The Gap-Shifted Boundary Method for solid mechanics problems: Linear elasticity, hyperelasticity and elastoplasticity

Immersed finite element methods significantly alleviate the burden of body-fitted mesh generation for complex geometries, yet they frequently suffer from algorithmic complexities associated with cut-cell numerical integration and matrix ill-conditioning. The Shifted Boundary Method (SBM) circumvents these pathologies by shifting the imposition of true boundary conditions to a surrogate boundary composed of full edges/faces of the grid. Boundary conditions are modified by means of Taylor expansions across the surrogate-to-true distance vectors. However, the standard primal-SBM produces sub-optimal error convergence in the L²[jls-end-space/]-norm on linear finite elements, when Neumann (traction) boundary conditions are present. To overcome this fundamental limitation and recover optimal accuracy, we develop the Gap-Shifted Boundary Method (Gap-SBM), which introduces approximate integration formulas to account for the virtual work contribution in the gap between the surrogate and true boundaries. These special quadratures avoid integration over cut cells. We comprehensively derive and implement the Gap-SBM for linear elasticity, finite-strain hyperelasticity, and path-dependent small-strain elastoplasticity. Numerical benchmarks on intricate topologies, including Triply Periodic Minimal Surfaces (TPMS) and rough fracture networks, demonstrate that Gap-SBM consistently delivers high-fidelity stress and yielding predictions, and facilitates the nonlinear Newton–Raphson convergence. The proposed framework establishes a robust, optimal, and geometrically flexible tool, bridging the gap between automated, unfitted meshing workflows and robust high-fidelity structural mechanics algorithms, providing foundations for extension to coupled Biot poromechanics, damage mechanics, and frictional-contact formulations on rough fracture surfaces.

Duke Scholars

Author Guglielmo Scovazzi Civil and Environmental Engineering