Hyperstatic and redundancy thresholds in truss topology optimization considering progressive collapse due to aleatory and epistemic uncertainties

Optimization leads to specialized structures which are not robust to disturbance events like unanticipated abnormal loading or human errors. Typical reliability-based and robust optimization mainly address objective aleatory uncertainties. To date, the impact of subjective epistemic uncertainties in optimal design has not been comprehensively investigated. In this paper, we use an independent parameter to investigate the effects of epistemic uncertainties in optimal design: the latent failure probability. Reliability-based and risk-based truss topology optimization are addressed. It is shown that optimal risk-based designs can be divided in three groups: (A) when epistemic uncertainty is small (in comparison to aleatory uncertainty), the optimal design is indifferent to it and yields isostatic structures; (B) when aleatory and epistemic uncertainties are relevant, optimal design is controlled by epistemic uncertainty and yields hyperstatic but nonredundant structures, for which expected costs of direct collapse are controlled; (C) when epistemic uncertainty becomes too large, the optimal design becomes redundant, as a way to control increasing expected costs of collapse. The three regions above are divided by hyperstatic and redundancy thresholds. The redundancy threshold is the point where the structure needs to become redundant so that its reliability becomes larger than the latent reliability of the simplest isostatic system. Simple truss topology optimization is considered herein, but the conclusions have immediate relevance to the optimal design of realistic structures subject to aleatory and epistemic uncertainties.     

An approach for the reliability based design optimization of laminated composites

This article aims at optimizing laminated composite plates taking into account uncertainties in the structural dimensions. As laminated composites require a global optimization tool, the Particle Swarm Optimization (PSO) method is employed. A new Reliability Based Design Optimization (RBDO) methodology based on safety factors is presented and coupled with PSO. Such safety factors are derived from the Karush–Kuhn–Tucker optimality conditions of the reliability index approach and eliminate the need for reliability analysis in RBDO. The plate weight minimization is the objective function of the optimization process. The results show that the coupling of the evolutionary algorithm with the safety-factor method proposed in this article successfully performs the RBDO of laminated composite structures.     

Enhanced similarity-based metamodel updating strategy for reliability-based design optimization

Metamodels are becoming increasingly popular for handling large-scale optimization problems in product development. Metamodel-based reliability-based design optimization (RBDO) helps to improve the computational efficiency and reliability of optimal design. However, a metamodel in engineering applications is an approximation of a high-fidelity computer-aided engineering model and it frequently suffers from a significant loss of predictive accuracy. This issue must be appropriately addressed before the metamodels are ready to be applied in RBDO. In this article, an enhanced strategy with metamodel selection and bias correction is proposed to improve the predictive capability of metamodels. A similarity-based assessment for metamodel selection (SAMS) is derived from the cross-validation and similarity theories. The selected metamodel is then improved by Bayesian inference-based bias correction. The proposed strategy is illustrated through an analytical example and further demonstrated with a lightweight vehicle design problem. The results show its potential in handling real-world engineering problems.     

A comparison of deterministic,reliability-based and risk-based structural optimization under uncertainty

In this paper, the effects of uncertainty and expected costs of failure on optimum structural design are investigated, by comparing three distinct formulations of structural optimization problems. Deterministic Design Optimization (DDO) allows one the find the shape or configuration of a structure that is optimum in terms of mechanics, but the formulation grossly neglects parameter uncertainty and its effects on structural safety. Reliability-based Design Optimization (RBDO) has emerged as an alternative to properly model the safety-under-uncertainty part of the problem. With RBDO, one can ensure that a minimum (and measurable) level of safety is achieved by the optimum structure. However, results are dependent on the failure probabilities used as constraints in the analysis. Risk optimization (RO) increases the scope of the problem by addressing the compromising goals of economy and safety. This is accomplished by quantifying the monetary consequences of failure, as well as the costs associated with construction, operation and maintenance. RO yields the optimum topology and the optimum point of balance between economy and safety. Results are compared for some example problems. The broader RO solution is found first, and optimum results are used as constraints in DDO and RBDO. Results show that even when optimum safety coefficients are used as constraints in DDO, the formulation leads to configurations which respect these design constraints, reduce manufacturing costs but increase total expected costs (including expected costs of failure). When (optimum) system failure probability is used as a constraint in RBDO, this solution also reduces manufacturing costs but by increasing total expected costs. This happens when the costs associated with different failure modes are distinct. Hence, a general equivalence between the formulations cannot be established. Optimum structural design considering expected costs of failure cannot be controlled solely by safety factors nor by failure probability constraints, but will depend on actual structural configuration.     

