A novel shape function approach of dynamic load identification for the structures with interval uncertainty

Aiming at uncertain structures, a computational inverse approach is proposed to identify the dynamic load on the basis of the shape function method and interval analysis. The forward model for an uncertain structure is established through the relationship between the uncertain load vector and the assembly matrix of the uncertain responses of the shape function loads in each discrete element in time domain. The uncertainty is characterized by the interval with a closed bounded set of uncertain parameters. On the basis of interval analysis method, the load identification for uncertain structures can be transformed into two kinds of deterministic inverse problems, namely the deterministic dynamic load identification and the first order derivatives of the unknown load to each parameter both at the midpoints of the uncertain parameters. In order to eliminate the ill-posedness of inversion, the regularization method is adopted to solve the deterministic equations. Two numerical examples demonstrates the effectiveness of the proposed method, and example one also gives the identified result using Monte Carlo method to compare with that using the proposed method.     

An interval approach to robust design with parameter uncertainty

In robust design, it is common to estimate empirical models that relate an output response variable to controllable input variables and uncontrollable noise variables from experimental data. However, when determining the optimal input settings that minimise output variability, parameter uncertainties in noise factors and response models are typically neglected. This article presents an interval robust design approach that takes parameter uncertainties into account through the confidence regions for these unknown parameters. To avoid obtaining an overly conservative design, the worst and best cases of mean squared error are both adopted to build an optimisation approach. The midpoint and radius of the interval are used to measure the location and dispersion performances, respectively. Meanwhile, a data-driven method is applied to obtain the relative weights of the location and dispersion performances in the optimisation approach. A simulation example and a case study using automobile manufacturing data from the dimensional tolerance design process are used to demonstrate the effectiveness of the proposed approach. The proposed approach of considering both uncertainties is shown to perform better than other approaches.     

Distributionally robust optimization for engineering design under uncertainty

This paper addresses the challenge of design optimization under uncertainty when the designer only has limited data to characterize uncertain variables. We demonstrate that the error incurred when estimating a probability distribution from limited data affects the out-of-sample performance (ie, performance under the true distribution) of optimized designs. We demonstrate how this can be mitigated by reformulating the engineering design problem as a distributionally robust optimization (DRO) problem. We present computationally efficient algorithms for solving the resulting DRO problem. The performance of the DRO approach is explored in a practical setting by applying it to an acoustic horn design problem. The DRO approach is compared against traditional approaches to optimization under uncertainty, namely, sample-average approximation and multiobjective optimization incorporating a risk reduction objective. In contrast with the multiobjective approach, the proposed DRO approach does not use an explicit risk reduction objective but rather specifies a so-called ambiguity set of possible distributions and optimizes against the worst-case distribution in this set. Our results show that the DRO designs, in some cases, significantly outperform those designs found using the sample-average or the multiobjective approach.     

A robust optimization approach to overall profit efficiency with data uncertainty: application on bank industry

In this paper, we consider a framework of data envelopment analysis (DEA) to measure the overall profit efficiency of decision-making units (DMUs) subject to inputs and outputs uncertainty. Under uncertain conditions, classic methods can lead to unrealistic solutions in practice. In this work, robust optimization is proposed to incorporate uncertainty into measuring the overall profit efficiency. In a robust optimization model, it is supposed that uncertain parameters belong to a specified set with a solution that is efficient for all possible uncertainty outcomes while it is not optimal for a given value of the parameters. We show that the overall profit efficiency score may not always occur in an optimistic case and the decision maker can obtain the overall profit efficiency score corresponding to a value in the uncertainty set. The results of the experiment on bank data show that a robust overall profit efficiency score provides a significant improvement in the performance, as the uncertainty increases.

Abbreviations: DEA: data envelopment analysis; DMUs: decision-making units; CRS: constant returns to scale; VRS: variable returns to scale; ROP: robust optimization problem; RC: robust counterpart; ROPE: robust overall profit efficiency; OOPE: optimistic overall profit efficiency; GAMS: generalized algebraic modeling system     

An enhanced fuzzy robust optimization model for regional solid waste management under uncertainty

In this study, an enhanced fuzzy robust optimization (EFRO) model is proposed for supporting regional solid waste management under uncertainty. This model is an extended version of robust optimization from a stochastic to a fuzzy environment, and novel in the following two aspects: (1) it uses multiple algorithms to tackle fuzzy constraints according to their characteristics; and (2) it incorporates fuzzy violation variables into the model, which could effectively reflect the trade-off between system economy and reliability. The regional waste management of the City of Dalian, China, was used as a case study for demonstration. A variety of solutions was obtained under various weight coefficients and confidence levels. From the case study, it was found that EFRO could help decision makers to design desired waste management alternatives under complex uncertainties. The successful application of EFRO in the studied real case is expected to be a good example for solid waste management in many other cities.     

