Research-Stack/3-Mathematical-Models/AMMR/WAVE_OVERHANGS_ADAPTER.md

Wave Overhangs Adapter

Purpose

This repository is treated as the wavefront toolpath-planning branch of the pressure / cavitation / vibration / diffusion catalog.

The adapter couples three layers:

wavefront path planning
    -> heat-2D thermal residue simulation
    -> calibrated warp-risk envelope

Core role:

unsupported overhang region
    -> wavefront / level-set path planning
    -> FDM deposition path
    -> thermal residue field
    -> warping or stable overhang

This file is a technical integration contract, not a claim that the slicer alone solves polymer mechanics.

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1. Native objects

Let:

Symbol	Meaning
Omega_o	unsupported overhang region to fill
partial Omega_s	supported / perimeter seed boundary
M(x,y)	printable geometry mask
T(x,y)	wavefront arrival / distance field
lambda	line spacing / wavelength
Gamma_k	kth generated wave track
x_nozzle(t)	time-parametrized nozzle trajectory
u(x,y,t)	thermal residue / cooling field
alpha(x,y)	effective diffusivity field
f(x,y,t)	deposition, reheat, and cooling source term

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2. Wavefront generation layer

A wavefront arrival field can be approximated by an Eikonal equation:

|\nabla T(x,y)| = \frac{1}{c(x,y)}

where c(x,y) is a geometry-dependent propagation weight.

Boundary condition:

T|_{\partial\Omega_s}=0

Blocked/outside region:

(x,y)\notin\Omega_o \Rightarrow M(x,y)=0

Wave tracks are level sets:

\Gamma_k = \{(x,y): T(x,y)=k\lambda\}\cap\Omega_o

A narrow-neck pruning gate should prevent fragile branches:

w_{neck}(x,y)<w_{min}\Rightarrow M(x,y)=0

where w_min corresponds to the slicer Minimum wave width setting.

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3. Toolpath ordering layer

The same level-set geometry can be emitted under different ordering policies:

Policy	Model interpretation	Expected thermal effect
Monotonic	one consistent direction per line	longer cooling interval between neighbors
Zig Zag	connected depth-first back-and-forth path	lower travel, higher local heat buildup risk
Smart	start from better-supported end	lower unsupported-start and anchor-risk penalty

Define an ordering functional:

\pi: \{\Gamma_k\}\rightarrow x_{nozzle}(t)

where pi maps unordered wavefront curves into a nozzle-time trajectory.

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4. Heat-2D coupling layer

A generated wave toolpath is treated as a moving thermal/material source:

f_{nozzle}(x,y,t)=Q_n\exp\left(-\frac{\|\mathbf{x}-\mathbf{x}_{nozzle}(t)\|^2}{2\sigma_n^2}\right)

Then solve the heat-2D compatible equation:

\partial_t u = \nabla\cdot(\alpha(x,y)\nabla u)+f_{nozzle}+f_{reheat}+f_{cool}

Cooling sink approximation:

f_{cool}(x,y,t)=-h_f(u-u_{air})

Reheat source from subsequent layers:

f_{reheat}(x,y,t)=\sum_{\ell=1}^{N}Q_{\ell}\exp\left(-\frac{\|\mathbf{x}-\mathbf{x}_{\ell}(t)\|^2}{2\sigma_{\ell}^{2}}\right)

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5. Slicer parameter bridge

WaveOverhangs setting	Adapter parameter	Effect
line spacing	lambda	wavefront interval / thermal line spacing
line width	sigma_n, bead width	source radius and material overlap
flow ratio	Q_n	deposited material and heat amplitude
print speed	`	dx_nozzle/dt
fan speed	h_f	cooling sink strength
perimeter overlap	A_anchor, boundary coupling	anchor conduction and edge attachment
minimum wave width	w_min	branch pruning threshold
monotonic / zig-zag / smart	pi	path-ordering and reheat pattern

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6. Warp-risk observables

6.1 Thermal-gradient proxy

G_T(x,y,t)=\|\nabla u(x,y,t)\|

Large gradients imply differential contraction risk.

