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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.

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.

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).

We propose new forms of low-order absorbing boundary conditions (ABC) for time-dependent elastic waves in isotropic and anisotropic media. The configuration considered is that of a two-dimensional elastic waveguide . The ABC shares with the widely-known Lysmer–Kuhlemeyer (LK) boundary condition the ease of implementation, especially in a finite-element (FE) setting, while offering gains in accuracy over the LK-like condition (or the LK condition in the isotropic case). The new conditions are proved to be energy-stable, even in the case when inverse modes are present, for which the normal components of the phase and group velocities have an opposite direction. The developed ABCs are particularly convenient to use in a Finite Element setting as they preserve the symmetry and positivity of the formulation and are simple to implement. The ABC features several parameters, which are optimized for accuracy. The optimization criterion is based on the notion of the energy-rate reflection coefficient . We test the performance of the scheme via a numerical example. While the gain in accuracy is found to be moderate (around 20% relative to the LK condition), the main result of this investigation lies in the development of ABCs for anisotropic elastodynamics that are proved to be stable from the outset, and are amenable for accuracy optimization. We plan to design in a future study analogous stable and optimized high-order ABCs using the methodology developed here.

The coupling of two-dimensional (2D) and one-dimensional (1D) models to form a single hybrid 2D–1D model is considered, for the time-dependent linear wave equation. The 1D model is used to represent a 2D computational domain where the solution behaves approximately in a 1D way. This hybrid model, if designed properly, is a more efficient way to solve the full 2D model for the entire problem. The paper focuses on the way the 2D–1D coupling is done, and on the coupling error generated. The Nitsche method is used for the mixed-dimensional coupling, thus extending previous work, which considered steady-state problems, to the time-dependent case. The hybrid formulation is derived, and the numerical accuracy and efficiency of the method is explored using numerical examples. The performance of this coupling method is compared to that of the simpler Panasenko method. It is shown that in some cases the former is more tolerant to a poor choice for the location of the 2D–1D interface.

We propose a computational method which improves the identification of sources and scatterers in a two-dimensional elastic medium using the Time Reversal (TR) technique. The identification procedure makes use of partial numerical data available at a small number of sensors. While in realistic applications these data are obtained by lab or field measurements, in the present work they are synthesized by solving the corresponding forward problem. We introduce an “augmentation” procedure that is utilized in addition to the TR technique and investigate its benefits. The augmentation procedure involves the solution of a suitable elliptic problem at selected time steps. In practice, one needs to solve, as a pre-process, a small number of “fundamental” elliptic problems which is equal to the number of measured quantities. A version of the augmentation procedure, which is easier to implement and involves almost no computational cost, is also presented. The local elastic energy in the domain is employed to decide whether the TR solution has properly refocused at the source. Cost functionals, whose minima are sought, are constructed for the source and scatterer identification problems. We conclude that augmentation improves the identification performance, in particular in the presence of noisy measurements. In some cases augmented TR succeeds in the identification whereas non-augmented TR fails.

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.

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.

The Double Absorbing Boundary (DAB) method was recently introduced as a new approach for solving wave problems in unbounded domains. It has common features to local high-order Absorbing Boundary Conditions (ABC) on one hand, and to Perfectly Matched Layers (PML) on the other, and has relative advantages with respect to both. In the DAB method, the unbounded domain is truncated to produce a finite computational domain, which is then enclosed by a thin layer, as in the PML. Local high-order ABC are then imposed on both the inner and outer boundaries of the layer. Auxiliary variables are defined on the two boundaries as well as inside the layer, and participate in the numerical scheme. Only low order derivatives appear in the formulation, which allows the use of standard discretization methods in space and time. In previous studies, the DAB was developed for acoustic waves which are solutions to the scalar wave equation, and for elastic waves in isotropic media. Here the approach is extended to time-dependent elastic waves in anisotropic media. The geometrical configuration assumed is that of a semi-infinite wave guide, truncated via the DAB layer. Standard Finite Elements (FE) are used for space discretization and the damped Newmark scheme is used for time discretization. The problems considered here are restricted to those with periodic lateral boundary conditions, and with anisotropic media which do not support inverse modes, since removing these limitations may lead to a numerical instability. The performance of the scheme is demonstrated via numerical examples, including uniform orthotropic and layered orthotropic media.

