A reliability-based multidisciplinary design optimization procedure based on combined probability and evidence theory

To address the reliability-based multidisciplinary design optimization (RBMDO) problem under mixed aleatory and epistemic uncertainties, an RBMDO procedure is proposed in this paper based on combined probability and evidence theory. The existing deterministic multistage-multilevel multidisciplinary design optimization (MDO) procedure MDF-CSSO, which combines the multiple discipline feasible (MDF) procedure and the concurrent subspace optimization (CSSO) procedure to mimic the general conceptual design process, is used as the basic framework. In the first stage, the surrogate based MDF is used to quickly identify the promising reliable regions. In the second stage, the surrogate based CSSO is used to organize the disciplinary optimization and system coordination, which allows the disciplinary specialists to investigate and optimize the design with the corresponding high-fidelity models independently and concurrently. In these two stages, the reliability-based optimization both in the system level and the disciplinary level are computationally expensive as it entails nested optimization and uncertainty analysis. To alleviate the computational burden, the sequential optimization and mixed uncertainty analysis (SOMUA) method is used to decompose the traditional double-level reliability-based optimization problem into separate deterministic optimization and mixed uncertainty analysis sub-problems, which are solved sequentially and iteratively until convergence is achieved. By integrating SOMUA into MDF-CSSO, the Mixed Uncertainty based RBMDO procedure MUMDF-CSSO is developed. The effectiveness of the proposed procedure is testified with one simple numerical example and one MDO benchmark test problem, followed by some conclusion remarks.     

A surrogate based multistage-multilevel optimization procedure for multidisciplinary design optimization

Optimization procedure is one of the key techniques to address the computational and organizational complexities of multidisciplinary design optimization (MDO). Motivated by the idea of synthetically exploiting the advantage of multiple existing optimization procedures and meanwhile complying with the general process of satellite system design optimization in conceptual design phase, a multistage-multilevel MDO procedure is proposed in this paper by integrating multiple-discipline-feasible (MDF) and concurrent subspace optimization (CSSO), termed as MDF-CSSO. In the first stage, the approximation surrogates of high-fidelity disciplinary models are built by disciplinary specialists independently, based on which the single level optimization procedure MDF is used to quickly identify the promising region and roughly locate the optimum of the MDO problem. In the second stage, the disciplinary specialists are employed to further investigate and improve the baseline design obtained in the first stage with high-fidelity disciplinary models. CSSO is used to organize the concurrent disciplinary optimization and system coordination so as to allow disciplinary autonomy. To enhance the reliability and robustness of the design under uncertainties, the probabilistic version of MDF-CSSO (PMDF-CSSO) is developed to solve uncertainty-based optimization problems. The effectiveness of the proposed methods is verified with one MDO benchmark test and one practical satellite conceptual design optimization problem, followed by conclusion remarks and future research prospects.     

A robust optimization approach based on multi-fidelity metamodel

Multi-fidelity (MF) metamodeling approaches have recently attracted a significant amount of attention in simulation-based design optimization due to their ability to conduct trade-offs between high accuracy and low computational expenses by integrating the information from high-fidelity (HF) and low-fidelity (LF) models. While existing MF metamodel assisted design optimization approaches may yield an inferior or even infeasible solution since they generally treat the MF metamodel as the real HF model and ignore the interpolation uncertainties from the MF metamodel. This situation will be more serious in non-deterministic optimization. Hence, in this work, a MF metamodel assisted robust optimization approach is developed, in which the interpolation uncertainty of the MF metamodel and design variable uncertainty are quantified and taken into consideration. To demonstrate the effectiveness and merits of the proposed approach, two numerical examples and a long cylinder pressure vessel design optimization problem are tested. Results show that for the test cases the proposed approach can obtain a solution that is both optimal and within the feasible region even with perturbation of the uncertain variables.     

