Abstract
The immersed boundary method (IBM) has been frequently utilized to simulate the motion and deformation of biological cells and capsules in various flow situations. Despite the convenience in dealing with flow-membrane interaction, direct implementation of membrane viscosity in IBM suffers severe numerical instability. It has been shown that adding an artificial elastic element in series to the viscous component in the membrane mechanics can efficiently improve the numerical stability in IBM membrane simulations. Recently Li and Zhang (Int J Numer Methods Biomed Eng 35:e3200, 2019) proposed a finite-difference method for calculating membrane viscous stress. In the present paper, two new schemes are developed based on the convolution integral expression of the Maxwell viscoelastic element. We then conduct several tests for the accuracy, stability, and efficiency performances of these three viscous stress schemes. By studying the behavior of a one-dimensional Maxwell element under sinusoidal deformation, we find that a good accuracy can be achieved by selecting an appropriate relaxation time. The twisting sphere tests confirm that, compared to the numerical errors induced by other components in capsule simulations, such as the finite element method for membrane discretization and IBM for flow-membrane interaction, the errors from the viscous stress calculation are negligible. Moreover, extensive simulations are conducted for the dynamic deformation of a spherical capsule in shear flow, using different numerical schemes and various combinations of the artificial spring constants and calculation frequency for the membrane viscous stress calculation. No difference is observed among the results from the three schemes; and these viscous stress schemes require very little extra computation time compared to other components in IBM simulations. The simulation results converge gradually with the increase in the artificial spring stiffness; however, a threshold value exists for the spring stiffness to maintain the numerical stability. The viscous stress calculation frequency has no apparent influence on the calculation results, but a large frequency number can cause the simulation to collapse. We therefore suggest to calculate the membrane viscous stress at each simulation time step, such that a better numerical stability can be achieved. The three numerical schemes have nearly identical performances in all aspects, and they can all be utilized in future IBM simulations of viscoelastic membranes.
Similar content being viewed by others
References
Barthes-Biesel D, Sgaier H (1985) Role of membrane viscosity in the orientation and deformation of a spherical capsule suspended in simple shear flow. J Fluid Mech 160:119
Bronzino JD (2000) The biomedical engineering handbook, 2nd edn. CRC Press, Boca Raton
Charrier JM, Shrivastava S, Wu R (1989) Free and constrained inflation of elastic membranes in relation to thermoforming—non-axisymmetric problems. J Strain Anal Eng Des 24:55
Christensen R (1980) A nonlinear theory of viscoelasticity for application to elastomers. J Appl Mech 47:762
Fai TG, Griffith BE, Mori Y, Peskin CS (2013) Immersed boundary method for variable viscosity and variable density problems using fast constant-coefficient linear solvers I: Numerical method and results. SIAM J Sci Comput 35:B1132
Gounley J, Peng Y (2015) Computational modeling of membrane viscosity of red blood cells. Commun Comput Phys 17:1073
Gounley J, Boedec G, Jaeger M, Leonetti M (2016) Influence of surface viscosity on droplets in shear flow. J Fluid Mech 791:464
Hochmuth RM, Waugh RE (1987) Erythrocyte membrane elasticity and viscosity. Annu Rev Physiol 49:209
Kim Y, Kim K, Park Y (2012) Measurement techniques for red blood cell deformability: recent advances. In: Moschandreou T (ed) Blood cell—an overview of studies in hematology. InTech, chap. 10
Kruger T, Varnik F, Raabe D (2011) Efficient and accurate simulations of deformable particles immersed in a fluid using a combined immersed boundary lattice Boltzmann finite element method. Comput Math Appl 61:3485
Li P, Zhang J (2019) A finite difference method with sub-sampling for immersed boundary simulations of the capsule dynamics with viscoelastic membranes. Int J Numer Methods Biomed Eng 35:e3200
Marques SPC, Creus GJ (2012) Computational viscoelasticity. Springer, Berlin
Mittal R, Iaccarino G (2005) Immersed boundary method. Annu Rev Fluid Mech 37:239
Neto MA, Amaro A, Cirne LRJ, Leal R (2017) Engineering computation of structures: the finite element method. Springer, Berlin
Oulaid O, Saad AW, Aires PS, Zhang J (2016) Effects of shear rate and suspending viscosity on deformation and frequency of red blood cells tanktreading in shear flows. Comput Methods Biomech Biomed Eng 19:648
Peskin CS (1972) Flow patterns around heart valves: a numerical method. J Comput Phys 10:252
Peskin CS (1977) Numerical analysis of blood flow in the heart. J Comput Phys 25:220
Peskin CS (2002) The immersed boundary method. Acta Numer 11:479
Pozrikidis C (1994) Effects of surface viscosity on the finite deformation of a liquid drop and the rheology of dilute emulsions in simple shearing flow. J Nonnewton Fluid Mech 5:161
Pozrikidis C (2010) Computational hydrodynamics of capsules and biological cells. CRC Press, Boca Raton
Secomb T, Skalak R (1982) Surface flow of viscoelastic membranes in viscous fluids. Q J Mech Appl Mech 35:233
Shrivastava SK, Tang J (1993) Large deformation finite element analysis of non-linear viscoelastic membranes with reference to thermoforming. J Strain Anal Eng Des 28:31
Skalak R, Tozeren A, Zarda RP, Chien S (1973) Strain energy function of red blood cell membranes. Biophys J 13:245
Tran-Son-Tay R, Sutera SP, Rao PR (1984) Deformation of red blood cell membrane viscosity from rheoscopic observations of tank-treading motion. Biophys J 46:65
Wan J, Forsyth AM, Stone HA (2011) Red blood cell dynamics: from cell deformation to ATP release. Integ Giology 3:972
Yazdani A, Bagchi P (2013) Influence of membrane viscosity on capsule dynamics in shear flow. J Fluid Mech 718:569
Yin X, Thomas T, Zhang J (2013) Multiple red blood cell flows through microvascular bifurcations: cell free layer, cell trajectory, and hematocrit separation. Microvasc Res 89:47
Zhang J (2011) Lattice Boltzmann method for microfluidics: models and applications. Microfluid Nanofluid 10:1
Zhang J, Johnson PC, Popel AS (2008) Red blood cell aggregation and dissociation in shear flows simulated by lattice Boltzmann method. J Biomech 41:47
Acknowledgements
The authors sincerely thank one anonymous reviewer of our previous publication (Li and Zhang 2019) for inspiring comments. This work was supported by the Natural Science and Engineering Research Council of Canada (NSERC). The calculations have been enabled by the use of computing resources provided by WestGrid (westgrid.ca), SHARCNet (sharcnet.ca), and Compute/Calcul Canada (computecanada.org). P.L. acknowledges the financial support from the Ontario Trillium Scholarship at Laurentian University.
Author information
Authors and Affiliations
Corresponding author
Additional information
Pulisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Li, P., Zhang, J. Finite-difference and integral schemes for Maxwell viscous stress calculation in immersed boundary simulations of viscoelastic membranes. Biomech Model Mechanobiol 19, 2667–2681 (2020). https://doi.org/10.1007/s10237-020-01363-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10237-020-01363-y