Imposition of Nonconforming Neumann Boundary Conditions in the High-Order Material Point Method with the Virtual Stress Boundary Method
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The virtual stress boundary (VSB) method has been shown to accurately and efficiently impose nonconforming Neumann boundary conditions in the material point method (MPM). Its primary advantage lies in eliminating the need for explicit boundary position information by transforming the original traction into an equivalent virtual stress field. This transformation enables the governing equations to be solved entirely through volume integrals, which are readily evaluated using particle-wise quadrature. This study extends the VSB framework to high-order MPM variants, including GIMP, B-Spline MPM, and LME MPM. The key modifications address the non-local support domain inherent to high-order shape functions and improve quadrature accuracy within fully filled cells. Numerical examples in 1D, 2D, and 3D validate the accuracy and versatility of the proposed approach.
