A new meshfree variational multiscale (VMS) method for essential boundary conditions

Meshfree methods offer significant flexibility in discretizing partial differential equations, including arbitrary smoothness uncoupled with the order of the approximation, and the ability to reconstruct connectivity on the fly, which enables extreme deformation capabilities for problems such as high-rate impact without erosion. Yet special care has to be taken to impose essential boundary conditions as shape functions generally do not interpolate field variables on the boundary. While various weak and strong methods are available, each has varying degrees of inherent difficulty, such as the selection of parameters, additional degrees of freedom, or sub-optimal convergence. As an alternative to these techniques, this work develops the variational multiscale (VMS) method for applying essential boundary conditions in meshfree methods. A primal formulation is developed such that no additional degrees of freedom are necessary. Stability is naturally derived from the fine-scale field rather than relying on a constant as in Nitsche's method. It is further shown that VMS is more accurate than Nitsche's method when material properties vary in space since the stabilization is local rather than global. In addition, we explore the idea of using VMS to inform the Nitsche method parameters for an improved solution. Numerical examples demonstrate the optimal convergence of the method and the insensitivity of the choice of fine-scale basis.