Reliability-based design optimization of knuckle component using conservative method of moving least squares meta-models

This paper discusses reliability-based design optimization (RBDO) of an automotive knuckle component under bump and brake loading conditions. The probabilistic design problem is to minimize the weight of a knuckle component subject to stresses, deformations, and frequency constraints in order to meet the given target reliability. The initial design is generated based on an actual vehicle specification. The finite element analysis is conducted using ABAQUS, and the probabilistic optimal solutions are obtained via the moving least squares method (MLSM) in the context of approximate optimization. For the meta-modeling of inequality constraint functions, a constraint-feasible moving least squares method (CF-MLSM) is used in the present study. The method of CF-MLSM based RBDO has been shown to not only ensure constraint feasibility in a case where the meta-model-based RBDO process is employed, but also to require low expense, as compared with both conventional MLSM and non-approximate RBDO methods.     

Reliability based design optimization using response surfaces in application to multidisciplinary systems

Traditionally, reliability based design optimization (RBDO) is formulated as a nested optimization problem. For these problems the objective is to minimize a cost function while satisfying the reliability constraints. The reliability constraints are usually formulated as constraints on the probability of failure corresponding to each of the failure modes or a single constraint on the system probability of failure. The probability of failure is usually estimated by performing a reliability analysis. The difficulty in evaluating reliability constraints comes from the fact that modern reliability analysis methods are themselves formulated as an optimization problem. Solving such nested optimization problems is extremely expensive for large scale multidisciplinary systems which are likewise computationally intensive. In this research, a framework for performing reliability based multidisciplinary design optimization using approximations is developed. Response surface approximations (RSA) of the limit state functions are used to estimate the probability of failure. An outer loop is incorporated to ensure that the approximate RBDO converges to the actual most probable point of failure. The framework is compared with the exact RBDO procedure. In the proposed methodology, RSAs are employed to significantly reduce the computational expense associated with traditional RBDO. The proposed approach is implemented in application to multidisciplinary test problems, and the computational savings and benefits are discussed.     

Decoupled approach to multidisciplinary design optimization under uncertainty

We propose solution methods for multidisciplinary design optimization (MDO) under uncertainty. This is a class of stochastic optimization problems that engineers are often faced with in a realistic design process of complex systems. Our approach integrates solution methods for reliability-based design optimization (RBDO) with solution methods for deterministic MDO problems. The integration is enabled by the use of a deterministic equivalent formulation and the first order Taylor’s approximation in these RBDO methods. We discuss three specific combinations: the RBDO methods with the multidisciplinary feasibility method, the all-at-once method, and the individual disciplinary feasibility method. Numerical examples are provided to demonstrate the procedure. Anukal Chiralaksanakul is currently a full-time lecturer in the Graduate School of Business Administration at National Institute of Development Administration (NIDA), Bangkok, Thailand.     

Reliability-based design optimization with progressive surrogate models

Reliability-based design optimization (RBDO) has traditionally been solved as a nested (bilevel) optimization problem, which is a computationally expensive approach. Unilevel and decoupled approaches for solving the RBDO problem have also been suggested in the past to improve the computational efficiency. However, these approaches also require a large number of response evaluations during optimization. To alleviate the computational burden, surrogate models have been used for reliability evaluation. These approaches involve construction of surrogate models for the reliability computation at each point visited by the optimizer in the design variable space. In this article, a novel approach to solving the RBDO problem is proposed based on a progressive sensitivity surrogate model. The sensitivity surrogate models are built in the design variable space outside the optimization loop using the kriging method or the moving least squares (MLS) method based on sample points generated from low-discrepancy sampling (LDS) to estimate the most probable point of failure (MPP). During the iterative deterministic optimization, the MPP is estimated from the surrogate model for each design point visited by the optimizer. The surrogate sensitivity model is also progressively updated for each new iteration of deterministic optimization by adding new points and their responses. Four example problems are presented showing the relative merits of the kriging and MLS approaches and the overall accuracy and improved efficiency of the proposed approach.     

The Design Space Root Finding method for efficient risk optimization by simulation

Reliability-Based Design Optimization (RBDO) is computationally expensive due to the nested optimization and reliability loops. Several shortcuts have been proposed in the literature to solve RBDO problems. However, these shortcuts only apply when failure probability is a design constraint. When failure probabilities are incorporated in the objective function, such as in total life-cycle cost or risk optimization, no shortcuts were available to this date, to the best of the authors knowledge. In this paper, a novel method is proposed for the solution of risk optimization problems. Risk optimization allows one to address the apparently conflicting goals of safety and economy in structural design. In the conventional solution of risk optimization by Monte Carlo simulation, information concerning limit state function behavior over the design space is usually disregarded. The method proposed herein consists in finding the roots of the limit state function in the design space, for all Monte Carlo samples of random variables. The proposed method is compared to the usual method in application to one and n-dimensional optimization problems, considering various degrees of limit state and cost function nonlinearities. Results show that the proposed method is almost twenty times more efficient than the usual method, when applied to one-dimensional problems. Efficiency is reduced for higher dimensional problems, but the proposed method is still at least two times more efficient than the usual method for twenty design variables. As the efficiency of the proposed method for higher-dimensional problems is directly related to derivative evaluations, further investigation is necessary to improve its efficiency in application to multi-dimensional problems.     

