Extended Hierarchical Kriging Method for Aerodynamic Model Generation Incorporating Multiple Low-Fidelity Datasets

Design approach for tilt propellers of UAM/eVTOLs for cruise and hover considering aerodynamic and aeroacoustic characteristics via a multi-fidelity model

Multi-fidelity surrogate modeling, by integrating high-fidelity (HF) and low-fidelity (LF) data, provides an efficient approximation framework for complex fluid dynamics problems, particularly those driven by numerical resolution discrepancies. However, under limited computational budgets, efficiently selecting HF samples remains a critical challenge. This study proposes a Nonlinear-Guided Active Learning Multi-Fidelity Surrogate (NG-AL-MFS) modeling method. The proposed framework employs a recursive Co-Kriging model combined with an adaptive sampling strategy to achieve hierarchical data fusion. In this approach, the adaptive sampling strategy introduces a local nonlinearity indicator derived from LF responses to identify complex regions and integrates a distance-based exploration criterion to balance exploitation and exploration during potential HF sampling. Throughout the process, NG-AL-MFS quantitatively extracts local nonlinear features from LF responses to iteratively and adaptively update HF samples, thereby constructing a multi-fidelity surrogate that achieves a superior trade-off between modeling accuracy and computational cost. The proposed framework is validated using three benchmark functions and two fluid engineering applications. Results demonstrate that NG-AL-MFS outperforms existing approaches in terms of prediction accuracy, convergence efficiency, and statistical robustness, offering notable advantages in multivariate nonlinear scenarios. Overall, NG-AL-MFS provides a data-efficient and reliable solution for budget-constrained surrogate modeling in fluid dynamics and other computation-intensive domains.

• Differentiable multi-fidelity inference: BMFIA enables gradient-based Bayesian inference for high-dimensional, non-differentiable multi-physics problems using low-fidelity model gradients. • Efficiency with minimal HF data: Accurate posterior estimates are achieved from only 100–300 HF–LF simulations with orders-of-magnitude speed-ups over HF-only inference. • Accuracy despite crude LF models: Robust posterior reconstruction is maintained even with weakly correlated or simplified low-fidelity models. High-dimensional Bayesian inverse analysis ( dim ≫ 100 ) is mostly unfeasible for computationally demanding, nonlinear physics-based high-fidelity (HF) models. Usually, the use of more efficient gradient-based inference schemes is impeded if the multi-physics models are provided by complex legacy codes. Adjoint-based derivatives are either exceedingly cumbersome to derive or nonexistent for practically relevant large-scale nonlinear and coupled multi-physics problems. Similarly, holistic automated differentiation w. r. t. primary variables of multi-physics codes is usually not yet an option and requires extensive code restructuring if not considered from the outset in the software design. This absence of differentiability further exacerbates the already present computational challenges. To overcome the existing limitations, we propose a novel inference approach called Bayesian multi-fidelity inverse analysis (BMFIA) , which leverages simpler and computationally cheaper lower-fidelity (LF) models that are designed to provide model derivatives. BMFIA learns a simple, probabilistic dependence of the LF and HF models, which is then employed in an altered likelihood formulation to statistically correct the inaccurate LF response. From a Bayesian viewpoint, this dependence represents a multi-fidelity (MF) conditional density (discriminative model). We demonstrate how this MF conditional density can be learned robustly in the small data regime from only a few HF and LF simulations (50 to 300), which would not be sufficient for naive surrogate approaches. The formulation is fully differentiable and allows the flexible design of a wide range of LF models. We demonstrate that BMFIA solves Bayesian inverse problems for scenarios that used to be prohibitive, such as finely-resolved and hence high-dimensional spatial reconstruction problems in two-dimensional Euclidean domains with static posteriors, given nonlinear and transient coupled poro-elastic media physics. We show that the resulting static MF posteriors are in excellent agreement with the (usually inaccessible) HF posteriors or ground-truth data and note that extending the framework to arbitrary three-dimensional domains is a natural and important direction for future work.

