Biomedical Engineering Reference
In-Depth Information
Table 1 Computation timings for some of the models considered in the convergence analysis. All
computations were performed on a single core of an Intel Core i7-930
Model
DOF
Matrix assembly (ms)
Linear solver (ms)
Total (ms)
Beam Tet10, 97 El.
714
27.36
10.13
37.49
Beam Tet10, 712 El.
4002
167.9
41.07
208.97
Beam Tet10, 1810 El.
9333
432.89
129.38
562,27
Beam Tet4, 1810 El.
1461
57.52
7.31
64.83
Beam Tet4, 8990 El.
5988
122.89
49.17
172.06
Beam Tet4, 19128 El.
11982
260.84
130.97
391.81
Cube Tet10, 577 El.
3021
151.17
30.25
181.42
Cube Tet10, 1025 El.
5172
267.29
64.35
331.64
Cube Tet10, 3128 El.
14553
774.21
313.9
1088.11
Cube Tet4, 3128 El.
2100
44.6
12.84
57.44
Cube Tet4, 11512 El.
6909
153.7
72.29
225.99
Cube Tet4, 25117 El.
14925
345.59
215.79
561.38
In contrast, the 10
10 cm cube is just stretched by 30% of its length. The
reference solution for all convergence plots is computed using the corotated FE
formulation with linear tetrahedral on a very high resolution mesh.
As expected, the quadratic formulation performs better for high resolution
meshes. The breakdown of the convergence rate for the linear formulation due to
volume locking is clearly visible in the cube stretching simulation. However, it is
interesting to note that in both examples the quadratic formulation outperforms the
lower order mesh even if few elements are used. For the beam bending simulation,
the quadratic mesh with 714 degrees of freedom (DOF) achieves a slightly better
accuracy as the linear mesh with 5988 DOF. In the cube stretching simulation, the
453DOF quadratic model achieves a better accuracy than all models with linear
shape function that were analyzed.
The computational effort per DOF depends primarily on two variables. For
higher resolution meshes, the linear solver is the most computational expensive
component of the algorithm. The solution time for the linear system of equations is
directly related to the number of DOF. Additionally, the computation time is
heavily influenced by the number of polar decompositions that have to be
performed for each time step. For the quadratic formulation we need four polar
decompositions per element, whereas the linear version needs only one. However, a
linear mesh typically has more than four times as many elements as a quadratic
mesh for the same number of DOF. Thus, we can expect that the linear and the
quadratic method have the same computational complexity per DOF.
Table
1
shows the execution time of the quadratic FEM and the linear corotated
FEM presented by Nesme et al. [
13
] for various model sizes. All computations were
performed on a single core of an Intel i7-930 CPU. It can be seen that the linear
formulation is approximately two times faster than the quadratic method for the
same numbers of DOF. Most of this discrepancy can be attributed to the faster polar
decomposition and matrix assembly of the linear version. This is due to fact that our
prototyping code isn't as optimized as the linear reference implementation.
10
