Scaling Properties of Multi-Fidelity Shape Optimization Algorithms

• 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.

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.

Research on multi-fidelity aerodynamic optimization methods

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

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

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

Computational fluid dynamics (CFD) simulations are a fundamental tool in aerodynamic design. Unfortunately, accurate, high-fidelity CFD models may be computationally too expensive to conduct the design using numerical optimization procedures. Recently, variable-fidelity optimization algorithms have attracted attention for their ability to reduce high CPU-cost related to the design process solely based on accurate CFD models. Low-fidelity simulation models are the most critical components of such algorithms. They normally employ the same CFD solver as the high-fidelity model but with reduced discretization density and reduced number of flow solver iterations. Typically, the selection of the appropriate model parameters has only been guided by the designer experience. In this chapter, an automated low-fidelity model selection technique is described. By defining the model setup task as a constrained nonlinear optimization problem, suitable grid and flow solver parameters are obtained. The approach is compared to two conventional methods of generating a family of variable-fidelity models. Comparison of the standard and the proposed approach is carried out in the context of aerodynamic design of transonic airfoils using a multi-level optimization algorithm. The results obtained for several test cases indicate that the automated model generation may lead to significant computational savings of the CFD-based airfoil design process. Illustration of the entire optimization cycle involving automated low-fidelity model preparation and B-spline-parameterized airfoil design using space mapping algorithm is also provided.

Multi-fidelity surrogate modeling (MFSM) methods are gaining recognition for their effectiveness in addressing simulation-based design challenges. Prior approaches have typically relied on recursive techniques, combining a limited number of high-fidelity (HF) samples with multiple low-fidelity (LF) datasets structured in hierarchical levels to generate a precise HF approximation model. However, challenges arise when dealing with non-level LF datasets, where the fidelity levels of LF models are indistinguishable across the design space. In such cases, conventional methods employing recursive frameworks may lead to inefficient LF dataset utilization and substantial computational costs. To address these challenges, this work proposes the extended hierarchical Kriging (EHK) method, designed to simultaneously incorporate multiple non-level LF datasets for improved HF model construction, regardless of minor differences in fidelity levels. This method leverages a unique Bayesian-based MFSM framework, simultaneously combining non-level LF models using scaling factors to construct a global trend model. During model processing, unknown scaling factors are implicitly estimated through hyperparameter optimization, resulting in minimal computational costs during model processing, regardless of the number of LF datasets integrated, while maintaining the necessary accuracy in the resulting HF model. The advantages of the proposed EHK method are validated against state-of-the-art MFSM methods through various analytical examples and an engineering case study involving the construction of an aerodynamic database for the KP-2 eVTOL aircraft under various flying conditions. The results demonstrated the superiority of the proposed method in terms of computational cost and accuracy when generating aerodynamic models from the given multi-fidelity datasets.

Computational fluid dynamics (CFD) models are powerful computational tools to simulate urban-landscape scale atmospheric dispersion events. They are proven to be very useful for security management and emergency response. Essential inputs to CFD models include landscape characteristics, which are often captured by various GIS data layers. While it is logical to couple GIS and CFD models to take advantage of available GIS data and the visualization and cartographic rendering capabilities of GIS, the integration of the two tools have been minimal. In this paper, we took the first step to evaluate the use of GIS data in CFD modeling. Specifically, we explore how efficient is to use GIS data in CFD models and how sensitive the CFD results are to different GIS data formats. Using campus topography and building data, and the FEFLO-URBAN CFD model, we performed atmospheric release simulations using topographic data in contour and raster formats. We found that using raster format was quite efficient and contour data required significant effort. Though the simulation outputs from the two data formats were not identical, their overall outcomes were similar and did not post alarming discrepancies. We concluded that using GIS data have tremendous potential for CFD modeling.

Expedited constrained multi-objective aerodynamic shape optimization by means of physics-based surrogates

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.

