Optimal design of composite laminated plates with the discreteness in ply angles and uncertainty in material properties considered

An algorithm is developed for optimizing laminated plate stacking sequences and determining thicknesses, which incorporates discrete ply angles and considers the uncertainties of material properties in a two-step optimization process. The branch and bound method was modified to handle discrete variables; and convex modeling was used to allow the consideration of variable material properties. The numerical results obtained show that the optimal stacking sequences are determined with fewer evaluations of the objective function than might be expected from considerations of the size of the design space. Our results also show that the optimal thickness increases when elastic moduli uncertainties are considered, which indicates that such uncertainties should not be ignored at the design stage.     

Efficient global sensitivity analysis with correlated variables

Existing methods for the computation of global sensitivity indices are challenged by both number of input-output samples required and the presence of dependent or correlated variables. First, a methodology is developed to increase the efficiency of sensitivity computations with independent variables by incorporating optimal space-filling quasi-random sequences into an existing importance sampling-based kernel regression sensitivity method. Two prominent situations where parameter correlations cannot be ignored, however, are (1) posterior distributions of calibrated parameters and (2) transient, coupled simulations. Therefore, the sensitivity methodology is generalized to dependent variables allowing for efficient post-calibration sensitivity analyses using input-output samples obtained directly from Bayesian calibration. These methods are illustrated using coupled, aerothermal simulations where it is observed that model errors and parameter correlations control the sensitivity estimates until coupling effects become dominant over time.     

Derivative based global sensitivity measures and their link with global sensitivity indices

A model function f(x₁,…,x_n) defined in the unit hypercube Hⁿ with Lebesque measure dx = dx₁…dx_n is considered. If the function is square integrable, global sensitivity indices provide adequate estimates for the influence of individual factors x_i or groups of such factors. Alternative estimators that require less computer time can also be used. If the function f is differentiable, functionals depending on ∂f/∂x_i have been suggested as estimators for the influence of x_i. The Morris importance measure modified by Campolongo, Cariboni and Saltelli μ  * is an approximation of the functional μi=_∫Hn|∂f/∂xi|dx${\mu }_i={\int }_{H^n}\left|\partial f\text{/}\partial x_i\right|dx$.     

Line sampling-based local and global reliability sensitivity analysis

Structural and Multidisciplinary Optimization - Local reliability sensitivity (RS) and global RS can provide useful information in reliability-based design optimization, but the algorithm for...     

Variational sensitivity analysis and design optimization

Variational method (VM) is employed to derive the co-state equations, boundary (transversality) conditions, and functional sensitivity derivatives. The converged solutions of the state equations together with the steady state solution of the co-state equations are integrated along the domain boundary to uniquely determine the functional sensitivity derivatives with respect to the design function. The application of the variational method to aerodynamic shape optimization problems is demonstrated on internal flow problems at supersonic Mach number range of 1.5. Optimization results for flows with and without shock phenomena are presented. The study shows that while maintaining the accuracy of aerodynamical objective function and constraint within the reasonable range for engineering prediction purposes, variational method provides a substantial gain in computational efficiency, i.e., computer time and memory, when compared with the finite difference computations.     

Surrogate-assisted global sensitivity analysis: an overview

Surrogate models are popular tool to approximate the functional relationship of expensive simulation models in multiple scientific and engineering discipli     

A global sensitivity analysis in structural mechanics

A methodology is developed to evaluate the response sensitivity of structural systems to variations in their design parameters. The sensitivity is evaluated by considering the global behavior of the system response when the parameters vary within a bounded region. The design parameters are characterized by means of baseline values plus fluctuating components, and the sensitivity of the system is measured in terms of the global variability of the response with respect to its baseline response. The methodology is then extended into the context of optimum redesign analysis of structural systems. Application of the method is made to a structural system defined by two-dimensional beam-column elements and to a system defined by plate elements. The numerical implementation of the global sensitivity approach is made by means of the finite element method. Several analyses are performed and the results are discussed. Finally, some extensions of the present work are presented.     

