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摘要
In this paper, we present a characteristics-based approach for solving elastic wave problems with time-dependent traction boundary conditions. A generalized mathematical model for this important class of problems is expressed as a set of first-order, linear, hyperbolic partial differential equations. We analyze the mathematical structure of this first-order linear system, verify its hyperbolicity, derive its characteristic form, and deduce its eigenvalues,.eigenvectors, and Riemann invariants. The eigenvalues correspond to the wave speeds, while the Riemann invariants are used to construct a solution by the method of characteristics. We benchmark the method of characteristics against several popular modal approaches. Two of these, which we refer to as the concentrated body force method (CBFM) and the homogeneous eigenfunction expansion method (HEEM), were developed to simplify the well-established but tedious Mindlin-Goodman method. To homogenize the boundary conditions and enable modal analysis, the CBFM and HEEM forgo the usual formalism of linear transformations (A la Mindlin-Goodman) in favor of intuitive modeling assumptions and postulated solution structures. We find, however, that these approaches introduce an artificial stress discontinuity at the forced boundary in their reformulated problems. When these reformulated problems are solved by modal analysis, spurious oscillations and significant overshoot, similar to the Gibbs phenomenon, emerge in the stress profile at the artificial discontinuity. We demonstrate that these oscillations and overshoot are physical manifestations of a series solution for stress, obtained from term-by-term differentiation, that is not uniformly convergent, as required by the formalism of mathematical analysis. The method of characteristics solution, on the other hand, is exact to within machine precision, yielding no artificial discontinuities, spurious oscillations, or unphysical overshoot. Unlike the modal approaches, the method of characteristics solves the first-order problem with time-dependent boundary conditions 'as is' without any reformulation, restructuring, or postulated solution structures. Further, its solutions require no post-processing: no a posteriori solution treatment like l'Hopital's rule to accommodate resonance (resonant behavior is inherently captured without the emergence of singularities), no term-by-term differentiation to deduce stress (stress is primitive in a first-order velocity-stress formulation of elastodynamics), and no convergence tests.
- 出版日期2014-2-3
