Application of Hyperelastic Nodal Force Method to Evaluation of Aortic Valve Cusps Coaptation: Thin Shell vs. Membrane Formulations
Abstract
:1. Introduction
2. Materials and Methods
2.1. Shell Kinematics
2.2. Constitutive Equations
2.3. Weak Formulation
2.4. Discretization
2.4.1. Discretization of the Membrane Part
2.4.2. Bending Part
2.4.3. Discretized Equilibrium Equations
3. Numerical Results
3.1. Benchmarks
3.1.1. Cantilever Subjected to End Shear Force
3.1.2. Slit Annular Plate Subjected to Lifting Line Force
3.2. Aortic Valve in Diastolic State
Algorithm 1 Solution of the static equilibrium equation and computation of the contact forces 

4. Discussion
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Model  Max. Coaptation  Free Edge  

Height, mm  Length, mm  
${\mathbf{h}}_{\mathbf{1}}$  ${\mathbf{h}}_{\mathbf{2}}$  ${\mathbf{h}}_{\mathbf{1}}$  ${\mathbf{h}}_{\mathbf{2}}$  
SVK  $E=1$ MPa, $\nu =0.5$ (membrane)  $3.51$  $3.45$  $23.20$  $23.15$ 
$E=1$ MPa, $\nu =0.5$ (shell)  $0.4$  $0.37$  $23.31$  $23.35$  
NeoHookean  $E=1$ MPa, $\mu =E/3$ (membrane)  $3.72$  $3.72$  $23.96$  $23.91$ 
$E=1$ MPa, $\mu =E/3$ (shell)  $0.73$  $0.76$  $23.75$  $23.77$  
Gent  $E=1$ MPa, $\mu =E/3$, ${J}_{m}=2.3$ (membrane)  $3.71$  $3.70$  $23.76$  $23.71$ 
$E=1$ MPa, $\mu =E/3$, ${J}_{m}=2.3$ (shell)  $0.58$  $0.6$  $23.61$  $23.61$  
SVK  $E=10$ MPa, $\nu =0.5$ (membrane)  $0.71$  $0.81$  $20.82$  $20.93$ 
$E=10$ MPa, $\nu =0.5$ (shell)  *  *  $20.94$  $20.99$  
NeoHookean  $E=10$ MPa, $\mu =E/3$ (membrane)  $0.67$  $0.66$  $20.93$  $20.99$ 
$E=10$ MPa, $\mu =E/3$ (shell)  *  *  $20.99$  $21.03$  
Gent  $E=10$ MPa, $\mu =E/3$, ${J}_{m}=2.3$ (membrane)  $0.66$  $0.67$  $20.91$  $20.90$ 
$E=10$ MPa, $\mu =E/3$, ${J}_{m}=2.3$ (shell)  *  *  $20.98$  $21.02$ 
Model  CPU Time, s  

SVK  $E=1$ MPa, $\nu =0.5$ (membrane)  220 
$E=1$ MPa, $\nu =0.5$ (shell)  3761  
NeoHookean  $E=1$ MPa, $\mu =E/3$ (membrane)  157 
$E=1$ MPa, $\mu =E/3$ (shell)  3940  
Gent  $E=1$ MPa, $\mu =E/3$, ${J}_{m}=2.3$ (membrane)  170 
$E=1$ MPa, $\mu =E/3$, ${J}_{m}=2.3$ (shell)  5047  
SVK  $E=10$ MPa, $\nu =0.5$ (membrane)  831 
$E=10$ MPa, $\nu =0.5$ (shell)  5736  
NeoHookean  $E=10$ MPa, $\mu =E/3$ (membrane)  810 
$E=10$ MPa, $\mu =E/3$ (shell)  5561  
Gent  $E=10$ MPa, $\mu =E/3$, ${J}_{m}=2.3$ (membrane)  832 
$E=10$ MPa, $\mu =E/3$, ${J}_{m}=2.3$ (shell)  4496 
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Vassilevski, Y.; Liogky, A.; Salamatova, V. Application of Hyperelastic Nodal Force Method to Evaluation of Aortic Valve Cusps Coaptation: Thin Shell vs. Membrane Formulations. Mathematics 2021, 9, 1450. https://doi.org/10.3390/math9121450
Vassilevski Y, Liogky A, Salamatova V. Application of Hyperelastic Nodal Force Method to Evaluation of Aortic Valve Cusps Coaptation: Thin Shell vs. Membrane Formulations. Mathematics. 2021; 9(12):1450. https://doi.org/10.3390/math9121450
Chicago/Turabian StyleVassilevski, Yuri, Alexey Liogky, and Victoria Salamatova. 2021. "Application of Hyperelastic Nodal Force Method to Evaluation of Aortic Valve Cusps Coaptation: Thin Shell vs. Membrane Formulations" Mathematics 9, no. 12: 1450. https://doi.org/10.3390/math9121450