Efficient spectral Galerkin framework for nonlinear transient heat transfer in finite domains
Abstract
Accurately modelling the temporal evolution of heterogeneous temperature fields requires resolving strong nonlinearities in the heat equation arising from temperature-dependent thermophysical properties, latent heats of transformation and any local heat sources/sinks. In this work, we present a spectral Galerkin (SG) framework to solve the fully nonlinear transient heat equation in finite domains to attain the accuracy of high-fidelity finite element (FE) simulations at considerably lower computational cost. The heat equation is reformulated into a linear reference problem with constant thermophysical properties and residual forcing terms. Solving the reference problem provides a complete three-dimensional orthonormal trigonometric basis, whose Galerkin projection reduces the heat equation to a set of modal ordinary differential equations (ODEs) in time; the reference operator is diagonal in the modal basis, and results in independent modal updates for a fixed nonlinear forcing. These ODEs can be integrated using exponential time differencing and iteratively corrected for nonlinearities. The SG method eliminates the global solve required by FE methods, and its use of structured grids allows efficient GPU parallelization. Applied to rapid laser-metal interactions, the SG solver reproduces high-fidelity FE temperature fields with less than 1% relative error while achieving 227-fold faster GPU runtimes. Applied to a partscale laser scanning study [Ramani et al., Additive Manufacturing 52 (2022) 102643], the method shows that accounting for evaporation and latent heat more than halves their proposed processing metric. The source code of the SG heat solver and some worked examples are available at https://github.com/manasvupadhyay/spectral_galerkin_heat under the Apache 2.0 license.
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BibTeX
@misc{andrieux2026efficient,
title = {{Efficient spectral Galerkin framework for nonlinear transient heat transfer in finite domains}},
author = {Théo Andrieux and Andreas Ntinos and Manas V Upadhyay},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22268},
eprint = {2609.22268},
archivePrefix = {arXiv},
url = {https://polytechnique.hal.science/hal-05743418v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Efficient spectral Galerkin framework for nonlinear transient heat transfer in finite domains
AU - Théo Andrieux
AU - Andreas Ntinos
AU - Manas V Upadhyay
PY - 2026
DA - 2026/09/09
PB - HAL
DO - 10.48550/arxiv.2609.22268
UR - https://polytechnique.hal.science/hal-05743418v1
AB - Accurately modelling the temporal evolution of heterogeneous temperature fields requires resolving strong nonlinearities in the heat equation arising from temperature-dependent thermophysical properties, latent heats of transformation and any local heat sources/sinks. In this work, we present a spectral Galerkin (SG) framework to solve the fully nonlinear transient heat equation in finite domains to attain the accuracy of high-fidelity finite element (FE) simulations at considerably lower computational cost. The heat equation is reformulated into a linear reference problem with constant thermophysical properties and residual forcing terms. Solving the reference problem provides a complete three-dimensional orthonormal trigonometric basis, whose Galerkin projection reduces the heat equation to a set of modal ordinary differential equations (ODEs) in time; the reference operator is diagonal in the modal basis, and results in independent modal updates for a fixed nonlinear forcing. These ODEs can be integrated using exponential time differencing and iteratively corrected for nonlinearities. The SG method eliminates the global solve required by FE methods, and its use of structured grids allows efficient GPU parallelization. Applied to rapid laser-metal interactions, the SG solver reproduces high-fidelity FE temperature fields with less than 1% relative error while achieving 227-fold faster GPU runtimes. Applied to a partscale laser scanning study [Ramani et al., Additive Manufacturing 52 (2022) 102643], the method shows that accounting for evaporation and latent heat more than halves their proposed processing metric. The source code of the SG heat solver and some worked examples are available at https://github.com/manasvupadhyay/spectral_galerkin_heat under the Apache 2.0 license.
ER -