Structural design of high-performance capacitive accelerometers using parametric optimization with uncertainties

Electrostatic or capacitive accelerometers are among the highest volume microelectromechanical systems (MEMS) products nowadays. The design of such devices is a complex task, since they depend on many performance requirements, which are often conflicting. Therefore, optimization techniques are often used in the design stage of these MEMS devices. Because of problems with reliability, the technology of MEMS is not yet well established. Thus, in this work, size optimization is combined with the reliability-based design optimization (RBDO) method to improve the performance of accelerometers. To account for uncertainties in the dimensions and material properties of these devices, the first order reliability method is applied to calculate the probabilities involved in the RBDO formulation. Practical examples of bulk-type capacitive accelerometer designs are presented and discussed to evaluate the potential of the implemented RBDO solver.     

Reliability-based design optimization with frequency constraints using a new safest point approach

In the reliability-based design optimization (RBDO) model, the mean values of uncertain variables are usually applied as design variables, and the cost is optimized subject to prescribed probabilistic constraints as defined by a nonlinear mathematical programming problem. Therefore, an RBDO solution that reduces the structural weight in non-critical regions provides not only an improved design, but also a higher level of confidence in the design. Solving such nested optimization problems is extremely expensive for large-scale multidisciplinary systems that are likewise computationally intensive. This article focuses on the study of a particular problem representing the failure mode of structural vibration analysis. A new method is proposed, called safest point, that can efficiently give the reliability-based optimum solution under frequency constraints, and then several probability distributions are developed, which are mathematically nonlinear functions, for the proposed method. Finally, the efficiency of the extended approach is demonstrated for probability distributions such as log-normal and uniform distributions, and its applicability to the design of structures undergoing fluid–structure interaction phenomena, especially the design process of aeroelastic structures, is also demonstrated.     

An integrated reliability-based design optimization of offshore towers

After recognizing the uncertainty in the parameters such as material, loading, geometry and so on in contrast with the conventional optimization, the reliability-based design optimization (RBDO) concept has become more meaningful to perform an economical design implementation, which includes a reliability analysis and an optimization algorithm. RBDO procedures include structural analysis, reliability analysis and sensitivity analysis both for optimization and for reliability. The efficiency of the RBDO system depends on the mentioned numerical algorithms. In this work, an integrated algorithms system is proposed to implement the RBDO of the offshore towers, which are subjected to the extreme wave loading. The numerical strategies interacting with each other to fulfill the RBDO of towers are as follows: (a) a structural analysis program, SAPOS, (b) an optimization program, SQP and (c) a reliability analysis program based on FORM. A demonstration of an example tripod tower under the reliability constraints based on limit states of the critical stress, buckling and the natural frequency is presented.     

Topology optimization for reinforcement of no-tension structures

In the paper, the reinforcement of no-tension structures by the application of superposed high-strength sheets, or by the insertion of tensile bars, has been considered with the purpose to set up a design path aiming at the positioning of the new material according to some optimal criterion. In detail, no-tension models are adopted which are recognized as an effective tool for analyzing a wide class of structures (e.g., masonry and reinforced concrete members), and the equilibrium and the failure analysis of the reinforced body are developed with particular reference to its ultimate limit state of collapse. Finally, an approach through the “topologic optimization” is proposed for the identification of the optimal distribution of the reinforcement, and some of the obtained results are shown.     

应变速率型RC梁柱显式分析单元及其在ABAQUS软件中的实现

为准确、高效地分析钢筋混凝土(RC)梁柱结构在强动载作用下的损伤破坏甚至倒塌,建立了能够描述大变形、大应变的空间纤维Timoshenko梁单元,并将其与考虑率相关效应的钢筋、混凝土材料模型相结合,通过用户显式单元子程序在ABAQUS中实现。基于所建立的应变速率型3D纤维梁显式单元,借助ABAQUS的前后处理及求解功能,对爆炸荷载作用下的RC梁柱构件的动态响应和破坏模式及RC框架结构的连续性倒塌进行分析。结果表明：建立的应变速率型3D纤维梁柱单元能够合理描述RC构件的变形特性及钢筋、混凝土的应变速率效应;可以模拟爆炸作用下RC构件的弯曲、弯剪及直剪破坏模式,以及RC框架的连续倒塌过程;将纤维梁柱单元与率相关模型相结合于ABAQUS软件提供了一种强动载作用下高效、精确的RC结构非线性分析方法。     

