Hierarchical Multiscale Modeling Research Articles

A temperature-related boundary Cauchy–Born method for multi-scale modeling of silicon nano-structures

Abstract The surface, edge and corner effects have significant influences in the electrical and optical properties of silicon nano-structures. In this paper, a novel hierarchical temperature-related multi-scale model is presented based on the boundary Cauchy–Born method to investigate not only the surface but also the edge and corner effects in thermal properties of diamond-like structures such as silicon nano-structures at finite temperature. A combined finite element method and molecular dynamics are respectively employed in macro- and micro-scale levels. The temperature-related Cauchy–Born rule is applied using the Helmholtz free energy, as the energy density of equivalent continua relating to the Tersoff inter-atomic potential. The model employs radial quadratures at the surface, edge and corner elements as an indicator of material behavior. The capability of computational algorithm is illustrated by numerical simulation of a nano-scale cube at finite temperature and the results are compared with the atomistic model.

A hierarchical multi-scale model for hexagonal materials taking into account texture evolution during forming simulation

Due to the absence of sufficient number of slip systems in hexagonal close packed (hcp) metals to accommodate arbitrary plastic deformation, mechanical twinning occupies an important role in the mechanical behavior of these metals. Twinning causes a significant and abrupt change in the orientation of crystals, whilst simultaneously affecting the hardening and plastic flow behavior of the material considerably. Modeling of forming processes of hexagonal close packed metals thus requires accounting for the evolution of texture and texture induced anisotropy, especially due to twinning. Additionally, the computational framework for the simulation of forming processes must be reliable and efficient in order to guarantee results in realistic time frames. In the present work, a phenomenological constitutive model consisting of an anisotropic yield function, associate flow rule and isotropic hardening, is coupled with the viscoplastic self-consistent polycrystal model in order to capture the slip and twinning activity, texture evolution and the evolving anisotropy during plastic deformation of hcp materials, similar to a hierarchical multi-scale modeling framework proposed by Van Houtte et al. [41] for cubic metals. To account for texture evolution during plastic deformation, the parameters involved in the anisotropic yield function are regularly updated based on mechanical test data obtained from the polycrystal model at the micro-scale. The parameter update is performed at discrete steps instead of every increment. The developed model is applied to describe the behaviors of pure zirconium at liquid nitrogen temperature. The results of the numerical simulations were found to be good agreement with experimental results, and at acceptable computation time.

A multiscale MD–FE model of diffusion in composite media with internal surface interaction based on numerical homogenization procedure

Mass transport by diffusion within composite materials may depend not only on internal microstructural geometry, but also on the chemical interactions between the transported substance and the material of the microstructure. Retrospectively, there is a gap in methods and theory to connect material microstructure properties with macroscale continuum diffusion characteristics. Here we present a new hierarchical multiscale model for diffusion within composite materials that couples material microstructural geometry and interactions between diffusing particles and the material matrix. This model, which bridges molecular dynamics (MD) and the finite element (FE) method, is employed to construct a continuum diffusion model based on a novel numerical homogenization procedure. The procedure is general and robust for evaluating constitutive material parameters of the continuum model. These parameters include the traditional bulk diffusion coefficients and, additionally, the distances from the solid surface accounting for surface interaction effects. We implemented our models to glucose diffusion through the following two geometrical/material configurations: tightly packed silica nanospheres, and a complex fibrous structure surrounding nanospheres. Then, rhodamine 6G diffusion analysis through an agarose gel network was performed, followed by a model validation using our experimental results. The microstructural model, numerical homogenization and continuum model offer a new platform for modeling and predicting mass diffusion through complex biological environment and within composite materials that are used in a wide range of applications, like drug delivery and nanoporous catalysts.

