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.     

Probabilistic optimal design of framed structures

Probabilistic structural design is a degree of difficulty greater than deterministic design and effectively is only tractable using a computer. This paper outlines one approach to the design of framed structures subject to random loading. In particular stochastic approximation is used iteratively to improve the value of the optimality criterion, leading to a guaranteed optimal solution in the limit. Two framed structures are used to illustrate the computations which are performed in conjunction with a random number generator to simulate the probabilistic aspects of the loading. The approach is independent of the probability model assumed for the loading. It is straightforward to implement and has good solution convergence characteristics.     

A sequential approximate programming strategy for reliability-based structural optimization

Although reliability-based structural optimization (RBSO) is recognized as a rational structural design philosophy that is more advantageous to deterministic optimization, most common RBSO is based on straightforward two-level approach connecting algorithms of reliability calculation and that of design optimization. This is achieved usually with an outer loop for optimization of design variables and an inner loop for reliability analysis. A number of algorithms have been proposed to reduce the computational cost of such optimizations, such as performance measure approach, semi-infinite programming, and mono-level approach. Herein the sequential approximate programming approach, which is well known in structural optimization, is extended as an efficient methodology to solve RBSO problems. In this approach, the optimum design is obtained by solving a sequence of sub-programming problems that usually consist of an approximate objective function subjected to a set of approximate constraint functions. In each sub-programming, rather than direct Taylor expansion of reliability constraints, a new formulation is introduced for approximate reliability constraints at the current design point and its linearization. The approximate reliability index and its sensitivity are obtained from a recurrence formula based on the optimality conditions for the most probable failure point (MPP). It is shown that the approximate MPP, a key component of RBSO problems, is concurrently improved during each sub-programming solution step. Through analytical models and comparative studies over complex examples, it is illustrated that our approach is efficient and that a linearized reliability index is a good approximation of the accurate reliability index. These unique features and the concurrent convergence of design optimization and reliability calculation are demonstrated with several numerical examples.     

Optimal Robust Control for Rigid Serial Manipulators: A Fuzzy Approach

We first investigate the fundamental properties of rigid serial manipulators as related to the control design. Then, a new optimal robust control is proposed for serial‐link mechanical manipulators with fuzzy uncertainty. Fuzzy set theory is used to describe the uncertainty. The desirable system performance is deterministic (assuring the bottom line) and also fuzzy (enhancing the cost consideration). The proposed control is deterministic and is not the usual if‐then rules‐based control. The resulting controlled system is uniformly bounded and uniformly ultimately bounded proved via the Lyapunov minimax approach. A performance index (the combined cost, which includes average fuzzy system performance and control effort) is proposed based on the fuzzy information. The optimal design problem associated with the control can then be solved by minimizing the performance index. The resulting control design is systematic and is able to guarantee the deterministic performance as well as minimizing the cost. In the end, a cylindrical robot manipulator is chosen for demonstration.     

Optimal design for robust control of uncertain flexible joint manipulators: a fuzzy dynamical system approach

A novel fuzzy dynamical system approach to the control design of flexible joint manipulators with mismatched uncertainty is proposed. Uncertainties of the system are assumed to lie within prescribed fuzzy sets. The desired system performance includes a deterministic phase and a fuzzy phase. First, by creatively implanting a fictitious control, a robust control scheme is constructed to render the system uniformly bounded and uniformly ultimately bounded. Both the manipulator modelling and control scheme are deterministic and not IF-THEN heuristic rules-based. Next, a fuzzy-based performance index is proposed. An optimal design problem for a control design parameter is formulated as a constrained optimisation problem. The global solution to this problem can be obtained from solving two quartic equations. The fuzzy dynamical system approach is systematic and is able to assure the deterministic performance as well as to minimise the fuzzy performance index.     

Comparison of evolutionary-based optimization algorithms for structural design optimization

In this paper, a comparison of evolutionary-based optimization techniques for structural design optimization problems is presented. Furthermore, a hybrid optimization technique based on differential evolution algorithm is introduced for structural design optimization problems. In order to evaluate the proposed optimization approach a welded beam design problem taken from the literature is solved. The proposed approach is applied to a welded beam design problem and the optimal design of a vehicle component to illustrate how the present approach can be applied for solving structural design optimization problems. A comparative study of six population-based optimization algorithms for optimal design of the structures is presented. The volume reduction of the vehicle component is 28.4% using the proposed hybrid approach. The results show that the proposed approach gives better solutions compared to genetic algorithm, particle swarm, immune algorithm, artificial bee colony algorithm and differential evolution algorithm that are representative of the state-of-the-art in the evolutionary optimization literature.     

