Multi-fidelity Models Research Articles

Automated Finite Element Analysis of Stiffened Tanks Using Multifidelity Modeling

Accurate structural analysis of stiffened thin-walled pressure vessels enables critical design decisions to be made during the conceptual and baseline design phases. Furthermore, having multifidelity finite element models that couple shell and solid representations facilitates exploring many design iterations quickly with sufficient accuracy. However, multifidelity model creation and utilization can be cumbersome to perform interactively, such that automated model creation, mesh generation, analysis submission, and results postprocessing are highly desirable. Of particular difficulty is generating multifidelity meshes robustly. A framework for automatic multifidelity model development using parameterized scripting in commercial finite element packages is developed to address these critical modeling needs. Issues associated with automated meshing of highly complex assemblies are examined, and a volume decomposition scheme based upon face partitions is proposed. Application is made to a complex stiffened thin-walled pressure vessel with full shell and solid/shell multifidelity model accuracies assessed by comparison against all solid models. Results indicate the proposed technique’s potential for design iteration improvements with significant reductions in interactive model preparation and postprocessing costs, as well as overall computational effort. Potential novel applications and improvements to the presented framework that can enable its application to complex problems are discussed.

Gradient-enhanced deep Gaussian processes for multifidelity modeling

Multifidelity models integrate data from multiple sources to produce a single approximator for the underlying process. Dense low-fidelity samples are used to reduce interpolation error, while sparse high-fidelity samples are used to compensate for bias or noise in the low-fidelity samples. Deep Gaussian processes (GPs) are attractive for multifidelity modeling as they are non-parametric, robust to overfitting, perform well for small datasets, and, critically, can capture nonlinear and input-dependent relationships between data of different fidelities. Many datasets naturally contain gradient data, most commonly when they are generated by computational models that have adjoint solutions or are built in automatic differentiation frameworks.Principally, this work extends deep GPs to incorporate gradient data. We demonstrate this method on an analytical test problem and two realistic aerospace problems: one focusing on a hypersonic waverider with an inviscid gas dynamics truth model and another focusing on the canonical ONERA M6 wing with a viscous Reynolds-averaged Navier-Stokes truth model.In both examples, the gradient-enhanced deep GP outperforms a gradient-enhanced linear GP model and their non-gradient-enhanced counterparts.

Precision Calibration in Wire-Arc-Directed Energy Deposition Simulations Using a Machine-Learning-Based Multi-Fidelity Model

Thermal simulation is essential in wire-arc-directed energy deposition (W-DED) to accurately estimate temperature distributions, impacting residual stress and distortion in components. Proper calibration of simulation models minimizes inaccuracies caused by varying material properties, machine settings, and environmental conditions. The lack of standardized calibration methods further complicates thermal predictions. This paper introduces a novel calibration method integrating both machine learning, as the high-fidelity (HF) model, and response surface modeling, as the low-fidelity (LF) model, within a multi-fidelity (MF) framework. The approach utilizes Bayesian optimization to effectively explore the search space for optimal solutions. A two-tiered model employs the LF model to identify feasible regions, followed by the HF model to refine calibration parameters, such as thermal efficiency (η), convection coefficient (h), and emissivity (ε), which are difficult to determine experimentally. A three-factor Box–Behnken design (BBD) is applied to explore the design space, requiring only thirteen parameter configurations, conserving resources and enabling robust model training. The efficacy of this MF model is demonstrated in multi-layer W-DED calibration, showing strong alignment between experimental and simulated temperatures, with a mean absolute error (MAE) of 7.47 °C. This method offers a replicable framework for broader additive manufacturing processes.

Concepts for Frequency Sweeps and Efficient Repeated Analysis in the Context of Vibroacoustic Optimization and Uncertainty Quantification

