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In this paper we deal with an implementation as well as numerical experiments for the coupling of interior and exterior problems of the elastodynamic wave equation with transparent boundary conditions in 3D as described in a previous paper by this author. In more detail, the FEM‐BEM‐coupling as well as the time discretization by using leapfrog and convolution quadrature is considered. Our aim is to provide an insight into the necessary steps of the implementation. Based on this, we present numerical experiments for a non‐convex domain and analyze the errors.
In this survey paper, we present a multiscale post-processing method in exploration. Based on a physically relevant mollifier technique involving the elasto-oscillatory Cauchy–Navier equation, we mathematically describe the extractable information within 3D geological models obtained by migration as is commonly used for geophysical exploration purposes. More explicitly, the developed multiscale approach extracts and visualizes structural features inherently available in signature bands of certain geological formations such as aquifers, salt domes etc. by specifying suitable wavelet bands.
FEM–BEM coupling for the thermoelastic wave equation with transparent boundary conditions in 3D
(2022)
We consider the thermoelastic wave equation in three dimensions with transparent boundary conditions on a bounded, not necessarily convex domain. In order to solve this problem numerically, we introduce a coupling of the thermoelastic wave equation in the interior domain with time-dependent boundary integral equations. Here, we want to highlight that this type of problem differs from other wave-type problems that dealt with FEM–BEM coupling so far, e.g., the acoustic as well as the elastic wave equation, since our problem consists of coupled partial differential equations involving a vector-valued displacement field and a scalar-valued temperature field. This constitutes a nontrivial challenge which is solved in this paper. Our main focus is on a coercivity property of a Calderón operator for the thermoelastic wave equation in the Laplace domain, which is valid for all complex frequencies in a half-plane. Combining Laplace transform and energy techniques, this coercivity in the frequency domain is used to prove the stability of a fully discrete numerical method in the time domain. The considered numerical method couples finite elements and the leapfrog time-stepping in the interior with boundary elements and convolution quadrature on the boundary. Finally, we present error estimates for the semi- and full discretization.
We deal with the reconstruction of inclusions in elastic bodies based on monotonicity methods and construct conditions under which a resolution for a given partition can be achieved. These conditions take into account the background error as well as the measurement noise. As a main result, this shows us that the resolution guarantees depend heavily on the Lamé parameter μ and only marginally on λ.
We deal with the shape reconstruction of inclusions in elastic bodies. For solving this inverse problem in practice, data fitting functionals are used. Those work better than the rigorous monotonicity methods from Eberle and Harrach (Inverse Probl 37(4):045006, 2021), but have no rigorously proven convergence theory. Therefore we show how the monotonicity methods can be converted into a regularization method for a data-fitting functional without losing the convergence properties of the monotonicity methods. This is a great advantage and a significant improvement over standard regularization techniques. In more detail, we introduce constraints on the minimization problem of the residual based on the monotonicity methods and prove the existence and uniqueness of a minimizer as well as the convergence of the method for noisy data. In addition, we compare numerical reconstructions of inclusions based on the monotonicity-based regularization with a standard approach (one-step linearization with Tikhonov-like regularization), which also shows the robustness of our method regarding noise in practice.