An efficient framework for optimal robust stochastic system design using stochastic simulation

The knowledge about a planned system in engineering design applications is never complete. Often, a probabilistic quantification of the uncertainty arising from this missing information is warranted in order to efficiently incorporate our partial knowledge about the system and its environment into their respective models. This leads to a robust stochastic design framework where probabilistic models of excitation uncertainties and system modeling uncertainties can be introduced; the design objective is then typically related to the expected value of a system performance measure, such as reliability or expected life-cycle cost. For complex system models, this expected value can rarely be evaluated analytically and so it is often calculated using stochastic simulation techniques, which involve an estimation error and significant computational cost. An efficient framework, consisting of two stages, is presented here for the optimization in such robust stochastic design problems. The first stage implements a novel approach, called stochastic subset optimization (SSO), for iteratively identifying a subset of the original design space that has high plausibility of containing the optimal design variables. The second stage adopts some other stochastic optimization algorithm to pinpoint the optimal design variables within that subset. The focus is primarily on the theory and implementation issues for SSO but also on topics related to the combination of the two different stages for overall enhanced efficiency. An illustrative example is presented that shows the efficiency of the proposed methodology; it considers the optimization of the reliability of a base-isolated structure considering future near-fault ground motions.     

A contribution to the real-time simulation of coupled finite element models of machine tools – A numerical comparison

In this paper the real-time simulation of finite element (FE) models of machine tools on a multi-processor architecture is presented. The simulation model is based on several FE component models that are connected by non-linear couplings. These couplings allow relative motions of the components in a wide range. The coupled linear FE models are decomposed at the non-linear coupling nodes and each component is solved locally. The linear structure of the components can be used for efficient simulation methods and the components can be distributed to several processors for a parallel computation. Methods that differ in numerical accuracy and stability, computational effort and real-time capacity will be presented. By means of a complex example, it will be illustrated that a parallel, stable computation can be realized time-deterministically.     

Robustness-based design optimization under data uncertainty

This paper proposes formulations and algorithms for design optimization under both aleatory (i.e., natural or physical variability) and epistemic uncertainty (i.e., imprecise probabilistic information), from the perspective of system robustness. The proposed formulations deal with epistemic uncertainty arising from both sparse and interval data without any assumption about the probability distributions of the random variables. A decoupled approach is proposed in this paper to un-nest the robustness-based design from the analysis of non-design epistemic variables to achieve computational efficiency. The proposed methods are illustrated for the upper stage design problem of a two-stage-to-orbit (TSTO) vehicle, where the information on the random design inputs are only available as sparse point data and/or interval data. As collecting more data reduces uncertainty but increases cost, the effect of sample size on the optimality and robustness of the solution is also studied. A method is developed to determine the optimal sample size for sparse point data that leads to the solutions of the design problem that are least sensitive to variations in the input random variables.     

Robust formulation for Reliability-based design optimization of structures

The reliability-based design optimization (RBDO) has been widely recognized as a powerful optimization tool under probabilistic constraints, through appropriate modeling of uncertainties. However, the drawback of RBDO is that it does not reflect the ability of the structure to comply with large data variations, unforeseen actions or deterioration mechanisms. On the other hand, the robust design optimization (RDO) reduces the variability of the structural performance, in addition to its mean level. However, RDO does not take direct advantage of the interaction between controllable (product design values) and noise variables (environmental random values), and the obtained results do not accurately indicate what parameter has the highest effect on the performance characteristics. The purpose of this paper is to propose a robust formulation for reliability-based design optimization (RRBDO) that combines the advantages of both optimization procedures and overcomes their weaknesses. The optimization model proposed overcomes the limitations of the existing models without compromising the reliability level, by considering a robust convex objective function and a performance variation constraint. The proposed formulation can consider the total cost of structures and can control structural parameter variations. It takes into account uncertainty and variability in the same mathematical formulation. A numerical solution procedure is also developed, for which results are analyzed and compared with RBDO for several examples of concrete and steel structures.     

