High-Order Finite Element Methods for Solving Non-Newtonian Biofluid Flow Problems

Abstract

This paper presents an analysis of high-order finite element methods (HO-FEM) for simulating non-Newtonian biofluid flows in complex physiological geometries. While traditional computational fluid dynamics approaches like finite volume methods and low-order finite elements have dominated non-Newtonian modeling, they face significant limitations in resolving boundary layer phenomena and capturing high shear rate gradients characteristic of biological flows. We develop a \(hp\)-adaptive framework that synergistically combines exponential convergence rates of spectral basis functions with robust stabilization techniques for viscoelastic flow instabilities. Through systematic comparison of six different constitutive models - including modified Casson, Quemada, and generalized Cross formulations - we establish quantitative relationships between polynomial enrichment and shear-thinning behavior prediction accuracy. The methodology incorporates novel tensor-product basis functions on hybrid meshes that maintain inf-sup stability for pressure-velocity coupling at Reynolds numbers up to 1,000. Extensive numerical experiments on cerebral aneurysm hemodynamics and synovial fluid dynamics demonstrate order-of-magnitude improvements in wall shear stress prediction compared to literature results. Our stabilization scheme reduces spurious oscillations in vortex cores by 72\% while maintaining temporal accuracy in unsteady flow separation. The results provide rigorous theoretical underpinning for clinical observations of method-dependent variability in computed thrombosis risk indices. This work establishes practical guidelines for polynomial order selection across different Peclet number regimes and presents scalable parallel implementation strategies for patient-specific simulations.