A Comparative study on multiobjective reliable and robust optimization for crashworthiness design of vehicle structure

Design optimization without considering uncertainties of system variables and parameters can be problematic in real life. In order to take into account the effect of uncertainties, reliable and robust design schemes have proven effective, but limited studies have been reported to compare their difference in a multiobjective framework. This paper takes a typical vehicle structure subject to offset frontal crashing scenario as an example to compare reliable and robust designs with their deterministic counterpart. The thicknesses of some key components of vehicle frontal structures were selected as design variables, the vehicle weight and energy absorption as the objectives, deceleration and firewall intrusion as the constraints. The deterministic multiobjective optimization problem was first solved by adopting Design of Experimental (DOE), metamodels and Non-dominated Sorting Genetic Algorithm II (NSGA-II). Take into account the uncertainties, a Monte Carlo Simulation (MCS) is adopted to generate random distributions of the objective and constraint functions for each design. For the reliability-based optimization the desired reliabilities of 90 %, 95 % and 99 % are considered, respectively. For the robustness-based optimization, two different formulation strategies are adopted. The optimization showed that the reliable and robust Pareto fronts are shifted away from their deterministic counterpart due to uncertainties. The different Pareto fronts yielded from the deterministic, reliable and robust designs are compared to provide some quantitative insights into how to apply these different design schemes for resolving uncertainty problems. It is shown that, compared with the baseline design, the optimizations enhance the crashworthiness of vehicle, though more conservative solutions could have been generated from the reliable and robust optimizations.     

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.     

On design optimization for structural crashworthiness and its state of the art

Optimization for structural crashworthiness and energy absorption has become an important topic of research attributable to its proven benefits to public safety and social economy. This paper provides a comprehensive review of the important studies on design optimization for structural crashworthiness and energy absorption. First, the design criteria used in crashworthiness and energy absorption are reviewed and the surrogate modeling to evaluate these criteria is discussed. Second, multiobjective optimization, optimization under uncertainties and topology optimization are reviewed from concepts, algorithms to applications in relation to crashworthiness. Third, the crashworthy structures are summarized, from generically novel structural configurations to industrial applications. Finally, some conclusions and recommendations are provided to enable academia and industry to become more aware of the available capabilities and recent developments in design optimization for structural crashworthiness and energy absorption.

     

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.     

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.     

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.     

Reliability-based multiobjective optimization for automotive crashworthiness and occupant safety

This paper presents a methodology for reliability-based multiobjective optimization of large-scale engineering systems. This methodology is applied to the vehicle crashworthiness design optimization for side impact, considering both structural crashworthiness and occupant safety, with structural weight and front door velocity under side impact as objectives. Uncertainty quantification is performed using two first order reliability method-based techniques: approximate moment approach and reliability index approach. Genetic algorithm-based multiobjective optimization software GDOT, developed in-house, is used to come up with an optimal pareto front in all cases. The technique employed in this study treats multiple objective functions separately without combining them in any form. It shows that the vehicle weight can be reduced significantly from the baseline design and at the same time reduce the door velocity. The obtained pareto front brings out useful inferences about optimal design regions. A decision-making criterion is subsequently invoked to select the “best” subset of solutions from the obtained nondominated pareto optimal solutions. The reliability, thus computed, is also checked with Monte Carlo simulations. The optimal solution indicated by knee point on the optimal pareto front is verified with LS-DYNA simulation results.     

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.     

Crashworthiness design for functionally graded foam-filled bumper beam

Automotive bumper beam is an important component to protect passenger and vehicle from injury and damage induced by severe collapse. Recent studies showed that foam-filled structures have significant advantages in light weight and high energy absorption. In this paper, a novel bumper beam filled with functionally graded foam (FGF) is considered here to explore its crashworthiness. To validate the FGF bumper beam model, the experiments at both component and full vehicle levels are conducted. Parametric study shows that gradient exponential parameter m that controls the variation of foam density has significant effect on bumper beam’s crashworthiness; and the crashworthiness of FGF-filled bumper beam is found much better than that of uniform foam (UF) filled and hollow bumper beam. The multiobjective optimization of FGF-filled bumper beam is also performed by considering specific energy absorption (SEA) and peak impact force as the design objectives, and the wall thickness t, foam densities ρ_f1 and ρ_f2 (foam densities at the end and at mid cross section, respectively) and gradient exponential parameter m as design variables. The Kriging surrogate modeling technique and multiobjective particle swarm optimization (MOPSO) algorithm were implemented to optimize the FGF-filled bumper beam. The optimized FGF-filled bumper beam is of great advantages and it can avoid the harmful local bending behavior and absorb more energy than UF filled and hollow bumper beam. Finally, the optimized FGF-filled bumper beam is installed to a passenger car model, and the results demonstrate that the FGF-filled bumper beam ensures the crashworthiness performance of the passenger car while reduces weight about 14.4% compared with baseline bumper beam.     

