In a time-harmonic setting, coupling of a high-dimensional [HighD-two-dimensional (2D) or three-dimensional (3D)] sub-model and a low-dimensional (LowD, 1D) sub-model (reduced from the former when circumstances allow such a reduction) is considered, forming a hybrid mixed-dimensional model. In particular, the coupling of 2D or 3D elasticity with a Bernoulli-Euler beam is under investigation for time-harmonic bending problems. A finite element scheme is applied, where the HighD part is modeled via standard isoparametric elements, and the LowD part is provided with C1 continuity by using Hermite-cubic elements. The paper proposes special procedures performed at the discrete level to couple the two sub-domains. Continuity of the displacements and bending rotations is partially enforced strongly and partially weakly. The performance of the method is illustrated by several numerical examples.

The solution of time-dependent inverse wave problems in nonlinear media is considered, using Full Waveform Inversion (FWI). The goal is the identification of parameters related to the nonlinear material behavior. The adjoint method is employed for the efficient calculation of the functional gradient. A general formulation of the method at the semi-discrete level is presented. Two specific models are used to experiment with this formulation: a miniature model that lacks physical meaning, and a model of the deflection of a membrane made of a softening material. The method is specialized to these two models, and numerical experiments demonstrate its performance in each case. Computational aspects that are considered include assessing the success of the identification, and evaluating the performance of the method in the presence of measurement noise.

Wave propagation is considered in a two-dimensional (2D) elastic structure, which includes a relatively small region whose behavior is fully 2D and a long and slender region whose bending behavior is like that of a Timoshenko beam. To save in computational effort, the latter region is reduced to a genuinely one-dimensional (1D) Timoshenko beam. The mathematical and computational problem posed then involves the coupling of the two regions, such that a well-posed, accurate, numerically stable and efficient hybrid 2D-elastic-Timoshenko-beam model is formed. The appropriate interface conditions are derived, and the well-posedness of the time-dependent problem is proved. A new computational coupling method is proposed, where the shape functions associated with the axial degrees of freedom on the interface of the elastic solid are modified in a special manner, to allow for the rotation continuity. The method results in a symmetric, positive and stable finite element formulation. Numerical examples are presented which demonstrate the performance of the scheme.

The problem of identifying the precise shape and material properties of a soft inclusion in a two-dimensional elastic medium is considered. The identification is performed based on time-dependent displacement response to a given wave source, measured by sensors at discrete locations. "Inclusion" here means an unknown local region whose material properties differ significantly from those of the background medium, which are assumed to be known. Applications include identification of unknown scatterers in solid earth geophysics and non-destructive testing of structures. The identification method is based on Full Waveform Inversion (FWI) and an adjoint scheme. In a previous publication, a precise shape identification method was developed for a cavity of an arbitrary shape in an acoustic medium (or a hole in a membrane). The present paper extends this publication in three ways. First, the medium is elastic rather than acoustic. Second, the scatterer is an inclusion rather then a cavity (the latter can be regarded as the limiting case of an inclusion). Third, an attempt is made to identify both the precise shape and the material properties of the inclusion, simultaneously. This attempt is only partly successful, and the reasons for this are analyzed. Numerical examples are used to demonstrate the proposed methodology.

The Hagstrom-Warburton (HW) boundary operators play an important role in the development of high-order computational schemes for problems in unbounded domains. They have been used on truncating boundaries in the formulation of a sequence of high-order local Absorbing Boundary Conditions (ABCs) and in the Double Absorbing Boundary (DAB) method. These schemes proved to be very accurate, efficient, and generalizable for various wave equations and complex media. Yet, Finite Element (FE) formulations incorporating such high-order ABCs or DAB lack symmetry and positivity. As a result, they suffer from some deficiencies, namely (a) they do not allow the use of an explicit time-stepping scheme, since the global mass matrix and/or damping matrix are non-symmetric, and lumping is unsafe, (b) their stability is difficult to control under certain conditions, and (c) they render the fully-discrete problem non- symmetric even if the original problem in the unbounded domain is self adjoint, hence prevent the use of a symmetric algebraic solver. In this paper the HW-ABC for the scalar wave equation is applied to 2D waveguide configurations. It is manipulated in such a way that it leads to a symmetric FE-ABC formulation with positive definite matrices. The new symmetric formulation is achieved by applying a number of operations to the HW condition: first, combining each pair of recursive relations into one relation, then using the wave equation for each auxiliary function, and finally integrating the resulting ABC in time. The latter is the crucial step in the new method. The proposed method is free from the deficiencies (a)-(c) mentioned above. The DAB method undergoes a similar treatment. In this case, one of the matrices is slightly asymmetric, but deficiencies (a) and (b) are still prevented. The stability and accuracy of the new formulations are discussed, and their performance is demonstrated via numerical examples.

