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A Novel Multi-fidelity Surrogate for Efficient Turbine Design Optimization
Qineng Wang, Liming Song, Zhendong Guo, Jun Li, Zhenping Feng
TL;DR
Turbine design optimization needs efficient methods that deliver strong final solutions despite costly high-fidelity evaluations. This paper proposes MSFO, which adaptively fuses multi- and local single-fidelity surrogates, and shows better solutions with faster convergence across tested optimization tasks.
Problem
Existing multi-fidelity optimization can converge quickly initially but may produce worse final turbine designs than high-fidelity-only single-fidelity optimization.
Method
MSFO uses DBSCAN to identify dense high-fidelity regions, builds a local kriging single-fidelity surrogate there, and adaptively ensembles it with a co-kriging multi-fidelity surrogate.
Results
MSFO achieved better solutions with much faster convergence than classic single- and multi-fidelity optimization algorithms across benchmark and turbine design tests.
Takeaways & Limitations
Fusing global multi-fidelity and promising-region single-fidelity predictions improves the effectiveness of surrogate-based turbine design optimization.
Takeaways & Limitations
Low-fidelity samples can impair local-trend prediction near the true optimum, so multi-fidelity optimization may yield worse final solutions than high-fidelity-only optimization.
Abstract
from arXiv · showhide
To solve the turbine design optimization problems efficiently, surrogate-based optimization (SBO) algorithms are frequently used. To further reduce the cost of turbine design, the multi-fidelity surrogate (MFS) based optimization is proposed by the researchers, who resort to augmenting the small number of expensive high-fidelity (HF) samples by a large portion of low-fidelity (LF) but cheap samples in surrogate modeling and optimization process. Nonetheless, according to our observations, the MFS based optimization sometimes can only have better convergence rate at the early stage of optimization process, but yielding worse final solution than the single-fidelity surrogate (SFS) based optimization that uses high-fidelity samples alone. The reason behind can be explained as follows. With the increase of HF samples in the optimization process, the LF samples can cause negative effect and therefore misleading the optimization search. To address the above issue, an ensemble weighted multi-fidelity surrogate (EMFS) is proposed. Specifically, the density-based spatial clustering of applications with noise (DBSCAN) is used to detect the region where the MFS cannot build a more accurate surrogate, and a local SFS is built there. Then, an EMFS is built by combining the MFS and SFS with adaptive weights, which is used to guide the optimization process. The related algorithm is named as multi- and single-fidelity surrogate fused optimization, i.e., MSFO. Through tests on GE-E3 blade optimization and the film cooling layout design of a turbine endwall, the effectiveness of proposed MSFO is well demonstrated.
NOMENCLATURE
This section defines the abbreviations and symbols used for turbine optimization, surrogate modeling, clustering, and design-variable notation.
- It specifies clustering terminology and objectives, including CO, DR, and DDR.
- It defines the optimization and surrogate acronyms, including SBO, SFS, MFS, EMFS, MSFO, CKO, EI, EGO, LHS, HF, and LF.
- It introduces notation for global and local quantities, the design variable vector x, and the function prediction Y.
1. INTRODUCTION
CFD-based turbine optimization is costly, motivating surrogate methods, but low-fidelity samples can improve early convergence while degrading final solutions. The proposed EMFS and MSFO address this issue by adaptively fusing multi- and single-fidelity surrogates, achieving better solutions with faster convergence in two turbine-design applications.
- Motivation: CFD accuracy increases turbine-design simulation costs, making high-fidelity optimization difficult within practical time and budget limits.Surrogate-based optimization replaces many CFD evaluations with cheaper approximations.
- Limitations of existing methods: MFS optimization often converges faster initially but is not guaranteed to outperform SFS optimization in final solution quality.As high-fidelity samples increase, low-fidelity information can mislead the search and negatively affect optimization.
- Proposed approach: EMFS combines co-kriging MFS and local kriging SFS predictions to reduce misleading low-fidelity effects and improve exploitation in promising regions.DBSCAN detects densely sampled high-fidelity regions, while a sigmoid density function adaptively weights the two surrogate predictions.
- Proposed approach: MSFO is proposed as a turbine-design optimization algorithm built on the ensemble weighted multi-fidelity surrogate.The method specifically addresses the negative effect of low-fidelity samples in MFS-based optimization.
- Results: Across turbine-blade aerodynamic and endwall film-cooling layout design, MSFO achieves better solutions with faster convergence than classic SFS- and MFS-based optimization.The method is also reported to be insensitive to the negative effect of low-fidelity samples.
2. PRELIMINARIES
This section introduces kriging, co-kriging, and DBSCAN as the foundations for constructing the ensemble weighted multi-fidelity surrogate (EMFS). Kriging models single-fidelity data, co-kriging combines high- and low-fidelity data, and DBSCAN identifies clusters and dense high-fidelity regions.
- 2.1 Kriging: Kriging predicts an unknown response as a trend function plus a Gaussian process, using sample correlations to compute mean prediction and uncertainty.The covariance matrix measures spatial correlations among training samples, while the correlation vector and process variance support prediction and uncertainty estimation; further details are cited in.
