Which method when

The smoother has three methods, Laplace smoothing, Winslow smoothing and an optimizer, and a schedule that combines Laplace and the optimizer. Each runs here on the same inputs with the same constraints, so the differences are the method’s own.

The four runs

method	call	runs	at most
Laplace only	laplace(passes=100, w=0.95)	one Laplace stage	100 Laplace passes
Winslow only	winslow(sweeps=200)	one Winslow stage at its defaults	200 Winslow sweeps
Optimizer only	optimize(iters=200)	one optimizer stage at the default delta=0.01	200 optimizer iterations
Combined	suggest_schedule(quality)	user_schedule() = 10 × [laplace(passes=3, w=0.97) + optimize(iters=20)] + 2 × laplace_local(passes=5, w=0.97); when a quad starts inverted, optimizer passes with a wide-to-narrow delta come first (untangle_schedule() = optimize(iters=60, delta=v) for v in 1, 0.3, 0.1, 0.03, 0.01)	30 Laplace passes, 200 optimizer iterations, 10 local Laplace passes; plus 300 optimizer iterations when a quad starts inverted

Side by side

One row per input, every panel on the row at one scale. The L-shape is a concave outline meshed as one block; its reflex corner is the point (1, 1).

worst-corner scaled J≤ 01input boundaryouter edge of a designed layer

annulus

input
min J 0.24

Laplace
min J 0.99

Winslow
min J 0.94

Optimizer
min J 0.94

Combined
min J 0.94

free disk

input
min J 0.27 · area +0.0 %

Laplace
min J 0.72 · area +0.0 %

Winslow
min J 0.85 · area +0.0 %

Optimizer
min J 0.86 · area +0.0 %

Combined
min J 0.86 · area +0.0 %

tangled square

input
min J -0.98 · 4 inverted

Laplace
min J 1.00 · 0 inverted

Winslow
min J 1.00 · 0 inverted

Optimizer
min J -0.98 · 4 inverted

Combined
min J 1.00 · 0 inverted

sphere

input
min J -0.32 · 3 inverted

Laplace
min J 0.83 · 0 inverted

Winslow
min J 0.87 · 0 inverted

Optimizer
min J 0.87 · 0 inverted

Combined
min J 0.87 · 0 inverted

graded layer

input
min J 0.46 · layer as meshed

Laplace
min J 0.86 · layer +229 %

Winslow
min J 0.86 · layer +207 %

Optimizer
min J 0.32 · layer +584 %

Combined
min J 0.36 · layer +746 %

L-shape

input
min J -0.32 · 10 inverted (9 off the corner)

Laplace
min J -0.95 · 6 inverted (5 off the corner)

Winslow
min J -0.68 · 1 inverted, at the corner

Optimizer
min J -0.35 · 9 inverted (8 off the corner)

Combined
min J -0.72 · 3 inverted (2 off the corner)

Scoreboard

The shape and size rows count the inputs (of 6) on which each method is best; the others are one input’s reading. Best in bold; readings within 0.1 % of each other tie, and a tie counts for each method.

	reading	Laplace only	Winslow only	Optimizer only	Combined schedule
shape	min scaled J	2	2	3	3
shape	mean phi	0	4	3	5
size	size ratio max	1	4	1	2
size	CV of area	3	1	1	3
tangled input	tangled squares repaired (of 8)	8	8	1	8
concave	L-shape, inverted quads left (input 10)	6	1	9	3
free boundary	area change, free disk (%)	+0.0	+0.0	+0.0	+0.0
grading	graded layer, thickness change (%)	+229	+207	+584	+746
cost	seconds per run, mean	0.03	0.15	0.39	0.44

What this shows

• Shape. On the 4 inputs every method leaves valid, the optimizer (alone or combined) has the lowest mean phi, the quantity it minimises. Winslow comes within 0.001 of it on the annulus and sphere, in 56 % of its time. By min scaled J, Laplace wins on the annulus and graded layer, the optimizer on the free disk and sphere: the scaled J reads angles only.

