High-fidelity Function Research Articles

Constrained multifidelity optimization using model calibration

Multifidelity optimization approaches seek to bring higher-fidelity analyses earlier into the design process by using performance estimates from lower-fidelity models to accelerate convergence towards the optimum of a high-fidelity design problem. Current multifidelity optimization methods generally fall into two broad categories: provably convergent methods that use either the high-fidelity gradient or a high-fidelity pattern-search, and heuristic model calibration approaches, such as interpolating high-fidelity data or adding a Kriging error model to a lower-fidelity function. This paper presents a multifidelity optimization method that bridges these two ideas; our method iteratively calibrates lower-fidelity information to the high-fidelity function in order to find an optimum of the high-fidelity design problem. The algorithm developed minimizes a high-fidelity objective function subject to a high-fidelity constraint and other simple constraints. The algorithm never computes the gradient of a high-fidelity function; however, it achieves first-order optimality using sensitivity information from the calibrated low-fidelity models, which are constructed to have negligible error in a neighborhood around the solution. The method is demonstrated for aerodynamic shape optimization and shows at least an 80% reduction in the number of high-fidelity analyses compared other single-fidelity derivative-free and sequential quadratic programming methods. The method uses approximately the same number of high-fidelity analyses as a multifidelity trust-region algorithm that estimates the high-fidelity gradient using finite differences.

複数精度関数値を用いたKriging応答曲面法による不確実性解析

In this paper, a variable fidelity Kriging model approach is applied to aerodynamic data modeling and uncertainty quantification of 2D airfoil aerodynamic performances. This approach enables to construct an accurate surrogate model efficiently by utilizing high and low fidelity function information simultaneously. The low fidelity functions are defined by utilizing coarser computational meshes in this research. The uncertainty quantification is executed by Monte-Carlo simulation on the surrogate model, which is often referred to as Inexpensive Monte-Carlo simulation approach. The developed uncertainty quantification approach showed comparable accuracy with full non-linear Monte-Carlo simulation results and was executed with much lower computational cost.

Gradient-based multifidelity optimisation for aircraft design using Bayesian model calibration

AbstractOptimisation of complex systems frequently requires evaluating a computationally expensive high-fidelity function to estimate a system metric of interest. Although design sensitivities may be available through either direct or adjoint methods, the use of formal optimisation methods may remain too costly. Incorporating low-fidelity performance estimates can substantially reduce the cost of the high-fidelity optimisation. In this paper we present a provably convergent multifidelity optimisation method that uses Cokriging Bayesian model calibration and first-order consistent trust regions. The technique is compared with a single-fidelity sequential quadratic programming method and a conventional first-order trust-region method on both a two-dimensional structural optimisation and an aerofoil design problem. In both problems adjoint formulations are used to provide inexpensive sensitivity information.

Combustor Design Optimization Using Co-Kriging of Steady and Unsteady Turbulent Combustion

In the gas turbine industry, computational fluid dynamics (CFD) simulations are often used to predict and visualize the complex reacting flow dynamics, combustion environment and emissions performance of a combustor at the design stage. Given the complexity involved in obtaining accurate flow predictions and due to the expensive nature of simulations, conventional techniques for CFD based combustor design optimization are often ruled out, primarily due to the limits on available computing resources and time. The design optimization process normally requires a large number of analyses of the objective and constraint functions which necessitates a careful selection of fast, reliable and efficient computational methods for the CFD analysis and the optimization process. In this study, given a fixed computational budget, an assessment of a co-Kriging based optimization strategy against a standard Kriging based optimization strategy is presented for the design of a 2D combustor using steady and unsteady Reynolds-averaged Navier Stokes (RANS) formulation. Within the fixed computational budget, using a steady RANS formulation, the Kriging strategy successfully captures the underlying response; however with unsteady RANS the Kriging strategy fails to capture the underlying response due to the existence of a high level of noise. The co-Kriging strategy is then applied to two design problems, one using two levels of grid resolutions in a steady RANS formulation and the other using steady and unsteady RANS formulations on the same grid resolution. With the co-Kriging strategy, the multifidelity analysis is expected to find an optimum design in comparatively less time than that required using the high-fidelity model alone since less high-fidelity function calls should be required. However, using the applied computational setup for co-Kriging, the Kriging strategy beats the co-Kriging strategy under the steady RANS formulation whereas under the unsteady RANS formulation, the high level of noise stalls the co-Kriging optimization process.

