Crashworthiness design optimization using successive response surface approximations

 Finite Element (FE) method is among the most powerful tools for crash analysis and simulation. Crashworthiness design of structural members requires repetitive and iterative application of FE simulation. This paper presents a crashworthiness design optimization methodology based on efficient and effective integration of optimization methods, FE simulations, and approximation methods. Optimization methods, although effective in general in solving structural design problems, loose their power in crashworthiness design. Objective and constraint functions in crashworthiness optimization problems are often non-smooth and highly non-linear in terms of design variables and follow from a computationally costly (FE) simulation. In this paper, a sequential approximate optimization method is utilized to deal with both the high computational cost and the non-smooth character. Crashworthiness optimization problem is divided into a series of simpler sub-problems, which are generated using approximations of objective and constraint functions. Approximations are constructed by using statistical model building technique, Response Surface Methodology (RSM) and a Genetic algorithm. The approximate optimization method is applied to solve crashworthiness design problems. These include a cylinder, a simplified vehicle and New Jersey concrete barrier optimization. The results demonstrate that the method is efficient and effective in solving crashworthiness design optimization problems. Received: 30 January 2002 / Accepted: 12 July 2002 Sponsorship for this research by the Federal Highway Administration of US Department of Transportation is gratefully acknowledged. Dr. Nielen Stander at Livermore Software Technology Corporation is also gratefully acknowledged for providing subroutines to create D-optimal experimental designs and the simplified vehicle model.     

Update-based evolution control: A new fitness approximation method for evolutionary algorithms

Evolutionary algorithms are robust optimization methods that have been used in many engineering applications. However, real-world fitness evaluations can be computationally expensive, so it may be necessary to estimate the fitness with an approximate model. This article reviews design and analysis of computer experiments (DACE) as an approximation method that combines a global polynomial with a local Gaussian model to estimate continuous fitness functions. The article incorporates DACE in various evolutionary algorithms, to test unconstrained and constrained benchmarks, both with and without fitness function evaluation noise. The article also introduces a new evolution control strategy called update-based control that estimates the fitness of certain individuals of each generation based on the exact fitness values of other individuals during that same generation. The results show that update-based evolution control outperforms other strategies on noise-free, noisy, constrained and unconstrained benchmarks. The results also show that update-based evolution control can compensate for fitness evaluation noise.     

An experimental and numerical study of the effects of length scale and strain state on the necking and fracture behaviours in sheet metals

Within sheet metal forming, crashworthiness analysis in the automotive industry and ship research on collision and grounding, modelling of the material failure/fracture, including the behaviour at large plastic deformations, is critical for accurate failure predictions. In order to validate existing failure models used in finite element (FE) simulations in terms of dependence on length scale and strain state, tests recorded with the optical strain measuring system ARAMIS have been conducted. With this system, the stress–strain behaviour of uniaxial tensile tests was examined locally, and from this information true stress–strain relations were calculated on different length scales across the necking region. Forming limit tests were conducted to study the multiaxial failure behaviour of the material in terms of necking and fracture. The failure criteria that were verified against the tests were chosen among those available in the FE software Abaqus and the Bressan–Williams–Hill (BWH) criterion proposed by Alsos et al, 2008. The experimental and numerical results from the tensile tests confirmed that Barba's relation is valid for handling stress–strain dependence on the length scale used for strain evaluation after necking. Also, the evolution of damage in the FE simulations was related to the processes ultimately leading to initiation and propagation of a macroscopic crack in the final phase of the tensile tests. Furthermore, numerical simulations using the BWH criterion for prediction of instability at the necking point showed good agreement with the forming limit test results. The effect of pre-straining in the forming limit tests and the FE simulations of them is discussed.     