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.     

Crashworthiness design of multi-component tailor-welded blank (TWB) structures

Crashworthiness of tailor-welded blank (TWB) structures signifies an increasing concern in lightweight design of vehicle. Although multiobjective optimization (MOO) has to a considerable extent been successfully applied to enhance crashworthiness of vehicular structures, majority of existing designs were restricted to single or uniform thin-walled components. Limited attention has been paid to such non-uniform components as TWB structures. In this paper, MOO of a multi-component TWB structure that involves both the B-pillar and inner door system subjected to a side impact, is proposed by considering the structural weight, intrusive displacements and velocity of the B-pillar component as objectives, and the thickness in different positions and the height of welding line of B-pillar as the design variables. The MOO problem is formulated by using a range of different metamodeling techniques, including response surface methodology (RSM), artificial neural network (ANN), radial basis functions (RBF), and Kriging (KRG), to approximate the sophisticated nonlinear responses. By comparison, it is found that the constructed metamodels based upon the radial basis function (RBF, especially multi-quadric model, namely RBF-MQ) fit to the design of experiment (DoE) checking points well and are employed to carry out the design optimization. The performance of the TWB B-pillar and indoor panel system can be improved by optimizing the thickness of the different parts and height of the welding line. This study demonstrated that the multi-component TWB structure can be optimized to further enhance the crashworthiness and reduce the weight, offering a new class of structural/material configuration for lightweight design.     

Structural optimization with crashworthiness constraints

An automated structural design methodology has been devised which simultaneously considers design criteria associated with both linear elastic and crashworthiness loading conditions. This method is developed within the context of a nonlinear mathematical programming based structural optimization capability using an efficient two-phased crashworthiness analysis technique. Specially constructed nonlinear approximations for the crashworthiness constraints are employed to further reduce the computational burden during the optimization process. This methodology is demonstrated on an automobile structural design problem. It is shown that more mass efficient designs can be obtained by simultaneously considering elastic and crashworthiness design criteria as compared to a sequential approach in which the structure is first designed for the elastic loads and then modified to satisfy the crashworthiness criteria.     

Robust optimization of foam-filled thin-walled structure based on sequential Kriging metamodel

Deterministic optimization has been successfully applied to a range of design problems involving foam-filled thin-walled structures, and to some extent gained significant confidence for the applications of such structures in automotive, aerospace, transportation and defense industries. However, the conventional deterministic design could become less meaningful or even unacceptable when considering the perturbations of design variables and noises of system parameters. To overcome this drawback, a robust design methodology is presented in this paper to address the effects of parametric uncertainties of foam-filled thin-walled structure on design optimization, in which different sigma criteria are adopted to measure the variations. The Kriging modeling technique is used to construct the corresponding surrogate models of mean and standard deviation for different crashworthiness criteria. A sequential sampling approach is introduced to improve the fitness accuracy of these surrogate models. Finally, a gradient-based sequential quadratic program (SQP) method is employed from 20 different initial points to obtain a quasi-global robust optimum solution. The optimal solutions were verified by using the Monte Carlo simulation. The results show that the presented robust optimization method is fairly effective and efficient, the crashworthiness and robustness of the foam-filled thin-walled structure can be improved significantly.     

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.     

Multivariable crashworthiness optimization of vehicle body by unreplicated saturated factorial design

Crashworthiness, influenced unequally by disparate factors such as the structural dimensions and the material parameters, represents a natural benchmark criterion to judge the passive safety quality of the automobile design. The unreplicated saturated factorial design has enjoyed a remarkable success in the factor screening of different industrial regions due to its huge benefits in the efficiency and accuracy. In order to single out the active factors which pose a profound impact on the crashworthiness, this paper introduces an unreplicated saturated factorial design to tackle the obstacle from the factor screening during the multivariable crashworthiness optimization design of the whole vehicle body. Three unreplicated saturated factorial design methods, including the normal or half-normal probability plot method, Dong93 method, and PSZ method, are employed to capture the active factors while D-optimal design is presented to obtain the design sampling points and to construct the response surface model for the crashworthiness optimization problem. Finally, multi-island genetic algorithm (MIGA) and sequential quadratic programming (SQP)-NLPQL are utilized to obtain the Pareto set of the optimal solution for the multivariable crashworthiness optimization design of the vehicle body under the full-scale frontal impact loading.     

