Analysis and Numerical Modeling of Faults Using the Boundary Element Method

One of the numerical methods in the mechanics of continuous environments is the boundary element method. In this method, the governing differential equations will be converted to integral equations and applied to the problem boundary. Then the boundary is divided into boundary parts and numerical integration is performed on the boundary elements, from the solution of which a single solution of the problem can be obtained. In the boundary element method, the partial differential equations defined within a space are converted to integral equations at the boundaries of that space, which reduces one dimension of the problem. For example, an elastodynamic problem defined in a two-dimensional space is replaced by an integral equation problem defined at its boundaries that has one dimension. If the problem defined in a three-dimensional space is replaced by a two-dimensional integral equation problem. Finally, the integral equations will be solved numerically by dividing the boundaries into a network of finite element discrete elements. The boundary element method can be easily applied to borders with complex geometry. Boundary element method or boundary integral equation (BIEM) is one of the numerical modeling methods that have many applications in numerical simulation of fault dynamics. Its results provide a broad view of the physics of earthquake rupture. To solve two-dimensional problems, the numerical technique of the boundary element method has been widely used. The boundary element method has been used to model the behavior of faults of overlapping centers, the growth of junction assemblies, veins and seismic analysis of topographic features. One form of BEM is based on separation, which is called the displacement discontinuity method. The theory of detachments in elastic materials has been widely used for more than half a century to evaluate the displacement, stress and strain fields around faults. By integrating Green's functions, the displacement field around the discontinuity surface can be calculated. These displacement fields are the Navier equations that are the governing equations of linear elastic theory. Strain components are obtained from the spatial derivatives of the displacement components, and the stress components can be calculated using Hooke's law for homogeneous and homogeneous elastic materials. Therefore, the mathematical tool of detachment theory is able to calculate the displacement, stress and strain fields around faults in half-elastic space, but it is less accurate compared to geophysical data. In this paper, numerical modeling of faults using the boundary element method has been reviewed, and studies conducted in the field of numerical modeling of faults using the boundary element method have been reviewed. Finally, the results show that the boundary element method is suitable for problems with complex boundaries such as fault geometry and problems with infinite boundaries. It is also possible to predict slip on the fault and surface deformation using numerical modeling using the boundary element method. The results of numerical modeling of the fault using the boundary element method and comparing these results with the observed values ​​show that the boundary element method is a suitable method for predicting issues such as slip distribution on the fault, surface displacement and slope instability. Also, the boundary element method is a suitable method for predicting the location of smaller and secondary faults. Therefore, it can be said that the boundary element method is a powerful tool for numerical modeling of earthquake or fault rupture dynamics. In the articles studied in this study, the slip distribution at the fault surface has been determined using the inversion of surface displacements resulting from the main earthquake. If aftershocks also cause surface displacements, then as a suggestion for future research, the inversion of surface displacements caused by aftershocks can also be used to determine the secondary slips created on the fault surface.