Rajan Filomeno Coelho

Multidisciplinary Design Optimization in Computational Mechanics

This book provides a comprehensive introduction to the mathematical and algorithmic methods for the Multidisciplinary Design Optimization (MDO) of complex mechanical systems such as aircraft or car engines. We have focused on the presentation of strategies efficiently and economically managing the different levels of complexity in coupled disciplines (e.g. structure, fluid, thermal, acoustics, etc.), ranging from Reduced Order Models (ROM) to full-scale Finite Element (FE) or Finite Volume (FV) simulations. Particular focus is given to the uncertainty quantification and its impact on the robustness of the optimal designs. A large collection of examples from academia, software editing and industry should also help the reader to develop a practical insight on MDO methods.

Multi-objective reliability-based optimization with stochastic metamodels

This paper addresses continuous optimization problems with multiple objectives and parameter uncertainty defined by probability distributions. First, a reliability-based formulation is proposed, defining the nondeterministic Pareto set as the minimal solutions such that user-defined probabilities of nondominance and constraint satisfaction are guaranteed. The formulation can be incorporated with minor modifications in a multiobjective evolutionary algorithm (here: the nondominated sorting genetic algorithm-II). Then, in the perspective of applying the method to large-scale structural engineering problems—for which the computational effort devoted to the optimization algorithm itself is negligible in comparison with the simulation—the second part of the study is concerned with the need to reduce the number of function evaluations while avoiding modification of the simulation code. Therefore, nonintrusive stochastic metamodels are developed in two steps. First, for a given sampling of the deterministic variables, a preliminary decomposition of the random responses (objectives and constraints) is performed through polynomial chaos expansion (PCE), allowing a representation of the responses by a limited set of coefficients. Then, a metamodel is carried out by kriging interpolation of the PCE coefficients with respect to the deterministic variables. The method has been tested successfully on seven analytical test cases and on the 10-bar truss benchmark, demonstrating the potential of the proposed approach to provide reliability-based Pareto solutions at a reasonable computational cost.

Adaptive sampling strategies for non‐intrusive POD‐based surrogates

Purpose – The purpose of this paper is to propose a novel contribution to adaptive sampling strategies for non‐intrusive reduced order models based on Proper Orthogonal Decomposition (POD). These strategies aim at reducing the cost of optimization by improving the efficiency and accuracy of POD data‐fitting surrogate models to be used in an online surrogate‐assisted optimization framework for industrial design.Design/methodology/approach – The effect of the strategies on the model accuracy is investigated considering the snapshot scaling, the design of experiment size and the truncation level of the POD basis and compared to a state‐of‐the‐art radial basis function network surrogate model on objectives and constraints. The selected test case is a Mach number and angle of attack domain exploration of the well‐known RAE2822 airfoil. Preliminary airfoil shape optimization results are also shown.Findings – The numerical results demonstrate the potential of the capture/recapture schemes proposed for adequately...

Metamodels for mixed variables based on moving least squares

Surrogate-based optimization has become a major field in engineering design, due to its capacity to handle complex systems involving expensive simulations. However, the majority of general-purpose surrogates (also called metamodels) are restricted to continuous variables, although versatile problems involve additional types of variables (discrete, integer, and even categorical to model technological options). Therefore, the main contribution of this paper consists in the development of metamodels specifically dedicated to handle mixed variables, in particular continuous and unordered categorical variables, and their comparison with state-of-the-art approaches. This task is performed in three steps: (i) considering an appropriate parametrization (integer mapping, regular simplex, dummy, effect codings) for the mixed variable design vector; (ii) defining metrics to compare pairs of design vectors; (iii) carrying out an ordinary or moving least square regression scheme based on the parametrization and metric previously defined. The proposed metamodels have been tested on six analytical benchmark test cases, and applied to the structural finite element analysis model of a rigid frame characterized by continuous and categorical variables. In particular, it is demonstrated that using a standard regular simplex representation for the nominal categorical variables usually outperforms a direct conversion of the nominal parameters to integer values, while offering an efficient and systematic way to encompass all types of variables in a common framework. It is also shown that the choice of a given variable representation has a higher impact on the results than the selected scheme (ordinary or moving least squares), or than the metric used for calculating distances between samples.

Probabilistic Dominance in Multiobjective Reliability-Based Optimization: Theory and Implementation

This paper is concerned with multiobjective evolutionary optimization under uncertainty modeled through probability distributions, with a focus on reliability-based approaches. The contribution is twofold. First, an in-depth study of the notion of probability of dominance is performed, including state-of-the-art multiobjective reliability-based formulations and their numerical calculation. In particular, the notion of dominance limit state function is defined and its properties are thoroughly investigated. Second, the assessment of the probability of dominance is proposed based on a first-order reliability method tailored for Pareto dominance and incorporated into a multiobjective evolutionary algorithm through a repairing mechanism. The analysis of the numerical results on five biobjective benchmark test cases (from two up to five design variables) by means of two adapted metrics (averaged Hausdorff distance and maximum Pareto front error) demonstrates the potential of the proposed approach to reach reliable nondominated fronts within a limited number of generations.

