Part-scale thermal process simulations play an important role in improving the part quality of the Laser Powder Bed Fusion (LPBF) process. The semi-analytical simulation method relies on the superposition of analytical fields to represent laser-induced heat sources in a semi-infinite space and a complementary temperature field to enforce boundary conditions. So far, boundary conditions have been imposed by analytical image fields for straight boundaries and numerically for non-straight boundaries. The latter requires considerable refinement on the spatial discretization, at least near the boundaries, and compromises the computational efficiency of the simulations. In this paper, we derive a closed-form solution for the image fields that can accurately enforce the boundary conditions for non-straight boundaries. A geometrically complex part boundary is represented by B-splines, and with the aid of an offset method and reparameterization, the positions of the image sources are determined. The image field's closed-form expression is then found using the boundary's local curvature calculated from the local tangent lines. Numerical examples on different levels of complexity revealed that the net heat lost along an adiabatic boundary vanishes when the novel image source solutions are used, and the thermal evolution of complex parts can be accurately predicted with high computational efficiency. Simulations involving multiple lasers can also be performed with no extra computational cost.