<p>A modular and integrated optimisation model for the design of underwater vehicles is presented. In the proposed optimisation model two modules (i.e. low fidelity and high fidelity) are incorporated and the basic geometric definition of computer aided design (CAD) is integrated with computational fluid dynamic (CFD) analysis. The hydrodynamic drag is considered as single objective with constraints on the geometric parameters of dimension, space and volume. The CAD model is implemented in MATLAB*TM and CFD model is implemented in Shipflow**TM. A real-world design example of an existing underwater vehicles is presented. The applicability of proposed optimisation model is shown. The presented results show that within given set or sets of constraints the application of optimisation model in design results into an efficient hull form.</p><p><strong>Defence Science Journal, Vol. 66, No. 1, January 2016, pp. 71-80, DOI: http://dx.doi.org/10.14429/dsj.66.8889</strong></p>

Atomization-based techniques are widely used in pharmaceutical industry for production of fine drug particles due to their versatility and adaptability. Key performance measure of such techniques is their ability to provide control over critical quality attributes (CQAs) of produced drug particles. CQAs of drug particles produced via atomization critically depend on fluid dynamics of sprays; resulting mixing, heat and mass transfer; distribution of supersaturation and subsequent nucleation and growth of particles. It is essential to develop and use computational fluid dynamics (CFD) models for adequate understanding of multi-scale transport processes ranging from molecular scale mixing and particle scale processes, and from atomizer nozzle to overall spray chamber scale establishing relationships between CQAs and design and operating parameters of spray nozzle and chamber. In this work, we critically review past and current research efforts on CFD modelling of pharmaceutical atomization-based processes with an objective to provide clear assessment of the state of the art and to provide recommendations. An overview of the key atomization-based methods for producing drug particles with desired CQAs is presented. Key underlying physical processes and relevant concepts are then outlined. This discussion is related to the demands on CFD models; and state of the art is then discussed with respect to the process needs. Recommendations are provided towards higher fidelity and more efficient models of atomized multiphase flow dynamics and turbulence, drying modelling for the produced particles, and validation approaches. We conclude by highlighting a perceived need for numerical atomization studies with a pharmaceutical context; then, we deliver an outlook on current promising active control and machine learning strategies to augment the shift towards quality-by-design approaches in pharmaceutical manufacturing.

Computational fluid dynamics (CFD) model has been widely used for prototyping data centers. Evolving it to high-fidelity {\em digital twin} is desirable for the management and operations of large-scale data centers. Manually calibrating CFD model parameters to achieve twin-class fidelity by specially trained domain expert is tedious and labor-intensive. To reduce manual efforts, existing automatic calibration approaches developed for various computational models apply heuristics to search model configurations within an empirically defined parameter bound. However, in the context of CFD, each search step requires long-lasting CFD model's iterated solving, rendering these approaches impractical with increased model complexity. This paper presents Kalibre, a knowledge-based neural surrogate approach that performs CFD model calibration by iterating four key steps of i) training a neural surrogate model based on CFD-generated data, ii) finding the optimal parameters at the moment through neural surrogate retraining based on sensor-measured data, iii) configuring the found parameters back to the CFD model, and iv) validating the CFD model using sensor-measured data as the ground truth. Thus, the parameter search is offloaded to the neural surrogate which is ultra-faster than CFD model's iterated solving. To speed up the convergence of Kalibre, we integrate prior knowledge of the twinned data center's thermophysics into the neural surrogate design to improve its learning efficiency. With about five hours computation on a 32-core processor, Kalibre achieves mean absolute errors (MAEs) of $0.81^o$C and $0.75^o$C in calibrating two CFD models for two production data halls hosting thousands of servers each while requires fewer CFD solving processes than existing baseline approaches.

Multi-fidelity (MF) metamodeling approaches have recently attracted a significant amount of attention in simulation-based design optimization due to their ability to conduct trade-offs between high accuracy and low computational expenses by integrating the information from high-fidelity (HF) and low-fidelity (LF) models. While existing MF metamodel assisted design optimization approaches may yield an inferior or even infeasible solution since they generally treat the MF metamodel as the real HF model and ignore the interpolation uncertainties from the MF metamodel. This situation will be more serious in non-deterministic optimization. Hence, in this work, a MF metamodel assisted robust optimization approach is developed, in which the interpolation uncertainty of the MF metamodel and design variable uncertainty are quantified and taken into consideration. To demonstrate the effectiveness and merits of the proposed approach, two numerical examples and a long cylinder pressure vessel design optimization problem are tested. Results show that for the test cases the proposed approach can obtain a solution that is both optimal and within the feasible region even with perturbation of the uncertain variables.