Optimization design of corrugated beam guardrail based on RBF-MQ surrogate model and collision safety consideration

As the main safety facility on the highway, a guardrail system is very essential for the highway traffics safety. In this paper, the Finite Element (FE) models of the vehicle and the corrugated beam guardrail system were created. Two types of widely used corrugated beam semi-rigid guardrails were simulated, which were the W-beam guardrail and the Thrie-beam guardrail. The collision between the corrugated beam guardrail systems and the vehicle body was analyzed. In the collision process, the snagging effect of the post to the vehicle body was also concerned. Under the considerations of the collision safety and the mechanism of the snagging effect, the multiobjective optimization problem was defined with dimensional sizes of guardrails to be the design variables. And the radial basis function (RBF) was applied to construct the regression models of the analytical objective, which increased the accuracy of fitting. The Pareto set and the optimal solution were obtained. After the optimization design, the W-beam guardrail and Thrie-beam guardrail were both greatly improved, that increased the collision safety between the corrugated beam guardrail and the vehicle body. This kind of analytical method can also be used for the crashworthiness optimization between any other cars and guardrails.     

A Novel Intelligent Multiobjective Simulated Annealing Algorithm for Designing Robust PID Controllers

This paper proposes an intelligent multiobjective simulated annealing algorithm (IMOSA) and its application to an optimal proportional integral derivative (PID) controller design problem. A well-designed PID-type controller should satisfy the following objectives: 1) disturbance attenuation; 2) robust stability; and 3) accurate setpoint tracking. The optimal PID controller design problem is a large-scale multiobjective optimization problem characterized by the following: 1) nonlinear multimodal search space; 2) large-scale search space; 3) three tight constraints; 4) multiple objectives; and 5) expensive objective function evaluations. In contrast to existing multiobjective algorithms of simulated annealing, the high performance in IMOSA arises mainly from a novel multiobjective generation mechanism using a Pareto-based scoring function without using heuristics. The multiobjective generation mechanism operates on a high-score nondominated solution using a systematic reasoning method based on an orthogonal experimental design, which exploits its neighborhood to economically generate a set of well-distributed nondominated solutions by considering individual and overall objectives. IMOSA is evaluated by using a practical design example of a super-maneuverable fighter aircraft system. An efficient existing multiobjective algorithm, the improved strength Pareto evolutionary algorithm, is also applied to the same example for comparison. Simulation results demonstrate high performance of the IMOSA-based method in designing robust PID controllers.     

Control-structure integrated multiobjective design for flexible spacecraft using MOEA/D

This paper proposes a multiobjective optimization method for the control-structure integrated design of flexible spacecraft to reduce the total mass and optimize the control performance. The equations of motion for flexible spacecraft are derived from the Lagrange’s principle and the assumed modes method. The design variables are the structural dimensions of the flexible structure and controller parameters. The objectives and constraints are derived from structure and control performance indexes. The objectives include total mass, control cost, and vibrational energy, and the constraints include the stability of the closed-loop system, settling time, overshoot, maximum control, and maximum vibrational displacement of the tip. A modified version of the multiobjective evolutionary algorithm based on decomposition (MOEA/D) with our proposed hybrid constraint handling method is proposed for optimization. As a case study, it has been applied to a spacecraft with symmetrically installed flexible appendages to find optimal tradeoffs in control-structure design. The simulation results show that the multiobjective optimization method for the control-structure integrated design of flexible spacecraft is feasible and effective, and could give an improvement of structural and control designs.     

Multi-objective robust design optimization of a novel NPR energy absorption structure for vehicles front ends to enhance pedestrian lower leg protection

The unique structure of negative Poisson’s ratio (NPR) materials provides a good energy absorption capacity that has found wide application in automotive and aeronautics industries. The present work proposes a novel NPR energy absorption structure installed between the bumper and anti-collision beam of automobiles for improving pedestrian lower leg protection. The performance of the proposed NPR structure based on tibia acceleration, knee bending angle, and knee shear displacement is evaluated by comparison with the performance of a conventional energy absorption structure. The performance of the proposed structure is further improved by conducting multi-objective design optimization with consideration for the perturbation induced by parameter uncertainties. For optimization, a parametric model of the NPR structure is first established based on the relationships between parameters. To promote computational efficiency, an optimal Latin hypercube sampling technique and the dual response surface method are combined to build surrogate models between responses and inputs. A multi-objective particle swarm optimization algorithm and six sigma criteria are then applied to obtain the optimal design parameters of the structure. The optimization results are validated by comparisons with the results of multi-objective deterministic optimization. The comparison results show that the proposed NPR energy absorption structure improves pedestrian lower leg protection significantly, and its performance is further improved by the proposed multi-objective robust design optimization. This study serves as a good example of the safety promotion provided by applying NPR structures with enhanced energy absorption capacity in vehicles.