Additive manufacturing has significantly transformed component production. However, anisotropic structural behaviour is commonly observed in additively manufactured components, despite the isotropic nature of the constituent materials. This behaviour can be attributed to the manufacturing process, which involves the extrusion and deposition of individual material trajectories or the powder-based melting of such trajectories. For instance, Fused Filament Fabrication (FFF) is a common technique used in polymer component production. Technological advancements have enabled continuous fibre reinforcement that can further strengthen the anisotropic material behaviour. Several computational models and approaches have been proposed to simulate and optimize (i.e. topology optimization and optimization of the material trajectories) additively manufactured components treated as an anisotropic continuum. Current methods rely on a finite element discretization of the continuum, where the material trajectories are assumed to be linear straight lines within a finite element. However, since the material trajectories are essentially arbitrary curves, a fine discretization is necessary to achieve realistic simulation. In this work, we propose a curvilinear material trajectory approach, where material trajectories at the element level are considered to be curvilinear. The mesh fineness of this concept is solely dependent on the stress gradients that need to be resolved. Several degrees of freedom can thus be saved and the simulation time reduced, which is of enormous importance especially for optimization tasks as well as non-linear analysis. Furthermore, curvilinear material trajectories represent the coordinate lines for describing anisotropy. As a result, the solution of the balance of linear momentum occurs within the local curvilinear coordinate system. We present the implementation of this approach within the finite element method, using an exemplary boundary value problem, and validate the simulation results using an optical measurement.