. Accordingly, an optimized aircraft using semi-empirical estimations and deterministic multi-disciplinary design optimization (MDO) approaches can be at risk of not being certifiable in the detailed design phase of the life cycle. The focus of this study was to implement a robust and efficient possibility-based design optimization (PBDO) method for the MDO of an eVTOL tilt-wing aircraft in the conceptual design phase, using existing conventional designs as an initial configuration. As implemented, the optimization framework utilizes a deterministic gradient-based optimizer, run sequentially with a possibility assessment algorithm, to select an optimal design. To achieve this, the uncertainties which arise from multi-fidelity calculations, such as semi-empirical methods, are considered and used to modify the final design such that its viability is guaranteed in the detailed design phase. With respect to various requirements, including trim, stability, and control behaviors, the optimized eVTOL tilt-wing aircraft design offers the preferred results which ensure that airworthiness criteria are met whilst complying with predefined constraints. The proposed approach may be used to revise currently available light aircraft and develop eVTOL versions from the original light aircraft. The resulting aircraft is not only an optimized layout but one where the stability of the eVTOL tilt-wing aircraft has been guaranteed.

Multiobjective Reliability-Based Design Optimization of the Fuzzy Logic Controller for MR Damper-Based Structures

To devise an optimum and robust fuzzy logic controller for MR damper-based structures subjected to earthquake ground motions, the multiobjective reliability-based design optimization (RBDO) using the adaptive Kriging model is performed to determine the parameters of the fuzzy logic controller. The optimization problem is formulated with two objective functions, namely, the minimization of interstory drift and average control force of the concerned structure, and subjected to a probability constraint on structural dynamic responses under the effects of random structural stiffness and stochastic earthquake loadings. To reduce the computational cost of reliability assessment, a global Kriging model is constructed in an augmented space as a surrogate for computational evaluations. Subsequently, the trained metamodel combined with the nondominated sorting genetic algorithm (NSGA-II) is integrated into the framework of RBDO for solving the fuzzy logic control (FLC) optimization problem. The feasibility and effectiveness of the multiobjective RBDO in the FLC design are finally validated by conducting numerical simulations on both linear and nonlinear structures. As demonstrated in the linear case, the fuzzy logic controllers obtained from the multiobjective RBDO show more robustness than those derived from the multiobjective deterministic design optimization (DDO). In the nonlinear case, using the multiobjective DDO to prelocate a coarse safety domain can significantly reduce the number of samples for training the metamodel and facilitate the implementation of the multiobjective RBDO; in addition, the controlled structural performance with a specified fuzzy logic controller can be further improved by considering MR damper distribution optimization.

Deterministic-based robust design optimization of composite structures under material uncertainty

We propose a new deterministic robust design optimization method for composite laminate structures under worst-case material uncertainty. The method is based on a simultaneous parametrization of topology and material and combines a design problem and a material uncertainty problem into a single min–max optimization problem which provides an efficient approach to handle variation of material properties in stiffness driven design optimization problems. An analysis is performed using a design problem based on a failure criterion formulation to evaluate the ability of the proposed method to generate robust composite designs. The design problem is solved using various loads, boundary conditions and manufacturing constraints. The designs generated with the proposed method have improved objective responses compared to the worst-case response of designs generated with nominal material properties and are less sensitive to the variation of material properties. The analysis indicates that the proposed method can be efficiently applied in a robust structural optimization framework.

Optimal performance and cost-effective design of BGA solder joints using a deterministic design optimization (DDO) under real operating conditions

Users of electronic devices expect efficiency, affordability, and long service life. From this perspective, the optimization of their performance while reducing costs is paramount. The objective of this paper is to find the best compromise between minimizing the volume and thus cost, and preserving the efficiency of the component. This work is based on coupling the finite element model on Ansys with an optimization algorithm on MATLAB. First, fatigue analysis tests were simulated to determine the geometric design variables' sensitivity, having an impact on the service life, then these variables are incorporated into the Deterministic Design Optimisation (DDO) loop. This efficient approach is used to identify the optimal configuration of the BGA (Ball Grid Array) component to increase operational life and efficiency under electro-thermomechanical conditions. The elaborated method constitutes a great impetus for electronic industries and researchers to successfully find the best combination of cost reduction and performance increase, allowing improved design.

