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 linear elasticity in two and three dimensions. The method explicitly reconstructs and integrates the geometric gap between the surrogate and true boundaries, while extending the active spline approximation into this region through complete high-order Taylor expansions. The gap cells are used exclusively for integration and introduce no additional degrees of freedom. In two dimensions, the gap is represented by curvilinear Coons patches, supplemented by hierarchical subdivision when the boundary geometry is under-resolved. In three dimensions, a robust pyramidal–tetrahedral decomposition is introduced, with curved geometric entities constructed consistently with the spline degree.Different strategies for prescribing boundary conditions on the reconstructed boundary are investigated, including direct imposition, interpolation, and enhanced SBM extrapolation. The latter substantially reduces the residual geometric error when the reconstructed and true boundaries do not coincide exactly, while direct imposition remains compatible with a symmetric Nitsche formulation.The numerical results demonstrate optimal convergence rates for immersed Dirichlet and Neumann boundary conditions in both two and three dimensions. The Gap–SBM retains the expected h⁻² condition-number scaling, with a lower conditioning constant than the standard SBM, and remains accurate for geometries containing characteristic length scales substantially smaller than the background mesh size. Applications to exact CAD surfaces and complex STL-derived geometries illustrate the robustness and flexibility of the proposed workflow. These benefits are obtained without increasing the algebraic problem size, at the expense of additional geometric preprocessing and quadrature costs, which become more significant for high-order three-dimensional reconstructions.

Duke Scholars

Author Guglielmo Scovazzi Civil and Environmental Engineering