Linear quadratic tracking problem for discrete‐time systems with multiple delays in single input channel

The linear quadratic tracking problem for discrete‐time systems with multiple delays in single input channel is considered. In this paper, we provide an approach without resorting to system state augmentation. The optimal tracking control is given in terms of the current state, the previous inputs, and the output of an auxiliary backward deterministic delay system which is formulated for the first time in this paper. The solution relies on a Riccati difference equation of the same dimension as the plant (ignoring the delays). The key to our development is the establishment of a duality between the optimal tracking control and the optimal smoothing estimation of an associated stochastic backward system as well as the introduction of the auxiliary backward deterministic delay system. An analysis of the computational complexity of the proposed approach and its comparison with that of the augmentation method, which is to incorporate the delayed inputs into the augmented state, are provided. An example is given to demonstrate the effectiveness of the results. Copyright © 2009 John Wiley & Sons, Ltd.     

Shape optimization for 2-D mixed-mode fracture using Extended FEM (XFEM) and Level Set Method (LSM)

Weight and service life are often the two most important considerations in design of structural components. This research incorporates a novel crack propagation analysis technique into shape optimization framework to support design of 2-D structural components under mixed-mode fracture for: (1) maximum service life, subject to an upper limit on volume, and (2) minimum weight subject to specified minimum service life. In both cases, structural performance measures are selected as constraints and CAD dimensions are employed as shape design variables. Fracture parameters, such as crack growth rate and crack growth direction are computed using extended finite element method (XFEM) and level set method (LSM). XFEM employs special enrichment functions to incorporate the discontinuity of structural responses caused by the crack surfaces and crack tip fields into finite element approximation. The LSM utilizes level set functions to track the crack during the crack propagation analysis. As a result, this method does not require highly refined mesh around the crack tip nor re-mesh to conform to the geometric shape of the crack when it propagates, which makes the method extremely attractive for crack propagation analysis. An accurate and efficient semi-analytical design sensitivity analysis (DSA) method is developed for calculating gradients of fracture parameters. Two different approaches—a batch-mode, gradient-based, nonlinear algorithm and an interactive what-if analysis—are used for optimization. An engine connecting rod example is used to demonstrate the feasibility of the proposed method.     

Robust optimization with convex model considering bounded constraints on performance variation

In robust optimization, double-looped structures are often adopted where the outer loop is used to seek for the optimal design and the optimization performed in the inner loop is for the robustness assessment of the candidate solutions. However, the double-looped techniques usually will lead to a significant increase in computational efforts. Therefore, in this paper, a new robustness index is developed to handle bounded constraints on performance variation where no optimization run is required for the robustness evaluation work in the inner loop. The computation of this new index is based on the sensitivity Jacobian matrix of the system performances with respect to the uncertainties and it can quantitatively measure the maximal allowable magnitude of system variations. By introducing this index, the robust design problem can be reformulated as a deterministic optimization with robustness indices requirements. Two numerical examples are tested to show the effectiveness and efficiency of the proposed approach, whose solutions and computational efforts are compared to those from a double-looped approach proposed in previous literature.     

Sequential optimization and reliability assessment for multidisciplinary systems design

With higher reliability and safety requirements, reliability-based design has been increasingly applied in multidisciplinary design optimization (MDO). A direct integration of reliability-based design and MDO may present tremendous implementation and numerical difficulties. In this work, a methodology of sequential optimization and reliability assessment for MDO is proposed to improve the efficiency of reliability-based MDO. The central idea is to decouple the reliability analysis from MDO with sequential cycles of reliability analysis and deterministic MDO. The reliability analysis is based on the first-order reliability method (FORM). In the proposed method, the reliability analysis and the deterministic MDO use two MDO strategies, the multidisciplinary feasible approach and the individual disciplinary feasible approach. The effectiveness of the proposed method is illustrated with two example problems.     

