Concurrent Multiscale Modeling Research Articles

A direct FE2 method for concurrent multilevel modeling of a coupled thermoelectric problem – Joule heating effect – in multiscale materials and structures

Coupled thermoelectric problem significantly affects the performance of electric devices, while there seems to be no method for concurrent multiscale modeling of such problems, which significantly limits the development of advanced electric devices consisting of multiscale structures. To this end, a Direct FE2 (D-FE2) method is proposed for concurrent multiscale modeling of a typical type of coupled thermoelectric problems, i.e., Joule heating effect, in heterogeneous materials and structures. To enforce energy equilibrium and realize information transfer between the macro- and meso‑scales, the Hill-Mandel homogenization condition was generalized to account for both thermal and electric energy, while periodic boundary conditions derived from kinematic constraints were prescribed to both electric potential and temperature to link meso‑scale RVEs to macro-elements in the D-FE2 model. The proposed D-FE2 method can be easily implemented using the available feature “Multiple Points Constraint (MPC)” in many commercial FE software. A series of numerical examples including steady-state analysis and transient analysis of fiber composites, porous materials and lattice structures validate the favorable accuracy and efficiency of the proposed D-FE2 method in modeling the Joule heating phenomenon in multiscale materials and structures. This also implies the promising potential of the proposed D-FE2 method in modeling and design of large-scale electric devices with multiscale structures.

A machine learning method of accelerating multiscale analysis for spatially varying microstructures

Multiscale computing for heterogeneous materials provides a powerful and fidelity approach to handling situations where a suitable macroscopic constitutive model is unavailable. However, there have been very few instances in the industrial sector where concurrent multiscale modeling techniques have been adopted, primarily because they require significant computational resources. In particular, when dealing with large numbers of representative volume elements (RVEs) with different microstructure morphologies throughout the macrostructure. In this paper, the novelty lies in providing an artificial neural network (ANN) model-based multiscale approach for accelerating multiscale analysis of the elliptical inclusion-reinforced composites with spatially varying microstructures, which give up the assumption of global periodicity. To tackle this challenge, firstly, a series of RVEs with different microstructure morphologies characterized by inclusion orientations and aspect ratios, under different load conditions, are simulated to generate training data. Second, the back propagation neural network (BPNN) is trained offline using data acquired from the RVE simulation for learning the underlying and possibly RVE morphologies. The trained BPNN replaces the complex microscopic mechanical solution problems over each macroscopic integration point at each iteration, to update the macroscopic stress-strain responses and it significantly reduces the computational cost of multiscale simulations. Then, the trained neural network is implemented on ABAQUS via user-defined materials (UMAT) and its effectiveness is verified by comparison with the classical multiscale finite element method (FE2) method. Finally, the outstanding computational capability of the proposed approach is further demonstrated through two numerical examples.

Investigation on the off-axis tensile failure behaviors of 3D woven composites through a coupled numerical-experimental approach

Computational models with high precision and efficiency are in urgent need for the damage analyses of 3D angle-interlock woven composites which are widely accepted in composites industry. A novel concurrent multi-scale damage evolution modeling scheme is established based on a meso-scale representative volume cell (RVC) model and the multiphase finite element method to characterize the off-axis tensile response of carbon/epoxy composites. Modified Puck criterion and maximum shear stress criterion are adopted for fiber yarns. Parabolic yield criterion is adopted for the matrix. The fiber breakage, the inter-fiber fracture and matrix cracking are considered at mesoscopic level. To validate the proposed multi-scale damage model, off-axis tensile test is conducted with the help of 3D Digital Image Correlation (DIC) method. A reasonably good agreement is achieved between numerical predictions and experimental observations. Furthermore, the validated damage model is used to predict the tensile response of the composites considering full range on-axis and off-axis angles.

CG-enriched concurrent multi-scale modeling of dynamic surface interactions between discrete particles and solid continua

CG-enriched concurrent multi-scale modeling of dynamic surface interactions between discrete particles and solid continua

Concurrent AtC Multiscale Modeling of Material Coupled Thermo-Mechanical Behaviors: A Review

Advances in the field of processing and characterization of material behaviors are driving innovations in materials design at a nanoscale. Thus, it is demanding to develop physics-based computational methods that can advance the understanding of material Multiphysics behaviors from a bottom-up manner at a higher level of precision. Traditional computational modeling techniques such as finite element analysis (FE) and molecular dynamics (MD) fail to fully explain experimental observations at the nanoscale because of the inherent nature of each method. Concurrently coupled atomic to the continuum (AtC) multi-scale material models have the potential to meet the needs of nano-scale engineering. With the goal of representing atomistic details without explicitly treating every atom, the AtC coupling provides a framework to ensure that full atomistic detail is retained in regions of the problem while continuum assumptions reduce the computational demand. This review is intended to provide an on-demand review of the AtC methods for simulating thermo-mechanical behavior. Emphasis is given to the fundamental concepts necessary to understand several coupling methods that have been developed. Three methods that couple mechanical behavior, three methods that couple thermal behavior, and three methods that couple thermo-mechanical behavior is reviewed to provide an evolutionary perspective of the thermo-mechanical coupling methods.

