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A Novel Multi-fidelity Surrogate for Turbomachinery Design Optimization

Qineng Wang, Liming Song, Zhendong Guo, Jun Li, Zhenping Feng

arXiv:2609.11111v1physics.flu-dyncs.CE

TL;DR

Turbomachinery optimization is costly, and conventional multi-fidelity optimization can become unreliable when sequential sampling creates local low-fidelity ineffectiveness. MSFO detects dense high-fidelity regions with DBSCAN and combines local single-fidelity with global multi-fidelity surrogates. Across numerical and turbomachinery cases, it is reported to improve local search and robustness against misleading low-fidelity information.

  • Problem

    Conventional MFO can have inadequate later-stage local search and fail to handle local low-fidelity ineffectiveness under uneven sequential sampling.

  • Method

    MSFO uses DBSCAN to detect dense high-fidelity regions and combines locally weighted SFS and MFS into an ensemble surrogate for sequential optimization.

  • Results

    MSFO was demonstrated on numerical benchmarks and two turbomachinery engineering problems, with reported improvements in local search and robustness to misleading low-fidelity information.

  • Takeaways & Limitations

    MSFO shows potential for turbomachinery design optimization when uneven sampling and misleading low-fidelity information challenge conventional MFO.

  • Takeaways & Limitations

    Prior multi-fidelity approaches commonly assume uniformly distributed samples and lack local adjustment for low-fidelity ineffectiveness.

Abstract

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Turbomachinery design optimization involves expensive black-box problems. Sample-efficient multi-fidelity optimization (MFO) offers an efficient solution. By utilizing multi-fidelity surrogates (MFS), the MFO algorithm can use fewer high-fidelity samples aided by low-fidelity samples to establish an accurate surrogate model. However, when MFS is used in sequential sampling optimization, it has been observed that the final optimal solution obtained by single-fidelity optimization (SFO) is better than that of MFO, even though MFO performs better at the early stages. This can be attributed to the assumption of an even and nested distribution of samples, which is incorrect when using a sequential adding strategy. To address these issues, we propose a novel algorithm called multi-single-fidelity optimization (MSFO) to overcome the limitations of the conventional MFO procedures. In the surrogate establishment of MSFO, we use the density-based spatial clustering of applications with noise (DBSCAN) method to detect local areas where low-fidelity samples are no longer effective. A combination of both global MFS and local single-fidelity surrogate model, built using high-fidelity samples alone, is used to establish an ensemble, which improves the anti-interference ability of the algorithm against misleading low-fidelity data. The effectiveness of the MSFO algorithm is verified first on numerical benchmark functions. Then, the algorithm is used to optimize the aerodynamic profile of a turbine and the film cooling layout design of a turbine endwall. Here, high-fidelity sample sources are obtained from fine-mesh CFD simulations, whereas low-fidelity sample sources are obtained from the same simulations run on a coarser mesh. The results demonstrate that our MSFO algorithm performs significantly better than the conventional SFO and MFO processes, with a higher level of robustness.

NOMENCLATURE

The paper uses abbreviations for single- and multi-fidelity surrogates, fidelity levels, the proposed optimizer, and its ensemble surrogate.

  • SFS denotes single-fidelity surrogate, MFS multi-fidelity surrogate, HF high-fidelity, and LF low-fidelity.
  • MSFO denotes multi-single-fidelity optimization, while EMFS denotes ensemble multi-fidelity surrogate.

1. INTRODUCTION

The introduction motivates MSFO by the cost of high-fidelity CFD and the limitations of conventional multi-fidelity optimization under uneven sequential sampling. MSFO locally combines single- and multi-fidelity surrogates to improve search and robustness.

  • High-fidelity CFD is computationally expensive, motivating surrogate-based optimization and adaptive sequential sampling to reduce simulation cost.Adaptive sampling adds points in regions of interest and can build an accurate surrogate with fewer samples than one-shot sampling.
  • MFO can converge faster than single-fidelity optimization yet produce worse final outcomes because local multi-fidelity surrogate accuracy may be poor.Poor local accuracy directly weakens later-stage local search.
  • Low-fidelity ineffectiveness occurs when low-fidelity data fails to improve, or decreases, surrogate accuracy relative to using high-fidelity data alone.Sequential sampling can make this problem local because samples concentrate near the optimum.
  • Prior approaches often assume uniformly distributed samples and therefore lack flexible local adjustment for low-fidelity ineffectiveness during sequential optimization.
  • MSFO uses DBSCAN to identify dense high-fidelity regions, combines MFS and SFS with reciprocal-variance weights, and guides subsequent sequential sampling with EMFS.
  • MSFO is reported to improve local search and robustness, allowing lower-relative-accuracy low-fidelity data to remain usable and expanding MFO application scope.The paper describes the method as accounting for uneven sample distributions in sequential adaptive sampling.

2. RESEARCH BACKGROUND

The research background introduces DBSCAN, kriging, and co-kriging as components of the proposed surrogate framework. DBSCAN identifies dense regions, while kriging and co-kriging model single- and multi-fidelity information.

