The benchmarks for this project may be found in the benchmarks folder.
Using the BUILD_BENCHMARKS option, CMake can be configured to build the
benchmarks instead of the unit tests. They can then be executed by hand or,
for non-MPI tests, sequentially using CTest. Timings are performed using
deal.II's Timer class.
These results were produced on a 2012 Macbook Pro Retina (MacBookPro10,1), which has a 2.6 GHz quad-core Intel Core i7 that supports (only) AVX1 instructions, and has 256 KB L2 Cache per core, 6 MB L3 cache and 16 GB of 1600 MHz DDR3 RAM. Given the age of this machine, it's not going to produce any record-setting performance results. But hopefully it'll provide some coarse indication as to what the performance deficit (or gains) with this implementation are and provide some insights as to whether its worth considering using this project within your own code.
Firstly, the parameters of the assessed benchmark test. Modifications of
step-44 were limited to the assembly loop, which was highly optimised in
its original implementation. Static condensation was not used (meaning that
linear operators were used as part of the linear solve process).
A summary of the various implementations are as follows:
step-44-variant_01a: Replication of "stockstep-44".step-44-variant_01d: Assembly using composite weak forms, with the Piola stress being used for the problem parameterisation (i.e. this is an application of the two-point formulation).step-44-variant_01h: Assembly using composite weak forms, with the Piola-Kirchhoff stress being used for the problem parameterisation (i.e. this is an application of the fully referential formulation).step-44-variant_01i: Assembly using composite weak forms, with the Kirchhoff stress is used for the problem parameterisation (i.e. the same parameterisation as the originalstep-44implementation).step-44-variant_02-energy_functional_sd_02: Assembly using a self-linearizing energy functional weak form in conjunction with symbolic differentiation. The internal energy is calculated by hand, and the external energy is also supplied. From this, all first and second derivatives are calculated automatically. For the symbolic batch optimiser, thelambdaoptimiser type is selected andallavailable optimisation methods are utilised.step-44-variant_02-energy_functional_sd_02a: Identical tostep-44-variant_02-energy_functional_sd_02, except that all SD calculations are cached between timesteps and Newton iterations.step-44-variant_02-energy_functional_sd_03: Identical tostep-44-variant_02-energy_functional_sd_02, except that the high performanceLLVMoptimiser type is selected.step-44-variant_02-energy_functional_sd_03a: Identical tostep-44-variant_02-energy_functional_sd_03, except that it also includes SD calculation caching.step-44-variant_02-residual_view_sd_02: Assembly using a self-linearizing residual weak form in conjunction with symbolic differentiation. The contributions to the residual are provided to the weak form, and from this its linearisation is calculated automatically. For the symbolic batch optimiser, thelambdaoptimiser type is selected andallavailable optimisation methods are utilised.step-44-variant_02-residual_view_sd_02a: Identical tostep-44-variant_02-residual_view_sd_02, except that all SD calculations are cached between timesteps and Newton iterations.step-44-variant_02-residual_view_sd_03: Identical tostep-44-variant_02-residual_view_sd_02, except that the high performanceLLVMoptimiser type is selected.step-44-variant_02-residual_view_sd_03a: Identical tostep-44-variant_02-residual_view_sd_03, except that it also includes SD calculation caching.
The four cases that were studied had the following discretisations:
| Benchmark | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Polynomial order | 1 | 2 | 1 | 2 |
| Quadrature order | 2 | 3 | 2 | 3 |
| Global refinements | 4 | 3 | 5 | 4 |
| Number of active cells | 4096 | 512 | 32768 | 49096 |
| Number of DoFs | 22931 | 18835 | 173347 | 140579 |
The performance ratio is defined as the ratio of the assembly time of the
nominated test versus that of the "manual" or "hand" implementation performed
in step-44-variant_01a.