Concurrent multi-scale modeling of granular materials: Role of coarse-graining in FEM-DEM coupling

The finite element method (FEM) is commonly used for modeling continuum media, while particle simulation methods like the so-called discrete element method (DEM) are used for discrete systems. Coupling the discrete (DEM) and continuum (FEM) methods is conventionally achieved through a direct mapping between discrete particles and finite elements. Coarse-graining (CG) is a micro–macro transition (discrete-to-continuum) method that maps discrete particle data onto smooth, differentiable fields that satisfy the continuum equations. By choosing an appropriate length scale (the coarse-graining width c), the coarse-grained fields are then homogenized and projected onto a FEM spatial discretization.This concept is utilized here to reformulate FEM-DEM coupling methods, both surface and volume, where in the limiting case of c→0, the classical coupling is recovered. For surface coupling, the discrete particle–surface contact forces are first mapped onto a continuous surface traction field (using CG) which is then coupled to the continuum FEM model. For volume coupling (also known as the Arlequin framework), the homogenization operators are enriched with CG functions, offering a non-local coupling approach between discrete particles, their continuum fields, and the finite elements.The CG enrichment represents a new strategy that consists of (1) a discrete-to-continuum mapping and (2) a continuum-to-continuum coupling based on “CG-enriched homogenization” (CGH). It is shown for surface coupling that the CG-enriched formulation not only leads to more accurate results, conserving symmetry, but also reduces energies generated by the coupling. For volume coupling, there is consistently less numerical dissipation with than without CG-enrichment, especially when the dynamic load contains high-frequency content. Finally, the optimal CG widths are identified for very simple test cases, with which the surface or volume coupling performs best.CGH can be potentially extended beyond the present examples, by considering other continuum fields (e.g., higher-order) and equations (e.g., multi-physics), and used to formulate other multi-scale modeling methods.

Machine learning-enabled self-consistent parametrically-upscaled crystal plasticity model for Ni-based superalloys

This paper introduces a concurrent multiscale modeling framework for developing parametrically-upscaled crystal plasticity models (PUCPM) for crystalline metals that are characterized by multiple phases in their intragranular microstructure. Specifically, Ni-based superalloys with distributions of γ′ precipitates in the γ matrix in their sub-grain microstructure are modeled in this study. The interaction of the γ−γ′ phases with nano-scale dislocation mechanisms determine the elasto-plastic behavior of the material across multiple scales. The PUCPM is designed to explicitly account for the morphological and configurational statistics of these γ−γ′ intragranular microstructures in its crystal plasticity constitutive coefficients. This approach provides a thermodynamically-consistent foundation to enable microstructure-aware material simulation at the higher scale. Establishing this multiscale characteristic requires an automated toolchain of computational methods to generate and embed heterogeneous, statistically-equivalent representative volume elements (SERVEs) into the concurrent multiscale simulation domains for self-consistent homogenization. The self-consistency condition is enforced through an optimization strategy invoking a series of coupled nonlinear finite element solutions. Supervised and unsupervised machine learning methods are integrated with the physics-based modeling at all stages of model development to overcome computational limitations and to provide the final, connecting map between PUCPM constitutive coefficients and γ−γ′ microstructural descriptors. The resulting PUCPM benefits from orders of magnitudes of speedup compared to the equivalent explicit representation of the lower scale microstructure. This advantage enables unique model capabilities for the multiscale analysis of deformation and failure in materials and location-specific design.

