Reliability-based robust assessment for multiobjective optimization design of improving occupant restraint system performance

Optimal performance of vehicle occupant restraint system (ORS) requires an accurate assessment of occupant injury values including head, neck and chest responses, etc. To provide a feasible framework for incorporating occupant injury characteristics into the ORS design schemes, this paper presents a reliability-based robust approach for the development of the ORS. The uncertainties of design variables are addressed and the general formulations of reliable and robust design are given in the optimization process. The ORS optimization is a highly nonlinear and large scale problem. In order to save the computational cost, an optimal sampling strategy is applied to generate sample points at the stage of design of experiment (DOE). Further, to efficiently obtain a robust approximation, the support vector regression (SVR) is suggested to construct the surrogate model in the vehicle ORS design process. The multiobjective particle swarm optimization (MPSO) algorithm is used for obtaining the Pareto optimal set with emphasis on resolving conflicting requirements from some of the objectives and the Monte Carlo simulation (MCS) method is applied to perform the reliability and robustness analysis. The differences of three different Pareto fronts of the deterministic, reliable and robust multiobjective optimization designs are compared and analyzed in this study. Finally, the reliability-based robust optimization result is verified by using sled system test. The result shows that the proposed reliability-based robust optimization design is efficient in solving ORS design optimization problems.     

Crashworthiness design of vehicle by using multiobjective robust optimization

Although deterministic optimization has to a considerable extent been successfully applied in various crashworthiness designs to improve passenger safety and reduce vehicle cost, the design could become less meaningful or even unacceptable when considering the perturbations of design variables and noises of system parameters. To overcome this drawback, we present a multiobjective robust optimization methodology to address the effects of parametric uncertainties on multiple crashworthiness criteria, where several different sigma criteria are adopted to measure the variations. As an example, a full front impact of vehicle is considered with increase in energy absorption and reduction of structural weight as the design objectives, and peak deceleration as the constraint. A multiobjective particle swarm optimization is applied to generate robust Pareto solution, which no longer requires formulating a single cost function by using weighting factors or other means. From the example, a clear compromise between the Pareto deterministic and robust designs can be observed. The results demonstrate the advantages of using multiobjective robust optimization, with not only the increase in the energy absorption and decrease in structural weight from a baseline design, but also a significant improvement in the robustness of optimum.     

Multiobjective robust design optimization of rail vehicle moving in short radius curved tracks based on the safety and comfort criteria

This work deals with the multiobjective robust design optimization of rail vehicle systems moving in short radius curved tracks. Two criteria are considered simultaneously, i.e., safety (considered by the derailment risk) and comfort given by noise level. The authors show that the deterministic optimal solutions, for the nominal design parameters, can be altered seriously by the design parameters uncertainty. The authors of this paper propose an original algorithm that combines Genetic Algorithms and Monte Carlo Simulation in order to be used for the robust multiobjective optimization of the rail vehicle design. The obtained solutions, presented by design vectors of the rail vehicle, are analyzed in terms of performances and robustness. The authors show that the robust multiobjective optimization can yield solutions less sensitive to the design parameters uncertainties.     

Robust Pareto active suspension design for vehicle vibration model with probabilistic uncertain parameters

Using the robust design of a vehicle vibration model considering uncertainties can elaborately show the effects of those unsure values on the performance of such a model. In this paper, probabilistic metrics, instead of deterministic metrics, are used for a robust Pareto multi-objective optimum design of five-degree of freedom vehicle vibration model having parameters with probabilistic uncertainties. In order to achieve an optimum robust design against probabilistic uncertainties existing in reality, a multi-objective uniform-diversity genetic algorithm (MUGA) in conjunction with Monte Carlo simulation is used for Pareto optimum robust design of a vehicle vibration model with ten conflicting objective functions. The robustness of the design obtained using such a probabilistic approach is shown and compared with that of the design obtained using deterministic approach.     

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.     

Recent advances in rapid multiobjective optimization of expensive simulation models in microwave and antenna engineering by Pareto front exploration

Practical engineering design problems are inherently multiobjective, that is, require simultaneous control of several (and often conflicting) criteria. In many situations, genuine multiobjective optimization is required to acquire comprehensive information about the system of interest. The most popular solution techniques are population‐based metaheuristics, however, they are not practical for handling expensive electromagnetic (EM)‐simulation models in microwave and antenna engineering. A workaround is to use auxiliary response surface approximation surrogates but it is challenging for higher‐dimensional problems. Recently, a deterministic approach has been proposed for expedited multiobjective design optimization of expensive models in computational EMs. The method relies on variable‐fidelity EM simulations, tracking the Pareto front geometry, as well as response correction. The algorithm sequentially generates Pareto‐optimal designs using a series of constrained single‐objective optimizations. The previously obtained design is used as a starting point for the next iteration. In this work, we review this technique and its modification based on space mapping surrogates. We also propose new variations exploiting adjoint sensitivities, as well as response features, which can be attractive depending on availability of derivatives or the characteristics of the system responses that need to be handled. We also discuss several case studies involving various antenna and microwave components.     

