Reliability-based design optimization of shells with uncertain geometry using adaptive Kriging metamodels

Optimal design under uncertainty has gained much attention in the past ten years due to the ever increasing need for manufacturers to build robust systems at the lowest cost. Reliability-based design optimization (RBDO) allows the analyst to minimize some cost function while ensuring some minimal performances cast as admissible failure probabilities for a set of performance functions. In order to address real-world engineering problems in which the performance is assessed through computational models (e.g., finite element models in structural mechanics) metamodeling techniques have been developed in the past decade. This paper introduces adaptive Kriging surrogate models to solve the RBDO problem. The latter is cast in an augmented space that "sums up" the range of the design space and the aleatory uncertainty in the design parameters and the environmental conditions. The surrogate model is used (i) for evaluating robust estimates of the failure probabilities (and for enhancing the computational experimental design by adaptive sampling) in order to achieve the requested accuracy and (ii) for applying a gradient-based optimization algorithm to get optimal values of the design parameters. The approach is applied to the optimal design of ring-stiffened cylindrical shells used in submarine engineering under uncertain geometric imperfections. For this application the performance of the structure is related to buckling which is addressed here by means of a finite element solution based on the asymptotic numerical method

Otimização eficiente global dirigida por metamodelos combinados : novos caminhos abertos pela aproximação por mínimos quadrados

Orientador: Alberto Luiz SerpaTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia MecânicaResumo: O presente trabalho representa a compilação dos resultados anteriores dessa pesquisa no campo de metamodelos combinados e otimização eficiente global (EGO), os quais foram sumetidos para publicação em periódicos especializados. Recentemente foi implementado nesse trabalho de doutorado o algoritmo LSEGO que é uma abordagem para conduzir algoritmos tipo EGO, baseando-se em metamodelos combinados através da aproximação por mínimos quadrados (metamodelos combinados LS). Através dos metamodelos combinados LS é possível estimar a incerteza da aproximação usando qualquer tipo de metamodelagem (e não somente do tipo kriging), permitindo estimar a função de expectativa de melhora para a função objetivo. Nos experimentos computacionais anteriores em problemas de otimização sem restrições, a abordagem LSEGO mostrou-se como uma alternativa viável para conduzir otimização eficiente global usando metamodelos combinados, sem se restringir a somente um ponto adicional por ciclo de otimização iterativa. Na presente tese o algoritmo LSEGO foi extendido de modo a tratar também problemas de otimização com restrições. Os resultados de testes numéricos com problemas analíticos e de referência e também em um estudo de caso de engenharia em escala industrial mostraram-se bastante promissores e competitivos em relação aos trabalhos similares encontrados na literaturaAbstract: In this work we review and compile the results of our previous research in the fields of ensemble of metamodels and efficient global optimization (EGO). Recently we implemented LSEGO that is an approach to drive EGO algorithms, based on LS (least squares) ensemble of metamodels. By means of LS ensemble of metamodels, it is possible to estimate the uncertainty of the prediction by using any kind of model (not only kriging) and provide an estimate for the expected improvement function. In previous numerical experiments with unconstrained optimization problems, LSEGO approach has shown to be a feasible alternative to drive efficient global optimization by using multiple or ensemble of metamodels, not restricted to kriging approximation or single infill point per optimization cycles. In the present work we extended the previous LSEGO algorithm to handle constrained optimization problems as well. Some numerical experiments were performed with analytical benchmark functions and also for industry scale engineering problems with competitive resultsDoutoradoMecanica dos Sólidos e Projeto MecanicoDoutor em Engenharia Mecânic

Single and Multiresponse Adaptive Design of Experiments with Application to Design Optimization of Novel Heat Exchangers

