Discontinuous coefficients in the Poisson equation lead to the weak discontinuity in the solution, e.g. the gradient in the field quantity exhibits a rapid change across an interface. In the real world, discontinuities are frequently found (cracks, material interfaces, voids, phase-change phenomena) and their mathematical model can be represented by Poisson type equation. In this study, the extended finite element method (XFEM) is used to solve the formulated discontinuous problem. The XFEM solution introduce the discontinuity through nodal enrichment function, and controls it by additional degrees of freedom. This allows one to make the finite element mesh independent of discontinuity location. The quality of the solution depends mainly on the assumed enrichment basis functions. In the paper, a new set of enrichments are proposed in the solution of the Poisson equation with discontinuous coefficients. The global and local error estimates are used in order to assess the quality of the solution. The stability of the solution is investigated using the condition number of the stiffness matrix. The solutions obtained with standard and new enrichment functions are compared and discussed.
In the paper, the extended finite element method (XFEM) is combined with a recovery procedure in the analysis of the discontinuous Poisson problem. The model considers the weak as well as the strong discontinuity. Computationally efficient low-order finite elements provided good convergence are used. The combination of the XFEM with a recovery procedure allows for optimal convergence rates in the gradient i.e. as the same order as the primary solution. The discontinuity is modelled independently of the finite element mesh using a step-enrichment and level set approach. The results show improved gradient prediction locally for the interface element and globally for the entire domain.
The evaluation accuracies of rock mass structures based on the ratings of the Rock Quality Designation (RQD) and discontinuity spacing (S) in the Rock Mass Rating (RMR) system are very limited due to the inherent restrictions of RQD and S. This study presents an improvement that replaces these two parameters with the modified blockiness index (Bz) in the RMR system. Before proceeding with this replacement, it is necessary for theoretical model building to make an assumption that the discontinuity network contains three sets of mutually orthogonal disc-shaped discontinuities with the same diameter and spacing of discontinuities. Then, a total of 35 types of theoretical DFN (Discrete Fracture Network) models possessing the different structures were built based on the International Society for Rock Mechanics (ISRM) discontinuity classification (ISRM, 1978). In addition, the RQD values of each model were measured by setting the scanlines in the models, and the Bz values were computed following the modified blockiness evaluation method. Correlations between the three indices (i.e., Bz, RQD and S) were explored, and the reliability of the substitution was subsequently verified. Finally, RMR systems based on the proposed method and the standard approach were applied to real cases, and comparisons between the two methods were performed. This study reveals that RQD is well correlated with S but is difficult to relate to the discontinuity diameter (D), and Bz has a good correlation with RQD/S. Additionally, the ratings of RQD and S are always far from the actual rock mass structure, and the Bz ratings are found to give better characterizations of rock mass structures. This substitution in the RMR system was found to be acceptable and practical.