- 2.2 Co-kriging: Co-kriging combines high- and low-fidelity samples by scaling the low-fidelity model and modeling their discrepancy with a Gaussian process.Its tuning first builds a low-fidelity kriging surrogate, then estimates the scale factor and discrepancy-process hyperparameters by likelihood maximization.
- 2.3 DBSCAN: DBSCAN clusters samples using neighborhood radius ε and minimum-neighbor count minPts, with core, directly density-reachable, and density-reachable relations defining clusters.The procedure determines parameters, computes distances and reachability relations, and groups points accordingly; the paper applies it to high-fidelity samples to detect regions with dense high-fidelity data.
3. PROPOSED METHOD
The proposed EMFS combines a global co-kriging MFS with a local HF-only kriging SFS, using DBSCAN and adaptive sigmoid weights to improve surrogate accuracy in dense HF regions. MSFO uses this ensemble with expected improvement to guide turbine-design optimization, and tests report improved optimization effectiveness across fidelity-correlation settings.
- EMFS construction: EMFS combines a global co-kriging MFS using all LF and HF samples with a local HF-only kriging SFS using adaptive sigmoid weights.The global surrogate captures the design-space trend, while the local surrogate targets a region containing densely distributed HF samples.
- HF-density detection: DBSCAN identifies the HF cluster with the greatest sample count, where local kriging can provide more accurate predictions than co-kriging MFS.Its radius and minimum-points parameters are defined according to problem dimension and the number of HF samples.
- MSFO algorithm: MSFO initializes LF and HF samples, builds EMFS when the densest HF region has at least D samples, and uses expected improvement to select new queries iteratively.If that region contains fewer than D HF samples, the algorithm uses co-kriging alone; otherwise, it builds both co-kriging MFS and kriging SFS.
- MSFO algorithm: Across 20 runs, MSFO demonstrates effective optimization across high- and low-fidelity similarity settings, while highly correlated fidelities let both CKO and MSFO outperform EGO.For corr = 0.87, both CKO and MSFO achieve better solutions with faster convergence than EGO; the paper also reports MSFO effectiveness when fidelity similarity is lower.
4. ENGINEERING OPTIMIZATION
Engineering tests apply MSFO to GE-E3 blade aerodynamic optimization and turbine-endwall cooling-layout aero-thermal optimization. In both applications, conventional MFS optimization converges faster initially but yields worse final solutions than EGO, while MSFO matches EGO’s final blade solutions with faster convergence.
- 4.1 GE-E3 blade aerodynamic optimization: For GE-E3 blade optimization, CKO converges faster early but attains a worse final solution than EGO.The comparison uses EGO, CKO, and MSFO; aerodynamic performance is further compared using the medians of their optimal solutions.
- 4.1 GE-E3 blade aerodynamic optimization: MSFO achieves final solutions similar to EGO with much faster convergence for GE-E3 blade-profile optimization.This result is based on averaged optimization results over 10 runs and is attributed to combining MFS with a local SFS using adaptive ensemble weighting.
- 4.1 GE-E3 blade aerodynamic optimization: The EGO, CKO, and MSFO optimal blade solutions reduce energy loss relative to the original GE-E3 blade and satisfy the mass-flow constraints.Their blade-loading distributions are aft-loaded compared with the original GE-E3 blade.
- 4.2 Endwall cooling-layout optimization: The endwall study uses a 6.35-million-node, experimentally validated HF mesh requiring 4 hours per run and a 0.8-million-node LF mesh requiring 40 minutes.Cooling-hole positions are optimized, with compound-angle directions adjusted according to local virtual-streamline tangential directions.
- 4.2 Endwall cooling-layout optimization: In turbine-endwall cooling-layout optimization, CKO again converges faster initially but produces a worse solution than EGO.The comparison evaluates MSFO, CKO, and EGO using the endwall cooling-layout objective.
5. CONCLUSION
The paper proposes EMFS to combine global trends from multi-fidelity surrogates with locally more accurate single-fidelity surrogates for turbine design optimization. The resulting MSFO algorithm achieves better solutions with faster convergence across benchmark and turbine-design tests.
- Conclusion: The paper proposes EMFS as a more efficient surrogate for turbine design optimization.The method addresses the limitations caused by purely combining HF and LF samples.
- Conclusion: LF samples help MFS capture the HF objective’s global trend but can impair local-trend modeling near the true optimum.Consequently, MFS optimization may converge faster initially yet produce a worse final solution than HF-only SFS optimization.
- Conclusion: EMFS combines a global MFS with a local SFS to improve prediction accuracy in promising regions near the true optimum.The local SFS is built where HF samples are relatively densely distributed, while the MFS captures the global function trend.
- Conclusion: MSFO achieved better solutions with much faster convergence in benchmark, turbine-blade aerodynamic-design, and turbine-endwall cooling-layout tests.These results demonstrate the effectiveness of the proposed optimization algorithm.