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.     

Cost-weight trades for modular composite structures

A design approach for airframe structures is formulated based on the concept of modularity allowing trade-offs and optimization between cost and weight. A modular structure can be created by replacing a collection of parts which all have a unique design by a collection of parts where the same design repeats multiple times. Structures with high levels of modularity have higher weight since it is harder to design a weight-efficient structure when the amount of design options is limited, but this weight increase might be worth the associated decrease in manufacturing cost. In modular design, cost reductions are achieved through learning curve effects and through reduction of the non-recurring cost, for example, due to lower tooling costs. Based on dynamic programming, an approach to determine the optimum number of repeating designs was determined and applied to a composite fuselage structure. Two examples are given where the cost-weight efficiency at different modularity levels is assessed for a composite airframe: the stringers and the frames in a fuselage. The corresponding cost-weight diagrams indicated that the modularity concept provides a useful methodology for designing more cost- weight efficient structures. In both cases it was possible to replace a large amount of designs and increase the level of modularity of the structure, yielding significant reductions in recurring and non-recurring manufacturing costs while keeping the associated weight increase of the structure to a minimum.     

Optimal design of underwater stiffened shells

The optimal design parameters of stiffened shells are determined using a rational multicriteria optimization approach. The adopted approach aims at simultaneously minimizing the shell vibration, associated sound radiation, weight of the stiffening rings as well as the cost of the stiffened shell. A finite element model is developed to determine the vibration and noise radiation from cylindrical shells into the surrounding fluid domain. The production cost as well as the life cycle and maintenance costs of the stiffened shells are computed using the Parametric Review of Information for Costing and Evaluation (PRICE) model. A Pareto/min-max multicriteria optimization approach is then utilized to select the optimal dimensions and spacing of the stiffeners. Numerical examples are presented to compare the vibration and noise radiation characteristics of optimally designed stiffened shells with the corresponding characteristics of plain un-stiffened shells. The obtained results emphasis the importance of the adopted multicriteria optimization approach in the design of quiet, low weight and low cost underwater shells which are suitable for various critical applications. Received September 14, 2000 Communicated by J. Sobieski     

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.     

Integrated product line design and supplier selection: A multi-objective optimization paradigm

Product line design is commonly used to provide higher product variety for satisfying diversified customer needs. To reduce the cost and development time and improve quality of products, companies quite often consider sourcing. Conventionally, product line design and supplier selection are dealt with separately. Some previous studies have been attempted to consider product line design and supplier selection simultaneously but two shortcomings were noted. First, the previous studies considered several objectives as a single objective function in the formulation of optimization models for the integrated problem. Second, positions of product variants to be offered in a product line in competitive markets are not clearly defined that would affect the formulation of marketing strategies for the product line. In this paper, a methodology for integrated product line design and supplier selection is proposed to address the shortcomings in which a multi-objective optimization model is formulated to determine their specifications and select suppliers for maximizing the profit, quality and performance as well as minimizing the cost of the product line. In addition, joint-spacing mapping is introduced to help estimate market share of products and indicate positions of product variants. The proposed methodology can provide decision makers with a better tradeoff among various objectives of product line design, and define market positions of product variants explicitly. The results generated based on the methodology could help companies develop product lines with higher profits, better product quality and larger market share to be obtained. A case study of a product line design of notebook computers was performed to illustrate the effectiveness of the proposed methodology. The results have shown that Pareto optimal product line designs and the specifications of product variants can be determined. Suppliers of components and modules can be selected with considerations of minimum sourcing cost, and maximum performance and quality of product variants. Prices and positions of the product variants can also be determined.     

Optimization and sensitivity of prestressed concrete beams

This paper is concerned with the optimum design of prestressed concrete beams. Both minimum weight and minimum cost optimization formulations are given for simply supported beams having three different sections. Sensitivity of the optimum designs, with respect to various design parameters, are also discussed. The formulation is programmed for interactive use on micro-computers. An example is given and results are discussed.     