A comparative study of numerical solutions to non-linear discrete crack modelling of concrete beams involving sharp snap-back

Numerical problems are often encountered in modelling crack propagation in concrete beams using non-linear finite element (FE) analysis, especially when sharp snap-back behaviour in load-displacement relations occurs. This paper firstly identifies 16 arc-length control based numerical strategies based on extensive literature review. They are then used to carefully model the structural behaviour of a four-point single notched shear beam using discrete crack modelling approach in which cracks are represented by interface elements with bilinear softening constitutive laws. Based on extensive FE analyses, detailed comparisons of the merits and demerits of these numerical algorithms are then made. The results indicate that the effectiveness and efficiency of different algorithms may vary considerably from one to another, with the local arc-length based procedures in conjunction with tangential stiffness strategy and reversible unloading model being the most robust.     

Multi-objective robust design of energy-absorbing components using coupled process–performance simulations

The stochastic uncertainties associated with the material, process and product are represented and propagated to process and performance responses. A finite element-based sequential coupled process–performance framework is used to simulate the forming and energy absorption responses of a thin-walled tube in a manner that both material properties and component geometry can evolve from one stage to the next for better prediction of the structural performance measures. Metamodelling techniques are used to develop surrogate models for manufacturing and performance responses. One set of metamodels relates the responses to the random variables whereas the other relates the mean and standard deviation of the responses to the selected design variables. A multi-objective robust design optimization problem is formulated and solved to illustrate the methodology and the influence of uncertainties on manufacturability and energy absorption of a metallic double-hat tube. The results are compared with those of deterministic and augmented robust optimization problems.     

基于目标应变分布的TC17合金双性能盘预成形形状优化设计

目的 研究TC17合金双性能盘目标应变分布下的预成形形状优化设计方法.方法 采用拉丁超立方试验设计方法对预成形形状设计变量抽样选取样本点,并通过Deform有限元数值模拟获得样本设计变量下的局部应变分布.以局部应变分布与目标应变分布之间的方差最小为目标函数,采用Kriging方程建立近似替代模型预测响应应变误差,并结合...     

Sheet forming optimization based on least square support vector regression and intelligent sampling approach

In this paper, a metamodel-based optimization method by integration of support vector regression (SVR) and intelligent sampling strategy is applied to optimize sheet forming design. Compared with other popular metamodeling techniques, the SVR is based on the principle of structure risk minimization (SRM) as opposed to the principle of the empirical risk minimization in conventional regression techniques. Thus, the accuracy and robust metamodel can be obtained. The intelligent sampling strategy is a kind of design of experiment (DOE) essentially. The characteristic of this method is to generate new sample automatically by responses of objective functions. Compared with traditional DOE methods, the number of samples isn’t constant according to different cases. Furthermore, the number of samples and size of design space can be well controlled according to the intelligent strategy. To minimize both objective functions of wrinkling, crack and thickness deformation efficiently, the proposed method is employed as a fast analysis tool to surrogate the time-consuming finite-element (FE) procedure in the iterations of optimization algorithm. An example is studied to illustrate the application of the approach proposed, and it is concluded that the proposed method is feasible for sheet forming optimization.     

Swarm algorithms for single- and multi-objective optimization problems incorporating sensitivity analysis

Swarm algorithms such as particle swarm optimization (PSO) are non-gradient probabilistic optimization algorithms that have been successfully applied for global searches in complex problems such as multi-peak problems. However, application of these algorithms to structural and mechanical optimization problems still remains a complex matter since local optimization capability is still inferior to general numerical optimization methods. This article discusses new swarm metaphors that incorporate design sensitivities concerning objective and constraint functions and are applicable to structural and mechanical design optimization problems. Single- and multi-objective optimization techniques using swarm algorithms are combined with a gradient-based method. In the proposed techniques, swarm optimization algorithms and a sequential linear programming (SLP) method are conducted simultaneously. Finally, truss structure design optimization problems are solved by the proposed hybrid method to verify the optimization efficiency.     