  • 2.1 DBSCAN algorithm: DBSCAN classifies sufficiently dense regions using a minimum-neighbor count minPts within radius ε, with density-connected neighborhoods forming clusters.Points not density-reachable from any core point are excluded from clusters.
  • 2.1 DBSCAN algorithm: Figure 1 illustrates DBSCAN’s clustering concepts, with minPts set to 4 and ε represented by circles.
  • 2.2 Kriging surrogate: Kriging predicts an unknown response as a trend function plus a normal random process describing local features around sampled points.The paper uses kriging as its single-fidelity surrogate technique.
  • 2.3 Co-kriging surrogate: Co-kriging extends kriging by leveraging samples from multiple fidelity sources to form one surrogate.
  • 2.3 Co-kriging surrogate: The co-kriging model links high- and low-fidelity responses through a scale factor ρ and discrepancy function Z_d(x), with ρ also adjusting low-fidelity weight.
  • 2.3 Co-kriging surrogate: The co-kriging predictor uses a correlation matrix C and correlation vector c.

3. PROPOSED METHOD

MSFO combines a global multi-fidelity surrogate with a locally weighted single-fidelity surrogate to address low-fidelity ineffectiveness caused by uneven sequential sampling. DBSCAN identifies dense high-fidelity regions, where reciprocal-variance weighting shifts the ensemble toward the high-fidelity-only model.

  • 3.1 Ensemble multi-fidelity surrogate: MSFO detects locally dense high-fidelity regions with DBSCAN and combines global MFS with local SFS through an ensemble surrogate.The method is designed for uneven sample distributions produced by sequential optimization.
  • 3.1 Ensemble multi-fidelity surrogate: The EMFS assigns region-dependent weights to global MFS and local SFS, increasing the SFS contribution where low-fidelity samples become ineffective.The global MFS and local SFS are denoted by the G and L subscripts, respectively.
  • 3.1 Ensemble multi-fidelity surrogate: Reciprocal-variance weighting gives greater weight to the more accurate model locally, with a sigmoid transformation smoothing spatial changes in the coefficients.The sigmoid parameters m and b are set to 10 and 5 in the described implementation.
  • 3.1 Ensemble multi-fidelity surrogate: The SFS uses high-density high-fidelity samples in a kriging model, whereas the MFS is constructed with co-kriging before weighted fusion.The two component surrogates are established separately and then combined into the EMFS.
  • 3.2 A 2-D illustrative example of the EMFS: In the Ackley illustration, 40 low-fidelity and 25 high-fidelity samples produce a model that remains globally MFS-like while using SFS locally for detail.DBSCAN identifies dense high-fidelity samples, and the local SFS weight approaches 1 in the low-fidelity-ineffectiveness region.
  • Numerical benchmark: Across three 5-dimensional benchmark cases, MSFO converges faster than EGO and conventional CKO, including when high- and low-fidelity similarity is 0.33.The cases use low-fidelity similarity settings corresponding to Pearson coefficients of 0.33, 0.57, and 0.87, with 20 repetitions per case.

4. ENGINEERING TEST CASES

The MSFO algorithm is evaluated on turbine blade aerodynamic and endwall cooling layout optimization problems using CFD-derived high- and low-fidelity data. Across both engineering cases, MSFO shows stronger local search, faster convergence, and better optimization results than the comparison methods.

  • Turbine blade design problem: For the GE-E3 blade case, MSFO achieves superior optimization results and stronger local search than CKO across all tested low-fidelity sources.The study compares MSFO with EGO and CKO using CFD data from fine and coarse meshes; each case repeats the calculation ten times.
  • Turbine endwall cooling layout design problem: MSFO maintains faster convergence and outperforms EGO throughout the endwall cooling optimization, whereas CKO stagnates after the sixth iteration.The initial MSFO and CKO convergence curves are nearly identical, but CKO later achieves a worse optimum than EGO.
  • Turbine endwall cooling layout design problem: The endwall overheating area A_h decreases from 324.0 cm2 to 111.8 cm2 after optimization.The optimized layout moves the circumferential cooling row toward the front edge and the axial row toward the pressure side, forming a larger high-effectiveness region.
  • Overall engineering results: Across the two engineering optimizations, MSFO shows significantly enhanced local search, faster initial convergence than EGO, and superior later-stage results.The engineering tests use 3d high-fidelity samples and 10d low-fidelity samples after Latin hypercube initial sampling.

5. CONCLUSION

The study identifies why conventional multi-fidelity surrogates lose local accuracy with uneven samples and proposes EMFS within MSFO to adjust fidelity contributions locally. MSFO combines DBSCAN-based detection and ensemble modeling to improve local search and robustness to misleading low-fidelity data.

  • Conventional multi-fidelity surrogates improve global accuracy but reduce local accuracy when samples are unevenly distributed because scale-factor adjustment alone is insufficient.This limitation explains inadequate local search in the late stage of multi-fidelity optimization.
  • EMFS uses DBSCAN to identify dense local high-fidelity regions, adjusts single- and multi-fidelity weights locally, and builds an ensemble surrogate.MSFO is formulated on this ensemble to address expensive black-box optimization with multiple sample sources.
  • MSFO demonstrates robust local search and resistance to misleading low-fidelity information across numerical and turbomachinery design problems.The method avoids late-stage stagnation while handling multiple sample sources in the evaluated optimization tasks.
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