ReviewiBem3D, a three-dimensional iterative boundary element method using angular dislocations for modeling geologic structures
Introduction
The rapid increase in the number of geologic, seismologic, and geodetic datasets with abundant and very precise spatial information on fault geometry and slip distributions promotes the development of more complex geometric and kinematic models of modern earthquake ruptures and paleoseismic events. These datasets indicate that faults commonly are composed of multiple discrete segments, each with a curved surface and curved tipline. Construction of model fault segments using multiple rectangular dislocations (Okada, 1985) introduces nonphysical gaps and overlaps with associated stress concentrations and irregularities in slip distributions that may differ significantly from those in nature (Maerten et al., 2005). Discretization of fault segments into a set of triangular dislocations enables one to approximate the curviplanar surfaces and curved tiplines to a precision that is consistent with the data (Jeyakumaran et al., 1992, Thomas, 1993, Maerten et al., 2005, Meade, 2007, Maerten et al., 2010, Maerten, 2010a, Maerten, 2010b).
The C computer code that was originally developed at Stanford University by Andrew Thomas (1993) in 1993 was called Poly3D. The idea of using the angular dislocation formalism to construct complex planar dislocations with constant displacement discontinuity was first used by Jeyakumaran et al. (1992). Since then, because of the rapid evolution of computer power and the constant demand for more complex and larger models, a new code has emerged, following the work of Jeyakumaran et al. (1992) for triangular elements. For the new code, iBem3D, the C++ object-oriented language was chosen, and an iterative solver now replaces the older direct solver (Gauss elimination). C++ allows modularity of the code (Maerten and Maerten, 2008, Maerten et al., 2010, Maerten, 2010a) while the iterative solver permits running larger models in a shorter time (Maerten et al., 2010). The strain field, given by the derivatives of the equations for the displacement field provided by Comninou and Dundurs (1975), was entirely rederived by hand for optimization considerations, whereas these derivatives in Poly3D were symbolically derived using a dedicated software. The call to the core equations now runs four times faster. Comparisons of Poly3D and iBem3D are summarized in Fig. 1, where the technological differences are highlighted.
In this paper, we summarize the theory behind iBem3D, along with verifications (Section 2), and present the latest improvements such as the implementation of material heterogeneity, static friction, optimizations, parallelization, linear-slip inversion, and paleostress recovery (Section 3). Finally, academic, research, and industrial applications are discussed in Section 4.
Section snippets
Theory behind iBem3D
The theory of dislocations in elastic materials has been used widely over the past half century to evaluate the displacement, strain, and stress fields around faults in Earth׳s lithosphere. Steketee, 1958b, Steketee, 1958a has discussed this theory and potential applications to geophysical problems in two papers. He reviewed Volterra׳s formulation for the dislocation problem and presented a method for the construction of Green׳s functions for the semi-infinite space containing a surface of
Enhancements to iBem3D
iBem3D incorporates several new enhancements thanks to the C++ object-oriented design.
Applications
Numerical models of rock deformation based on continuum mechanics can provide significant computational tools for the understanding of geologic structures and phenomena in the context of theoretical research, teaching, hydrocarbon exploration and production, as well as civil engineering.
Conclusions
The C computer program Poly3D has been applied to a wide variety of problems in academic and industrial structural geology since 1993, with over 130 published papers. iBem3D provides a new formulation of the 3D problem of multiple triangular dislocations arranged to model faults and fractures in an elastic whole- or half-space using the boundary element method. It offers significant enhancements over Poly3D, including modularity, an iterative solver, greater model size and complexity,
Acknowledgments
The authors thank M. David Barnett and Huajian Gao of Stanford University, Department of Material Sciences for their help in understanding the dislocation theory. Yann Lagalay is also greatly acknowledged for helping us to identify and correct the “shadow effect”. We also thank all the iBem3D and Poly3D users who have contributed for over two decades to making this code more stable by reporting problems. The two anonymous reviewers are also acknowledged.
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