A Hierarchical Multiscale Model for Microfluidic Fuel Cells with Porous Electrodes

In this paper a hierarchical multiscale model for microfluidic fuel cells with porous electrodes is developed for the first time. An example of the hierarchical multiscale model for all-vanadium microfluidic fuel cells is presented in this study, in which the diffusion coefficient is used as a bridge between the microscale, mesoscale, and macroscale models. Three-level theories are used to describe the microfluidic fuel cells systems. The development of the model in different time and length scales is emphasized. Results from each scale model are compared with the corresponding experimental and other reasonable data in the literature. We demonstrate that the multiscale model is able to predict the characteristics of microfluidic fuel cells more accurately than before. The hierarchical multiscale model can be a potential tool for developing, designing, and optimizing microfluidic fuel cells.

Interface energy determination for the fully coherent β″ phase in Al–Mg–Si: making a case for a first principles based hybrid atomistic modelling scheme

We introduce a first principles based hierarchical multi-scale model scheme with application to a system comprising a fully coherent precipitate, immersed in a host lattice environment. As a test case, the needle-shaped main hardening phase β″ in the Al–Mg–Si alloy system has been examined. Calculations were confined to a cross-section slab, where the coherency of the interfaces is well established experimentally. The scheme couples a density functional theory (DFT) based description of the interface vicinity to a linear elasticity theory (LET) based description of the larger surroundings as well as the precipitate interior. The establishing link between these descriptions is purely structural, and LET based. At the boundary between the DFT and LET regions, subsystem distortion energies were compared using both formalisms, revealing only weak differences. On the basis of the modelling results, the need for a multi-scale model scheme over a full DFT analysis has been quantified through analysis of the β″ strain field decay. In the interface vicinity, and for a system comprising a β″–Mg5Al2Si4 precipitate with 4 × 8 unit cells along aP, cP, respectively, the calculated interface energy of 2.36 kJ per mol exceeds predictions as obtained with currently available alternative model schemes by ≈20%. Model system changes, required in order to approach a more reliable DFT–LET coupling and clarify the above described interface energy sensitivity, have been discussed. In comparison with alternative frameworks, our scheme offers additional flexibility when addressing interface configuration stabilities. This may allow for the study of more realistic configurations in the future.

A negative stiffness metamaterial inclusion via small-scale curved structural elements

This work presents a means to produce negative stiffness metamaterial units without employing inherent nonlinear material behavior or dynamic effects, but rather through large deformations of tailored microscale geometry. Specifically, the work of Qiu et al. [J. Microelectromech. Syst. 13, 137–146 (2004)] is generalized to create metamaterial units displaying non-monotonic stress-strain relationships with regions of negative stiffness behavior. These cells are designed to be embedded into a continuous matrix material and negative stiffness is elicited through pre-strain of the metamaterial unit cells brought about by changes in ambient pressure. This work presents a nonlinear hierarchical multiscale material model to estimate the macroscopic stiffness and loss of a composite material containing these pre-strained microscale structured inclusions. The multiscale model consists of two scale transition models: (i) an energy-based nonlinear finite element (FE) method to determine the anisotropic tangent moduli of the inclusion, and (ii) an analytical micromechanical model to determine the effective stiffness and loss tensors due to small perturbations about the local strain state of the metamaterial inclusions. Models and results for various volume fractions are presented and discussed.

Patient-Specific Computational Modeling and Magnetic Nanoconstructs: Tools for Maximizing the Efficacy of Stem Cell-Based Therapies

Stem cell transplantation has the potential to restore heart function following myocardial infarction. However, the success of any stem cell-based therapy is critically linked to the effective homing and early engraftment of the injected cells at the infarcted site. Here, a hierarchical multiscale computational model is proposed for predicting the patient-specific vascular transport and intratissue homing and migration of stem cells injected either systemically or locally. Starting with patient-specific data, such as the vascular geometry, blood flow, and location of the infarcted area, the computational model can be used to perform parametric analysis to identify optimal injection conditions in terms of administration route, injection site, catheter type, and infusion velocity. In addition to this, a new generation of magnetic nanoconstructs is introduced for labeling stem cells and monitoring their behavior in vivo via magnetic resonance imaging. These nanoconstructs also can be used for multimodal imaging, merging MRI and nuclear imaging, and the intracellular delivery of active agents to support stem cell differentiation. The convergence of computational modeling and novel nanoconstructs for stem cell labeling could improve our understanding in cell homing and early engraftment at the infarcted site and thus pave the way to more effective stem cell-based therapies.

