Dislocation transport using an explicit Galerkin/least-squares formulation

Abstract

An explicit Galerkin/least-squares formulation is introduced for a quasilinear transport equation in field dislocation mechanics (FDM) and applied to the study of the kinematics of dislocation density evolution in the following physical contexts: annihilation of dislocations, expansion of a polygonal dislocation loop and simulation of a Frank–Read source. Stability analysis is carried out for the corresponding linear one-dimensional (1D) case. The formulation reduces to the Lax–Wendroff finite difference scheme for the 1D equation when equal weighting is used for the Galerkin and least-squares terms and the shape functions are linear. This conditionally stable method leads to a symmetric well-conditioned system of equations with constant coefficients, making it attractive for large-scale problems.It is shown that the transport equation, in the contexts mentioned above simplifies to the Hamilton–Jacobi equations governing geometrical optics and level-set methods. The weak solutions to these equations are not unique, and the numerical method is able to capture solutions corresponding to shock as well as rarefraction waves by appropriate algorithmic modifications.

On particle size effects: An internal length mean field approach using field dislocation mechanics

“…It is of hyperbolic character, and conventional Galerkin finite element simulations lead to numerically unstable approximate solutions. In the forthcoming simulations, stable solutions are provided by using an explicit Galerkin-least squares finite element method [33]. At a mesoscopic scale, the plastic distortion rate receives contributions from the mobility of both GND and SSD densities:…”

Continuum Mechanics of the Interaction of Phase Boundaries and Dislocations in Solids

“…Significant components of such problems have been dealt with computationally in our prior work e.g. [RA05,VBAF06,FTC11,TCF13a], and detailed considerations for the present model will be the subject of future work.…”

Untitled

“…The finite-element discretization for the system of equations above is discussed in [Roy and Acharya 2006]. Here we only summarize the finite-element discretization of (7), which has an extra term in the weak formulation corresponding to the least-squares finite-element discretization of the inflow boundary condition on α [Varadhan et al 2006].…”