Reliability-based design optimization for crashworthiness of vehicle side impact

With the advent of powerful computers, vehicle safety issues have recently been addressed using computational methods of vehicle crashworthiness, resulting in reductions in cost and time for new vehicle development. Vehicle design demands multidisciplinary optimization coupled with a computational crashworthiness analysis. However, simulation-based optimization generates deterministic optimum designs, which are frequently pushed to the limits of design constraint boundaries, leaving little or no room for tolerances (uncertainty) in modeling, simulation uncertainties, and/or manufacturing imperfections. Consequently, deterministic optimum designs that are obtained without consideration of uncertainty may result in unreliable designs, indicating the need for Reliability-Based Design Optimization (RBDO).Recent development in RBDO allows evaluations of probabilistic constraints in two alternative ways: using the Reliability Index Approach (RIA) and the Performance Measure Approach (PMA). The PMA using the Hybrid Mean Value (HMV) method is shown to be robust and efficient in the RBDO process, whereas RIA yields instability for some problems. This paper presents an application of PMA and HMV for RBDO for the crashworthiness of a large-scale vehicle side impact. It is shown that the proposed RBDO approach is very effective in obtaining a reliability-based optimum design.     

A variable-size local domain approach for increased design confidence in simulation-based optimization

Simulation-based design optimization utilizes computational models that rely on assumptions and approximations. There is a need therefore, to ensure that the obtained designs will exhibit the desired behavior as anticipated given the model predictions. The common approach to accomplish that is to validate the utilized computational models prior to the design optimization process. However, this is practically an impossible task especially for design problems with high-dimensional design and parameter spaces. We have recently proposed a different approach for maximizing confidence in the designs generated during a sequential simulation-based optimization process based on calibrating the computational models when necessary and within local subdomains of the design space. In that work, the size of the local domains was held fixed and not linked to uncertainty, and the confidence in designs was quantified using Bayesian hypothesis testing. In this article, we present an improved methodology. Specifically, we use a statistical methodology to account for uncertainty and to determine the size of the local domains at each stage of the sequential design optimization process using parametric bootstrapping that involves maximum likelihood estimators of model parameters. The sequential process continues until the local domain does not change from stage to stage during the design optimization process, ensuring convergence to an optimal design. The proposed methodology is illustrated with the design of a thermal insulator using one-dimensional, linear heat conduction in a solid slab with heat flux boundary conditions.     

Hydroelastic optimization of a keel fin of a sailing boat: a multidisciplinary robust formulation for ship design

The paper presents a formulation for multidisciplinary design optimization of vessels, subject to uncertain operating conditions. The formulation couples the multidisciplinary design analysis with the Bayesian approach to decision problems affected by uncertainty. In the present context, the design specifications are no longer given in terms of a single operating design point, but in terms of probability density function of the operating scenario. The optimal configuration is that which maximizes the performance expectation over the uncertain parameters variation. In this sense, the optimal solution is “robust” within the stochastic scenario assumed. Theoretical and numerical issues are addressed and numerical results in the hydroelastic optimization of a keel fin of a sailing yacht are presented.     

Maximum variation analysis based analytical target cascading for multidisciplinary robust design optimization under interval uncertainty

Analytical target cascading (ATC) is a generally used hierarchical method for deterministic multidisciplinary design optimization (MDO). However, uncertainty is almost inevitable in the lifecycle of a complex system. In engineering practical design, the interval information of uncertainty can be more easily obtained compared to probability information. In this paper, a maximum variation analysis based ATC (MVA-ATC) approach is developed. In this approach, all subsystems are autonomously optimized under the interval uncertainty. MVA is used to establish an outer-inner framework which is employed to find the optimal scheme of system and subsystems. All subsystems are coordinated at the system level to search the system robust optimal solution. The accuracy and validation of the presented approach are tested using a classical mathematical example, a heart dipole optimization problem, and a battery thermal management system (BTMS) design problem.     

Multidisciplinary design of a small satellite launch vehicle using particle swarm optimization

In the present paper, particle swarm optimization, a relatively new population based optimization technique, is applied to optimize the multidisciplinary design of a solid propellant launch vehicle. Propulsion, structure, aerodynamic (geometry) and three-degree of freedom trajectory simulation disciplines are used in an appropriate combination and minimum launch weight is considered as an objective function. In order to reduce the high computational cost and improve the performance of particle swarm optimization, an enhancement technique called fitness inheritance is proposed. Firstly, the conducted experiments over a set of benchmark functions demonstrate that the proposed method can preserve the quality of solutions while decreasing the computational cost considerably. Then, a comparison of the proposed algorithm against the original version of particle swarm optimization, sequential quadratic programming, and method of centers carried out over multidisciplinary design optimization of the design problem. The obtained results show a very good performance of the enhancement technique to find the global optimum with considerable decrease in number of function evaluations.     

