Bayesian Quality-Diversity approaches for constrained optimization problems with mixed continuous, discrete and categorical variables

Complex engineering design problems, such as those involved in aerospace, civil, or energy engineering, require the use of numerically costly simulation codes in order to predict the behavior and performance of the system to be designed. To perform the design of the systems, these codes are often embedded into an optimization process to provide the best design while satisfying the design constraints. Recently, new approaches, called Quality-Diversity, have been proposed in order to enhance the exploration of the design space and to provide a set of optimal diversified solutions with respect to some feature functions. These functions are interesting to assess trade-offs. Furthermore, complex engineering design problems often involve mixed continuous, discrete, and categorical design variables allowing to take into account technological choices in the optimization problem. In this paper, a new Quality-Diversity methodology based on mixed continuous, discrete and categorical Bayesian optimization strategy is proposed. This approach allows to reduce the computational cost with respect to classical Quality - Diversity approaches while dealing with discrete choices and constraints. The performance of the proposed method is assessed on a benchmark of analytical problems as well as on an industrial design optimization problem dealing with aerospace systems

Decoupled UMDO formulation for interdisciplinary coupling satisfaction under uncertainty

International audienceAt early design phases, taking into account uncertainty for the optimization of a multidisciplinary system is essential to establish the optimal system characteristics and performances. Uncertainty Multidisciplinary Design Optimization (UMDO) formulations have to eciently organize the dierent disciplinary analyses, the uncertainty propagation, the optimization, but also the handling of interdisciplinary couplings under uncertainty. A decoupled UMDO formulation (Individual Discipline Feasible - Polynomial Chaos Expansion) ensuring the coupling satisfaction for all the instantiations of the uncertain variables is presented in this paper. Ensuring coupling satisfaction in instantiations is essential to ensure the equivalence between the coupled and decoupled UMDO problem formulations. The proposed approach relies on the iterative construction of surrogate models based on Polynomial Chaos Expansion in order to represent at the convergence of the optimization problem, the coupling functional relations as a coupled approach under uncertainty does. The performances of the proposed formulation is assessed on an analytic test case and on the design of a new Vega launch vehicle upper stage

Multi-fidelity modeling with different input domain definitions using Deep Gaussian Processes

Multi-fidelity approaches combine different models built on a scarce but accurate data-set (high-fidelity data-set), and a large but approximate one (low-fidelity data-set) in order to improve the prediction accuracy. Gaussian Processes (GPs) are one of the popular approaches to exhibit the correlations between these different fidelity levels. Deep Gaussian Processes (DGPs) that are functional compositions of GPs have also been adapted to multi-fidelity using the Multi-Fidelity Deep Gaussian process model (MF-DGP). This model increases the expressive power compared to GPs by considering non-linear correlations between fidelities within a Bayesian framework. However, these multi-fidelity methods consider only the case where the inputs of the different fidelity models are defined over the same domain of definition (e.g., same variables, same dimensions). However, due to simplification in the modeling of the low-fidelity, some variables may be omitted or a different parametrization may be used compared to the high-fidelity model. In this paper, Deep Gaussian Processes for multi-fidelity (MF-DGP) are extended to the case where a different parametrization is used for each fidelity. The performance of the proposed multifidelity modeling technique is assessed on analytical test cases and on structural and aerodynamic real physical problems

Overview of Gaussian process based multi-fidelity techniques with variable relationship between fidelities, application to aerospace systems

International audienceThe design process of complex systems such as new configurations of aircraft or launch vehicles is usually decomposed in different phases which are characterized by the depth of the analyses in terms of number of design variables and fidelity of the physical models. At each phase, the designers have to deal with accurate but computationally intensive models as well as cheap but inaccurate models. Multi-fidelity modeling is a way to merge different fidelity models to provide engineers with accurate results with a limited computational cost. Within the context of multi-fidelity modeling, approaches based on Gaussian Processes emerge as popular techniques to fuse information between the different fidelity models. The relationship between the fidelity models is a key aspect in multi-fidelity modeling. This paper provides an overview of Gaussian process-based multi-fidelity modeling techniques for variable relationship between the fidelity models (e.g., linearity, non-linearity, variable correlation). Each technique is described within a unified framework and the links between the different techniques are highlighted. All approaches are numerically compared on a series of analytical test cases and four aerospace related engineering problems in order to assess their benefits and disadvantages with respect to the problem characteristics