Coupling of a two-dimensional (2D) sub-model and a one-dimensional (1D) sub-model, to form a hybrid mixed-dimensional model, is considered for the linear scalar wave equation. A Dirichlet-to-Neumann (DtN) method is used to perform this coupling, which is based on enforcing the continuity of the DtN map across the 2D-1D interface. The DtN map relates the primary variable to the flux, on each side of the interface. A previously proposed DtN-coupling formulation was monolithic, namely the simulation had to be simultaneously performed with the combined 2D and 1D sub-models. In contrast, here a new sequential formulation is proposed, in which the 2D problem is solved separately, after a fundamental 1D problem is solved. Thus, in the new method the 2D-1D coupling is one-way. This has a clear advantage over the monolithic formulation, as the 2D and 1D problems can be solved using different codes, and with space and time discretizations that do not have to match at the interface. The proposed method is applied here in conjunction with a Finite Element (FE) formulation. Numerical examples demonstrate the performance of the method.

The inverse problem of identifying a small unknown inclusion or cavity in an elastic medium is considered. Important applications are structural damage identification, medical imaging and geophysical exploration; the latter is the focus of the present investigation. It is assumed that the (isotropic but possibly heterogeneous) material properties of the medium are known, except for the presence of a small inclusion of a different material or of a small cavity. The goal is to find the location, size and shape of the inclusion or cavity. Identification is performed using full waveform inversion (FWI) and the adjoint method for the efficient calculation of the gradient of the misfit function. The identification is accomplished by considering a single unknown material-property field. The inclusion manifests itself in the inverse solution as a local region where the unknown material property becomes significantly different than the known background property. The limiting case of cavity identification requires special treatment in the minimization process to avoid failure, as Burchner et al. showed empirically for the scalar wave equation. Their & rho;$$ \rho $$-scaling approach, whose success is explained mathematically here for the first time, is extended to elastodynamics. The performance of the proposed method is demonstrated via numerical examples, involving two geophysical models: a homogeneous model and the heterogeneous Marmousi model, which is a standard testing model in geophysical research.

We combine vision transformers with operator learning to solve diverse inverse problems described by partial differential equations (PDEs). Our approach, named ViTO, combines a U-Net based architecture with a vision transformer. We apply ViTO to solve inverse PDE problems of increasing complexity, namely for the wave equation, the Navier-Stokes equations and the Darcy equation. We focus on the more challenging case of super-resolution, where the input dataset for the inverse problem is at a significantly coarser resolution than the output. The results we obtain are comparable or exceed the leading operator network benchmarks in terms of accuracy. Furthermore, ViTO`s architecture has a small number of trainable parameters (less than 10% of the leading competitor), resulting in a performance speed-up of over 5x when averaged over the various test cases.

Kirchhoff Migration (KM), sometimes called Arrival (or Travel) Time Imaging, is a basic and popular imaging technique based on the arrival time of waves from given sources to given sensors. It is commonly used in the fields of underwater acoustics and solid earth geophysics, for both subsurface structure analysis and for identifying unknown local obstacles (scatterers) in the medium. The present paper concentrates on the latter application. For acoustics, the KM algorithm is extremely simple and efficient, although it usually produces a rather crude image, which is the reason for its use as the method of choice when high resolution is not needed, or as a fast technique to produce an initial guess for a more sophisticated imaging method. For elasticity, KM is much more involved, as the arrival-time algorithm is not obvious, mainly since there is more than one wave speed at each spatial point. In this paper, a new KM scheme is proposed for obstacle identification in an isotropic piecewise-homogeneous elastic medium. The scheme is based on measuring two quantities that are second-order operators of the displacement field, which are related to P and S waves, and applying the acoustic KM algorithm to each of them, with the appropriate wave speed. It is demonstrated numerically that the operator related to S waves results in very good identification in many cases. The fact that measurements based on the S-related operator are preferred over those based on the P-related operator is an empirical observation, and awaits full analysis, although a partial explanation is given here.

The problem of non-destructively identifying the damage severity in a known damage region of a structure is considered. The damage is modeled as a reduction in Young’s modulus of the material in a local region. In the proposed model-based identification technique, wave sources are activated on the boundary of the structure, which generate traveling elastic waves. These waves travel from the sources to the damage region, and from there reach sensors, also located on the structure’s boundary. The sensors measure the time-dependent response of the structure, by recording certain displacement components at many discrete times. In the current work, all the data are generated synthetically, with an added artificial measurement noise. Based on these data, the damage severity in the various damage regions is estimated by solving an inverse minimization problem, involving a non-convex misfit functional. The minimization is performed using the Method of Moving Asymptotes. It is demonstrated, via several numerical examples, that it is possible to achieve accurate identification using this technique, sometimes even with a single sensor which is remote from the damage region and measures a single displacement component. Of course, to prove the practicality of the proposed method, laboratory experimentation is required, which is beyond the scope of this paper.

