Factor screening and multivariable crashworthiness optimization for vehicle side impact by factorial design

This paper demonstrates the application of factor screening to multivariable crashworthiness design of the vehicle body subjected to the side impact loading. Crashworthiness, influenced unequally by disparate factors such as the structural dimensions and material parameters, represents a natural benchmark criterion to judge the passive safety quality of the automobile design. In order to single out the active factors which pose a profound influence on the crashworthiness of vehicle bodies subjected to the side impact loading, the unreplicated saturated factorial design is adopted to tackle the obstacle from the factor screening due to its huge benefits in the efficiency and accuracy. In this paper, two different kinds of vehicles are analyzed by the unreplicated saturated factorial design for multivariable crashworthiness and the optimization results enhance the crashworthiness of vehicle. This method overcomes the limitations of design variables selection which depends on experience, and solves the in-efficiency problems caused by the direct optimization design without the selection of variables. It will shorten the design cycles, decrease the development costs and will have a certain reference value for the improvement of the vehicle’s crashworthiness performance.     

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.     

Reliability-based design optimization using probabilistic sufficiency factor

A probabilistic sufficiency factor approach is proposed that combines safety factor and probability of failure. The probabilistic sufficiency factor approach represents a factor of safety relative to a target probability of failure. It provides a measure of safety that can be used more readily than the probability of failure or the safety index by designers to estimate the required weight increase to reach a target safety level. The probabilistic sufficiency factor can be calculated from the results of Monte Carlo simulation with little extra computation. The paper presents the use of probabilistic sufficiency factor with a design response surface approximation, which fits it as a function of design variables. It is shown that the design response surface approximation for the probabilistic sufficiency factor is more accurate than that for the probability of failure or for the safety index. Unlike the probability of failure or the safety index, the probabilistic sufficiency factor does not suffer from accuracy problems in regions of low probability of failure when calculated by Monte Carlo simulation. The use of the probabilistic sufficiency factor accelerates the convergence of reliability-based design optimization.     

Reliability-based design optimization of aeroelastic structures

Aeroelastic phenomena are most often either ignored or roughly approximated when uncertainties are considered in the design optimization process of structures subject to aerodynamic loading, affecting the quality of the optimization results. Therefore, a design methodology is proposed that combines reliability-based design optimization and high-fidelity aeroelastic simulations for the analysis and design of aeroelastic structures. To account for uncertainties in design and operating conditions, a first-order reliability method (FORM) is employed to approximate the system reliability. To limit model uncertainties while accounting for the effects of given uncertainties, a high-fidelity nonlinear aeroelastic simulation method is used. The structure is modelled by a finite element method, and the aerodynamic loads are predicted by a finite volume discretization of a nonlinear Euler flow. The usefulness of the employed reliability analysis in both describing the effects of uncertainties on a particular design and as a design tool in the optimization process is illustrated. Though computationally more expensive than a deterministic optimum, due to the necessity of solving additional optimization problems for reliability analysis within each step of the broader design optimization procedure, a reliability-based optimum is shown to be an improved design. Conventional deterministic aeroelastic tailoring, which exploits the aeroelastic nature of the structure to enhance performance, is shown to often produce designs that are sensitive to variations in system or operational parameters.     

System reliability based vehicle design for crashworthiness and effects of various uncertainty reduction measures

Reliability-based design optimization of automobile structures for crashworthiness has been studied by many researchers by using either single component probabilistic constraints or single failure mode based probabilistic constraints, while system reliability considerations are mostly disregarded. In this paper, we perform system reliability based design optimization (SRBDO) of an automobile for crashworthiness and analyze the effect of reliability allocation in different failure modes. In addition, effects of various uncertainty reduction measures (e.g., reducing variability in material properties, reducing error of finite element analysis) are investigated and tradeoff plots of uncertainty reduction, system reliability and structural weight are generated. These types of tradeoff plots can be used by a company manager to decide whether to allocate the company resources for employing uncertainty reduction measures or allocating the resources for the excess weight to protect against the unreduced uncertainties. Furthermore, relative importance of automobile structural members in different crash scenarios is quantified. Submitted for publication in the Structural and Multidisciplinary Optimization (SMO).     

Multidisciplinary design optimization of a vehicle system in a scalable, high performance computing environment

Multidisciplinary Design Optimization of a vehicle system for safety, NVH (noise, vibration and harshness) and weight, in a scalable HPC environment, is addressed. High performance computing, utilizing several hundred processors in conjunction with approximation methods, formal MDO strategies and engineering judgement are effectively used to obtain superior design solutions with significantly reduced elapsed computing times. The increased computational complexity in this MDO work is due to addressing multiple safety modes including frontal crash, offset crash, side impact and roof crush, in addition to the NVH discipline, all with detailed, high fidelity models and analysis tools. The reduction in large-scale MDO solution times through HPC is significant in that it now makes it possible for such technologies to impact the vehicle design cycle and improve the engineering productivity.     

Extensions of design potential concept for reliability-based design optimization to nonsmooth and extreme cases

The reliability-based design optimization (RBDO) can be described by the design potential concept in a unified system space, where the probabilistic constraint is identified by the design potential surface of the reliability target that is obtained analytically from the first-order reliability method (FORM). This paper extends the design potential concept to treat nonsmooth probabilistic constraints and extreme case design in RBDO. In addition, refinement of the design potential surface, which yields better optimum design, can be obtained using more accurate second-order reliability method (SORM). By integrating performance probability analysis into the iterative design optimization process, the design potential concept leads to a very effective design potential method (DPM) for robust system parameter design. It can also be applied effectively to extreme case design (ECD) by directly representing a probabilistic constraint in terms of the system performance function. Received July 25, 2000     

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.     