Deterministic and stochastic seismic analysis of buildings with uncertain-but-bounded mass distribution

The actual and the computed seismic response of buildings are in general different due to the unavoidable uncertainties involved in the definition of the mass and stiffness distributions as well as in the estimation of the ground motion spatial variability. The discrepancies are accounted for by building codes through the so-called accidental eccentricity, which defines the bounds of the actual position of the centre of mass for each floor. Therefore, the seismic analysis problem is posed as the finding of the response of a structural system with uncertain-but-bounded parameters forced by deterministic or stochastic loads. In this paper, in the framework of the interval perturbation method, a procedure for determining upper and lower bounds of the dynamic response of structures with uncertain-but-bounded mass distribution vibrating under either deterministic or stochastic input is proposed. The procedure requires the definition of a unique structural model so reducing the number of analyses to be performed. Moreover, by the proposed approach all the possible permutations are implicitly considered so to include the worst condition. Numerical results showed a very good accuracy of the proposed procedure for all the cases analyzed.     

Reduction of calibration effort in FEM-based optimization via numerical and experimental data fusion

In this paper a fusion metamodeling approach is suggested as a method for reducing the experimental and computational effort generally required for calibrating the parameters of FEM simulations models. The metamodel is used inside an optimization routine for linking data coming from two different sources: simulations and experiments. The method is applied to a real problem: the optimal design of a metal foam filled tube to be used as an anti-intrusion bar in vehicles. The model is hierarchical, in the sense that one set of data (the experiments) is considered to be more reliable and it is labeled as “high-fidelity” and the other set (the simulations) is labeled as “low-fidelity”. In the proposed approach, Gaussian models are used to describe results of computer experiments because they are flexible and they can easily interpolate data coming from deterministic simulations. Since the results of experiments are obviously fully accurate, but aleatory, a second stage (“linkage”) model is used, which adjusts the prediction provided by the first model to more accurately represent the real experimental data. In the paper, the modeling and prediction ability of the method is first demonstrated and explained by means of artificially generated data and then applied to the optimization of foam filled tubular structures. The fusion metamodel yields comparable predictions (and optimal solution) if built over calibrated simulations vs. non-calibrated FEM models.     

Sensor network design based on system-wide reliability criteria. Part II: Formulations and applications

In part I of this series of articles, the concept of overall system reliability was presented for two applications: reliable estimation of variables for steady state linear flow processes, and reliable fault detection and diagnosis for any process. In this part, systematic generation of the proposed system-wide reliability expression is discussed. In particular, an approach for generating the system reliability using the sum of disjoint product method is presented. This serves as the objective function to be maximized in various constrained optimization formulations for sensor network design, which are also proposed in this work for both applications. A heuristic is proposed to solve the resulting nonlinear integer programming problems. The sensor network design formulations are applied to two benchmark case studies: (1) Tennessee Eastman process, (2) Steam metering process. Results indicate the utility of the proposed sensor network design approach in designing optimal sensor networks that maximize system reliability.     

不确定性机械系统的最优鲁棒控制设计: 模糊法

针对不确定性机械系统,提出了一种新的最优鲁棒控制方法.本文用模糊法去描述机械系统中的不确定性.机械系统的性能要求是确定的(保证最低要求),同时也是模糊的(成本控制里用到).所提出的控制方法是确定的,而不是基于假设的规则.经过严格的理论证明,控制系统最终可达到理想的性能指标.基于模糊信息,本文设计了一个性能指标(综合成本,包括系统的平均模糊性能和控制成本).通过最小化此性能指标,可解决控制的最优设计问题.这种最优设计方法可得到唯一的解析形式的最优解.总的来说,这种最优鲁棒控制方法较为系统,能够保证确定的系统性能得以实现,同时控制成本最小.最后,本文选了一个机械系统作为例子.     

Maximum variation analysis based analytical target cascading for multidisciplinary robust design optimization under interval uncertainty

Analytical target cascading (ATC) is a generally used hierarchical method for deterministic multidisciplinary design optimization (MDO). However, uncertainty is almost inevitable in the lifecycle of a complex system. In engineering practical design, the interval information of uncertainty can be more easily obtained compared to probability information. In this paper, a maximum variation analysis based ATC (MVA-ATC) approach is developed. In this approach, all subsystems are autonomously optimized under the interval uncertainty. MVA is used to establish an outer-inner framework which is employed to find the optimal scheme of system and subsystems. All subsystems are coordinated at the system level to search the system robust optimal solution. The accuracy and validation of the presented approach are tested using a classical mathematical example, a heart dipole optimization problem, and a battery thermal management system (BTMS) design problem.     

