Arbitrary Lagrangian-Eulerian formulation for element-free Galerkin method

Arbitrary Lagrangian-Eulerian (ALE) formulation of the Element Free Galerkin (EFG) method is presented. EFG is a meshless method for solving partial differential equations in which the trial and test functions employed in the discretization process result from moving least square interpolants. The most significant advantage of the method is that it requires only nodes and a description of internal and external boundaries and interfaces, such as cracks, of the model: no element connectivity is needed. However, as for any discretization method, acceptable solutions can only be obtained for a sufficiently refined discretization. In dynamic fracture problems, where the crack path can be arbitrary, and is thus a priori unknown, this necessitates a refined discretization in large parts of the computational domain which can lead to prohibitive computation costs. ALE formulation allows to continuously relocate nodes on the computational domain. By combining EFG with ALE, it is thus possible, in a crack propagation problem, to refine locally the spatial discretization in the neighborhood of a propagating crack-tip. Results are presented for a wave propagation problem as well as for 2-D dynamic crack propagation problems.