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arxiv: 2604.21250 · v1 · submitted 2026-04-23 · 🧮 math-ph · math.MP· physics.app-ph

How it cools? Studying the heat flow out of a semi-infinite slab in welding: An analytical approach

Fawzi Aly , Alex Kitt , Luke Mohr This is my paper

Pith reviewed 2026-05-08 13:51 UTC · model grok-4.3

classification 🧮 math-ph math.MPphysics.app-ph
keywords semi-infinite slabheat transferLaplace transformFourier seriesNewton's law of coolingweldingadditive manufacturinganalytical solutions
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The pith

Closed-form solutions for temperature profiles in semi-infinite slabs are derived using Laplace transforms and Fourier series with Newton's cooling boundary conditions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops exact analytical expressions for how heat dissipates from a semi-infinite slab during welding and additive manufacturing, incorporating surface cooling via Newton's law. Existing approximations like Rosenthal's solutions cannot readily include cooling effects, finite-time transients, or certain source shapes without heavy computation. The authors apply two independent techniques, Laplace transform inversion and Fourier series expansion, and prove they produce identical results for both transient and steady-state cases across Gaussian, ellipsoidal, double-ellipsoidal, and switched heat sources. These closed forms allow direct evaluation of temperature fields, which supports faster prediction of stresses and distortions while enabling synthetic data generation for machine-learning models. The work positions the solutions as a scalable alternative to purely numerical simulations for process optimization.

Core claim

By applying the Laplace transform and Fourier series expansions to the heat equation in a semi-infinite domain with convective cooling at the surface, closed-form solutions are obtained for both transient and steady-state temperature distributions under Gaussian, ellipsoidal, double-ellipsoidal, and time-dependent on/off heat sources, with direct numerical comparisons confirming agreement.

What carries the argument

Dual Laplace transform and Fourier series solution of the heat conduction equation subject to Newton's law of cooling boundary condition, yielding explicit temperature expressions for multiple source models.

If this is right

  • Direct evaluation of temperature fields without spatial discretization reduces computational cost for repeated thermal calculations in manufacturing design.
  • Closed-form expressions enable generation of large synthetic datasets to train machine-learning predictors of heat distribution.
  • Time-dependent source solutions allow modeling of pulsed or intermittent welding processes and their effect on cooling.
  • Explicit dependence on source parameters supports sensitivity analysis for minimizing residual stresses and cracking.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The equivalence proof between Laplace and Fourier routes suggests hybrid schemes could handle more complex source motions or material properties analytically in limited regions.
  • The same transform techniques could be adapted to model heat flow in other semi-infinite or layered manufacturing processes such as laser cladding or thermal spraying.
  • Quick analytical estimates might be embedded in real-time process control loops for additive manufacturing to adjust power or speed on the fly.

Load-bearing premise

The semi-infinite slab geometry combined with Newton's law of cooling boundary conditions accurately captures heat flow in real welding and additive manufacturing.

What would settle it

Temperature readings from embedded sensors in a thick metal plate under a controlled moving heat source that deviate systematically from the analytical predictions would show the model does not capture the actual physics.

Figures

Figures reproduced from arXiv: 2604.21250 by Alex Kitt, Fawzi Aly, Luke Mohr.

Figure 1
Figure 1. Figure 1: FIG. 1: Comparison between heat profile evolution for view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: This figure illustrates a semi-infinite slab (gray) view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: A schematic of the Bromwich contour integral view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Comparative analysis of temperature profile evo view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Contour plot of integrand of ( view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: This figure compares two approaches for view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: This figure compares temperature profiles derived from the implementation of the ILT and truncated FS view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: A comparison of temperature profile evolution view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: This figure illustrates that our cooling model, view at source ↗
read the original abstract

Additive manufacturing and welding processes are highly sensitive to heat dissipation, where improper thermal management leads to residual stresses, distortions, and cracking. Existing heat transfer models, such as Rosenthal's solutions, fail to handle finite 3D geometries, cooling effects, or transient behavior, limiting their accuracy. We overcome these limitations by developing an analytical framework that incorporates cooling boundary conditions mimicking Newton's Law of Cooling. Using two different and proven-equivalent approaches, Laplace transform and Fourier series, we derive closed-form solutions for transient and steady-state temperature profiles under various heat sources, including Gaussian, ellipsoidal, double-ellipsoidal, and time-dependent on/off switch sources. We compare our analytical solutions to numerical implementations, demonstrating strong agreement while providing deeper physical insight. This approach significantly reduces computational cost and experimental requirements, making it a scalable tool for optimizing thermal predictions and mitigating residual stresses in metal-based manufacturing. Additionally, our framework enables the generation of synthetic datasets for machine learning models to predict heat distribution efficiently.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript develops an analytical framework for transient and steady-state heat conduction in a semi-infinite slab with Robin boundary conditions representing Newton's law of cooling. It derives closed-form temperature profiles for Gaussian, ellipsoidal, double-ellipsoidal, and time-dependent on/off heat sources by applying Laplace transforms (in time) combined with Fourier transforms (in space) and, separately, Fourier series expansions; the two routes are asserted to be equivalent. Solutions are compared to numerical implementations, with claims of strong agreement, and positioned as computationally efficient tools for welding and additive manufacturing that also enable synthetic ML datasets.