Boundary integral equation method for shape optimization of elastic structures

A general method for shape design sensitivity analysis as applied to plane elasticity problems is developed with a direct boundary integral equation formulation, using the material derivative concept and adjoint variable method. The problem formulation is very general and a complete consideration is given to describing the boundary variation by including the tangential component of the velocity field. The method is then applied to obtain the sensitivity formula for a general stress constraint imposed over a small part of the boundary. The accuracy of the design sensitivity analysis is studied with a fillet and an elastic ring design problem. Among the several numerical implementations tested, the second order boundary elements with a cubic spline representation of the moving boundary have shown the best accuracy. A smooth characteristic function is found to be better than a plateau function for localization of the stress constraint. Optimal shapes for the two problems are presented to show numerical applications.     

A PDE-constrained optimization approach for topology optimization of strained photonic devices

Recent studies have demonstrated the potential of using tensile-strained, doped Germanium as a means of developing an integrated light source for (amongst other things) future microprocessors. In this work, a multi-material phase-field approach to determine the optimal material configuration within a so-called Germanium-on-Silicon microbridge is considered. Here, an “optimal” configuration is one in which the strain in a predetermined minimal optical cavity within the Germanium is maximized according to an appropriately chosen objective functional. Due to manufacturing requirements, the emphasis here is on the cross-section of the device; i.e. a so-called aperture design. Here, the optimization is modeled as a non-linear optimization problem with partial differential equation and manufacturing constraints. The resulting problem is analyzed and solved numerically. The theory portion includes a proof of existence of an optimal topology, differential sensitivity analysis of the displacement with respect to the topology, and the derivation of first- and second-order optimality conditions. For the numerical experiments, an array of first- and second-order solution algorithms in function-space are adapted to the current setting, tested, and compared. The numerical examples yield designs for which a significant increase in strain (as compared to an intuitive empirical design) is observed.     

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.     

A robust optimization approach for the capacitated vehicle routing problem with demand uncertainty

In this paper we introduce a robust optimization approach to solve the Vehicle Routing Problem (VRP) with demand uncertainty. This approach yields routes that minimize transportation costs while satisfying all demands in a given bounded uncertainty set. We show that for the Miller-Tucker-Zemlin formulation of the VRP and specific uncertainty sets, solving for the robust solution is no more difficult than solving a single deterministic VRP. Our computational results on benchmark instances and on families of clustered instances show that the robust solution can protect from unmet demand while incurring a small additional cost over deterministic optimal routes. This is most pronounced for clustered instances under moderate uncertainty, where remaining vehicle capacity is used to protect against variations within each cluster at a small additional cost. We compare the robust optimization model with classic stochastic VRP models for this problem to illustrate the differences and similarities between them. We also observe that the robust solution amounts to a clever management of the remaining vehicle capacity compared to uniformly and non-uniformly distributing this slack over the vehicles.     

A shape optimization approach for simulating contact of elastic membranes with rigid obstacles

The obstacle problem consists in computing equilibrium shapes of elastic membranes in contact with rigid obstacles. In addition to the displacement u of the membrane, the interface Γ on the membrane demarcating the region in contact with the obstacle is also an unknown and plays the role of a free boundary. Numerical methods that simulate obstacle problems as variational inequalities share the unifying feature of first computing membrane displacements and then deducing the location of the free boundary a posteriori. We present a shape optimization-based approach here that inverts this paradigm by considering the free boundary to be the primary unknown and compute it as the minimizer of a certain shape functional using a gradient descent algorithm. In a nutshell, we compute Γ then u, and not u then Γ. Our approach proffers clear algorithmic advantages. Unilateral contact constraints on displacements, which render traditional approaches into expensive quadratic programs, appear only as Dirichlet boundary conditions along the free boundary. Displacements of the membrane need to be approximated only over the noncoincidence set, thereby rendering smaller discrete problems to be resolved. The issue of suboptimal convergence of finite element solutions stemming from the reduced regularity of displacements across the free boundary is naturally circumvented. Most importantly perhaps, our numerical experiments reveal that the free boundary can be approximated to within distances that are two orders of magnitude smaller than the mesh size used for spatial discretization. The success of the proposed algorithm relies on a confluence of factors- choosing a suitable shape functional, representing free boundary iterates with smooth implicit functions, an ansatz for the velocity of the free boundary that helps realize a gradient descent scheme and triangulating evolving domains with universal meshes. We discuss these aspects in detail and present numerous examples examining the performance of the algorithm.     

