Concurrent Multiscale Methods Research Articles

Iterated learning and multiscale modeling of history-dependent architectured metamaterials

Neural network based models have emerged as a powerful tool in multiscale modeling of materials. One promising approach is to use a neural network based model, trained using data generated from repeated solution of an expensive small scale model, as a surrogate for the small scale model in application scale simulations. Such approaches have been shown to have the potential accuracy of concurrent multiscale methods like FE2, but at the cost comparable to empirical methods like classical constitutive models or parameter passing. A key question is to understand how much and what kind of data is necessary to obtain an accurate surrogate. This paper examines this question for history dependent elastic–plastic behavior of an architected metamaterial modeled as a truss. We introduce an iterative approach where we use the rich arbitrary class of trajectories to train an initial model, but then iteratively update the class of trajectories with those that arise in large scale simulation and use transfer learning to update the model. We show that such an approach converges to a highly accurate surrogate, and one that is transferable.

A review of concurrent multiscale methods for the analysis of fine scale discontinuity problems

A review of concurrent multiscale methods for the analysis of fine scale discontinuity problems

Multiscale modeling of nanoindentation and nanoscratching by generalized particle method

Most concurrent multiscale methods that expand atomistic region by continuum domain suffer from inconsistent material constitutive properties, which affect the integrity of model in the interface of atomistic and continuum domains. In this paper, the Generalized Particle (GP) method is employed to simulate nanoindentation and nanoscratching of a single-crystal aluminum sample. The main advantage of the GP method lies in its ability to extend the simulation model while maintaining consistent atomic properties across all scales. Coarsening of the atomic domain has been conducted through two-scale and three-scale GP model. The results showed a strong consistency between the results of full atomic simulations and those achieved through the GP method for both nanoindentation and nanoscratch simulations. Also, wave reflections were not seen at the interfaces. The study revealed that an increase in the number of distinct particle domains led to a reduction in the accuracy of multiscale simulations.

Three-dimensional multiscale modeling of nanoindentation

Concurrent multiscale methods have been developed to reduce the degrees of freedom and reduce the effects of boundary conditions on the results of atomic simulations. In this paper, two simplified concurrent multiscale methods, one with handshake region and another without handshake region are used to investigate the nanoindentation process on a single crystal of Al at room temperature. The multiscale models are validated by observing reasonably well similarities in the load–depth curves obtained from multiscale and full MD simulations. Refining the element size down to atomic spacing resulted in high computational efforts while the analysis results do not improve significantly. Also, it is shown that by defining the thermostat in the atomistic part, wave reflections are eliminated at the interface of atomic and continuum domains. It is shown that by selecting appropriate dimensions of the atomic domain, there is no need to use nonlinear elasticity in the continuum region. Also, hardness is more affected by sample size than the elastic modulus.

A Bayesian multiscale CNN framework to predict local stress fields in structures with microscale features

Multiscale computational modelling is challenging due to the high computational cost of direct numerical simulation by finite elements. To address this issue, concurrent multiscale methods use the solution of cheaper macroscale surrogates as boundary conditions to microscale sliding windows. The microscale problems remain a numerically challenging operation both in terms of implementation and cost. In this work we propose to replace the local microscale solution by an Encoder-Decoder Convolutional Neural Network that will generate fine-scale stress corrections to coarse predictions around unresolved microscale features, without prior parametrisation of local microscale problems. We deploy a Bayesian approach providing credible intervals to evaluate the uncertainty of the predictions, which is then used to investigate the merits of a selective learning framework. We will demonstrate the capability of the approach to predict equivalent stress fields in porous structures using linearised and finite strain elasticity theories.

Phonon spectrum and phonon focusing in coarse-grained atomistic simulations

This work investigates the accuracy, efficiency, and applicability of coarse-grained (CG) atomistic methods in simulation of phonon dynamics. First, we compute and compare phonon dispersion relations in CG models with those in atomically resolved models, using the concurrent atomistic-continuum (CAC). The CG atomistic models using the CAC method are shown to reproduce long-wavelength phonons with great accuracy, while capturing the dynamics of some short-wavelength phonons that are usually inaccessible to CG methods. We then present CG simulation results of the propagation of heat pulses in Si with the interaction between atoms being modelled with the Stillinger-Weber potential; the experimentally observed phonon-focusing patterns in the (1 0 0) and (1 1 1) planes of Si crystals are reproduced. The accuracy and efficiency of the CAC method in CG simulation of acoustic and optical phonon branches are quantified with respect to atomically-resolved molecular dynamics simulations. The applicability and limitations of concurrent multiscale methods in the simulation of phonon transport across atomistic-continuum interface are investigated. Possible ways to overcome the limitations are discussed.