Concurrent n-scale modeling for non-orthogonal woven composite

Concurrent analysis of composite materials can provide the interaction among scales for better composite design, analysis, and performance prediction. A data-driven concurrent n-scale modeling approach (\(\text {FExSCA}^\text {n-1}\)) is adopted in this paper for woven composites utilizing a mechanistic reduced order model (ROM) called Self-consistent Clustering Analysis (SCA). We demonstrated this concurrent multiscale modeling theory with a \(\text {FExSCA}^2\) approach to study the 3-scale woven carbon fiber reinforced polymer (CFRP) laminate structure. \(\text {FExSCA}^2\) significantly reduced expensive 3D nested composite representative volume element (RVE) computation for woven and unidirectional (UD) composite structures by developing a material database. The modeling procedure is established by integrating the material database into a woven CFRP structural numerical model, formulating a concurrent 3-scale modeling framework. This framework provides an accurate prediction for the structural performance (e.g., nonlinear structural behavior under tensile load), as well as the woven and UD physics field evolution. The concurrent modeling results are validated against physical tests that link structural performance to the basic material microstructures. The proposed methodology provides a comprehensive predictive modeling procedure applicable to general composite materials aiming to reduce laborious experiments needed.

Multiscale modeling of fracture in 2D materials

Research on two-dimensional (2D) materials, such as graphene and molybdenum disulfide (MoS2), now involves researchers worldwide, implementing cutting edge technology to study them. However, when considering using 2D materials in such promising applications, one of the major concerns is the mechanical failure, which remains heavily underexplored. In this work, we demonstrate the use of sequential and concurrent multiscale modeling to study the effect of various fracture modes on graphene and molybdenum disulfide (MoS2). Some of the fracture modes explored are pure tensile loading, shear loading, and a combination of tensile and shear loading and cyclic loading to investigate fatigue.

Concurrent multiscale modeling of in-plane micro-damage evaluation in Z-pinned composite laminates

Most of the unit cell models for Z-pinned composite laminates (ZCL) are mesoscopic in scale, where the fiber reinforced zone is modeled as homogeneous but without detailed characterization and failure analysis of the fibers and matrix. This paper proposes a new concurrent multiscale unit cell model to investigate the in-plane micro-damage evolution of ZCL. The novelty worth mentioning is the proposed element-based method to generate random microbuckling spatial fibers in the fiber reinforced zone, where the microbuckling is caused by fiber waviness and fiber crimp, and then the mesh of fiber, Z-pin, matrix, and the interface between Z-pin and matrix is regenerated. In-plane tensile tests (ASTM D3039) are conducted for verification. There is good agreement between experiment and simulation, only with deviations of −0.97% and 2.69% for tensile modulus and fracture initiation stress, respectively. The research indicates that in-plane micro-damage initiates from the interface between Z-pin and matrix. The stress concentration and higher local fiber volume fraction, caused by the microbuckling spatial fibers clustered near the Z-pin, play a major role in the evolution of interface damage, crack initiation and growth.

Multiscale modeling for fire induced spalling in concrete tunnel linings based on the superposition-based phase field fracture model

In view of the global–local property and multiscale characteristics of spalling behavior in tunnel lining, superposition-based phase field fracture model is introduced within a framework of concurrent multiscale modeling for fire-induced spalling process in tunnel lining. After formulating the governing equations for the coupled thermo-mechanical-phase field model, the numerical solutions by the superposition method or the so-called s-version method are provided. Subsequently, taking the tunnel segment as the demonstration component, the s-version model for the tunnel lining is developed, and the spalling behavior is observed at both macroscopic and mesoscopic scales. On the basis of the s-version model of tunnel segment, the single-scale and multiscale stratum-structure models are further developed, by taking into account the interaction between tunnel structure and surrounding stratum, the heterogeneous microstructure of concrete material, and the formation of local damage in the critical region.

Multiscale modeling in smart cities: A survey on applications, current trends, and challenges

A smart city model views the city as a complex adaptive system consisting of services, resources, and citizens that learn through interaction and change in both the spatial and temporal domains. The characteristics of dynamic evolution and complexity are key issues for megacity planners and require a new systematic and modeling approach. Multiscale models involved in smart cities and megacities have recently become a popular topic because they can understand complex adaptive systems and efficiently solve complex problems at multiple scales (i.e., micro, meso, and macro) to improve system efficiency and reduce computational complexity and cost. However, there are numerous opportunities to improve this interdisciplinary field considering the lack of applicability of the multiscale modeling approach in megacities and smart cities, and the potential of multiscale modeling in various complex systems within smart cities. Therefore, a review that summarizes the state-of-art researches and opens opportunities around the theme of multiscale modeling participating in megacities and smart cities is warranted. This study, therefore, provides a comprehensive review covering the introduction of megacities, their current challenges, and their emergence in smart cities. Then, the introduction of the smart city along with its characteristics and different generations is disclosed. Moreover, we shed light on multiscale modeling, its categories (i.e., sequential multiscale modeling and concurrent multiscale modeling), and the need for multiscale modeling in megacities and smart cities along with its emerging applications. Finally, based on a literature review, the study highlights the current challenges and future directions related to multiscale modeling in megacities and smart cities, which provide a roadmap for the optimized operation of megacities and smart city systems.