Robust least squares methods under bounded data uncertainties

We study the problem of estimating an unknown deterministic signal that is observed through an unknown deterministic data matrix under additive noise. In particular, we present a minimax optimization framework to the least squares problems, where the estimator has imperfect data matrix and output vector information. We define the performance of an estimator relative to the performance of the optimal least squares (LS) estimator tuned to the underlying unknown data matrix and output vector, which is defined as the regret of the estimator. We then introduce an efficient robust LS estimation approach that minimizes this regret for the worst possible data matrix and output vector, where we refrain from any structural assumptions on the data. We demonstrate that minimizing this worst-case regret can be cast as a semi-definite programming (SDP) problem. We then consider the regularized and structured LS problems and present novel robust estimation methods by demonstrating that these problems can also be cast as SDP problems. We illustrate the merits of the proposed algorithms with respect to the well-known alternatives in the literature through our simulations.     

A computational procedure for response statistics-based optimization of stochastic non-linear FE-models

An approach for an efficient solution of response statistics-based optimization problems of non-linear FE systems under stochastic loading is presented. A sequential approximate optimization approach, where approximate stochastic analyses are used during portions of the optimization process, is implemented in the proposed formulation. In this approach, analytical approximations of the performance functions in terms of the design variables are considered during the optimization process. The analytical approximations are constructed by combining a mixed linearization approach with a stochastic response sensitivity analysis. The state of the system is defined in terms of the statistical second-moment characteristics of the structural response. The stochastic loading and the response of the system are represented by an orthogonal series expansion of the corresponding covariance matrices. In particular, a truncated Karhunen-Loève (K-L) expansion is applied. The system of non-linear equations is replaced by a statistical equivalent linear system. The evaluation of the K-L vectors is carried out by an efficient procedure that combines local linearization, modal analysis and static response of higher structural modes. An illustrative example is presented that shows the efficiency of the proposed methodology: it considers a building finite element model enforced with non-linear hysteretic devices and subject to a stochastic ground acceleration. Two types of problems are considered: a minimum structural weight design problem and an optimal non-linear device design problem.     

Efficient strategies for reliability-based design optimization of variable stiffness composite structures

This study investigates efficient design optimization frameworks for composite structures with uncertainties related to material properties and loading. The integration of two decoupled reliability-based design optimization methodologies with a decoupled discrete material optimization is proposed to determine material and fiber orientation for three-dimensional composite structures. First, a deterministic and decoupled discrete material optimization is used for baseline comparison. The objective is to minimize the cost of composite structures with the design variables comprising of the piecewise patch orientations and material properties of the fiber reinforced composites. The reliability-based design optimization includes a hybrid method, and also the sequential optimization and reliability assessment method. In the sequential optimization and reliability assessment method, the inverse reliability analysis is evaluated using a stochastic response surface method and a first order reliability approach. Comparing the methods based on the optimal material and fiber orientations, the uncertainties in loads and material properties lead to different optimal layouts compared to the deterministic solutions. The numerical results also reveal that the hybrid method applied in reliability based designs results in negligible additional computational cost.     

Integrated Structural–Architectural Design for Interactive Planning

Traditionally, building floor plans are designed by architects with their usability, functionality and architectural aesthetics in mind; however, the structural properties of the distribution of load‐bearing walls and columns are usually not taken into account at this stage. In this paper, we propose a novel approach for the design of architectural floor plans by integrating structural layout analysis directly into the planning process. In order to achieve this, we introduce a planning tool which interactively enforces checks for structural stability of the current design, and which on demand proposes how to stabilize it if necessary. Technically, our solution contains an interactive architectural modelling framework as well as a constrained optimization module where both are based on respective architectural rules. Using our tool, an architect can predict already in a very early planning stage whose designs are structurally sound such that later changes due to stability reasons can be prevented. We compare manually computed solutions with optimal results of our proposed automated design process in order to show how much our proposed system can help architects to improve the process of laying out structural models optimally.