The heterogeneous multiscale method applied to inelastic polymer mechanics.

Mechanisms emerging across multiple scales are ubiquitous in physics and methods designed to investigate them are becoming essential. The heterogeneous multiscale method (HMM) is one of these, concurrently simulating the different scales while keeping them separate. Owing to the significant computational expense, developments of HMM remain mostly theoretical and applications to physical problems are scarce. However, HMM is highly scalable and is well suited for high performance computing. With the wide availability of multi-petaflop infrastructures, HMM applications are becoming practical. Rare applications to mechanics of materials at low loading amplitudes exist, but are generally confined to the elastic regime. Beyond that, where history-dependent, irreversible or nonlinear mechanisms occur, not only computational cost but also data management issues arise. The micro-scale description loses generality, developing a specific microstructure based on the deformation history, which implies inter alia that as many microscopic models as discrete locations in the macroscopic description must be simulated and stored. Here, we present a detailed description of the application of HMM to inelastic mechanics of materials, with emphasis on the efficiency and accuracy of the scale-bridging methodology. The method is well suited to the estimation of macroscopic properties of polymers (and derived nanocomposites) starting from knowledge of their atomistic chemical structure. Through application of the resulting workflow to polymer fracture mechanics, we demonstrate deviation in the predicted fracture toughness relative to a single-scale molecular dynamics approach, thus illustrating the need for such concurrent multiscale methods in the predictive estimation of macroscopic properties.This article is part of the theme issue ‘Multiscale modelling, simulation and computing: from the desktop to the exascale’.

Large-scale simulations of quasi-brittle microcracking in realistic highly heterogeneous microstructures obtained from micro CT imaging

We present large-scale simulations of microcracking initiation and propagation using a continuum mechanics description in realistic, voxel-based microstructures of heterogeneous quasi-brittle materials obtained from CT imaging techniques. The phase field method is used to describe the evolution of the complex microcracks networks in both uniaxial tension and compression of a sub-volume of a lightweight concrete, where all pores and sand grains are explicitly described. A description of the meshing techniques for such complex voxel-based models is provided, and a convergence study of the tensile failure strength with respect to the sample size is carried out. Such large-scale simulations have high potential to be used either within recent concurrent multiscale methods or in approaches combining in-situ experiments with 3D imaging techniques and simulations for inverse identification of microstructural damage models.

Benchmarking, validation and reproducibility of concurrent multiscale methods are still needed

Benchmarking, validation and reproducibility of concurrent multiscale methods are still needed

Mesh refinement schemes for the concurrent atomistic-continuum method

Most concurrent multiscale methods that employ domain decomposition divide the simulation domain into atomistic and continuum subdomains such that nanoscale defects are described at atomistic resolution while a continuum treatment is employed elsewhere. An example is the concurrent atomistic-continuum (CAC) method. While dislocations can nucleate and migrate in the continuum domain in CAC, they are restricted to prescribed slip planes along interelement boundaries. In this paper, we extend the original CAC method by implementing mesh refinement schemes in a straightforward manner to accommodate a 3-D dislocation network and curved dislocation migration. CAC simulation results are validated by comparing to their equivalent fully atomistic simulations. Augmented with an adaptive mesh refinement scheme, a brittle-to-ductile transition in dynamic fracture response of single crystalline FCC Cu is accurately predicted using CAC simulations. Differing from most multiscale methods, the complexity of the geometric topologies during mesh refinement is avoided in this approach because CAC does not require interelement compatibility. The extended CAC method is expected to apply in multiscale modeling of more complex dislocation activities in crystalline materials up to μm scale.

