Efficient thermal simulation in metal additive manufacturing via semi-analytical isogeometric analysis

TL;DR

The paper tackles LPBF heat transfer modeling by marrying a semi-analytical temperature decomposition with an isogeometric boundary-correction solver. By representing the laser input as an analytical point-source field and solving the boundary-correction for the remaining domain with a high-continuity NURBS discretization, the approach eliminates the need for scan-wise remeshing and handles complex geometries with fewer resources. The Crank–Nicolson–type time integration and exact geometry representation enable accurate temperature predictions while reducing degrees of freedom compared to conventional FEM, demonstrated across simple and complex geometries. This framework offers a scalable, high-fidelity tool for part-scale thermal simulations in LPBF, with potential extensions to phase change, multi-laser systems, and process optimization.

Abstract

Thermal modeling of Laser Powder Bed Fusion (LPBF) is challenging due to steep, rapidly moving thermal gradients induced by the laser, which are difficult to resolve accurately with conventional Finite Element Methods. Highly refined, dynamically adaptive spatial discretization is typically required, leading to prohibitive computational costs. Semi-analytical approaches mitigate this by decomposing the temperature field into an analytical point-source solution and a complementary numerical field that enforces boundary conditions. However, state-of-the-art implementations either necessitate extensive mesh refinement near boundaries or rely on restrictive image source techniques, limiting their efficiency and applicability to complex geometries. This study presents a novel reformulation of the semi-analytical framework using Isogeometric Analysis. The laser heat input is captured by the analytical point-source solution, while the complementary correction field, which imposes boundary conditions, is solved using a spline-based IGA discretization. The governing heat equation for the correction field is cast in a weak form, discretized with NURBS basis functions, and advanced in time using an implicit $θ$-scheme. This approach leverages IGA's key advantages: exact geometry representation, higher-order continuity, and superior accuracy per degree of freedom. These features unlock efficient thermal modeling of realistic parts with complex contours. Our strategy eliminates the need for scan-wise remeshing and robustly handles intricate geometric features like sharp corners and varying cross-sections. Numerical examples demonstrate that the proposed semi-analytical IGA method delivers accurate temperature predictions and achieves substantial computational efficiency gains compared to standard FEM, establishing it as a powerful new tool for high-fidelity thermal simulation in LPBF.

Figures (17)

• Figure 1: Schematic illustration of the image source: (a) Single straight boundary, for imaging through a simple straight boundary $\partial V_1$, the adiabatic boundary conditions can be easily satisfied by introducing a single image source $J$. (b) Multiple connected boundaries, in cases involving multiple connected boundaries, the image source $J_1$ reflected from one boundary $\partial V_1$ may also influence another boundary $\partial V_2$, and similarly, the image source $J_2$ reflected from $\partial V_2$ may affect $\partial V_1$. (c) Two adjacent solid boundaries, when imaging between two neighboring solid boundaries, the image source $J$ reflected by one boundary $\partial V_1$ may be placed within the solid domain associated with the other boundary. (d) Cross-sectional variation along the building direction ($x_3$), when the cross section varies along the $x_3$ direction, the image source $J$ can only satisfy the adiabatic boundary conditions on the top plane. In this case, the physical source $I$ and its image $J$ are separated by an equal distance $d$ to the boundary only on the top plane.
• Figure 2: Schematic of the LPBF process: (a) Body $V$ submerged in the powder bed, with bottom $\partial V_{\mathrm{bot}}$ fused to the build platform and a thin powder layer on $\partial V_{\mathrm{top}}$; (b) Boundary decomposition of $V$, showing $\partial V_{\mathrm{bot}}$, $\partial V_{\mathrm{lat}}$, and $\partial V_{\mathrm{top}}$.
• Figure 3: A curvilinear (contour) laser scan discretized into a sequence of point sources. The total temperature field is obtained by superimposing the semi-infinite point-source solutions with the complementary field that enforces the finite-part boundary conditions.
• Figure 4: (a) A point source (red) positioned near the curved boundary of a $2mm\times2mm\times2mm$ cubic domain, where a quarter-cylinder of radius $R_c=1mm$ has been removed. (b) Top view of the part. The shortest distance from the point source to the arc is $100µm$, located at $\theta=\pi/4$ relative to the $x_2$-axis.
• Figure 5: IGA mesh and Finite Element (FE) mesh for computing $\hat{T}$. Small elements of approximately 50µm are used to refine the boundary near the point source.