A nonlinear negative stiffness metamaterial unit cell and small-on-large multiscale material model

A persistent challenge in the design of composite materials is the ability to fabricate materials that simultaneously display high stiffness and high loss factors for the creation of structural elements capable of passively suppressing vibro-acoustic energy. Relevant recent research has shown that it is possible to produce composite materials whose macroscopic mechanical stiffness and loss properties surpass those of conventional composites through the addition of trace amounts of materials displaying negative stiffness (NS) induced by phase transformation [R. S. Lakes et al., Nature 410, 565-567 (2001)]. The present work investigates the ability to elicit NS behavior without employing physical phenomena such as inherent nonlinear material behavior (e.g., phase change or plastic deformation) or dynamic effects, but rather the controlled buckling of small-scale structural elements, metamaterials, embedded in a continuous viscoelastic matrix. To illustrate the effect of these buckled elements, a nonlinear hierarchical multiscale material model is derived, which estimates the macroscopic stiffness and loss of a composite material containing pre-strained microscale structured inclusions. The multiscale model consists of two scale transition models, the first being an energy-based nonlinear finite element (FE) method to determine the tangent modulus of the metamaterial unit cell, and the other a classical analytical micromechanical model to determine the effective stiffness and loss tensors of a heterogeneous material for small perturbations from the local strain state of the unit cells. The FE method enables the estimation of an effective nonlinear anisotropic stiffness tensor of a buckled microstructure that produces NS and is sufficiently general to consider geometries different from those given in this work.

Hierarchical Multiscale Modeling of Polymer–Solid Interfaces: Atomistic to Coarse-Grained Description and Structural and Conformational Properties of Polystyrene–Gold Systems

A hierarchical simulation approach was developed in order to study polystyrene films sandwiched between two parallel Au(111) surfaces. The coarse-grained potentials describing the interaction of polystyrene with the gold surface were developed systematically using constrained all-atom molecular simulations of a styrene trimer on the Au(111) surface. The model was validated by studying a 5 nm film of short (10-mer) atactic polystyrene chains using all-atom and coarse-grained molecular dynamics simulations. The density, structure, and conformational properties of coarse-grained films were found to be in excellent agreement with all-atom ones. The coarse-grained model was then used to study the structural and conformational properties of roughly 10 and 20 nm thick films with 10-, 50-, 100-, and 200-mer chains. The width of the interphase region of the polymer films is property specific. The density profiles reached the bulk value around 1.5 nm from the interface, for all chain lengths. An estimate of the width of the interphase region based on the conformation tensor profile indicates that the interphase width is around 2–3 times the radius of gyration, which is proportional to the square root of the chain length (number of monomers), and for 200-mer chains is approximately 6–10 nm.

Hierarchical multiscale model for biomechanics analysis of microfilament networks

The mechanisms of force generation and transference via microfilament networks are crucial to the understandings of mechanobiology of cellular processes in living cells. However, there exists an enormous challenge for all-atom physics simulation of real size microfilament networks due to scale limitation of molecular simulation techniques. Following biophysical investigations of constitutive relations between adjacent globular actin monomers on filamentous actin, a hierarchical multiscale model was developed to investigate the biomechanical properties of microfilament networks. This model was validated by previous experimental studies of axial tension and transverse vibration of single F-actin. The biomechanics of microfilament networks can be investigated at the scale of real eukaryotic cell size (10 μm). This multiscale approach provides a powerful modeling tool which can contribute to the understandings of actin-related cellular processes in living cells.