A new variance-based global sensitivity analysis technique

A new set of variance-based sensitivity indices, called W$W$-indices, is proposed. Similar to the Sobol’s indices, both main and total effect indices are defined. The W$W$-main effect indices measure the average reduction of model output variance when the ranges of a set of inputs are reduced, and the total effect indices quantify the average residual variance when the ranges of the remaining inputs are reduced. Geometrical interpretations show that the W$W$-indices gather the full information of the variance ratio function, whereas, Sobol’s indices only reflect the marginal information. Then the double-loop-repeated-set Monte Carlo (MC) (denoted as DLRS MC) procedure, the double-loop-single-set MC (denoted as DLSS MC) procedure and the model emulation procedure are introduced for estimating the W$W$-indices. It is shown that the DLRS MC procedure is suitable for computing all the W$W$-indices despite its highly computational cost. The DLSS MC procedure is computationally efficient, however, it is only applicable for computing low order indices. The model emulation is able to estimate all the W$W$-indices with low computational cost as long as the model behavior is correctly captured by the emulator. The Ishigami function, a modified Sobol’s function and two engineering models are utilized for comparing the W$W$- and Sobol’s indices and verifying the efficiency and convergence of the three numerical methods. Results show that, for even an additive model, the W$W$-total effect index of one input may be significantly larger than its W$W$-main effect index. This indicates that there may exist interaction effects among the inputs of an additive model when their distribution ranges are reduced.     

Structural modelling and sensitivity analysis of shape optimization

It is presented in this paper that the structural modelling of shape optimization is composed of, in general cases, four distinct processes on geometry, design, analysis and perturbation models. The relationships between these models are discussed. An integrated modelling approach based on geometric shape parameterization and automatic mesh generation is proposed. In cooperation with this modelling approach, the semi-analytic sensitivity analysis has been effectively employed. These techniques join shape optimization with FEM and CAD packages and apply it versatilely to optimum designs of general structures. The implementation and applications of the integrated modelling approach and semi-analytic sensitivity analysis to shape optimization of structures with coupling of stress and temperature fields are illustrated.Presented at NATO ASI Optimization of Large Structural Systems, held in Berchtesgaden, Germany, Sept. 23 — Oct. 4, 1991     

Structural design sensitivity using boundary elements and polynomial response function

This main issue of this paper is a conjunction of the structural design sensitivity analysis using the Boundary Element Method with the polynomial response function determination. The procedure is so general that it enables sensitivity analysis for potential and elasticity problems within both homogeneous and heterogeneous plane and 3D problems. The essential difference with respect to the previous approaches like the Direct Differentiation Method or the Adjoint Variable Method is in discrete evaluation of the structural response using the response polynomials of some state parameters and design variable as the independent parameter. Such a determination is carried out via the several solutions of the given boundary value problem, where design parameter mean value is regularly perturbed in each of the solutions to cover the closest neighborhood of this mean value. Those few solutions make it possible to recover the polynomial response function from node-to node within the boundary elements, so that further symbolic differentiation using MAPLE returns the sensitivity gradients particular values. The entire procedure is tested here twice—first example deals with the homogeneous cantilever beam, where comparison against pure analytical differentiation is done and, separately, for two-component composite cantilever, where such a comparison is made against the central difference method linked with the same BEM solution.     

Extending a global sensitivity analysis technique to models with correlated parameters

The identification and representation of uncertainty is recognized as an essential component in model applications. One important approach in the identification of uncertainty is sensitivity analysis. Sensitivity analysis evaluates how the variations in the model output can be apportioned to variations in model parameters. One of the most popular sensitivity analysis techniques is Fourier amplitude sensitivity test (FAST). The main mechanism of FAST is to assign each parameter with a distinct integer frequency (characteristic frequency) through a periodic sampling function. Then, for a specific parameter, the variance contribution can be singled out of the model output by the characteristic frequency based on a Fourier transformation. One limitation of FAST is that it can only be applied for models with independent parameters. However, in many cases, the parameters are correlated with one another. In this study, we propose to extend FAST to models with correlated parameters. The extension is based on the reordering of the independent sample in the traditional FAST. We apply the improved FAST to linear, nonlinear, nonmonotonic and real application models. The results show that the sensitivity indices derived by FAST are in a good agreement with those from the correlation ratio sensitivity method, which is a nonparametric method for models with correlated parameters.     