I-VFRP: An interval-valued fuzzy robust programming approach for municipal waste-management planning under uncertainty

In this study, an interval-valued fuzzy robust programming (I-VFRP) model has been developed and applied to municipal solid-waste management under uncertainty. The I-VFRP model can explicitly address system uncertainties with multiple presentations, and can directly communicate the waste manager's confidence gradients into the optimization process, facilitating the reflection of weak or strong confidence when subjectively estimating parameter values. Parameters in the I-VFRP model can be represented as either intervals or interval-valued fuzzy sets. Thus, variations of the waste manager's confidence gradients over defining parameters can be effectively handled through interval-valued membership functions, leading to enhanced robustness of the optimization efforts. The results of a theoretical case study indicate that useful solutions for planning municipal solid-waste-management practices can be generated. The waste manager's confidence gradients over various subjective judgments can be directly incorporated into the modeling formulation and solution process. The results also suggest that the proposed methodology can be applied to practical problems that are associated with complex and uncertain information.     

Optimisation approach to target costing under uncertainty with application to ICT-service

Target costing is a modern approach applied during product development that defines cost targets for products and its components. These cost targets are driven by customer requirements and achievable revenues. The intention of this paper is the integration of target costing with modern concepts of modelling uncertainty and management of risk based on optimisation. Contrary to the traditional focus of target costing on cost targets, this paper prefers a strategy for achieving a target profit. Moreover, in this paper target costing is understood as a continuous process with incremental changes of cost drivers, product and component design as well as product prices. Therefore, the change in costs and profit with respect to aforementioned control parameters is modelled by linear approximations. Hence, improved decisions concerning design and prices are derived by linear programming models. In practice, information concerning product and component costs, demand or customer preferences are not given with certainty. Therefore, we apply a stochastic programming approach to manage the risk inherent in the target costing process. After a general presentation, we apply our approach to the provision of an information and communication technology service where the level of uncertainty is considerable.     

Level-set topology optimization for robust design of structures under hybrid uncertainties

This paper will develop a new robust topology optimization (RTO) method based on level sets for structures subject to hybrid uncertainties, with a more efficient Karhunen-Loève hyperbolic Polynomial Chaos–Chebyshev Interval method to conduct the hybrid uncertain analysis. The loadings and material properties are considered hybrid uncertainties in structures. The parameters with sufficient information are regarded as random fields, while the parameters without sufficient information are treated as intervals. The Karhunen-Loève expansion is applied to discretize random fields into a finite number of random variables, and then, the original hybrid uncertainty analysis is transformed into a new process with random and interval parameters, to which the hyperbolic Polynomial Chaos–Chebyshev Interval is employed for the uncertainty analysis. RTO is formulated to minimize a weighted sum of the mean and standard variance of the structural objective function under the worst-case scenario. Several numerical examples are employed to demonstrate the effectiveness of the proposed RTO, and Monte Carlo simulation is used to validate the numerical accuracy of our proposed method.     

A computational approach for robust nondestructive test design maximizing characterization capabilities for solids and structures subject to uncertainty

A robust approach to nondestructive test (NDT) design for material characterization and damage identification in solids and structures is presented and numerically evaluated. The generally applicable approach combines maximization of test sensitivity with minimization of test information redundancy, while simultaneously minimizing the effects of uncertain system parameters to determine optimal NDT parameters for robust nondestructive evaluation. In addition, to maintain reasonable computational expense while also allowing for general applicability, a stochastic collocation technique is presented for the quantification of uncertainty in the robust design metrics. The robust NDT design approach was tested through simulated case studies based on the characterization of localized variations in Young's modulus distributions in aluminum structural components utilizing steady‐state dynamic surface excitation and localized measurements of displacement and compared with a purely deterministic NDT design approach. The robust design approach is shown to produce substantially different NDT designs than the analogous deterministic strategy. More importantly, the robust NDT designs are shown to provide significant improvements in the ability to accurately nondestructively evaluate structural properties for the cases considered, when there is significant uncertainty in system parameters and/or aspects of the NDT implementation. Copyright © 2015 John Wiley & Sons, Ltd.     

