Computational Modeling of Individual Red Blood Cell Dynamics Using Discrete Flow Composition and Adaptive TimeStepping Strategies
Abstract
:1. Introduction
2. Mathematical Framework
2.1. Preliminaries
2.2. Membrane Model
2.3. Interface Tracking: LevelSet Representation
2.4. NonNewtonian Fluid Model
Dimensionless Nonlinear Coupled Problem
3. Composition Technique Applied to the SecondOrder BDF Scheme
3.1. SecondOrder Backward Differentiation BDF2
 Step 1:
 $({y}_{n1/2},{y}_{n1})={{\rm Y}}_{{a}_{1}\Delta {t}_{n},{\gamma}_{i}}^{\mathrm{BDF}2}({y}_{n1},{y}_{n2}),$
 Step 2:
 $({y}_{n},{y}_{n1})=\mathrm{Re}\left({{\rm Y}}_{{a}_{2}\Delta {t}_{n},{\omega}_{i}}^{\mathrm{BDF}2}\circ {{\rm Y}}_{{a}_{1}\Delta {t}_{n},{\gamma}_{i}}^{\mathrm{BDF}2}({y}_{n1},{y}_{n2})\right),$
3.2. Step 1: Calculation of an Intermediate Solution
3.3. Step 2 and Solution Method
3.4. Algorithm of Composed BDF2 Scheme with Fixed Time Step
Algorithm 1 Composed BDF2 with fixed time step. 

3.5. Algorithm for Composing BDF2 with Adaptive Time Step
Algorithm 2 Composed BDF2 with adaptive time steps. 

4. Numerical Approximation of the Fluid/Membrane Problem
4.1. Time Discretization of the Fluid Problem
4.2. Penalty Method
4.3. Level Set Problem
5. Numerical Results
5.1. Example 1: OneDimensional Test Case—AccuracyOrder Analysis
5.2. Example 2: Membrane Dynamics in a Newtonian Fluid under Simple Shear Flow
5.2.1. TankTreading Regime
5.2.2. Tumbling Regime
5.2.3. Calibration of the Penalty Parameter
5.2.4. Quantitative Validation with Respect to Existing Results
5.3. Example 3: Membrane Dynamics in a Casson Shear Flow
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A
Note on the Composition Technique for the LevelSet Problem
References
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$1/\Delta t$  20  40  80  160  320  720 
${{\rm Y}}_{\Delta t}^{\mathrm{BDF}2}$  1.65  1.83  1.92  1.96  1.98  1.99 
${{\rm Y}}_{{a}_{2}\Delta t}^{\mathrm{BDF}2}\circ {{\rm Y}}_{{a}_{1}\Delta t}^{\mathrm{BDF}2}$  2.777  2.925  2.975  2.991  2.996  2.998 
${\mathit{\epsilon}}_{\mathit{\lambda}}$  ${\mathit{h}}^{1}$  ${\mathit{h}}^{1.2}$  ${\mathit{h}}^{1.4}$  ${\mathit{h}}^{1.6}$  ${\mathit{h}}^{1.8}$  ${\mathit{h}}^{2}$  ${\mathit{h}}^{2.3}$ 

$\beta =1$  0.0022  0.021  0.0138  0.0082  0.03692  0.0512  0.0675 
$\beta =10$  0.036  0.0013  0.0027  0.0057  0.0199  0.0309  0.0458 
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Laadhari, A.; Deeb, A. Computational Modeling of Individual Red Blood Cell Dynamics Using Discrete Flow Composition and Adaptive TimeStepping Strategies. Symmetry 2023, 15, 1138. https://doi.org/10.3390/sym15061138
Laadhari A, Deeb A. Computational Modeling of Individual Red Blood Cell Dynamics Using Discrete Flow Composition and Adaptive TimeStepping Strategies. Symmetry. 2023; 15(6):1138. https://doi.org/10.3390/sym15061138
Chicago/Turabian StyleLaadhari, Aymen, and Ahmad Deeb. 2023. "Computational Modeling of Individual Red Blood Cell Dynamics Using Discrete Flow Composition and Adaptive TimeStepping Strategies" Symmetry 15, no. 6: 1138. https://doi.org/10.3390/sym15061138