A review of the FE² method for composites

Abstract

Composite materials and structures are inherently inhomogeneous and anisotropic across multiple scales. Multiscale modelling offers opportunities to understand the coupling of material behaviour and characteristics from the micro- to meso- and macro-scales, critical to the optimal design of composite structures for lightweighting and mechanical performance. FE² is an increasingly popular class of multiscale methods because of its versatility to model heterogeneous material behaviour across multiple scales. In classical FE² analysis, two finite elements (FE) calculations are carried out in a nested manner, one at the macroscale and the other at the microscale. Unlike conventional analysis, the macroscale FE analysis does not require homogenized constitutive properties because these are derived from the microscale FE simulations at the representative volume element (RVE) level. This has exciting significance for composite mechanics because properties characterized and defined at the microscale can be directly transferable to higher scales and validated with experiments. For example, failure criteria for composites need only be formulated at the microscale level of fibers and matrix. However, FE² analysis is computationally expensive and the generally more complex classical nested implementation of FE² is disadvantageous. This paper presents a review of the FE² method to model various phenomena in the mechanics of composite materials and discusses various implementations. Recently, the Direct FE² method, a variant of the FE² method, has been shown to be particularly easy to implement in commercial FE codes, which also means that it has the additional advantage of ready access to inbuilt constitutive models library and other advanced features of the commercial code. We conclude with future directions for multiscale modelling of composites using FE².

1 Introduction

Composites comprise a wide range of materials such as particle reinforced polymers, short fibre reinforced polymer, and continuous/long fibre reinforced polymers. All these composites are heterogeneous materials with clearly distinguishable scales of constituents. When properly designed, composites exhibit better properties than those of the individual constituents separately. It is therefore essential to understand the behaviour of these composites to design them for a defined application. Multiscale modelling of composites enables us to better understand their overall mechanical behaviour. Experimentally observed macroscopic material behaviour and phenomena emerge from the mechanics and physics of the underlying material microstructures, which motivates the development and application of multiscale methods for composites modelling.

Multiscale high-fidelity modelling methods are gaining popularity among the research community to quicken the pace of composite materials discovery, development and virtual testing. Multiscale methods provide physically derived bottom-up explanations of inelastic mechanical behaviour (Feyel and Chaboche 2001; Nezamabadi et al. 2015; Herwig and Wagner 2018; Xu et al. 2018) and their interactions with non-mechanical phenomena (Schröder et al. 2016) observed in heterogeneous material structures, which can be provided only at the scale where dissipative (inelastic) mechanisms operate in the material. This scale, generally at a much lower length scale than the structural length scale, can be the micro-scale (length scale of the material micro-structure), nanoscale or the molecular scale. This review excludes multi-scale methods at the nano or molecular scales; it focuses on multi-scale methods with lengths spanning from microscale (defined as an order of microns in dimension) to macroscale (in the order of meters or tens of meters).

A widely adopted multiscale modelling method is FE² (Fig. 1)—a versatile computational homogenization approach that is generally applicable to many material systems (Hernández et al. 2014). The term FE² was first used by Feyel et al. (2000) and since then has been applied for studying material and structural behaviours and phenomena (Feyel and Chaboche 2001; Özdemir et al. 2008a; Regener et al. 2011; Nezamabadi et al. 2015; Schröder et al. 2016; Herwig and Wagner 2018; Xu et al. 2018). FE² uses FE calculations at both scales with information constantly being exchanged between both scales (e.g. deformation gradient from macro to micro scales; local macroscopic stress and local macroscopic consistent tangent from micro stresses and microstructural stiffness respectively) as shown in Fig. 1. Although the idea of coupling constitutive equations across scales had been known for quite some time (Suquet et al. 1985; Renard and Marmonier 1987), major developments by several researchers occurred about a decade later (Guedes and Kikuchi 1990; Ghosh et al. 1995; Moulinec and Suquet 1998; Smit et al. 1998; Miehe et al. 1999; Terada et al. 2000; Kouznetsova et al. 2001; Terada and Kikuchi 2001; Miehe and Koch 2002; Miehe and Bayreuther 2007). Some of the bottlenecks to wide adoption of the FE² method in the research community are its computational intensity and difficulty in its implementation.

Fig. 1

First-order computational homogenization of continua—FE²: P_M is macroscale stress; F_M is macroscale deformation gradient and B.V.P is Boundary Value Problem (Geers et al. 2010)

Computational homogenization and the FE² method literature have also been reviewed by several authors in the recent past (Geers et al. 2010; Schröder 2014; Saeb et al. 2016; Matouš et al. 2017; Otero et al. 2018). These review articles mainly discuss FE², its implementation and general application. The present article presents a review of modelling composite materials with FE², focussing on applications to modelling fiber reinforced polymer composites and various implementations in addition to the classical FE².

