Expensive High-fidelity Model Research Articles

Investigation of impact of shoreline alteration on coastal hydrodynamics using Dimension REduced Surrogate based Sensitivity Analysis

To inform the decision-making of coastal protection against sea level rise (SLR), we have to estimate the impact of shoreline alterations on the hydrodynamics. This task involves estimating of a large number of shoreline decision combinations. Here we present a sensitivity analysis based approach to understand how the variation in the shoreline, for example, due to construction of seawalls at different location along the shoreline, would impact the water levels over the coastal region under SLR. To facilitate efficient sensitivity analysis for expensive high-fidelity numerical models with high-dimensional outputs, we propose a Dimension REduced Surrogate based Sensitivity Analysis (DRESSA) method. DRESSA uses Principal Component Analysis (PCA) to exploit the correlation in the high-dimensional outputs to find a low-dimensional latent output representation, then builds a surrogate model for the latent outputs based on a small number of runs of the high-fidelity numerical model. In the end, DRESSA first establishes relevant covariance matrices for the low-dimensional latent outputs using the surrogate model, and then directly establishes sensitivity indexes for the high-dimensional outputs using these covariance matrices and the derived transformation between sensitivity in latent space and original space. We applied this method to generate sensitivity maps and investigate the impact of different containment strategies on peak water level (PWL) over the entire San Francisco Bay under SLR.

Crystal Plasticity Model Calibration for Ti-7Al Alloy with a Multi-fidelity Computational Scheme

The present work addresses a Gaussian process-based multi-fidelity computational scheme to enable the crystal plasticity modeling of Ti-7Al alloy. The crystal plasticity simulations are performed by using computational techniques that lead to two different solution fidelities. The first technique involves the use of a one-point probability descriptor, orientation distribution function (ODF), which measures the volume fractions of crystals in different orientations. The ODF is posed as the low-fidelity method in the multi-fidelity scheme since it neglects the effects of the microstructural correlations and grain shapes to the macro-scale stress-strain response of the material. For the high-fidelity computational technique, crystal plasticity finite element method (CPFEM) is utilized. This is because the CPFEM resolves better in grain-level microstructural features. However, the CPFEM is a computationally expensive technique and it is not feasible to be utilized for computationally costly problems. Therefore, a multi-fidelity modeling scheme that improves the low-fidelity ODF simulation data with the high-fidelity CPFEM simulations is utilized to obtain the crystal plasticity parameters. The presented approach uses more samples from the computationally less expensive low-fidelity model and less samples from the computationally expensive high-fidelity model to build a numerical framework that satisfies both accuracy and computational time expectations. The results of the presented multi-fidelity scheme show a significant improvement on the accuracy of the crystal plasticity modeling of Ti-7Al compared to the results of the previous low-fidelity solution.

Adaptive metamodel-assisted shape optimization for springback in metal forming processes

This paper aims to propose a shape optimization strategy for springback shape in metal forming processes based on the adaptive metamodel of the spatial field that can reduce the computational time and cost. The first stage includes solving metal forming problem using finite element simulation and building of Reduction Order Model of spatial field based on Proper Orthogonal Decomposition. Then, the metamodel of spatial field is built using the combination of Reduction Order Model and Kriging method to replace the expensive high fidelity model. In the second stage, the metamodel of spatial field is used within the iterative optimization procedure to find the optimal design for the final shape after springback in the metal forming processes. This strategy allows reducing computational cost to achieve the optimal design with respect to the approach based on the traditional metamodel. The proposed methodology is illustrated with the “U-shape bending” from the Numisheet2011 benchmark problem. Two parameters: the blank holder force and the die radius are chosen to optimize the springback effect.