Multi-objective design optimization using cascade evolutionary computations

The consideration of uncertainties in conjunction with the probability of violation of the constraints imposed by the design codes is examined in the framework of structural optimization. The optimum design achieved based on a deterministic formulation is compared, in terms of the optimum weight, the probability of violation of the constraints and the probability of failure, with the optimum designs achieved through a robust design formulation where the variance of the response is considered as an additional criterion. The stochastic finite element problem is solved using the Monte Carlo Simulation method, combined with the Latin Hypercube Sampling technique for improving its computational efficiency. A non-dominant cascade evolutionary algorithm-based methodology is adopted for the solution of the multi-objective optimization problem encountered, in order to obtain the global Pareto front curve.     

An efficient procedure for structural reliability-based robust design optimization

This paper focuses on the development of an optimization tool with the aim to obtain robust and reliable designs in short computational time. The robustness measures considered here are the expected value and standard deviation of the performance function involved in the optimization problem. When using these robustness measures combined, the search of optimal design appears as a robust multiobjective optimization (RMO) problem. Reliable design addresses uncertainties to restrict the structural probability of failure. The mathematical formulation for the reliability based robust design optimization (RBRDO) problem is obtained by adding a reliability based constraint into the RMO problem. As both, statistics calculations and the reliability analysis could be very costly, approximation technique based on reduced-order modeling (ROM) is also incorporated in our procedure. The selected ROM is the proper orthogonal decomposition (POD) method, with the aim to produce fast outputs considering structural non-linear behavior. Moreover, to obtain RBRDO designs with reduced CPU time we propose others developments to be added in the integrated tool. They are: Probabilistic Collocation Method (PCM) to evaluate the statistics of the structural responses and, also, an approximated reliability constraints procedure based on the Performance Measure Approach (PMA) for reliability constraint assessment. Finally, Normal-Boundary Intersection (NBI) or Normalized Normal-Constraint (NNC) multiobjective optimization techniques are employed to obtain fast and even spread Pareto robust designs. To illustrate the application of the proposed tool, optimization studies are conducted for a linear (benchmark) and nonlinear trusses problems. The nonlinear example consider different loads level, exploring the material plasticity. The integrated tool prove to be very effective reducing the computational time by up to five orders of magnitude, when compared to the solutions obtained via classical standard approaches.     

Multiobjective firefly algorithm for continuous optimization

Design problems in industrial engineering often involve a large number of design variables with multiple objectives, under complex nonlinear constraints. The algorithms for multiobjective problems can be significantly different from the methods for single objective optimization. To find the Pareto front and non-dominated set for a nonlinear multiobjective optimization problem may require significant computing effort, even for seemingly simple problems. Metaheuristic algorithms start to show their advantages in dealing with multiobjective optimization. In this paper, we extend the recently developed firefly algorithm to solve multiobjective optimization problems. We validate the proposed approach using a selected subset of test functions and then apply it to solve design optimization benchmarks. We will discuss our results and provide topics for further research.     

LOcal Uncertainty Processing (LOUP) method for multidisciplinary robust design optimization

In this paper, we develop an easy-to-implement approximate method to take uncertainties into account during a multidisciplinary optimization. Multidisciplinary robust design usually involves setting up a full uncertainty propagation within the system, requiring major modifications in every discipline and on the shared variables. Uncertainty propagation is an expensive process, but robust solutions can be obtained more easily when the disciplines affected by uncertainties have a significant effect on the objectives of the problem. A heuristic method based on local uncertainty processing (LOUP) is presented here, allowing approximate solving of specific robust optimization problems with minor changes in the initial multidisciplinary system. Uncertainty is processed within the disciplines that it impacts directly, without propagation to the other disciplines. A criterion to verify a posteriori the applicability of the method to a given multidisciplinary system is provided. The LOUP method is applied to an aircraft preliminary design industrial test case, in which it allowed to obtain robust designs whose performance is more stable than the one of deterministic solutions, relatively to uncertain parameter variations.     

