Performance¶

This page documents the scaling behavior of torch-fem on canonical benchmark problems and shows how to reproduce the results on your own hardware. All benchmarks measure the same three phases per run:

• Setup — mostly computing the sparsity pattern.
• Forward solve — assembly and sparse linear system solve.
• Backward solve — reverse-mode AD through the solve via autograd.

Peak RAM is tracked by polling the child-process RSS every 50 ms; peak VRAM is sampled at the driver level.

Structural benchmark: cube extension¶

A unit cube is subjected to one-dimensional extension:

• Discretised with \(N \times N \times N\) linear hexahedral (Hexa1) elements (\(3N^3\) degrees of freedom).
• Material: isotropic linear elasticity, \(E = 1000\), \(\nu = 0.3\).
• Boundary conditions: fully clamped at \(x = 0\); prescribed displacement \(u_x = 0.1\) at \(x = 1\).
• Backward pass: gradient of u.sum() w.r.t. nodal forces.

Thermal benchmark: SIMP heated slab¶

A quasi-2D heated slab with SIMP-penalized conductivity, mirroring the thermal-mesh problem of the mosaic benchmark suite, which compares differentiable solvers (including torch-fem) across frameworks:

• Domain \([0,2] \times [0,1] \times [0,1]\), discretised with a single layer of \(N \times N/2 \times 1\) linear hexahedral (Hexa1) elements — one temperature degree of freedom per node.
• Material: isotropic conductivity with SIMP penalization \(k(\rho) = k_\min + (k_\max - k_\min)\,\rho^3\), \(k_\min = 10^{-3} k_\max\), at uniform density \(\rho = 0.5\).
• Boundary conditions: \(T = 0\) at \(x = 0\); uniform heat flux \(Q = 1\) on the face at \(x = 2\).
• Backward pass: gradient of the thermal compliance \(C = \oint_{\Gamma_N} q_n T \, d\Gamma\) w.r.t. the per-element densities \(\rho\) — the adjoint sensitivity used in topology optimization.

Hyperelastic benchmark: Neo-Hookean large stretch¶

A unit cube is stretched to twenty times its original length, mirroring the large stretch example:

• Discretised with \((N-1) \times (N-1) \times (N-1)\) linear hexahedral (Hexa1) elements (\(3N^3\) degrees of freedom, \(N\) odd).
• Material: Neo-Hookean strain energy \(\psi(\mathbf{C}) = \frac{\mu}{2}(\textrm{tr}(\mathbf{C}) - 3) - \mu \ln J + \frac{\lambda}{2} (\ln J)^2\) with \(E = 1000\), \(\nu = 0.3\).
• Boundary conditions: \(u_x = 0\) at \(x = 0\); prescribed displacement \(u_x = 19\) at \(x = 1\) (stretch \(\lambda = 20\)); symmetry constraints on the center planes.
• Load applied in 21 increments that are geometric in the stretch, with full Newton iterations (nlgeom=True) per increment.
• Backward pass: gradient of the total reaction force w.r.t. the Lamé parameters \((\mu, \lambda)\) — the adjoint used in material parameter calibration.

The forward solution matches the analytical uniaxial Neo-Hookean response.

Reproducing the results¶

The scripts live in benchmarks/ at the repository root.

1. Run the benchmark:

# All benchmarks on CPU (default)
python benchmarks/run.py

# Structural benchmark on CUDA
python benchmarks/run.py -problem cube -device cuda --label rtx5090_cuda --hardware "RTX 5090"

# Thermal benchmark
python benchmarks/run.py -problem thermal --label m1_pro_cpu --hardware "Apple M1 Pro"

The label identifies the machine; results are written to benchmarks/results/<problem>_<label>.json.

2. Regenerate the plots:

python benchmarks/plot.py

This reads all JSON files in benchmarks/results/, groups them by problem, and writes the timing, backward, and RAM plots to docs/images/benchmark_*.png (structural) and docs/images/benchmark_thermal_*.png (thermal).