On the occasion of 60 years to a seminal paper by the Swedish mathematician Germund Dahlquist, this review paper starts by discussing the celebrated Dahlquist's barriers, which are theoretical limitations on the accuracy and stability properties of a broad and important class of time-stepping methods, i.e., Linear Multistep (LMS) methods, for the solution of initial value problems. Perhaps the two most dramatic Dahlquist barriers are the one that precludes an explicit LMS method from being unconditionally stable, and the one that precludes a high-order LMS method from being unconditionally stable. We discuss Dahlquist's barriers and also later barriers proved by other authors. We then discuss some time-stepping methods which seemingly break at least one of Dahlquist's barriers. Of course, the explanation of this “paradox” is that these are not LMS methods, so the barriers do not necessarily apply to them. We relate to two types of barrier breakers: those that break stability barriers (explicit unconditionally-stable methods), and those that break order barriers (high-order explicit or unconditionally-stable methods). We also review some current advanced time-integration techniques which significantly deviate from the LMS format.

This paper deals with the inverse problem of accurately identifying the shape, size and location of 2D objects or voids in a given medium via the measurement of waves scattered from them. Examples of scatterers include an obstacle in a 2D acoustic medium or a small hole in a membrane. To fix ideas, the latter example, governed by the scalar wave equation, serves here in demonstrating the method. In the proposed method, a known source generates waves that propagate in the membrane, and these time-dependent waves are measured by a small number of sensors located at chosen points. Based on these measurements, an iterative process is performed to find the closed curve that represents the boundary of the scatterer. Mathematically, the iterative process aims at minimizing a cost functional that represents the difference between the measured wave signals and the wave signals obtained in the presence of a candidate scatterer. The gradient of this functional, which represents its sensitivity to the scatterer geometry, is calculated efficiently using the adjoint method. Finite elements are used to discretize the spatial domain, the Newmark method is used for time-stepping, and a smoothed gradient method is used for the functional minimization. The unknown boundary of the scatterer is discretized from the outset, which allows a simpler derivation of the adjoint procedure compared to a fully variational treatment. The unknown curve is defined by a parametric representation, that is completely general and does not make any preliminary assumptions about the scatterer geometry. Multiple sources are used to enhance the identification. The performance of the proposed method is demonstrated via numerical examples. It is shown that excellent identification is obtained even in the presence of noise, with a small number of sensors, and with no need for regularization (other than that induced by the discretization).

The paper falls into the category of computational methods for inverse scattering techniques for the identification of scatterers. We consider a linear elastodynamic problem and compare two popular methods for identifying a scatterer in the domain. Finite elements are employed with each of the two methods for spatial discretization. One method considered is Full Waveform Inversion using a gradient-based optimization and the adjoint method. In the adjoint procedure for calculating the gradient, we use the variant of discretizing the unknown parameters from the outset while all other variables remain continuous. Gradient optimization is performed in the examples using a quasi-Newton method. The other method compared is the computational Time Reversal technique, which is used in combination with an augmentation procedure to enhance performance. advantages and limitations of the two methods are outlined, and their performance is compared through an example from geophysics.

The coupling of three-dimensional (3D) and plane (2D) finite element (FE) models to form a single hybrid 3D–2D model is considered for linear elastodynamic problems. The 2D model is used to represent a large thin 3D computational domain where the solution behaves approximately in a 2D (plane-elasticity) way. Problems where the normal displacement is important in the thin region (bending) are excluded. The hybrid model, if designed properly, is a more efficient way to solve the full 3D model over the entire problem. This paper focuses on the way the 3D–2D coupling is performed, and on the coupling error generated. The Panasenko technique is used to couple the 3D and 2D models. This method has been used previously for mixed-dimensional coupling in steady-state problems, as well as for 2D–1D coupling of acoustic waves. Here it is being used for the first time for the 3D–2D coupling of time-dependent elastic problems. The hybrid formulation is derived, and is shown to be well-posed. It is shown that the Panasenko method is extremely simple to implement in the framework of FE analysis, yet the resulting coupling yields a rather small error level, provided the 3D–2D interface is sensibly placed. The numerical accuracy and efficiency of the method are explored for a couple of example problems. In particular, it is shown that spurious reflections from the interface back into the 3D region are negligible. • A hybrid 3D–2D model is constructed for elastodynamics. • The 3D–2D coupling is performed using the Panasenko technique. • The hybrid formulation is shown to be well posed. • Numerical experimentation is performed with a few configurations. • Excellent agreement is obtained between the proposed and reference models.

We approximate the underwater acoustic wave problem for locating sources in that medium. We create a time dependent synthetic data-set of sensor recorded pressures, based on a small set of sensors placed in the domain, and perturb this data with high random multiplicative noise. We show that reference time-reversal based method struggles with high noise, and a naive deep-learning method also fails. We propose a method, based on physically-informed neural-networks and time-reversal, for approximating the source location even in the presence of high sensors noise.

