Efficient computation of order and mode of corner singularities in 3D-elasticity

A general numerical procedure is presented for the efficient computation of corner singularities, which appear in the case of non-smooth domains in th ree-dimensional linear elasticity. For obtaining the order and mode of sing ularity, a neighbourhood of the singular point is considered with only loca l boundary conditions. The weak formulation of the problem is approximated by a Galerkin-Petrov finite element method. A quadratic eigenvalue problem (P + lambdaQ + lambda R-2) u = 0 is obtained, with explicitly analytically defined matrices P,Q,R. Moreover, the three matrices are found to have opti mal structure, so that P,R are symmetric and Q is skew symmetric, which can serve as an advantage in the following solution process. On this foundatio n a powerful iterative solution technique based on the Arnoldi method is su bmitted. For not too large systems this technique needs only one direct fac torization of the banded matrix P for finding all eigenvalues in the interv al Re(lambda) is an element of (-0.5, 1.0) (no eigenpairs can be 'lost') as well as the corresponding eigenvectors, which is a great improvement in co mparison with the normally used determinant method. For large systems a var iant of the algorithm with an incomplete factorization of P is implemented to avoid the appearance of too much fill-in. To illustrate the effectivenes s of the present method several new numerical results are presented. In gen eral, they show the dependence of the singular exponent on different geomet rical parameters and the material properties. Copyright (C) 2001 John Wiley & Sons, Ltd.