Research on multi-fidelity aerodynamic optimization methods

The scope of this paper is to apply a model-correction-based strategy for efficient reliability analysis of uncertain dynamical systems based on a low-fidelity (LF) model whose outcomes are corrected in a probabilistic sense to represent the more realistic outcomes of a high-fidelity (HF) model. In the model-correction approach utilized, the LF model is calibrated to the HF model close to the so-called most probable point in standard normal space, which allows a more realistic assessment of the considered complex dynamical system. Since only few expensive limit state function evaluations of the HF model are required, an efficient reliability analysis is enabled. In an application example, the LF model describes an existing single-span railway bridge modelled as simply supported Euler–Bernoulli beam subjected to moving single forces representing the axle loads of a moving train. The HF modelling approach accounts for the bridge–train interaction by modelling the passing train as mass-spring-damper system, however increasing the computational effort of the limit state function evaluations. Failure probabilities evaluated with the model-correction approach are contrasted and discussed with failure probabilities of the sophisticated bridge–train interaction model evaluated with the first-order reliability method (FORM). It is demonstrated that the efficiency of the method depends on the correlation between the LF and the HF model. A comparison of the results of FORM and the model-correction-based approach shows that the latter provides reliable failure probability prediction of the HF model while leading to a significant reduction in computational effort.

Scaling Properties of Multi-Fidelity Shape Optimization Algorithms

A reanalysis-based multi-fidelity (RBMF) surrogate framework for efficient structural optimization

Verification and validation of simulation models are critical steps in engineering. This paper aims at verifying the suitability of reduced order aerodynamic models used in an aeroservoelastic framework designed to analyze the flight dynamics of flexible aircraft, known as the Cranfield Accelerated Aircraft Loads Model. This framework is designed for rapid assessment of aircraft configurations at the conceptual design stage. Therefore, it utilizes or relies on methods that are of relatively low fidelity for high computational speeds, such as modified strip theory coupled with Leishmann–Beddoes unsteady aerodynamic model. Hence, verification against higher order methods is required. Although low fidelity models are widely used for conceptual design and loads assessments, the open literature still lacks a comparison against higher fidelity models. This work focuses on steady-trimmed flight conditions and investigates the effect of aerodynamic wing deformation under such loads on aerodynamic performance. Key limitations of the reduced order models used, namely fuselage and interference effects, are discussed. The reasons for the overall agreement between the two approaches are also outlined.

Many researchers have studied Multi-fidelity (MF) models to obtain the optimum solution efficiently when the time of analysis and evaluation is long. The MF model is a meta-model that combines High fidelity (HF) data, which requires large computational cost for evaluation; and Low fidelity (LF) data, which needs small computation cost but has low accuracy. Therefore, integrating HF and LF data with a scale factor and correction model is highly important in MF modeling. This study investigates the performance of the MF model in terms of definition and estimation. First, three different MF models are built: An LF model multiplied by the scale factor, an LF model combined with a correction model, and an LF model with both scale factor and correction model. Second, the effects of scale factor on the MF model are analyzed in terms of constant vs. linear and maximum likelihood estimation vs. least squares estimation. Finally, a correction model is built using two methods, namely, interpolation and regression, to assess the influence of the correction model on the MF model. To evaluate the MF model performance, several test problems are applied, and the root mean square error of each model is estimated as the accuracy measure. In conclusion, the characteristics of different types of MF models are summarized and guidelines for the generation of MF models are proposed to approximate closely the precise models.

Application of deep learning based multi-fidelity surrogate model to robust aerodynamic design optimization

A multifidelity surrogate (MFS) model is a data fusion method for the enhanced prediction of less intensively sampled primary variables of interest (i.e., high-fidelity (HF) samples) with the assistance of intensively sampled auxiliary variables (i.e., low-fidelity (LF) samples). In this article, an MFS model based on the gradient-enhanced radial basis function, termed gradient-enhanced multifidelity surrogate based on the radial basis function (GEMFS-RBF), is proposed to establish a mapping relationship between HF and LF samples. To identify the scaling factor and the undetermined coefficients in GEMFS-RBF, an expanded correlation matrix is constructed by considering the correlations between the acquired samples, the correlations between the gradients, and the correlations between the samples and their corresponding gradients. To evaluate the prediction accuracy of the GEMFS-RBF model, it is compared with the co-Kriging model, multifidelity surrogate based on the radial basis function (MFS-RBF) model, and two single-fidelity surrogate models. The influences of key factors (i.e., the correlations between the HF and LF functions, the subordinations between the sample sets) and the effect of the cost ratio on the performance of GEMFS-RBF are also investigated. It is observed that GEMFS-RBF presents a more acceptable accuracy rate and is less sensitive to the aforementioned factors than the other benchmark models in most cases in this article, which illustrates the practicability and robustness of the proposed GEMFS-RBF model.

