Numerical analysis of singularities in two‐dimensions part 1: Computation of eigenpairs

A numerical method is described for the computation of eigenpairs which characterize the exact solution of linear second‐order elliptic partial differential equations in two dimensions in the vicinity of singular points. The singularities may be caused by re‐entrant corners and abrupt changes in boundary conditions or material properties. Such singularities are of great interest from the point of view of failure initiation: The eigenpairs characterize the straining modes and their amplitudes quantify the amount of energy residing in particular straining modes. For this reason, failure theories directly or indirectly involve the eigenpairs and their amplitudes. This paper addresses the problem of determining the eigenpairs numerically on the basis of the Steklov formulation. Numerical results are presented for several cases. Importantly, the method is applicable to three‐dimensional cases.