Superconvergent second derivative recovery technique and its application in a nonlocal damage mechanics model

• A nonlocal finite element technique is proposed that is based on the superconvergent second derivative recovery methods.
• A way to construct element-based patches is proposed that is suitable for unstructured meshes.
• The proposed technique is implemented with a simplified Hessian matrix computation process.
• The proposed technique is verified by the recovery of the equivalent strains in a nonlocal damage mechanics model.

This study proposes a nonlocal finite element technique that is based on the super-convergent patch (SCP) second derivative recovery methods. The SCP second derivative recovery methods are proven effective in evaluating the Laplacian of the equivalent strains at C0 linear elements. The element-based patches and their construction criteria are defined. Other patch-based techniques either are limited to structured meshes or only utilize nodal based patches for first derivative recoveries. The SCP technique has not previously been used to recover higher order (2nd and higher) strain derivatives. The proposed technique can deal with unstructured meshes with various element types and the patches are constructed in a fashion that greatly accelerates the evaluation of Hessian matrices. A numerical example of a strain gradient damage mechanics problem is given at the end. The results show that the proposed technique is capable of evaluating the Laplacian of the equivalent strains and has the potential for even higher order derivative recovery.