Goal oriented error estimation for fracture mechanics

Numerical methods introduce an error in the solution due to the approximations used to solve the problem, thus, it becomes necessary to quantify this error in order to guarantee the quality of the results. Goal oriented error estimation and adaptive procedures are essential for the accurate and efficient evaluation of finite element numerical simulations that involve complex domains. By locally improving the approximation quality, for example, by using the extended finite element method (XFEM), we can solve expensive problems which could result intractable otherwise.

In the EPSRC project EP/G042705/1 we have worked towards the development of an error estimation technique for enriched finite element approximations that is based on an equilibrated recovery method. The technique considers the stress intensity factor, typical of fracture mechanics problems,  as the quantity of interest. The locally equilibrated superconvergent patch recovery is used to obtain enhanced stress fields for the primal and dual problems defined to evaluate the error estimate. In particular, to perform the recovery of the primal and dual solutions we consider three main ideas: (i) enforcement of the internal equilibrium equation, (ii) enforcement of boundary equilibrium and (iii) splitting of the stress field into singular and smooth parts.

FE (left) and recovered (right) stress-y for the primal problem

FE (left) and recovered (right) stress-y for the primal problem

FE (left) and recovered fields (right) stress-y for the dual problem

FE (left) and recovered fields (right) stress-y for the dual problem

Distribution of the estimated error in the quantity of interest K

Distribution of the estimated error in the quantity of interest K

In a different context, but following similar ideas, we have investigated methods to validate numerical results obtained using other non standard FE approximations besides XFEM.  We have presented a technique to control the error in Smooth Finite Element (SFEM) approximations. Finite element techniques based on strain smoothing were shown to provide significant advantages compared to conventional finite element approximations. In particular, a widely cited strength of such methods is improved accuracy for the same computational cost. The proposed method is a recovery technique, closely related to the superconvergent patch recovery (SPR), which yielded good results for local and global effectivity  of the error indicator.

Cylinder under internal pressure. Distribution of the local effectivity index.

Cylinder under internal pressure. Distribution of the local effectivity index.

In another work, we worked in the development of an error indicator for arbitrary polygonal finite element formulations, using a moving least squares approach. PolyFEM provides more flexibility in the mesh generation and adaptivity processes. We adapted the XMLS recovery technique, previously proposed for standard FEM approximations, to the context of arbitrary polygonals with good results in the error indicator, both at the local and global level.

This work was done in collaboration with researchs groups in Universitat Politecnica de Valencia, Spain, Cardiff University, UK,  and CENAERO, Belgium.

 

Related publications

1. González-Estrada, O. A., Ródenas, J. J., Bordas, S. P. A., Nadal, E., Kerfriden, P., & Fuenmayor, F. J. (2015). Locally equilibrated stress recovery for goal oriented error estimation in the extended finite element method. Computers & Structures, 152, 1–10. http://doi.org/10.1016/j.compstruc.2015.01.015

2. González-Estrada, O. A., Nadal, E., Ródenas, J. J., Kerfriden, P., Bordas, S. P. A., & Fuenmayor, F. J. (2013). Mesh adaptivity driven by goal-oriented locally equilibrated superconvergent patch recovery. Computational Mechanics, (in press), 1–20. doi:10.1007/s00466-013-0942-8

3. Ródenas, J. J., González-Estrada, O. A., Fuenmayor, F. J., & Chinesta, F. (2013). Enhanced error estimator based on a nearly equilibrated moving least squares recovery technique for FEM and XFEM. Computational Mechanics, 52, 321–344. doi:10.1007/s00466-012-0814-7.

4. González-Estrada, O. A., Natarajan, S., Ródenas, J. J., Nguyen-Xuan, H., & Bordas, S. P. A. (2013). Efficient recovery-based error estimation for the smoothed finite element method for smooth and singular linear elasticity. Computational Mechanics, 52(1), 37–52. doi:10.1007/s00466-012-0795-6

5. González-Estrada, O. A., Natarajan. S., Ródenas, J. J., Bordas, S. P. A.,Heaney, C. Recovery based error estimation for the polygonal finite element method for smooth and singular linear elasticity. (2014). 11th World Congress on Computational Mechanics (WCCM 2014), Barcelona, Spain. CINME