Numerical considerations for advection-diffusion problems in cardiovascular hemodynamics.
Neumann inflow boundary condition
backflow stabilization
cardiovascular simulation
consistent flux boundary condition
discontinuity-capturing operator
scalar advection diffusion
Journal
International journal for numerical methods in biomedical engineering
ISSN: 2040-7947
Titre abrégé: Int J Numer Method Biomed Eng
Pays: England
ID NLM: 101530293
Informations de publication
Date de publication:
09 2020
09 2020
Historique:
received:
26
01
2020
revised:
21
04
2020
accepted:
07
06
2020
pubmed:
24
6
2020
medline:
9
11
2021
entrez:
24
6
2020
Statut:
ppublish
Résumé
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.
Types de publication
Journal Article
Research Support, Non-U.S. Gov't
Research Support, U.S. Gov't, Non-P.H.S.
Langues
eng
Sous-ensembles de citation
IM
Pagination
e3378Subventions
Organisme : American Heart Association-American Stroke Association
ID : 20POST35220004
Pays : United States
Organisme : Wellcome Trust
ID : 204823/Z/16/Z
Pays : United Kingdom
Organisme : Wellcome Trust
ID : 203148/Z/16/Z
Pays : United Kingdom
Informations de copyright
© 2020 John Wiley & Sons, Ltd.
Références
Oshima M, Torii R, Kobayashi T, Taniguchi N, Takagi K. Finite element simulation of blood flow in the cerebral artery. Comput Methods Appl Mech Eng. 2001;191(6-7):661-671.
Perktold K, Rappitsch G. Computer simulation of local blood flow and vessel mechanics in a compliant carotid artery bifurcation model. J Biomech. 1995;28(7):845-856.
Vignon-Clementel IE, Figueroa CA, Jansen KE, Taylor CA. Outflow boundary conditions for three-dimensional finite element modeling of blood flow and pressure in arteries. Comput Methods Appl Mech Eng. 2006;195(29-32):3776-3796.
Kim HJ, Figueroa C, Hughes T, Jansen K, Taylor C. Augmented Lagrangian method for constraining the shape of velocity profiles at outlet boundaries for three-dimensional finite element simulations of blood flow. Comput Methods Appl Mech Eng. 2009;198(45-46):3551-3566.
Moghadam ME, Bazilevs Y, Hsia TY, Vignon-Clementel IE, Marsden AL. Others . A comparison of outlet boundary treatments for prevention of backflow divergence with relevance to blood flow simulations. Comput Mech. 2011;48(3):277-291.
Fouchet-Incaux J. Artificial boundaries and formulations for the incompressible Navier-Stokes equations: applications to air and blood flows. SeMA J. 2014;64(1):1-40.
Bruneau CH, Fabrie P. Effective downstream boundary conditions for incompressible Navier-Stokes equations. Int J Num Method Fluid. 1994;19(8):693-705.
Bruneau CH, Fabrie P. New efficient boundary conditions for incompressible Navier-stokes equations: a well-posedness result. ESAIM Math Model Num Anal. 1996;30(7):815-840.
Lanzendörfer M, Stebel J. On pressure boundary conditions for steady flows of incompressible fluids with pressure and shear rate dependent viscosities. Appl Math. 2011;56(3):265-285.
Feistauer M, Neustupa T. On the existence of a weak solution of viscous incompressible flow past a cascade of profiles with an arbitrarily large inflow. J Math Fluid Mech. 2013;15:701-715.
Dong S, Karniadakis GE, Chryssostomidis C. A robust and accurate outflow boundary condition for incompressible flow simulations on severely-truncated unbounded domains. J Comput Phys. 2014;261:83-105.
Braack M, Mucha PB. Directional do-nothing condition for the Navier-Stokes equations. J Comput Math. 2014;32(5):507-521.
Porpora A, Zunino P, Vergara C, Piccinelli M. Numerical treatment of boundary conditions to replace lateral branches in hemodynamics. Int J Num Method Biomed Eng. 2012;28(12):1165-1183.
Dong S, Shen J. A pressure correction scheme for generalized form of energy-stable open boundary conditions for incompressible flows. J Comput Phys. 2015;291:254-278.
Dong S. A convective-like energy-stable open boundary condition for simulations of incompressible flows. J Comput Phys. 2015;302:300-328.
Ni N, Yang Z, Dong S. Energy-stable boundary conditions based on a quadratic form: applications to outflow/open-boundary problems in incompressible flows. J Comput Phys. 2019;391:179-215.
Gravemeier V, Comerford A, Yoshihara L, Ismail M, Wall WA. A novel formulation for Neumann inflow boundary conditions in biomechanics. Int J Num Method Biomed Eng. 2012;28(5):560-573.
Bertoglio C, Caiazzo A, Bazilevs Y, et al. Benchmark problems for numerical treatment of backflow at open boundaries. Int J Num Method Biomed Eng. 2018;34(2):e2918.
Hughes TJ, Wells GN. Conservation properties for the Galerkin and stabilised forms of the advection-diffusion and incompressible Navier-Stokes equations. Comput Methods Appl Mech Eng. 2005;194(9-11):1141-1159.
Bertoglio C, Caiazzo A. A tangential regularization method for backflow stabilization in hemodynamics. J Comput Phys. 2014;261:162-171.
Pérez CE, Thomas JM, Blancher S, Creff R. The steady Navier-Stokes/energy system with temperature-dependent viscosity-part 1: analysis of the continuous problem. Int J Num Method Fluid. 2008;56(1):63-89.
