Accurately resolving temperature evolution in laser powder bed fusion (LPBF) remains computationally demanding due to the steep and rapidly moving thermal gradients generated by the scanning laser. In a recent study published on arXiv, researchers from Delft University of Technology present a semi-analytical thermal simulation framework that combines analytical point-source solutions with isogeometric analysis (IGA), enabling efficient temperature prediction for geometrically complex metal parts without scan-wise adaptive meshing.
Thermal modeling is a central component of LPBF process simulation because temperature histories govern melt-pool behavior, solidification rates, microstructure evolution, residual stress, and geometric distortion. Conventional finite element method (FEM) simulations struggle to balance accuracy and computational cost. Capturing laser-induced thermal gradients typically requires element sizes on the order of the laser spot radius, often tens of micrometers, leading to very large meshes. While adaptive remeshing strategies can reduce the global mesh size, scan-wise remeshing introduces significant overhead due to frequent updates of the mesh, conductivity matrix, and load vector, limiting practical applicability at part scale.
The Delft study builds on semi-analytical approaches that decompose the temperature field into two components: an analytical solution that captures the localized thermal response to the laser, and a complementary numerical field that enforces boundary conditions associated with the finite part geometry. In the proposed framework, the laser scan path is discretized into a sequence of instantaneous point heat sources. The analytical solution represents the transient temperature rise from each source in a semi-infinite domain, inherently resolving steep thermal gradients without numerical refinement.
Boundary conditions are enforced through a complementary temperature field solved numerically. Instead of using FEM or analytical image-source techniques, the authors employ isogeometric analysis to compute this correction field. IGA represents both geometry and solution fields using spline-based basis functions, such as non-uniform rational B-splines, allowing exact reproduction of the CAD geometry and higher-order continuity across elements. The correction problem is formulated in weak form and integrated in time using an implicit Crank–Nicolson scheme, while the analytical temperature field is updated explicitly as the laser advances.
This reformulation addresses limitations of earlier semi-analytical methods that relied on image sources to satisfy adiabatic boundary conditions. Image-source approaches work well for simple geometries with single, planar boundaries but break down for realistic LPBF components featuring sharp corners, connected boundaries, or varying cross-sections. In such cases, enforcing boundary conditions requires an infinite series of reflections or can introduce unphysical heating in adjacent regions. By solving the boundary-correction problem numerically with IGA, the new framework avoids these geometric constraints while maintaining a coarse, fixed discretization.
The authors evaluate the method using a sequence of numerical examples that compare FEM and IGA performance. In a single point-source benchmark near a curved boundary, IGA achieves accuracy comparable to a highly refined FEM reference solution while using substantially fewer degrees of freedom. FEM requires minimum element sizes smaller than half the laser spot radius to maintain acceptable error levels, whereas IGA maintains relative errors on the order of 10% even when element sizes exceed the laser spot radius by more than an order of magnitude. Heat-flux evaluations along the boundary confirm that the IGA-based correction field accurately compensates the analytical solution to satisfy adiabatic conditions.
The study then examines continuous laser scanning along a curved contour. In this case, an IGA discretization with a minimum element size of 100 micrometers produces temperature fields closely matching FEM results obtained with 10 micrometer elements. Despite the coarser mesh, the IGA simulations exhibit comparable or lower integrated boundary heat-loss errors at selected time steps during the scan. Temperature contour lines remain orthogonal to part boundaries throughout the process, indicating consistent enforcement of boundary conditions.
To test robustness under geometric complexity, the framework is applied to a butterfly-shaped part with cross-sections that vary nonlinearly along the build direction. Such geometries pose significant challenges for image-source methods and scan-wise remeshing strategies. Using IGA alone, the authors simulate a continuous contour scan with stable boundary enforcement on both the top surface and subsurface cross-sections. Temperature iso-surfaces remain orthogonal to boundaries at all times, and subsurface temperature fields exhibit physically consistent attenuation relative to the top layer. The authors note that an equivalent FEM simulation would require prohibitively large meshes to achieve comparable resolution.
Across all test cases, the results show that the semi-analytical IGA framework eliminates the need for laser-following remeshing while remaining insensitive to geometric complexity. By capturing laser-scale thermal gradients analytically and enforcing boundary conditions numerically using spline-based discretization, the method enables part-scale thermal simulations with significantly reduced computational cost compared to conventional FEM-based approaches.
Future work will focus on coupling the thermal solution with phase-change and melt-pool fluid dynamics, extending the approach to multi-laser systems, improving efficiency through hierarchical spline refinement, and validating predictions against in-situ temperature measurements across different materials and geometries. The study was conducted by Yang Yang, Ye Ji, Matthias Möller, and Can Ayas from Delft University of Technology’s Faculty of Mechanical Engineering and Delft Institute of Applied Mathematics.
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Featured image shows Schematic of the LPBF process. Image via arXiv.
