The displacement model of the hybrid-Trefftz finite element is formulated for elastodynamic problems defined on unsaturated soils. The mathematical formulation is based on the theory of mixtures with interfaces. The model considers the full coupling between the solid, fluid and gas phases, including the effects of relative (seepage) accelerations. The hyperbolic problem is integrated in time using a step-by-step implicit scheme that transforms it into a series of elliptic problems in space. The free-field solutions of these problems are derived in cylindrical coordinates and used to construct the domain approximation of the hybrid-Trefftz displacement element. This builds relevant physical information in the approximation basis, increasing the convergence of the elements under p-refinement and their robustness to wide variations of the frequency of the propagating wave.
SUMMARYA new indirect approach to the problem of approximating the particular solution of non-homogeneous hyperbolic boundary value problems is presented. Unlike the dual reciprocity method, which constructs approximate particular solutions using radial basis functions, polynomials or trigonometric functions, the method reported here uses the homogeneous solutions of the problem obtained by discarding all timederivative terms from the governing equation. Nevertheless, what typifies the present approach from a conceptual standpoint is the option of not using these trial functions exclusively for the approximation of the particular solution but to fully integrate them with the (Trefftz-compliant) homogeneous solution basis. The particular solution trial basis is capable of significantly improving the Trefftz solution even when the original equation is genuinely homogeneous, an advantage that is lost if the basis is used exclusively for the recovery of the source terms. Similarly, a sufficiently refined Trefftz-compliant basis is able to compensate for possible weaknesses of the particular solution approximation. The method is implemented using the displacement model of the hybrid-Trefftz finite element method. The functions used in the particular solution basis reduce most terms of the matrix of coefficients to boundary integral expressions and preserve the Hermitian, sparse and localized structure of the solving system that typifies hybrid-Trefftz formulations. Even when domain integrals are present, they are generally easy to handle, because the integrand presents no singularity.
The equations that govern the dynamic response of saturated porous media are first discretized in time to define the boundary value problem that supports the formulation of the hybrid-Trefftz stress element. The (total) stress and pore pressure fields are directly approximated under the condition of locally satisfying the domain conditions of the problem. The solid displacement and the outward normal component of the seepage displacement are approximated independently on the boundary of the element. Unbounded domains are modelled using either unbounded elements that locally satisfy the Sommerfeld condition or absorbing boundary elements that enforce that condition in weak form. As the finite element equations are derived from first-principles, the associated energy statements are recovered and the sufficient conditions for the existence and uniqueness of the solutions are stated. The performance of the element is illustrated with the time domain response of a biphasic unbounded domain to show the quality of the modelling that can be attained for the stress, pressure, displacement and seepage fields using a high-order, wavelet-based time integration procedure.
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