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Computer Methods in Applied Mechanics and Engineering

Continuous and discontinuous finite element methods for a peridynamics model of mechanics☆

Abstract

In contrast to classical partial differential equation models, the recently developed peridynamic nonlocal continuum model for solid mechanics is an integro-differential equation that does not involve spatial derivatives of the displacement field. As a result, the peridynamic model admits solutions having jump discontinuities so that it has been successfully applied to fracture problems. The peridynamic model features a horizon which is a length scale that determines the extent of the nonlocal interactions. Based on a variational formulation, continuous and discontinuous Galerkin finite element methods are developed for the peridynamic model. Discontinuous discretizations are conforming for the model without the need to account for fluxes across element edges. Through a series of simple, one-dimensional computational experiments, we investigate the convergence behavior of the finite element approximations and compare the results with theoretical estimates. One issue addressed is the effect of the relative sizes of the horizon and the grid. For problems with smooth solutions, we find that continuous and discontinuous piecewise-linear approximations result in the same accuracy as that obtained by continuous piecewise-linear approximations for classical models. Piecewise-constant approximations are less robust and require the grid size to be small with respect to the horizon. We then study problems having solutions containing jump discontinuities for which we find that continuous approximations are not appropriate whereas discontinuous approximations can result in the same convergence behavior as that seen for smooth solutions. In case a grid point is placed at the locations of the jump discontinuities, such results are directly obtained. In the general case, we show that such results can be obtained through a simple, automated, abrupt, local refinement of elements containing the discontinuity. In order to reduce the number of degrees of freedom while preserving accuracy, we also briefly consider a hybrid discretization which combines continuous discretizations in regions where the solution is smooth with discontinuous discretizations in small regions surrounding the jump discontinuities.

Introduction

Nonlocal theories in continuum mechanics have been introduced since at least the 1970s [17], [18], [23] and have recently become topical again [1], [2], [3], [7], [19], [21], [22], [24], [33], [34]. The peridynamics model [24], [30] is one such nonlocal theory formulated to describe the formation of discontinuities, e.g., cracks and fractures due to deformations, in the displacement field. In contrast to the classical local theory and also to most other nonlocal approaches, the peridynamic equation of motion is free of any spatial derivatives of the displacement. The peridynamic model has been applied in several settings; see, e.g., [4], [5], [6], [14], [15], [25], [27], [28], [29]. Modeling and theoretical studies regarding the peridynamic model are found in, e.g., [8], [9], [11], [16], [24], [26], [30], [31], [32], [35], [36], [37], [38], [39] whereas its computational solution is considered in, e.g., [6], [11], [13], [14], [15], [20], [28], [36], [37]. In particular, finite element discretizations are considered in [13], [20], [37].

In this paper, we study the use of continuous and discontinuous Galerkin finite element methods for discretizing a specific linear peridynamics model. We study the performance of straightforward finite element discretizations, showing that they fail to adequately treat problems having discontinuous solutions. We show, using an one-dimensional setting, that several variants of the straightforward approaches that naturally suggest themselves do not completely ameliorate the inadequacies of the latter. We then develop a finite element method that uses local refinement and that obtains the same optimal convergence rates for problems having solutions containing jump discontinuities as it does for problems having smooth solutions. We note that previous studies of finite element methods for the peridynamics model [13], [20], [37] do not consider the same issues, and in fact, deal with problems having relatively smooth solutions; certainly, problems with solutions having jump discontinuities are not treated.

This paper is the first to systematically study, in the peridynamics setting, many algorithmic and implementation issues related to finite element methods, and especially discontinuous Galerkin methods. For this reason and for the sake of keeping the exposition simple, we only consider the bond-based peridynamics model [24] which treats only central forces between material points. However, more general peridynamics models have been developed. In [24], a preliminary version of the state-based peridynamics is introduced that allows one to consider shear forces. Then, in [30], the peridynamics state-based model is fully developed through the introduction of the notion of a non-ordinary peridynamic material so that the interaction between two points is not collinear with the line between the two points. See also [26] and especially the review paper [30]. The algorithms introduced in this paper can be extended to peridynamics states in a fairly straightforward manner.