This paper aims at passive noise control for vibroacoustic problems, which are analyzed by finite and boundary element techniques. The author distinguishes interior and exterior problems mainly because of the quantities used as the objective function to assess the acoustic quality. For interior problems, it is common to use local quantities such as the sound pressure at a field point or, in rare cases, energy density at a field point. The situation is different for exterior problems where the radiated sound power accounts for a suitable and global quantity to assess the emission from a vibrating structure. For most engineering purposes, the assessment requires frequency sweeps in which the problem needs to be solved at many discrete frequencies. In vibroacoustic optimization and in sampling based uncertainty quantification, it is very common that structural parameters are varied, while the acoustic field remains the same throughout the entire process. We will review concepts and recent developments of efficient frequency sweeps and repeated analysis with unmodified fluid domain. For many practical cases, the situation for interior problems is rather simple to survey. Either the authors have applied a modal analysis and used a modal superposition for the frequency sweep and repeated analysis, or the concept of unmodified acoustic transfer vectors is applied. Both concepts are quite successful as long as certain conditions are fulfilled. For exterior problems, a modal superposition is possible but, so far, only for a limited number of cases practically applicable as discussed herein. The concept of using acoustic transfer vectors becomes inefficient since the evaluation of the radiated sound power asanintegral over a closed enveloping surface would require an excessively high amount of storage capacity. Therefore, other concepts are being followed. For frequency sweeps, a number of methods are using a frequency interpolation based on a limited number of discrete sample frequencies. Often, these techniques are used together with Krylov–subspace model order reduction techniques. Further recently published approaches investigate low-rank approximations, greedy algorithms, and deflated Krylov subspace techniques. A completely different kind of approach is based on multi–fidelity models and Gaussian processes. The field of efficient repeated analysis shows some interesting developments, which can be easily applied to sampling based uncertainty quantification but do not seem to be easily and generally applied to optimization.

Multi-Fidelity Modelling of the Effect of Combustor Traverse on High-Pressure Turbine Temperatures

As turbine entry temperatures of modern jet engines continue to increase, additional thermal stresses are introduced onto the high-pressure turbine rotors, which are already burdened by substantial levels of centrifugal and gas loads. Usually, for modern turbofan engines, the temperature distribution upstream of the high-pressure stator is characterized by a series of high-temperature regions, determined by the circumferential arrangement of the combustor burners. The position of these high-temperature regions, both radially and circumferentially in relation to the high-pressure stator arrangement, can have a strong impact on their subsequent migration through the high-pressure stage. Therefore, for a given amount of thermal power entering the turbine, a significant reduction in maximum rotor temperatures can be achieved by adjusting the inlet temperature distribution. This paper is aimed at mitigating the maximum surface temperatures on a high-pressure turbine rotor from a modern commercial turbofan engine by conducting a parametric analysis and optimization of the inlet temperature field. The parameters considered for this study are the circumferential position of the high-temperature spots, and the overall bias of the temperature distribution in the radial direction. High-fidelity unsteady (phase-lag) and conjugate heat transfer simulations are performed to evaluate the effects of inlet clocking and radial bias on rotor metal temperatures. The optimized inlet distribution achieved a 100 K reduction in peak high-pressure rotor temperatures and 7.5% lower peak temperatures on the high-pressure stator vanes. Furthermore, the optimized temperature distribution is also characterized by a significantly more uniform heat load allocation on the stator vanes, when compared to the baseline one.

Adaptive-Quadratic-Neural-Network-Based Multifidelity Modeling Approach for Buckling of Stiffened Panels

This paper presents a method for predicting the buckling load of stiffened panels using multifidelity modeling based on quadratic neural networks (QNNs) with adaptive activation functions. The effectiveness of the proposed approach is demonstrated through a series of simulations on a range of stiffened panel configurations, and the results are compared to those obtained from traditional multifidelity and high-fidelity models in terms of accuracy and computational efficiency. Numerical experiments demonstrate that the model can accurately and efficiently predict the buckling load of stiffened panels while significantly reducing the computational cost of evaluating the surrogate model. In particular, the proposed adaptive quadratic neural networks (AQNNs) model achieves convergence approximately three times faster and four times less trainable parameters compared to traditional artificial neural networks while maintaining the same validation loss. This approach can significantly improve the design and optimization of aerospace structures by easily and quickly exploring various design configurations and finding stable and efficient configurations. This study highlights the potential of a new multifidelity modeling framework for predicting the buckling load and collapse load of aerospace structures by enhancing convergence speed and prediction accuracy while reducing the computational complexity of neural networks.