An optimality criterion method for dynamic optimization of structures

This paper presents an optimality criterion method for the determination of the least weight design of a structural system which satisfies a specific frequency requirement plus upper and lower bounds on the design variables. The design algorithm is an iterative solution of the Kuhn–Tucker optimality criterion based on choosing a single value of the Lagrange multiplier which minimizes the sum of the squares of residuals. The method has been applied to a variety of structures. Results assert that the method is capable of locating the optimal design in a small number of redesign cycles. The method avoids the scaling of design variables. It can treat non-structural masses and is applicable to. structural elements with a wide variety of size-stiffness. The procedure has been completely automated in a computer program on an IBM-PC microcomputer.     

Horsetail matching: a flexible approach to optimization under uncertainty*

It is important to design engineering systems to be robust with respect to uncertainties in the design process. Often, this is done by considering statistical moments, but over-reliance on statistical moments when formulating a robust optimization can produce designs that are stochastically dominated by other feasible designs. This article instead proposes a formulation for optimization under uncertainty that minimizes the difference between a design's cumulative distribution function and a target. A standard target is proposed that produces stochastically non-dominated designs, but the formulation also offers enough flexibility to recover existing approaches for robust optimization. A numerical implementation is developed that employs kernels to give a differentiable objective function. The method is applied to algebraic test problems and a robust transonic airfoil design problem where it is compared to multi-objective, weighted-sum and density matching approaches to robust optimization; several advantages over these existing methods are demonstrated.     

Life-cycle reliability-based robust design optimization for GP model with response uncertainty

Reliability-based robust design optimization (RBRDO) is a crucial tool for life-cycle quality improvement. Gaussian process (GP) model is an effective alternative modeling technique that is widely used in robust parameter design. However, there are few studies to deal with reliability-based design problems by using GP model. This article proposes a novel life-cycle RBRDO approach concerning response uncertainty under the framework of GP modeling technique. First, the hyperparameters of GP model are estimated by using the Gibbs sampling procedure. Second, the expected partial derivative expression is derived based on GP modeling technique. Moreover, a novel failure risk cost function is constructed to assess the life-cycle reliability. Then, the quality loss function and confidence interval are constructed by simulated outputs to evaluate the robustness of optimal settings and response uncertainty, respectively. Finally, an optimization model integrating failure risk cost function, quality loss function, and confidence interval analysis approach is constructed to find reasonable optimal input settings. Two case studies are given to illustrate the performance of the proposed approach. The results show that the proposed approach can make better trade-offs between the quality characteristics and reliability requirements by considering response uncertainty.     

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.     

An approach to robust optimization of impact problems using random samples and meta-modelling

Conventionally optimized structures may show a tendency to be sensitive to variations, for instance in geometry and loading conditions. To avoid this, research has been carried out in the field of robust optimization where variations are taken into account in the optimization process. The overall objective is to create solutions that are optimal both in the sense of mean performance and minimum variability. This work presents an alternative approach to robust optimization, where the robustness of each design is assessed through multiple sampling of the stochastic variables at each design point. Meta-models for the robust optimization are created for both the mean value and the standard deviation of the response. Furthermore, the method is demonstrated on an analytical example and an example of an aluminium extrusion with quadratic cross-section subjected to axial crushing. It works well for the chosen examples and it is concluded that the method is especially well suited for problems with a large number of random variables, since the computational cost is essentially independent of the number of random variables. In addition, the presented approach makes it possible to take into consideration variations that cannot be described with a variable. This is demonstrated in this work by random geometrical perturbations described with the use of Gaussian random fields.     

A multi-objective optimization model for hub network design under uncertainty: An inexact rough-interval fuzzy approach

This article proposes a multi-objective mixed-integer model to optimize the location of hubs within a hub network design problem under uncertainty. The considered objectives include minimizing the maximum accumulated travel time, minimizing the total costs including transportation, fuel consumption and greenhouse emissions costs, and finally maximizing the minimum service reliability. In the proposed model, it is assumed that for connecting two nodes, there are several types of arc in which their capacity, transportation mode, travel time, and transportation and construction costs are different. Moreover, in this model, determining the capacity of the hubs is part of the decision-making procedure and balancing requirements are imposed on the network. To solve the model, a hybrid solution approach is utilized based on inexact programming, interval-valued fuzzy programming and rough interval programming. Furthermore, a hybrid multi-objective metaheuristic algorithm, namely multi-objective invasive weed optimization (MOIWO), is developed for the given problem. Finally, various computational experiments are carried out to assess the proposed model and solution approaches.