Finite element formulations for exterior problems: application to hybrid methods, non-reflecting boundary conditions, and infinite elements

Harari, Isaac; Barbone, Paul E.; Montgomery, Joshua M.

doi:10.1002/(sici)1097-0207(19970815)40:15<2791::aid-nme191>3.0.co;2-w

Cited by 26 publications

(3 citation statements)

References 30 publications

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“…4 In the context of wave propagation problems, most ABCs have been developed in the framework of the classical (local) theories. 18 The extension to peridynamic (PD) wave-type equations is challenging as the interactions between material points are nonlocal and the corresponding nonlocal operators are associated with volume constrained boundary conditions. [19][20][21] For this case, the calculation of the kernel function from the Laplace transform becomes complicated due to the nonlocality.…”

Section: Introductionmentioning

confidence: 99%

“…An important advantage of ABCs is that they are directly constructed in the time and space domains (in contrast to PMLs) and, in the case of high‐order ABCs, up to any desired order 4 . In the context of wave propagation problems, most ABCs have been developed in the framework of the classical (local) theories 18 . The extension to peridynamic (PD) wave‐type equations is challenging as the interactions between material points are nonlocal and the corresponding nonlocal operators are associated with volume constrained boundary conditions 19‐21 .…”

Section: Introductionmentioning

confidence: 99%

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Dirichlet‐type absorbing boundary conditions for peridynamic scalar waves in two‐dimensional viscous media

Hermann

Shojaei

Seleson

et al. 2023

Numerical Meth Engineering

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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.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Dirichlet‐type absorbing boundary conditions for peridynamic scalar waves in two‐dimensional viscous media

Hermann

Shojaei

Seleson

et al. 2023

Numerical Meth Engineering

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“…With the attractiveness of embedded interface method and Nitsche's formulation to weakly and stably enforce interface constraints, we intend to investigate the possibility of using such approach for steady‐state acoustic problems with material interfaces. Similar formulations were proposed for hybrid asymptotic‐numerical methods in scattering, non‐reflecting boundary conditions, and finite elements for exterior acoustics problems . It is well known that the Helmholtz equation leads to an indefinite variational form.…”

Section: Introductionmentioning

confidence: 99%

Nitsche's method for Helmholtz problems with embedded interfaces

Zou

Aquino

Harari

2016

Numerical Meth Engineering

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SUMMARY In this work, we use Nitsche’s formulation to weakly enforce kinematic constraints at an embedded interface in Helmholtz problems. Allowing embedded interfaces in a mesh provides significant ease for discretization, especially when material interfaces have complex geometries. We provide analytical results that establish the well-posedness of Helmholtz variational problems and convergence of the corresponding finite element discretizations when Nitsche’s method is used to enforce kinematic constraints. As in the analysis of conventional Helmholtz problems, we show that the inf-sup constant remains positive provided that the Nitsche’s stabilization parameter is judiciously chosen. We then apply our formulation to several 2D plane-wave examples that confirm our analytical findings. Doing so, we demonstrate the asymptotic convergence of the proposed method and show that numerical results are in accordance with the theoretical analysis.

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Boundary infinite elements for the Helmholtz equation in exterior domains

Harari

Barbone

Slavutin

et al. 1998

Int. J. Numer. Meth. Engng.

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A novel approach to the development of infinite element formulations for exterior problems of time‐harmonic acoustics is presented. This approach is based on a functional which provides a general framework for domain‐based computation of exterior problems. Special cases include non‐reflecting boundary conditions (such as the DtN method). A prominent feature of this formulation is the lack of integration over the unbounded domain, simplifying the task of discretization. The original formulation is generalized to account for derivative discontinuities across infinite element boundaries, typical of standard infinite element approximations. Continuity between finite elements and infinite elements is enforced weakly, precluding compatibility requirements. Various infinite element approximations for two‐dimensional configurations with circular interfaces are presented. Implementation requirements are relatively simple. Numerical results demonstrate the good performance of this scheme. © 1998 John Wiley & Sons, Ltd.

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Finite element formulations for exterior problems: application to hybrid methods, non-reflecting boundary conditions, and infinite elements

Cited by 26 publications

References 30 publications

Dirichlet‐type absorbing boundary conditions for peridynamic scalar waves in two‐dimensional viscous media

Dirichlet‐type absorbing boundary conditions for peridynamic scalar waves in two‐dimensional viscous media

Nitsche's method for Helmholtz problems with embedded interfaces

Boundary infinite elements for the Helmholtz equation in exterior domains

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