Peridynamic elastic waves in two-dimensional unbounded domains: Construction of nonlocal Dirichlet-type absorbing boundary conditions

Crack propagation in a semi-unbounded domain with two initial vertical pre-notches due to a rigid projectile impact.
Crack propagation in a semi-unbounded domain with two initial vertical pre-notches due to a rigid projectile impact. Left: simulation with the proposed absorbing boundary conditions. Right: reference simulation.
2023: Q2


This paper presents a new approach for constructing accurate absorbing boundary conditions for (nonlocal) peridynamics problems in 2D involving propagating cracks. 

Significance and Impact

Modeling elastic wave propagation in unbounded and semi-unbounded media plays a key role in various fields of engineering and natural science, notably for design of structures in civil and geotechnical engineering, with numerous applications in soil-structure interaction analysis, earthquake and ground motion, and monitoring of unbounded cracked media for assessing the failure risk of structures. A way to cope with unbounded-domain problems is truncating the problem domain at a certain distance from the region of interest and applying absorbing boundary conditions (ABCs) to the truncating boundary. Peridynamics is a nonlocal continuum theory that has been widely exploited to solve problems in mechanics and physics, achieving great success in modeling spontaneous formation of cracks in solids; however, most peridynamics studies have focused on bounded-domain problems. This paper presents a simple, accurate, and stable strategy to construct ABCs for peridynamics problems, resulting in significant computational cost savings.

Research Details

In this paper, a new approach to construct nonlocal absorbing boundary conditions (ABCs) for bond-based peridynamics (PD) is introduced. The proposed ABCs are derived from a semi-analytical solution of the PD equation of motion at the exterior domain (far field). This solution is constructed by a series of plane-wave modes that satisfy the dispersion relations of PD and effectively transmit the waves from the near field to the far field. At the continuum level and in the limit of vanishing nonlocality, we demonstrated that the PD modes recover those of the linear elastic wave equation corresponding to classical continuum mechanics (CCM). The proposed ABCs are entirely developed in the time and space domains. The advantage of this appealing feature is twofold. First, the construction of the proposed ABCs is free from any Fourier or Laplace transforms coping with which is cumbersome for the PD nonlocal operators. Second, the proposed ABCs can be easily applied to problems with material nonlinearity and dynamic fracture. The ABCs are derived from an approximate far-field solution in the form of time-dependent Dirichlet-type boundary conditions. Therefore, their imposition neither require differential operators nor auxiliary field variables which are not straightforward to deal with even for the case of CCM. We demonstrated that the proposed ABCs do not introduce surface effects, as opposed to bounded-domain simulations. At the discrete level, the modes satisfy the numerical dispersion relations of wave using the same meshfree discretization scheme of the near field. This strategy makes the ABCs matched/consistent with that of the near field and avoids undesired numerical instabilities due to combination of two different numerical techniques in the computational domain. At each absorbing node on the boundary layer, the unknown coefficients of the far-field semi-analytical solution are computed in terms of the nodal values within a cloud (centered at the absorbing node) via a simple collocation procedure in space and time. This results in two updating vectors (related to the displacement and velocity fields) for each absorbing node. Multiplication of the updating vectors by the nodal values of their corresponding cloud at each time results in the proposed Dirichlet-type ABCs. Since the boundary values are found individually for each absorbing node, they are easy to implement and free from any distribution procedure over the boundary layer. We scrutinized the performance of the proposed ABCs in the solution of unbounded- and semi-unbounded-domain problems. The numerical examples are set such that the propagating waves are highly dispersive. Our investigation shows that the proposed method yields a proper level of accuracy and performs stably in the case of long-term computations.


The focus of this paper is on application of peridynamics (PD) to propagation of elastic waves in unbounded domains. We construct absorbing boundary conditions (ABCs) derived from a semi-analytical solution of the PD governing equation at the exterior region. This solution is made up of a finite series of plane waves, as fundamental solutions (modes), which satisfy the PD dispersion relations. The modes are adjusted to transmit the energy from the interior region (near field) to the exterior region (far field). The corresponding unknown coefficients of the series are found in terms of the displacement field at a layer of points adjacent to the absorbing boundary. This is accomplished through a collocation procedure at subregions (clouds) around each absorbing point. The proposed ABCs offer appealing advantages, which facilitate their application to PD. They are of Dirichlet-type, hence their implementation is relatively simple as no derivatives of the field variables are required. They are constructed in the time and space domains and thus application of Fourier and Laplace transforms, cumbersome for nonlocal models, is not required. At the discrete level, the modes satisfy the same numerical dispersion relations of the near field, which makes the far-field solution compatible with that of the near field. We scrutinize the performance of the proposed ABCs through several examples. Our investigation shows that the proposed ABCs perform stably in time with an appropriate level of accuracy even in problems characterized by highly-dispersive propagating waves, including crack propagation in semi-unbounded brittle solids.

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Last Updated: February 26, 2023 - 5:10 pm