Piezoresistive Microphone Design Pareto Optimization: Tradeoff Between Sensitivity and Noise Floor

This paper addresses tradeoffs between pressure sensitivity and electronic noise floor in optimizing the performance of a piezoresistive microphone. A design optimization problem is formulated to find the optimum dimensions of the diaphragm, the piezoresistor geometry and location for two objective functions: maximum pressure sensitivity and minimum electronic noise floor. The Pareto curve of optimum designs for both objectives is generated. The minimum detectable pressure (MDP) was also employed as an objective function, generating a point on the Pareto curve that may be the best compromise between the original two objectives. The results indicated that this application the critical constraints are the linearity and power consumption. The minimized MDP design was less sensitive to uncertainty in design variables. The optimization methodology presented in this paper as well as some of the conclusions are applicable to other types of piezoresistive sensors     

Truss topology optimization including bar properties different for tension and compression

The problem of optimum truss topology design based on the ground structure approach is considered. It is known that any minimum weight truss design (computed subject to equilibrium of forces and stress constraints with the same yield stresses for tension and compression) is—up to a scaling—the same as a minimum compliance truss design (subject to static equilibrium and a weight constraint). This relation is generalized to the case when different properties of the bars for tension and for compression additionally are taken into account. This situation particularly covers the case when a structure is optimized which consists of rigid (heavy) elements for bars under compression, and of (light) elements which are hardly/not able to carry compression (e.g. ropes). Analogously to the case when tension and compression is handled equally, an equivalence is established and proved which relates minimum weight trusses to minimum compliance structures. It is shown how properties different for tension and compression pop up in a modified global stiffness matrix now depending on tension and compression. A numerical example is included which shows optimal truss designs for different scenarios, and which proves (once more) the big influence of bar properties (different for tension and for compression) on the optimal design.     

Multi‐objective design optimization of antennas for reflection,size, and gain variability using kriging surrogates and generalized domain segmentation

Cost‐efficient multi‐objective design optimization of antennas is presented. The framework exploits auxiliary data‐driven surrogates, a multi‐objective evolutionary algorithm for initial Pareto front identification, response correction techniques for design refinement, as well as generalized domain segmentation. The purpose of this last mechanism is to reduce the volume of the design space region that needs to be sampled in order to construct the surrogate model, and, consequently, limit the number of training data points required. The recently introduced segmentation concept is generalized here to allow for handling an arbitrary number of design objectives. Its operation is illustrated using an ultra‐wideband monopole optimized for best in‐band reflection, minimum gain variability, and minimum size. When compared with conventional surrogate‐based approach, segmentation leads to reduction of the initial Pareto identification cost by over 20%. Numerical results are supported by experimental validation of the selected Pareto‐optimal antenna designs.     

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.     

Optimum energy-based design of BRB frames using nonlinear response history analysis

In this paper, an optimum design method for buckling restrained brace frames subjected to seismic loading is presented. The multi-objective charged system search is developed to optimize costs and damages caused by the earthquake for steel frames. Minimum structural weight and minimum seismic energy which including seismic input energy divided by maximum hysteretic energy of fuse members are selected as two objective functions to find a Pareto solutions that copes with considered preferences. Also, main design constraints containing allowable amount of the inter-story drift and plastic rotation of beam, column members and plastic displacement of buckling restrained braces are controlled. The results of optimum design for three different frames are obtained and investigated by the developed method.     

Soft computing based multi-objective optimization of steam cycle power plant using NSGA-II and ANN

In this paper a steam turbine power plant is thermo-economically modeled and optimized. For this purpose, the data for actual running power plant are used for modeling, verifying the results and optimization. Turbine inlet temperature, boiler pressure, turbines extraction pressures, turbines and pumps isentropic efficiency, reheat pressure as well as condenser pressure are selected as fifteen design variables. Then, the fast and elitist Non-dominated Sorting Genetic Algorithm (NSGA-II) is applied to maximize the thermal efficiency and minimize the total cost rate (sum of investment cost, fuel cost, and maintenance cost) simultaneously. The results of the optimal design are a set of multiple optimum solutions, called ‘Pareto optimal solutions’. The optimization results in some points show 3.76% increase in efficiency and 3.84% decrease in total cost rate simultaneously, when it compared with the actual data of the running power plant. Finally as a short cut to choose the system optimal design parameters a correlation between two objectives and fifteen decision variables with acceptable precision are presented using Artificial Neural Network (ANN).     