Summary The LATIN (acronym of LArge Time INcrement) method was originally devised as a non-incremental procedure for the solution of quasi-static problems in continuum mechanics with material nonlinearity. In contrast to standard incremental methods like Newton and modified Newton, LATIN is an iterative procedure applied to the entire loading path. In each LATIN iteration, two problems are solved: a local problem, which is nonlinear but algebraic and miniature, and a global problem, which involves the entire loading process but is linear. The convergence of these iterations, which has been shown to occur for a large class of nonlinear problems, provides an approximate solution to the original problem. In this paper, the LATIN method is presented from a different viewpoint, taking advantage of the causality principle. In this new view, LATIN is an incremental method, and the LATIN iterations are performed within each load step, similarly to the way that Newton iterations are performed. The advantages of the new approach are discussed. In addition, LATIN is extended for the solution of time-dependent wave problems. As a relatively simple model for illustrating the new formulation, lateral wave propagation in a flat membrane made of a nonlinear material is considered. Numerical examples demonstrate the performance of the scheme, in conjunction with finite element discretization in space and the Newmark trapezoidal algorithm in time. Copyright © 2017 John Wiley & Sons, Ltd.

The computational inverse problem of identifying a scatterer in a time-dependent wave field is considered. The wave speed of the background medium and the wave source are assumed to be known. Wave measurements, possibly noisy, are given at chosen discrete points in space (sensor locations) and time. The goal is to find scatterer parameters such as location, size and shape. The computational solution procedure consists of two steps. In the first step, a standard fast Arrival-Time Imaging (ATI) algorithm is employed. This results in a rough image which provides possible regions for the location of the scatterer. In the second step an optimization scheme based on Time Reversal (TR) is used to determine the location, size and shape of the scatterer. The preliminary ATI step has the effect of reducing considerably the search space for the TR optimization. Also, proposed here is an improved definition of the objective function used for the optimization, which tends to eliminate spurious solutions. Numerical experiments, based on a finite element discretization in space and an explicit Newmark time-stepping, show the identification capability of the proposed scheme, for a model problem involving the linear scalar wave equation in a bounded domain. Two types of scatterers are considered: a crack with a known orientation, whose location and size are sought, and a rectangular scatterer whose location, aspect ratio and size are sought. The performance of the scheme in the presence of measurement noise is also demonstrated.

In this paper, we consider the numerical solution of the time-dependent wave equation in a semi-infinite waveguide. We employ the Double Absorbing Boundary (DAB) method, by introducing two parallel artificial boundaries on the side where waves are outgoing. In contrast to the original implementation of the DAB, where the numerical solution involved either a low-order finite difference scheme or a low-order finite element scheme, here we incorporate the DAB into a high-order spectral element formulation, which provides us with very accurate solutions of wave problems in unbounded domains. This is demonstrated by numerical experiments. While the method is highly accurate, it suffers from long-time instability. We show how to postpone the onset of the instability by a prudent choice of the computational parameters.

We model the inverse physical problem of identifying an underwater obstacle by using the acoustic wave equation. Measurements are collected in a set of sensors placed in the medium. We use this partial information to approximate a segmentation of the obstacle in its correct location. This is an ill-posed problem. We propose a novel deep learning architecture that takes as input the sensor data and computes an approximate segmentation map of the obstacle.

A new computational approach for the mixed-dimensional modeling of time-harmonic waves in elastic structures is proposed. A two-dimensional (2D) structure is considered, which includes a part that is assumed to behave in a one-dimensional (1D) way. The 2D and 1D structural regions are discretized by using 2D and 1D Finite Element (FE) formulations. The hybrid model, if designed properly, is much more efficient than the standard 2D model taken for the entire problem. One important issue related to such hybrid 2D–1D models is the way the 2D–1D coupling is done, and the coupling error generated. Here, three closely-related coupling methods are considered. They are all based on the Dirichlet-to-Neumann (DtN) map associated with the 1D problem, on the interface. The first coupling method is the DtN method in its usual form, where the DtN map is calculated numerically, and where the 1D and 2D problems are solved separately. The second coupling method is the one devised by Carka, Mear and Landis (CML), which is equivalent to the former one, but in which the 2D and 1D interface solutions are solved for simultaneously. In the third coupling method, the DtN map is enforced on the interface iteratively. Direct application of these three methods results in low accuracy, as a recent study shows. Therefore, these methods are used here in conjunction with a Boundary Stress Recovery (BSR) technique, originally proposed by Hughes, which provides the same order of accuracy for the stress as for the primary variable. The performance of the three methods is demonstrated and they are compared via numerical examples. Conclusions are drawn on their relative merit.