Multiscale surrogate modelling of the elastic response of thick composite structures with embedded defects and features

This paper presents a multiscale modelling approach for thick composite structures containing internal defects and features. The proposed approach was developed using a surrogate model to represent the composite response on the meso-scale. A set of Representative Volume Element (RVE) models under periodic boundary conditions were used to sample the response at specified locations across the composite design space. As an example of its application, wrinkle defects of various severities were introduced to the RVE models to assess the defect contribution to the composite response. The homogenized responses from the meso-scale RVE models were then used as input to the surrogate model. To link the macro and meso scales, a set of 3D lamination parameters representing the composite layup were developed. A surrogate model using the 3D lamination parameters and the defect severity as input was built to link the macro-model to the meso-scale responses. The proposed multi-scale approach was verified against a set of high fidelity models with different levels of wrinkle defect severity. Good agreement was found between the new multi-scale approach and the more computationally expensive high-fidelity models.

A Multi-Fidelity Model Approach for Simultaneous Scheduling of Machines and Vehicles in Flexible Manufacturing Systems

Driven by sensor technologies and Internet of Things, massive real-time data from highly interconnected devices are available, which enables the improvement of decision-making quality. Scheduling of such production systems can be challenging as it must incorporate the latest data and be able to re-plan quickly. In this research, a multi-fidelity model for simultaneous scheduling problem of machines and vehicles at flexible manufacturing system has been proposed. In order to improve the computational efficiency, we extend the framework, called multi-fidelity optimization with ordinal transformation and optimal sampling, with combining with the K-means method. The proposed framework enables the benefits of both fast and inexpensive low-fidelity models with accurate but more expensive high-fidelity models. Results show that this approach can significantly decrease computational cost compared with other algorithms in the literature.

Convergence analysis of multifidelity Monte Carlo estimation

The multifidelity Monte Carlo method provides a general framework for combining cheap low-fidelity approximations of an expensive high-fidelity model to accelerate the Monte Carlo estimation of statistics of the high-fidelity model output. In this work, we investigate the properties of multifidelity Monte Carlo estimation in the setting where a hierarchy of approximations can be constructed with known error and cost bounds. Our main result is a convergence analysis of multifidelity Monte Carlo estimation, for which we prove a bound on the costs of the multifidelity Monte Carlo estimator under assumptions on the error and cost bounds of the low-fidelity approximations. The assumptions that we make are typical in the setting of similar Monte Carlo techniques. Numerical experiments illustrate the derived bounds.

Multifidelity Preconditioning of the Cross-Entropy Method for Rare Event Simulation and Failure Probability Estimation

Accurately estimating rare event probabilities with Monte Carlo can become costly if for each sample a computationally expensive high-fidelity model evaluation is necessary to approximate the syste...

Adaptive response prediction for aerodynamic shape optimization

PurposeThe purpose of this paper is to reduce the overall computational time of aerodynamic shape optimization that involves accurate high-fidelity simulation models.Design/methodology/approachThe proposed approach is based on the surrogate-based optimization paradigm. In particular, multi-fidelity surrogate models are used in the optimization process in place of the computationally expensive high-fidelity model. The multi-fidelity surrogate is constructed using physics-based low-fidelity models and a proper correction. This work introduces a novel correction methodology – referred to as the adaptive response prediction (ARP). The ARP technique corrects the low-fidelity model response, represented by the airfoil pressure distribution, through suitable horizontal and vertical adjustments.FindingsNumerical investigations show the feasibility of solving real-world problems involving optimization of transonic airfoil shapes and accurate computational fluid dynamics simulation models of such surfaces. The results show that the proposed approach outperforms traditional surrogate-based approaches.Originality/valueThe proposed aerodynamic design optimization algorithm is novel and holistic. In particular, the ARP correction technique is original. The algorithm is useful for fast design of aerodynamic surfaces using high-fidelity simulation data in moderately sized search spaces, which is challenging using conventional methods because of excessive computational costs.