Improvement of three-phase flow numerical simulations by means of novel hybrid methods

Numerical simulations of three-phase flows are facing the challenge that their mathematical models include a lot of assumptions and the equation systems often deliver controversial solutions. The object of this study is the improvement of numerical simulations of a three-phase (solid, gas, liquid) flow according to the four-way coupling Eulerian-Eulerian frame. Following the strategy of incorporating a priori knowledge in a system, initial velocity information achieved by several experimental and numerical techniques is implemented in the numerical simulations. Particle image velocimetry (PIV) data are employed in a numeroexperimental hybrid and artificial neural network (ANN) data in numeroneuronal and neuroexperimental hybrids, where the ANNs are trained with numerical or PIV data, respectively. The employment of the three presented hybrid methods affords better convergence of the numerical simulations, delivers more accurate numerical results and enables saving of computational time, thus, more precise information about the behaviour of the fluid mechanical system is faster achieved.     

Modal effective electromechanical coupling approximate evaluations and simplified analyses: numerical and experimental assessments

This contribution presents numerical and experimental assessments of the modal effective electromechanical coupling coefficient (EMCC) using popular approximate evaluations and simplified analyses of piezoelectric structures. For this purpose, first, a common benchmark, consisting of a cantilever Aluminum (Al) beam with symmetrically surface-bonded two pairs of large piezoceramic (PZT) patches, is retained for the assessment of EMCC different evaluation formulas and plane strain (PStrain) and plane stress (PStress) two-dimensional (2D) analyses using ANSYS \({^\circledR}\) coupled piezoelectric three-dimensional (3D) and 2D finite elements (FE). Then, similarly, an experimental assessment is conducted on two benchmarks consisting of Al long and short cantilevers equipped symmetrically with pairs of small and large PZT patches. It is found that, in order to get EMCC accurate approximate numerical evaluation, it is crucial to consider the patches electrodes equipotential constraints and, in order to get EMCC accurate 2D results with regard to 3D calculations, it is necessary to use PStress kinematics for approximate 2D analysis. Besides, 3D FE and experimental frequencies are shown to be bounded from below by PStress and from above by PStrain 2D FE results. Moreover, EMCC 2D PStress results are found closer to 3D FE and experimental results than PStrain 2D FE ones.     

Thermoeconomic optimization of a single effect water/LiBr vapour absorption refrigeration system

In this paper, the thermoeconomic theory is applied to the economic optimization of a single effect water/LiBr vapour absorption refrigeration system for air-conditioning application, aimed at minimizing its overall operation and amortization cost. The mathematical and numerical techniques based optimization of thermal system is not always possible due to plant complexities. Therefore, a simplified cost minimization methodology is applied to evaluate the economic costs of all the internal flows and products of the system by formulating exergoeconomic cost equations. Once these costs are determined, the system is thermoeconomically evaluated to identify the effects of design variables on costs and enables to suggest values of design variables that would make the overall system cost effective. Finally, an approximate optimum design configuration is obtained by means of sequential local optimization of the system, carried out unit by unit. The result compares this optimum with the base case and shows percentage variations in the system's operation and amortization cost.     

Crash reconstruction of pedestrian accidents using optimization techniques

Numerical simulations of vehicle-to-pedestrian crash (VPC) are frequently used to develop a detailed understanding of how pedestrian injuries relate to documented vehicle damage. Given the complexity of the event, modeling the interactions typically involves subjective evaluations of the pre-impact conditions using a limited number of simulations. The goal of this study is to develop a robust methodology for obtaining the pre-impact pedestrian posture and vehicle speed utilizing multi-body simulations and optimization techniques. First, a continuous sequence of the pedestrian gait based on the literature data and simulations was developed for use as a design parameter during the optimization process. Then, the robustness and efficiency of three optimization algorithms were evaluated in a mock (idealized) crash reconstruction. The pre-impact parameters of the pedestrian and the vehicle models were treated as unknown design variables for the purpose of validating the optimization technique. While all algorithms found solutions in close vicinity of the exact solution, a genetic algorithm exhibited the fastest convergence. The response surfaces of the objective function showed higher sensitivities to the pedestrian posture and its relative position with respect to the vehicle than to the vehicle speed for the chosen design space. After validating the methodology with the mock reconstruction, a real-world vehicle-to-pedestrian accident was reconstructed using the data obtained from the field investigation and the optimization methodology. A set of pedestrian and vehicle initial conditions capable of matching all observed contact points was determined. Based on the mock and real-world reconstructions, this study indicates that numerical simulations coupled with optimization algorithms can be used to predict pedestrian and vehicle pre-impact conditions.     

