Enabling Detailed, Biophysics-Based Skeletal Muscle Models on HPC Systems

Figures

• 

  Figure 1. (Left) Schematic view of a 3D muscle domain that contains a given number of nx × ny muscle fibers per 3D partition, ex × ey × ez finite elements for the 3D model (7a), and sx nodes per fiber for (7b) and (7c). (Right) Schematic view of the multi-scale time stepping scheme based on a Godunov splitting of the monodomain equation.

• 

  Figure 2. Visualization of pillar-like (Left) and cuboid (Right) domain decomposition approaches. Both depicted approaches partition the same domain into 16 subdomains with px, py, and pz subdivisions in x-, y-, and z-direction, respectively.

• 

  Figure 3. (A) Relative error dependency on the number K of 0D time steps in [0, dt1D]. The error of Euler's and Heun's method shows the expected O(K-1) and O(K-2) behavior. (B) Dependency of the runtime on the required accuracy for explicit Euler and Heun. We varied the time step dt0D between 5·20 and 5·2−12μs for Euler and between 5·20 and 5·2−11μs for Heun.

• 

  Figure 4. (A) Relative error dependency on the 1D time step size dt1D. The error of the Godunov- and Strang-splitting scheme shows the expected O(dt1D) and O(dt1D2) behavior, respectively. (B) Efficiency of different splitting schemes. Each scheme is performed for dt1D = 0.5, 1, 2 and 4 μs.

• 

  Figure 5. Comparison of the runtime for different linear solvers. A single fiber is considered for the time t ∈ [0, 3 ms].

• 

  Figure 6. Runtime for a simulated time interval t ∈ [0, 1 ms] with a varying number of elements per fiber. 2 × 2 × 2 3D elements, 6 × 6 1D fibers. Solid lines: baseline implementation, Dashed lines: implementation with improvements of this paper.

• 

  Figure 7. Strong scaling measurements—Experiment #1: Scenario with 12 × 12 × 8 = 1,152 3D elements, 24 × 24 fibers (i.e., 13,824 1D elements), the cube shaped partitioning strategy computed with 1–4 nodes with 12 processes per node. (Top) Total runtime and runtimes of solvers for the 0D, 1D and 3D problem, Tp, (solid lines); projected runtimes for optimal scaling, Tp, opt = T12·p/12 (dashed thin lines). (Bottom) Parallel efficiency Ep = Tp, opt/Tp.

• 

  Figure 8. Weak scaling measurements–experiment #2: Total runtime as well as individual runtimes (solver for the 0D, 1D, and 3D problem) for the cuboid partitioning (solid lines) and pillar-like partitions (dashed lines). The error bars indicate estimated standard deviation of runtimes on all involved processes of a single simulation run.

• 

  Figure 9. Weak scaling measurements—experiment #3: Runtimes for different model components. The results for cuboid and pillar-like partitions are depicted by solid and dashed lines, respectively. Different runtime components are encoded in colors, i. e., the total runtime in black, 0D solver in yellow, the 1D solver in red, the 3D solver in green, the interpolation in light blue and the homogenization in dark blue.

• 

  Figure 10. Weak scaling measurements—experiment #3: Total memory consumption per process at the end of the runtime. The total memory consumption is depicted in magenta and the average number of 3D ghost layer elements per process in black. Again, the solid lines represent cuboid partitioning and the dashed line piller-like partitioning.

• 

  Figure 11. Dependency between runtime and partition shape—experiment #4: Runtime in dependence on the average boundary area of the partitions. We show the accumulated total computational time and the runtimes of the sub-problems as in the previous studies.

• 

  Figure 12. The discretized 1D muscle fibers are rendered as continuous tubes to show the characteristics and implicit geometry (with the distinct fiber directions of the superficial and deep part of the Tibialis Anterior) of the individual strands. The color coding shows the distribution of parameter values along the fibers (local membrane potential; red: low, blue: high).