Combining multiple surrogate models to accelerate failure probability estimation with expensive high-fidelity models

In failure probability estimation, importance sampling constructs a biasing distribution that targets the failure event such that a small number of model evaluations is sufficient to achieve a Monte Carlo estimate of the failure probability with an acceptable accuracy; however, the construction of the biasing distribution often requires a large number of model evaluations, which can become computationally expensive. We present a mixed multifidelity importance sampling (MMFIS) approach that leverages computationally cheap but erroneous surrogate models for the construction of the biasing distribution and that uses the original high-fidelity model to guarantee unbiased estimates of the failure probability. The key property of our MMFIS estimator is that it can leverage multiple surrogate models for the construction of the biasing distribution, instead of a single surrogate model alone. We show that our MMFIS estimator has a mean-squared error that is up to a constant lower than the mean-squared errors of the corresponding estimators that uses any of the given surrogate models alone—even in settings where no information about the approximation qualities of the surrogate models is available. In particular, our MMFIS approach avoids the problem of selecting the surrogate model that leads to the estimator with the lowest mean-squared error, which is challenging if the approximation quality of the surrogate models is unknown. We demonstrate our MMFIS approach on numerical examples, where we achieve orders of magnitude speedups compared to using the high-fidelity model only.

A VF-SLP framework using least squares hybrid scaling for RBDO

Using computationally cheap low-fidelity (LF) model and more accurate but expensive high-fidelity (HF) model, variable fidelity (VF) model has been widely used in engineering design to replace the actual computationally expensive experiments or computer simulations. To further extend the application of VF to reliability-based design optimization (RBDO), a new framework based on sequential linear programming (SLP) is proposed in this paper. Combining the advantages of additive scaling method and multiplicative scaling method, a hybrid scaling method based on least squares (LSHS) is developed. In LSHS method, the VF model is introduced to replace the implicit performance function in RBDO by using the HF function values and gradient values at all evaluated points around the current design. With the failure probability and its gradient calculated by Monte Carlo Simulation (MCS) at current design, SLP is adopted to calculate the next design. A novel method which considers the target reliability index and the influence domain at the current design is also developed to determine the step size in every sub-optimization problem. Two numerical examples and the shape optimization problem of a curved beam are analyzed in order to demonstrate the performance of the proposed methodology. The comparison results show that the proposed method is very accurate and efficient.

Evaluation of multiple reduced-order models to enhance confidence in global sensitivity analyses

Variance-based global sensitivity analysis (e.g., the Sobol’ sensitivity index) can be used to identify the important parameters over the entire parameter space. However, one often cannot afford the computational costs of sampling-based approaches in combination with expensive high-fidelity forward models. Reduced-order models (ROM) can substantially accelerate calculation of these sensitivities. However, it is usually difficult to determine what type of ROM should be used and how accurately the ROM represents the high-fidelity model (HFM) results. In this paper, we propose to concurrently use multiple ROMs as a way to assess the robustness of the model-reduction method. Two sets of HFM simulations are needed, one set for building ROMs and the other for validating ROMs. Our goal is to keep the total number of HFM simulations to a minimum. Ideally some of the HFM simulations in the first set can be shared by different ROMs. Based on validation results, the ROMs can be combined with different schemes. We demonstrate that we can achieve the goal by using four different ROMs and still considerably save computational time compared to using traditional HFM simulation for calculating sensitivity indices. We apply the approach to an example problem of a large-scale geological carbon dioxide storage system, in which the objective is to calculate a sensitivity index to identify important parameters. For this problem, the locally best ROM provides better estimates than the weighted average from all ROMs.

Optimal Model Management for Multifidelity Monte Carlo Estimation

This work presents an optimal model management strategy that exploits multifidelity surrogate models to accelerate the estimation of statistics of outputs of computationally expensive high-fidelity models. Existing acceleration methods typically exploit a multilevel hierarchy of surrogate models that follow a known rate of error decay and computational costs; however, a general collection of surrogate models, which may include projection-based reduced models, data-fit models, support vector machines, and simplified-physics models, does not necessarily give rise to such a hierarchy. Our multifidelity approach provides a framework to combine an arbitrary number of surrogate models of any type. Instead of relying on error and cost rates, an optimization problem balances the number of model evaluations across the high-fidelity and surrogate models with respect to error and costs. We show that a unique analytic solution of the model management optimization problem exists under mild conditions on the models. Ou...