Distance correlation-based method for global sensitivity analysis of models with dependent inputs

Structural and Multidisciplinary Optimization - Global sensitivity analysis (GSA) plays an important role to quantify the relative importance of uncertain parameters to the model response. However,...     

Product platform design through sensitivity analysis and cluster analysis

Scale-based product platform design consists of platform configuration to decide which variables are shared among which product variants, and selection of the optimal values for platform (shared) and non-platform variables for all product variants. The configuration step plays a vital role in determining two important aspects of a product family: efficiency (cost savings due to commonality) and effectiveness (capability to satisfy performance requirements). Many existing product platform design methods ignore it, assuming a given platform configuration. Most approaches, whether or not they consider the configuration step, are single-platform methods, in which design variables are either shared across all product variants or not shared at all. In multiple-platform design, design variables may be shared among variants in any possible combination of subsets, offering opportunities for superior overall design but presenting a more difficult computational problem. In this work, sensitivity analysis and cluster analysis are used to improve both efficiency and effectiveness of a scale-based product family through multiple-platform product family design. Sensitivity analysis is performed on each design variable to help select candidate platform design variables and to provide guidance for cluster analysis. Cluster analysis, using performance loss due to commonization as the clustering criterion, is employed to determine platform configuration. An illustrative example is used to demonstrate the merits of the proposed method, and the results are compared with existing results from the literature.     

Non-parametric methods for global sensitivity analysis of model output with dependent inputs

This paper addresses the issue of performing global sensitivity analysis of model output with dependent inputs. First, we define variance-based sensitivity indices that allow for distinguishing the independent contributions of the inputs to the response variance from their mutual dependent contributions. Then, two sampling strategies are proposed for their non-parametric, numerical estimation. This approach allows us to estimate the sensitivity indices not only for individual inputs but also for groups of inputs. After testing the accuracy of the non-parametric method on some analytical test functions, the approach is employed to assess the importance of dependent inputs on a computer model for the migration of radioactive substances in the geosphere.     

Application of global sensitivity analysis to a tire model with correlated inputs

When a vehicle equipped with tire is manoeuvred on the ground, the tires are submitted to a number of forces – longitudinal force when driving or braking torque is applied to the wheel and/or lateral force when the wheel is steered to turn at a corner. Pacejka model describes these forces that represent the reaction of the road onto the tire. This nonlinear model depends on correlated parameters such as the friction coefficient, the vertical load, and the cornering stiffness, which have to be identified from some measurements. The sensitivity of Pacejka model to these correlated parameters are studied using an approach based on polynomial chaos. It consists in decorrelating the parameters using the Nataf transformation and then, in expanding the model output onto polynomial chaos. The sensitivity indices are then obtained straightforwardly from the algebraic expression of the coefficients of the polynomial expansion.     

Efficient sampling methods for global reliability sensitivity analysis

An important problem in structure reliability analysis is how to reduce the failure probability. In this work, we introduce a main and total effect indices framework of global reliability sensitivity. By decreasing the uncertainty of input variables with high main effect indices, the most reduction of failure probability can be obtained. By decreasing the uncertainty of the input variables with small total effect indices (close to zero), the failure probability will not be reduced significantly. The efficient sampling methods for evaluating the main and total effect indices are presented. For the problem with large failure probability, a single-loop Monte Carlo simulation (MCS) is derived for computing these sensitivity indices. For the problem with small failure probability, the single-loop sampling methods combined with the importance sampling procedure (IS) and the truncated importance sampling procedure (TIS) respectively are derived for improving the calculation efficiency. Two numerical examples and one engineering example are introduced for demonstrating the efficiency and precision of the calculation methods and illustrating the engineering significance of the global reliability sensitivity indices.     

