Optimized Finite Difference Schemes for Wave Propagation in High Loss Viscoelastic Material

An efficient time domain method is needed to analyze problems containing high loss viscoelastic materials with complex geometry. One presently available method for low loss, constant , materials is the finite difference method with memory variables attributed to Carcione et al., and Blanch et al. [2, 5]. In this method, the constitutive relation is approximated by a sum of decaying exponential functions with matched or optimized relaxation spectra. This allows the time domain convolutions, appearing in the constitutive relation, to be eliminated at the cost of additional field variables or "memory variables". The total number of field variables is roughly doubled for both 2D and 3D models with attendant increases in computer speed and storage requirements. Greater efficiencies are needed to make this method more competitive. In this thesis, I investigate the idea of cancelling the error due to the constitutive relation approximations (optimization error) with the error due to the finite difference time domain model (discretization error) by a process referred to as reoptimiztion. Error cancellation is possible because the discretization error is completely predictable and can be accounted for during the optimization procedure. To this end, the work of Blanch et al. is extended to high loss factor materials with } = O (1) by matching both real and imaginary parts of the complex modulus with experimental data and implemented in a finite difference model using a Predictor-Corrector scheme with 3-point centered spatial differencing. Using this model, the reoptimization process is compared to the optimization process for three cases: 1.) high loss factor, narrow band, moderate Courant number; 2.) high loss factor, wide band, moderate Courant number; and 3.) high loss factor, wide band, low Courant number. Results show that, for a given accuracy, if the reoptimization process is used, then the temporal and spatial step sizes of the finite difference model are roughly doubled for all three cases. This represents a decrease in model run time by 8 times in 2D and 16 times in 3D. The corresponding reduction in storage requirements is 4 times in 2D and 8 times in 3D. Acknowledgments I would like to first thank my family, without their support and encouragement, I simply would not be here. Special thanks to my sister, Maryhelen, for grammatical editing of this beastly thing. m sure there are errors, still, but you don't need a new career. Thanks to my advisor, Rob Fricke, for his patient …