Pérez CE, Thomas JM, Blancher S, Creff R. The steady Navier-stokes/energy system with temperature-dependent viscosity-part 2: the discrete problem and numerical experiments. Int J Num Method Fluid. 2008;56(1):91-114.
Neustupa T. The weak solvability of the steady problem modelling the flow of a viscous incompressible heat-conductive fluid through the profile cascade. Int J Num Method Heat Fluid Flow. 2017;27(7):1451-1466.
Ceretani AN, Rautenberg CN. The Boussinesq system with mixed non-smooth boundary conditions and do-nothing boundary flow. Zeitschrift für Angewandte Mathematik Und Physik. 2019;70.1-24.
Liu X, Xie Z, Dong S. On a simple and effective thermal open boundary condition for convective heat transfer problems. Int J Heat Mass Transf. 2020;151:119355.
Hansen KB, Arzani A, Shadden SC. Finite element modeling of near-wall mass transport in cardiovascular flows. Int J Num Method Biomed Eng. 2019;35(1):1-15.
Arzani A, Gambaruto AM, Chen G, Shadden SC. Lagrangian wall shear stress structures and near-wall transport in high-Schmidt-number aneurysmal flows. J Fluid Mech. 2016;790:158-172.
Farghadan A, Arzani A. The combined effect of wall shear stress topology and magnitude on cardiovascular mass transport. Int J Heat Mass Transf. 2019;131:252-260.
Biasetti J, Spazzini PG, Swedenborg J, Gasser TC. An integrated fluid-chemical model toward modeling the formation of intra-luminal thrombus in abdominal aortic aneurysms. Front Physiol. 2012;3:266.
Ford MD, Stuhne GR, Nikolov HN, et al. Virtual angiography for visualization and validation of computational models of aneurysm hemodynamics. IEEE Trans Med Imaging. 2005;24(12):1586-1592.
Leiderman K, Fogelson AL. Grow with the flow: a spatial-temporal model of platelet deposition and blood coagulation under flow. Math Med Biol. 2011;28(1):47-84.
Yazdani A, Li H, Bersi MR, et al. Data-driven modeling of hemodynamics and its role on thrombus size and shape in aortic dissections. Sci Rep. 2018;8(1):2515.
Griffiths DF. The ‘no boundary condition’ outflow boundary condition. Int J Num Method Fluid. 1997;24(4):393-411.
Hughes TJ, Mallet M. A new finite element formulation for computational fluid dynamics: IV. A discontinuity-capturing operator for multidimensional advective-diffusive systems. Comput Methods Appl Mech Eng. 1986;58(3):329-336.
Codina R. A discontinuity-capturing crosswind-dissipation for the finite element solution of the convection-diffusion equation. Comput Methods Appl Mech Eng. 1993;110(3-4):325-342.
De Sampaio P, Coutinho ALGA. A natural derivation of discontinuity capturing operator for convection-diffusion problems. Comput Methods Appl Mech Eng. 2001;190(46-47):6291-6308.
Brooks AN, Hughes TJ. Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput Methods Appl Mech Eng. 1982;32(1-3):199-259.
Papanastasiou TC, Malamataris N, Ellwood K. A new outflow boundary condition. Int J Num Method Fluid. 1992;14(5):587-608.
Renardy M. Imposing ‘no’ boundary condition at outflow: why does it work? Int J Num Method Fluid. 1997;24(4):413-417.
Le Beau GJ, Ray SE, Aliabadi SK, Tezduyar TE. SUPG finite element computation of compressible flows with the entropy and conservation variables formulations. Comput Methods Appl Mech Eng. 1993;104(3):397-422.
CRIMSON. www.crimson.software.
van Bakel TM, Arthurs CJ, van Herwaarden JA, et al. A computational analysis of different endograft designs for Zone 0 aortic arch repair. Eur J Cardiothorac Surg. 2018;54(2):389-396.
Hathcock JJ. Flow effects on coagulation and thrombosis. Arterioscler Thromb Vasc Biol. 2006;26(8):1729-1737.
Tarbell JM. Mass transport in arteries and the localization of atherosclerosis. Annu Rev Biomed Eng. 2003;5(1):79-118.
Coppola G, Caro C. Arterial geometry, flow pattern, wall shear and mass transport: potential physiological significance. J Roy Soc Interf. 2009;6(35):519-528.
Kaazempur-Mofrad MR, Wada S, Myers JG, Ethier CR. Mass transport and fluid flow in stenotic arteries: axisymmetric and asymmetric models. Int J Heat Mass Transf. 2005;48(21-22):4510-4517.
Perktold K, Leuprecht A, Prosi M, et al. Fluid dynamics, wall mechanics, and oxygen transfer in peripheral bypass anastomoses. Ann Biomed Eng. 2002;30(4):447-460.
Vignon-Clementel I, Figueroa C, Jansen K, Taylor C. Outflow boundary conditions for 3D simulations of non-periodic blood flow and pressure fields in deformable arteries. Comput Methods Biomech Biomed Engin. 2010;13(5):625-640.
Arthurs CJ, Agarwal P, John AV, Dorfman AL, Grifka RG, Figueroa CA. Reproducing patient-specific hemodynamics in the Blalock-Taussig circulation using a flexible multi-domain simulation framework: applications for optimal shunt design. Front Pediatr. 2017;5.78-78.
Catabriga L, Coutinho ALGA. Improving convergence to steady state of implicit SUPG solution of Euler equations. Commun Num Method Eng. 2002;18(5):345-353.