Meshless method for linear elastostatics based on a path integral formulation

A Path Integral (PI) formulation of linear elastostatics is "rst presented. For this, Navier equations are modi"ed by adding a "ctitious &timeʹ derivative of displacements so that equilibrium corresponds to the steady state of the resulting di!usion-like equations. The evolution of displacement is then represented as the propagation, through the "ctitious time co-ordinate, of an initial displacement "eld corresponding to the unloaded state. The resulting procedure somehow mimics the well-known Feynman path integral of quantum mechanics, which is equivalent to the di!erential formulation embodied in SchroK dinger equation. However, the path integral for elastostatics is formulated in terms of in"nitesimal propagators of local support. In its simplest form, the formulation can be used as a relaxation method of solution, by updating displacements until convergence. This may be advantageous for problems involving a very large number of unknowns. On the other hand, by equating the updated displacement "eld to the actual one a direct method of solution is obtained, which leads to non-symmetric (but sparse and banded) discrete equations. Unlike variational principles this formulation does not require integration over the whole domain, e!ectively eliminating the need of a background mesh for integration. Also, it only requires continuity of the displacement "eld on the propagatorʹs support. As a consequence, the formulation lends itself to very #exible meshless implementations. To demonstrate this we describe a simple numerical method in which displacements around each node are approximated by quadratic bivariate polynomials, which is the simplest approximation technique. The feasibility of the method is assessed through a number of numerical examples and comparisons with analytical solutions and other meshless methods.