This paper is concerned with the development of simple, yet accurate, absorbing boundaries that can be incorporated readily into existing finite element programs directly in the time domain. The wave equation in a two‐dimensional exterior domain is considered as a prototype situation. For this problem a family of approximate absorbing boundary elements is constructed by an asymptotic expansion procedure. Each element is completely determined by a pair of local, symmetric, stiffnesslike and dampinglike constant matrices, coupled only through adjoining nodes. This makes it possible to combine the new boundary elements with standard finite elements used to represent the interior domain via ordinary assembly procedures, while preserving the symmetry and bandwidth of the global matrices. Results of numerical experiments for a circular geometry confirm that accuracy increases with the order of the approximations, with the radius of the absorbing boundary, and with the dominant frequencies of the excitation. This implies that the higher the frequency of the excitation the smaller the size of the buffer annulus required to attain a desired accuracy.

We discuss a total wavefield-based inversion approach for the reconstruction of the material profile of heterogeneous semi-infinite domains, directly in the time domain, based on surficial measurements of the domain's response to prescribed wave illumination. The ability to recover the in-depth profile of moduli/wave velocities typically associated with geotechnical site characterization applications is of particular interest. We address four key issues associated with the wavefield-based inversion: (a) to limit the semi-infinite extent of the heterogeneous physical domain, a perfectly-matched-layer (PML) is introduced at the truncation interface; (b) to account for the introduction of the PML, we use a mixed unsplit-field PML formulation for the coupled PML-regular-domain problem; (c) to tackle the inversion, we adopt a partial-differential-equation (PDE)-constrained optimization framework that formally leads to a classic Karush–Kuhn–Tucker (KKT) system comprising the initial-value state, final-value adjoint and time-independent control problems; and (d) to narrow the feasibility space and alleviate the inherent solution multiplicity, we discuss Tikhonov and Total Variation regularization schemes, endowed with a regularization factor continuation algorithm. We also limit the total observation time to optimally account for the domain's heterogeneity during inversion iterations. We report on the theory and results that lead efficiently to the reconstruction of both sharp and smooth profiles in one dimension.