Reliability-based design optimization using a moment method and a kriging metamodel

Reliability-based design optimization (RBDO) has been used for optimizing engineering systems with uncertainties in design variables and system parameters. RBDO involves reliability analysis, which requires a large amount of computational effort, so it is important to select an efficient method for reliability analysis. Of the many methods for reliability analysis, a moment method, which is called the fourth moment method, is known to be less expensive for moderate size problems and requires neither iteration nor the computation of derivatives. Despite these advantages, previous research on RBDO has been mainly based on the first-order reliability method and relatively little attention has been paid to moment-based RBDO. This article considers difficulties in implementing the moment method into RBDO; they are solved using a kriging metamodel with an active constraint strategy. Three numerical examples are tested and the results show that the proposed method is efficient and accurate.     

Bidirectional evolutionary structural optimization for nonlinear structures under dynamic loads

This study aims to develop efficient numerical optimization methods for finding the optimal topology of nonlinear structures under dynamic loads. The numerical models are developed using the bidirectional evolutionary structural optimization method for stiffness maximization problems with mass constraints. The mathematical formulation of topology optimization approach is developed based on the element virtual strain energy as the design variable and minimization of compliance as the objective function. The suitability of the proposed method for topology optimization of nonlinear structures is demonstrated through a series of two- and three-dimensional benchmark designs. Several issues relating to the nonlinear structures subjected to dynamic loads such as material, geometric, and contact nonlinearities are addressed in the examples. It is shown that the proposed approach generates more reliable designs for nonlinear structures.     

Retracted: Stability analysis of blocky structure system using discontinuity layout optimization

The collapse mechanism identification and limit load calculation of block composite structures are essential tasks in practical engineering. In this work, the discontinuity layout optimization (DLO) is utilized to simulate a stable blocky system structure under static and pseudostatic loading by considering soil–structure interaction effects. The program refers to the discretization of the system under consideration by utilizing the potential slip lines to connect nodes. Then, linear programming is applied to identify the critical layout. The DLO procedure is extended to mimic rotations in the approximate simulation of rotational and translational failures along boundaries, thus simulating the mechanism on the rotating block. The simulated failure mechanism and collapse load are consistent with the experimental observations and numerical modeling results. Furthermore, the procedure is applied to two potential practical applications: the stability analysis of a jointed rock slope and concrete masonry retaining wall. The results show that the DLO is a simple but scientific method for identifying the mechanism of the critical failure of blocky structures.     

Thermomechanical topology optimization of shape-memory alloy structures using a transient bilevel adjoint method

We present a novel method for computational design of adaptive shape-memory alloy (SMA) structures via topology optimization. By optimally distributing a SMA within the prescribed design domain, the proposed algorithm seeks to tailor the two-way shape-memory effect (TWSME) and pseudoelasticity response of the SMA materials. Using a phenomenological material model, the thermomechanical response of the SMA structure is solved through inelastic finite element analysis, while assuming a transient but spatially uniform temperature distribution. The material distribution is parameterized via a SIMP formulation, with gradient-based optimization used to perform the optimization search. We derive a transient, bilevel adjoint formulation for analytically computing the design sensitivities. We demonstrate the proposed design framework using a series of two-dimensional thermomechanical benchmark problems. These examples include design for optimal displacement due to the TWSME, and design for maximum mechanical advantage while accounting for pseudoelasticity.     

Topology optimization of compliant mechanisms using the homogenization method

A procedure to obtain a topology of an optimal structure considering flexibility is presented. The methodology is based on a mutual energy concept for formulation of flexibility and the homogenization method. A multi-objective optimization problem is formulated as an application of compliant mechanism design. Some examples of the design of compliant mechanisms for plane structures are presented. © 1998 John Wiley & Sons, Ltd.     

An accurate and efficient reliability-based design optimization using the second order reliability method and improved stability transformation method

The first order reliability method has been extensively adopted for reliability-based design optimization (RBDO), but it shows inaccuracy in calculating the failure probability with highly nonlinear performance functions. Thus, the second order reliability method is required to evaluate the reliability accurately. However, its application for RBDO is quite challenge owing to the expensive computational cost incurred by the repeated reliability evaluation and Hessian calculation of probabilistic constraints. In this article, a new improved stability transformation method is proposed to search the most probable point efficiently, and the Hessian matrix is calculated by the symmetric rank-one update. The computational capability of the proposed method is illustrated and compared to the existing RBDO approaches through three mathematical and two engineering examples. The comparison results indicate that the proposed method is very efficient and accurate, providing an alternative tool for RBDO of engineering structures.