Multifidelity importance sampling

Estimating statistics of model outputs with the Monte Carlo method often requires a large number of model evaluations. This leads to long runtimes if the model is expensive to evaluate. Importance sampling is one approach that can lead to a reduction in the number of model evaluations. Importance sampling uses a biasing distribution to sample the model more efficiently, but generating such a biasing distribution can be difficult and usually also requires model evaluations. A different strategy to speed up Monte Carlo sampling is to replace the computationally expensive high-fidelity model with a computationally cheap surrogate model; however, because the surrogate model outputs are only approximations of the high-fidelity model outputs, the estimate obtained using a surrogate model is in general biased with respect to the estimate obtained using the high-fidelity model. We introduce a multifidelity importance sampling (MFIS) method, which combines evaluations of both the high-fidelity and a surrogate model. It uses a surrogate model to facilitate the construction of the biasing distribution, but relies on a small number of evaluations of the high-fidelity model to derive an unbiased estimate of the statistics of interest. We prove that the MFIS estimate is unbiased even in the absence of accuracy guarantees on the surrogate model itself. The MFIS method can be used with any type of surrogate model, such as projection-based reduced-order models and data-fit models. Furthermore, the MFIS method is applicable to black-box models, i.e., where only inputs and the corresponding outputs of the high-fidelity and the surrogate model are available but not the details of the models themselves. We demonstrate on nonlinear and time-dependent problems that our MFIS method achieves speedups of up to several orders of magnitude compared to Monte Carlo with importance sampling that uses the high-fidelity model only.

Fast Low-fidelity Wing Aerodynamics Model for Surrogate-based Shape Optimization

Variable-fidelity optimization (VFO) can be efficient in terms of the computational cost when compared with traditional approaches, such as gradient-based methods with adjoint sensitivity information. In variable-fidelity methods, the direct optimization of the expensive high-fidelity model is replaced by iterative re-optimization of a physics-based surrogate model, which is constructed from a corrected low-fidelity model. The success of VFO is dependent on the reliability and accuracy of the low-fidelity model. In this paper, we present a way to develop a fast and reliable low-fidelity model suitable for aerodynamic shape of transonic wings. The low-fidelity model is component based and accounts for the zero-lift drag, induced drag, and wave drag. The induced drag can be calculated by a proper method, such lifting line theory or a panel method. The zero-lift drag and the wave drag can be calculated by two-dimensional flow model and strip theory. Sweep effects are accounted for by simple sweep theory. The approach is illustrated by a numerical example where the induced drag is calculated by a vortex lattice method, and the zero-lift drag and wave drag are calculated by MSES (a viscous-inviscid method). The low-fidelity model is roughly 320 times faster than a high-fidelity computational fluid dynamics models which solves the Reynolds-averaged Navier-Stokes equations and the Spalart-Allmaras turbulence model. The responses of the high-and low-fidelity models compare favorably and, most importantly, show the same trends with respect to changes in the operational conditions (Mach number, angle of attack) and the geometry (the airfoil shapes).

Reduced order modeling in iTOUGH2

The inverse modeling and uncertainty quantification capabilities of iTOUGH2 are augmented with reduced order models (ROMs) that act as efficient surrogates for computationally expensive high fidelity models (HFMs). The implementation of the ROM capabilities involves integration of three main computational components. The first component is the ROM itself. Two response surface approximations are currently implemented: Gaussian process regression (GPR) and radial basis function (RBF) interpolation. The second component is a multi-output adaptive sampling procedure that determines the sample points used to construct the ROMs. The third component involves defining appropriate error measures for the adaptive sampling procedure, allowing ROMs to be constructed efficiently with limited user intervention. Details in all three components must complement one another to obtain an accurate approximation. The new capability and its integration with other analysis tools within iTOUGH2 are demonstrated in two examples. The results from using the ROMs in an uncertainty quantification analysis and a global sensitivity analysis compare favorably with the results obtained using the HFMs. GPR is more accurate than RBF, but the difference can be small and similar conclusion can be deduced from the analyses. In the second example involving a realistic numerical model for a hypothetical industrial-scale carbon storage project in the Southern San Joaquin Basin, California, USA, significant reduction in computational effort can be achieved when ROMs are used to perform a rigorous global sensitivity analysis.