Multi-level Reliability-Based Design Optimization (RBDO) study for electronic cooling: Application of heat sink based on multiple phase change materials enhanced with carbon nanoparticles

Phase change materials represent efficient performances in the thermal management field. However, the low thermal conductivity poses a serious restraint for electronic cooling applications. Recently, the use of multiple phase change materials and the dispersion of nanoparticles are the most recommended techniques to improve the performance of the heat sink. In this study, a developed heat sink filled with multiple phase change materials (PCMs) enhanced with carbon nanoparticles was proposed. First, a parametric study was carried out to determine the optimal pair of phase change materials and nanoparticles. Then, deterministic design optimization and various reliability-based design optimization methods are performed. The optimization and reliability problems are solved by coupling the MATLAB optimization toolbox and the ANSYS Finite Element Model. The optimization-reliability results prove the efficiency of the Robust Hybrid Method in proposing a reliable and optimal heat sink design filled with multiple nano-enhanced phase change materials. The temperature is reduced by 25 % (16,5 °C) and the discharge time is reduced by 8,69 % (14 min) compared to the baseline heat sink.

Ship design optimization with mixed uncertainty based on evidence theory

With the continuous development of ship design and application requirements, the deterministic design optimization based on design conditions has been difficult to meet the current demands of ship design. Some parameters regarded as fixed values in ship design inevitably fluctuate when sailing in the real marine environment, and the changes of these uncertainties will directly affect the performance of the ship scheme. Therefore, higher requirements are proposed by the designers for the robustness and reliability of the ship scheme under the influence of uncertainty. Generally, uncertainty can be divided into two types, which are aleatory uncertainty an epistemic uncertainty. In most researches, due to the modeling of epistemic uncertainty is not mature, uncertain parameters which might belong to epistemic uncertainty are often modeled as the stochastic distribution belonging to aleatory uncertainty, thus insufficient distribution information will lead to false modeling and result in wrong analysis. Also, few studies have taken the influence of aleatory and epistemic uncertainty into consideration simultaneously in ship design. Therefore, in this research, evidence theory is used to model interval variables of epistemic uncertainty, and the influence of mixed uncertainty on constraints and objective functions is analyzed. By deriving the failure probability of the constraint function and the expectation and standard deviation of the objective function in the framework of evidence theory, and the reliability and robustness expression under mixed uncertainty are completed. Finally, a unified analysis and propagation method of mixed uncertainty is proposed based on evidence theory. By applying this method to the engineering optimization of a bulk carrier, the optimal scheme under mixed uncertainty is obtained, which has better reliability and robustness than the deterministic optimal scheme.

Deterministic vs. robust design optimization using DEM-based metamodels

In design optimization of bulk handling equipment (BHE) we generally focus on the mean performance of the equipment. However, granular materials behave stochastic due to irregularities in particle shape and size which leads to stochastic performance of the equipment. To include the stochastic performance we propose robust metamodel-based design optimization (MBDO). The used metamodels are trained with stochastic performance data from randomly repeated discrete element method (DEM) simulations and predict mean and variance of the equipment performance. This method is compared to the conventional deterministic optimization method by means of a case study of a discharging hopper including verification and validation. The robust MBDO shows more distinctive optimal designs compared to the deterministic approach. In addition, the DEM-based metamodel is a relatively accurate method to predict DEM-model simulation results. However, the validation indicates that differences between DEM-model and experimental results highly affect the reliability of the found optima.

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.

Adjoint-based robust optimization design of laminar flow wing under flight condition uncertainties

It is an inherent uncertainty problem that the application of laminar flow technology to the wing of large passenger aircraft is affected by flight conditions. In order to seek a more robust natural laminar flow control effect, it is necessary to develop an effective optimization design method. Meanwhile, attention must be given to the impact of crossflow (CF) instability brought on by the sweep angle. This paper constructs a robust optimization design framework based on discrete adjoint methods and non-intrusive polynomial chaos. Transition prediction is implemented by coupled Reynolds-Averaged Navier-Stokes (RANS) and simplified eN method, which can consider both Tollmien-Schlichting (TS) wave and crossflow vortex instability. We have performed gradient enhancement processing on the general Polynomial Chaos Expansion (PCE), which is advantageous to reduce the computational cost of single uncertainty propagation. This processing takes advantage of the gradient information obtained by solving the coupled adjoint equations considering transition. The statistical moment gradient solution used for the robust optimization design also uses the derivatives of coupled adjoint equations. The framework is applied to the robust design of a 25° swept wing with infinite span in transonic flow. The uncertainty quantification and sensitivity analysis on the baseline wing shows that the uncertainty quantification method in this paper has high accuracy, and qualitatively reveals the factors that dominate in different flow field regions. By the robust optimization design, the mean and standard deviation of the drag coefficient can be reduced by 29% and 45%, respectively, and compared with the deterministic optimization design results, there is less possibility of forming shock waves under flight condition uncertainties. Robust optimization results illustrate the trade-off between the transition delay and the wave drag reduction.

