Boundary locking induced by penalty enforcement of essential boundary conditions in mesh-free methods

Mesh-free methods continue to have problems in enforcing the essential boundary conditions because of the absence of the Kronecker delta property in mesh-free shape functions, constructed from moving least squares method, reproducing kernel particle method, and others. In this paper, this problem in dealing with the essential boundary conditions is examined. Boundary locking, induced by the penalty enforcement of the essential boundary conditions in mesh-free methods, is first explored, and the reason for obtaining an over-constrained solution is explained theoretically. To justify these observations, various numerical examples are also presented. Additionally, based on further theoretical investigations, a remedy to circumvent this problem of the over-constraint phenomena, caused by the penalty enforcement of essential boundary conditions, is proposed.