Search

We consider the Galerkin and Collocation Isogeometric approximations of the acoustic wave equation with absorbing boundary conditions, while the time discretization is based on second-order Newmark schemes. A numerical study investigates the properties of the two IGA methods as concerns stability thresholds, convergence errors, accuracy, computational time, and sparsity of the stiffness matrices varying the polynomial degree p , mesh size h , regularity k , and time step Δ t. In order to compare the two IGA methods, we focus on two meaningful examples in the framework of wave propagation simulations: a test problem with an oscillatory exact solution having increasing wave number, and the propagation of two interfering Ricker wavelets. Numerical results show that the IGA Collocation method retains the convergence and stability properties of IGA Galerkin. In all examples considered, IGA Collocation is in general less accurate when we adopt the same choices of parameters p , k , h , and Δ t. On the other hand, regarding the computational cost and the amount of memory required to achieve a given accuracy, we observe that the IGA Collocation method often outperforms the IGA Galerkin method, especially in the case of maximal regularity k = p − 1 with increasing NURBS degree p. [ABSTRACT FROM AUTHOR]

In this paper, we address the approximation of the coupling problem for the wave equation and Maxwell's equations of electromagnetism in the time domain in terms of electric field by means of a nodal linear finite element discretization in space, combined with a classical explicit finite difference scheme for time discretization. Our study applies to a particular case where the dielectric permittivity has a constant value outside a subdomain, whose closure does not intersect the boundary of the domain where the problem is defined. Inside this subdomain, Maxwell's equations hold. Outside this subdomain, the wave equation holds, which may correspond to Maxwell's equations with a constant permittivity under certain conditions. We consider as a model the case of first-order absorbing boundary conditions. First-order error estimates are proven in the sense of two norms involving first-order time and space derivatives under reasonable assumptions, among which lies a CFL condition for hyperbolic equations. The theoretical estimates are validated by numerical computations, which also show that the scheme is globally of the second order in the maximum norm in time and in the least-squares norm in space. [ABSTRACT FROM AUTHOR]

The conditioning and spectral properties of the mass and stiffness matrices for acoustic wave problems are here investigated when isogeometric analysis (IGA) collocation methods in space and Newmark methods in time are employed. Theoretical estimates and extensive numerical results are reported for the eigenvalues and condition numbers of the acoustic mass and stiffness matrices in the reference square domain with Dirichlet, Neumann, and absorbing boundary conditions. This study focuses in particular on the spectral dependence on the polynomial degree p, mesh size h, regularity k, of the IGA discretization and on the time step size Δ t and parameter β of the Newmark method. Results on the sparsity of the matrices and the eigenvalue distribution with respect to the number of degrees of freedom d.o.f. and the number of nonzero entries nz are also reported. The results show that the spectral properties of the IGA collocation matrices are comparable with the available spectral estimates for IGA Galerkin matrices associated with the Poisson problem with Dirichlet boundary conditions, and in some cases, the IGA collocation results are better than the corresponding IGA Galerkin estimates, in particular for increasing p and maximal regularity k = p - 1 . [ABSTRACT FROM AUTHOR]

When addressing large-scale electromagnetic problems using the finite element method (FEM), the resulting matrices are typically sparse, necessitating numerous sparse matrix-vector multiplication (SpMV) operations. To handle this efficiently, research has focused on leveraging large-scale parallel processing with graphics processing units (GPUs). These GPUs can be controlled directly using NVIDIA’s Compute Unified Device Architecture (CUDA). In this paper, we analyze electromagnetic scattering for dielectric and dielectric-coated structures using iterative solvers with FEM. To accelerate the handling of the large-scale matrices generated during this process, we employ compressed sparse row (CSR) format, various preconditioners, and CUDA-based GPU parallelization. We verify the accuracy of our results by comparing them with those obtained using the commercial electromagnetic software High Frequency Structure Simulator (HFSS) and our custom-developed MATLAB-based FEM code. Performance improvements are assessed by comparing these results with those from MATLAB’s backslash direct solver under single-core processing conditions.