Multi-objective topology optimization through GA-based hybridization of partial solutions

In a recent project the authors have developed an approach to assist the identification of the optimal topology of a technical system, capable of overcoming geometrical contradictions that arise from conflicting design requirements. The method is based on the hybridization of partial solutions obtained from mono-objective topology optimization tasks. In order to investigate efficiency, effectiveness and potentialities of the developed hybridization algorithm, a comparison among the proposed approach and traditional topology optimization techniques such as Genetic Algorithms (GAs) and gradient-based methods is presented here. The benchmark has been performed by applying the hybridization algorithm to several case studies of multi-objective optimization problems available in literature. The obtained results demonstrate that the proposed approach is definitely less expensive in terms of computational requirements, than the conventional application of GAs to topology optimization tasks, still keeping the same effectiveness in terms of searching the global optimum solution. Moreover, the comparison among the hybridized solutions and the solutions obtained through GAs and gradient-based optimization methods, shows that the proposed algorithm often leads to very different topologies having better performances.     

Numerical investigation of non-hierarchical coordination for distributed multidisciplinary design optimization with fixed computational budget

This paper presents a numerical investigation of the non-hierarchical formulation of Analytical Target Cascading (ATC) for coordinating distributed multidisciplinary design optimization (MDO) problems. Since the computational cost of the analyses can be high and/or asymmetric, it is beneficial to understand the impact of the number of ATC iterations required for coordination and the number of iterations required for disciplinary feasibility on the quality of the obtained MDO solution. At each “outer” ATC iteration, the disciplinary optimization subproblems are solved for a predefined maximum number of “inner” loop iterations. The numerical experiments consider different numbers of maximum outer iterations while keeping the total computational budget of analyses constant. Solution quality is quantified by optimality (objective function value) and consistency (violation of coordination-related consistency constraints). Since MDO problems are typically simulation-based (and often blackbox) problems, we compare implementations of the mesh-adaptive direct search optimization algorithm (a derivative-free method with convergence properties) to the gradient-based interior-point algorithm implementation of the popular Matlab optimization toolbox. The impact of the values of two parameters involved in the alternating directions updating scheme of the augmented Lagrangian penalty functions (aka method of multipliers) on solution quality is also investigated. Numerical results are provided for a variety of MDO test problems. The results indicate consistently that a balanced modest number of outer and inner iterations is more effective; moreover, there seems to be a specific combination of parameter value ranges that yield better results.     

Model reduction for multidisciplinary optimization - application to a 2D wing

Multidisciplinary optimization (MDO) is a growing field in engineering, with various applications in aerospace, aeronautics, car industry, etc. However, the presence of multiple disciplines leads to specific issues, which prevent MDO to be fully integrated in industrial design methodology. In practice, the key issues in MDO lie in the management of the interconnections between disciplines, along with the high number of simulations required to find a feasible multidisciplinary (optimal) solution. Therefore, in this paper, a novel approach is proposed, combining proper orthogonal decomposition to decrease the amount of data exchanged between disciplines, with surrogate models based on moving least squares to reduce disciplines. This method is applied to an original 2D wing demonstrator involving two disciplines (fluid and structure). The numerical results obtained for an optimization task show its benefits in diminishing both the interfaces between disciplines and the overall computational time.     

Biochemical parameter estimation vs. benchmark functions: A comparative study of optimization performance and representation design

Computational Intelligence methods, which include Evolutionary Computation and Swarm Intelligence, can efficiently and effectively identify optimal solutions to complex optimization problems by exploiting the cooperative and competitive interplay among their individuals. The exploration and exploitation capabilities of these meta-heuristics are typically assessed by considering well-known suites of benchmark functions, specifically designed for numerical global optimization purposes. However, their performances could drastically change in the case of real-world optimization problems. In this paper, we investigate this issue by considering the Parameter Estimation (PE) of biochemical systems, a common computational problem in the field of Systems Biology. In order to evaluate the effectiveness of various meta-heuristics in solving the PE problem, we compare their performance by considering a set of benchmark functions and a set of synthetic biochemical models characterized by a search space with an increasing number of dimensions. Our results show that some state-of-the-art optimization methods – able to largely outperform the other meta-heuristics on benchmark functions – are characterized by considerably poor performances when applied to the PE problem. We also show that a limiting factor of these optimization methods concerns the representation of the solutions: indeed, by means of a simple semantic transformation, it is possible to turn these algorithms into competitive alternatives. We corroborate this finding by performing the PE of a model of metabolic pathways in red blood cells. Overall, in this work we state that classic benchmark functions cannot be fully representative of all the features that make real-world optimization problems hard to solve. This is the case, in particular, of the PE of biochemical systems. We also show that optimization problems must be carefully analyzed to select an appropriate representation, in order to actually obtain the performance promised by benchmark results.     