Computationally efficient multiscale modeling for probabilistic analysis of CFRP composites with micro-scale spatial randomness

The physics of spatial material randomness in CFRP composites is incorporated in a concurrent multiscale modeling framework. A sampling based non-intrusive approach is adopted to demonstrate natural variability in the material. The key outcome is the computational efficiency achieved by the proposed numerical model. Improvement in computational cost is achieved by the strategic use of parallel computing tools. Multi-directional fiber composite laminates are numerically modeled to predict progressive damage. The accuracy of the framework is demonstrated by comparing with experimental results of unnotched and open-hole quasi-isotropic laminates.

On-the-fly construction of surrogate constitutive models for concurrent multiscale mechanical analysis through probabilistic machine learning

Concurrent multiscale finite element analysis (FE2) is a powerful approach for high-fidelity modeling of materials for which a suitable macroscopic constitutive model is not available. However, the extreme computational effort associated with computing a nested micromodel at every macroscopic integration point makes FE2 prohibitive for most practical applications. Constructing surrogate models able to efficiently compute the microscopic constitutive response is therefore a promising approach in enabling concurrent multiscale modeling. This work presents a reduction framework for adaptively constructing surrogate models for FE2 based on statistical learning. The nested micromodels are replaced by a machine learning surrogate model based on Gaussian Processes (GP). The need for offline data collection is bypassed by training the GP models online based on data coming from a small set of fully-solved anchor micromodels that undergo the same strain history as their associated macroscopic integration points. The Bayesian formalism inherent to GP models provides a natural tool for online uncertainty estimation through which new observations or inclusion of new anchor micromodels are triggered. The surrogate constitutive manifold is constructed with as few micromechanical evaluations as possible by enhancing the GP models with gradient information and the solution scheme is made robust through a greedy data selection approach embedded within the conventional finite element solution loop for nonlinear analysis. The sensitivity to model parameters is studied with a tapered bar example with plasticity and the framework is further demonstrated with the elastoplastic analysis of a plate with multiple cutouts and with a crack growth example for mixed-mode bending. Although not able to handle non-monotonic strain paths in its current form, the framework is found to be a promising approach in reducing the computational cost of FE2, with significant efficiency gains being obtained without resorting to offline training.

The time response of plasmonic sensors due to binary adsorption: analytical versus numerical modeling

In order to allow for multiscale modeling of complex systems, we focus on various approaches to modeling binary adsorption. We consider multiple methods of modeling the temporal response of general plasmonic sensors. We start from the analytical approach. The kinetics of adsorption and desorption is modeled both as a first order reaction and as a second-order reaction. The criteria for their validity and the choice between them in the case of two-component adsorption are established. Due to the nonlinearities of the second-order reactions and the lack of their analytical solutions, computer-aided modeling is considered next, also in multiple ways: the employment of numerical solvers, fitting of experimental results, the stochastic simulation algorithms and the employment of artificial neural networks (ANN). The examples we present illustrate the advantages and disadvantages of the particular approaches. The goal is to aid the concurrent multiscale modeling of adsorption-based devices. Machine learning in ANN performed here is used to estimate the equilibrium values of adsorbed quantities. The obtained results show that to train an ANN for the estimation of the equilibrium adsorption quantities the Levenberg–Marquardt and the Bayesian regularization algorithms are less efficient than the quasi-Newton BFGS (Broyden–Fletcher–Goldfarb–Shanno) algorithm.

Predicting Young’s modulus of agglomerated graphene/polymer using multi-scale modeling

A novel combination of concurrent and hierarchical multi-scale modeling approach is developed to predict Young’s modulus of graphene/polymer composites. In this novel approach, laminate analogy and generalized method of cells are integrated into the same framework. The former takes into account orientation, while the latter is utilized to address the effect of graphene dispersions on the results. A phenomenological technique is employed for capturing various graphene dispersion inside the matrix avoiding heavy stochastic modeling. Although stochastic modeling is preferred, due to the randomness of agglomeration and orientation of embedded graphene nanosheets in the matrix, deterministic modeling is developed in this research to prevent huge computational efforts. The results are compared with previously developed stochastic modeling and more precision is observed. The developed modeling strategy is validated using published experimental observations taken from the literature by highlighting the accuracy of the proposed technique.

