The u‐p approximation versus the exact dynamic equations for anisotropic fluid‐saturated solids. I. Hyperbolicity

The numerical solution of dynamic problems for porous fluid-saturated solids is often performed with the use of simplified equations known as the u-p approximation. The simplification of the equations consists in neglecting some acceleration terms, which is justified for a certain class of problems related, in particular, to geomechanics and earthquake engineering. There exist two u-p approximations depending on how many acceleration terms are neglected. All comparative studies of the exact and u-p formulations are focused on the question of how well the u-p solutions approximate those obtained with the exact equations. In this paper, the equations are compared from a different point of view, addressing the question of well-posedness of boundary value problems. The exact equations must be hyperbolic and satisfy the corresponding hyperbolicity conditions for the boundary value problems to be well posed. The u-p equations are not of the form to which the conventional definition of hyperbolicity applies. A slight extension of the approach makes it possible to derive hyperbolicity conditions as necessary conditions for well-posedness for the u-p approximations. ... mehrThe hyperbolicity conditions derived in this paper for the u-p approximations are formulated in terms of the acoustic tensor of the skeleton. They differ essentially from the hyperbolicity conditions for the exact equations.