Multi-objective optimization of expensive electromagnetic simulation models

Vast majority of practical engineering design problems require simultaneous handling of several criteria. For the sake of simplicity and through a priori preference articulation one can turn many design tasks into single-objective problems that can be handled using conventional numerical optimization routines. However, in some situations, acquiring comprehensive knowledge about the system at hand, in particular, about possible trade-offs between conflicting objectives may be necessary. This calls for multi-objective optimization that aims at identifying a set of alternative, Pareto-optimal designs. The most popular solution approaches include population-based metaheuristics. Unfortunately, such methods are not practical for problems involving expensive computational models. This is particularly the case for microwave and antenna engineering where design reliability requires utilization of CPU-intensive electromagnetic (EM) analysis. In this work, we discuss methodologies for expedited multi-objective design optimization of expensive EM simulation models. The solution approaches that we present here rely on surrogate-based optimization (SBO) paradigm, where the design speedup is obtained by shifting the optimization burden into a cheap replacement model (the surrogate). The latter is utilized for generating the initial approximation of the Pareto front representation as well as further front refinement (to elevate it to the high-fidelity EM simulation model level). We demonstrate several application case studies, including a wideband matching transformer, a dielectric resonator antenna and an ultra-wideband monopole antenna. Dimensionality of the design spaces in the considered examples vary from six to fifteen, and the design optimization cost is about one hundred of high-fidelity EM simulations of the respective structure, which is extremely low given the problem complexity.     

A new approach for online multiobjective optimization of mechatronic systems

We present a new concept for online multiobjective optimization and its application to the optimization of the operating point assignment for a doubly-fed linear motor. This problem leads to a time-dependent multiobjective optimization problem. In contrast to classical optimization where the aim is to find the (global) minimum of a single function, we want to simultaneously minimize k objective functions. The solution to this problem is given by the set of optimal compromises, the so-called Pareto set. In the case of the linear motor, there are two conflicting aims which both have to be maximized: the degree of efficiency and the inverter utilization factor. The objective functions depend on velocity, force and power, which can be modeled as time-dependent parameters. For a fixed point of time, the entire corresponding Pareto set can be computed by means of a recently developed set-oriented numerical method. An online computation of the time-dependent Pareto sets is not possible, because the computation itself is too complex. Therefore, we combine the computation of the Pareto set with numerical path following techniques. Under certain smoothness assumptions the set of Pareto points can be characterized as the set of zeros of a certain function. Here, path following allows to track the evolution of a given solution point through time.     

Optimum design of sensor arrays via simulation-based multivariate calibration

This paper is concerned with the optimum design of sensor arrays (i.e., electronic noses or tongues) using simulation experiments. The proposed design method considers multiple criteria simultaneously for the evaluation of sensor arrays, and a multiple objective tabu search algorithm was adapted to search for a number of Pareto optimum array designs in terms of those criteria. The evaluation of a candidate sensor array is based on its multivariate calibration model, which is efficiently estimated from well-designed simulation experiments. The method can be used to optimize the design of both linear and nonlinear sensor arrays.     

Uncertainty quantifications of Pareto optima in multiobjective problems

Design is a multi-objective decision-making process considering manufacturing, cost, aesthetics, usability among many other product attributes. The set of optimal solutions, the Pareto set, indicates the trade-offs between objectives. Decision-makers generally select their own optima from the Pareto set based on personal preferences or other judgements. However, uncertainties from manufacturing processes and from operating conditions will change the performances of the Pareto optima. Evaluating the impacts of uncertainties on Pareto optima requires a large amount of data and resources. Comparing multiple Pareto solutions under uncertainty are also very costly. In this work, local Pareto set approximation is integrated with uncertainty propagation technique to quantify design variations in the objective space. An optimality influence range is proposed using linear combinations of objective functions that creates a more accurate polygon objective variation subspace. A set of ‘virtual samples’ is then generated to form two quantifications of the objective variation subspace, namely an influence noise to indicate how a design remains optimal, and an influence range that quantifies the overall variations of a design. In most engineering practices, a Pareto optimum with a smaller influence noise and a smaller influence range is preferred. We also extend the influence noise/range concept to nonlinear Pareto set with the second-order approximation. The quadratic local Pareto approximation method in the literature is also extended in this work to solve multi-objective engineering problems with black-box functions. The usefulness of the proposed quantification method is demonstrated using a numerical example as well as using an engineering problem in structural design.     