Automatic routine tuning to represent landform attributes on multicore and multi-GPU systems

Auto-tuning techniques have been used in the design of routines in recent years. The goal is to develop routines which automatically adapt to the conditions of the computational system in such a way that efficient executions are obtained independently of the end-user experience. This paper aims to explore programming routines that can be automatically adapted to the computational system conditions, making possible to use auto-tuning to represent landform attributes on multicores and multi-GPU systems using high- performance computing techniques for efficient solution of two-dimensional polynomial regression models that allow large problem instances to be addressed.     

A sequential approximate programming strategy for reliability-based structural optimization

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

Influent generator for probabilistic modeling of nutrient removal wastewater treatment plants

The availability of influent wastewater time series is crucial when using models to assess the performance of a wastewater treatment plant (WWTP) under dynamic flow and loading conditions. Given the difficulty of collecting sufficient data, synthetic generation could be the only option. In this paper a hybrid of statistical (a Markov chain-gamma model for stochastic generation of rainfall and two different multivariate autoregressive models for stochastic generation of air temperature and influent time series in dry conditions) and conceptual modeling techniques is proposed for synthetic generation of influent time series. The time series of rainfall and influent in dry weather conditions are generated using two types of statistical models. These two time series serve as inputs to a conceptual sewer model for generation of influent time series. The application of the proposed influent generator to the Eindhoven WWTP shows that it is a powerful tool for realistic generation of influent time series and is well-suited for probabilistic design of WWTPs as it considers both the effect of input variability and total model uncertainty.     

Sequential probabilistic analytical target cascading method for hierarchical multilevel optimization under uncertainty

Probabilistic Analytical Target Cascading (PATC) is a methodology for hierarchical multilevel optimization under uncertainty. In PATC, the statisticalmoments of the stochastic interrelated responses are matched between neighbouring levels to ensure the consistency of the solution. When the interrelated response is far from normal distribution, high order moments may need to be matched in the PATC formulation, which results in great computational difficulty. To overcome this disadvantage, a sequential PATC (SPATC) approach is proposed in this paper. SPATC firstly decouples the original probabilistic design problem into deterministic optimization problem and probabilistic analysis, and then hierarchically decomposes them into subproblems. The statistical information matching between neighbouring levels in the existing PATC framework is eliminated in SPATC. All in one probabilistic analysis and hierarchical probabilistic analysis are established to calculate the probabilistic characteristic of the interrelated responses and linking variables. Three examples are used to demonstrate the effectiveness and efficiency of the proposed SPATC approach. The results show that the SPATC approach is more efficient and accurate than PATC, especially for the multilevel design problem with non-normal interrelated responses.     

Possibilities and comparison of structural component models for the seismic assessment of modern unreinforced masonry buildings

This paper addresses the possibilities of structural component models for the seismic assessment of masonry buildings using pushover analysis. The ultimate goal is to allow efficient design of new unreinforced masonry buildings in seismic areas. Different modelling strategies are presented and the implementation of a simple design tool based on structural component models is described in detail. The different approaches are applied to a two-storey building with regular plan and elevation, allowing for a comparison between them. The results indicate that macro-modelling using structural component models and pushover analysis are adequate approaches for the seismic assessment and design of unreinforced masonry buildings, as the tools require very low computational resources, allow easy interpretation of results and provide satisfactory accuracy.     

Reliability-based structural optimization with probability and convex set hybrid models

For structural systems exhibiting both probabilistic and bounded uncertainties, it may be suitable to describe these uncertainties with probability and convex set models respectively in the design optimization problem. Based on the probabilistic and multi-ellipsoid convex set hybrid model, this paper presents a mathematical definition of reliability index for measuring the safety of structures in presence of parameter or load uncertainties. The optimization problem incorporating such reliability constraints is then mathematically formulated. By using the performance measure approach, the optimization problem is reformulated into a more tractable one. Moreover, the nested double-loop optimization problem is transformed into an approximate single-loop minimization problem by considering the optimality conditions and linearization of the limit-state function, which further facilitates efficient solution of the design problem. Numerical examples demonstrate the validity of the proposed formulation as well as the efficiency of the presented numerical techniques.     