Simulation of a Thick Plate Forming Benchmark Using a Multi Scale Texture Evolution and Anisotropic Plasticity Model

Thick plate and sheet materials are often characterised by an inhomogeneous distribution of properties such as yield strength and anisotropy throughout their thickness. Forming of these materials involves further heterogeneous evolution of these properties. A recently developed computational framework [1, now allows these heterogeneities to be modelled via a hierarchical multi-scale material modelling scheme: the evolution of texture and plastic anisotropy can be tracked and individually updated at every integration point in a finite element model, in a computationally efficient manner. In this paper we present the application of this multi-scale model to a benchmark forming simulation, the three point bending test of thick plate steels. A number of hot rolled high strength low alloy steels were considered, two of which are presented here. The results of the simulations are validated against experimental results. Comparison is made between computed and experimental deformed shapes and strain fields, using data acquired by digital image correlation. Predictions of heterogeneously evolved textures are compared with experimental macro-textures, acquired by XRD, at key locations in the final deformed samples. Such models for plate steel forming simulations that are able to provide accurate predictions of deformation textures and derived quantities in the entire volume of the material can be crucial to study further processing steps and properties of the final product.

NUMERICAL MODELING OF DIFFUSION IN COMPLEX MEDIA WITH SURFACE INTERACTION EFFECTS

Diffusion in natural, technological and biological systems is very common and most important process. Within these systems, which contain complex media, diffusion may depend not only on internal geometry, but also on the chemical interactions between solid phase and transported particles. Modeling remains a challenge due to this complexity. Here we first present a new hierarchical multiscale microstructural model for diffusion within complex media that incorporates both the internal geometry of complex media and the interaction between diffusing particles and surfaces of microstructures. Hierarchical modeling approach, which was introduced in [1], is employed to construct a continuum diffusion model based on a novel numerical homogenization procedure. Using this procedure, we evaluate constitutive material parameters of the continuum model, which include: equivalent bulk diffusion coefficients and the equivalent distances from the solid surface. Here, we examined diffusion of glucose through water using the following two geometrical/material configurations: silica nanofibers, and a complex internal structure consisting of randomly placed nanospheres and nanofibers. This new approach, consisting of microstructural model, numerical homogenization and continuum model, offer a new platform for modeling diffusion within complex media, capable of connecting micro and macro scales.

NUMERICAL MODELING OF DIFFUSION IN COMPLEX MEDIA WITH SURFACE INTERACTION EFFECTS

Diffusion in natural, technological and biological systems is very common and most important process. Within these systems, which contain complex media, diffusion may depend not only on internal geometry, but also on the chemical interactions between solid phase and transported particles. Modeling remains a challenge due to this complexity. Here we first present a new hierarchical multiscale microstructural model for diffusion within complex media that incorporates both the internal geometry of complex media and the interaction between diffusing particles and surfaces of microstructures. Hierarchical modeling approach, which was introduced in [1], is employed to construct a continuum diffusion model based on a novel numerical homogenization procedure. Using this procedure, we evaluate constitutive material parameters of the continuum model, which include: equivalent bulk diffusion coefficients and the equivalent distances from the solid surface. Here, we examined diffusion of glucose through water using the following two geometrical/material configurations: silica nanofibers, and a complex internal structure consisting of randomly placed nanospheres and nanofibers. This new approach, consisting of microstructural model, numerical homogenization and continuum model, offer a new platform for modeling diffusion within complex media, capable of connecting micro and macro scales.

An Atomistic-Based Hierarchical Multiscale Examination of Age Hardening in an Al-Cu Alloy

A large class of modern structural alloys derives its strength from precipitation hardening. Precipitates obstruct the motion of dislocations and thereby increase alloy strength. This paper examines the process using an atomistic-based hierarchical multiscale modeling framework. Atomistic modeling is employed to (1) compute solute-dislocation interaction energies for input into a semi-analytic solute hardening model and (2) evaluate precipitate strengths for use in dislocation line tension simulations. The precipitate microstructure in the dislocation line tension simulations is obtained from simple analytic precipitation kinetics relations. Fitting only the rate constants in the precipitation kinetics model, the macroscopic strength predictions of the hierarchical multiscale model are found to correspond reasonably well with experiments. By analyzing the potential sources of discrepancy between the model’s macroscopic predictions and experiments, this work illuminates the importance of specific atomic-scale processes and highlights important challenges that remain before truly predictive mechanism-based plasticity modeling can be realized.