Multi-Objective Multidisciplinary Design Optimization Approach for Partially Reusable Launch Vehicle Design

International audienceReusability of the first stage of launch vehicles may offer new perspectives to lower the cost of payload injection into orbit if sufficient reliability and efficient refurbishment can be achieved. One possible option that may be explored is to design the vehicle first stage for both reusable and expendable uses, in order to increase the flexibility and adaptability to different target missions. This paper proposes a multilevel multidisciplinary design optimization (MDO) approach to design aerospace vehicles addressing multimission problems. The proposed approach is focused on the design of a family of launchers for different missions sharing commonalities using multi-objective MDO to account for the computational cost associated with the discipline simulations. The multimission problem addressed considers two missions: 1) a reusable configuration for a sun synchronous orbit with a medium payload range and recovery of the first stage using a gliding-back strategy; 2) an expendable configuration for a medium payload injected into a geostationary transfer orbit. A dedicated MDO formulation introducing couplings between the missions is proposed in order to efficiently solve such a coupled problem while limiting the number of calls to the exact multidisciplinary analysis thanks to the use of Gaussian processes and multi-objective efficient global optimization

A Vitual-Force Based Swarm Algorithm for Balanced Circular Bin Packing Problems

Balanced circular bin packing problems consist in positioning a given number of weighted circles in order to minimize the radius of a circular container while satisfying equilibrium constraints. These problems are NP-hard, highly constrained and dimensional. This paper describes a swarm algorithm based on a virtual-force system in order to solve balanced circular bin packing problems. In the proposed approach, a system of forces is applied to each component allowing to take into account the constraints and minimizing the objective function using the fundamental principle of dynamics. The proposed algorithm is experimented and validated on benchmarks of various balanced circular bin packing problems with up to 300 circles. The reported results allow to assess the effectiveness of the proposed approach compared to existing results from the literature.Comment: 23 pages including reference

A survey of rare event simulation methods for static input–output models

International audienceCrude Monte-Carlo or quasi Monte-Carlo methods are well suited to characterize events of which associated probabilities are not too low with respect to the simulation budget. For very seldom observed events, such as the collision probability between two aircraft in airspace, these approaches do not lead to accurate results. Indeed, the number of available samples is often insufficient to estimate such low probabilities (at least 10^6 samples are needed to estimate a probability of order 10^-4with 10% relative error with Monte-Carlo simulations). In this article, one reviewed different appropriate techniques to estimate rare event probabilities that require a fewer number of samples. These methods can be divided into four main categories: parameterization techniques of probability density function tails, simulation techniques such as importance sampling or importance splitting, geometric methods to approximate input failure space and finally, surrogate modeling. Each technique is detailed, its advantages and drawbacks are described and a synthesis that aims at giving some clues to the following question is given: “which technique to use for which problem?”

Surrogate model-based multi-objective MDO approach for partially Reusable Launch Vehicle design

International audienceReusability of the first stage of launch vehicles may offer new perspectives to lower the cost of payload injection into orbit if sufficient reliability and low refurbishment costs can be achieved. One possible option that may be explored is to design the launch vehicle first stage for both reusable and expendable uses, in order to increase the flexibility and adaptability to different target missions. This paper proposes a multi-level MDO approach to design aerospace vehicles addressing multi-mission problems. The proposed approach is focused on the design of a family of launchers for different missions sharing commonalities using multi-objective Bayesian Optimization to account for the computational cost associated with the discipline simulations. The multi-mission problem addressed in this paper considers two missions: a reusable configuration for a SSO orbit with a medium payload range and recovery of the first stage using a glider strategy; and an expendable configuration for a medium payload injected into a Geostationary Transfer Orbit (GTO). A dedicated MDO formulation introducing couplings between the missions is proposed in order to efficiently solve the multi-objective MDO problem while limiting the number of calls to the exact MDA thanks to the use of Gaussian Processes and multi-objective Efficient Global Optimization