A model-based non destructive testing (NDT) method is proposed for damage identification in elastic structures, incorporating computational time reversal (TR) analysis. Identification is performed by advancing elastic wave signals, measured at discrete sensor locations, backward in time. In contrast to a previous study, which was purely numerical and employed only synthesized data, here an experimental system with displacement sensors is used to provide physical measurements at the sensor locations. The performance of the system is demonstrated by considering two problems of a thin metal plate in a plane stress state. The first problem, which represents passive damage identification, consists in finding the location of a small impact region from remote measurements. The second problem is the identification of the location of a square hole in the plate. The difficulties one encounters in applying this identification method and ways to overcome them are described. It is concluded that while this is a viable model-based identification method, which may lead, after further development, to a practical NDT procedure, one must be careful when drawing conclusions about its performance based solely on numerical experiments with synthesized data.

A computational approach for the mixed-dimensional modeling of time-harmonic waves in elastic structures is proposed. A two-dimensional (2D) structure is considered, that includes a part which is assumed to behave in a one-dimensional (1D) way. The 2D and 1D structural regions are discretized using 2D and 1D finite element formulations. The coupling of the 2D and 1D regions is performed weakly, by using the Nitsche method. The advantage of using the Nitsche method to impose boundary and interface conditions has been demonstrated by various authors; here this advantage is shown in the context of mixed-dimensional coupling. The computational aspects of the method are discussed, and it is compared to the slightly simpler penalty method, both theoretically and numerically. Numerical examples are presented in various configurations: where the 1D model is either confined laterally or laterally free, and where the 2D part is either simply connected or doubly connected. The performance is investigated for various wave numbers and various extents of the 1D region. Varying material properties and distributed loads in the 1D and 2D parts are also considered. It is concluded that the Nitsche method is a viable technique for mixed-dimensional coupling of elliptic problems of this type.

A general computational methodology is proposed for identifying cracks in structures. It is based on a time reversal (TR) technique and on the notion of refocusing. In the proposed procedure, a known source generates waves in the structure, and the time-varying response of the structure is measured only at certain points and times. In an industrial application this step is performed experimentally, but in the present study it is emulated numerically. Relying on a computational model of the structure and on the measured signals, a TR solution is obtained for each assumed set of crack parameters. This amounts to evolving the solution backward in time, till the initiation time of the original source. The crack identification is based on seeking, among all crack candidates, the crack which yields the best wave refocusing at the true source location. To test the proposed methodology, a simple rectangular membrane model governed by the 2D time-dependent scalar wave equation is employed. Finite element discretization of the structure and an explicit time-stepping scheme are used. The performance of the method is tested under various conditions and with various amounts of partial information. Its sensitivity to noise and to perturbations in the material properties is also investigated.

A new approach is devised for solving wave problems in unbounded domains. It has common features to each of two types of existing techniques: local high-order Absorbing Boundary Conditions (ABC) and Perfectly Matched Layers (PML). However, it is different from both and enjoys relative advantages with respect to both. The new method, called the Double Absorbing Boundary (DAB) method, is based on truncating the unbounded domain to produce a finite computational domain @W, and on applying a local high-order ABC on two parallel artificial boundaries, which are a small distance apart, and thus form a thin non-reflecting layer. Auxiliary variables are defined on the two boundaries and inside the layer bounded by them, and participate in the numerical scheme. The DAB method is first introduced in general terms, using the 2D scalar time-dependent wave equation as a model. Then it is applied to the 1D Klein-Gordon equation, using finite difference discretization in space and time, and to the 2D wave equation in a wave guide, using finite element discretization in space and dissipative time stepping. The computational aspects of the method are discussed, and numerical experiments demonstrate its performance.