Assessment of Multifidelity Surrogate Approaches for Expedient Loads Prediction in High-Speed Flows

Fast and accurate predictions about the flowfield surrounding a hypersonic flight vehicle are necessary for both early design phases and autonomous vehicle control. Data-driven model reduction presents an opportunity to harness the accuracy of high-fidelity techniques for expedient online application. Surrogate modeling is one approach that seeks to emulate the response of a system with evaluation times much faster than the data source. Multifidelity surrogate modeling seeks to reduce the amount of high-fidelity data needed for a sufficiently accurate online model. The objective of this paper is to assesses the tradeoffs between a kriging-based multifidelity model and a neural-network-based multifidelity model, applied to high-speed flow past a canonical flight geometry. All models studied in this work reproduce the results of the benchmark simulations with less than 100 training cases. The kriging models marginally outperform the neural-network-based models in accuracy, but requirements for training and storing the kriging models scale exponentially as the training data gets larger. Kriging models offer built-in uncertainty quantification and require significantly less decision making about hyperparameters and architecture choices. This study also indicates that the inclusion of data from classical engineering methods can improve the prediction accuracy of surrogate models at practically no cost.

An efficient multi-fidelity simulation approach for performance prediction of adaptive cycle engines

Adaptive cycle engine (ACE) with multi-stream, modulated bypass ratio, and high-thrust/high-efficiency mode provides the capability of multiple mission adaptation for the next-generation aviation propulsion system. The low-fidelity zero-dimensional (0D) performance simulation method is commonly adopted in conventional engines such as turbofan and turbojet. However, it is hard to accommodate well to ACE with complex variable geometry schemes and strong interaction between components. This paper presents an efficient multi-fidelity simulation approach, often referred to as component zooming, for predicting the performance of ACE with the three-stream configuration. The one-dimensional (1D) mean-line models of the adaptive fan and low-pressure turbine (LPT) are integrated into the 0D ACE model by the iteratively-coupled method. The performance predicted by the multi-fidelity ACE models with different component zooming strategies is compared. The maximum deviations of thrust, specific fuel consumption, turbine inlet temperature, and bypass ratio between the 0D/1D Adaptive Fan/1D LPT coupled model and 0D ACE model are −10.8%, −4.4%, −5.4% (−75 K), and 1.6%, respectively, which are considerable and non-negligible for the application in the design stage of ACE. The computing time of all the multi-fidelity models is less than 2 minutes. The effects of the key aerodynamic parameters of the adaptive fan and LPT on the engine performance are also evaluated. The proposed approach provides a generic and efficient solution for multi-fidelity ACE performance prediction with acceptable computing resource requirements and time cost, which is applicable in the engine conceptual and preliminary design stage.

Multi-fidelity surrogate with heterogeneous input spaces for modeling melt pools in laser-directed energy deposition

Multi-fidelity (MF) modeling is a powerful statistical approach that can intelligently blend data from varied fidelity sources. This approach finds a compelling application in predicting melt pool geometry for laser-directed energy deposition (L-DED). One major challenge in using MF surrogates to merge a hierarchy of melt pool models is the variability in input spaces. To address this challenge, this paper introduces a novel approach for constructing an MF surrogate for predicting melt pool geometry by integrating models of varying complexity, that operate on heterogeneous input spaces. The first thermal model incorporates five input parameters i.e., laser power, scan velocity, powder flow rate, carrier gas flow rate, and nozzle height. In contrast, the second thermal model can only handle laser power and scan velocity. A mapping is established between the heterogeneous input spaces so that the five-dimensional space can be morphed into a pseudo two-dimensional space. Predictions are then blended using a Gaussian process-based co-kriging method. The resulting heterogeneous multi-fidelity Gaussian process (Het-MFGP) surrogate not only improves predictive accuracy but also offers computational efficiency by reducing evaluations required from the high-dimensional, high-fidelity thermal model. The tested Het-MFGP yields an R2 of 0.975 for predicting melt pool depth. This surpasses the comparatively modest R2 of 0.592 achieved by a GP trained exclusively on high-dimensional, high-fidelity data. Similarly, in the prediction of melt pool width, the Het-MFGP excels with an R2 of 0.943, outshining the GP's performance, which registers a lower R2 of 0.588. The results underscore the benefits of employing Het-MFGP for modeling melt pool behavior in L-DED. The framework successfully demonstrates how to leverage multimodal data and handle scenarios where certain input parameters may be difficult to model or measure.