An efficient method for stochastic optimal control with joint chance constraints for nonlinear systems

Stochastic model predictive control hinges on the online solution of a stochastic optimal control problem. This paper presents a computationally efficient solution method for stochastic optimal control for nonlinear systems subject to (time‐varying) stochastic disturbances and (time‐invariant) probabilistic model uncertainty in initial conditions and parameters. To this end, new methods are presented for joint propagation of time‐varying and time‐invariant probabilistic uncertainty and the nonconservative approximation of joint chance constraint (JCC) on the system state. The proposed uncertainty propagation method relies on generalized polynomial chaos and conditional probability rules to obtain tractable expressions for the state mean and covariance matrix. A moment‐based surrogate is presented for JCC approximation to circumvent construction of the full probability distribution of the state or the use of integer variables as required when using the sample average approximation. The proposed solution method for stochastic optimal control is illustrated on a nonlinear semibatch reactor case study in the presence of probabilistic model uncertainty and stochastic disturbances. It is shown that the proposed solution method is significantly superior to a standard random sampling method for stochastic optimal control in terms of computational requirements. Furthermore, the moment‐based surrogate for the JCC is shown to be substantially less conservative than the widely used distributionally robust Cantelli‐Chebyshev inequality for chance constraint approximation.     

Sampling-based RBDO using the stochastic sensitivity analysis and Dynamic Kriging method

This paper presents a sampling-based RBDO method using surrogate models. The Dynamic Kriging (D-Kriging) method is used for surrogate models, and a stochastic sensitivity analysis is introduced to compute the sensitivities of probabilistic constraints with respect to independent or correlated random variables. For the sampling-based RBDO, which requires Monte Carlo simulation (MCS) to evaluate the probabilistic constraints and stochastic sensitivities, this paper proposes new efficiency and accuracy strategies such as a hyper-spherical local window for surrogate model generation, sample reuse, local window enlargement, filtering of constraints, and an adaptive initial point for the pattern search. To further improve computational efficiency of the sampling-based RBDO method for large-scale engineering problems, parallel computing is proposed as well. Once the D-Kriging accurately approximates the responses, there is no further approximation in the estimation of the probabilistic constraints and stochastic sensitivities, and thus the sampling-based RBDO can yield very accurate optimum design. In addition, newly proposed efficiency strategies as well as parallel computing help find the optimum design very efficiently. Numerical examples verify that the proposed sampling-based RBDO can find the optimum design more accurately than some existing methods. Also, the proposed method can find the optimum design more efficiently than some existing methods for low dimensional problems, and as efficient as some existing methods for high dimensional problems when the parallel computing is utilized.     

An adaptive region segmentation combining surrogate model applied to correlate design variables and performance parameters in a transonic axial compressor

Surrogate models have been widely applied to correlate design variables and performance parameters in turbomachinery optimization applications. With more design variables and uncertain factors taken into account in an optimization design problem, the mathematical relations between the design variables and the performance parameters might present linear, low-order nonlinear or even high-order nonlinear characteristics, and are usually analytically unknown. Therefore, it is required that surrogate models have high adaptability and prediction accuracy for both the linear and nonlinear characteristics. The paper mainly investigates the effectiveness of an adaptive region segmentation combining surrogate model based on support vector regression and kriging model applied to a transonic axial compressor to approximate the complicated relationships between geometrical variables and objective performance outputs with different sampling methods and sizes. The purpose is to explore the prediction accuracy and computational efficiency of this adaptive surrogate model in real turbomachinery applications. Three different sampling techniques are studied: (1) uniform design; (2) Latin hypercube sampling method; (3) Sobol quasi-random design. For the low dimensional case with five variables, the adaptive region segmentation combining surrogate model performs better (not worse) than the single component surrogate in terms of prediction accuracy and computational efficiency. In the meanwhile, it is also noted that the uniform design applied to the adaptive surrogate model has more advantages over the Latin hypercube sampling method especially for the small sample size cases, both performing better than the Sobol quasi-random design. Moreover, a high dimensional case with 12 variables is also utilized to further validate the prediction advantage of the adaptive region segmentation combining surrogate model over the single component surrogate, and the computational results favor it. Overall, the adaptive region segmentation combining surrogate model has produced acceptable to high prediction accuracy in presenting complex relationships between the geometrical variables and the objective performance outputs and performed robustly for a transonic axial compressor problem.