Wave problems in unbounded domains are often treated numerically by truncating the domain to produce a finite computational domain. The double absorbing boundary (DAB) method, which was invented recently as an alternative to methods of high-order absorbing boundary conditions and to the perfectly matched layer, is investigated here for problems in acoustics and elastodynamics. The paper offers two main contributions. The first one pertains to the well-posedness of the DAB scheme for the acoustics problem written in second-order form. The energy method is employed to obtain uniform-in-time estimates of the norm of the solution and the auxiliary functions, thus establishing the well-posedness and asymptotic stability of the DAB formulation. The second part pertains to the extension of the DAB to isotropic elastodynamics, written in first-order conservation form. Numerical experiments for an elastic wave guide demonstrate the performance of the scheme.

Two new high-order local absorbing boundary conditions (ABCs) are devised for use on an artificial boundary for time-dependent elastic waves in unbounded domains, in two dimensions. The elastic medium in the exterior domain is assumed to be homogeneous and isotropic. The first ABC is written directly with respect to the displacement vector field, using an operator involving high derivatives. The second ABC is applied to the problem written in a first-order conservation form, using stresses and velocities as variables, and is formulated as a sequence of recursive relations using auxiliary variables. The two ABCs are not equivalent but are closely related. In each case, the order of the ABC determines its accuracy and can be chosen to be arbitrarily high. Both ABCs involve a product of first-order differential operators; all of them are of the Higdon type, except one which is of the Lysmer–Kuhlemeyer type. The stability of both ABCs is analyzed theoretically. The second (stress–velocity) ABC is implemented, employing a finite difference discretization scheme in space and time. Numerical experiments demonstrate the performance of the scheme. To the best of the authors’ knowledge, these are the first existing local high-order ABCs for elastodynamics which are long-time stable.

Time reversal is a well-known procedure in application fields involving wave propagation. Among other uses, it can be applied as a computational tool for solving certain inverse problems. The procedure is based on advancing the solution of the relevant wave problem “backward in time”. One important use of numerical time-reversal is that of refocusing, where a reverse run is performed to recover the location of a source applied at an initial time based on measurements at a later time. Usually, only partial, noisy, information is available, at certain measurement locations, on the field values that serve as data for the reverse run. In this paper, the question concerning the amount and characterization of the available data needed for a successful refocusing is studied for the scalar wave equation. In particular, a simple procedure is proposed which exploits multiple measurement times, and is shown to be very beneficial for refocusing. A tradeoff between availability of spatial and temporal information is discussed. The effect of measurement noise is studied, and the technique is shown to be quite robust, sometimes even in the presence of very high noise levels. The use of the technique as a basis for scatterer identification is also discussed. A numerical study of these effects is presented, employing finite elements in space and a standard explicit marching scheme in time. In contrast to some previous studies, the propagation medium is taken to be homogeneous.

c = wave speed f = wave source g = boundary loading h = mesh parameter k = wave number L = global length scale M = number of frequencies of interest N = number of spatial degrees of freedom NT = number of quasi periods T = basic period t = time U = time-dependent wave field u = wave amplitude x = spatial location t = time-step size = number of floating-point operations = wave length ! = angular frequency

Thin layers with material properties which differ significantly from those of the adjacent media appear in a variety of applications, as in the form of fiber coatings in composite materials. Fully modeling of such thin layers by standard finite element (FE) analysis is often associated with difficult meshing and high computational cost. Asymptotic procedures which model such thin domains by an interface of no thickness on which appropriate interface conditions are devised have been known in the literature for some time. The present paper shows how the first-order asymptotic interface model proposed by Bovik in 1994, and later generalized by Benveniste, can be incorporated in a FE formulation, to yield an accurate and efficient computational scheme for problems involving thin layers. This is done here for linear scalar elliptic problems in two dimensions, prototyped by steady-state heat conduction. Moreover, it is shown that by somewhat modifying the formulation of the Bovik–Benveniste asymptotic model, the proposed formulation is made to preserve the self-adjointness of the original three-phase problem, thus leading to a symmetric FE stiffness matrix. Numerical examples are presented that demonstrate the performance of the method, and show that the proposed scheme is more cost-effective than the full standard FE modeling of the layer.

