University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Faculty Publications from the Department of Mechanical & Materials Engineering, Engineering Mechanics Department of 2009 Convergence, adaptive refinement, and scaling in 1D peridynamics Florin Bobaru Ph.D. University of Nebraska at Lincoln, [email protected] Mijia Yabg Ph.D. University of Nebraska at Lincoln Leonardo F. Alves M.S. University of Nebraska at Lincoln Stewart A. Silling Ph.D. Sandia National Laboratories Ebrahim Askari Ph.D. Boeing Co., Bellevue, WA See next page for additional authors Follow this and additional works at: https://digitalcommons.unl.edu/engineeringmechanicsfacpub Part of the Applied Mechanics Commons, Computational Engineering Commons, Engineering Mechanics Commons, and the Mechanics of Materials Commons Bobaru, Florin Ph.D.; Yabg, Mijia Ph.D.; Alves, Leonardo F. M.S.; Silling, Stewart A. Ph.D.; Askari, Ebrahim Ph.D.; and Xu, Jifeng Ph.D., "Convergence, adaptive refinement, and scaling in 1D peridynamics" (2009). Faculty Publications from the Department of Engineering Mechanics. 68. https://digitalcommons.unl.edu/engineeringmechanicsfacpub/68 This Article is brought to you for free and open access by the Mechanical & Materials Engineering, Department of at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Faculty Publications from the Department of Engineering Mechanics by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Authors Florin Bobaru Ph.D., Mijia Yabg Ph.D., Leonardo F. Alves M.S., Stewart A. Silling Ph.D., Ebrahim Askari Ph.D., and Jifeng Xu Ph.D. This article is available at DigitalCommons@University of Nebraska - Lincoln: https://digitalcommons.unl.edu/ engineeringmechanicsfacpub/68 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2009; 77:852–877 Published online 26 August 2008 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.2439 Convergence, adaptive refinement, and scaling in 1D peridynamics Florin Bobaru1,∗,†,‡, Mijia Yang1,4, Leonardo Frota Alves1, Stewart A. Silling2, Ebrahim Askari3 and Jifeng Xu3 1Department of Engineering Mechanics, University of Nebraska-Lincoln, Lincoln, NE, U.S.A. 2Multiscale Dynamic Material Modeling Department, Sandia National Laboratories, Albuquerque, NM, U.S.A. 3Boeing Co., Bellevue, WA, U.S.A. 4Department of Civil and Environmental Engineering, University of Texas - San Antonio, San Antonio, TX, U.S.A. SUMMARY We introduce here adaptive refinement algorithms for the non-local method peridynamics, which was proposed in (J. Mech. Phys. Solids 2000; 48:175–209) as a reformulation of classical elasticity for discontinuities and long-range forces. We use scaling of the micromodulus and horizon and discuss the particular features of adaptivity in peridynamics for which multiscale modeling and grid refinement are closely connected. We discuss three types of numerical convergence for peridynamics and obtain uniform convergence to the classical solutions of static and dynamic elasticity problems in 1D in the limit of the horizon going to zero. Continuous micromoduli lead to optimal rates of convergence independent of the grid used, while discontinuous micromoduli produce optimal rates of convergence only for uniform grids. Examples for static and dynamic elasticity problems in 1D are shown. The relative error for the static and dynamic solutions obtained using adaptive refinement are significantly lower than those obtained using uniform refinement, for the same number of nodes. Copyright q 2008 John Wiley & Sons, Ltd. Received 21 December 2007; Revised 3 June 2008; Accepted 8 July 2008 KEY WORDS: peridynamics; non-local methods; adaptive refinement; convergence; multiscale modeling ∗Correspondence to: Florin Bobaru, Department of Engineering Mechanics, University of Nebraska-Lincoln, Lincoln, NE, U.S.A. †E-mail: [email protected] ‡Associate Professor. Contract/grant sponsor: Boeing Co.; contract/grant number: DE-AC04-94AL85000 Contract/grant sponsor: Computer Science Research Foundation and Computer Science Research Institute at Sandia National Laboratories Copyright q 2008 John Wiley & Sons, Ltd. CONVERGENCE, ADAPTIVE REFINEMENT, AND SCALING IN 1D PERIDYNAMICS 853 1. INTRODUCTION 1.1. Literature review Adaptive grid refinement techniques have been introduced in numerical simulations in order to gain both accuracy and efficiency. Capturing stress and strain concentrations near cracks, for example, requires dense computational grids. Adaptive methods permit reduction of grid density away from steep gradients thus rendering the computational problems tractable. Error estimators are required to determine where grid refinement is necessary. Adaptive methods for finite element methods (FEM) have been studied for the past three