Simulation-driven design using surrogate-based optimization and variable-resolution computational fluid dynamic models

Reliable and robust computational fluid dynamics (CFD) solvers are nowadays commonly utilized in the design and analyses of various engineering systems, such as aircraft, turbomachinery, ships, and automotives. Although this has resulted in a drastic decrease of the number of prototype and experimental testing, the use of CFD in the design automation process is still limited. In practice, the geometry parameters that ensure satisfaction of assumed performance requirements are often obtained by repetitive CFD simulations guided by engineering experience. This is a tedious process which does not guarantee optimal results. On the other hand, straight forward automation attempts by employing the CFD solvers directly in the optimization loop are typically impractical, even when using adjoint sensitivity information, because high-fidelity CFD simulations tend to be computationally very expensive. In this paper, we describe a surrogate-based design optimization methodology that shifts the computational burden from the accurate and expensive high-fidelity CFD model to its fast and yet reasonably accurate surrogate. As the surrogate models are computationally much cheaper than the high-fidelity ones, the cost of the design process is greatly reduced. Here, the surrogates are constructed using low-fidelity CFD models and response correction techniques. Application examples, involving the design of axisymmetric hulls in subsonic flow, and airfoils in both subsonic and transonic flows, are presented.

Bayesian assimilation of multi-fidelity finite element models

A complex system can be modeled using various fidelities with the finite element method. A high-fidelity model is expected to be more computationally expensive compared to a low-fidelity model and in general may contain more degrees of freedom and more elements. This paper proposes a novel multi-fidelity approach to solve boundary value problems using the finite element method. A Bayesian approach based on Gaussian process emulators in conjunction with the domain decomposition method is developed. Using this approach one can seamlessly assimilate a low-fidelity model with a more expensive high-fidelity model. The idea is illustrated using elliptic boundary value problems.

Variable-fidelity aerodynamic shape optimisation of single-element airfoils at high-lift conditions

A computationally efficient simulation-driven variable-fidelity design methodology for single-element airfoils at high-lift conditions is described. Direct optimisation of an accurate but computationally expensive high-fidelity simulation model based on the Reynolds-averaged Navier-Stokes equations and a one equation turbulence model is replaced by iterative updating and re-optimisation of a cheap surrogate. The surrogate model exploits a low-fidelity model (a polynomial approximation of the high-fidelity model data) and appropriate correction that aim at aligning its corresponding airfoil surface pressure distribution and skin friction distribution with that of the high-fidelity model using a shape-preserving response prediction technique. The test cases include lift coefficient maximisation of single-element airfoils at high-lift conditions, subject to a constraint on the drag coefficient. Over 83% reduction in the number of high-fidelity model simulations is demonstrated when compared to high-fidelity model optimisation using a pattern-search algorithm.

Parallel evolutionary algorithms for optimization problems in aerospace engineering

This paper presents the recent developments in hierarchical genetic algorithms (HGAs) to speed up the optimization of aerodynamic shapes. It first introduces HGAs, a particular instance of parallel GAs based on the notion of interconnected sub-populations evolving independently. Previous studies have shown the advantages of introducing a multi-layered hierarchical topology in parallel GAs. Such a topology allows the use of multiple models for optimization problems, and shows that it is possible to mix fast low-fidelity models for exploration and expensive high-fidelity models for exploitation. Finally, a new class of multi-objective optimizers mixing HGAs and Nash Game Theory is defined. These methods are tested for solving design optimization problems in aerodynamics. A parallel version of this approach running a cluster of PCs demonstrate the convergence speed up on an inverse nozzle problem and a high-lift problem for a multiple element airfoil.

Design optimisation of a semi‐solid tyre using response surface approximations

A general approach for the construction of global approximations to structural behaviour using response surface methodology is presented. The computation and use of these approximations are demonstrated using a semi‐solid tyre example. The use of these global approximations to the responses made it viable to utilise the capabilities of non‐linear analysis software in design optimisation. The insight gained from a preliminary low fidelity model was utilized in a two‐stage approach to achieve the maximum benefit from a more expensive high fidelity model. The resulting high‐accuracy approximations greatly reduced the cost of subsequent design calculations such as multidisciplinary and discrete optimisation.