     

An ANOVA approach for accounting for life-cycle uncertainty reduction measures in RBDO: the FSAE brake pedal case study

Accounting for uncertainty reduction measures (URMs) is critical to maximize the potential benefits of probabilistic design methods such as reliability-based design optimization (RBDO) and tackle the challenges in the design and construction of lightweight, high quality and reliable products. This work formulates and solves the RBDO of a Formula SAE (FSAE) brake pedal model with two failure modes (stress-S_max and buckling-f_buck) accounting for uncertainty reduction measures (URMs) throughout the product lifecycle while establishing the URMs global relative contributions to weight savings (expected value and variability) and computational expense. Given a set of URMs such as number of coupon tests, mesh refinement and manufacturing control, the solution approach includes: i) modeling structural analysis errors, ii) construction of surrogate models for the functions of interest, e.g., mass-M, S_max, f_buck and the corresponding error functions, iii) modeling pre-design and post-design URMs, such as material property density functions from coupon tests, and manufacturing tolerances (quality control), iv) solving the RBDO problems associated with each of the entries in a DOE with replication, and v) using ANOVA to compute main effects of most significant URMs on selected performance measures, i.e., mean and standard deviation of brake pedal mass, and computational expense. Results show that in the context of the brake pedal case study: the adoption of URMs led to reductions of up to 15 and 85% of mass mean and standard deviation, respectively, design and post-design URMs were responsible for 77 and 19% of the maximum mass reduction, respectively, and it was possible to set preliminary guidelines for URMs allocation and meet a particular performance objective under alternative URMs.     

Variable selection using Gaussian process regression-based metrics for high-dimensional model approximation with limited data

In recent years, the importance of computationally efficient surrogate models has been emphasized as the use of high-fidelity simulation models increases. However, high-dimensional models require a lot of samples for surrogate modeling. To reduce the computational burden in the surrogate modeling, we propose an integrated algorithm that incorporates accurate variable selection and surrogate modeling. One of the main strengths of the proposed method is that it requires less number of samples compared with conventional surrogate modeling methods by excluding dispensable variables while maintaining model accuracy. In the proposed method, the importance of selected variables is evaluated using the quality of the model approximated with the selected variables only. Nonparametric probabilistic regression is adopted as the modeling method to deal with inaccuracy caused by using selected variables during modeling. In particular, Gaussian process regression (GPR) is utilized for the modeling because it is suitable for exploiting its model performance indices in the variable selection criterion. Outstanding variables that result in distinctly superior model performance are finally selected as essential variables. The proposed algorithm utilizes a conservative selection criterion and appropriate sequential sampling to prevent incorrect variable selection and sample overuse. Performance of the proposed algorithm is verified with two test problems with challenging properties such as high dimension, nonlinearity, and the existence of interaction terms. A numerical study shows that the proposed algorithm is more effective as the fraction of dispensable variables is high.

     