Significance. If the derivations are rigorous and the claimed equivalence is established without hidden approximations, the work would meaningfully extend classical solutions (e.g., Rosenthal) by incorporating surface cooling and transients in closed form. This could reduce reliance on expensive numerical simulations for process optimization and provide exact expressions usable for generating training data, representing a practical advance in the field.

major comments (1)
  1. [Abstract and §2] Abstract and §2 (Methods): The central claim that the Laplace-transform and Fourier-series approaches are 'proven-equivalent' and both yield identical closed-form transient/steady-state profiles requires an explicit demonstration of the L → ∞ limit for the eigenfunction expansion while preserving term-by-term identity with the direct Laplace-Fourier solution for distributed, possibly moving sources. Without this limiting procedure or a side-by-side identity proof, the dual-method validation strategy remains unsubstantiated and load-bearing for the paper's novelty assertion.
minor comments (2)
  1. [Figure captions and §4] Figure captions and §4 (Results): Several comparison plots lack error metrics (e.g., L2 norms or pointwise maximum deviations) between analytical and numerical solutions; adding these would strengthen the 'strong agreement' claim.
  2. [Notation] Notation: The definition of the heat-source terms (Gaussian vs. ellipsoidal) should be collected in a single table or subsection to avoid repeated re-derivation of the same integrals in different sections.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need to substantiate the claimed equivalence between the two analytical approaches. We address this point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract and §2] Abstract and §2 (Methods): The central claim that the Laplace-transform and Fourier-series approaches are 'proven-equivalent' and both yield identical closed-form transient/steady-state profiles requires an explicit demonstration of the L → ∞ limit for the eigenfunction expansion while preserving term-by-term identity with the direct Laplace-Fourier solution for distributed, possibly moving sources. Without this limiting procedure or a side-by-side identity proof, the dual-method validation strategy remains unsubstantiated and load-bearing for the paper's novelty assertion.

    Authors: We agree that an explicit demonstration of the L → ∞ limit would strengthen the paper and remove any ambiguity regarding equivalence. In the revised manuscript we will add a new subsection (or appendix) in §2 that carries out this limiting procedure in detail. For a slab of finite thickness L with the Robin condition at z = 0, the eigenfunction expansion in z is discrete; as L → ∞ the eigenvalues become dense and the sum over modes passes to the continuous Fourier integral that appears in the direct Laplace-Fourier solution. We will show term-by-term that the resulting expressions for the temperature field coincide for a general distributed source Q(x,y,z,t). The same identity holds when the source is moving (e.g., a translating Gaussian or double-ellipsoidal beam), because the source term enters identically in both formulations and the spatial transforms are unaffected by the limit. The numerical validations already reported in the manuscript are consistent with this analytic equivalence, but the explicit limiting argument will now be provided. revision: yes

Circularity Check

0 steps flagged

Standard Laplace and Fourier methods applied to heat equation with no self-referential reductions or fitted predictions

full rationale

The abstract and provided context describe direct application of Laplace transforms and Fourier series to the heat equation on a semi-infinite domain with Robin boundary conditions. No equations are shown that define a quantity in terms of itself, rename a fit as a prediction, or rely on load-bearing self-citations for uniqueness. The claimed equivalence of the two methods is asserted but does not reduce any derived profile to an input by construction; it remains an independent verification step against numerical solutions. This matches the default expectation of no significant circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on standard mathematical transforms applied to the heat equation under a semi-infinite geometry and Newton's cooling boundary condition. No free parameters or invented entities are mentioned.

axioms (3)
  • domain assumption The workpiece is modeled as a semi-infinite slab
    Geometry chosen to simplify the heat flow problem in welding.
  • domain assumption Surface cooling obeys Newton's law of cooling
    Boundary condition introduced to account for heat loss to the environment.
  • standard math Laplace transform and Fourier series solutions are equivalent for this problem
    Claimed as proven-equivalent in the abstract.

pith-pipeline@v0.9.0 · 5479 in / 1448 out tokens · 62313 ms · 2026-05-08T13:51:18.138042+00:00 · methodology

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Reference graph

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