Robust topology optimization of frame structures under geometric or material properties uncertainties

A robust topology optimization algorithm is proposed for frame structures in the presence of geometric or material properties uncertainties. While geometric uncertainties were modeled with uncorrelated random variables expressing the node locations of the structure, material properties uncertainties were modeled with a correlated random field of the material Young’s modulus with an exponentially decaying correlation structure throughout the domain. The proposed algorithm uses stochastic perturbation method for propagating these uncertainties to the structural response level, measured in terms of compliance, and optimizes the expected value plus multiple factors of the standard deviation of the response. A comparison between the resulting robust designs and deterministic designs is made, and changes to the final topologies are discussed. Moreover, using Monte Carlo simulation, it was shown that the robust designs outperform the deterministic designs under real-world situations that are accompanied with uncertainties.     

Reliability-based structural optimization of frame structures for multiple failure criteria using topology optimization techniques

Topology optimization methods using discrete elements such as frame elements can provide useful insights into the underlying mechanics principles of products; however, the majority of such optimizations are performed under deterministic conditions. To avoid performance reductions due to later-stage environmental changes, variations of several design parameters are considered during the topology optimization. This paper concerns a reliability-based topology optimization method for frame structures that considers uncertainties in applied loads and nonstructural mass at the early conceptual design stage. The effects that multiple criteria, namely, stiffness and eigenfrequency, have upon system reliability are evaluated by regarding them as a series system, where mode reliabilities can be evaluated using first-order reliability methods. Through numerical calculations, reliability-based topology designs of typical two- or three-dimensional frames are obtained. The importance of considering uncertainties is then demonstrated by comparing the results obtained by the proposed method with deterministic optimal designs.     

A design optimization strategy for creating devices that traverse the Pareto frontier over time

In some instances, the performance or function that is needed by a product naturally and predictably changes over time. Providing solutions that anticipate, account for, and allow for these changes is a significant challenge to manufacturers and design engineers. In this paper, a multiobjective optimization design method involving the strategic use of a series of optimization formulations is introduced to design products that adapt to changing needs by moving from one location on the Pareto frontier to another through the addition of a module. The design of a simple unmanned air vehicle is used to demonstrate implementation of the method, and results in the development of one aircraft platform and two module designs that adapt the aircraft to perform optimally for the particular mission at hand, thus optimally satisfying all three different mission profiles individually. The authors conclude that the developed method provides a new and general framework for selecting platform and module designs, and is capable of providing a set of designs based on predicted changes in needs.     

Reliability-based robust Pareto design of linear state feedback controllers using a multi-objective uniform-diversity genetic algorithm (MUGA)

In this paper, fuzzy threshold values, instead of crisp threshold values, have been used for optimal reliability-based multi-objective Pareto design of robust state feedback controllers for a single inverted pendulum having parameters with probabilistic uncertainties. The objective functions that have been considered are, namely, the normalized summation of rising time and overshoot of cart (S_RO–C) and the normalized summation of rising time and overshoot of pendulum (S_RO–P) in the deterministic approach. Accordingly, the probabilities of failure of those objective functions are also considered in the reliability-based design optimization (RBDO) approach. A new multi-objective uniform-diversity genetic algorithm (MUGA) is presented and used for Pareto optimum design of linear state feedback controllers for single inverted pendulum problem. In this way, Pareto front of optimum controllers is first obtained for the nominal deterministic single inverted pendulum using the conflicting objective functions in time domain. Such Pareto front is then obtained for single inverted pendulum having probabilistic uncertainties in its parameters using the statistical moments of those objective functions through a Monte Carlo simulation (MCS) approach. It is shown that multi-objective reliability-based Pareto optimization of the robust state feedback controllers using MUGA with fuzzy threshold values includes those that may be obtained by various crisp threshold values of probability of failures and, thus, remove the difficulty of selecting suitable crisp values. Besides, the multi-objective Pareto optimization of such robust feedback controllers using MUGA unveils some very important and informative trade-offs among those objective functions. Consequently, some optimum robust state feedback controllers can be compromisingly chosen from the Pareto frontiers.     

Multiobjective optimization for crash safety design of vehicles using stepwise regression model

In automotive industry, structural optimization for crashworthiness criteria is of special importance. Due to the high nonlinearities, however, there exists substantial difficulty to obtain accurate continuum or discrete sensitivities. For this reason, metamodel or surrogate model methods have been extensively employed in vehicle design with industry interest. This paper presents a multiobjective optimization procedure for the vehicle design, where the weight, acceleration characteristics and toe-board intrusion are considered as the design objectives. The response surface method with linear and quadratic basis functions is employed to formulate these objectives, in which optimal Latin hypercube sampling and stepwise regression techniques are implemented. In this study, a nondominated sorting genetic algorithm is employed to search for Pareto solution to a full-scale vehicle design problem that undergoes both the full frontal and 40% offset-frontal crashes. The results demonstrate the capability and potential of this procedure in solving the crashworthiness design of vehicles.     