Atomistic aspects of fracture

Any fracture process ultimately involves the rupture of atomic bonds. Processes at the atomic scale therefore critically influence the toughness and overall fracture behavior of materials. Atomistic simulation methods including large-scale molecular dynamics simulations with classical potentials, density functional theory calculations and advanced concurrent multiscale methods have led to new insights e.g. on the role of bond trapping, dynamic effects, crack-microstructure interactions and chemical aspects on the fracture toughness and crack propagation patterns in metals and ceramics. This review focuses on atomistic aspects of fracture in crystalline materials where significant advances have been achieved over the last ten years and provides an outlook on future perspectives for atomistic modelling of fracture.

A computational library for multiscale modeling of material failure

We present an open-source software framework called PERMIX for multiscale modeling and simulation of fracture in solids. The framework is an object oriented open-source effort written primarily in Fortran 2003 standard with Fortran/C++ interfaces to a number of other libraries such as LAMMPS, ABAQUS, LS-DYNA and GMSH. Fracture on the continuum level is modeled by the extended finite element method (XFEM). Using several novel or state of the art methods, the piece software handles semi-concurrent multiscale methods as well as concurrent multiscale methods for fracture, coupling two continuum domains or atomistic domains to continuum domains, respectively. The efficiency of our open-source software is shown through several simulations including a 3D crack modeling in clay nanocomposites, a semi-concurrent FE-FE coupling, a 3D Arlequin multiscale example and an MD-XFEM coupling for dynamic crack propagation.

A three-layer-mesh bridging domain for coupled atomistic–continuum simulations at finite temperature: Formulation and testing

Although concurrent multiscale methods have been well developed for zero-temperature simulations, improvements are needed to meet challenges pertaining to finite-temperature simulations. Bridging domain method (BDM) is one of the most efficient and widely-used multiscale atomistic–continuum techniques. It is recently revealed that the BDM coupling algorithm has a cooling effect on the atoms in the bridging domain (BD), and application of thermostats to rectify the cooling effect in the original BDM formulation is unstable. We propose improvement of the BDM formulation for finite-temperature simulations by employing a three-layer mesh structure in the BD, consisting of coarse, meso, and atomic meshes. The proposed method uses a mesh-independent physics-based discrimination between thermal and mechanical waves to define and introduce a meso mesh that is independent of the finite-element (FE) mesh. Temperature stability in the BD is achieved by constraining only the mechanical part of atomic motion to the FE displacements while unconstrained thermal vibrations are thermostatted using local thermostats in the BD. The new architecture of three-layer-mesh BD effectively mitigates the temperature cooling effect encountered by the conventional BDM as well as suppresses the spurious mechanical wave reflection. Numerical simulations have shown the robustness and accuracy of the proposed multiscale method at finite temperature.

Comparison of Concurrent Multiscale Methods in the Application of Fracture in Nickel

This paper presents a study of fracture in nickel using multiscale modeling. A comparison of six concurrent multiscale methods was performed in their application to a common problem using a common framework in order to evaluate each method relative to each other. Each method was compared in both a quasi-static case of crack tip deformation as well as a dynamic case in the study of crack growth. Each method was compared to the fully atomistic model with similarities and differences between the methods noted and reasons for these provided. The results showed a distinct difference between direct and handshake coupling methods. In general, for the quasi-static case, the direct coupling methods took longer to run compared to the handshake coupling methods but had less error with respect to displacement and energy. In the dynamic case, the handshake methods took longer to run, but had reduced error most notably when wave dissipation at the atomistic/continuum region was an issue. Comparing each method under common conditions showed that many similarities exist between each method that may be hidden by their original formulation. The comparison also showed the dependency on the application as well as the simulation techniques used in determining the performance of each method.

Bridging the scales in nano engineering and science

This review article describes various multiscale approaches, development of which was spurred by the emergence of nanotechnology. The multiscale approaches are grouped into two main categories: information-passing and concurrent. In the concurrent multiscale methods both, the discrete and continuum scales are simultaneously resolved, whereas in the information-passing schemes, the discrete scale is modelled and its gross response is infused into the continuum scale. Most of the information-passing approaches provide sublinear computational complexity, (i.e., scales sublinearly with the cost of solving a fine scale problem), but the quantities of interest are limited to or defined only on the coarse scale. The issues of appropriate scale selection and uncertainty quantification are also reviewed.