When solving open-region electromagnetic scattering problems using the standard finite element method, the infinite region exterior to the scatterer must be truncated with an artificial outer boundary. In the interior or finite part of the domain, finite elements can be used, but a boundary condition must be introduced at this artificial boundary to obtain a unique finite element solution. One class of boundary conditions designed for this purpose is called absorbing boundary conditions (ABCs). Several ABCs have been reported in the literature. These can be broadly classified into two categories: local and non-local. In this paper, we present new closed-form expressions for Nth-order local ABCs, equivalent to the well-known sequences of the Bayliss, Gunzburger, and Turkel boundary operators. To the best of our knowledge, these expressions have not yet been reported in the literature. Two- and three-dimensional problems are considered in this study. We also discuss the problem of incorporating custom ABCs into commercial finite element method programs, which are usually closed-source software packages. Two 2D electromagnetic scattering and radiation problems with known analytical solutions are analyzed. [ABSTRACT FROM AUTHOR]

The stability and convergence analysis of high-order numerical approximations for the one- and two-dimensional nonlocal wave equations on unbounded spatial domains are considered. We first use the quadrature-based finite difference schemes to discretize the spatially nonlocal operator, and apply the explicit difference scheme to approximate the temporal derivative to achieve a fully discrete infinity system. After that, we construct the Dirichlet-to-Neumann (DtN)-type absorbing boundary conditions (ABCs), to reduce the infinite discrete system into a finite discrete system. To do so, we first adopt the idea in Du et al. (2018, Commun. Comput. Phys. , 24 , 1049–1072) and Du et al. (2018, SIAM J. Sci. Comp. , 40 , A1430–A1445) to derive the Dirichlet-to-Dirichlet (DtD)-type mappings for one- and two-dimensional cases, respectively. We then use the discrete nonlocal Green's first identity to achieve the discrete DtN-type mappings from the DtD-type mappings. The resulting DtN-type mappings make it possible to perform the stability and convergence analysis of the reduced problem. Numerical experiments are provided to demonstrate the accuracy and effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

We propose a simulator for time-dependent Maxwell's equations with linear computational cost. We employ B-spline basis functions as considered in the isogeometric analysis (IGA). We focus on non-stationary Maxwell's equations defined on a regular patch of elements. We employ the idea of alternating-directions splitting (ADS) and employ a second-order accurate time-integration scheme for the time-dependent Maxwell's equations in a weak form. After discretization, the resulting stiffness matrix exhibits a Kronecker product structure. Thus, it enables linear computational cost LU factorization. Additionally, we derive a formulation for absorbing boundary conditions (ABCs) suitable for direction splitting. We perform numerical simulations of the scattering problem (traveling pulse wave) to verify the ABC. We simulate the radiation of electromagnetic (EM) waves from the dipole antenna. We verify the order of the time integration scheme using a manufactured solution problem. We then simulate magnetotelluric measurements. Our simulator is implemented in a shared memory parallel machine, with the GALOIS library supporting the parallelization. We illustrate the parallel efficiency with strong and weak scalability tests corresponding to non-stationary Maxwell simulations. [ABSTRACT FROM AUTHOR]

Construction of absorbing boundary conditions (ABCs) for nonlocal models is generally challenging, primarily due to the fact that nonlocal operators are commonly associated with volume constrained boundary conditions. Moreover, application of Fourier and Laplace transforms, which are essential for the majority of available methods for ABCs, to nonlocal models is complicated. In this paper, we propose a simple method to construct accurate ABCs for peridynamic scalar wave‐type problems in viscous media. The proposed ABCs are constructed in the time and space domains and are of Dirichlet type. Consequently, their implementation is relatively simple, since no derivatives of the wave field are required. The proposed ABCs are derived at the continuum level, from a semi‐analytical solution of the exterior domain using harmonic exponential basis functions in space and time (plane‐wave modes). The numerical implementation is done using a meshfree collocation approach employed within a boundary layer adjacent to the interior domain boundary. The modes satisfy the peridynamic numerical dispersion relation, resulting in a compatible solution of the interior region (near‐field) with that of the exterior region (far‐field). The accuracy and stability of the proposed ABCs are demonstrated with several numerical examples in two‐dimensional unbounded domains. [ABSTRACT FROM AUTHOR]

