Discrete-continuum Coupling Method to Simulate Highly Dynamic Multi-scale Problems: Simulation of Laser-induced Damage in Silica Glass, Volume 2

Preface

Smart materials, added value manufacturing and factories for the future are key technological subjects for the future product developments and innovation. One of the key challenges is to play with the microstructure of the material to not only improve its properties but also to find new properties. Another key challenge is to define micro- or nanocomposites in order to mix physical properties. This allows enlarging the field of possible innovative material design. The other key challenge is to define new manufacturing processes to realize these materials and new factory organization to produce the commercial product. From the material to the product, the numerical design tools must follow all these evolutions from the nanoscopic scale to the macroscopic scale (simulation and optimization of the factory). If we analyze the great amount of numerical tool development in the world, we find a great amount of development at the nanoscopic to the microscopic scales, typically linked to ab initio calculations and molecular dynamics. We also find a great amount of numerical approaches used at the millimeter to the meter scales. The most famous one in the field of engineering is the finite element method. However, there is a numerical death valley to pass through, from micrometers to several centimeters. This scale corresponds to the need for taking into account discontinuity or microstructures in the material behavior at the sample scale or component scale (several centimeters). Since the 2000s, some attempts have been carried out to apply the discrete element method (DEM) for simulation of continuous materials. This method has been developed historically for true granular materials, such as sand, civil engineering grains and pharmaceutical powders. Some recent developments give new and simple tools to simulate quantitatively continuous materials and to pass from microscopic interactions at the material scale to the classical macroscopic properties at the component scale (stress and strain, thermal conductivity, cracks, damages, electrical resistivity, etc.).

In this set of books on discrete element model and simulation of continuous materials, we propose to present and explain the main advances in this field since 2010. The first book explained in a clear and simple manner the numerical way to build a DEM simulation that gives the right (same) macroscopic material properties, e.g. Young’s modulus, Poisson’s ratio, thermal conductivity, etc. Then, it showed how this numerical tool offers a new and powerful method for analysis and modeling of cracks, damages and finally failure of a component. In this second book, we present the coupling (bridging) between the DEM method and continuum numerical methods, such as the constrained natural element method. This allows us to focus DEM in the parts where the microscopic properties and discontinuities lead the behavior and leave continuum calculation where the material can be considered as continuous and homogeneous. Coupling scales for highly dynamic problem has been a challenge for a long time. This book shows how to choose the coupling parameters properly to avoid spurious wave reflection and to allow the passage of all the dynamic information both from fine to coarse model and from coarse to fine model. The second part demonstrates the ability of the coupling method to simulate a highly nonlinear dynamical problem: the laser shock processing of silica glass.

A further book in this set presents the numerical code developed under the free License GPL ‘GranOO’: www.granoo.org. All the presented developments are implemented in a simple way on this platform. This allows scientists and engineers to test and contribute to improving the presented methods in a simple and open way.

Now, dear readers let us open this book and welcome in the DEM community for the material of future development …

Ivan IORDANOFF

Bordeaux, France

August, 2015

Introduction

I.1. Bridging the scales in science and engineering

Over the past few decades, numerical simulation has firmly established itself as a partner to experiment with unraveling the fundamental principles behind continuous material behaviors. Starting from the 1960s, this approach received strong scientific interest which led to the development of a great number of numerical methods. These methods can be divided into continuum methods (CMs) and discrete methods (DMs). Undoubtedly, the CMs are the most commonly used to solve problems at the engineering (macroscopic) scale, at which the mechanical behavior of materials can generally be described by continuum mechanics. However, their application to investigate microscopic effects, which can have a profound impact on what happens at larger space and time scales, faces several difficulties. Although solutions that are more or less reliable have been proposed in the literature to get over these difficulties, an accurate description of numerous engineering problems remains very challenging for CMs. Some difficulties, associated with reliance of these methods on a predefined mesh and/or unsuitability in dealing with discontinuities, are still not adequately ironed out. In contrast, the DMs naturally provide solutions for most of these outstanding difficulties. These are based on discrete mechanics and do not rely on any kind of mesh. Using DMs, the studied domain is modeled by a set of discrete bodies allowing discontinuities to be naturally taken into account. Although these methods were originally developed to study naturally discrete problems, their features have been proven to be very attractive for several continuous problems involving complex microscopic effects, e.g. damage, fracture and fragmentation. Application of such methods to overcome the CM limitations is then well worth exploring. Nevertheless, the lack of theoretical framework allowing these methods to properly model continua has restricted their application on this kind of problem until very recently.

