Variational sensitivity analysis and changes in the material configuration

This contribution is concerned with some aspects of variational design sensitivity analysis in the physical and material configuration. Sensitivity analysis is a branch of structural optimization, e.g. shape or topology optimization. In these disciplines we consider variations of the material configuration and we are interested in the change of the state variables and the objective functional due to these variations. In the context of structural optimization this is termed as design sensitivity analysis. The sensitivities are required in order to solve the corresponding Lagrangian equation within standard nonlinear programming algorithms. In many engineering applications, the energy functional of the problem is used as the objective functional. In this paper, we consider variations of the energy with respect to the state and the design and we investigate sensitivity relations for the physical and material problem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Probabilistic sensitivity of limit states of structures. The Monte Carlo simulation

The main issue of the paper is the probabilistic sensitivity of the limit states of structures with respect to selected input design variables. Attempt to the problem is done by the dedicated Monte Carlo simulation procedure. Basic design variables are random variables of given probability distributions, presented in the form of random numbers. Uni-parametrical increment of the dominant basic variable (basic variables) is done, finally achieving the structural limit state. The simulation procedure restuls in a set of limit multipliers. Statistical analysis leads to the estimate of the probability density function of the limit state. Thus the numerical image is presented of the probabilistic sensitivity of the structural limit state. Reliability or the probability of failure are to be estimated, as statistical parameters of the histogram. Numerical examples of engineering structures illustrate the method introduced in the paper, conclusions are formulated eventually. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Design space exploration based on variational sensitivity analysis

This contribution deals with variational sensitivity analysis of a non-linear solid shell in context of shape optimisation. A technique to explore the structural design is outlined. It is based on the analytical derivation and efficient computation of the Fréchet derivatives of the state functions with respect to the full space of all possible design parameters. This overhead of sensitivity information is examined by a principal component analysis (PCA) in order to detect design changes with major and minor influence on the structural response, the objectives and constraints. This knowledge enables the engineers to understand and improve structural optimisation models and to detect possible numerical defects in the early stage. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Eigenvalue sensitivity and parametric optimization of the large rotating systems

The contribution presents the modal synthesis method of the mathematical modelling of the large rotating systems. The condensed mathematical model is used for deriving analytical formula for eigenvalue sensitivity calculation. According to the sensitivity analysis suitable design parameters are chosen and the optimization process for the purpose of steady state response minimization in a certain operating speed area is performed. The theory is applied to a simple test‐gearbox. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Forward-reverse switch between density-based and regional sensitivity analysis

Global sensitivity analysis is a widely used tool for uncertainty apportionment and is very useful for decision making, risk assessment, model simplification, optimal design of experiments, etc. Density-based sensitivity analysis and regional sensitivity analysis are two widely used approaches. Both of them can work with a given sample set of model input-output pairs. One significant difference between them is that density-based sensitivity analysis analyzes output distributions conditional on input values (forward), while regional sensitivity analysis analyzes input distributions conditional on output values (reverse). In this paper, we study the relationship between these two approaches and show that regional sensitivity analysis (reverse), when focusing on probability density functions of input, converges towards density-based sensitivity analysis (forward) as the number of classes for conditioning model outputs in the reverse method increases. Similar to the existing general form of forward sensitivity indices, we derive a general form of the reverse sensitivity indices and provide the corresponding reverse given-data method. Due to the shown equivalence, the reverse given-data method provides an efficient way to approximate density-based sensitivity indices. Two test examples are used to verify this connection and compare the results. Finally, we use the reverse given-data method to perform sensitivity analysis in a carbon dioxide storage benchmark problem with multiple outputs, where forward analysis of density-based indices would be impossible due to the high-dimensionality of its model outputs.     

Feedback control optimization of nonlinear systems under random excitations

This paper uses stochastic averaging method to design an optimal feedback control for nonlinear stochastic systems. The method of stochastic averaging is used to reduce the dimension of the state space and to derive the Itô stochastic differential equation for the response amplitude process. Two approaches to optimization, namely, with the exact steady state probability density function of the amplitude process and the Rayleigh approximation are compared. The cost function is a steady state response measure. Numerical examples are studied to demonstrate the performance of the control both in transient and steady-state. The effect of the control on the system response and control performance is studied. The regions where the controls are conservative and unconservative are pointed out.     

Design of electronic assembly lines: An analytical framework and its application

The design of component assembly lines in Printed Circuit Board (PCB) manufacturing environments is a challenging problem faced by many firms in the electronics industry. The main design approaches to such component assembly lines are the Mini-Line, Flexible Flow Line, and Hybrid Line designs. In this paper, we discuss the operational trade-offs associated with these design alternatives and present a mathematical programming framework that captures relevant system design issues. Each of the design alternatives can be viewed as a special case of the stated mathematical programming model. We develop effective algorithms to solve these mathematical programs. We have used the framework in a specific PCB manufacturing environment to advise managers on the best configuration of their lines. The models were used as sensitivity analysis tools. The results of our computational experiments, combined with qualitative comparisons of different design approaches developed by a crossfunctional team (engineers, manufacturing and product managers), have led to the development of a set of managerial guidelines for the selection of the design plan for component assembly lines in the studied environment.     