Hierarchical multi-scale model reduction in the simulation of catalytic converters

This paper presents a methodology for multi-scale modelling using pre-computed data for the small scales. By pre-computing the required data at the micro- and meso-scales and storing them in a look-up table, these small-scale effects can be captured in a macro-scale model that executes with practicable execution times. The methodology is presented in the context of modelling a washcoated monolith reactor for the catalytic combustion of methane using a detailed multistep mechanistic model. The modelling is illustrated as a sequence of steps of increasing model simplification. In the first instance, only the reaction rates are pre-computed. Then the washcoat diffusion is included using a 2D axisymmetric model with a washcoat of uniform thickness, then with a non-symmetrical channel with a washcoat of non-uniform thickness. It is shown how a simple 1D single channel model can accurately account for the presence of a non-uniform washcoat. Finally, the external transport resistance is included in the pre-computation step, and it is shown that a simple one dimensional pseudo-homogeneous single channel model can be used provided the effective average rates are appropriately calculated. Execution times are shown to decrease by several orders of magnitude at minimal loss of solution accuracy. The use of pre-computed data allows for the inclusion of complex heat and mass transfer phenomena, as well as complex kinetics in full scale converter models at relatively small computational cost. The implementation of look-up tables into full scale CFD models is described, and optimal strategies proposed.

Nanomechanics of collagen microfibrils

Collagen constitutes one third of the human proteome, providing mechanical stability, elasticity and strength to organisms and is thus the prime construction material in biology. Collagen is also the dominating material in the extracellular matrix where its stiffness controls cell differentiation, growth and pathology. We use atomistic-based hierarchical multiscale modeling to describe this complex biological material from the bottom up. This includes the use and development of large-scale computational modeling tools to investigate several aspects related to collagen-based tissues, including source of visco-elasticity and deformation mechanisms at the nanoscale level. The key innovation of this research is that until now, collagen materials have primarily been described at macroscopic scales, without explicitly understanding the mechanical contributions at the molecular and fibrillar levels. The major impact of this research will be the development of fundamental models of collagenous tissues, important to the design of new scaffolding biomaterials for regenerative medicine as well as for the understanding of collagen-related diseases.

Hierarchical multiscale modeling of macromolecules and their assemblies

Soft materials (e.g., enveloped viruses, liposomes, membranes and supercooled liquids) simultaneously deform or display collective behaviors, while undergoing atomic scale vibrations and collisions. While the multiple space-time character of such systems often makes traditional molecular dynamics simulation impractical, a multiscale approach has been presented that allows for long-time simulation with atomic detail based on the co-evolution of slowly-varying order parameters (OPs) with the quasi-equilibrium probability density of atomic configurations. However, this approach breaks down when the structural change is extreme, or when nearest-neighbor connectivity of atoms is not maintained. In the current study, a self-consistent approach is presented wherein OPs and a reference structure co-evolve slowly to yield long-time simulation for dynamical soft-matter phenomena such as structural transitions and self-assembly. The development begins with the Liouville equation for N classical atoms and an ansatz on the form of the associated N-atom probability density. Multiscale techniques are used to derive Langevin equations for the coupled OP-configurational dynamics. The net result is a set of equations for the coupled stochastic dynamics of the OPs and centers of mass of the subsystems that constitute a soft material body. The theory is based on an all-atom methodology and an interatomic force field, and therefore enables calibration-free simulations of soft matter, such as macromolecular assemblies.

