Stress Estimation in Viscous Flows Using an Iterative Tikhonov Regularized Stokes Inverse Model

Yuanhao Gao (), Yang Wang and Jizhou Zhang
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Yuanhao Gao: Department of Fintech, Shanghai Normal University Tianhua College, Shanghai 201815, China
Yang Wang: School of Finance and Business, Shanghai Normal University, Shanghai 200234, China
Jizhou Zhang: School of Finance and Business, Shanghai Normal University, Shanghai 200234, China

Mathematics, 2025, vol. 13, issue 11, 1-21

Abstract: In this paper, we propose and develop a stationary Stokes Inverse Model (SIM) to estimate the stress distributions that are difficult to measure directly in flows. We estimate the driving stresses from the velocities by solving the inverse problem governed by Stokes equations under iterative Tikhonov (IT) regularization. We investigate the heuristic L-curve criterion to determine the proper regularization parameter. The solution existence and uniqueness for the Stokes inverse problem have been analyzed. We also conducted convergence analysis and error estimation for perturbed data, providing a fast and stable convergence. The finite element method is applied to the numerical approach. Following the theoretical investigation and formulation, we validate the model and demonstrate that the velocity data closely match the velocity fields that were reconstructed using the computed stress distributions. In particular, the proposed SIM can be used to reliably derive the stress distributions for the flows governed by the Stokes equations with small Reynolds number. Additionally, the model is robust to a certain number of perturbations, which enables the precise and effective estimation of the stress distributions. The proposed stationary SIM may be widely applicable in the estimation of stresses from experimental velocity fields in engineering and biological applications.

Keywords: stokes flow; inverse problem; iterative Tikhonov regularization; L-curve; finite element scheme (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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