Reliability-based design optimization for elastoplastic mechanical structures

In this paper, two special formulations to carry out a reliability-based design optimization of elastoplastic mechanical structures are introduced. The first approach is based on a well-known two-level method where the first level involves the optimization for the design parameters whereas the evaluation of the probabilistic constraints is carried out in a sub-optimization level. Because the evaluation of the probabilistic constraints in a sub-optimization level causes non-convergence behavior for some problems as indicated in the literature, an alternative formulation based on one-level is developed considering the optimality conditions of the β-computation by which the probabilistic constraint appears in the first level reliability-based design optimization formulation. In both approaches, an explicit parameter optimization problem is proposed for the computation of a design point for elastoplastic structures.Three examples in this paper demonstrate that the one-level reliability-based design optimization formulation is superior in terms of convergence to an optimal design than the two-level reliability-based design optimization formulation.     

A multipoint version of space mapping optimization applied to vehicle crashworthiness design

In this paper, space mapping (SM) technique is combined with response surface methodology (RSM). SM is an optimization method well suited for very costly problems to find an improved design with fulfilled constraints. The SM technique use less costly models, which complements the correct models. The theory is established and compared to the corrected RSM. A multipoint version of SM is presented, where a separate evaluation is done in each iteration to improve the mapping function. Using this additional evaluation to update the mapping function, generally, the number of iterations to find the optimum solution can be reduced. Thus, the elapsed time to solve the optimization problem can be reduced if a parallel computer is utilized. Finally, one engineering optimization problem is solved to illustrate the application of SM in vehicle crashworthiness structural optimization.     

An investigation of structural optimization in crashworthiness design using a stochastic approach

In this paper the response surface methodology (RSM) and stochastic optimization (SO) are compared with regard to their efficiency and applicability in crashworthiness design. Optimization of simple analytic expressions and optimization of a front rail structure are the applications used to assess the respective qualities of both methods. A low detail vehicle structure is optimized to demonstrate the applicability of the methods in engineering practice. The investigations reveal that RSM is better compared to SO for fewer than 10–15 design variables. The convergence behaviour of SO improves compared to RSM when the number of design variables is increased. A novel zooming method is proposed which improves the convergence behaviour. A combination of both the RSM and the SO is efficient, stochastic optimization could be used in order to determine appropriate starting points for an RSM optimization, which continues the optimization. Two examples are investigated using this combined method.     

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.     

Topology optimization design of crushed 2D-frames for desired energy absorption history

The present work deals with topology optimization for obtaining a desired energy absorption history of a crushed structure. The optimized energy absorbing structures are used to improve the crashworthiness of transportation vehicles. The ground structure consists of rectangular 2D-beam elements with plastic hinges. The elements can undergo large rotations, so the analysis accommodates geometric nonlinearities. A quasi-static nonlinear finite element solution is obtained with an implicit backward Euler algorithm, and the analytical sensitivities are computed by the direct differentiation method.     

Polynomial genetic programming for response surface modeling Part 2: adaptive approximate models with probabilistic optimization problems

This is the second in a series of papers. The first deals with polynomial genetic programming (PGP) adopting the directional derivative-based smoothing (DDBS) method, while in this paper, an adaptive approximate model (AAM) based on PGP is presented with the partial interpolation strategy (PIS). The AAM is sequentially modified in such a way that the quality of fitting in the region of interest where an optimum point may exist can be gradually enhanced, and accordingly the size of the learning set is gradually enlarged. If the AAM uses a smooth high-order polynomial with an interpolative capability, it becomes more and more difficult for PGP to obtain smooth polynomials, whose size should be larger than or equal to the number of the samples, because the order of the polynomial becomes unnecessarily high according to the increase in its size. The PIS can avoid this problem by selecting samples belonging to the region of interest and interpolating only those samples. Other samples are treated as elements of the extended data set (EDS). Also, the PGP system adopts a multiple-population approach in order to simultaneously handle several constraints. The PGP system with the variable-fidelity response surface method is applied to reliability-based optimization (RBO) problems in order to significantly cut the high computational cost of RBO. The AAMs based on PGP are responsible for fitting probabilistic constraints and the cost function while the variable-fidelity response surface method is responsible for fitting limit state equations. Three numerical examples are presented to show the performance of the AAM based on PGP.     

Reliability-based design optimization of problems with correlated input variables using a Gaussian Copula

The reliability-based design optimization (RBDO) using performance measure approach for problems with correlated input variables requires a transformation from the correlated input random variables into independent standard normal variables. For the transformation with correlated input variables, the two most representative transformations, the Rosenblatt and Nataf transformations, are investigated. The Rosenblatt transformation requires a joint cumulative distribution function (CDF). Thus, the Rosenblatt transformation can be used only if the joint CDF is given or input variables are independent. In the Nataf transformation, the joint CDF is approximated using the Gaussian copula, marginal CDFs, and covariance of the input correlated variables. Using the generated CDF, the correlated input variables are transformed into correlated normal variables and then the correlated normal variables are transformed into independent standard normal variables through a linear transformation. Thus, the Nataf transformation can accurately estimates joint normal and some lognormal CDFs of the input variable that cover broad engineering applications. This paper develops a PMA-based RBDO method for problems with correlated random input variables using the Gaussian copula. Several numerical examples show that the correlated random input variables significantly affect RBDO results.