Heuristic sensitivity analysis in a combinatoric environment: An exposition and case study

Sensitivity orpost-optimality analysis investigates the effect of parametric changes on heuristic robustness and solution quality. This approach is relatively unexplored for combinatorial optimization problems, and yet is of considerable interest in analyzing performance characteristics of heuristic approaches. The purpose of this paper is to: (1) develop the semantics and rationale of parametric analysis within the combinatoric environment; (2) present as an example the design and implementation of sensitivity analysis procedures for a newly developed heuristic — theVariable-Depth-Search Heuristic (VDSH) — to solve the Generalized Assignment Problem (GAP). The concepts and methodology discussed in this paper may as well be applied to other heuristics, or in developing a heuristic sensitivity analysis procedure for a large-scale optimization method.     

Sensitivity of parameter estimates to observations,system identification,and optimal inputs

The sensitivity of parameter estimates to observation noise or disturbances affects the accuracy of the parameter estimates in system identification. The (parameter estimate)/observation sensitivity is derived for a scalar nonlinear differential system. In the design of optimal inputs, the sensitivity of the state variable to the unknown parameter is maximized. It is shown that the (parameter estimate)/observation sensitivity tends to be lowered for optimal inputs. A numerical example is given.     

An airfoil optimization technique for wind turbines

Optimization algorithms coupled with computational fluid dynamics are used for wind turbines airfoils design. This differs from the traditional aerospace design process since the lift-to-drag ratio is the most important parameter and the angle of attack is large. Computational fluid dynamics simulations are performed with the incompressible Reynolds-averaged Navier–Stokes equations in steady state using a one equation turbulence model. A detailed validation of the simulations is presented and a computational domain larger than suggested in literature is shown to be necessary. Different approaches to parallelization of the computational code are addressed. Single and multiobjective genetic algorithms are employed and artificial neural networks are used as a surrogate model. The use of artificial neural networks is shown to reduce computational time by almost 50%.     

Preconditioning of One-Shot Pseudo-Timestepping Methods for Shape Optimization

One-shot pseudo-timestepping [1, 2] has proven to be a powerful approach for the solution of optimal design problems in engineering applications. It blends well in within pre-existing pseudo-timestepping simulation solvers. The goal of this approach is to transform the equations for the necessary optimality conditions of the optimal design problem into an ordinary differential equation (ODE) and then solve the pseudo-time embedded state, co-state and design equations all at once. Similar approaches have been considered in [4, 5]. An application of this method to a real-world example within the MEGADESIGN-Project has been presented in [3]. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On-line robust nonlinear state estimators for nonlinear bioprocess systems

This paper presents the design of a new robust nonlinear estimator for estimation of states of nonlinear systems. Two approaches are considered based on the state-dependent Riccati equation formulation and the technique of H-infinity control design. The proposed method differs from other well-known state estimators, because not only nonlinear dynamics but also the robustness is taken into account. The proposed method is implemented and tested on a biological wastewater system. The simulation study compares the Extended Kalman Estimator (EKE), the State-Dependent Riccati Estimator (SDRE), and the Extended H-infinity Estimator (EHE) with a new proposed State Dependent H-infinity Estimator (SDHE). The results are compared for different weather conditions, i.e. dry, rain and storm, showing a superior performance of the proposed method.     

Economic design of an attribute control chart for monitoring mean life based on multiple deferred state sampling

In this paper, we design an attribute np control chart using multiple deferred state (MDS) sampling under Weibull distribution based on time truncated life test. This chart is constructed for monitoring the variation of mean life of the product in a manufacturing process. The optimal parameters of MDS sampling and the control limit coefficients are determined so that the in‐control average run length (ARL) is as close as to the target ARL. The optimal parameters of MDS sampling are sample size and number of successive subgroups required for declaring the current state of process. Out‐of‐control ARL is considered as a measure of the performance of proposed chart and reported with determined optimal parameters for various shift constants. The out‐of‐control ARL of the proposed chart obtained under various distributions is compared with each other. The performance of proposed control chart is compared with the performance of the existing control chart designed under single sampling. In addition, the economic design of proposed chart using variable sampling interval scheme is discussed, and sensitivity analysis on expected costs is also investigated.     

