An efficient adaptive analysis procedure for certified solutions with exact bounds of strain energy for elasticity problems

We present an efficient adaptive analysis procedure to obtain certified solutions of desired accuracy with bounds to the exact solution in energy norm for elasticity problems. The procedure makes the use of the recent finding that the upper bound to the exact strain energy can be obtained using the linearly conforming point interpolation method (LC-PIM), and the well-known fact that the lower bound can be obtained using the standard displacement-based fully compatible finite element method (FEM). To perform the adaptive analysis, a residual error-based error indicator and a simple h-type refinement scheme are employed and the relative error of the computed strain energy is used as the global stopping criteria. A number of numerical examples, including problems with singularity, have been studied to demonstrate the effectiveness and efficiency of the present procedure. The numerical results have been found converging very fast to the exact solution, and the bounds to the exact strain energy can be obtained efficiently at any stage in the adaptive process whenever required.