• Concave outline. The L-shape’s Coons patch starts with 10 inverted quads. Laplace leaves 6, the optimizer alone 9, the combined schedule 3; Winslow leaves 1, only at the reflex corner, where two quads share a 270-degree turn.

• Size. No method evens element size everywhere: Laplace has the lowest spread of area (CV) on the annulus, free disk and L-shape, another method on the tangled square, sphere and graded layer.

• Tangled squares. Of 8 tangled squares, Laplace repaired 8, Winslow 8, the optimizer alone 1 (at the default delta it stops after 1 iteration), the combined schedule 8.

• Free boundary. The disk has boundary="tangent": a boundary vertex slides along the boundary and never moves inward, so the area is kept (Laplace +0.0 %, Winslow +0.0 %, optimizer +0.0 %). The optimizer works there too: it runs 200 iterations to min scaled J 0.857, against a frozen boundary’s 0.857 and boundary="slide"’s 0.856. With boundary="free" Laplace pulls the boundary onto its chords (-15.7 %) and the optimizer lets the rim cells grow (+61.7 %). Geometry you know goes on the boundary as OnCurve (Constraints and projection).

• Grading. Every method thickens the graded layer: Laplace +229 %, Winslow +207 % (its guard stopped it after 3 sweeps), the optimizer +584 %; see Boundary layers and grading.

• Cost. A Laplace pass costs 0.4 ms, a Winslow sweep 0.9 ms, an optimizer iteration 4.3 ms with its line search. Medians over the inputs; single runs on a laptop, running serial.

Every reading, per input

annulus

reading	input	Laplace only	Winslow only	Optimizer only	Combined schedule
min scaled J	0.238	0.986	0.944	0.940	0.940
mean scaled J	0.812	0.996	0.987	0.986	0.986
mean phi	4.915	1.504	1.447	1.446	1.447
size ratio max	1.88	1.11	1.07	1.08	1.08
CV of area	0.350	0.295	0.419	0.424	0.426
area change (%)	+0.0	+0.0	+0.0	+0.0	+0.0
inverted	0	0	0	0	0
seconds	–	0.04	0.13	0.44	0.39

free disk

reading	input	Laplace only	Winslow only	Optimizer only	Combined schedule
min scaled J	0.266	0.725	0.855	0.857	0.857
mean scaled J	0.863	0.970	0.979	0.978	0.978
mean phi	3.522	1.604	1.063	1.060	1.059
size ratio max	1.72	1.31	1.45	1.41	1.41
CV of area	0.477	0.316	0.695	0.643	0.650
area change (%)	+0.0	+0.0	+0.0	+0.0	+0.0
inverted	0	0	0	0	0
seconds	–	0.04	0.19	0.71	0.41

tangled square

reading	input	Laplace only	Winslow only	Optimizer only	Combined schedule
min scaled J	-0.983	0.999	1.000	-0.983	1.000
mean scaled J	0.832	1.000	1.000	0.832	1.000
mean phi	1.1e+04	1.001	1.000	1.2e+11	1.000
size ratio max	4.34	1.02	1.00	4.33	1.00
CV of area	0.283	0.019	0.000	0.283	0.000
area change (%)	+0.0	+0.0	+0.0	+0.0	+0.0
inverted	4	0	0	4	0
seconds	–	0.03	0.04	0.00	0.26
tangled squares repaired (of 8)	0	8	8	1	8

sphere

reading	input	Laplace only	Winslow only	Optimizer only	Combined schedule
min scaled J	-0.319	0.826	0.868	0.868	0.868
mean scaled J	0.787	0.973	0.973	0.973	0.973
mean phi	3.786	1.067	1.059	1.060	1.060
size ratio max	3.56	1.31	1.26	1.24	1.24
CV of area	0.468	0.234	0.204	0.193	0.193
area change (%)	+0.0	+0.7	+0.6	+0.7	+0.7
inverted	3	0	0	0	0
seconds	–	0.04	0.36	0.44	0.70

