I. Explore tip trajectories with rotating anisotropy;

Transcription

1 Below is the summary of the recent work. Updates for V1.1: I. Added irregular behavior tables for Nz = 31. II. Added some comments of the irregular behaviors of scroll waves. III. Added the corrected phase diagram for Barkley 2D model. IV. Uploaded density plot, iso-surface plot videos and filament dynamic video for the model with larger value of Nz (Nz = 31) to media4 server under jianfeng/barkley/mono3drotating/nz31 I. Explore tip trajectories with rotating anisotropy; 1. When the parameters a and b of Barkley model are a=0.94, b=0.17 respectively. Without rotating anisotropy, the tips trajectory in monodomain model is where Δt = (t. u.), loops = 10000, T = Δt x loops = 20 (t. u.), Nx = Ny = 127, Nz = 7, Z axis Layer 1;

2 D perp / D parallel = 0.9 where Δt = (t. u.), loops = 10000, T = Δt x loops = 20 (t. u.), Nx = Ny = 127, Nz = 7, Z axis Layer 1; D perp / D parallel = 0.7 where Δt = (t. u.), loops = 10000, T = Δt x loops = 20 (t. u.), Nx = Ny = 127, Nz = 7, Z axis Layer 1;

3 D perp / D parallel = 0.5 where Δt = (t. u.), loops = 10000, T = Δt x loops = 20 (t. u.), Nx = Ny = 127, Nz = 7, Z axis Layer 1; D perp / D parallel = 0.3 where Δt = (t. u.), loops = 10000, T = Δt x loops = 20 (t. u.), Nx = Ny = 127, Nz = 7, Z axis Layer 1;

4 D perp / D parallel = 0.1 where Δt = (t. u.), loops = 10000, T = Δt x loops = 20 (t. u.), Nx = Ny = 127, Nz = 7, Z axis Layer 1; Thoughts: anisotropy affects the tips trajectories. Especially when the D perp / D parallel ratio is small (= 0.1). Even if the trajectory is closed in the case of isotropy, the trajectories may not be closed when anisotropy is introduced. In the case that a=0.94, b=0.17, the trajectory patterns for D perp / D parallel = 0.1 and 0.3 show irregular results. This inspires a closer look at the scroll wave dynamics.

5 II. Irregular scroll wave behavior of Barkley model in monodomain 3D; The a and b parameters in Barkley model were chosen as a = 0.94, b=0.17 based on previous results. Nx and Ny were increased to 255 to accommodate the scroll wave when the D perp / D parallel ratio is small (= 0.1). With rotating angel = 120 o, anisotropy ratio D perp / D parallel = 0.1, the export image of tecplot results are below: Density plot at time = 20 (t. u.)

6 iso-surface plot at time = 20 (t. u.)

7 Slice plot for Z axis layer 1 at time 20 (t. u.) where in the plots above Δt = (t. u.), loops = 10000, T = Δt x loops = 20 (t. u.), Nx = Ny = 255, Nz = 7;

8 The filament number, filament length and filament dynamic plots are below:

9 Filament dynamic plot at time 20 (t. u.) Filament dynamic plot at around time 12 (t. u.)

10 From the filament dynamic plots, we can see that the filament is heavily twisted and broken into parts. The iso-surface plots shows a scroll wave fragment and the main portion of the scroll wave at time 20 (t. u.). I have uploaded the movies of density plot, iso-surface plot, slice plot and filament dynamic onto the media4 server under jianfeng/barkley/mono3drotating. III. Verification 1. Using the same parameter set [Nx = Ny = 255, Nz = 7, Δx = 0.25, a = 0.94, b = 0.17 ] but reduce the time step Δt to 0.001, increase the loop number to The result is the same as Δt = The irregular behavior of the scroll wave is not due to the numerical instability. The density plot, iso-surface plot, slice plot and filament dynamic plot for Δt = at time 20 (t. u.) are given below.

11

12 The filament number and filament length plots are:

13 2. Tips trajectories verification in monodomain 3D, with 2D comparison. The 3D results came from an isotropic monodomain model without fiber rotation. Fixing b = 0.17, the values of a vary from 0.85 to 1.03 with step size The range of the values of a cover all 3 regions in the phase diagram (no spirals, periodic and meandering patterns). Both 2D and 3D model share the same space step (0.25) and time step (0.002). The domain size of 2D model is 127 x 127 grid points. The domain size of 3D model is 127 x 127 x 7 grid points. Below I list a selection of side-by-side comparison of the results from 2D and 3D with the same a and b combination.