In this paper, we address the approximation of the coupling problem for the wave equation and Maxwell’s equations of electromagnetism in the time domain in terms of electric field by means of a nodal linear finite element discretization in space, combined with a classical explicit finite difference scheme for time discretization. Our study applies to a particular case where the dielectric permittivity has a constant value outside a subdomain, whose closure does not intersect the boundary of the domain where the problem is defined. Inside this subdomain, Maxwell’s equations hold. Outside this subdomain, the wave equation holds, which may correspond to Maxwell’s equations with a constant permittivity under certain conditions. We consider as a model the case of first-order absorbing boundary conditions. First-order error estimates are proven in the sense of two norms involving first-order time and space derivatives under reasonable assumptions, among which lies a CFL condition for hyperbolic equations. The theoretical estimates are validated by numerical computations, which also show that the scheme is globally of the second order in the maximum norm in time and in the least-squares norm in space.

When solving open-region electromagnetic scattering problems using the standard finite element method, the infinite region exterior to the scatterer must be truncated with an artificial outer boundary. In the interior or finite part of the domain, finite elements can be used, but a boundary condition must be introduced at this artificial boundary to obtain a unique finite element solution. One class of boundary conditions designed for this purpose is called absorbing boundary conditions (ABCs). Several ABCs have been reported in the literature. These can be broadly classified into two categories: local and non-local. In this paper, we present new closed-form expressions for Nth-order local ABCs, equivalent to the well-known sequences of the Bayliss, Gunzburger, and Turkel boundary operators. To the best of our knowledge, these expressions have not yet been reported in the literature. Two- and three-dimensional problems are considered in this study. We also discuss the problem of incorporating custom ABCs into commercial finite element method programs, which are usually closed-source software packages. Two 2D electromagnetic scattering and radiation problems with known analytical solutions are analyzed.

• The enriched degree of freedom method is proposed to solve the problems with absorbing boundaries. • The proposed method is independent of the load frequency. • The proposed method can be coupled with the XFEM to solve the problem with multiple cracks. • The proposed method can be used to simulate the field-scale and laboratory-scale models. For elastodynamic problems, significant effort has been exerted by numerical researchers to eliminate the reflection of outgoing elastic waves at the boundary of the computational domain. Eliminating these reflection is very important for correctly analyzing the dynamic response of various structures. In this paper, the enriched degree of freedom method is first proposed to absorb outgoing elastic waves; degrees of freedom are added to the nodes in the absorbing domain by using a user-defined function. Next, the gradually increasing mass matrix is derived to further dampen the elastic waves based on the theory of the perfectly matched layer. Then, a novel damping method is proposed based on the enriched degree of freedom method, and it is appropriately coupled with the XFEM and the modified Newmark algorithm to solve elastodynamic problems with infinite boundaries. Finally, the numerical results show that the proposed method is feasible and applicable for absorbing the outgoing elastic waves. Importantly, the proposed method is independent of the frequency; it can be used to simulate the field-scale model and the laboratory-scale model, and it is effective in solving problems with multiple cracks. [ABSTRACT FROM AUTHOR]

The present study formulates the dynamic analysis of concrete arch damreservoir systems using FE-(FE-TE) approach. In this technique, dam and reservoir are discretized by solid and fluid finite elements. Moreover, the GN high-order condition is imposed at the reservoir truncation boundary. This task is formulated by employing a truncation element at that boundary. It is emphasized that reservoir far-field is excluded from the discretized model. The formulation is initially explained in detail. Subsequently, the response of the idealized Morrow Point arch dam-reservoir system is obtained for two conditions of fully reflective and absorptive reservoir bottom/sidewalls for all three types of excitations. Different orders of GN condition are considered and convergence process is evaluated. Furthermore, the results are compared against exact solutions which are based on rigorous FE-(FE-HE) approach. It is shown that the technique reaches convergence prior to the beginning of instability problems known to exist for high orders in GN condition. It must be emphasized that although time harmonic analysis is considered in the present study, the main part of formulation is explained in the context of time domain. Therefore, the approach can easily be extended for transient type of analysis. [ABSTRACT FROM AUTHOR]

