Summary Double‐scale numerical methods constitute an effective tool for simultaneously representing the complex nature of geomaterials and treating real‐scale engineering problems such as a tunnel excavation or a pressuremetre at a reasonable numerical cost. This paper presents an approach coupling discrete elements (DEM) at the microscale with finite elements (FEM) at the macroscale. In this approach, a DEM‐based numerical constitutive law is embedded into a standard FEM formulation. In this regard, an exhaustive discussion is presented on how a 2D/3D granular assembly can be used to generate, step by step along the overall computation process, a consistent Numerically Homogenised Law. The paper also focuses on some recent developments including a comprehensive discussion of the efficiency of Newton‐like operators, the introduction of a regularisation technique at the macroscale by means of a second gradient framework, and the development of parallelisation techniques to alleviate the computational cost of the proposed approach. Some real‐scale problems taking into account the material spatial variability are illustrated, proving the numerical efficiency of the proposed approach and the benefit of a particle‐based strategy.

SUMMARYIn this work, the consequences of using several different discrete element granular assemblies for the representation of the microscale structure, in the framework of multiscale modeling, have been investigated. The adopted modeling approach couples, through computational homogenization, a macroscale continuum with microscale discrete simulations. Several granular assemblies were used depending on the location in the macroscale finite element mesh. The different assemblies were prepared independently as being representative of the same material, but their geometrical differences imply slight differences in their response to mechanical loading. The role played by the micro-assemblies, with weaker macroscopic mechanical properties, on the initiation of strain localization in biaxial compression tests is demonstrated and illustrated by numerical modeling of different macroscale configurations.

Summary This paper presents a multiscale model based on a FEM×DEM approach, a method that couples discrete elements at the microscale and finite elements at the macroscale. FEM×DEM has proven to be an effective way to treat real‐scale engineering problems by embedding constitutive laws numerically obtained using discrete elements into a standard finite element framework. This proposed paper focuses on some numerical open issues of the method. Given the nonlinearity of the problem, Newton's method is required. The standard full Newton method is modified by adopting operators different from the consistent tangent matrix and by developing ad‐hoc solution strategies. The efficiency of several existing operators is compared, and a new and original strategy is proposed, which is shown to be numerically more efficient than the existing propositions. Furthermore, a shared memory parallelization framework using OpenMP directives is introduced. The combination of these enhancements allows to overcome the FEM×DEM computational limitations, thus making the approach competitive with classical FEM in terms of stability and computational cost.

The purpose of this work is to determine, via a homogenization technique and in the framework of small strains, the macroscopic poroelastic properties of a saturated, deformable, cracked porous medium. The poroelastic matrix is assumed to be homogeneous and the cracks to be periodically distributed, the size of the period being small with respect to the size of the sample. The considered up-scaling method (based on asymptotic expansions) will provide two uncoupled mechanical and hydraulic problems describing the overall behavior of the material. The degradation of the mechanical properties due to damage is then introduced. Damage depends on cracks' opening, thus making the problem non-linear. A numerical solution of the problem is provided using finite elements. Stress-strain time-histories corresponding to an oedometric and a biaxial test are used as numerical examples. The numerical solution allows the exploration of the non-linear anisotropic behavior along with the bifurcation phenomenon.

The paper presents a multi-scale modeling of Boundary Value Problem (BVP) approach involving cohesive-frictional granular materials in the FEM × DEM multi-scale framework. On the DEM side, a 3D model is defined based on the interactions of spherical particles. This DEM model is built through a numerical homogenization process applied to a Volume Element (VE). It is then paired with a Finite Element code. Using this numerical tool that combines two scales within the same framework, we conducted simulations of biaxial and pressuremeter tests on a cohesive-frictional granular medium. In these cases, it is known that strain localization does occur at the macroscopic level, but since FEMs suffer from severe mesh dependency as soon as shear band starts to develop, the second gradient regularization technique has been used. As a consequence, the objectivity of the computation with respect to mesh dependency is restored.

It is very common for natural or synthetic materials to be characterized by a periodic or quasi-periodic micro-structure. This micro-structure, under the different loading conditions may play an important role on the apparent, macroscopic behaviour of the material. Although, fine, detailed information can be implemented at the micro-structure level, it still remains a challenging task to obtain experimental metrics at this scale. In this work, a constitutive law obtained by the asymptotic homogenization of a cracked, damageable, poroelastic medium is first evaluated for multi-scale use. For a given range of micro-scale parameters, due to the complex mechanical behaviour at micro-scale, such multi-scale approaches are needed to describe the (macro) material’s behaviour. To overcome possible limitations regarding input data, meta-heuristics are used to calibrate the micro-scale parameters targeted on a synthetic failure envelope. Results show the validity of the approach to model micro-fractured materials such as coal or crystalline rocks.

Continuum media from classical mechanics cannot appropriately reproduce the evolution of materials exhibiting strong heterogeneities in the strain field, e.g. strain localization. Models without a microscale representation cannot properly reproduce the microscale mechanisms that trigger the strain localization, in addition, first gradient relations don't present any length parameter in the formulation. This results in a model without a characteristic length that cannot exhibit any objective band width. In this paper, techniques to introduce an internal length will be enumerated. Microstuctured materials will be retained and in particular Second Gradient model will be exposed and used along with a FEMxDEM approach. Numerical results showing the abilities of the enriched model will conclude the text.

Recent improvements in micro-scale material descriptions allow to build increasingly refined multiscale models in geomechanics. This often comes at the expense of computational cost which can eventually become prohibitive. Among other characteristics, the non-determinism of a micro-scale response makes its replacement by a surrogate particularly challenging. Machine Learning (ML) is a promising technique to substitute physics-based models, nevertheless existing ML algorithms for the prediction of material response do not integrate non-determinism in the learning process. Is it possible to use the numerical output of the latest micro-scale descriptions to train a ML algorithm that will then provide a response at a much lower computational cost? A series of ML algorithms with different levels of depth and supervision are trained using a data-driven approach. Gaussian Process Regression (GPR), Self-Organizing Maps (SOM) and Generative Adversarial Networks (GANs) are tested and the latter retained because of its superior results. A modified GANs with lower network depth showed good performance in the generation of failure probability maps, with good reproduction of the non-deterministic micro-scale response. The trained generator can be incorporated into existing multiscale models allowing to, at least partially, bypass the costly micro-scale computations.