graded layer

reading	input	Laplace only	Winslow only	Optimizer only	Combined schedule
min scaled J	0.465	0.862	0.858	0.317	0.358
mean scaled J	0.915	0.995	0.994	0.925	0.903
mean phi	18.544	7.490	8.349	4.070	3.997
size ratio max	3.61	1.97	1.92	2.55	2.49
CV of area	0.690	0.547	0.564	0.397	0.368
area change (%)	+0.0	+0.0	+0.0	+0.0	+0.0
inverted	0	0	0	0	0
seconds	–	0.03	0.01	0.74	0.74
layer thickness change, max (%)	0	229	207	584	746

L-shape

reading	input	Laplace only	Winslow only	Optimizer only	Combined schedule
min scaled J	-0.316	-0.948	-0.678	-0.352	-0.722
mean scaled J	0.806	0.832	0.887	0.806	0.818
mean phi	2.0e+12	1.0e+12	2.471	4.5e+11	768.719
size ratio max	8.06	3.12	2.24	9.79	4.29
CV of area	0.530	0.483	0.493	0.538	0.515
area change (%)	+0.0	+0.0	+0.0	+0.0	+0.0
inverted	10	6	1	9	3
seconds	–	0.03	0.17	0.01	0.14
inverted away from the reflex corner	9	5	0	8	2

Trade-offs

method	good at	weak at	cost	use when
Laplace only	cheap; never adds an inverted quad; best min scaled J on simple planar patches; repaired every tangled square here	blind to phi; folds stay on a concave outline (6 left on the L-shape); relaxes grading	0.4 ms a pass	a cheap first pass, or a valid result is enough
Winslow only	a concave outline: only the reflex-corner quad stays folded (1 left on the L-shape); the optimizer’s shape on valid, ungraded input in 56 % of its time; repaired every tangled square	erases grading once its guard lets it run (+778 % with max_sj_drop=1; pair with KeepLayer); can drift on a closed surface over many sweeps	0.9 ms a sweep	Laplace fails: a concave outline, a thin curved block (Winslow smoothing)
Optimizer only	lowest phi; weights and constraints act inside it	stalls on a tangle at the default delta; grows a "free" boundary; undoes grading	4.3 ms an iteration	a valid input whose shape is the goal
Combined	optimizer passes with a wide-to-narrow delta first, then its shape	the optimizer’s cost, grading and boundary behaviour	0.44 s a run (mean)	the default; any damaged or unknown input

In practice

• Attach geometry first (Constraints and projection), or use boundary="slide" when there is none: "free" shrinks a convex boundary under Laplace and grows it under the optimizer; "tangent" holds it for every method.

• Start from s.suggest(): when a quad starts inverted it leads with optimizer passes over a wide-to-narrow delta.

• Add a Winslow stage by hand where Laplace fails: a concave outline, a thin curved block (Winslow smoothing).

• Protect designed grading with KeepLayer (Boundary layers and grading); Winslow needs it most.

• Judge by more than one reading: the scaled Jacobian is blind to edge ratio, size and grading.

References

• K. Mittal, P. Fischer, “Mesh smoothing for the spectral element method”, J. Sci. Comput. 78(2), 1152-1173 (2019), doi:10.1007/s10915-018-0812-9. Used for: The approach this smoother follows for spectral-element meshes. Cited as a record; its method is not summarised here.

• K. Mittal, “Spectral element mesh generation and improvement methods”, M.S. thesis, University of Illinois at Urbana-Champaign (2016), hdl:2142/95509. Used for: With the paper above, the mesh-improvement approach this smoother follows for spectral-element meshes.

• S. A. Canann, J. R. Tristano, M. L. Staten, “An approach to combined Laplacian and optimization-based smoothing for triangular, quadrilateral, and quad-dominant meshes”, 7th International Meshing Roundtable, Sandia National Laboratories (1998); no DOI. Used for: A combination of constrained Laplacian smoothing with an optimization-based smoother for triangle, quad and mixed meshes: the idea of the combined schedule.