Reliability-based design optimization strategies based on FORM: a review

In deterministic optimization, the uncertainties of the structural system (i.e. dimension, model, material, loads, etc) are not explicitly taken into account. Hence, resulting optimal solutions may lead to reduced reliability levels. The objective of reliability based design optimization (RBDO) is to optimize structures guaranteeing that a minimum level of reliability, chosen a priori by the designer, is maintained. Since reliability analysis using the First Order Reliability Method (FORM) is an optimization procedure itself, RBDO (in its classical version) is a double-loop strategy: the reliability analysis (inner loop) and the structural optimization (outer loop). The coupling of these two loops leads to very high computational costs. To reduce the computational burden of RBDO based on FORM, several authors propose decoupling the structural optimization and the reliability analysis. These procedures may be divided in two groups: (i) serial single loop methods and (ii) unilevel methods. The basic idea of serial single loop methods is to decouple the two loops and solve them sequentially, until some convergence criterion is achieved. On the other hand, uni-level methods employ different strategies to obtain a single loop of optimization to solve the RBDO problem. This paper presents a review of such RBDO strategies. A comparison of the performance (computational cost) of the main strategies is presented for several variants of two benchmark problems from the literature and for a structure modeled using the finite element method

Nesterov-aided Stochastic Gradient Methods using Laplace Approximation for Bayesian Design Optimization

Finding the best setup for experiments is the primary concern for Optimal Experimental Design (OED). Here, we focus on the Bayesian experimental design problem of finding the setup that maximizes the Shannon expected information gain. We use the stochastic gradient descent and its accelerated counterpart, which employs Nesterov's method, to solve the optimization problem in OED. We adapt a restart technique, originally proposed for the acceleration in deterministic optimization, to improve stochastic optimization methods. We combine these optimization methods with three estimators of the objective function: the double-loop Monte Carlo estimator (DLMC), the Monte Carlo estimator using the Laplace approximation for the posterior distribution (MCLA) and the double-loop Monte Carlo estimator with Laplace-based importance sampling (DLMCIS). Using stochastic gradient methods and Laplace-based estimators together allows us to use expensive and complex models, such as those that require solving partial differential equations (PDEs). From a theoretical viewpoint, we derive an explicit formula to compute the gradient estimator of the Monte Carlo methods, including MCLA and DLMCIS. From a computational standpoint, we study four examples: three based on analytical functions and one using the finite element method. The last example is an electrical impedance tomography experiment based on the complete electrode model. In these examples, the accelerated stochastic gradient descent method using MCLA converges to local maxima with up to five orders of magnitude fewer model evaluations than gradient descent with DLMC.Comment: 36 pages, 14 figure

AN HYBRID STOCHASTIC-DETERMINISTIC OPTIMIZATION ALGORITHM FOR STRUCTURAL DAMAGE IDENTIFICATION

This paper presents a hybrid stochastic/deterministic optimization algorithm to solve the target optimization problem of vibration-based damage detection. The use of a numerical solution of the representation formula to locate the region of the global solution, i.e., to provide a starting point for the local optimizer, which is chosen to be the Nelder-Mead algorithm (NMA), is proposed. A series of numerical examples with different damage scenarios and noise levels was performed under impact and ambient vibrations. To test the accuracy and efficiency of the optimization algorithm, its results were compared to previous procedures available in the literature, which employed different solutions such as the genetic algorithm (GA) and the harmony search algorithm (HS). The performance of the proposed optimization scheme was more accurate and required a lower computational cost than the GA and HS algorithms, emphasizing the capacity of the proposed methodology for its use in damage diagnosis and assessment

Optimization in the presence of uncertainties

L’optimisation est un sujet très important dans tous les domaines. Cependant, parmi toutes les applications de l’optimisation, il est difficile de trouver des exemples de systèmes à optimiser qui ne comprennent pas un certain niveau d'incertitude sur les valeurs de quelques paramètres. Le thème central de cette thèse est donc le traitement des différents aspects de l’optimisation en présence d’incertitudes. Nous commençons par présenter un bref état de l’art des méthodes permettant de prendre en compte les incertitudes dans l’optimisation. Cette revue de la littérature a permis de constater une lacune concernant la caractérisation des propriétés probabilistes du point d’optimum de fonctions dépendant de paramètres aléatoires. Donc, la première contribution de cette thèse est le développement de deux méthodes pour approcher la fonction densité de probabilité (FDP) d’un tel point : la méthode basée sur la Simulation de Monte Carlo et la méthode de projection en dimension finie basée sur l’Approximation par polynômes de chaos. Les résultats numériques ont montré que celle-ci est adaptée à l’approximation de la FDP du point optimal du processus d'optimisation dans les situations étudiées. Il a été montré que la méthode numérique est capable d’approcher aussi des moments d'ordre élevé du point optimal, tels que l’aplatissement et l’asymétrie. Ensuite, nous passons au traitement de contraintes probabilistes en utilisant l’optimisation fiabiliste. Dans ce sujet, une nouvelle méthode basée sur des coefficients de sécurité est développée. Les exemples montrent que le principal avantage de cette méthode est son coût de calcul qui est très proche de celui de l’optimisation déterministe conventionnelle, ce qui permet son couplage avec un algorithme d’optimisation globale arbitraire.The optimization is a very important tool in several domains. However, among its applications, it is hard to find examples of systems to be optimized that do not possess a certain uncertainty level on its parameters. The main goal of this thesis is the treatment of different aspects of the optimization under uncertainty. We present a brief review of the literature on this topic, which shows the lack of methods able to characterize the probabilistic properties of the optimum point of functions that depend on random parameters. Thus, the first main contribution of this thesis is the development of two methods to eliminate this lack: the first is based on Monte Carlo Simulation (MCS) (considered as the reference result) and the second is based on the polynomial chaos expansion (PCE). The validation of the PCE based method was pursued by comparing its results to those provided by the MCS method. The numerical analysis shows that the PCE method is able to approximate the probability density function of the optimal point in all the problems solved. It was also showed that it is able to approximate even high order statistical moments such as the kurtosis and the asymmetry. The second main contribution of this thesis is on the treatment of probabilistic constraints using the reliability based design optimization (RBDO). Here, a new RBDO method based on safety factors was developed. The numerical examples showed that the main advantage of such method is its computational cost, which is very close to the one of the standard deterministic optimization. This fact makes it possible to couple the new method with global optimization algorithms

