Simulating Extraocular Muscle Dynamics. A Comparison between Dynamic Implicit and Explicit Finite Element Methods
Abstract
:1. Introduction
2. Materials and Methods
2.1. Muscle Contraction Model
2.2. Principle of Virtual Work and Finite Element Discretization
2.3. Implicit Solution Method
- 1.
- Initialize , and
- 2.
- Compute the mass matrix
- 3.
- For each time step increment:
- (a)
- Initialize the displacement increment and the internal force
- (b)
- Iterations for finding “dynamic equilibrium” within the time step increment:
- Compute tangential stiffness matrix:
- Compute the algorithmic stiffness matrix:
- Compute
- Solve the linear system:
- Update the displacement increments:
- For each integration point k:
- -
- Compute the strain increment:
- -
- Compute the stress increment:
- -
- Compute the total stress:
- Compute internal force:
- Compute accelerations:
- Compute residual:
- Check convergence: if , with the convergence tolerance, go to Step (c).
- (c)
- Compute the velocities and displacements at the end of the time step:
- -
- Velocities:
- -
- Displacements
2.4. Explicit Solution Method
- 1.
- Initialize and
- 2.
- Compute the mass matrix
- 3.
- Compute
- 4.
- For each time step increment:
- (a)
- Solve for total displacements:
- (b)
- Compute the displacement increment:
- (c)
- For each integration point j:
- Compute the strain increment:
- Compute the stress increment:
- Compute the total stress:
- (d)
- Compute the internal force vector:
- (e)
- Solve for the new accelerations:
- (f)
- Compute the velocities at new mid-time:
2.5. Development of User Material Subroutine
2.6. Eyeball and EOM Finite Element Model
3. Results
4. Discussion
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Part | Nodes | Elements | Volume (mm3) | Mass (g) |
---|---|---|---|---|
Eyeball | 2686 | 2211 | 8134 | |
Lateral EOM | 2573 | 1920 | 488 | |
Inferior EOM | 2291 | 1680 | 576 | |
Medial EOM | 1953 | 1440 | 419 | |
Superior EOM | 1377 | 936 | 461 |
Parameters | |
---|---|
Passive behavior | = MPa |
MPa | |
MPa | |
MPa | |
MPa | |
Maximum isometric stress | MPa |
Force length relationship | = 1 |
= 1 | |
= | |
Force time relationship | |
= variable |
Mass Added (%) | Calculation Time (%) | Maximum Displacement Reduction (%) | |
---|---|---|---|
Implicit | 0 | 5 | |
Explicit no m.s. | 0 | 100 | 0 |
Explicit m.s. factor | 1 | 99 | |
Explicit m.s. factor | 10 | 96 | |
Explicit m.s. factor 4 | 300 | 53 | |
Explicit m.s. factor 6 | 500 | 35 | |
Explicit m.s. factor 10 | 900 | 30 | |
Explicit m.s. factor 100 | 9900 | 11 |
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Grasa, J.; Calvo, B. Simulating Extraocular Muscle Dynamics. A Comparison between Dynamic Implicit and Explicit Finite Element Methods. Mathematics 2021, 9, 1024. https://doi.org/10.3390/math9091024
Grasa J, Calvo B. Simulating Extraocular Muscle Dynamics. A Comparison between Dynamic Implicit and Explicit Finite Element Methods. Mathematics. 2021; 9(9):1024. https://doi.org/10.3390/math9091024
Chicago/Turabian StyleGrasa, Jorge, and Begoña Calvo. 2021. "Simulating Extraocular Muscle Dynamics. A Comparison between Dynamic Implicit and Explicit Finite Element Methods" Mathematics 9, no. 9: 1024. https://doi.org/10.3390/math9091024