Multi-fidelity design framework integrating compositional kernels to facilitate early-stage design exploration of complex systems

Abstract Early-stage design of complex systems is considered by many to be one of the most critical design phases because that is where many of the major decisions are made. The design process typically starts with low-fidelity tools, such as simplified models and reference data, but these prove insufficient for novel designs, necessitating the introduction of high-fidelity tools. This challenge can be tackled through the incorporation of multi-fidelity models. The application of MF models in the context of design optimization problems represents a developing area of research. This study proposes incorporating compositional kernels into the autoregressive scheme (AR1) of Multi-Fidelity Gaussian Processes, aiming to enhance the predictive accuracy and reduce uncertainty in design space estimation. The effectiveness of this method is assessed by applying it to 5 benchmark problems and a simplified design scenario of a cantilever beam. The results demonstrate significant improvement in the prediction accuracy and a reduction in the prediction uncertainty. Additionally, the paper offers a critical reflection on scaling up the method and its applicability in early-stage design of complex engineering systems, providing valuable insights into its practical implementation and potential benefits.

A machine learning assisted multifidelity modelling methodology to predict 3D stresses in the vicinity of design features in composite structures

Multifidelity global–local finite element (FE) analyses are typically used to predict damage initiation hotspots around repetitive design features in large composite structures, such as composite airframes. We propose the use of machine learning (ML) methods to accelerate these analyses. We demonstrate this ML assisted framework for the stress analysis of a hole in plate feature in an aerospace C-spar structure. To enable this framework, we develop the following original features: a computationally efficient sampling scheme; a work-equivalent boundary condition homogenisation scheme; a volume averaged ply-by-ply stress approach; and a sequential long-short term memory neural network reformulated from a time basis to a stacking sequence basis with further bi-directionality customisation. Overall, we show that the developed method results in high-accuracy prediction of 3D stresses, with over two orders of magnitude reduction in modelling and simulation time compared to FE analyses.

Global sensitivity analysis of ultrasonic testing simulations of slot-like defects with multifidelity modeling

Abstract In this paper, an efficient global sensitivity analysis (GSA) method for simulation-based ultrasonic testing (UT) of slot-like defects using multifidelity modeling with novel termination criterion is proposed. GSA quantifies the effect of quantities of interest with variability (e.g., position, height, and angle) on the output (e.g., amplitude). GSA with Sobol' indices requires the use of Monte Carlo simulations (MCS) when dealing with nonlinear problems having many parameters. It is impractical to perform GSA directly on high-fidelity physics-based models due to their long evaluation times and the large number of required samples. Multifidelity methods construct surrogate models based on data from an accurate high-fidelity model (HFM) and fast low-fidelity models (LFMs). The multifidelity surrogates evaluate quickly and can be used in lieu of the HFM to accelerate the GSA. Conventional multifidelity methods construct the surrogate to meet a prespecified error metric before using it within an analysis. This requires a separate set of testing data and an often arbitrary error metric threshold. To avoid these, a novel multifidelity modeling termination criterion for GSA is proposed that is based on the absolute relative change of the Sobol' indices. The proposed approach is demonstrated on a simulated UT case inspecting a slot-like defect with three uncertainty variables. The results show a potential for significant reduction in computational cost compared with conventional approaches.

GP+: A Python library for kernel-based learning via Gaussian processes

In this paper we introduce GP+, an open-source library for kernel-based learning via Gaussian processes (GPs) which are powerful statistical models that are completely characterized by their parametric covariance and mean functions. GP+ is built on PyTorch and provides a user-friendly and object-oriented tool for probabilistic learning and inference. As we demonstrate with a host of examples, GP+ has a few unique advantages over other GP modeling libraries. We achieve these advantages primarily by integrating nonlinear manifold learning techniques with GPs’ covariance and mean functions. As part of introducing GP+, in this paper we also make methodological contributions that (1) enable probabilistic data fusion and inverse parameter estimation, and (2) equip GPs with parsimonious parametric mean functions which span mixed feature spaces that have both categorical and quantitative variables. We demonstrate the impact of these contributions in the context of Bayesian optimization, multi-fidelity modeling, sensitivity analysis, and calibration of computer models.