An iterative numerical method for time-harmonic acoustic multiple scattering from complexly shaped obstacles is devised. The classical formulation of multiple scattering problem is transformed into a set of single scattering boundary value problems (BVPs) coupled at their scatterer boundaries. By introducing iterative procedures, the single scattering BVPs are decoupled and a convenient domain truncation technique, employing artificial boundaries, reduces the unbounded domain of the original BVPs to relatively small bounded domain around each scatterer. Then, high order Karp's farfield expansions absorbing boundary conditions (KFE) are imposed at the artificial boundaries of each scatterer. An overall high order numerical method is constructed by combining these absorbing boundary conditions (ABC) with a high order isogeometric finite element method (IGA) in the interior of the computational domains. We performed ample numerical experiments for complex configuration of scatterers, using the isogeometric finite elements and also finite differences techniques combined with the high order KFE. They show accuracies close to machine precision and high order rate of convergence at relatively low computational costs. • Multiple Scattering method coupling IGA with high order farfield expansion ABC. • Iterative technique leads to a system of single scattering boundary value problems. • Solution anywhere can be obtained from computation in relatively small domains • Only one matrix inversion needed in the iteration procedure for each scatterer. • Convergence rate increases by adding terms to the farfield expansion for high order IGA. [ABSTRACT FROM AUTHOR]

Metal materials will produce fatigue cracks under long-term cyclic loading. It is difficult to fully image the topography of defects with the current mainstream ultrasound imaging technology. The Reverse time migration (RTM), which has extensive application in geophysical exploration, can image complex geological structures. This research introduces the RTM into the field of ultrasonic non-destructive testing. Through numerical simulation, it is concluded that the RTM based on no absorbing boundary can make the defect contour clearer, but the RTM based on the absorbing boundary can make the image contrast metric higher. In order to obtain the RTM results with high contrast metric and clear defect contours, the solution given in this study is to increase the migration aperture. Then this research discusses the influence of different cross-correlation imaging condition on the imaging quality. And compared with the prevailing total focusing methods (TFM), it shows the superiority of the RTM to the bottom opening crack (BOC) imaging. Finally, the RTM was used to complete the imaging of the curved and bifurcated cracks with complex structures. And after reducing the number of ultrasonic transducers, there are still good imaging results, which can reduce the waste of resources and energy.

In this paper, we present a fully discrete second-order finite-difference scheme with fast evaluation of the convolution involved in the absorbing boundary conditions to solve the one-dimensional linearized Green-Naghdi system. The Padé expansion of the square-root function in the complex plane is used to implement the fast convolution. By introducing a constant damping parameter into the governing equations, the convergence analysis is developed when the damping term fulfills some conditions. In addition, the scheme is stable and leads to a highly reduced computational cost and low memory storage. A numerical example is provided to support the theoretical analysis and to illustrate the performance of the fast numerical scheme. [ABSTRACT FROM AUTHOR]

We study a mean field game system introduced by Chan and Sircar (Appl Math Optim 71:533–569, 2015) to model production of an exhaustible resource. In particular, we study the sensitivity of the solution with respect to a parameter ε , which measures the degree to which producers are interchangeable. We prove that on some interval [ 0 , ε 0 ] , where ε 0 > 0 , the solution is infinitely differentiable with respect to ε . The result is based on a set of new a priori estimates for forward-backward systems of linear partial differential equations. [ABSTRACT FROM AUTHOR]

We study the optimal control problem over an infinite time horizon for the third-order JMGT equation, defined on a 3-d bounded domain Ω with L 2 (0 , ∞ ; L 2 (Γ 0)) -Robin boundary control on one part Γ 0 of the boundary Γ , and damping in the Neumann BC on the complementary part Γ 1 . The pathology present in the corresponding abstract model impacts on the final theory. It results in several new features including: (i) a non-standard pointwise feedback representation of the optimal control in terms of the optimal solution that involves a non-trivial inverse and (ii) a new Riccati operator that satisfies a non-standard algebraic Riccati equation with unbounded coefficients. [ABSTRACT FROM AUTHOR]