Nonlinear multiscale modeling approach to characterize elastoplastic behavior of CNT/polymer nanocomposites considering the interphase and interfacial imperfection

A hierarchical multiscale modeling approach to characterize the elastic and plastic behavior of carbon nanotube (CNT)-reinforced polymer nanocomposites is proposed via molecular dynamics simulations and a continuum nonlinear micromechanics based on the secant moduli method. Even though the importance of the densified interphase zone formed between the CNT and polymer matrix has been demonstrated by many related studies for elastic properties, studies on how to identify the behavior and contribution of the interfacial condition and interphase zone in the overall elastoplastic behavior of nanocomposites is still an open issue. Different from conventional micromechanics approaches that homogenize overall elastoplastic behavior of heterogeneous structures from known behaviors of its constituent phases, the present study focuses on the identification of local elastoplastic behavior of the interphase region from the known elastoplastic behavior of nanocomposites through a hierarchical domain decomposition method. Firstly, the overall elastoplastic behavior of the CNT-reinforced nanocomposites is obtained from molecular dynamics (MD) simulations which are based on an ab initio force field. Due to a weak van der Waals interaction between the pristine CNT and the matrix polymer, the elastoplastic behavior of the nanocomposites clearly shows a weakened interface condition, while the matrix molecular structure in the vicinity of the CNT confirms the existence of the interphase zone. In upper level analysis, an effective matrix concept is adopted, and its elastoplastic behavior is inversely identified by equating the MD simulation result to a two-phase nonlinear micromechanics model that can consider imperfect interfacial condition. Then, the effective matrix domain is again decomposed into the interphase and pure matrix polymer regions in lower level analysis, and the elastoplastic behavior of the interphase is again identified through the same method. Using the constitutive relation of the interphase obtained from the proposed multiscale model, the overall elastoplastic behavior of the nanocomposites is obtained and compared with some available experimental results and an additional MD simulation result to validate the applicability and physical rigorousness of the proposed nonlinear multiscale approach.

A PERIDYNAMICS FORMULATION BASED HIERARCHICAL MULTISCALE MODELING APPROACH BETWEEN CONTINUUM SCALE AND ATOMISTIC SCALE

In this paper, a multiscale modeling framework has been established between peridynamics and atomistic models. Peridynamics (PD) formulation is based on continuum theory implying nonlocal force based interactions. Peridynamics (PD) and molecular dynamics (MD) have similarities since both use nonlocal force based interaction. It means continuum points in PD and MD atoms are separated by finite distance and exert force upon each other. In this work PD based continuum model of epoxy polymer is defined by meshless Lagrangian particles. MD is coupled with PD based continuum model through a hierarchical multiscale modeling framework. In this framework, PD particles at coarse scale interact with fine scale PD particles by transferring pressure, displacements and velocities among each other. Based on the same hierarchical coupling method, fine scale PD model is seamlessly interfaced with molecular model through an intermediate mesoscale region i.e. coarse-grain atomic model. At the end of this hierarchical downscaling, the information — such as deformation, energy and other important parameters — were captured in the atomistic region under the applied force at micro and macro regions. A two-dimensional plate of neat epoxy was considered for demonstration of such multiscale simulation platform. The region of interest in the 2D plate was interfaced with atomistic model by applying the proposed hierarchical coupling method. The results show reasonable consistency between PD and MD simulations.

Hierarchical multi-scale modeling of texture induced plastic anisotropy in sheet forming

In this paper we present a Hierarchical Multi-Scale (HMS) model of coupled evolutions of crystallographic texture and plastic anisotropy in plastic forming of polycrystalline metallic alloys. The model exploits the Finite Element formulation to describe the macroscopic deformation of the material. Anisotropy of the plastic properties is derived from a physics-based polycrystalline plasticity micro-scale model by means of virtual experiments. The homogenized micro-scale stress response given by the micro-scale model is approximated by an analytical plastic potential function. The methods to reconstruct the plastic potential upon a sufficient change of the crystallographic texture are discussed. A dedicated stress integration scheme is elaborated to utilize the hierarchy of the models. Cup drawing from circular blanks is considered as an example of sheet forming process. The results from the HMS simulations with updating of texture and anisotropy are compared to the outcomes of the FE model assuming constant anisotropy. Experimental verification of the HMS model is provided for two steel grades and one aluminum alloy. An assessment of the texture evolution calculated by the HMS model reveals very high reliability of the predictions. Macroscopic cup profiles obtained by the HMS model show good agreement with the experiment. A substantial improvement of the predicted cup profile is found for one of the investigated steels. The origin of this enhancement is discussed in the context of evolving plastic anisotropy.