A Framework for Developing Systematic Testbeds for Multi-Fidelity Optimization Techniques

Abstract Multi-fidelity (MF) models abound in simulation-based engineering fields. Many MF strategies have been proposed to improve the efficiency in engineering processes, especially in design optimization. When it comes to assessing the performance of MF optimization techniques, existing practice usually relies on test cases involving contrived MF models of seemingly random math functions, due to limited access to real-world MF models. While it is acceptable to use contrived MF models, these models are often manually written up rather than created in a systematic manner. This gives rise to the potential pitfall that the test MF models may be not representative of general scenarios. We propose a framework to generate test MF models systematically and characterize tested MF optimization methods' performances comprehensively. In our framework, the MF models are generated based on given high-fidelity (HF) model and come with two parameters to control their fidelity levels and allow model randomization. In our testing process, MF case problems are systematically formulated using our model creation method. Running the given MF optimization technique on these problems produces what we call “savings curve” that characterizes the method's performance similarly to how ROC curves characterize machine learning classifiers. Our test results also allow plotting “optimality curves” that serve similar functionality to savings curves in certain types of problems. The flexibility of our MF model creation facilitates the development of testing processes for general MF problem scenarios, and our framework can be easily extended to other MF applications than optimization.

Federated Gaussian Process: Convergence, Automatic Personalization and Multi-fidelity Modeling.

In this paper, we propose FGPR: a Federated Gaussian process ( GP) regression framework that uses an averaging strategy for model aggregation and stochastic gradient descent for local computations. Notably, the resulting global model excels in personalization as FGPR jointly learns a shared prior across all devices. The predictive posterior is then obtained by exploiting this shared prior and conditioning on local data, which encodes personalized features from a specific dataset. Theoretically, we show that FGPR converges to a critical point of the full log-marginal likelihood function, subject to statistical errors. This result offers standalone value as it brings federated learning theoretical results to correlated paradigms. Through extensive case studies on several regression tasks, we show that FGPR excels in a wide range of applications and is a promising approach for privacy-preserving multi-fidelity data modeling.

SEAHOWL: Partitioned Multiphysics and Multifidelity Modelling of Wind Turbines with Monolithically Coupled Elastodynamics

We introduce SEAHOWL (Servo-Elasto-Aero-Hydro Offshore Wind Lab), a novel multiphysics multifidelity simulation framework for large-scale onshore, offshore, and floating wind turbines. For structural dynamics, multibody and finite element problems are coupled monolithically and solved within a single system of equations through Project Chrono, providing high numerical stability and optimal coupling accuracy. Combined with partitioned multiphysics, modularity is at the heart of the framework with various numerical methods and solvers (internal or external) available for each physics. This flexibility allows for the selection a target trade-off between numerical accuracy and computational efficiency on a use-case basis. Simulations showcased here have been selected through the prism of industrial R&D challenges, covering: controller response and loads over the operational wind range, 3P effect for different levels of fidelity for the blades, floating wind turbine response under irregular waves and turbulent wind, and innovative multibody application for 3P fatigue mitigation. SEAHOWL is cross-validated against the reference OpenFAST whole-turbine simulator from NREL, showing overall good agreement while differences between the two frameworks are highlighted.

Central cone film cooling scheme of turbofan engine based on multi-fidelity simulation

To improve the computational efficiency of multi-fidelity simulation, an improved fully coupled method was proposed, which was applied to the coupled simulation of a 3-D model of an exhaust system and a 0-D model of a turbofan engine. Different from conventional approach, by “truncating” the engine model, the CFD calculation of the exhaust system can be decoupled from the solution of nonlinear equations in the engine model, and an outer iteration equation set was constructed to solve the multi-fidelity model. It was found that the improved fully coupled method has better computational efficiency than traditional methods while maintaining equivalent computational results. Then, the improved fully coupled method was applied to the design of the central cone film cooling system, the influence of cooling gas bleed mechanism, the hole open area ratio of the central cone and hole density, hole direction, hole shape on the infrared characteristics was investigated. The impact of central cone film cooling on the overall performance of the engine was quantitatively evaluated. Combined with various favorable factors, when maintaining the HP rotor speed unchanged, the integrated infrared radiation intensity of the central cone under supersonic cruise conditions is reduced by 93 % compared to the uncooled state, while the engine thrust increases by 0.3 % and fuel consumption rate increases by 0.18 %. The multi-fidelity simulation method proposed in this paper and the conclusions obtained are of great significance for the structural design of low-infrared exhaust systems.