Multi-fidelity design optimization of transonic airfoils using physics-based surrogate modeling and shape-preserving response prediction

A computationally efficient design methodology for transonic airfoil optimization has been developed. In the optimization process, a numerically cheap physics-based low-fidelity surrogate (the transonic small-disturbance equation) is used in lieu of an accurate, but computationally expensive, high-fidelity (the compressible Euler equations) simulation model. Correction of the low-fidelity model is achieved by aligning its corresponding airfoil surface pressure distribution with that of the high-fidelity model using a shape-preserving response prediction technique. The resulting method requires only a single high-fidelity simulation per iteration of the design process. The method is applied to airfoil lift maximization in two-dimensional inviscid transonic flow, subject to constraints on shock-induced pressure drag and airfoil cross-sectional area. The results showed that more than a 90% reduction in high-fidelity function calls was achieved when compared to direct high-fidelity model optimization using a pattern-search algorithm.

Multi-fidelity design optimization of transonic airfoils using shape-preserving response prediction

A computationally efficient methodology for transonic airfoil design optimization is presented. Our approach exploits a corrected physics-based low-fidelity surrogate that replaces, in the optimization process, an accurate but computationally expensive highfidelity airfoil model. Correction of the low-fidelity model is achieved by aligning its corresponding airfoil surface pressure distributions with that of the high-fidelity model using a shape-preserving response prediction technique. The presented method is applied to airfoil lift maximization in two-dimensional inviscid transonic flow, subject to constraints on shock induced pressure drag and airfoil cross-sectional area. More than a 90% reduction in high-fidelity function calls is achieved when compared to direct highfidelity model optimization.

Surrogate-Based Optimization Using Multifidelity Models with Variable Parameterization and Corrected Space Mapping

Surrogate-based-optimization methods provide a means to achieve high-fidelity design optimization at reduced computational cost by using a high-fidelity model in combination with lower-fidelity models that are less expensive to evaluate. This paper presents a provably convergent trust-region model-management methodology for variable-parameterization design models: that is, models for which the design parameters are defined over different spaces. Corrected space mapping is introduced as a method to map between the variable-parameterization design spaces. It is then used with a sequential-quadratic-programming-like trust-region method for two aerospace-related design optimization problems. Results for a wing design problem and a flapping-flight problem show that the method outperforms direct optimization in the high-fidelity space. On the wing design problem, the new method achieves 76% savings in high-fidelity function calls. On a bat-flight design problem, it achieves approximately 45% time savings, although it converges to a different local minimum than did the benchmark.

Hybrid Variable Fidelity Optimization by Using a Kriging-Based Scaling Function

Solving design problems that rely on very complex and computationally expensive calculations using standard optimization methods might not be feasible given design cycle time constraints. Variable fidelity methods address this issue by using lower-fidelity models and a scaling function to approximate the higher-fidelity models in a provably convergent framework. In the past, scaling functions have mainly been either first-order multiplicative or additive corrections. These are being extended to second order. In this investigation variable metric approaches for calculating second-order scaling information are developed. A kriging-based scaling function is introduced to better approximate the high-fidelity response on a more global level. An adaptive hybrid method is also developed in this investigation. The adaptive hybrid method combines the additive and multiplicative approaches so that the designer does not have to determine which is more suitable prior to optimization. The methodologies developed in this research are compared to existing methods using two demonstration problems. The first problem is analytic, whereas the second involves the design of a supercritical high-lift airfoil. The results demonstrate that the krigingbased scaling methods improve computational expense by lowering the number of high-fidelity function calls required for convergence. The results also indicate the hybrid method is both robust and effective.

An Overview of First-Order Model Management for Engineering Optimization

First-order approximation/model management optimization (AMMO) is a rigorous methodology for solving high-fidelity optimization problems with minimal expense in high-fidelity function and derivative evaluation. AMMO is a general approach that is applicable to any derivative based optimization algorithm and any combination of high-fidelity and low-fidelity models. This paper gives an overview of the principles that underlie AMMO and puts the method in perspective with other similarly motivated methods. AMMO is first illustrated by an example of a scheme for solving bound-constrained optimization problems. The principles can be easily extrapolated to other optimization algorithms. The applicability to general models is demonstrated on two recent computational studies of aerodynamic optimization with AMMO. One study considers variable-resolution models, where the high-fidelity model is provided by solutions on a fine mesh, while the corresponding low-fidelity model is computed by solving the same differential equations on a coarser mesh. The second study uses variable-fidelity physics models, with the high-fidelity model provided by the Navier-Stokes equations and the low-fidelity model—by the Euler equations. Both studies show promising savings in terms of high-fidelity function and derivative evaluations. The overview serves to introduce the reader to the general concept of AMMO and to illustrate the basic principles with current computational results.