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

Abstract

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. KEYWORDSextended finite element method, fast Fourier transform, fracture, method of difference potentials, rate of convergence INTRODUCTIONThe method of difference potentials (DPM) was originally developed by Ryaben'kii 1 and Ryaben'kii and Tsynkov 2 and has been applied to boundary value problems (BVPs) in fluid dynamics, 3 acoustics, 4-6 and many other applications. 1 The methodology is based on the same boundary integral equation as that used in boundary element methods. 7-11 Within the boundary element method, this boundary integral equation is numerically integrated given the knowledge of a suitable Green's function. The DPM avoids any need for Green's functions by embedding the problem within a well-defined auxiliary problem. The auxiliary problem is then solved numerically, avoiding the solution of the integral equation.A boundary system of equations must be constructed to calculate appropriate grid densities at the embedded boundary. This requires the auxiliary problem to be solved a multiple number of times, where the number of solutions required increases with required accuracy. Definition of the auxiliary problem can be flexible provided that it is well defined and completely encloses the embedded domain and its boundary. The auxiliary problem is therefore chosen as to facilitate the most efficient solution method. Previously, a square domain with a regular grid has been used along with the use of finite difference methods, which can be particularly efficient for regular problems such as this. In this paper, an adjusted Int J Numer Methods Eng. 2018;115:75-98.wileyonlinelibrary.com/journal/nme