A GUI platform for uncertainty quantification of complex dynamical models

Uncertainty quantification (UQ) refers to quantitative characterization and reduction of uncertainties present in computer model simulations. It is widely used in engineering and geophysics fields to assess and predict the likelihood of various outcomes. This paper describes a UQ platform called UQ-PyL (Uncertainty Quantification Python Laboratory), a flexible software platform designed to quantify uncertainty of complex dynamical models. UQ-PyL integrates different kinds of UQ methods, including experimental design, statistical analysis, sensitivity analysis, surrogate modeling and parameter optimization. It is written in Python language and runs on all common operating systems. UQ-PyL has a graphical user interface that allows users to enter commands via pull-down menus. It is equipped with a model driver generator that allows any computer model to be linked with the software. We illustrate the different functions of UQ-PyL by applying it to the uncertainty analysis of the Sacramento Soil Moisture Accounting Model. We will also demonstrate that UQ-PyL can be applied to a wide range of applications.     

A hybrid memory-based dragonfly algorithm with differential evolution for engineering application

The dragonfly algorithm (DA) is a swarm-based stochastic algorithm which possesses static and dynamic behavior of swarm and is gaining meaningful popularity due to its low computational cost and fast convergence in solving complex optimization problems. However, it lacks internal memory and is thereby not able to keep track of its best solutions in previous generations. Furthermore, the solution also lacks in diversity and thereby has a propensity of getting trapped in the local optimal solution. In this paper, an iterative-level hybridization of dragonfly algorithm (DA) with differential evolution (DE) is proposed and named as hybrid memory-based dragonfly algorithm with differential evolution (DADE). The reason behind selecting DE is for its computational ability, fast convergence and capability in exploring the solution space through the use of crossover and mutation techniques. Unlike DA, in DADE the best solution in a particular iteration is stored in memory and proceeded with DE which enhances population diversity with improved mutation and accordingly increases the probability of reaching global optima efficiently. The efficiency of the proposed algorithm is measured based on its response to standard set of 74 benchmark functions including 23 standard mathematical benchmark functions, 6 composite benchmark function of CEC2005, 15 benchmark functions of CEC2015 and 30 benchmark function of CEC2017. The DADE algorithm is applied to engineering design problems such as welded beam deign, pressure vessel design, and tension/compression spring design. The algorithm is also applied to the emerging problem of secondary user throughput maximization in an energy-harvesting cognitive radio network. A comparative performance analysis between DADE and other most popular state-of-the-art optimization algorithms is carried out and significance of the results is deliberated. The result demonstrates significant improvement and prominent advantages of DADE compared to conventional DE, PSO and DA in terms of various performance measuring parameters. The results of the DADE algorithm applied on some important engineering design problems are encouraging and validate its appropriateness in the context of solving interesting practical engineering challenges. Lastly, the statistical analysis of the algorithm is also performed and is compared with other powerful optimization algorithms to establish its superiority.

     

Development and validation of a dynamic metamodel based on stochastic radial basis functions and uncertainty quantification

A dynamic radial basis function (DRBF) metamodel is derived and validated, based on stochastic RBF and uncertainty quantification (UQ). A metric for assessing metamodel efficiency is developed and used. The validation includes comparisons with a dynamic implementation of Kriging (DKG) and static metamodels for both deterministic test functions (with dimensionality ranging from two to six) and industrial UQ problems with analytical and numerical benchmarks, respectively. DRBF extends standard RBF using stochastic kernel functions defined by an uncertain tuning parameter whose distribution is arbitrary and whose effects on the prediction are determined using UQ methods. Auto-tuning based on curvature, adaptive sampling based on prediction uncertainty, parallel infill, and multiple response criteria are used. Industrial problems are two UQ applications in ship hydrodynamics using high-fidelity computational fluid dynamics for the high-speed Delft catamaran with stochastic operating and environmental conditions: (1) calm water resistance, sinkage and trim with variable Froude number; and (2) mean value and root mean square of resistance and heave and pitch motions with variable regular head wave. The number of high-fidelity evaluations required to achieve prescribed error levels is considered as the efficiency metric, focusing on fitting accuracy and UQ variables. DKG is found more efficient for fitting low-dimensional test functions and one-dimensional UQ, whereas DRBF has a greater efficiency for fitting higher-dimensional test functions and two-dimensional UQ.     