This paper is a basic tutorial on the adjoint method when used in a computational scheme for solving an inverse problem. The adjoint method is a technique for the efficient calculation of the gradient of the functional which is to be minimized in the solution process. The method is presented in a slightly non-standard way, which is believed to be simpler and less abstract than the common presentation found in most books and papers, yet equally general. More specifically, the unknown parameters are discretized from the outset while all other variables remain continuous. The adjoint method is applied here both at the continuous level and at the discrete level. Both steady-state (elliptic) and time-dependent problems are considered. Various computational aspects are discussed.

Coupling of a 2D model and a 1D model, to form a hybrid mixed-dimensional model, is considered in the context of elastic wave propagation. A Dirichlet-to-Neumann (DtN) method is used to perform this coupling. This approach is an extension of previous work (which was applied to steady-state wave problems) to the time-dependent regime. It is based on enforcing the continuity of the DtN map, relating the displacements to the tractions, on the 2D-1D interface. To apply the DtN map, the approach of discretization in time first (the Rothe method) is adopted, resulting in an elliptic problem at each time step. The more typical case, where longitudinal waves dominate in the 1D sub-domain, is considered first. Then the more general case is considered, where transverse waves are present as well, and several ways to handle it are discussed. The proposed DtN approach is compared to the simpler Panasenko semi-weak approach, and is shown to be advantageous, in particular in the presence of transverse waves. (c) 2021 Elsevier Inc. All rights reserved.

In this classroom note, the old and well-known plane-stress elastic class of problems is revisited, using an analysis technique which is different than that commonly found in the literature, and with a pedagogical benefit. An asymptotic analysis is applied to problems of thin linear elastic plates, made of a homogeneous and rather general anisotropic material, under the plane stress assumption. It is assumed that there are no body forces, that the boundary conditions are uniform over the thickness, and that the material (hence also the solution) is symmetric about the middle plane. The small parameter in this analysis is $\epsilon =t/D$ where $t$ is the (uniform) thickness of the plate and $D$ is a measure of its overall size. The goal of this analysis is to show how the three-dimensional (3D) problem of this type is reduced asymptotically to a sequence of essentially two-dimensional (2D) problems for a small $\epsilon $ . As expected, the leading problem in this sequence is shown to be the classical plane-stress problem. The solutions of the higher-order problems are corrections to the plane-stress solution. The analysis also shows that all six 3D compatibility equations are satisfied as $\epsilon $ goes to zero, and that the error incurred by the plane stress assumption is $O(\epsilon ^{2})$ . For the special case of an isotropic in-the-plane material, the second-order solution is shown to be the exact solution of the 3D problem, up to an $O(\epsilon ^{2})$ error in the close vicinity of the edge (which agrees with a well-known result for an isotropic material).

The problem of identifying an obstacle (scatterer) in the form of a cavity in a 2D geophysical medium is considered. This is posed as an inverse wave problem, where the location of the cavity is sought based on measurements of the elastic waves recorded by sensors located at certain points in the domain. The sensor measurements are noisy, and are generated synthetically as a first step. The inverse problem is solved by seeking the minimum of a specially designed cost functional, based on a computational time reversal (TR) procedure, where waves are radiated back in time from the sensors. The cost functional is defined to measure, for each scatterer candidate in the search space, the quality of the refocusing of the backward‐propagating waves on the given wave source at the end of the TR process. While the basic idea has appeared in previous publications, here it is applied to a 2D heterogeneous geophysical model (albeit a relatively simple one), and is enhanced by using several auxiliary techniques. These include (a) an “augmentation” technique, which is used to strengthen the coherent information (and thus to weaken the noncoherent information) by solving an elliptic problem at each time step; (b) experimentation with both instantaneous (impact) sources and time‐harmonic sources of finite duration; (c) combining the identification results of several sources and several source wavelengths to enhance identification; (d) reducing the size of the search space by a special “zooming in” technique; and (e) defining a performance index to assess the method's success and to provide a measure of confidence in the identification result in each specific case. Several numerical experiments are presented that demonstrate the performance of the proposed schemes. The sensitivity of the identification process to various parameters of the scatterer, the measurements, and the medium is investigated.

The Double Absorbing Boundary (DAB) is a recently proposed absorbing layer used to truncate an unbounded domain with high-order accuracy. While it was originally designed for time-dependent acoustics and elastodynamics, here the DAB construction is adapted and applied to the 2D Helmholtz equation. Both wave-guide and corner configurations are considered. A high-order spectral finite element scheme is used in order to match the discretization accuracy to the accuracy of the DAB. The DAB scheme is analyzed, and numerical experiments demonstrate its performance.