Simulation-Based Reliability Design Optimization Method for Industrial Robot Structural Design

Robots are main elements in Industry 4.0. Research on the design optimization of robots has a great significance in manufacturing industries. There inevitably exist various uncertainties in robot design that have an important influence on the reliability of robots. At present, the design optimization of robots considering the uncertainties is mainly focused on joints design and trajectory optimization. However, for the structural design of robots, deterministic design optimization still plays a leading role. In this paper, a simulation-based reliability design optimization method is proposed to improve the reliability of robots’ structural design. In the proposed method, the Latin hypercube sampling (LHS), computer simulation, response surface method (RSM) and SORA (Sequential Optimization and Reliability Assessment) algorithm are integrated to complete the structural design of the robot. Firstly, samples of the uncertainty design variables were obtained by LHS, and then, the reliability performance constraint functions were firstly constructed through the RSM in which the joint simulation of MTLAB and ANSYS was adopted. Afterwards, the reliability design optimization model was established on the basis of the probabilistic reliability theory. At last, the SORA algorithm was employed to realize the optimization. The design optimization problems of the big arm and the small arm of a 6 Kg industrial robot were considered to verify the proposed method. The results showed that the weights of the big arm and the small arm were, respectively, reduced by 7.73% and 25.70% compared with those of the original design, and the design was more effective in ensuring the reliability requirements compared with the deterministic optimization. Moreover, the results also demonstrated that the proposed method has a better computational efficiency compared with the reliability design optimization of the double-loop method.

Enriched global–local multi‑objective optimization scheme for fuzzy logic controller-driven magnetorheological damper-based structural system

To construct an optimal and reliable fuzzy logic controller (FLC) for magnetorheological (MR) damper-based structural systems, a reliability-based design optimization (RBDO) problem is formulated considering two competing objectives, namely, minimizing structural responses and control cost. This work incorporates a two-stage enrichment scheme with the non-dominated sorting genetic algorithm (NSGA-II)-based multi-objective optimization method to solve the RBDO problem. In the global stage of the proposed method (denoted as TS-MO method), an initial Kriging surrogate model is constructed in the augmented space and updated by sequentially adding enrichment points within the whole design space to obtain a coarse Kriging surrogate. In the local stage, within the multi-objective optimization iterations, the surrogate model is further refined using enrichment points selected in the vicinity of the current set of design points near the limit state function based on a distance-refined U learning function. Two convergence criteria with dynamic thresholds are employed to ensure the local precision of the trained surrogate model. The efficacy of the TS-MO method is demonstrated by conducting comparison works with two previous methods, i.e., the deterministic design optimization and global Kriging-based multi-objective optimization method (DDO-GK-MO) and the unified method.

A decoupled double-loop method with the adaptive allowable limits for probabilistic performance-based design optimization

In this study, inspired by the idea of the sequential optimization and reliability assessment (SORA) method, a new decoupled double-loop method for probabilistic performance-based design optimization (PPBDO) has been proposed. The highlight of the proposed method is the independence of this method from the type of problem and the limit state function; in such a way that it can be used in non-linear problems with multiple probabilistic constraints with the implicit form of the limit state function. However, it requires less computational effort than conventional double-loop methods. In each cycle of this method, deterministic performance-based design optimization was performed and then the reliability of the optimal design was evaluated. If the results of the reliability assessment are not desirable, the allowable limit of the constraints in the deterministic optimization problem will be changed based on the results of the reliability assessment. This process repeats until the results of the reliability assessment become desirable. The metaheuristic enhanced vibrating particle system algorithm and the Monte Carlo simulation method were used for optimization process and reliability analysis, respectively. To evaluate the efficiency of the proposed method, the optimization problem was defined so as to reduce the structural weight while meeting the acceptance criteria for performance-based design of steel moment frames according to ASCE 41-13. The PPBDO of three- and nine-story steel moment frames were investigated and the results showed that the proposed method was able to reduce the volume of calculations and direct the search path toward a reliable space.