decades (see, e.g. [1, 2]). Automatic and goal-oriented h, p,andhp-adaptive strategies for FEM have been introduced and discussed in [3–6]. While many applications of adaptive refinement methods in FEM have been developed for static problems, adaptivity for dynamic problems has also been studied (e.g. [7–10]). In 1D time-dependent problems, several additional issues have to be dealt with, such as (see [7, 8]) spurious reflections of waves over non-uniform grids. A local error estimator and an adaptive time-stepping procedure for dynamic analysis are proposed in [9]. Dispersion effects due to FE spatial discretization are also discussed in [11]. Adaptive finite element approximations for the acoustic wave equation using a global duality argument and Galerkin orthogonality are presented in [12], while in [10] automatic energy conserving space–time refinement techniques for dynamic FEM are shown. The method presented in [10] performs automatic refining and coarsening for the wave propagation problem in 1D. Adaptive discontinuous Galerkin FEM for dynamic problems are developed in [13]. In multiple dimensions, the grid generation for adaptive refinement becomes more complex and special algorithms have been proposed. One of the simplest is using quad tree structures (in 2D) and has been used in the context of finite volume method in [14–16], for FEM in [17], and with the meshfree element-free Galerkin method [18], where level-1 quad-tree structures are employed even though this is not explicitly mentioned in the paper. The level-1 (or 1-irregular) grid refinement has been originally introduced in [1]. In the present study we introduce algorithms and show convergence results regarding adap- tive grid refinement algorithms for static and dynamic elasticity problems in 1D for a non-local method, the peridynamic formulation. Non-local methods are necessary in modeling of localization phenomena, such as fracture, damage, shear bands, for reasons described in detail in the review paper [19]. Adaptive refinement for other non-local methods applied to damage problems and discretized using the finite element method has appeared in, for example, [20–23]. These methods cannot be directly applied to peridynamics because the peridynamic formulation, first introduced in [24], is different from other non-local methods. We discuss this topic briefly below. 1.2. Overview of peridynamics The peridynamic method, introduced in [24], is a non-local formulation of continuum mechanics in which each material point is connected (via peridynamic bonds) with nodes in a certain region around it and not only with its nearest neighbors. The method is particularly well suited for dealing with cracks and damage in solid mechanics (see [25, 26]) especially in situations where the crack path, for example, is not known in advance. The method also allows for easy introduction of long- range forces, such as van der Waals interactions and examples of deformation and damage at the nanoscale are treated in [26, 27]. In fact, all forces in the peridynamic formulation are long-range. Other non-local methods have been proposed over the years: see, e.g. [28–33]. For a recent review Copyright q 2008 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2009; 77:852–877 DOI: 10.1002/nme 854 F. BOBARU ET AL. of other non-local methods, we refer the reader to [19]. Peridynamics differs from other integral non-local methods in at least two crucial aspects: (1) spatial derivatives are eliminated while other methods average infinitesimal stresses over a certain area and (2) the way damage and fracture are introduced in the method (see [26]). A description of a meshfree numerical implementation for the peridynamic formulation is given in [25], where bond failure is related to the classical energy release rate in brittle fracture. The method is used in [26] to simulate the tearing of non-linear membranes and failure of nanofiber networks (see also [27]). Spontaneous fracture initiation and dynamic propagation are captured and membrane wrinkling and the influence of Van der Waals forces in nanofiber networks are discussed. Classical continuum mechanics has been applied to explain the behavior of structural bodies which remain continuous as they deform under external loads. However, the effectiveness of compu- tational methods based on classical continuum mechanics, such as finite elements, in modeling material failure has lagged far behind their capabilities in traditional stress analysis. This difficulty arises because these methods solve the classical equations of continuum mechanics, which are a set of partial differential equations. These equations cannot be applied directly across disconti- nuities resulted from material