**1. Introduction**

Through better engineering design we can build superior aircraft engines with higher efficiency and lower weight, wind turbines that can produce more energy at

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*Design and Manufacturing*

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a reduced cost, and steam turbines that can reach higher efficiencies, to mention a few examples. The importance of a fast design process which produces global optima with the fewest amount of resources is obviously fundamentally crucial. This process, for the purpose of this chapter, can be thought of as a multi-objective optimization problem. Consider as an example the design of aerodynamic airfoil shapes. First, the performance is produced, followed by a mechanical and aeromechanics assessment. Aeromechanical feedback and reactive aerodynamic redesign rely heavily on the domain expertise of the design engineers. It is not atypical to cycle through 50 of these iterations to obtain a design that satisfies mechanical (stress, creep, fatigue to mention a few) and aeromechanical (say, clean Campbell and flutter resistant) requirements. During these cycles, data from expensive computational codes and/or real-world experiments are collected and the design cycle continues in a direction suggested by this information. Generally speaking, as some examples of the key contributions toward high resource requirements are expensive computer simulations such as computational fluid dynamics (CFD) and ANSYS. In some cases, real-world experiments need to be performed, e.g., when it comes to passing FAA certification.

the efficiency and to minimize the mass flow rate but in general, we may have *N* objectives. When *N*>1, we are perform multi-objective optimization, and later in this chapter, we explore a set of sampling strategies suitable for various values of *N*. The set of all points in objective space form a *response surface* which is assumed to be extremely costly to exhaustively explore putting the search for global optima at risk. Toward reducing the cost of the overall design process regardless of whether the data is obtained by real-world experiments or complex computer codes, a key ingredient is meta-/surrogate modeling where an approximation to the response surface is obtained by generalizing its behavior under certain assumptions (such as smoothness) based on few observed instances of said surface. This is typically done by querying a set of initial points on which the surrogate model is built. Following this, the meta-model is extremely cheap to query in comparison to the surface itself. These meta-models require, typically, only a handful of data to construct an accurate representation of the response surface. An important question however arises on which points to pick, i.e., on how to form the design [10, 12–14]. Given a budget

*Industrial Applications of Intelligent Adaptive Sampling Methods for Multi-Objective…*

on the total number of data points, how are new points added to this design sequentially? This is where adaptive sampling comes into play and GEBHM is used in conjunction with a powerful technology called intelligent design and analysis of

In industry applications, it is not uncommon for the data to be multidimensional, noisy, highly non-linear, and expensive to collect. On a day-to-day basis, we address the challenge of enabling a robust and uncertainty certified design utilizing both limited expansive simulations data and noisy field measurements. GE Research has an in-house software framework for advanced Bayesian modeling and machine learning named GEBHM, sometimes we shall refer to this as simply BHM, which enables the combination of multiple numerical simulations and experimental sources of data in one unified workflow. As shown in **Figure 1**, GEBHM capabilities are: uncertainty propagation and quantification, sensitivity analysis, full Bayesian model calibration, meta-modeling, multi-fidelity analysis, and adaptive design of numerical experiments. The theoretical framework of the GEBHM is based on Kennedy O'Hagans approach to modeling and fusing simulation and experimental data with associated uncertainties using GPs [6]. The noisy highfidelity model is represented as Gaussian process aggregated from a linear combi-

nation of a low-fidelity model and a model discrepancy function *δ*ð Þ� as

as possible. The measurement error is denoted by ϵ.

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where *y*ð Þ� is the (high-fidelity numerical model or experimentally measured) response. The low-fidelity model is *η*ð Þ �*;* � and discrepancy term *δ*ð Þ� are modeled by separate GPs. The design variable is denoted by *xi*, while the calibration parameters are denoted by *θ*. GEBHM allows for calibration of a set of model parameters in the low-fidelity model. For example, this could be parameters in a CFD simulation that we want to tune/calibrate in order to match real-world experimental runs as closely

The GP hyperparameters are learned using a Markov Chain Monte Carlo (MCMC) technique based on an Metropolis-Hastings-within-Gibbs algorithm [27, 28] with univariate proposal distributions for the posterior distribution

*y x*ð Þ¼*<sup>i</sup> η xi* ð Þþ *; θ δ*ð Þþ *xi* ϵ*,* for *i* ¼ 1*,* …*, n,* (1)

details and application coverage can be found in Refs. [7, 11, 15–26].

In this section, we provide an overview of the mathematical framework of the GEBHM and GE-IDACE technologies introduced in Section 1. Further theoretical

computer experiments (GE-IDACE/IDACE) [15, 16].

*DOI: http://dx.doi.org/10.5772/intechopen.88213*

**2.1 Bayesian hybrid modeling (GEBHM/BHM)**

With this, it should also be clear that engineering design is performed under very strict budgets. Each datum obtained whether from a simulation, physical experiment, or an expert needs to be as informative toward the goals we are trying to accomplish as possible. In some cases, it can take weeks or months to evaluate a single datum. In this case, a meta-model is built on a small representative set of data. This can be Gaussian processes (GPs) [1–5], Bayesian hybrid modeling as used at GE [6, 7], or polynomial chaos expansions (PCEs) [8, 9].

One of the key goals of this chapter is to present the state-of-the-art industrial tools toward achieving the best possible optima under strict budget constraints. Specifically at GE we are regularly seeing a reduction in cost needed to obtain the same level of information on the order of 30–90%. This leaves more room in the budget for finding even better, more competitive, designs as hitherto possible. As a consequence of the technologies covered in this chapter, we are building better aircraft engines, improving our steam turbines, and harvesting more wind energy because of this. There is still a lot more to be invented and improved, but the following sections will give an idea of where we currently stand.
