Efficient model-correction-based reliability analysis of uncertain dynamical systems

• 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.

In this work, a method for the application of multi-fidelity modeling to the reliability analysis of 2D elastostatic structures using the boundary element method (BEM) is proposed. Reliability analyses were carried out on a rectangular plate with a center circular hole subjected to uniaxial tension using Monte Carlo simulations (MCS), the first-order reliability method (FORM), and the second-order reliability method (SORM). Two BEM models were investigated, a low-fidelity model (LFM) of 20 elements and a high-fidelity model (HFM) of 100 elements. The response of these models at several design points was used to create multi-fidelity models (MFMs) utilizing second-order polynomial response surfaces and their reliability, alongside that of the LFM and the HFM, was evaluated. Results show that the MFMs that directly called the LFM were significantly superior in terms of accuracy to the LFM, achieving very similar levels of accuracy to the HFM, while also being of similar computational cost to the LFM. These direct MFMs were found to provide good substitutes for the HFM for MCS, FORM, and SORM.

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.

History matching is a critical step within the reservoir management process to synchronize the simulation model with the production data. The history-matched model can be used for planning optimum field development and performing optimization and uncertainty quantifications. We present a novel history matching workflow based on a Bayesian framework that accommodates subsurface uncertainties. Our workflow involves three different model resolutions within the Bayesian framework: 1) a coarse low-fidelity model to update the prior range, 2) a fine low-fidelity model to represent the high-fidelity model, and 3) a high-fidelity model to re-construct the real response. The low-fidelity model is constructed by a multivariate polynomial function, while the high-fidelity model is based on the reservoir simulation model. We firstly develop a coarse low-fidelity model using a two-level Design of Experiment (DoE), which aims to provide a better prior. We secondly use Latin Hypercube Sampling (LHS) to construct the fine low-fidelity model to be deployed in the Bayesian runs, where we use the Metropolis-Hastings algorithm. Finally, the posterior is fed into the high-fidelity model to evaluate the matching quality. This work demonstrates the importance of including uncertainties in history matching. Bayesian provides a robust framework to allow uncertainty quantification within the reservoir history matching. Under uniform prior, the convergence of the Bayesian is very sensitive to the parameter ranges. When the solution is far from the mean of the parameter ranges, the Bayesian introduces bios and deviates from the observed data. Our results show that updating the prior from the coarse low-fidelity model accelerates the Bayesian convergence and improves the matching convergence. Bayesian requires a huge number of runs to produce an accurate posterior. Running the high-fidelity model multiple times is expensive. Our workflow tackles this problem by deploying a fine low-fidelity model to represent the high-fidelity model in the main runs. This fine low-fidelity model is fast to run, while it honors the physics and accuracy of the high-fidelity model. We also use ANOVA sensitivity analysis to measure the importance of each parameter. The ranking gives awareness to the significant ones that may contribute to the matching accuracy. We demonstrate our workflow for a geothermal reservoir with static and operational uncertainties. Our workflow produces accurate matching of thermal recovery factor and produced-enthalpy rate with physically-consistent posteriors. We present a novel workflow to account for uncertainty in reservoir history matching involving multi-resolution interaction. The proposed method is generic and can be readily applied within existing history-matching workflows in reservoir simulation.

What is Transferred and How Much is Retained? A Simulation Study of Complex Surgical Skills

The interaction of trailing vortices with lifting surfaces is investigated using two levels of modeling fidelity. An overset mesh-based Reynolds-averaged Navier–Stokes solver is considered as the high-fidelity computational model. A lower-fidelity model is developed by combining a vortex panel method with a propeller aerodynamic model and slipstream theory. The high-fidelity model is first validated against available experimental data obtained from the interaction of a trailing vortex generated by an upstream wing with a downstream wing. The ability of both models to represent the development of the vortex wake and integrated loads is assessed for a number of parametric configurations, including a case in which the vortex core directly impinges on the wing surface. Following this, isolated propeller and wing-mounted propeller configurations are studied. In all of these cases, the high-fidelity model is effective in predicting the details of the flow and integrated airloads. The low-fidelity model, although less accurate, is shown to accurately predict interactional air loads and performance at orders of magnitude less cost than the high-fidelity model, justifying its role as a viable tool in design and trajectory-planning applications.

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.

