# Efficient Mesh Deformation Based on Randomized RBF Solvers

Augustin Parret-Fréaud, Wael Bader, Sébastien da Veiga, Youssef Mesri

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Augustin Parret-Fréaud, Wael Bader, Sébastien da Veiga, Youssef Mesri. Efficient Mesh Deformation Based on Randomized RBF Solvers. 2024 SIAM Conference on Parallel Processing for Scientific Computing, Mar 2024, Baltimore (United States), United States. hal-04888424

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Submitted on 15 Jan 2025

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# Efficient Mesh Deformation Based on Randomized RBF Solvers

A. Parret-Fréaud*, W. Bader, S. Da Veiga, Y. Mesri

* Safran Tech, France,
Email: augustin.parret-freaud@safrangroup.com

## ABSTRACT

Mesh deformation methods have been widely used for the past decades in various fields such as fluid-structure interaction, aerodynamic shape optimization, unsteady and aeroelastic computational fluid dynamics [3-10]. Among the existing methods, radial basis functions interpolation (RBF) is particularly suitable for unstructured mesh applications due to its simplicity and the high quality of the resulting mesh. Such approach requires solving dense linear systems, generally symmetric positive definite (SPD), which tends to be computationally expensive and memory demanding, which is a major drawback when dealing with large-scale meshes [1].

In this work, we aim to speed-up RBF mesh deformation procedure using methods coming from probabilistic linear algebra [2] to solve the associated dense linear system. Indeed, such methods start to emerge into several fields, including numerical linear algebra and optimization, exploiting its spectral properties as an efficient alternative to reduce the complexity of solving large scale linear system. We will address the question of how to create an approximation to the RBF matrix by projecting the initial large scale operator onto a smaller subspace that exhibits specific properties such as a better sparsity and investigate various solving strategies.

The proposed approach will be discussed on the basis of 2D and 3D applications.

## REFERENCES

[1] R. Alomairy, W. Bader, H. Ltaief, Y. Mesri, D. Keyes, High-performance 3D Unstructured Mesh Deformation Using Rank Structured Matrix Computations, ACM Trans. Parallel Comput. 9(1), 2022, pp. 1-23

[2] N. Halko, P.G. Martinsson, J. Tropp, Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions., SIAM Review, 53(2), 2011, pp. 217-288.

[3] W. Daldoul, E. Hachem, Y. Mesri, A ‘R-to-H’ Mesh Adaptation Approach for Moving Immersed Complex Geometries Using Parallel Computers., International Journal of Computational Fluid Dynamics 34 (7-8), 2020, pp. 597-609.

[4] Y. Mesri, H. Guillard, Coupez T. Automatic coarsening of three dimensional anisotropic unstructured meshes for multigrid applications. Appl Math Comput., 218(21), 2012, pp. 10500-10519.

[5] Y. Mesri, W. Zerguine, H. Digonnet, L. Silva, T. Coupez, Dynamic parallel adaption for three dimensional unstructured meshes: Application to interface tracking, in: Proceedings of the 17th International Meshing Roundtable, Springer, 2008, pp. 195-212.

[6] A Bazile, E Hachem, JC Larroya-Huguet, Y Mesri, Variational Multiscale error estimator for anisotropic adaptive fluid mechanic simulations: application to convection-diffusion problems., CMAME 331, 2018, pp. 94-115.

[7] Y. Yang, Y. Mesri, Learning by neural networks under physical constraints for simulation in fluid mechanics., Comp. & Fluids 248, 2022, pp. 105632.

[8] J Sari, F Cremonesi, M Khalloufi, F Cauneau, P Meliga, Y. Mesri, E. Hachem. Anisotropic adaptive stabilized finite element solver for RANS models., IJNMF, 86(11), 2018, pp. 717-736

[9] Y. Mesri, J.-M. Gratien, O. Ricois, R. Gayno. Parallel Adaptive Mesh Refinement for Capturing Front Displacements: Application to Thermal EOR Processes. SPE, SPE-166058, 2013

[10] Mesri, Y., Digonnet, H., Coupez, T.: Hierarchical adaptive multi-mesh partitioning algorithm on heterogeneous systems. Lect. Notes Comput. Sci. Eng. n74, 299-306 (2010)