A reduced-order model of finite periodic structures with absorbing boundary conditions (ABCs) and localized time-dependent excitations is proposed. 1D-periodic structures whose cells can represent any 2D or 3D arbitrary substructures are considered. The key steps for modeling an ABC in the time domain are: (i) expression of the impedance matrix with the wave finite element (WFE) method; (ii) partial rational decomposition of the impedance matrix; (iii) introduction of vectors of supplementary variables to describe the ABC in the time domain. In this paper, a model reduction approach is proposed to speed up the computation of the ABCs. The focus is on the reduction of the number of internal degrees of freedom of substructures, and the reduction of the number of degrees of freedom at the substructure interfaces. The approach allows the consideration of complex substructures with large FE models, and the computation of the time response of periodic structures at a low computational cost. Numerical experiments are carried out to highlight the relevance of the proposed approach. Specifically, the analysis of a metamaterial structure containing nonlinear resonant substructures is carried out with the proposed approach. Results show that, when compared to the linear case, band gaps in periodic structures with nonlinear substructures can be significantly improved. • The time response of periodic structures with localized excitations is computed. • Absorbing boundaries are built using supplementary variables and reduced DOFs. • Solutions are obtained by time integration of a finite element dynamic system. • Examples of infinite periodic structures with a large number of DOFs are given. • The influence of nonlinearities on band gaps effects is illustrated. [ABSTRACT FROM AUTHOR]

Full-Waveform Inversion (FWI) is a high-resolution numerical technique for seismic waves used to estimate the physical characteristics of a subsurface region. The continuous problem involves solving an inverse problem on an infinite domain, which is impractical from a computational perspective. In limited area models, absorbing boundaries conditions (ABCs) are usually imposed, to avoid wave reflections. Several relevant ABCs have been proposed, with extensive literature on their effectiveness on the direct wave problem. Here, we investigate and compare the theoretical and computational characteristics of several ABCs in the full inverse problem. After a brief review of the most widely used ABCs, we derive their formulations in their respective adjoint problems. The different ABCs are implemented in a highly optimized domain-specific language (DLS) computational framework, Devito, which targets seismic modeling problems. We evaluate the effectiveness, computational efficiency, and memory requirements of the ABC methods, considering from simple models to realistic ones. Our findings reveal that, even though the popular Perfectly Matching Layers (PMLs) are effective in avoiding wave reflections on the boundaries, they can be computationally more demanding than less used Hybrid ABCs. We show here that a proposed Hybrid ABC formulation, with nested Higdon's boundary conditions, is the most cost-effective method among the methods considered here, being as effective, or more, as PML and other schemes, but being computationally more efficient. [ABSTRACT FROM AUTHOR]

• A non-local absorbing boundary conditions is constructed in peridynamic theory. • Motion equation of peridynamic theory is re-derived considering material damping. • Algorithm to realize the non-local absorbing boundary conditions in peridynamics is proposed. • Elastic wave propagation in semi-unbounded and unbounded solids are simulated. In order to simulate the elastic wave propagation in unbounded solids, this paper introduces a non-local method to construct absorbing boundary conditions in the bond-based peridynamic theory. To construct non-local absorbing boundary conditions in the form of absorbing layers with increasing damping, we first re-derived the particle motion equilibrium equation of the bond-based peridynamic theory, based on the principle of virtual work and Lagrange's equation on the premise of considering material damping. We also give the expression for the inter-particle damping force density which satisfies the balance laws of linear momentum and angular momentum. This paper also provides a process and algorithm for realizing the non-local absorbing boundary conditions in the bond-based peridynamic theory, and adopts the Fortran language for computer program compilation. Finally, we simulate the elastic wave propagation in semi-unbounded and unbounded solids as numerical examples. The results show that such non-local absorbing boundary conditions can stably absorb and attenuate incident elastic waves. Since it is derived in the time domain, this method does not require frequency-domain operations, such as Fourier and Laplace transformations, nor does it require wave field splitting. [ABSTRACT FROM AUTHOR]

The perfectly matched layer (PML) technique is a popular truncation method to model wave propagation in unbounded elastic media. Both numerical efficiency and high stability are important improvement areas in the field. In this study, we extend the optimal PML, previously proposed for acoustic media, to elastic and poroelastic media, which turns out to be more efficient and flexible than the classical PML. We investigate the accuracy and stability of the optimal PML by comparing it with the classical PML in several scenarios. First, the effectiveness of the optimal PML is studied using frequency-domain and time-domain simulations for isotropic and homogeneous elastic solids. The efficiency and accuracy of the optimal PML and classical PML are then compared across a wide range of Poisson's ratios of elastic media. The stability of the optimal PML and the classical PML is also compared taking into account the effect of the outer boundary conditions of the PML as well as the heterogeneity of the geological model. Moreover, the optimal PML is applied to poroelastic media to address the instability problem of the classical PML. Comprehensive analyses of numerical results show that the optimal PML can absorb the outgoing waves. This can be done using thinner layers and with higher accuracy than for classical PMLs. In addition, the long-time stability of optimal PML increases. [ABSTRACT FROM AUTHOR]