In Section 2, we provide a short overview of bond-based peridynamics models including, in Section 2.2, the specific linearized peridynamics model for proportional microelastic materials. In Section 3, we first provide a variational formulation of the one-dimensional, linear peridynamics model. Then, we introduce the three finite element discretization methods used in our computational studies; these are based on continuous piecewise-linear, discontinuous piecewise-constant, and discontinuous piecewise-linear finite element spaces. That section also includes a brief discussion of theoretical error estimates. In Section 4, we list the data and exact solutions for the two one-dimensional problems we use to test the accuracy of the discretization methods we develop. One of these problems has a polynomial solution so it is representative of problems having smooth solutions whereas the second problem has a solution containing a jump discontinuity. The results of computational experiments are found in Sections 5 Computational results for smooth solutions, 6 Computational results for discontinuous solutions with grid points coinciding with points of discontinuity, 7 Computational results for discontinuous solutions with grid points not coinciding with points of discontinuity. In these sections, we compare the relative advantages and disadvantages of the three methods and study their convergence behaviors. A particular focus of our studies is the effects that different choices for the horizon, a parameter that appears in the definition of the model, have on the accuracy of the resulting approximations. In Section 5 results for smooth solutions are given whereas, in Section 6, discontinuous exact solutions are considered for the case of having a grid point located exactly at the point where a jump discontinuity in the exact solution occurs. In Section 6, we also consider a hybrid method that combines discontinuous basis functions near the discontinuity with continuous basis functions away from the discontinuity. In Section 7, we again consider discontinuous exact solutions but in the more general situation of not having grid points coinciding with points of discontinuity of the exact solution. Ultimately, we develop a locally refined method that, when using discontinuous piecewise-linear finite element approximations, results in the same accuracy for nearly the same cost for problems having discontinuous solutions as it does for smooth solutions. In Section 8, we provide concluding remarks describing current and future work.

Section snippets

The general bond-based model

We follow the presentation of [24], the paper which introduced the peridynamics model. The equation of motion at any point x in the reference configuration at time t is given by ρ u ¨ ( x , t ) = ∫ H x f ( u ( x ′ , t ) - u ( x , t ) , x ′ - x ) d V x ′ + b ( x , t ) , where H x denotes a neighborhood of x, u the displacement vector field, b a prescribed body force density field, ρ the mass density in the reference configuration, and f a pairwise function whose value is the force density per unit volume that the particle located at x′ (in

Galerkin finite element discretizations

Let S(Ω′) denote a Banach space3 such that S(Ω′)   ⊂ L 2(Ω′). Define the affine space S g (Ω′)   =   {u(x)   ∈ S(Ω′)∣u(x)   = g(x) a.e. on Γ} and the subspace S 0  =   {v(x)   ∈ S(Ω′)∣v(x)   =   0 on Γ}. We then pose the Galerkin variational problem given b ( x ) ∈ L 2 ( Ω ) and g ( x ) ∈ L 2 ( Γ ) , seek u ( x ) ∈ S g ( Ω ′ ) satisfying 1 δ 2 ∫ α β v ( x ) ∫ x - δ x + δ u ( x ) - u ( x ′ ) | x - x ′ | d x ′ d x = ∫ α β b ( x ) v ( x ) d x , ∀ v ( x ) ∈ S 0 ( Ω ′ ) , corresponding to (8). The well-posedness of the variational problem (9) is shown in

Model problems used in the computational experiments

For the computational experiments, we choose α  =   0 and β  =   1 so that Ω  =   (0,   1) and Ω′   =   (−δ,   1   + δ).

We use the method of manufactured solutions to define problems for which the exact solutions are known. In some cases, this construction is facilitated by the following observations. If we assume that u(x) is sufficiently smooth and apply Taylor's theorem to the left-hand side of (8), we obtain 1 δ 2 ∫ x - δ x + δ u ( x ) - u ( x ′ ) | x - x ′ | d x ′ = - 1 2 u ″ ( x ) - 1 48 u ⁗ ( x ) δ 2 + ⋯ which we can use to determine the right-hand side b(x) of (8)

Computational results for smooth solutions

We present results of computational experiments for the model problems having the smooth solution (25) and for each of the three discretization schemes discussed in Sections 3.1 Continuous piecewise-linear finite element spaces, 3.2 Discontinuous piecewise-constant finite element spaces, 3.3 Discontinuous piecewise-linear finite element spaces. In the tables and figures, we provide the L 2(Ω) and L ∞(Ω) norms and, where appropriate, the H 1(Ω) semi-norms of the error and the corresponding rates of

Computational results for discontinuous solutions with grid points coinciding with points of discontinuity

In this section, computational results are presented for the discontinuous exact solution (26) for the "best-case scenario" in which a grid point is placed exactly at the location of the point at which the jump discontinuity occurs.15 The main reason we use this "best-case scenario" is to compare the different finite element methods we study in a setting for which the accuracy is not compromised

Computational results for discontinuous solutions with grid points not coinciding with points of discontinuity

We have shown that finite element discretizations of the peridynamic model for materials has the potential of producing accurate solutions, even for discontinuous solutions. In particular, discontinuous piecewise-linear approximations are apparently robustly optimally accurate with respect to choices for the model parameter, i.e., the horizon δ, and the smoothness of the solution, so long as one places a grid point at any point at which a jump discontinuity of the exact solution occurs. Of

Concluding remarks

Current and future work addresses implementations of the finite element methodology in two and three dimensions. Extension of the methodology to peridynamics state-based models and to nonlinear problems are also being considered. Of particular interest are adaptive grid refinement strategies for problems having solutions containing jump discontinuities. Given the estimates in [38], a related question of interest is the effects that local refinements have on the condition numbers of the

Acknowledgements

The authors thank the referees for their careful reading of this paper and for their many suggestions that resulted in substantial improvements.

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