3D concurrent multiscale model for crack propagation in concrete

A new approach for concurrent multiscale modeling of three-dimensional crack propagation in concrete is proposed. A macroscopic model with homogenized elastic parameters is adopted in the regions where the material behaves elastically. For regions where cracks are expected to occur, a mesoscopic model based on a mesh fragmentation technique is used to represent the concrete as a heterogeneous three-phase material composed of mortar matrix, coarse aggregates and interfacial transition zone. In this technique, standard finite elements with high aspect ratio are inserted in between all regular finite elements of the mortar matrix and in between the mortar matrix and aggregate elements in order to describe the crack initiation and propagation process by using an appropriate tensile damage constitutive model. Coarse aggregates with regular shapes are generated from a grading curve and placed into the mortar matrix randomly, using the “take-and-place” method. Coupling finite elements are used for connecting the non-matching meshes corresponding to the macro and mesoscale regions, without increasing the total number of degrees of freedom of the problem. Realistic predictions of crack formation and propagation were obtained for different tests, replicating accurately the observed experimental patterns.

FFT-based homogenization of hypoelastic plasticity at finite strains

Fast Fourier Transform (FFT) based methods are becoming increasingly popular for modeling the texture evolution and local mechanical response of polycrystalline materials within the representative volume element (RVE). Originally, the FFT-based method was formulated through the Lippman–Schwinger (L–S) equation in terms of a homogeneous elastic reference medium. More recently, the Fourier–Galerkin method was proposed to discretize the RVE weak form independently of the reference medium using trigonometric polynomials. The Fourier–Galerkin method, albeit efficient and robust, has not been extended to solve crystal plasticity (CP) problems with spatial heterogeneity and fine resolution. Also, its algorithmic homogenization has not yet been investigated, which is essential for concurrent multiscale modeling. In this paper, a general framework is proposed that connects the FFT-based method and objective rate constitutive models, in particular a generalized implementation of crystal plasticity. Consistent linearization is achieved by pulling back the tangent stiffness from unrotated configuration to reference configuration, which improves the convergence behavior of the Newton–Krylov solver. Applying the inexact Newton method further improves the numerical efficiency. Also, the algorithmic homogenized tangent stiffness for the Fourier–Galerkin method is derived to enable mixed boundary conditions and concurrent multiscale modeling. Lastly, the Fourier–Galerkin method’s accuracy and efficiency for solving CP problems are studied in a parallelized environment, and significant speedup is observed versus the finite element method and L–S based FFT method.

Concurrent multiscale modeling of microstructural effects on localization behavior in finite deformation solid mechanics

The heterogeneity in mechanical fields introduced by microstructure plays a critical role in the localization of deformation. To resolve this incipient stage of failure, it is therefore necessary to incorporate microstructure with sufficient resolution. On the other hand, computational limitations make it infeasible to represent the microstructure in the entire domain at the component scale. In this study, the authors demonstrate the use of concurrent multiscale modeling to incorporate explicit, finely resolved microstructure in a critical region while resolving the smoother mechanical fields outside this region with a coarser discretization to limit computational cost. The microstructural physics is modeled with a high-fidelity model that incorporates anisotropic crystal elasticity and rate-dependent crystal plasticity to simulate the behavior of a stainless steel alloy. The component-scale material behavior is treated with a lower fidelity model incorporating isotropic linear elasticity and rate-independent $$J_{2}$$ plasticity. The microstructural and component scale subdomains are modeled concurrently, with coupling via the Schwarz alternating method, which solves boundary-value problems in each subdomain separately and transfers solution information between subdomains via Dirichlet boundary conditions. In this study, the framework is applied to model incipient localization in tensile specimens during necking.

High-Fidelity, High-Performance Computational Algorithms for Intrasystem Electromagnetic Interference Analysis of IC and Electronics

Ever-increasing complexity in high-speed electronic devices and systems presents significant computational challenges in the numerical analysis in terms of desired accuracy, efficiency, and scalable parallelism. The objective of this paper is to investigate high-resolution, high-performance full-wave field solvers for scalable electromagnetic simulations of product-level integrated circuits (ICs) and electronics. The emphasis is placed on advancing parallel algorithms that are provably scalable facilitating a design-through-analysis paradigm, and enabling concurrent multiscale modeling and computation. The capability and the benefits of the algorithms are validated and illustrated through complex 3-D IC and electronics applications.