Economic design of inspection procedures using guard band when measurement errors are present

One of the most important aspects in designing production systems is implementing an adequate inspection procedure to ensure the product quality. Our literature study indicates that two important questions still remain unanswered. First, most of the previous studies on inspection procedures are mainly concerned with how to evaluate and analyze the adequacy of inspection procedures during a production stage. Although the issue of the adequacy of inspection procedures at this point is clearly important, many production processes may better benefit from designing enhanced inspection procedures during an early design stage. Second, the majority of previous studies do not incorporate possible economic impacts of measurement errors when inspection procedures need to be designed. More extensive efforts in designing inspection procedures need to be investigated based on economic considerations. Two approaches, the use of guard band and the selection of inspection precision level, are simultaneously examined for designing economic inspection procedures. By incorporating these two approaches, this paper proposes an optimization scheme for economic design of inspection procedures. A numerical example along with sensitivity analysis demonstrates the applicability of the proposed model.     

Sensitivity analysis of release time of software reliability models incorporating testing effort with multiple change-points

To accurately model software failure process with software reliability growth models, incorporating testing effort has shown to be important. In fact, testing effort allocation is also a difficult issue, and it directly affects the software release time when a reliability criteria has to be met. However, with an increasing number of parameters involved in these models, the uncertainty of parameters estimated from the failure data could greatly affect the decision. Hence, it is of importance to study the impact of these model parameters. In this paper, sensitivity of the software release time is investigated through various methods, including one-factor-at-a-time approach, design of experiments and global sensitivity analysis. It is shown that the results from the first two methods may not be accurate enough for the case of complex nonlinear model. Global sensitivity analysis performs better due to the consideration of the global parameter space. The limitations of different approaches are also discussed. Finally, to avoid further excessive adjustment of software release time, interval estimation is recommended for use and it can be obtained based on the results from global sensitivity analysis.     

假期中顾客以概率θ进人的Geo/G/1(E,SV)排队系统的队长分布

考虑单重休假的Geo/G/1离散时间排队系统,其中在服务员休假期间到达的顾客以概率θ(0<θ≤1)进入系统.通过引入"服务员忙期"和使用全概率分解技术,从任意初始状态出发,研究了队长的瞬态和稳态性质,导出了在任意时刻n瞬态队长分布的z-变换的递推表达式和稳态队长分布的递推表达式,以及稳态队长的随机分解.最后,通过数值实例,讨论了稳态队长分布对系统参数的敏感性,并阐述了获得便于计算的稳态队长分布的表达式在系统容量设计中有重要的价值.     

The Sensitivity of the Number of Clusters in a Gaussian Mixture Model to Prior Distributions

One of the main advantages of Bayesian approaches is that they offer principled methods of inference in models of varying dimensionality and of models of infinite dimensionality. What is less widely appreciated is how the model inference is sensitive to prior distributions and therefore how priors should be set for real problems. In this paper prior sensitivity is considered with respect to the problem of inference in Gaussian mixture models. Two distinct Bayesian approaches have been proposed. The first is to use Bayesian model selection based upon the marginal likelihood; the second is to use an infinite mixture model which ‘side steps’ model selection. Explanations for the prior sensitivity are given in order to give practitioners guidance in setting prior distributions. In particular the use of conditionally conjugate prior distributions instead of purely conjugate prior distributions are advocated as a method for investigating prior sensitivity of the mean and variance individually.     

An adaptive multi-start graph partitioning algorithm for structuring cellular networks

In mobile network design, the problem of assigning network elements to controllers when defining network structure can be modeled as a graph partitioning problem. In this paper, a comprehensive analysis of a sophisticated graph partitioning algorithm for grouping base stations into packet control units in a mobile network is presented. The proposed algorithm combines multi-level and adaptive multi-start schemes to obtain high quality solutions efficiently. Performance assessment is carried out on a set of problem instances built from measurements in a live network. Overall results confirm that the proposed algorithm finds solutions better than those obtained by the classical multi-level approaches and much faster than classical multi-start approaches. The analysis of the optimization surface shows that the best local minima values follow a Gumbel distribution, which justifies the stagnation of naive multi-start approaches after a few attempts. Likewise, the analysis shows that the best local minima share strong similarities, which is the reason for the superiority of adaptive multi-start approaches. Finally, a sensitivity analysis shows the best internal parameter settings in the algorithm.     

Sensitivity analysis and model optimization for reaction-diffusion systems

This paper presents a computational framework for the optimization and sensitivity analysis of a process whose state depends upon several parameter functions. Assuming that the process is described by a system of quasilinear, parabolic, partial differential equations, we show how determining the problem parameters so as to improve an associated objective functional is directly related to knowing the state function sensitivities. An expression for the gradient of the objective functional in terms of the solutions of an adjoint system enables one to bypass the calculation of state function sensitivities. These concepts are illustrated for a simple model of cooperative processes in chemical kinetics. Since sensitivity analysis and model optimization are important tools for investigating parameter dependence and validating mathematical models, research developments in such diverse fields as optimal design theory, chemical kinetics, and parameter identification are important motivations for this paper.This author would like to gratefully acknowledge Dr. M. Delle Donne, EGG, for several helpful discussions.This author was partially supported by NSF Grant No. CMS-80-05677.