Engineering design optimization often involves complex computer simulations. Optimization with such simulation models can be time consuming and sometimes computationally intractable. In order to reduce the computational burden, the use of approximation-assisted optimization is proposed in the literature. Approximation involves two phases, first is the Design of Experiments (DOE) phase, in which sample points in the input space are chosen. These sample points are then used in a second phase to develop a simplified model termed as a metamodel, which is computationally efficient and can reasonably represent the behavior of the simulation response. The DOE phase is very crucial to the success of approximation assisted optimization. This dissertation proposes a new adaptive method for single and multiresponse DOE for approximation along with an approximation-based framework for multilevel performance evaluation and design optimization of air-cooled heat exchangers. The dissertation is divided into three research thrusts. The first thrust presents a new adaptive DOE method for single response deterministic computer simulations, also called SFCVT. For SFCVT, the problem of adaptive DOE is posed as a bi-objective optimization problem. The two objectives in this problem, i.e., a cross validation error criterion and a space-filling criterion, are chosen based on the notion that the DOE method has to make a tradeoff between allocating new sample points in regions that are multi-modal and have sensitive response versus allocating sample points in regions that are sparsely sampled. In the second research thrust, a new approach for multiresponse adaptive DOE is developed (i.e., MSFCVT). Here the approach from the first thrust is extended with the notion that the tradeoff should also consider all responses. SFCVT is compared with three other methods from the literature (i.e., maximum entropy design, maximin scaled distance, and accumulative error). It was found that the SFCVT method leads to better performing metamodels for majority of the test problems. The MSFCVT method is also compared with two adaptive DOE methods from the literature and is shown to yield better metamodels, resulting in fewer function calls. In the third research thrust, an approximation-based framework is developed for the performance evaluation and design optimization of novel heat exchangers. There are two parts to this research thrust. First, is a new multi-level performance evaluation method for air-cooled heat exchangers in which conventional 3D Computational Fluid Dynamics (CFD) simulation is replaced with a 2D CFD simulation coupled with an e-NTU based heat exchanger model. In the second part, the methods developed in research thrusts 1 and 2 are used for design optimization of heat exchangers. The optimal solutions from the methods in this thrust have 44% less volume and utilize 61% less material when compared to the current state of the art microchannel heat exchangers. Compared to 3D CFD, the overall computational savings is greater than 95%

Solving optimisation problems in metal forming using FEM: A metamodel based optimisation algorithm

During the last decades, Finite Element (FEM) simulations of metal forming processes have\ud become important tools for designing feasible production processes. In more recent years,\ud several authors recognised the potential of coupling FEM simulations to mathematical opti-\ud misation algorithms to design optimal metal forming processes instead of only feasible ones.\ud This report describes the selection, development and implementation of an optimisa-\ud tion algorithm for solving optimisation problems for metal forming processes using time\ud consuming FEM simulations. A Sequential Approximate Optimisation algorithm is pro-\ud posed, which incorporates metamodelling techniques and sequential improvement strate-\ud gies for enhancing the e±ciency of the algorithm. The algorithm has been implemented in\ud MATLABr and can be used in combination with any Finite Element code for simulating\ud metal forming processes.\ud The good applicability of the proposed optimisation algorithm within the ¯eld of metal\ud forming has been demonstrated by applying it to optimise the internal pressure and ax-\ud ial feeding load paths for manufacturing a simple hydroformed product. Resulting was\ud a constantly distributed wall thickness throughout the ¯nal product. Subsequently, the\ud algorithm was compared to other optimisation algorithms for optimising metal forming\ud by applying it to two more complicated forging examples. In both cases, the geometry of\ud the preform was optimised. For one forging application, the algorithm managed to solve\ud a folding defect. For the other application both the folding susceptibility and the energy\ud consumption required for forging the part were reduced by 10% w.r.t. the forging process\ud proposed by the forging company. The algorithm proposed in this report yielded better\ud results than the optimisation algorithms it was compared to

MULTI-DIMENSIONAL SURROGATE BASED AFT FORM OPTIMIZATION OF SHIPS USING HIGH FIDELITY SOLVERS

Surrogate (metamodel) based optimization has numerous potential applications in the field of naval architecture. It is aimed here to establish a methodology for the aft form optimization for minimum viscous resistance, thus the present study is focused on the aft form where the viscous effects become dominant. It is necessary to solve this problem within acceptable time span from practical naval architectural point of view which requires metamodeling techniques currently under investigation. Accordingly, the present paper investigates the metamodeling ability of the Kriging interpolation and attempts to explore its capabilities and limitations in the aft form optimization from viscous resistance point of view. As metamodeling techniques become more widely used, their constraints are more apparent. Especially in highly nonlinear design spaces, the effect of dimensionality should be taken into consideration. Taking all those factors into account, the present paper is to examine the capabilities of Kriging and to establish the learning performance in terms of RMS error, correlation coefficient and required number of training points according to selected optimization algorithm for multidimensional ship design problem. The results show that, at least 5% reduction in viscous pressure drag can be attained by the present optimization methodology

