[pastewka]

Why

Once the solves got faster (muGrid #251/#252, #37), the element-wise sensitivity work became the largest item after them. In the 4-rank MI300A run at 512³ (optimization 842 s):

	time
hv sensitivity	207 s	1.7 s per load case and Hessian product
hessian_vector_product (other)	56 s	nodal gather/scatter, material derivatives
sensitivity (objective)	20 s

All of it ran in NumPy on the host. Per load case, that meant four device-to-host copies of the sensitivity fields, and several full-size host temporaries per accumulation, on ~33 M elements per rank.

What changes

These now live in the fields' array module (CuPy on a GPU), in double precision:

• the element density;
• the SIMP derivatives;
• the gradient and Hessian-product accumulators;
• the per-case Hessian cache (g_shear, g_vol, rho_e).

Only the design comes in through the host, and the gradient or Hessian product goes out.

• Homogenization.xp: the fields' array module. set_density accepts device arrays and interpolates on the device.
• NodalElementMap.gather_mean(..., on_device=True) returns the element average on the device, and scatter_mean accepts device arrays.
• The accumulations are now float64 also for single-precision fields; for the nodal design they used to follow the field precision.
• The regularization (~2% of the run) stays on the host.

Memory: the per-case Hessian cache (2 fields per load case) moves from host to device memory. That's the same physical memory on an APU, and on a discrete GPU it's 12 extra scalar fields.

Tests

• The full suite passes on the CPU (249), including the finite-difference checks of gradient and Hessian. There the array module is NumPy, so this checks the arithmetic but not CuPy-specific behaviour, and muTopOpt has no GPU CI.
• I went through the GPU path by hand: no np.asarray on device arrays; the material derivatives use NumPy ufuncs, which dispatch to CuPy; the nodal-map weights are NumPy scalars times device arrays, as before.
• Please check on MI300A: a timing run like the last one. I expect hv sensitivity to drop from ~1.7 s to tens of ms per load case and product, and hessian_vector_product (other) to shrink as well.

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