Innovative seismic design optimization with reliability constraints

Performance-Based Design (PBD) methodologies is the contemporary trend in designing better and more economic earthquake-resistant structures where the main objective is to achieve more predictable and reliable levels of safety and operability against natural hazards. On the other hand, reliability-based optimization (RBO) methods directly account for the variability of the design parameters into the formulation of the optimization problem. The objective of this work is to incorporate PBD methodologies under seismic loading into the framework of RBO in conjunction with innovative tools for treating computational intensive problems of real-world structural systems. Two types of random variables are considered: Those which influence the level of seismic demand and those that affect the structural capacity. Reliability analysis is required for the assessment of the probabilistic constraints within the RBO formulation. The Monte Carlo Simulation (MCS) method is considered as the most reliable method for estimating the probabilities of exceedance or other statistical quantities albeit with excessive, in many cases, computational cost. First or Second Order Reliability Methods (FORM, SORM) constitute alternative approaches which require an explicit limit-state function. This type of limit-state function is not available for complex problems. In this study, in order to find the most efficient methodology for performing reliability analysis in conjunction with performance-based optimum design under seismic loading, a Neural Network approximation of the limit-state function is proposed and is combined with either MCS or with FORM approaches for handling the uncertainties. These two methodologies are applied in RBO problems with sizing and topology design variables resulting in two orders of magnitude reduction of the computational effort.     

Computational Methods for Verification of Stochastic Hybrid Systems

Stochastic hybrid system (SHS) models can be used to analyze and design complex embedded systems that operate in the presence of uncertainty and variability. Verification of reachability properties for such systems is a critical problem. Developing sound computational methods for verification is challenging because of the interaction between the discrete and the continuous stochastic dynamics. In this paper, we propose a probabilistic method for verification of SHSs based on discrete approximations focusing on reachability and safety problems. We show that reachability and safety can be characterized as a viscosity solution of a system of coupled Hamilton-Jacobi-Bellman equations. We present a numerical algorithm for computing the solution based on discrete approximations that are derived using finite-difference methods. An advantage of the method is that the solution converges to the one for the original system as the discretization becomes finer. We also prove that the algorithm is polynomial in the number of states of the discrete approximation. Finally, we illustrate the approach with two benchmarks: a navigation and a room heater example, which have been proposed for hybrid system verification.     

Quick energy prediction and comparison of options at the early design stage

The energy-efficient building design requires building performance simulation (BPS) to compare multiple design options for their energy performance. However, at the early stage, BPS is often ignored, due to uncertainty, lack of details, and computational time. This article studies probabilistic and deterministic approaches to treat uncertainty; detailed and simplified zoning for creating zones; and dynamic simulation and machine learning for making energy predictions. A state-of-the-art approach, such as dynamic simulation, provide a reliable estimate of energy demand, but computationally expensive. Reducing computational time requires the use of an alternative approach, such as a machine learning (ML) model. However, an alternative approach will cause a prediction gap, and its effect on comparing options needs to be investigated. A plugin for Building information modelling (BIM) modelling tool has been developed to perform BPS using various approaches. These approaches have been tested for an office building with five design options. A method using the probabilistic approach to treat uncertainty, detailed zoning to create zones, and EnergyPlus to predict energy is treated as the reference method. The deterministic or ML approach has a small prediction gap, and the comparison results are similar to the reference method. The simplified model approach has a large prediction gap and only makes only 40% comparison results are similar to the reference method. These findings are useful to develop a BIM integrated tool to compare options at the early design stage and ascertain which approach should be adopted in a time-constraint situation.     

A general purpose adaptivity driver for FE software

This paper describes a tool that serves as an automatic mesh adaptivity driver program for general purpose finite element (FE) software packages. Many commercially available FE programs lack a feature to control the numerical solution's accuracy properly. Our tool implements a mesh adaptive method that, in conjunction with separate finite element software, allows one to fully automatically improve the quality of the numerical solution up to a user specified accuracy. We demonstrate the use of the package with selected computational examples performed with a commercial FE package, ANSYS, and with our FE program FEMEngine. Copyright © 2003 John Wiley & Sons, Ltd.     

基于响应面和空间映射技术的模面设计优化

阐述了响应面近似模型和空间映射技术用于快速模面设计优化的基本原理．为避免数值噪声干扰和求解隐函数敏度,提出通过多项式响应面方法构造真实目标和约束的逼近曲面,以光滑响应进行全局最优;并利用空间映射技术的代理模型确定新的响应面及其梯度,经精细模型修正优化方向和设计子域,使计算成本降低．算法结合有限元模拟实施,通过对金钣拉延成形和回弹补偿的分析证明,该算法具有很高的效率和鲁棒性,适用于模面设计优化．