A three‐dimensional hybrid finite element — spectral boundary integral method for modeling earthquakes in complex unbounded domains

Gabriele Albertini,Ahmed E Elbanna,David S Kammer

doi:10.1002/nme.6816

Abstract

AbstractWe present a 3D hybrid method which combines the finite element method (FEM) and the spectral boundary integral method (SBIM) to model nonlinear problems in unbounded domains. The flexibility of FEM is used to model the complex, heterogeneous, and nonlinear part— such as the dynamic rupture along a fault with near fault plasticity—and the high accuracy and computational efficiency of SBIM is used to simulate the exterior half spaces perfectly truncating all incident waves. The exact truncation allows us to greatly reduce the domain of spatial discretization compared to a traditional FEM approach, leading to considerable savings in computational time and memory requirements. The coupling of FEM and SBIM is achieved by the exchange of traction and displacement boundary conditions at the computationally defined boundary. The method is suited to implementation on massively parallel computers. We validate the developed method by means of a benchmark problem. Three more complex examples with a low velocity fault zone, low velocity off‐fault inclusion, and interaction of multiple faults, respectively, demonstrate the capability of the hybrid scheme in solving problems of very large sizes. Finally, we discuss potential applications of the hybrid method for problems in geophysics and engineering.

Highlights

Earthquakes are a prime example of complex natural processes with far-from-equilibrium nonlinear dynamics at multiple scales. 9 The lack of quantitative data on timescales capturing multiple large earthquake cycles is a fundamental impediment for progress in the field
Physics-based simulations provide the only path for overcoming the lack of data and elucidating the multi-scale 11 dynamics and spatio-temporal patterns that extend the knowledge beyond sporadic case studies and regional statistical laws
The current work extends recent 62 work by the authors and their groups over the past few years which first developed the hybrid scheme for the 2D dynamic anti63 plane problem combining finite difference and spectral boundary integral methods, 28 and the 2D dynamic in-plane problem 64 using the finite element method for bulk discretization in the hybrid scheme

Summary

Introduction

Earthquakes are a prime example of complex natural processes with far-from-equilibrium nonlinear dynamics at multiple scales. 9 The lack of quantitative data on timescales capturing multiple large earthquake cycles is a fundamental impediment for progress in the field. Numerical methods based on bulk discretization such as the finite difference (FD) and finite element methods have been used in simulating earthquake ruptures since mid-1970s and early 1980s with the pioneering works of Boore et al, 5 Andrews, 6 Das & Aki, 7 Archuleta & Day, 8 Day, 9 Virieux & Madariaga, 10 and others These methods are more flexible than the boundary integral approaches in handling heterogeneities, nonlinearities, and fault geometry complexities (see Fig. 1a&b). 27 This is partially due to the high spatial discretization cost and the lack of a systematic approach to handle both dynamic and quasi-dynamic calculations in the same framework which is required for simulating both earthquake ruptures and intersesismic slow deformations Another challenge in these methods is defining fault loading. The current work extends recent work by the authors and their groups over the past few years which first developed the hybrid scheme for the 2D dynamic anti plane problem combining finite difference and spectral boundary integral methods, 28 and the 2D dynamic in-plane problem 64 using the finite element method for bulk discretization in the hybrid scheme. 29 Prior work has demonstrated the accuracy and 65 computational efficiency of the coupled approach and its potential for modeling dynamic ruptures with high resolution fault a infinite medium with known Green's function b c absorbing boundary

Methods

Results

Discussion

Conclusion