Load and resistance convex models for optimum design

This paper is concerned with the optimal design of structures that are affected by uncertainties present in the loads applied to the structure, and by uncertainties affecting the internal resistance of the structural members. The magnitude of the applied loads and the modulus of elasticity of the structural members are assumed to vary within deterministic bounds. These uncertainties are idealized using a nonprobabilistic method, the convex model. The two types of uncertainties are considered simultaneously by employing the Cartesian product of convex sets. Two different convex models are examined to account for the uncertainties: the ellipsoidal convex model, and the uniform bound convex model. The optimum designs of a truss using the two convex models are compared to a worst case scenario optimum design in order to evaluate their performance. It is shown that it is not possible to identify a single worst case scenario that would be able to account for all possible combinations of uncertainties. However, both the ellipsoidal and the uniform bound convex model designs are found to be superior to the worst case scenario design in terms of constraint violations.     

Multi-objective evolutionary optimization of polynomial neural networks for modelling and prediction of explosive cutting process

In this paper, evolutionary algorithms (EAs) are deployed for multi-objective Pareto optimal design of group method of data handling (GMDH)-type neural networks which have been used for modelling an explosive cutting process using some input–output experimental data. In this way, multi-objective EAs (non-dominated sorting genetic algorithm, NSGA-II) with a new diversity-preserving mechanism are used for Pareto optimization of such GMDH-type neural networks. The important conflicting objectives of GMDH-type neural networks that are considered in this work are, namely, training error (TE), prediction error (PE), and number of neurons (N) of such neural networks. Different pairs of theses objective functions are selected for 2-objective optimization processes. Therefore, optimal Pareto fronts of such models are obtained in each case which exhibit the trade-off between the corresponding pair of conflicting objectives and, thus, provide different non-dominated optimal choices of GMDH-type neural networks models for explosive cutting process. Moreover, all the three objectives are considered in a 3-objective optimization process, which consequently leads to some more non-dominated choices of GMDH-type models representing the trade-offs among the training error, prediction error, and number of neurons (complexity of network), simultaneously. The overlay graphs of these Pareto fronts also reveal that the 3-objective results include those of the 2-objective results and, thus, provide more optimal choices for the multi-objective design of GMDH-type neural networks in terms of minimum training error, minimum prediction error, and minimum complexity.     

An interactive fuzzy multi-objective optimization method for engineering design

The coupling of performance functions due to common design variables and uncertainties in an engineering design process will result in difficulties in optimization design problems, such as poor collaboration among design objectives and poor resolution of design conflicts. To handle these problems, a fuzzy interactive multi-objective optimization model is developed based on Pareto solutions, where the metric function and some additional constraints are added to ensure the collaboration among design objectives. The trade-off matrix at the Pareto solutions was developed, and the method for selecting weighting coefficients of optimization objectives is also provided. The proposed method can generate a Pareto optimal set with the maximum satisfaction degree and the minimum distance from ideal solution. The favorable optimal solution can be then selected from the Pareto optimal set by analyzing the trade-off matrix and collaborative sensitivity. Two examples are presented to illustrate the proposed method.     

Generalized Pareto ranking bisection for computationally feasible multiobjective antenna optimization

Multiobjective optimization (MO) allows for obtaining comprehensive information about possible design trade‐offs of a given antenna structure. Yet, executing MO using the most popular class of techniques, population‐based metaheuristics, may be computationally prohibitive when full‐wave EM analysis is utilized for antenna evaluation. In this work, a low‐cost and fully deterministic MO methodology is introduced. The proposed generalized Pareto ranking bisection algorithm permits identifying a set of Pareto optimal sets of parameters representing the best trade‐offs between considered objectives. The subsequent designs are found by iterative partitioning of the intervals connecting previously obtained designs and executing Pareto‐ranking‐based poll search. The initial approximation of the Pareto front found using the bisection procedure is subsequently refined to the level of the high‐fidelity EM model of the antenna at hand using local optimization. The proposed framework overcomes a serious limitation of the original, recently reported, bisection algorithm, which was only capable of considering two objectives. The generalized version proposed here allows for handling any number of design goals. An improved poll search procedure has also been developed and incorporated. Our algorithm has been demonstrated using two examples of UWB monopole antennas with four figures of interest taken into account: structure size, reflection response, total efficiency, and gain variability.