Wavelet-enhanced adaptive meshing: Optimizing finite element simulations for structural and fluid dynamics

Mohamed Ayari; Zeineb Klai; Atef Gharbi; Omar Kahouli; Sulaiman A. Almohaimeed; Fahd Alhamazani; Nasser Albalwi; Mohamed Ayari; Zeineb Klai; Atef Gharbi; Omar Kahouli; Sulaiman A. Almohaimeed; Fahd Alhamazani; Nasser Albalwi

doi:10.3934/math.20261188

AIMS Mathematics

2026, Volume 11, Issue 9: 29934-29977. doi: 10.3934/math.20261188

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Research article

Wavelet-enhanced adaptive meshing: Optimizing finite element simulations for structural and fluid dynamics

1.
Department of Information Technology, Faculty of Computing and Information Technology, Northern Border University, Kingdom of Saudi Arabia
2.
Department of Computer Sciences, Faculty of Computing and Information Technology, Northern Border University, Kingdom of Saudi Arabia
3.
Department of Information Systems, Faculty of Computing and Information Technology, Northern Border University, Kingdom of Saudi Arabia
4.
Department of Electronics Engineering, Applied College, University of Ha'il, Ha'il 2440, Saudi Arabia
5.
Department of Electrical Engineering, College of Engineering, Qassim University, Saudi Arabia

Received: 21 November 2025 Revised: 07 August 2026 Accepted: 21 August 2026 Published: 15 September 2026
MSC : 65N30, 65T60, 65M50, 76M10

This paper presents a practical integration of wavelet analysis with the finite element method (FEM) for adaptive mesh refinement in structural and fluid-dynamics simulations. At each adaptive cycle, the computed finite element solution is decomposed using a multiresolution wavelet transform, and the resulting detail coefficients are converted into element-level refinement indicators. Elements associated with the largest localized variations are refined, after which the finite element matrices and load vectors are reassembled on the updated mesh. The proposed procedure was evaluated using a simply supported vibrating beam and a two-dimensional flat-plate boundary-layer problem at $ R{e}_{L} = 1000 $. For the beam benchmark, the wavelet-adapted mesh uses 60 elements, compared with 160 elements for a uniformly refined mesh of similar modal accuracy, corresponding to an element-count reduction of approximately 62%. The modal-solution time decreases from 4.40 to 1.90 s. For the flat-plate problem, the adapted mesh uses 4096 elements instead of 16,384 elements and reduces the time required to reach the steady solution from 162 to 68 s. The adapted beam solution predicts the first three natural frequencies with a maximum relative error below 4%, while the flat-plate solution reproduces the end-of-plate skin-friction coefficient and integrated drag coefficient within approximately 4% and 2% of the corresponding laminar correlations, respectively. The results indicate that wavelet detail coefficients can provide useful localized refinement information when they are consistently coupled with element marking, mesh updating, and finite element reassembly. The computational overhead, parameter sensitivity, applicability, and limitations of the approach are also discussed.
- wavelet transform,
- adaptive mesh refinement,
- finite element method,
- structural dynamics,
- boundary-layer flow,
- multiresolution analysis,
- computational efficiency
Citation: Mohamed Ayari, Zeineb Klai, Atef Gharbi, Omar Kahouli, Sulaiman A. Almohaimeed, Fahd Alhamazani, Nasser Albalwi. Wavelet-enhanced adaptive meshing: Optimizing finite element simulations for structural and fluid dynamics[J]. AIMS Mathematics, 2026, 11(9): 29934-29977. doi: 10.3934/math.20261188

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Abstract

This paper presents a practical integration of wavelet analysis with the finite element method (FEM) for adaptive mesh refinement in structural and fluid-dynamics simulations. At each adaptive cycle, the computed finite element solution is decomposed using a multiresolution wavelet transform, and the resulting detail coefficients are converted into element-level refinement indicators. Elements associated with the largest localized variations are refined, after which the finite element matrices and load vectors are reassembled on the updated mesh. The proposed procedure was evaluated using a simply supported vibrating beam and a two-dimensional flat-plate boundary-layer problem at $ R{e}_{L} = 1000 $. For the beam benchmark, the wavelet-adapted mesh uses 60 elements, compared with 160 elements for a uniformly refined mesh of similar modal accuracy, corresponding to an element-count reduction of approximately 62%. The modal-solution time decreases from 4.40 to 1.90 s. For the flat-plate problem, the adapted mesh uses 4096 elements instead of 16,384 elements and reduces the time required to reach the steady solution from 162 to 68 s. The adapted beam solution predicts the first three natural frequencies with a maximum relative error below 4%, while the flat-plate solution reproduces the end-of-plate skin-friction coefficient and integrated drag coefficient within approximately 4% and 2% of the corresponding laminar correlations, respectively. The results indicate that wavelet detail coefficients can provide useful localized refinement information when they are consistently coupled with element marking, mesh updating, and finite element reassembly. The computational overhead, parameter sensitivity, applicability, and limitations of the approach are also discussed.

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