The paper aims to evaluate the performance of the Lagrange-based finite element method and the non-uniform rational B-splines isogeometric analysis of time-harmonic acoustic exterior scattering problems using high-order local absorbing boundary conditions, in particular based on the Karp's and Wilcox's far-field expansions. The analysis of accuracy and convergence of both methods is achieved by observing the effect of the order of the approximating polynomial, the number of degrees of freedom, the wave number, and the absorbing boundary conditions tuning parameters. It is concluded that, regardless of the polynomial order, IGA provides a higher accuracy per degree of freedom compared to the traditional Lagrange-based finite element method. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A convolution perfectly matched layer (CPML) can efficiently absorb boundary reflection in numerical simulation. However, the CPML is suitable for the first-order elastic wave equation and is difficult to apply directly to the second-order elastic wave equation. In view of this, based on the first-order CPML absorbing boundary condition, we propose a new CPML (NCPML) boundary which can be directly applied to the second-order wave equation. We first systematically extend the first-order CPML technique into second-order wave equations, neglecting the space-varying characteristics of the partial damping coefficient in the complex-frequency domain, avoiding the generation of convolution in the time domain. We then transform the technique back to the time domain through the inverse Fourier transform. Numerical simulation indicates that the space-varying characteristics of the attenuation factor have little influence on the absorption effect and increase the memory at the same time. A number of numerical examples show that the NCPML proposed in this study is effective in simulating elastic wave propagation, and this algorithm is more efficient and requires less memory allocation than the conventional PML absorbing boundary. [ABSTRACT FROM AUTHOR]

This paper aims to solve numerically the linearized Korteweg-de Vries equation. We begin by deriving suitable boundary conditions then approximate them using finite difference method. The methodology of derivation, used in this paper, yields to Non-Standard Boundary Conditions (NSBC) that perfectly absorb wave reflections at the boundary. In addition, these NSBC are exact and local in time and space for non necessarily supported initial data and source terms. We finish with numerical examples that show the absorbing quality of these boundary conditions. Further comparisons are made using standard boundary conditions like, Dirichlet, Neumann and a variant of absorbing boundary conditions called discrete artificial ones. [ABSTRACT FROM AUTHOR]

Wrapping a computation domain with a perfectly matched layer (PML) is one of the most effective methods of imitating/approximating the radiation boundary condition in the Maxwell and wave equation solvers. Many PML implementations often use a smoothly increasing attenuation coefficient to increase the absorption for a given layer thickness, and, at the same time, to reduce the numerical reflection from the interface between the computation domain and the PML. In discontinuous Galerkin time-domain (DGTD) methods, using a PML coefficient that varies within a mesh element requires a different mass matrix to be stored for every element and therefore significantly increases the memory footprint. In this work, this bottleneck is addressed by applying a weight-adjusted approximation to these mass matrices. The resulting DGTD scheme has the same advantages as the scheme that stores individual mass matrices, namely, higher accuracy (due to reduced numerical reflection) and increased meshing flexibility (since the PML does not have to be defined layer by layer), but it requires significantly less memory. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We consider a nonconforming hp‐finite element approximation of a variational formulation of the time‐harmonic Maxwell equations with impedance boundary conditions proposed by Costabel et al. The advantages of this formulation is that the variational space is embedded in H1 as soon as the boundary is smooth enough (in particular it holds for domains with an analytic boundary) and standard shift theorem can be applied since the associated boundary value problem is elliptic. Finally in order to perform a wavenumber explicit error analysis of our problem, a splitting lemma and an estimation of the adjoint approximation quantity are proved by adapting to our system the results from Melenk and Sauter obtained for the Helmholtz equation. Some numerical tests that illustrate our theoretical results are also presented. Analytic regularity results with bounds explicit in the wavenumber of the solution of a general elliptic system with lower order terms depending on the wavenumber are needed and hence proved. [ABSTRACT FROM AUTHOR]

