Extended Mixed B‐Spline Material Point Method With Nitsche's Method for Quasi‐Compressible Soft Materials

ABSTRACT Soft materials, including elastomers, gels, and biological tissues, undergo extreme, complex deformations under near‐incompressibility, posing significant challenges for numerical modeling. The hybrid Eulerian–Lagrangian particle‐based approach, the Material Point Method (MPM), offers advantages for modeling these behaviors, but challenges remain, including prescribing essential boundary conditions, volumetric locking, cut‐cell instabilities, and cell crossing errors. In this work, we introduce an extended mixed B‐spline MPM that integrates subdivision‐stabilized displacement–pressure interpolations with the Nitsche method for weak enforcement of essential boundary conditions. The subdivision‐based mixed formulation ensures smooth, oscillation‐free pressure and stress fields throughout the simulations while alleviating volumetric locking in quasi‐compressible problems. The extended B‐spline approach improves numerical stability at domain boundaries by mitigating ill‐conditioning in the stiffness matrix, while higher‐order interpolations reduce cell‐crossing errors. The symmetric Nitsche method preserves the symmetry of the stiffness matrix, enabling efficient mechanical stability analysis in implicit MPM. Numerical examples—including torsional buckling, pore collapse, and soft metamaterial compression—demonstrate the method's robustness in capturing highly nonlinear mechanical responses under extreme deformations. The developed framework provides a stable and efficient particle‐based approach for simulating soft materials and can be extended to multi‐field coupled problems.