Robust global sensitivity analysis under deep uncertainty via scenario analysis

Complex social-ecological systems models typically need to consider deeply uncertain long run future conditions. The influence of this deep (i.e. incalculable, uncontrollable) uncertainty on model parameter sensitivities needs to be understood and robustly quantified to reliably inform investment in data collection and model refinement. Using a variance-based global sensitivity analysis method (eFAST), we produced comprehensive model diagnostics of a complex social-ecological systems model under deep uncertainty characterised by four global change scenarios. The uncertainty of the outputs, and the influence of input parameters differed substantially between scenarios. We then developed sensitivity indicators that were robust to this deep uncertainty using four criteria from decision theory. The proposed methods can increase our understanding of the effects of deep uncertainty on output uncertainty and parameter sensitivity, and incorporate the decision maker's risk preference into modelling-related activities to obtain greater resilience of decisions to surprise.     

Structural design of composite nonlinear feedback control for linear systems with actuator constraint

The performance of the composite nonlinear feedback (CNF) control law relies on the selection of the linear feedback gain and the nonlinear function. However, it is a tough task to select an appropriate linear feedback gain and appropriate parameters of the nonlinear function because the general design procedure of CNF control just gives some simple guidelines for the selections. This paper proposes an operational design procedure based on the structural decomposition of the linear systems with input saturation. The linear feedback gain is constructed by two linear gains which are designed independently to stabilize the unstable zero dynamics part and the pure integration part of the system respectively. By investigating the influence of these two linear gains on transient performance, it is flexible and efficient to design a satisfactory linear feedback gain for the CNF control law. Moreover, the parameters of the nonlinear function are tuned automatically by solving a minimization problem. The proposed design procedure is illustrated by applying it to design a tracking control law for the inverted pendulum on a cart system. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Eulerian shape design sensitivity analysis and optimization with a fixed grid

Conventional shape optimization based on the finite element method uses Lagrangian representation in which the finite element mesh moves according to shape change, while modern topology optimization uses Eulerian representation. In this paper, an approach to shape optimization using Eulerian representation such that the mesh distortion problem in the conventional approach can be resolved is proposed. A continuum geometric model is defined on the fixed grid of finite elements. An active set of finite elements that defines the discrete domain is determined using a procedure similar to topology optimization, in which each element has a unique shape density. The shape design parameter that is defined on the geometric model is transformed into the corresponding shape density variation of the boundary elements. Using this transformation, it has been shown that the shape design problem can be treated as a parameter design problem, which is a much easier method than the former. A detailed derivation of how the shape design velocity field can be converted into the shape density variation is presented along with sensitivity calculation. Very efficient sensitivity coefficients are calculated by integrating only those elements that belong to the structural boundary. The accuracy of the sensitivity information is compared with that derived by the finite difference method with excellent agreement. Two design optimization problems are presented to show the feasibility of the proposed design approach.     

Shape design sensitivity analysis of kinematical boundaries

A new approach is used in this paper to derive the design sensitivity formulation with kinematical design boundaries. By employing the concept of the conventional finite difference approach, the variation of structural response due to change of the kinematic design boundary can be represented by the perturbed structure under a set of kinematical boundary conditions. Parameterization of the design variation with respect to the design variable enables us to transform the design sensitivity into the solutions of a boundary value problem with perturbation displacements on the design boundary. The perturbation diplacements can be evaluated from the stress and displacement fields of the initial problem. This approach can be treated as a special case of the general direct formulation, but the derivation using the finite difference procedure gives a strong physical meaning of the method, and the formulation derived provides an explicit form for design sensitivity calculation. The numerical implementation of this approach based on the boundary element method is discussed, and a few numerical examples are used to verify the proposed formulation.