On computing upper and lower bounds on the outputs of linear elasticity problems approximated by the smoothed finite element method

Verification of the computation of local quantities of interest, e.g. the displacements at a point, the stresses in a local area and the stress intensity factors at crack tips, plays an important role in improving the structural design for safety. In this paper, the smoothed finite element method (SFEM) is used for finding upper and lower bounds on the local quantities of interest that are outputs of the displacement field for linear elasticity problems, based on bounds on strain energy in both the primal and dual problems. One important feature of SFEM is that it bounds the strain energy of the structure from above without needing the solutions of different subproblems that are based on elements or patches but only requires the direct finite element computation. Upper and lower bounds on two linear outputs and one quadratic output related with elasticity – the local reaction, the local displacement, and the J-integral – are computed by the proposed method in two different examples. Some issues with SFEM that remain to be resolved are also discussed.