A gradient based optimization procedure for finding axle weights in probabilistic bridge weigh-in-motion method

Bridge weight in motion (BWIM) consists in the use of sensors on bridges to assess the loads of passing vehicles. Probabilistic Bridge Weight in Motion (pBWIM) is an approach for solving the inverse problem of finding vehicle axle weights based on deformation information. The pBWIM approach uses a probabilistic influence line and seeks the most probable axle weights, given in-situ measurements. To compute such weights, the original pBWIM employed a grid search, which may lead to computational complexity, specially when applied to vehicles with high number of axles. Hence, this note presents an improved version of pBWIM, modifying how the most probable weights are sough. Here, a gradient based optimization procedure is proposed for replacing the computationally expensive grid-search of the original algorithm. The required gradients are fully derived and validated in numerical examples. The proposed modification is shown to highly decrease the computational complexity of the problem.The accepted manuscript in pdf format is listed with the files at the bottom of this page. The presentation of the authors' names and (or) special characters in the title of the manuscript may differ slightly between what is listed on this page and what is listed in the pdf file of the accepted manuscript; that in the pdf file of the accepted manuscript is what was submitted by the author

A backtracking search algorithm for the simultaneous size, shape and topology optimization of trusses

This paper presents a Backtracking Search Optimization algorithm (BSA) to simultaneously optimize the size, shape and topology of truss structures. It focuses on the optimization of these three aspects since it is well known that the most effective scheme of truss optimization is achieved when they are simultaneously considered. The minimization of structural weight is the objective function, imposing displacement, stress, local buckling and/or kinematic stability constraints. The effectiveness of the BSA at solving this type of optimization problem is demonstrated by solving a series of benchmark problems comparing not only the best designs found, but also the statistics of 100 independent runs of the algorithm. The numerical analysis showed that the BSA provided promising results for the analyzed problems. Moreover, in several cases, it was also able to improve the statistics of the independent runs such as the mean and coefficient of variation values

Optimisation en présence d'incertitudes

L optimisation est un sujet très important dans tous les domaines. Cependant, parmi toutes les applications de l optimisation, il est difficile de trouver des exemples de systèmes à optimiser qui ne comprennent pas un certain niveau d'incertitude sur les valeurs de quelques paramètres. Le thème central de cette thèse est donc le traitement des différents aspects de l optimisation en présence d incertitudes. Nous commençons par présenter un bref état de l art des méthodes permettant de prendre en compte les incertitudes dans l optimisation. Cette revue de la littérature a permis de constater une lacune concernant la caractérisation des propriétés probabilistes du point d optimum de fonctions dépendant de paramètres aléatoires. Donc, la première contribution de cette thèse est le développement de deux méthodes pour approcher la fonction densité de probabilité (FDP) d un tel point : la méthode basée sur la Simulation de Monte Carlo et la méthode de projection en dimension finie basée sur l Approximation par polynômes de chaos. Les résultats numériques ont montré que celle-ci est adaptée à l approximation de la FDP du point optimal du processus d'optimisation dans les situations étudiées. Il a été montré que la méthode numérique est capable d approcher aussi des moments d'ordre élevé du point optimal, tels que l aplatissement et l asymétrie. Ensuite, nous passons au traitement de contraintes probabilistes en utilisant l optimisation fiabiliste. Dans ce sujet, une nouvelle méthode basée sur des coefficients de sécurité est développée. Les exemples montrent que le principal avantage de cette méthode est son coût de calcul qui est très proche de celui de l optimisation déterministe conventionnelle, ce qui permet son couplage avec un algorithme d optimisation globale arbitraire.The optimization is a very important tool in several domains. However, among its applications, it is hard to find examples of systems to be optimized that do not possess a certain uncertainty level on its parameters. The main goal of this thesis is the treatment of different aspects of the optimization under uncertainty. We present a brief review of the literature on this topic, which shows the lack of methods able to characterize the probabilistic properties of the optimum point of functions that depend on random parameters. Thus, the first main contribution of this thesis is the development of two methods to eliminate this lack: the first is based on Monte Carlo Simulation (MCS) (considered as the reference result) and the second is based on the polynomial chaos expansion (PCE). The validation of the PCE based method was pursued by comparing its results to those provided by the MCS method. The numerical analysis shows that the PCE method is able to approximate the probability density function of the optimal point in all the problems solved. It was also showed that it is able to approximate even high order statistical moments such as the kurtosis and the asymmetry. The second main contribution of this thesis is on the treatment of probabilistic constraints using the reliability based design optimization (RBDO). Here, a new RBDO method based on safety factors was developed. The numerical examples showed that the main advantage of such method is its computational cost, which is very close to the one of the standard deterministic optimization. This fact makes it possible to couple the new method with global optimization algorithms.ROUEN-INSA Madrillet (765752301) / SudocSudocFranceF