6.2 Hot-time exposure

t_{hot}(x,y)=\int \mathbf{1}_{u(x,y,t)>T_g}\,dt

where T_g is the glass-transition / mobility threshold proxy.

6.3 Reheat activation proxy

R_{reheat}(x,y)=\max_t \mathbf{1}_{u(x,y,t)>T_g}

This estimates where earlier strands may re-enter a mobile polymer state.

6.4 Anchor distance penalty

A_{anchor}(x,y)=\exp\left(-\frac{d(x,y,\partial\Omega_s)}{\ell_a}\right)

where ell_a is an anchor coupling length.

6.5 Span-size proxy

C_{span}\propto L_{unsupported}^{2}

Large unsupported spans should be considered high-risk until calibrated print data says otherwise.

6.6 Curl-risk score

C_{curl}=w_1\max G_T+w_2\max R_{reheat}+w_3\max t_{hot}+w_4C_{span}-w_5\max A_{anchor}

Interpretation:

higher C_curl -> higher predicted warp/failure risk
lower C_curl  -> better candidate for unsupported wave overhang

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7. Calibration receipts

Each print test should log:

Receipt	Required data
geometry receipt	STL/model name, overhang area, unsupported span, holes/concavity flags
slicer receipt	line spacing, width, flow ratio, pattern, speed, fan, overlap, min width
material receipt	polymer, filler, nozzle temp, bed temp, chamber temp, cooling duct
simulation receipt	grid size, timestep, alpha field, source amplitude, boundary conditions
outcome receipt	pass/fail, max curl, detachment, bead break, surface roughness, photo/hash

Minimum calibration target:

C_{curl}\uparrow \Rightarrow P(failure)\uparrow

Do not promote the score beyond CALIBRATED_ENGINEERING_DELTA until this monotonic relationship is measured over multiple geometries and materials.

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8. Test cases

Test A: single wave stripe

Purpose: validate moving-source thermal trail.

Expected behavior:

• smooth trail behind nozzle,
• peak temperature decays after source passes,
• stronger fan coefficient shortens hot lifetime.

Test B: adjacent wave lines

Purpose: test line spacing / width / flow-ratio coupling.

Expected behavior:

• tighter spacing increases overlap,
• higher flow ratio increases heat accumulation,
• slower print speed increases dwell and bonding but may reheat earlier lines.

Test C: monotonic vs zig-zag ordering

Purpose: compare ordering policies.

Expected behavior:

• monotonic gives neighboring lines more cooling time,
• zig-zag lowers travel but can increase local heat accumulation,
• smart ordering should reduce unsupported-start risk.

Test D: large unsupported span

Purpose: detect when the wave strategy should fall back to support.

Expected behavior:

• larger spans increase curl-risk proxy,
• two-sided cooling reduces vertical-gradient risk if modeled in a layered extension,
• fiber-filled material proxy should use higher conductivity, lower expansion risk, and higher stiffness in downstream mechanical model.

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9. Failure gates

Gate	Trigger	Action
thermal-gradient gate	max G_T > theta_G	slow print / raise fan / change path order
hot-time gate	max t_hot > theta_t	increase line interval cooling time
span gate	L_unsupported > theta_L	fall back to support / bridge / hybrid support
anchor gate	A_anchor < theta_A at line start	use smart start or add perimeter overlap
branch gate	w_neck < w_min	split/prune branch before propagation

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10. Scope warning

This adapter does not claim wave overhangs are fully solved by heat diffusion.

Pure heat diffusion does not model:

• bead sag,
• nozzle pressure deformation,
• shape-memory polymer behavior,
• viscoelastic stress relaxation,
• crystallization,
• full 3D strand mechanics.

Correct role:

wavefront toolpath geometry -> thermal afterimage -> calibrated warping-risk proxy

Downstream mechanical models should consume this field instead of being hidden inside the diffusion solver.

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11. Catalog placement

This branch belongs beside:

pressure / cavitation / vibration / diffusion
    -> heat-2D residue model
    -> passive fluidic geometry
    -> rocket cooling geometry
    -> wavefront toolpath geometry

The shared abstraction is:

geometry + source + diffusion/propagation law -> residual field -> failure envelope