Abstract
Adapting an unstructured CFD mesh to the modified geometry, in accordance with the updated value-set of design parameters at the end of each cycle, is a must in CFD–based shape optimization loops. Mesh adaptation is a nice alternative to remeshing procedures which might become expensive and, also, hinder the initialization of new simulations from previous results. Mesh morphing, based on Radial Basis Functions (RBF) network, has been widely used in the past to smoothly propagate boundary nodal displacements into the volume mesh while preserving its validity and quality. To precisely capture even small design changes, all surface mesh nodes must be used as interpolation nodes which, in case of large meshes for real-world application, leads to excessive computational cost and memory requirements. This paper introduces a cost reduction strategy for mesh adaptation, by proposing a new two-step RBF interpolation employing the Sparse Approximate Inverse (SPAI) preconditioner and the Fast Multipole Method (FMM). The software is demonstrated in the aerodynamic shape optimization of a turbomachinery row. The purpose of this paper is not to solve the optimization problem itself; emphasis is laid on the way the proposed method may handle large displacements and, for this reason, Evolutionary Algorithms (EA) which allow great variations in the values of the design variables were first used. Adjoint-based optimization follows; its role is to perform the refinement of the best solution obtained by the EA-based search.
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Notes
- 1.
\(||\ldots ||\) is the Euclidean norm.
- 2.
Under certain conditions explained in Buhmann (2009).
- 3.
An integer lattice is tessellation of the \(\mathrm {I\!R}^3\) euclidean space in bricks. It is equivalent to a level of an octree.
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Acknowledgements
The authors thank Dr. Varvara G. Asouti and Dimitrios H. Kapsoulis for their assistance assistance during the EA-based optimization with the s/w EASY (Evolutionary Algorithm System) (2008) and Dr. S. Xenofon Trompoukis for his assistance with the in-house flow and adjoint solvers (Asouti et al. 2011).
This research was funded from the People Programme (ITN Marie Curie Actions) of the European Union’s H2020 Framework Programme (MSCA-ITN-2014-ETN) under REA Grant Agreement no. 642959 (IODA project). The first author is an IODA Early Stage Researcher.
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Gagliardi, F., Tsiakas, K.T., Giannakoglou, K. (2019). A Two–Step Mesh Adaptation Tool Based on RBF with Application to Turbomachinery Optimization Loops. In: Andrés-Pérez, E., González, L., Periaux, J., Gauger, N., Quagliarella, D., Giannakoglou, K. (eds) Evolutionary and Deterministic Methods for Design Optimization and Control With Applications to Industrial and Societal Problems. Computational Methods in Applied Sciences, vol 49. Springer, Cham. https://doi.org/10.1007/978-3-319-89890-2_9
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