The objective of this paper is to efficiently perform the reliability assessment of shell and tube heat exchangers. Although inexpensive empirical/analytical heat exchanger models exist and could be used for brute force Monte-Carlo simulations (MCS) based reliability analysis, they typically do not characterize the shell side flow accurately. Hence, higher fidelity models are often needed for predicting the shell and tube heat exchanger performance with a desired level of accuracy. These higher fidelity models are generally associated with higher computational costs, making them impractical for MCS-based reliability analysis. To circumvent this problem, a multi-fidelity technique leveraging the lower and higher fidelity models is proposed to locally construct the failure boundaries using support vector machines (SVM). For this purpose, an adaptive sampling scheme, which explores the regions of inconsistency between failure boundaries from lower and higher fidelity models, is developed.

Recently, surrogate modeling methods have been explored for structural acoustics applications. These often involve evaluation of an “expensive” high-fidelity computational model to obtain training data. However, in many applications, models of varying fidelity and computational cost are available. In such situations, one can leverage multi-fidelity surrogate modeling, where the training data from models of varying fidelity are combined and simultaneously used to produce a surrogate model. A particularly popular class of multi-fidelity surrogate modeling techniques is known as co-Kriging, where simulation output from both “expensive” and “cheap” computational models are correlated and a correction process is obtained that maps between the results of these models of varying fidelity. This talk will review co-Kriging and demonstrate its utility on a canonical structural acoustics problem. [Work supported by the Office of Naval Research.]

This paper presents a methodology to move toward reliable real-time structural health monitoring (SHM). The proposed procedure relies upon surrogate modeling based on a multi-fidelity (MF) deep neural network (DNN), conceived to map damage and operational parameters onto sensor recordings. Within a stochastic framework, the MF-DNN is adopted by a Markov chain Monte Carlo sampling procedure to update the probability distribution of the structural state, conditioned on noisy observations. The MF-DNN enables to locate and possibly quantify the presence of damage, and its multi-fidelity configuration effectively blends datasets featuring different fidelities without any prior assumption. The training datasets are generated with physics-based models of the monitored structure: high fidelity (HF) and low fidelity (LF) models are considered to simulate the structural response under varying operational conditions, respectively in the presence or absence of a structural damage. The MF-DNN is a composition of a fully-connected LF-DNN, which mimics sensor recordings in the undamaged condition, and of a long short-term memory HF-DNN, which is exploited to enrich the LF approximation for the considered damaged scenarios. By framing the model updating strategy as an incremental or residual modeling problem, the MF-DNN is reported to provide numerous advantages over single-fidelity based models for SHM purposes.KeywordsBayesian model updatingDeep learningMarkov chain Monte CarloStructural health monitoringMulti-fidelity methodsReduced-order modelingReal-time damage identification

When choosing an appropriate hydrodynamic model, there is always a compromise between accuracy and computational cost, with high fidelity models being more expensive than low fidelity ones. However, when assessing uncertainty, we can use a multifidelity approach to take advantage of the accuracy of high fidelity models and the computational efficiency of low fidelity models. Here, we apply the multilevel multifidelity Monte Carlo method (MLMF) to quantify uncertainty by computing statistical estimators of key output variables with respect to uncertain inputs, using the high fidelity hydrodynamic model XBeach and the lower fidelity coastal flooding model SFINCS. The multilevel aspect opens up the further advantageous possibility of applying each of these models at multiple resolutions. This work represents the first application of MLMF in the coastal zone and one of its first applications in any field. For both idealised and real-world test cases, MLMF can significantly reduce computational cost for the same accuracy compared to both the standard Monte Carlo method and to a multilevel approach utilising only a single model (the multilevel Monte Carlo method). In particular, here we demonstrate using the case of Myrtle Beach, USA, that this improvement in computational efficiency allows in-depth uncertainty analysis to be conducted in the case of real-world coastal environments – a task that would previously have been practically unfeasible. Moreover, for the first time, we show how an inverse transform sampling technique can be used to accurately estimate the cumulative distribution function (CDF) of variables from the MLMF outputs. MLMF based estimates of the expectations and the CDFs of the variables of interest are of significant value to decision makers when assessing risk.