A high-order absorbing boundary condition (ABC) is devised on an artificial boundary for time-dependent elastic waves in unbounded domains. The configuration considered is that of a two-dimensional elastic waveguide. In the exterior domain, the unbounded elastic medium is assumed to be isotropic and homogeneous. The proposed ABC is an extension of the Hagstrom–Warburton ABC which was originally designed for acoustic waves, and is applied directly to the displacement field. The order of the ABC determines its accuracy and can be chosen to be arbitrarily high. The initial boundary value problem including this ABC is written in second-order form, which is convenient for geophysical finite element (FE) analysis. A special variational formulation is constructed which incorporates the ABC. A standard FE discretization is used in space, and a Newmark-type scheme is used for time-stepping. A long-time instability is observed, but simple means are shown to dramatically postpone its onset so as to make it harmless during the simulation time of interest. Numerical experiments demonstrate the performance of the scheme.

The problem of optimal reduction of a linear dynamic subsystem is revisited. The subsystem may represent, for example, a complex but minor part of a large elastic structure. The goal is to drastically reduce the number of degrees of freedom of a subsystem attached through an interface to a main system in such a way as to affect the dynamic behavior of the main system in the least possible way. In a recent publication, the Optimal Modal Reduction (OMR) algorithm was developed to this end. This algorithm seeks a reduction of the subsystem that will have minimal effect, in the L2 norm, on the Dirichlet-to-Neumann (DtN) map on the interface. Here this algorithm is extended and improved in a number of ways. First, a family of alternative formulations are derived for the DtN map, which lead to alternative OMR algorithms; one of them, called OMR j=2, is shown to yield better results than the original formulation. Second, the OMR algorithm, which was originally developed for undamped subsystems, is extended to subsystems undergoing Rayleigh damping. In addition, an enhanced derivation of the original OMR algorithm is presented, and the good pointwise performance of OMR is explained by relating to minimization with respect to the H1 norm. The extensions mentioned above are discussed theoretically as well as demonstrated via numerical examples. Experiments and discussion include comparison of the OMR algorithm to simple coarsening of the subsystem discretization. In all cases, central finite difference discretization in space and explicit time-stepping are employed to solve the scalar wave equation. Copyright © 2010 John Wiley & Sons, Ltd.

The solution of the time-dependent wave equation in a semi-infinite wave guide is considered. An artificial boundary ℬ is introduced, which encloses a finite computational domain. On ℬ the Hagstrom–Warburton (H–W) local high-order absorbing boundary condition (ABC) is imposed. In contrast to previous studies, which involved either finite difference schemes or low-order finite element schemes, here the way to incorporate the H–W ABC into a spectral element formulation is shown. To this end, the Seriani–Priolo spectral element is used in space and a 4th-order Runge–Kutta scheme is employed in time. This leads to exponential convergence in both the polynomial order of the spectral element, and in the order of the ABC, and thus avoids the convergence rate inconsistency which is otherwise present. The combined ABC-spectral formulation enables one to obtain extremely accurate solutions of wave problems in unbounded domains. This is demonstrated via a number of numerical examples. Copyright © 2008 John Wiley & Sons, Ltd.

We develop complete plane wave expansions for time-dependent waves in a half-space and use them to construct arbitrary order local radiation boundary conditions for the scalar wave equation and equivalent first order systems. We demonstrate that, unlike other local methods, boundary conditions based on complete plane wave expansions provide nearly uniform accuracy over long time intervals. This is due to their explicit treatment of evanescent modes. Exploiting the close connection between the boundary condition formulations and discretized absorbing layers, corner compatibility conditions are constructed which allow the use of polygonal artificial boundaries. Theoretical arguments and simple numerical experiments are given to establish the accuracy and efficiency of the proposed methods.

Global–regional model interaction is considered for two-dimensional linear time dependent waves in a dispersive non-uniform medium with a continuously varying wave speed. The setup, which is sometimes called ‘one-way nesting,’ arises in Numerical Weather Prediction (NWP) as well as in other fields concerning waves in very large domains. The Carpenter scheme for this type of problem is revisited, in the context of the dispersive wave equation with a variable wave speed. The original Carpenter scheme is based on the Sommerfeld radiation operator, and thus is associated with low-order accuracy. By replacing the Sommerfeld operator with the high-order Hagstrom–Warburton absorbing operator, a modified Carpenter open boundary condition emerges which possesses high-order accuracy. This is demonstrated via a numerical example in a wave guide with a wave speed which varies linearly in the cross section.