Comparison of MDO methods with mathematical examples

Recently, engineering systems are quite large and complicated. The design requirements are fairly complex and it is not easy to satisfy them by considering only one discipline. Therefore, a design methodology that can consider various disciplines is needed. Multidisciplinary design optimization (MDO) is an emerging optimization method that considers a design environment with multiple disciplines. Seven methods have been proposed for MDO. They are Multiple-discipline-feasible (MDF), Individual-discipline-feasible (IDF), All-at-once (AAO), Concurrent subspace optimization (CSSO), Collaborative optimization (CO), Bi-level integrated system synthesis (BLISS), and Multidisciplinary design optimization based on independent subspaces (MDOIS). Through several mathematical examples, the performances of the methods are evaluated and compared. Specific requirements are defined for comparison and new types of mathematical problems are defined based on the requirements. All the methods are coded and the performances of the methods are compared qualitatively and quantitatively.     

Shape Optimization by Free-Form Deformation: Existence Results and Numerical Solution for Stokes Flows

Shape optimization problems governed by PDEs result from many applications in computational fluid dynamics. These problems usually entail very large computational costs and require also a suitable approach for representing and deforming efficiently the shape of the underlying geometry, as well as for computing the shape gradient of the cost functional to be minimized. Several approaches based on the displacement of a set of control points have been developed in the last decades, such as the so-called free-form deformations. In this paper we present a new theoretical result which allows to recast free-form deformations into the general class of perturbation of identity maps, and to guarantee the compactness of the set of admissible shapes. Moreover, we address both a general optimization framework based on the continuous shape gradient and a numerical procedure for solving efficiently three-dimensional optimal design problems. This framework is applied to the optimal design of immersed bodies in Stokes flows, for which we consider the numerical solution of a benchmark case study from literature.     

Sensitivity analysis of optimum solutions by updating active constraints: application in aircraft structural design

Often the parameters considered as constants in an optimization problem have some uncertainty and it is interesting to know how the optimum solution is modified when these values are changed. The only way to continue having the optimal solution is to perform a new optimization loop, but this may require a high computational effort if the optimization problem is large. However, there are several procedures to obtain the new optimal design, based on getting the sensitivities of design variables and objective function with respect to a fixed parameter. Most of these methods require obtaining second derivatives which has a significant computational cost. This paper uses the feasible direction-based technique updating the active constraints to obtain the approximate optimum design. This procedure only requires the first derivatives and it is noted that the updating set of active constraints improves the result, making possible a greater fixed parameter variation. This methodology is applied to an example of very common structural optimization problems in technical literature and to a real aircraft structure.     

Uncertainty quantification of MEMS using a data-dependent adaptive stochastic collocation method

This paper presents a unified framework for uncertainty quantification (UQ) in microelectromechanical systems (MEMS). The goal is to model uncertainties in the input parameters of micromechanical devices and to quantify their effect on the final performance of the device. We consider different electromechanical actuators that operate using a combination of electrostatic and electrothermal modes of actuation, for which high-fidelity numerical models have been developed. We use a data-driven framework to generate stochastic models based on experimentally observed uncertainties in geometric and material parameters. Since we are primarily interested in quantifying the statistics of the output parameters of interest, we develop an adaptive refinement strategy to efficiently propagate the uncertainty through the device model, in order to obtain quantities like the mean and the variance of the stochastic solution with minimal computational effort. We demonstrate the efficacy of this framework by performing UQ in some examples of electrostatic and electrothermomechanical microactuators. We also validate the method by comparing our results with experimentally determined uncertainties in an electrostatic microswitch. We show how our framework results in the accurate computation of uncertainties in micromechanical systems with lower computational effort.