Modeling continuous problems with DMs mainly faces two significant challenges. The first challenge concerns the choice of the cohesive links between the neighboring discrete bodies and the identification of their microscopic parameters so as to ensure the expected macroscopic mechanical behavior. The second challenge concerns the construction of the discrete domain which must take into account the structural properties of the original problem domain, e.g. homogeneity and isotropy, and must ensure independence of the macroscopic mechanical behavior on the discrete bodies number. The first book of this set, Discrete Element Method to Model 3D Continuous Materials [JEB 15], tried to tackle these challenges and to provide a comprehensive methodology allowing for correct discrete element modeling of continuous materials. This methodology was developed for a particular discrete element method (DEM) in which a given material is modeled by a set of rigid spheres in interaction with each other by three-dimensional (3D) cohesive beam bonds. As shown in [JEB 15], several conditions must be satisfied to properly model continua using the proposed DEM variation. The development of this DEM variation, in addition to the ever-increasing power and affordability of fast computers, has brought discrete element modeling of continuous material within reach. Nowadays, such a method presents a prominent tool for elucidating complex mechanical behaviors of continuous materials [AND 12b, JEB 15]. It was successfully applied to investigate several challenging problems that cannot be easily treated by CMs [AND 13, AND 12a, TER 13, JEB 13a, JEB 13b]. The major drawback of this method is that it is very time-consuming compared to CMs and the computation time can quickly become crippling, especially in the case of a large studied domain. However, in modern material science and engineering, real materials usually exhibit phenomena requiring multi-scale analysis. These phenomena require on one scale a very accurate and computationally expensive description to capture the complex effects at this scale and on another scale a coarser description is sufficient and, in fact, necessary to avoid prohibitively large computation. Therefore, in a view of expanding the scope of proposed DEM and alleviating its limitations, it would be beneficial to couple this approach with a CM, such that the computation effort can be distributed as needed.

In many mechanical problems, the notion of multi-scale modeling arises quite naturally. Indeed, most of the material behaviors at the macroscopic scale, which is the scale of interest for engineering applications, are determined by microscopic interactions between atoms. This is why such a notion has become a special area of interest for many scientists. Consequently, several multi-scale coupling methods have been developed over the last three decades. In a pioneer work, Ben Dhia [BEN 01, BEN 05, BEN 98] developed the Arlequin approach as a general framework which allows the intermixing of various mechanical models for structural analysis and computation. Abraham et al. [ABR 98, BRO 99] developed a methodology that couples the tight-bending quantum mechanics with molecular dynamics (MD) such that the two Hamiltonians are averaged in a bridging region. A damping was used in this region to reduce spurious reflections at the interface between the two models. Nevertheless, the choice of the damping coefficient remains difficult. Smirnova et al. [SMI 99] proposed a combined MD and finite element method (FEM) model with a transition zone in which the FEM nodes coincide with the positions of the particles in the MD region. The particles in the transition zone interact with the MD region via the interaction potential. At the same time, they experience the nodal forces due to the FEM grid. Belytschko and Xiao [BEL 03, XIA 04] developed a coupling method between the molecular dynamics and continuum mechanics models based on the bridging domain technique. In this method, the two models are overlaid at the interface and constrained with a Lagrange multiplier model in the bridging region. Fish et al. [FIS 07] formulated an atomistic-continuum coupling method based on a blend of the continuum stress and the atomistic force. In terms of equations, this method is very similar to the Arlequin approach [BEN 01, BEN 05, BEN 98]. In an interesting work, Chamoin et al. [CHA 10] analyzed the main spurious effects in the atomic-to-continuum coupling approaches and they proposed a corrective method based on the computation and injection of dead forces in the Arlequin formulation to offset these effects. Aubertin et al. [AUB 10] applied the Arlequin approach to couple the extended finite element method (X-FEM) with MD to study dynamic crack propagation. Bauman et al. [BAU 09] developed a 3D multi-scale method, based on the Arlequin approach, between highly heterogeneous particle models and nonlinear elastic continuum models. For more details, several papers reviewing these methods can be found in the literature [LU 05, XU 09, JEB 14, CUR 03]. Based on these papers, three approaches can mainly be used to couple DEM with CMs: the hierarchical, concurrent and hybrid hierarchical-concurrent coupling