Numerical considerations for advection‐diffusion problems in cardiovascular hemodynamics

• Published in: International journal for numerical methods in biomedical engineering Vol. 36; no. 9; pp. e3378 - n/a
• Main Authors: Lynch, Sabrina R., Nama, Nitesh, Xu, Zelu, Arthurs, Christopher J., Sahni, Onkar, Figueroa, C. Alberto
• Format: Journal Article
• Language: English
• Published: Hoboken, USA John Wiley & Sons, Inc 01.09.2020; Wiley Subscription Services, Inc
• Subjects: Advection; backflow stabilization; Biological Transport; Boundaries; Boundary conditions; cardiovascular simulation; Cardiovascular System; Computer applications; Computer simulation; consistent flux boundary condition; Diffusion; discontinuity‐capturing operator; Fluctuations; Flux; Hemodynamics; Humans; Mass transport; Neumann inflow boundary condition; scalar advection diffusion; Transport processes; backflow stabilization; consistent flux boundary condition; discontinuity-capturing operator; scalar advection diffusion; cardiovascular simulation; Neumann inflow boundary condition
• ISSN: 2040-7939; 2040-7947
• DOI: 10.1002/cnm.3378

Numerical simulations of cardiovascular mass transport pose significant challenges due to the wide range of Péclet numbers and backflow at Neumann boundaries. In this paper we present and discuss several numerical tools to address these challenges in the context of a stabilized finite element computational framework. To overcome numerical instabilities when backflow occurs at Neumann boundaries, we propose an approach based on the prescription of the total flux. In addition, we introduce a “consistent flux” outflow boundary condition and demonstrate its superior performance over the traditional zero diffusive flux boundary condition. Lastly, we discuss discontinuity capturing (DC) stabilization techniques to address the well‐known oscillatory behavior of the solution near the concentration front in advection‐dominated flows. We present numerical examples in both idealized and patient‐specific geometries to demonstrate the efficacy of the proposed procedures. The three contributions discussed in this paper successfully address commonly found challenges when simulating mass transport processes in cardiovascular flows. In this work we present a stabilized finite element framework to study scalar mass transport in realistic cardiovascular geometries. Our framework includes the following key features: (a) a backflow stabilization technique, (b) a “consistent flux” boundary condition that minimally disturbs the local physics of the problem, and (c) a front‐capturing stabilization technique to regularize the solution near the wavefront in the case high Péclet numbers. We illustrate the efficacy of these features in both idealized and patient‐specific geometries. Novelty Statement The presented study is to the best of our knowledge the first implementation of backflow stabilization for 3D scalar mass transport problems. In addition, this paper is the first analysis of the “consistent flux” boundary condition in 3D patient‐specific geometries. The novelty of our study is the implementation of backflow stabilization, the consistent flux boundary condition, and discontinuity‐capturing stabilization in a unified scalar mass transport framework that can be applied to study the cardiovascular system.