Co-evolutionary optimization for multi-objective design under uncertainty

This paper focuses on multi-objective optimization under uncertainty for mechanical design, through a reliability-based formulation referring to the concept of probabilistic nondominance. To address this problem, the implementation of a co-evolutionary strategy is advocated, consisting of the concurrent evolution of two intertwined populations optimized according to coupled subproblems: the upper level optimizer handles the design variables, whereas the corresponding values of the probabilistic thresholds for the objectives (namely the reliable nondominated front) are retrieved at the lower stage. The proposed methodology is successfully applied to six analytical test cases, as well as to the sizing optimization of two truss structures, demonstrating an improved capacity to cover wider ranges of the reliable nondominated front in comparison with all-at-once strategies tackling all types of variables simultaneously.

Proper orthogonal decomposition with high number of linear constraints for aerodynamical shape optimization

Shape optimization involving finite element analysis in engineering design is frequently hindered by the prohibitive cost of function evaluations. Reduced-order models based on proper orthogonal decomposition (POD) constitute an economical alternative. However the truncation of the POD basis implies an error in the calculation of the global values used as objectives and constraints which in turn affects the optimization results. In our former contribution (Xiao and Breitkopf, 2013), we have introduced a constrained POD projector allowing for exact linear constraint verification for a reduced order model. Nevertheless, this approach was limited a to relatively low numbers constraints. Therefore, in the present paper, we propose an approach for a high number of constraints. The main idea is to extend the snapshot POD by introducing a new constrained projector in order to reduce both the physical field and the constraint space. This allows us to search for the Pareto set of best compromises between the projection and the constraint verification errors thereby enabling fine-tuning of the reduced model for a particular purpose. We illustrate the proposed approach with the reduced order model of the flow around an airfoil parameterized with shape variables.

Graph theory in evolutionary truss design optimization

This chapter explores how to integrate graph theory in evolutionary algorithms for efficient truss optimization. It turns out that truss assemblies fit well with graph-based representation through a real-value matrix encoding, which is demonstrated to be a sound and powerful alternative to vector encoding. Because the structural component sizes and the system connectivity are expressed in a compact formalism, sizing and topology optimization are naturally carried out simultaneously. Evolutionary operators are subsequently adapted and repair procedures to prevent unstable configurations are readily derived. The baseline evolutionary algorithm described hereafter is presented in its generic form so that the reader can adapt the material content in various ways (hybrid algorithm, bilevel optimization, and so on). Applications to three-dimensional structures illustrate the methodology.

PAMUC II for multicriteria optimization of mechanical designs with expert rules

This study addresses the problem of optimizing mechanical components during the first stage of the design process. While a previous work focused on parametrized designs with fixed configurations - which led to the development of the PAMUC (Preferences Applied to Multiobjectivity and Constraints) method, for solving multicriteria constrained problems within evolutionary algorithms (EAs), the models analyzed in this work are enriched by the presence of topological variables enabling to consider simultaneously different configurations. Therefore, in order to create optimal but also realistic designs, i.e. fulfilling not only technical requirements but also technological constraints (e.g. related to the machining or the assembly), which are more naturally expressed in terms of rules, an original approach is proposed, named PAMUC II. It consists in integrating an inference engine within the EA, and repairing the individuals (with a given probability of replacement) violating the technological constraints (written as 0-order logical rules). PAMUC II is illustrated on a mechanical benchmark and an industrial case: the multicriteria optimization of a poppet valve design from the VINCI engine of launcher Ariane 5. Results show the efficiency of the proposed method to provide at once optimal and feasible designs.

An Automated Structural Optimisation Methodology for Scissor Structures Using a Genetic Algorithm

We developed a fully automated multiobjective optimisation framework using genetic algorithms to generate a range of optimal barrel vault scissor structures. Compared to other optimisation methods, genetic algorithms are more robust and efficient when dealing with multiobjective optimisation problems and provide a better view of the search space while reducing the chance to be stuck in a local minimum. The novelty of this work is the application and validation (using metrics) of genetic algorithms for the shape and size optimisation of scissor structures, which has not been done so far for two objectives. We tested the feasibility and capacity of the methodology by optimising a 6źm span barrel vault to weight and compactness and by obtaining optimal solutions in an efficient way using NSGA-II. This paper presents the framework and the results of the case study. The in-depth analysis of the influence of the optimisation variables on the results yields new insights which can help in making choices with regard to the design variables, the constraints, and the number of individuals and generations in order to obtain efficiently a trade-off of optimal solutions.