Ranking multi-fidelity model performances in reproducing internal and external wake impacts at neighbouring offshore wind farms

Performances of multi-fidelity numerical models in reproducing the impacts of internal and external wakes on power production are evaluated against SCADA data from two operational offshore wind farms in the North Sea. Results obtained for two engineering wake models, Jensen and Cumulative Curl (default and tuned settings), and a higher-fidelity Reynolds-averaged Navier-Stokes (RANS) setup are presented here. Tuned engineering models were configured against RANS results for an idealized wind farm layout consisting of several turbine setups. Input data for the model evaluation scenarios was obtained from a met mast located between the wind farms and binned according to atmospheric stability, wind speed and direction, and turbulence intensity. Results show that the higher fidelity of RANS is most advantageous for inflows that generate more complex internal wakes (e.g., deep array effects) and in replicating external wake impacts. The tuned setups of the engineering models, modified according to stability, outperform their default setups. Default model properties are defined in accordance with developer recommendations. One of the tuned models shows close farm-level agreements with RANS and even exceeds its performances in some of the cases of lower flow complexity.

Data-driven hierarchical collaborative optimization method with multi-fidelity modeling for aerodynamic optimization

The aerodynamic optimization of aircraft based on evolutionary algorithms generally requires thousands or even tens of thousands of iterations. Moreover, it faces challenges such as high-dimensional design variables and high computational costs, leading to low efficiency in algorithm search and poor optimization results. In order to enhance the efficiency of aerodynamic optimization, a data-driven collaborative optimization and multi-fidelity hierarchical modeling method is proposed. Firstly, a Clustering-Enhanced Teaching-Learning-Based Optimization (CETLBO) algorithm is introduced to improve its exploration and exploitation capabilities. By employing a collaborative optimization strategy, a divide-and-conquer paradigm is utilized to efficiently solve high-dimensional, nonlinear, and complex engineering optimization problems. Secondly, a two-stage hierarchical aerodynamic optimization framework is designed to reduce computational costs by integrating data from different fidelity levels. Stage 1 adopts a data-driven approximate optimization method to provide prior knowledge for the high-fidelity optimization of stage 2. Finally, the proposed method is applied to the optimization design of supersonic wing shapes. Compared to traditional optimization design systems, it demonstrates high efficiency in high-dimensional aerodynamic optimization problems, yielding better optimization results.

Steam generator efficiency optimization under uncertainty through multi-fidelity modeling

The steam generator is an essential equipment in power plants, and understanding its behavior with the aid of mathematical models allows for improved decision-making aiming toward better efficiency. Well-calibrated models can help identify new operating conditions that lead to efficiency enhancement. Unfortunately, due to the complexity of steam generators, computationally efficient models based solely on energy-mass balance usually present limiting simplifications. Alternatively, computational fluid dynamics can solve reacting flow phenomena but require a great amount of computational time. Purely data-driven models often extrapolate poorly as they do not necessarily contain the physical equations that drive the system’s behavior. In this scenario, the multi-fidelity approach is an attractive solution because it combines physics-grounded representation with a low computational footprint and allows for mapping the efficiency of the steam generator considering uncertainty propagation. This paper presents a multi-fidelity model that simulates the efficiency of a steam generator from the PECÉM power plant. The model is assembled by the sum of two Gaussian processes, trained using one high-fidelity and one low-fidelity dataset. The high-fidelity dataset is formed by the observed operation of the actual power plant at varying operating conditions. These same operating parameters are the input for the energy-mass balance model built using the EBSILON software to generate the low-fidelity dataset. The observed efficiency of the PECÉM power plant ranged from 82.89% to 84.37%, with a mean value of 83.58%. The resulting multi-fidelity model presents a root mean square error of 0.19%, which is six times lower than the error obtained by the energy-mass balance model. This model is then used for global sensitivity analysis and robust optimization. A Pareto front mapping expected efficiency and its confidence level is built by exploring not yet experimented operational points. Finally, data clustering allowed the identification of efficiencies around 86%, which adds 1.63% to the power plant’s maximum level.