The convective wave equation deals with wave propagation in a moving media. We focus on the underwater acoustic wave equation where the convective element is the flow of water inside a river, along its length. The main thrust of this paper is the ill-posed “refocusing” problem. The initial condition simulates an explosion in a small compact region and the response is recorded over time at several microphones. Having only partial and noisy information we expect that small perturbations will destroy the ability to recover the complete initial data. We use the time reversal (TR) technique to determine the location of the original explosion, given limited spatial observations. We test the effectiveness of this scheme under conditions including dissipation, dispersion, etc. We use finite differences and implement absorbing boundary conditions to simulate an unbounded region.

PurposeThe interaction of a global model (GM) and a local (regional) model (LM) of heat flow is considered under the framework of so‐called “one‐way nesting”. In this framework, the GM is constructed in a large domain with coarse discretization in space and time, while the LM is set in a small subdomain with fine discretization.Design/methodology/approachThe GM is solved first, and its results are then used via some boundary transfer operator (BTO) on the GM–LM interface in order to solve the LM. Past experience in various fields of application has shown that one has to be careful in the choice of BTO to be used on the GM–LM interface, since this choice affects both the stability and accuracy of the computational scheme. Here the problem is first theoretically analyzed for the linear heat equation, and stable BTOs are identified. Then numerical experiments are performed with one‐way nesting in a two‐dimensional channel for heat flow with and without radiation emission and linear reaction, using four different BTOs.FindingsAmong other conclusions, it is shown that the “negative Robin” BTO is unstable, whereas the Dirichlet, Neumann and “positive Robin” BTO are all stable. It is also shown that in terms of accuracy, the Neumann and “positive Robin” BTOs should be preferred over the Dirichlet BTO.Originality/valueThis study may be the first step in analyzing BTO accuracy and stability for more general atmospheric systems.

A computational framework is developed in which cracks in two-dimensional structures are identified, in conjunction with non-destructive testing of specimens. As opposed to a previous study by the authors, which was based on time-harmonic excitation with a single frequency, here the transient response of the structure to a short-duration signal is measured along part of the external boundary. Crack detection is performed using the solution of an inverse time-dependent problem. It is shown that the arrival time of the input signal to the points of measurement is a good criterion for crack identification in the time domain. The inverse problem of identification is solved using a genetic algorithm, while each forward problem is solved by the time-dependent extended finite element method (XFEM). The XFEM scheme is efficient in that it allows the use of a single regular mesh for a large number of forward time response problems with different crack geometries. Numerical examples involving a crack in a flat membrane are presented. Identification based on ‘arrival time’ is shown to perform better than that based on time-harmonic response. Copyright © 2008 John Wiley & Sons, Ltd.

A two-dimensional global–regional model interaction problem for linear time-dependent waves is considered. The setup, which is sometimes called ‘one-way nesting,’ arises in numerical weather prediction as well as in other fields concerning waves in very large domains. It involves the interaction of a coarse global model and a fine limited-area (regional) model through an ‘open boundary.’ The multiscale nature of this general problem is described. The Carpenter scheme, originally proposed in a note by K. M. Carpenter in 1982 for this type of problem, is then revisited, in the context of the linear scalar wave equation. The original Carpenter scheme is based on the Sommerfeld radiation operator and thus is associated with low-order accuracy. By replacing the Sommerfeld operator with the high-order Hagstrom–Warburton absorbing operator, a modified Carpenter open-boundary condition emerges, which possesses high-order accuracy. This boundary condition is incorporated in a computational scheme, which uses finite element discretization in space and Newmark time-stepping. Error analysis and numerical tests for wave guides demonstrate the performance of the modified scheme for combinations of incoming and outgoing waves. Copyright © 2008 John Wiley & Sons, Ltd.

The solution of the time-dependent wave equation in an unbounded domain is considered. An artificial boundary B is introduced which encloses a finite computational domain. On B an absorbing boundary condition (ABC) is imposed. A formulation of local high-order ABCs recently proposed by Hagstrom and Warburton and based on a modification of the Higdon ABCs, is further developed and extended in a number of ways. First, the ABC is analyzed in new ways and important information is extracted from this analysis. Second, The ABCs are extended to the case of a dispersive medium, for which the Klein-Gordon wave equation governs. Third, the case of a stratified medium is considered and the way to apply the ABCs to this case is explained. Fourth, the ABCs are extended to take into account evanescent modes in the exact solution. The analysis is applied throughout this paper to two-dimensional wave guides. Two numerical algorithms incorporating these ABCs are considered: a standard semi-discrete finite element formulation in space followed by time-stepping, and a high-order finite difference discretization in space and time. Numerical examples are provided to demonstrate the performance of the extended ABCs using these two methods.

