Developments of the method of difference potentials for linear elastic fracture mechanics problems

Recently, the method of difference potentials has been extended to linear elastic fracture mechanics. The solution was calculated on a grid boundary belonging to the domain of an auxiliary problem, which must be solved multiple times. Singular enrichment functions, such as those used within the extended finite element method, were introduced to improve the approximation near the crack tip leading to near-optimal convergence rates. Now, the method is further developed by significantly reducing the computation time. This is achieved via the implementation of a system of basis functions introduced along the physical boundary of the problem. The basis functions form an approximation of the trace of the solution at the physical boundary. This method has been proven efficient for the solution of problems on regular (Lipschitz) domains. By introducing the singularity into the finite element space, the approximation of the crack can be realised by regular functions. Near-optimal convergence rates are then achieved for the enriched formulation. A solution algorithm using the fast Fourier transform is provided with the aim of further increasing the efficiency of the method.