A Damped Dynamic Finite Difference Approach for Modeling Static Stress–Strain Fields
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Modeling dynamic and static responses of an elastic medium often employs different numerical schemes. By introducing damping into the system, we show how the widely used time-marching staggered finite difference (FD) approach in solving elastodynamic wave equation can be used to model time-independent elastostatic problems. The damped FD method can compute elastostatic stress and strain fields of a model subject to the influence of an external field via prescribed boundary conditions. We also show how to obtain an optimized value for the damping factor. We verified the damped FD approach by comparing results against the analytical solutions for a borehole model and a laminated model. We also validated our approach numerically for an inclusion model by comparing the results computed by a finite element method. The damped FD showed excellent agreement with both the analytical results and the finite element results.
KeywordsFinite difference static stress–strain fields numerical modeling damped wave equation effective medium
We thank Professor Leon Thomsen for discussions about the physical meaning of the damping parameter. We thank Dr. Nishank Saxena for pointing to the finite element code. We appreciate the financial support from UH and SUSTC. We appreciate the UH Xfrac group for providing clusters for computations. We also appreciate the comments from Dr. Hao Hu and Dr. Chen Qi. R. Lin and Y. Zheng are partially supported by NSF EAR-1833058. X. Fang is supported by National Natural Science Foundation of China (grant 41704112) and National Key R&D Program of China (2018YFC0310105).
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