A Statistical Approach to Surface Metrology for 3D-Printed Stainless Steel

Figures & data

Fig. 1 Illustration of the problem statement. Here, the actual geometries of a small number of 3D-printed steel components, each notionally a flat panel, are provided and the aim is to construct a generative statistical model for the actual geometry of a 3D-printed steel component whose notional geometry is cylindrical.

Table 1 Summary statistics traditionally used in surface metrology; here $z:[0,L]\to \mathbb{R}$ is the profile through the surface of the material and μ is its average or intercept as per (1).

Parameter	Parameter
name	type	Description	Mathematical definition
Rq	Amplitude	Root mean square roughness	Rq $:=\sqrt{\frac{1}{L}{\int }_0^L{[z(x)-\mu ]}^2\mathrm{d}x}$
Ra	Amplitude	Average roughness	Ra $:=\frac{1}{L}{\int }_0^L|z(x)-\mu |\mathrm{d}x$
HSC	Texture	Number of local peaks per unit length	(NA—this depends on definition of a peak)
${\mathrm{R}}_L$	Texture	Ratio of developed length of a profile compared to nominal length	${\mathrm{R}}_L:=\frac{1}{L}{\int }_0^L\sqrt{1+z\prime {(x)}^2}\mathrm{d}x$

NOTE: Adapted from (Thomas 1999, chap. 7 and 8) .

Fig. 2 Laser scan of 3D-printed steel sheet. (a) Photograph of the handheld scanning equipment. (b) Orthographic projection of (a portion of) the scanned sheet, which will be called a panel. The notional thickness of the panel is 3.5mm. [Observe the residual stress induces a slight curvature in the notionally flat panel.]

Fig. 3 Data are conceptualized as a two-dimensional vector field u on an oriented manifold $\mathcal{M}$. The value $u^{(1)}(x)$ represents the height of the upper surface of the material above $x\in \mathcal{M}$, while the value $u^{(2)}(x)$ represents the height of the lower surface below $x\in \mathcal{M}$.

Table 2 Model selection based on the training dataset ${\mathcal{D}}_1$.

Isotropic	Stationary	Noise	Log-likelihood
			# Parameters	Surrogate	Exact	AIC
✓	✓	White	4	21,187	1,041,766	–2,083,524
✓	✓	Smoother	5	21,775	1,154,950	–2,309,890
✓	✓	sm. oscil.	6	22,752	1,208,810	–2,417,608
✓	✗	White	100	24,404	774,539	–1,548,878
✓	✗	Smoother	101	25,900	1,023,561	–2,046,920
✓	✗	sm. oscil.	102	29,118	1,042,737	–2,085,270
✗	✓	White	6	22,005	1,100,358	–2,200,704
✗	✓	Smoother	7	23,166	1,164,952	–2,329,890
✗	✓	sm. oscil.	8	25,151	1,233,936	–2,467,856
✗	✗	White	102	25,385	914,913	–1,829,622
✗	✗	Smoother	103	26,336	1,056,735	–2,113,264
✗	✗	sm. oscil.	104	28,150	1,111,449	–2,222,690

NOTE: The first three columns indicate whether the differential operator $\mathcal{L}$ was isotropic and stationary and the driving noise process Z that was used. The remaining columns indicate the number of free parameters in each model, the approximately maximized values of the surrogate and exact log-likelihood and the Akaike information criterion (AIC). [The log-likelihood and AIC values are rounded to the nearest integer.]

Fig. 4 (a) Laser scan of a section from a real 3D-printed cylinder. (b) Sample from the fitted model $M^{*}$ generated based on a cylindrical manifold. (c) The distribution over the thickness of the wall of the real (blue, shaded) and the simulated (red, solid) 3D-printed cylinder.

Fig. 5 Predicting the outcome of a compressive test on a 3D-printed cylinder, using the fitted generative model $M^{*}$ for the geometry of the cylinder and a finite element simulation of the experiment. (a)–(c) Samples from the generative model for the geometry of the cylinder after the simulated compressive test. (d) and (e) Buckled cylinders, experimentally produced. (Reproduced with permission from Buchanan, Wan, and Gardner 2018).

Thomas, T. R. (1999), Rough Surfaces, World Scientific Publishing Co. Pte. Ltd.

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Buchanan, C., Wan, W. Y. H., and Gardner, L. (2018), “Testing of Wire and Arc Additively Manufactured Stainless Steel Material and Cross-Sections,” in Proceedings of the 9th International Conference on Advances in Steel Structures.

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