Artificial viscosity penalty-projection finite element approximations of non-Newtonian fluid flow for aneurysmal disease

Keqing Feng; Yingchun Shan; Guang-an Zou; Keqing Feng; Yingchun Shan; Guang-an Zou

doi:10.3934/era.2025304

Electronic Research Archive

2025, Volume 33, Issue 11: 6885-6921. doi: 10.3934/era.2025304

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Artificial viscosity penalty-projection finite element approximations of non-Newtonian fluid flow for aneurysmal disease

1.
School of Mathematics and Statistics, Henan University, Kaifeng 475000, China
2.
College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China

Received: 30 September 2025 Revised: 30 October 2025 Accepted: 05 November 2025 Published: 17 November 2025

In this paper, we propose a new formulation of the artificial-viscosity penalty finite element methods for solving the mathematical model of non-Newtonian blood flow, treating blood as a pseudo-plastic fluid obeying the power-law Navier-Stokes equations. The proposed schemes are based on the artificial-viscosity mixed finite element method in the spatial direction, combined with a backward-Euler scheme and penalty-projection scheme for temporal discretization. It is rigorously derived the stability and error estimates for the fully discrete schemes. Numerical examples are presented to validate the theoretical ideas and to demonstrate the effectiveness of our proposed strategies. The comparison shows that the artificial viscosity penalty-projection scheme demands less central processing unit time, as well as a slightly higher-order precision. Finally, the penalty-projection finite element scheme is successfully applied to study the influence of blood viscosity on hemodynamics in aortic aneurysms. Moreover, numerical investigations for flow patterns based on patient-specific geometric models of cerebral aneurysms through segmentation of magnetic resonance imaging have been performed. The present study clearly illustrates the importance of taking the non-Newtonian properties of blood flow within aneurysms into account when studying the risk of aneurysm rupture.
- power-law Navier-Stokes equations,
- artificial viscosity method,
- penalty-projection scheme,
- finite element method,
- aneurysmal disease
Citation: Keqing Feng, Yingchun Shan, Guang-an Zou. Artificial viscosity penalty-projection finite element approximations of non-Newtonian fluid flow for aneurysmal disease[J]. Electronic Research Archive, 2025, 33(11): 6885-6921. doi: 10.3934/era.2025304

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Abstract

In this paper, we propose a new formulation of the artificial-viscosity penalty finite element methods for solving the mathematical model of non-Newtonian blood flow, treating blood as a pseudo-plastic fluid obeying the power-law Navier-Stokes equations. The proposed schemes are based on the artificial-viscosity mixed finite element method in the spatial direction, combined with a backward-Euler scheme and penalty-projection scheme for temporal discretization. It is rigorously derived the stability and error estimates for the fully discrete schemes. Numerical examples are presented to validate the theoretical ideas and to demonstrate the effectiveness of our proposed strategies. The comparison shows that the artificial viscosity penalty-projection scheme demands less central processing unit time, as well as a slightly higher-order precision. Finally, the penalty-projection finite element scheme is successfully applied to study the influence of blood viscosity on hemodynamics in aortic aneurysms. Moreover, numerical investigations for flow patterns based on patient-specific geometric models of cerebral aneurysms through segmentation of magnetic resonance imaging have been performed. The present study clearly illustrates the importance of taking the non-Newtonian properties of blood flow within aneurysms into account when studying the risk of aneurysm rupture.

References

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