Finite element mesh improvement using an a priori local p-refinement for stress analysis of underground excavations

Abstract

As our understanding and modeling capabilities evolve, an ever-increasing complexity of models representing the behaviour of geologic medium are analyzed. One way to evaluate these substantial problems is to optimize the underlying discretization of the governing differential equations by concentrating finite elements where solution accuracy counts the most. This paper develops and evaluates the performance of a priori local p-refinement method for finite element mesh improvement for stress analysis of underground excavations. This type of refinement entails a mesh with higher-order elements near the region of interest and lower-order elements elsewhere. The focus of the paper is the automated insertion of transitional elements at the interface of the two regions. The method relies on transitional finite elements in order to connect a mesh of quadratic interpolation order elements with a mesh of linear interpolation order elements. Four types of transitional elements were considered (4-node and 5-node triangles, 5-node and 7-node quadrilaterals). These were incorporated into a finite element code, and their performance was tested using representative problems such as a pressurized cavity or tunnelling through rock. For these problems the global stiffness matrix size was reduced on average by 85% and by 81% for the models using triangles and quadrilaterals, respectively, as a result, the calculation times were considerably shortened as well. While the average percentage of error with respect to the models without improvement, measured at critical points, was 0.04% and 0.02% in the case of triangular and quadrilateral elements, respectively.