NEAR FIELD TRANSIENT AXISYMMETRIC WAVES IN LAYERED STRUCTURES: EFFECTS OF WEAK COUPLING

Axisymmetric stress wave transmission through the leading layers of layered structures of infinite radial but finite axial extent is numerically studied by employing two different computational approaches: a technique based on the numerical inversion of Double Integral Transformations (DIT), and a Finite Element (FE) analysis. Considering the first approach, careful selections of the limits of the numerical inversions and the sampling rates are required in order to overcome inherent numerical instabilities associated with exponential dichotomy. This type of numerical instability is more evident in layered media with weak coupling. In such systems, direct multiplications of layer transfer matrices are avoided by employing a global scheme to assemble well-conditioned global transfer matrices. Moreover, the specific structure of the propagation and attenuation zones of the structure are taken into account for increasing the efficiency and effectiveness of the transfer matrix manipulations. Satisfactory agreement between the DIT and FE numerical results is observed, at least for early times. Close to the region of application of the external pressure, the FE simulations suffer from the discretization of the applied load, node-to-node oscillations and reflections from ‘infinite’ elements (‘silent boundaries’). Using the aforementioned numerical techniques, transient wave transmission in two-layered systems (one with weak and one with strong interlayer coupling) is considered, and the effects of weak coupling on the wave transmission is studied. We show that at early times, weak coupling results in stress localization in the region close to the applied pressure, a result which can have potential application in the use of layered media as shock isolators. © 1997 by John Wiley & Sons, Ltd.     

Optimal design of gating systems by gradient search methods

A numerical optimization technique based on gradient-search is applied to obtain an optimal design of a typical gating system used for the gravity process to produce aluminum parts. This represents a novel application of coupling nonlinear optimization techniques with a foundry process simulator, and it is motivated by the fact that a scientifically guided search for better designs based on techniques that take into account the mathematical structure of the problem is preferred to commonly found trial-and-error approaches. The simulator applies the finite volume method and the VOF algorithm for CFD analysis. The direct gradient optimization algorithm, sequential quadratic programming (SQP), was used to solve both a 2D and a 3D gating system design problems using two design variables. The results clearly show the effectiveness of the proposed approach for finding high quality castings when compared with current industry practices.     

Finite element solution of Fokker–Planck equation of nonlinear oscillators subjected to colored non-Gaussian noise

Nonlinear oscillators subjected to colored Gaussian/non-Gaussian excitations are modelled through a set of three coupled first-order stochastic differential equations by representing the excitation as a first-order filtered white noise. A 3-D finite element (FE) formulation is developed to solve the corresponding 3-D Fokker Planck (FP) equations. The joint probability density functions of the state variables, obtained as a solution of the FP equation, are typically non-Gaussian and are used for computing the crossing statistics of the response – an essential metric for time variant reliability analysis. The method is illustrated through a noisy Lorenz attractor and a Duffing oscillator subjected to additive colored noise. The increase in state-space dimension when the Duffing oscillator is additionally excited with a parametric Gaussian noise is effectively handled by using stochastic averaging to reduce the state-space dimension. Investigations are carried out to examine the accuracy of the FE method vis-a-vis Monte Carlo simulations. The proposed method is observed to be computationally significantly cheaper for these three problems.     

