The isogeometric Gap-Shifted Boundary Method

High-order immersed methods offer significant advantages for the simulation of complex geometries, but their practical performance is often limited by a trade-off between accuracy, robustness, and numerical conditioning. In this work, we present a high-order isogeometric extension of the Gap–Shifted Boundary Method (Gap–SBM) for linearelasticity that overcomes these limitations. The method explicitly accounts for the geometric gap between the surrogate and true boundaries by extending active finite elements into the gap region through high-order Taylor expansions, thereby avoiding cut-cell integration while enabling accurate enforcement of Neumann boundary conditions. A highorder integration strategy for the gap region is introduced based on curvilinear gap elements constructed via Coons patch parametrizations and enhanced, when necessary, by hierarchical internal subdivisions. Several strategies for prescribing boundary conditions on approximated boundaries are also investigated, including interpolation- and shift- based extrapolation, showing that appropriate boundary data reconstruction can improve accuracy by several orders of magnitude when geometric approximation errors dominate. Extensive numerical experiments demonstrate that the proposed Gap–SBM achieves optimal convergence rates for both Dirichlet and Neumann boundary conditions,with minimal sensitivity to the position of the surrogate boundary. Moreover, the formulation remains stable and accurate for geometries featuring characteristic length scales far below the mesh resolution, well beyond the nominal theoretical range of validity of shifted-boundary methods.