The finite-element (FE) approximation of linear elliptic eigenvalue problems is considered. An analysis based on a number of known estimates leads to the simple formula M=r0ed/(2p)N relating the total number of degrees of freedom N, the maximum relative error level e desired for the eigenvalues, and the number of ‘reliable’ modes M. (Here d is the spatial dimension and p is the polynomial degree of the FE space.) Moreover, a rough estimate for the numerical value of the constant r0 for a given application is found. This result supports a well-known rule of thumb. Copyright © 2008 John Wiley & Sons, Ltd.

SUMMARY A shallow water model with linear time-dependent dispersive waves in an unbounded domain is considered. The domain is truncated with artificial boundaries B where a sequence of high-order non-reflecting boundary conditions (NRBCs) proposed by Higdon are applied. Methods devised by Givoli and Neta that afford easy implementation of Higdon NRBCs are refined in order to reduce computational expenses. The new refinement makes the computational effort associated with the boundary treatment quadratic rather than exponential (as in the original scheme) with the order. This allows for implementation of NRBCs of higher orders than previously. A numerical example for a semi-infinite channel truncated on one side is presented. Finite difference schemes are used throughout. Copyright q 2007 John Wiley & Sons, Ltd.

The scalar wave equation in a two-dimensional semi-infinite wave guide is considered. The recently proposed Hagstrom–Warburton (H–W) local high-order absorbing boundary conditions (ABCs), which are based on a modification of the Higdon ABCs, are presented in this context. The P-order ABC involves the free parameters 0 < aj ≤ 1, for j = 0, 1, …, P, which have to be chosen. The choice aj = 1 for all j is shown to be satisfactory, in general, although not necessarily optimal. The optimal choice of the parameters is discussed via both theoretical analysis and numerical experiments. In addition, an adaptive scheme which controls the time-varying values of P and aj is presented and tested.

A new computational tool is developed for the accurate detection and identification of cracks in structures, to be used in conjunction with non-destructive testing of specimens. It is based on the solution of an inverse problem. Based on some measurements, typically along part of the boundary of the structure, that describe the response of the structure to vibration in a chosen frequency or a combination of frequencies, the goal is to estimate whether the structure contains a crack, and if so, to find the parameters (location, size, orientation and shape) of the crack that produces a response closest to the given measurement data in some chosen norm. The inverse problem is solved using a genetic algorithm (GA). The GA optimization process requires the solution of a very large amount of forward problems. The latter are solved via the extended finite element method (XFEM). This enables one to employ the same regular mesh for all the forward problems. Performance of the method is demonstrated via a number of numerical examples involving a cracked flat membrane. Various computational aspects of the method are discussed, including the a priori estimation of the ill-posedness of the crack identification problem. Copyright © 2007 John Wiley & Sons, Ltd.

The solution of the fully non-linear time-dependent two-dimensional shallow water equations is considered. Dispersive effects due to the Coriolis forces are taken into account. Such effects are of major importance in geophysical fluid dynamics applications. The recently proposed string gradient weighted moving finite element method is extended for this class of problems. This method simultaneously determines, at each time step, the solution of the governing partial differential equations and an optimal location of the finite element nodes. It has previously been applied to non-dispersive wave problems; here its performance under the demanding conditions of large Coriolis forces, inducing large mesh and field rotation, is studied. Optimal rates of convergence are obtained. Results for some example problems of water hump release are presented. Non-linear and linearized solutions are compared.

The Hagstrom–Warburton high-order absorbing boundary conditions (ABCs) are considered. They are based on a high-order form of the Higdon ABCs using auxiliary variables and constitute a modification of the previously proposed Givoli–Neta ABCs. Here the Hagstrom–Warburton ABCs, which were originally used in a finite difference scheme, are incorporated into a finite element formulation. Exterior time-dependent problems are considered with rectangular computational domains. Special corner conditions are used in conjunction with the ABCs to make the truncated problem well-posed. The properties of the Hagstrom–Warburton and Givoli–Neta formulations are compared, and the relations between the two formulations are established. Numerical examples demonstrate the performance of the Hagstrom–Warburton finite element scheme.