A new graphical visualization of n-dimensional Pareto front for decision-making in multiobjective optimization

New challenges in engineering design lead to multiobjective (multicriteria) problems. In this context, the Pareto front supplies a set of solutions where the designer (decision-maker) has to look for the best choice according to his preferences. Visualization techniques often play a key role in helping decision-makers, but they have important restrictions for more than two-dimensional Pareto fronts. In this work, a new graphical representation, called Level Diagrams, for n-dimensional Pareto front analysis is proposed. Level Diagrams consists of representing each objective and design parameter on separate diagrams. This new technique is based on two key points: classification of Pareto front points according to their proximity to ideal points measured with a specific norm of normalized objectives (several norms can be used); and synchronization of objective and parameter diagrams. Some of the new possibilities for analyzing Pareto fronts are shown. Additionally, in order to introduce designer preferences, Level Diagrams can be coloured, so establishing a visual representation of preferences that can help the decision-maker. Finally, an example of a robust control design is presented - a benchmark proposed at the American Control Conference. This design is set as a six-dimensional multiobjective problem.     

Pareto-based negotiation in distributed multidisciplinary design

The process of distributed engineering design calls for a methodology making use of the most recent advances in optimization-based design including multidisciplinary and multiobjective optimization. In distributed design, the participating teams do not have access to the design problems of other teams but may exchange limited information about their own current designs, making negotiation among themselves a key mechanism to reach a desired compromise which, nevertheless, is also a Pareto design to the original problem. A mathematical model of this distributed but decomposable design process is posed and solved using Lagrangian relaxation, while Pareto optimality is equivalently converted to single-objective optimality by means of multicriteria decision making strategies. The proposed coordination algorithm allows negotiation among the teams (subproblems) by sharing only limited information that is restricted to values of optimization quantities. The proposed modeling and solution scheme is applied to a numerical example representing the design of vehicle subsystems and components.     

Dynamic Multiple Swarms in Multiobjective Particle Swarm Optimization

A multiple-swarm multiobjective particle swarm optimization (PSO) algorithm, named dynamic multiple swarms in multiobjective PSO, is proposed in which the number of swarms is adaptively adjusted throughout the search process via the proposed dynamic swarm strategy. The strategy allocates an appropriate number of swarms as required to support convergence and diversity criteria among the swarms. Additional novel designs include a PSO updating mechanism to better manage the communication within a swarm and among swarms and an objective space compression and expansion strategy to progressively exploit the objective space during the search process. Comparative study shows that the performance of the proposed algorithm is competitive in comparison to the selected algorithms on standard benchmark problems. In particular, when dealing with test problems with multiple local Pareto fronts, the proposed algorithm is much less computationally demanding. Sensitivity analysis indicates that the proposed algorithm is insensitive to most of the user-specified design parameters.     

Accelerated multiobjective design of miniaturized microwave components by means of nested kriging surrogates

Design of microwave components is an inherently multiobjective task. Often, the objectives are at least partially conflicting and the designer has to work out a suitable compromise. In practice, generating the best possible trade‐off designs requires multiobjective optimization, which is a computationally demanding task. If the structure of interest is evaluated through full‐wave electromagnetic (EM) analysis, the employment of widely used population‐based metaheuristics algorithms may become prohibitive in computational terms. This is a common situation for miniaturized components, where considerable cross‐coupling effects make traditional representations (eg, network equivalents) grossly inaccurate. This article presents a framework for accelerated EM‐driven multiobjective design of compact microwave devices. It adopts a recently reported nested kriging methodology to identify the parameter space region containing the Pareto front and to render a fast surrogate, subsequently used to find the first approximation of the Pareto set. The final trade‐off designs are produced in a separate, surrogate‐assisted refinement process. Our approach is demonstrated using a three‐section impedance matching transformer designed for the best matching and the minimum footprint area. The Pareto set is generated at the cost of only a few hundred of high‐fidelity EM simulations of the transformer circuit despite a large number of geometry parameters involved.     

Robust optimization using a gradient index: MEMS applications

This paper discusses a simple and effective robust optimization formulation and illustrates its application to MicroElectroMechanical Systems (MEMS) devices. The proposed formulation improves robustness of the objective function by minimizing a gradient index (GI), defined as a function of gradients of performance functions with respect to uncertain variables. The level of constraint feasibility is also enhanced by adding a term determined by a constraint value and the gradient index. In the robust optimal design procedure, a deterministic optimization for performance improvement is followed by a sensitivity analysis with respect to uncertainties such as MEMS fabrication errors and changes of material properties. During the process of the deterministic optimization and sensitivity analysis, dominant performances and critical uncertain variables are identified to define the GI. Our approach for robust design requires no statistical information on the uncertainties and yet achieves robustness effectively. Two MEMS application examples including a micro accelerometer and a resonant-type micro probe are presented.