Research on multi-fidelity aerodynamic optimization methods

Active learning with multifidelity modeling for efficient rare event simulation

Multidisciplinary design optimization has great potential to support the turbomachinery development process by improving designs at reduced time and cost. As part of the industrial compressor design process, we seek for a rotor blade geometry that minimizes stresses without impairing the aerodynamic performance. However, the presence of structural mechanics, aerodynamics, and their interdisciplinary coupling poses challenges concerning computational effort and organizational integration. In order to reduce both computation times and the required exchange between disciplinary design teams, we propose an inter- instead of multidisciplinary design optimization approach tailored to the studied optimization problem. This involves a distinction between main and side discipline. The main discipline, structural mechanics, is computed by accurate high-fidelity finite element models. The side discipline, aerodynamics, is represented by efficient low-fidelity models, using Kriging and proper-orthogonal decomposition to approximate constraints and the gas load field as coupling variable. The proposed approach is shown to yield a valid blade design with reasonable computational effort for training the aerodynamic low-fidelity models and significantly reduced optimization times compared to a high-fidelity multidisciplinary design optimization. Especially for expensive side disciplines like aerodynamics, the multi-fidelity interdisciplinary design optimization has the potential to consider the effects of all involved disciplines at little additional cost and organizational complexity, while keeping the focus on the main discipline.

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.

In this paper tunnel probabilistic structural analysis (TuPSA) was performed using the first order reliability method (FORM). In TuPSA, a tunnel performance function is defined according to the boundary between the structural stability and instability. Then the performance function is transformed from original space into the standard normal variable space to obtain the design point, reliability index, and also the probability of tunnel failure. In this method, it is possible to consider the design factors as the dependent or independent random parameters with arbitrary probability distributions. A software code is developed to perform the tunnel probabilistic structural analysis (TuPSA) using the FORM. For validation and verification of TuPSA, a typical tunnel example with random joints orientations as well as mechanical properties has been studied. The results of TuPSA were compared with those obtained from Monte-Carlo simulation. The results show, in spite of deterministic analysis which indicates that the rock blocks are stable, that TuPSA resulted in key-blocks failure with certain probabilities. Comparison between probabilistic and deterministic analyses results indicates that probabilistic results, including the design point and probability of failure, are more rational than deterministic factor of safety.

The configurations of instruments fielded on an experiment affect the amount of information captured and the quality of subsequent inference. We investigate the problem of optimizing plasma x-ray radiation detectors in a magneto-inertial fusion experiment at Sandia National Laboratories. It is impossible to directly measure properties such as the temperature of the thermonuclear fusion plasma produced in these experiments because of the extreme environment and destructive nature of the experiment. Among other diagnostics, several detectors are placed with significant standoff from the fusion target to capture the x-rays emitted by the fusion plasma, which can be used to infer some of its properties. To optimize the configuration of these detectors, a high-fidelity model (HFM) is used for simulating outputs and a low-fidelity model (LFM) is used for inference. We develop methods based on A- and L-optimality criteria that are efficient to compute while explicitly accounting for the discrepancy between the HFM and the LFM. The method allows us to find detector configurations that perform similarly to or better than the configuration obtained using an existing sampling-based optimization method while decreasing computational time by a factor of 50. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

The aim of this paper is to study the effects of mistuning on fan flutter and to compare the prediction of two numerical models of different fidelity. The high fidelity model used here is a three-dimensional, whole assembly, time-accurate, viscous, finite-volume compressible flow solver. The Code used for this purpose is AU3D, written in Imperial College and validated for flutter computations over many years. To the best knowledge of authors, this is the first time such computations have been attempted. This is due to the fact that, such non-linear aeroelastic computations with mistuning require large amount of CPU time and cannot be performed routinely and consequently, faster (low fidelity) models are required for this task. Therefore, the second model used here is the aeroelastic fundamental mistuning model (FMM) and it based on an eigenvalue analysis of the linearized modal aeroelastic system with the aerodynamic matrix calculated from the aerodynamic influence coefficients. The influence coefficients required for this algorithm are obtained from the time domain non-linear Code by shaking one blade in the datum (tuned) frequency and mode. Once the influence coefficients have been obtained, the computations of aero damping require minimal amount of CPU time and many different mistuning patterns can be studied. The objectives of this work are to: 1. Compare the results between the two models and establish the capabilities/limitations of aeroelastic FMM, 2. Check if the introduction of mistuning would bring the experimental and computed flutter boundaries closer, 3. Establish a relationship between mistuning and damping. A rig wide-chord fan blade, typical of modern civil designs, was used as the benchmark geometry for this study. All the flutter analyses carried out in this paper are with frequency mistuning, but the possible consequences of mistuned mode shapes are briefly discussed at the end of this paper. Only the first family of modes (1F, first flap) is considered in this work. For the frequency mistuning analysis, the 1F frequency is varied around the annulus but the 1F mode shapes remain the same for all the blades. For the mode shape mistuning computations, an FE analysis of the whole assembly different mass blades is performed. The results of this work clearly show the importance of mistuning on flutter. It also demonstrates that when using rig test data for aeroelastic validation of CFD codes, the amount mistuning present must be known. Finally, it should be noted that the aim of this paper is the study of mistuning and not steady/unsteady validation of a CFD code and therefore minimal aerodynamic data are presented.