<p>We propose an element concept that facilitates quadrature in Floating Isogeometric Analysis (FLIGA) by aligning the numerical integration points to the discontinuity lines of the basis functions. FLIGA is an isogeometric solid mechanics method we devised for the computational study of history-dependent problems with severe but oriented deformations, with extrusion-based additive manufacturing as target application. Its key feature is a new mathematical operation called floating regulation, that is active along the extrusion axis in order to constantly down-regulate mesh distortions by the Lagrangian analysis. The existing enhancements, such as automated floating regulation and adaptive refinement, are included in the novel element formulation, but the manual initialization of one large material point set is no longer required. We also extend FLIGA to NURBS discretizations. The accuracy and robustness of element-based FLIGA is investigated on the polymer extrusion problem (with viscoelastic effects modeled in integral-type representation) as well as on benchmarks in solid mechanics.</p>