14 a = 0.85, b = D 3D a = 0.89, b = D 3D

15 a = 0.93, b = D 3D a = 0.99, b = D 3D

16 a = 1.05, b = D 3D a = 1.11, b = D 3D

17 a = 1.13, b = D 3D We can see just a slight shift of the boundaries between the regions from 2D to 3D. But the difference is small. The irregular behavior is more convincing to explain as introducing by rotating anisotropy than the difference originated from switching 2D to 3D model. IV. Explore the parameter space for irregular scroll wave behavior. There are 6 parameters may affect the irregular behavior of scroll wave using Barkley model. Three of them, ε, a and b, relate to the Barkley model. The rest are rotating angle of the fiber direction, anisotropy ratio and the lenght on Z direction. For now we leave the Barkley parameters fixed as ε=0.02, a = 0.94 and b = Below is a quick look at the boundary of the irregular behavior. The conclusion of the irregular is based on the density plot results. Using Nx = Ny = 255 (grid points), Δx = 0.25, Δt = With the anisotropy ratio and the Nz fixed, the irregular scroll wave behavior disappears when the rotation angle reduce below 40 o. D perp / D parallel Rot. Angle (degree) Nz (layers) irregular comment Yes Yes Yes Yes Yes Yes No

18 Now if the rotational angle and Nz are fixed ( rot. angle = 120 o, Nz = 7), by varying the anisotropy ratio, we obtained, The irregular behavior disappears when the anisotropy ratio D perp / D parallel is larger than 0.5 with the tested setting. D perp / D parallel Rot. Angle (degree) Nz (layers) irregular comment Yes Yes No No No The irregular behaviors of the scroll wave for Nz = 15 layers. The tables look the same as those tables with Nz = 7 layers. D perp / D parallel Rot. Angle (degree) Nz (layers) irregular comment Yes Yes Yes Yes Yes Yes No D perp / D parallel Rot. Angle (degree) Nz (layers) irregular comment Yes Yes No No No When Nz is increased to 31 layers. The results are listed below. They are the same as the two group of tables shown above. D perp / D parallel Rot. Angle (degree) Nz (layers) irregular comment Yes Yes Yes Yes Yes Yes No D perp / D parallel Rot. Angle (degree) Nz (layers) irregular comment Yes Yes No No No

19 In general, the irregular behavior goes away when the rotational angle is reduced and D perp / D parallel is increased. It is consistent with intuitive thinking. The filament number and length plots have been checked and they agree with the results in the tables above. The conclusion so far is, the anisotropy plays the most import role in the irregular behaviors of the scroll waves. As we can see from the tables above, when the anisotropic ratio D perp / D parallel is small (equals to 0.1), the scroll waves show irregular behaviors even if the rotational angle is reduced to 30 o. When we reduce the anisotropic effect, for example, increasing D perp / D parallel to 0.5, the irregular behaviors of the scroll waves go away. The rotation angle is also an important factor that affects the dynamic of the scroll wave, although it carries less weight than the anisotropy when comes to the role in the irregular behavior of scroll wave. Keeping the anisotropic ratio unchanged, we can observe the irregular behaviors of the scroll waves become less significant when we reduce the rotational angle. That is to say, when we reduce the rotation angel to 60 o or smaller, although we can still observe the irregular behaviors of the scroll waves, the scroll waves are stable most of the time. This can be explained better with a series video of the plots. Another observation is that it is the interaction between the anisotropy and rotation angle cause a great impact on the scroll wave behavior. When the anisotropic ratio D perp / D parallel is small and the rotation angle is large, for example, D perp / D parallel = 0.1, Rot. Angle = 120 o, the horizontal distance of the two ends of the core filament is small. However the twisting caused by the anisotropy and rotation is restricted in a small vertical distance, so the middle part of the core filament is stretched and broken easily, causing waves fragments. When the rotation angle is decreased, the effect of the anisotropy becomes dominant. The lower end of the core filament starts drifting away from the upper end of the filament. The horizontal distance between these two ends of the filament increases. The result is that there will be more room for the middle part of the filament to stretch and twist. Therefore the irregular behaviors of the scroll waves become less significant. The effect of thickness of the model doesn't manifest in the study so far. This may be due to the dominant effect of the anisotropy and rotation. Some test runs are current being done using Nz = 63. IV Below is the updated phase diagram for Barkley 2D The boundaries of the regions are different than the diagram shown in Barkley's articles. But the big picture is the same.

20 This is the diagram in Barkley's paper for comparison

21 V Work to be done: 1. Run bidomain model for model size Nx = 255, Ny = 255, Nz = 7. The model size with Nz increase to 15 or 31 is too large that the speed of a 20,000 run is too slow. 2. Complete the monodomain 3D phase diagram for Barkley model. 3. For those show no irregular behavior, run longer time see if they become unstable.