Diffusion-type problems in (nearly) unbounded domains play important roles in various fields of fluid dynamics, biology, and materials science. The aim of this paper is to construct accurate absorbing boundary conditions (ABCs) suitable for classical (local) as well as nonlocal peridynamic (PD) diffusion models. The main focus of the present study is on the PD diffusion formulation. The majority of the PD diffusion models proposed so far are applied to bounded domains only. In this study, we propose an effective way to handle unbounded domains both with PD and classical diffusion models. For the former, we employ a meshfree discretization, whereas for the latter the finite element method (FEM) is employed. The proposed ABCs are time-dependent and Dirichlet-type, making the approach easy to implement in the available models. The performance of the approach, in terms of accuracy and stability, is illustrated by numerical examples in 1D, 2D, and 3D. [ABSTRACT FROM AUTHOR]

A new simple formulation for incorporating a higher-order perfectly matched layer (PML) stretching function within a convolution PML (CPML) implementation in finite-difference time-domain (FDTD) electromagnetic modeling codes is developed. Obtaining in closed form the corresponding time domain impulse response of the inverse of a number of higher-order PML stretching functions enables the efficient and simple implementation of such higher-order PMLs using recursive convolution, in the same way as it was introduced initially for the complex frequency shifted (CFS) PML. This new higher-order CPML exhibits excellent performance that is comparable to the performance shown by other higher-order PML formulations whilst it retains the advantage of a relatively simpler implementation. [ABSTRACT FROM AUTHOR]

The aim of this study is to design and implement an asynchronous computational scheme for solving the acoustic wave propagation equation with absorbing boundary conditions (ABCs) in the context of seismic imaging applications. While the convolutional perfectly matched layer (CPML) is typically used for ABCs in the oil and gas industry, its formulation further stresses memory accesses and decreases the arithmetic intensity at the physical domain boundaries. The challenges with CPML are twofold: (1) the strong, inherent data dependencies imposed on the explicit time-stepping scheme render asynchronous time integration cumbersome and (2) the idle time is further exacerbated by the load imbalance introduced among processing units. In fact, the CPML formulation of the ABCs requires expensive synchronization points, which may hinder the parallel performance of the overall asynchronous time integration. In particular, when deployed in conjunction with the multicore-optimized wavefront diamond temporal blocking (MWD-TB) approach for the inner domain points, it results in a major performance slow down. To relax CPML's synchrony and mitigate the resulting load imbalance, we embed CPML's calculation into MWD-TB's inner loop and carry on the time integration with fine-grained computations in an asynchronous, holistic way. This comes at the price of storing transient results to alleviate dependencies from critical data hazards while maintaining the numerical accuracy of the original scheme. Performance and scalability results on various x86 architectures demonstrate the superiority of MWD-TB with CPML support against the standard spatial blocking on various grid sizes. To our knowledge, this is the first practical study that highlights the consolidation of CPML ABCs with asynchronous temporal blocking stencil computations. [ABSTRACT FROM AUTHOR]

The compact finite difference scheme is designed to numerically solve the nonlinear Schrödinger equation and coupled nonlinear Schrödinger equations on an unbounded domain in this paper. The original problem on an unbounded domain is reduced to an initial boundary value problem defined on a bounded computational domain by applying the artificial boundary method. Then, the reduced problem on the bounded computational domain is solved by an efficient semi-implicit compact finite difference scheme, which is a fourth-order scheme with respect to spatial variable. The scheme efficiently avoids the time-consuming iteration procedure necessary for the nonlinear scheme and thus is relatively time-saving. Finally, the stability of the proposed scheme is rigorously analyzed. Numerical examples are given to illustrate the accuracy and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