| Benchmark | Assembly time (s) | Solver time (s) | Total time (s) | Assembly time (% of total) | Performance ratio |
|---|---|---|---|---|---|
| step-44-variant_01a | 4.072 | 250 | 255.5 | 1.59% | 1.00 |
| step-44-variant_01d | 8.876 | 251.2 | 261.5 | 3.39% | 2.18 |
| step-44-variant_01h | 15.26 | 248.7 | 265.5 | 5.75% | 3.75 |
| step-44-variant_01i | 9.224 | 240.4 | 251 | 3.67% | 2.27 |
| step-44-variant_02-energy_functional_sd_02 | 44.04 | 251.2 | 296.8 | 14.84% | 10.82 |
| step-44-variant_02-energy_functional_sd_02a | 47.42 | 259.1 | 308.1 | 15.39% | 11.65 |
| step-44-variant_02-energy_functional_sd_03 | 38.1 | 255.2 | 294.8 | 12.92% | 9.36 |
| step-44-variant_02-energy_functional_sd_03a | 30.61 | 255.4 | 287.5 | 10.65% | 7.52 |
| step-44-variant_02-residual_view_sd_02 | 113.7 | 247.4 | 362.9 | 31.33% | 27.92 |
| step-44-variant_02-residual_view_sd_02a | 109.6 | 256.4 | 367.8 | 29.80% | 26.92 |
| step-44-variant_02-residual_view_sd_03 | 76.27 | 255.2 | 333.3 | 22.88% | 18.73 |
| step-44-variant_02-residual_view_sd_03a | 34.07 | 258.3 | 294 | 11.59% | 8.37 |
| Benchmark | Assembly time (s) | Solver time (s) | Total time (s) | Assembly time (% of total) | Performance ratio |
|---|---|---|---|---|---|
| step-44-variant_01a | 15.5 | 296.3 | 313 | 4.95% | 1.00 |
| step-44-variant_01d | 16.39 | 287.5 | 305.1 | 5.37% | 1.06 |
| step-44-variant_01h | 26.11 | 288.2 | 315.5 | 8.28% | 1.68 |
| step-44-variant_01i | 13.34 | 295.3 | 309.9 | 4.30% | 0.86 |
| step-44-variant_02-energy_functional_sd_02 | 37.39 | 286.6 | 325.3 | 11.49% | 2.41 |
| step-44-variant_02-energy_functional_sd_02a | 37.4 | 287.8 | 326.5 | 11.45% | 2.41 |
| step-44-variant_02-energy_functional_sd_03 | 35.52 | 288.9 | 325.7 | 10.91% | 2.29 |
| step-44-variant_02-energy_functional_sd_03a | 30.09 | 288.7 | 320 | 9.40% | 1.94 |
| step-44-variant_02-residual_view_sd_02 | 65.52 | 278.9 | 345.9 | 18.94% | 4.23 |
| step-44-variant_02-residual_view_sd_02a | 58.39 | 280.2 | 340.1 | 17.17% | 3.77 |
| step-44-variant_02-residual_view_sd_03 | 58.73 | 298.3 | 358.6 | 16.38% | 3.79 |
| step-44-variant_02-residual_view_sd_03a | 29.48 | 312.9 | 343.7 | 8.58% | 1.90 |
| Benchmark | Assembly time (s) | Solver time (s) | Total time (s) | Assembly time (% of total) | Performance ratio |
|---|---|---|---|---|---|
| step-44-variant_01a | 35.11 | 5920 | 5964 | 0.59% | 1.00 |
| step-44-variant_01i | 83.72 | 5918 | 6010 | 1.39% | 2.38 |
| step-44-variant_02-energy_functional_sd_03a | 262.8 | 5852 | 6123 | 4.29% | 7.49 |
| step-44-variant_02-residual_view_sd_03a | 287.9 | 5897 | 6194 | 4.65% | 8.20 |
| Benchmark | Assembly time (s) | Solver time (s) | Total time (s) | Assembly time (% of total) | Performance ratio |
|---|---|---|---|---|---|
| step-44-variant_01a | 131.4 | 6998 | 7135 | 1.84% | 1.00 |
| step-44-variant_01i | 106.9 | 6781 | 6894 | 1.55% | 0.81 |
| step-44-variant_02-energy_functional_sd_03a | 254.6 | 7092 | 7352 | 3.46% | 1.94 |
| step-44-variant_02-residual_view_sd_03a | 250.7 | 7170 | 7427 | 3.38% | 1.91 |
Considering the tabulated data, here are some preliminary observations and comments on the comparative performance of the current implementation:
- The assembly time of the four "hand implementation" variants,
01a,01d,01hand01iare relatively similar. The01dand01hvariants have some performance deficit due to the additional pull-back operations (some could) be removed with a re-statement of the constitutive laws. The test01his quite a bit more expensive due to the push-forward of shape functions and additional split of the material and geometric elastic tangents. - Symbolic computations using an energy functional or a residual view are always
more costly than hand calculations. Using the
LLVMJIT compiler in conjunction with symbolic expression caching (i.e. the03avariant) always increases the overall performance. - For polynomial order
2, the weak form implementation01iis actually 15%-20% quicker than the original implementation. This is likely due to the additional parallelisation that vectorisation (across the quadrature loop) provides. One might speculate than the benefit might be greater at even higher FE degrees. - For polynomial order
2, the performance deficit of the automatic linearisation strategies is minimised at around1.9xthe manual implementation. The increase in performance of the03abenchmarks when compared to the lowest order FE discretisation is quite substantial. This suggests that increasing the polynomial order offsets the costs of symbolic computations. This aligns with the expectations from the implementation, as the symbolic calculations are done on a quadrature-point level and is then more heavily reused as the number of quadrature points per cell increases.
In the end, there is certainly some overhead to using this (symbolic) weak forms implementation. It is the judgement of the user as to whether the trade off of raw performance versus any convenience that it provides is worthwhile.