Reliability-based design of offshore structures for oil and gas applications

Offshore structures are complex in their structural and functional form and operate in a harsh and uncertain environment with complex interactions between ocean variables. Consequently, the ocean environment presents a high risk to these structures hence the need to develop an efficient and reliable design. Therefore, the need for a design that effectively: captures complex ocean parameter interactions, reduces the computational burden in structural response determination, quantifies the structure's ability to bounce back when faced with disruptive events, and minimizes cost under uncertainty at the desired safety levels of the asset is critical. A robust offshore structural design under uncertainty is essential for the safety of life, asset, and the environment during oil and gas exploration and production activities. This thesis presents improved methods for the effective reliability-based design of offshore structures. First, a framework is developed to capture the dependency of multivariate environmental ocean variables using vine copula and its impact on the reliability assessment of offshore structural systems. The model was tested using a cantilever beam and applied to an offshore jacket structure. The comparative results from the jacket structure and cantilever problem reveals that failure probability considering dependence between ocean variables is closer to the reference value than when variables are independent or modeled with a Gaussian copula. The outcome shows the importance of capturing nonlinearity and tail dependence between ocean variables in reliability evaluation. Secondly, the effectiveness of a hybrid metamodel, which is a combination of two commonly and independently used methods, Kriging and Polynomial Chaos Expansions (PCE), is investigated for offshore structural response determination and reliability studies. The hybrid metamodel herein, called (APCKKm-MCS) is constructed from an adaptive process with multiple enrichment of Experimental Design (ED). The hybrid approach was tested on simple non-linear functions, a truss bar, and an offshore deepwater Steel Catenary Riser (SCR). The study's outcome revealed that APCKKm-MCS produced a high predictive response capacity, reduced model evaluation, and shorter computing time during reliability evaluation than the single enrichment case (APCK-MCS) and the adaptive ordinary Kriging case (AK-MCS) considered. In addition, a novel framework is developed for the resilience quantification of offshore structures in terms of their time-varying reliability, adaptability, and maintainability. The developed framework was demonstrated using an internally corroded pipeline segment subject to disruptive events of leak, burst, and rupture. The framework captured the resilience index of the natural gas pipeline for its design life, and its sensitivity analysis revealed the influence of the pipe wall thickness and corrosion depth growth rate on the resilience of the pipeline. The framework provides a quantitative approach to determine the resilience of offshore structures and ascertain their critical influencing parameters for effective decision-making. Finally, a methodology for optimal structural design under uncertainty considering the dependency of environmental variables with the implementation of a hybrid metamodel in the inner loop of a nested optimization problem is presented and demonstrated on a steel column function and a segmented SCR. The study showed different decision outcomes for various vine tree configurations in the dependence modeling for the steel column function noting the importance of choosing the appropriate variable order in the vine tree for optimal design under uncertainty. Also, the research reveals the suitability of adaptive PCK for the inner loop reliability phase for a double-loop structural optimization due to its high predictive capacity and observed relatively low cross-validation error. The method shows the importance of effective dependence modeling of environmental ocean variables in structural cost minimization and selecting optimal structural design variables under uncertainty. From the research outcomes, considering multivariate dependence between ocean variables using vine copula and utilizing multiple enrichment hybrid metamodels in response evaluation for reliability and optimal design assessment of offshore structures could better predict their failure probability and enhance a safer structural design. In addition, the resilience quantification framework developed provides a vital decision-making tool for offshore structural systems' design and integrity management. The research into high dimensional dependence modeling of offshore structures using vine copula, comparative study of sampling strategies required for the hybrid (Kriging and PCE) metamodel construction, dependence-based structural resilience quantification, and multiobjective dependence-based structural optimization under uncertainty are among areas proposed for future investigation

Statistical Tests for Cross-Validation of Kriging Models

We derive new statistical tests for leave-one-out cross-validation of Kriging models. Graphically, we present these tests as scatterplots augmented with confi…dence intervals. We may wish to avoid extrapolation, which we de…fine as prediction of the output for a point that is a vertex of the convex hull of the given input combinations. Moreover, we may use bootstrapping to estimate the true variance of the Kriging predictor. The resulting tests (with or without extrapolation or bootstrapping) have type-I and type-II error probabilities, which we estimate through Monte Carlo experiments. To illustrate the application of our tests, we use an example with two inputs and the popular borehole example with eight inputs