The stability and error analysis of a second-order fast approximation are considered for the one-dimensional local and nonlocal diffusion equations in the unbounded spatial domain. We first use the conventional central difference scheme to discretize the local second-order spatial derivative operator and use an asymptotically compatible difference scheme to discretize the spatial nonlocal diffusion operator, and apply second-order backward differentiation formula (BDF2) to approximate the temporal derivative to achieve a fully discrete infinity system. To solve the resulting fully discrete systems, we develop a unified framework that is applicable to the discretization of both local and nonlocal problems. A key ingredient is to derive Dirichlet-to-Neumann (DtN)-type absorbing boundary conditions (ABCs). To do so, we apply the z-transform and solve an exterior problem using an iteration technique to derive a Dirichlet-to-Dirichlet (DtD)-type mapping as exact ABCs. After that, we use the Green formula to reformulate the DtD-type mapping equivalently as the DtN-type mapping. The resulting DtN-type mapping allows us to reduce the infinity discrete system into a finite discrete system in a truncated computational domain of interest, and also make it possible to present the stability and convergence analysis of the reduced problem under some open but reasonable assumptions. To efficiently implement the exact ABCs, we further develop a fast convolution algorithm based on approximation of the contour integral induced by the inverse z-transform. The stability and error analysis of the reduced finite discrete system based on the fast algorithm for exact ABCs are also established, and numerical examples are provided to demonstrate the effectiveness of our proposed approach. [ABSTRACT FROM AUTHOR]

Summary: This article is devoted to the efficient numerical solution of the Helmholtz equation in a two‐ or three‐dimensional (2D or 3D) rectangular domain with an absorbing boundary condition (ABC). The Helmholtz problem is discretized by standard bilinear and trilinear finite elements on an orthogonal mesh yielding a separable system of linear equations. The main key to high performance is to employ the fast Fourier transform (FFT) within a fast direct solver to solve the large separable systems. The computational complexity of the proposed FFT‐based direct solver is 𝒪(NlogN) operations. Numerical results for both 2D and 3D problems are presented confirming the efficiency of the method discussed. [ABSTRACT FROM AUTHOR]

Summary: In this paper, we explore a combination of the reciprocal absorbing boundary condition with perfectly matched discrete layers for the time‐domain analysis of wave motion in the full plane. Coupling of these absorbing boundaries leads to solutions for out‐of‐plane as well as in‐plane excitations of the full plane. Together with numerical transient simulations, harmonic analysis is also carried out for the purpose of verifying the results by comparing with available exact frequency‐domain solutions. The proposed combination is investigated through several numerical examples, which demonstrate its accuracy and efficiency. [ABSTRACT FROM AUTHOR]

We present a method for the automatic generation of netlists describing general three-dimensional electrothermal and electromagnetic field problems. Using a pair of structured orthogonal grids as spatial discretisation, a one-to-one correspondence between grid objects and circuit elements is obtained by employing the finite integration technique. The resulting circuit can then be solved with any standard available circuit simulator, circumventing the need for the implementation of a custom time integrator and dealing with non-linearities. Additionally, the approach straightforwardly allows for field-circuit coupling simulations by appropriately stamping the circuit description of lumped devices. As the computational domain in wave propagation problems must be finite, stamps representing absorbing boundary conditions are developed as well. Representative numerical examples are used to verify the approach. The results obtained by circuit simulation on the generated netlists are compared with appropriate reference solutions. [ABSTRACT FROM AUTHOR]

The focus of this work is on the analysis of the Westervelt equation modeling nonlinear propagation of high intensity ultrasound, in the practically relevant setting of a truncated computational domain with absorbing boundary conditions. We especially consider the zero and first order nonlinear absorbing boundary conditions devised in [38] in one and two space dimensions. As a matter of fact, the energy identities and estimates presented here were crucial for designing these absorbing boundary conditions in such a way that the desired energy dissipation through the boundary is guaranteed. Under the hypothesis of small initial data, we establish local well-posedness and provide higher order energy estimates, that we expect to be of additional use in boundary control and stabilization. [ABSTRACT FROM AUTHOR]

As a significant contribution, we present a novel formulation of the complex frequency-shifted perfectly matched layer (CFS-PML) for differential forms, introducing the innovative concept of a directional incidence matrix. This paper explores the application of differential forms in simulating wave propagation problems in three-dimensional domains. To confine the unbounded domain, we utilize a discretization technique based on the CFS-PML. Additionally, we provide an illustrative example that involves a system comprising a GPR (Ground Penetrating Radar) horn antenna, air, lossy soil, and a buried sphere. [ABSTRACT FROM AUTHOR]