NEW DEVELOPMENTS IN STRUCTURAL OPTIMIZATION USING ADAPTIVE MESH REFINEMENT AND MULTIPOINT APPROXIMATIONS

Combining Shape Optimization (SO) with Adaptive Mesh Refinement (AMR) potentially offers a higher accuracy and higher computational efficiency, especially if the applied target error for AMR is reduced in the course of the optimization process. The disadvantage of that approach is that the rate of convergence of the corresponding optimization processes can be significantly lower as compared to processes which apply a fixed target error for AMR. In the present paper the so-called Multipoint Approximation Method (MAM) is used as a basis for SO in conjunction with AMR. Several techniques for improvement of the rates of convergence are presented and investigated. Firstly, alternative algorithms for determining the approximation functions using a weighted least squares method are investigated. The focus is on weights which depend on the discretization errors. Secondly, different strategies for moving and resizing the search sub-regions in the space of design variables are presented. The proposed methods are illustrated by means of several optimization problems in which the effect of AMR with changing discretization errors is modelled by artificially introduced numerical noise.     

Springback reduction with control of punch speed and blank holder force via sequential approximate optimization with radial basis function network

This paper proposes a springback reduction technique with the control of punch speed and blank holder force (BHF) via sequential approximate optimization (SAO). Springback is one of the major defects in sheet forming and its reduction is a crucial issue for improving product quality. Computer-aided-engineering is one of the helpful tools for predicting springback and widely used in automotive industries. Various approaches are considered for springback reduction, and we optimize the punch speed as well as BHF (variable BHF). Sheet forming simulation is generally costly and time-consuming, and the SAO with the radial basis function network is used to determine the optimal punch speed and variable BHF. The U-shaped forming in NUMISHEET’93 is used in the numerical simulation. The standard deviation of the bending moment is minimized subject tearing evaluated with the forming limit diagram. The punch speed and the variable BHF are taken as the design variables. The validity is examined through numerical simulation.     

Performance of some SQP algorithms on structural design problems

This paper describes the basic ideas of sequential quadratic programming (SQP) algorithms for design optimization. There are two fundamental differences between the various algorithms: (i) the difinition of the QP subproblem solved at each iteration, and (ii) the descent function used during step size determination. The performances of the algorithms can change dramatically depending on how the two steps are executed. Numerical implementation details of various computational steps are discussed. Three programs based on SQP algorithms are used to solve 17 structural design problems having 7 to 96 design variables and 10 to 1051 performance constraints besides design variable bounds. Based on the performance of these programs, efficient procedures to execute various steps of the SQP methods are determined. It is concluded that the potential constraint strategy, where only a subset of the constraints is used to define the QP subproblem, is essential for large scale engineering design applications. With this strategy the SQP methods are quite robust and have great potential for routine application in engineering design.     

A polynomial chaos efficient global optimization approach for Bayesian optimal experimental design

This paper proposes a global optimization framework to address the high computational cost and non convexity of Optimal Experimental Design (OED) problems. To reduce the computational burden and the presence of noise in the evaluation of the Shannon expected information gain (SEIG), this framework proposes the coupling of Laplace approximation and polynomial chaos expansions (PCE). The advantage of this procedure is that PCE allows large samples to be employed for the SEIG estimation, practically vanishing the noisy introduced by the sampling procedure. Consequently, the resulting optimization problem may be treated as deterministic. Then, an optimization approach based on Kriging surrogates is employed as the optimization engine to search for the global solution with limited computational budget. Four numerical examples are investigated and their results are compared to state-of-the-art stochastic gradient descent algorithms. The proposed approach obtained better results than the stochastic gradient algorithms in all situations, indicating its efficiency and robustness in the solution of OED problems.     

Predictive modelling for optimization of textile composite forming

Wrinkling often occurs during textile composite forming and is a major problem for manufacturers. The prediction of this defect is, therefore, of major importance for the design and optimization of textile composite structures. Numerical simulations of forming for textile composites over a hemisphere have been conducted using a rate/temperature-dependent hybrid FE model. The hybrid FE model incorporates a fully predictive multi-scale energy model which determines the shear resistance of the textile composite sheet. The effects of varying the normal force distribution across the edges of the blank and blank size, together with the effect of changes in forming temperature on the final fibre pattern and wrinkling behaviour, are investigated. Predictions are evaluated against press-formed components. The results from the simulation and the experiments have good correlation and show that wrinkling can be minimized by optimizing the force distribution around the edge of the manufacturing tool and by careful choice of forming temperature.