Section 2 discusses various composite modelling techniques using continuous fibre polymer reinforced composites highlighting the different scales of modelling in these materials. Section 3 briefly reviews multi-scale methods. The FE² method is introduced and elaborated in Sect. 4. Alternative FE² approaches are discussed in Sect. 5 and Sect. 6 concludes with potential future directions.

2 Modelling composite materials and structures

Multiscale virtual and experimental testing is exemplified by the qualification of a typical airframe, which requires around 10,000 tests of material specimens, along with tests of components and structures up to entire tails, wing boxes, and fuselages, to achieve safety certification (Cox and Yang, 2006; Fawcett, 1997). As carbon-fibre composites are increasingly used in large commercial aircraft structures, the cost-saving to partially replace experimental tests by virtual testing is substantial (e.g., the pyramid test model(Rabois et al 2015) adopted in Airbus for testing and certifying aircraft components). The staggering experimental requirement has inspired a roadmap for virtual testing of these materials, shown in Fig. 2(a)–(c). The figure shows microscale (ply), mesoscale (laminate) and macroscale (component) levels for composite modelling.

Fig. 2

Virtual testing of composites: a microscale: representative volume element (RVE) within an individual ply made of constituents—fibres and matrix; b mesoscale: Laminate level made of individual homogenized plies; c macroscale: component level made of homogenized laminates

Modelling of non-linear behaviour involves capturing various damage mechanisms with an accurate physics-based model. The preferred length scale for modelling composites is highly dependent on the scale at which the numerous sources of non-linearities and failure mechanisms are active. The most favoured way of modelling long fibre reinforced composites is modelling laminates and the structures with individual plies (layers/laminae) or the laminate as a homogenized material, depending on the required resolution and availability of computational resources. In the following sub-sections, research carried out and limitations in modelling composites at each of these levels are reviewed.

2.1 Microscale models

Microscale modelling (Fig. 2(a)) refers to modelling of the individual constituents of the composite in an appropriate representative volume element (RVE)—a geometrical description and a FE model of the unit cell in an individual ply. Generally, it is the first step of a hierarchical multiscale modelling methodology. Microscale modelling of composites has been carried out by numerous researchers (Grufman and Ellyin, 2008; Melro et al., 2013; Xia et al., 2000; Zhang et al., 2005) to obtain a homogenized ply level constitutive model. It is at the microscale that individual material constituent damage such as fibre fracture, matrix cracking or fibre debond initiates. At this level, the interplay between intra- and inter-ply damage modes cannot be captured since only individual ply constituents are being modelled. Also, in-situ behaviour of a ply (Camanho et al. 2006) is not captured since only constituents of a single-ply are being modelled. In general, it is unfeasible to use microscale models to predict damage evolution even at modestly larger scales (Forghani et al. 2015), although they have been successful at predicting the elastic properties of undamaged materials and the onset of damage. This is because the RVE for the virgin material is not the same as the unit cell of an extensively damaged material at larger scales. Smearing and homogenization cannot be performed at the same scale for both the damaged and undamaged material.

2.2 Mesoscale models

At the next scale, mesoscale models shown in Fig. 2(b) are used to predict the composite laminate behaviour with individual plies as the basic building block (Kashtalyan and Soutis, 2005; Ladevèze et al., 2000; Maimí et al., 2007; Pinho et al., 2013; Talreja, 1985; van der Meer et al., 2012; Williams and Vaziri, 1995, 2001). The mesoscale models, in addition to referring to ply-level models for continuous fibre reinforced composites, can also include models at the yarn architecture scale (i.e. explicit modelling of yarn sizes and weave architecture with the surrounding matrix material) for woven composites.

Generally, individual unidirectional ply properties such as ply elastic moduli, (E₁₁, E₂₂, E₃₃, v₁₂, v₁₃, v₂₃, G₁₂, G₁₃, G₂₃) and ply strengths (X_t, X_c, Y_t, Y_c, S_L, S_t) are the inputs required for mesoscale models. These ply properties are usually obtained experimentally by testing coupons or assumed. These homogenized plies are further used to model laminates with consolidated plies. For woven composites, an additional step in the homogenization is often necessary, where the weave characteristics are modelled explicitly in a meso-level RVE.

At this level, all damage modes can be accounted for both interlaminar and intralaminar failure mechanisms with smeared crack material degradation/damage models and cohesive zone models (CZM) respectively. Modelling delamination is a vast field in itself (Tay 2003). Much work has been carried out to model delamination and damage in composites (Maimí et al. 2007; Ridha et al. 2014; Abir et al. 2017; Hu et al. 2018a). Yuan and Fish (2016) developed a dual-purpose damage model that can simultaneously model intraply and interply damage as an alternative to the conventional practice of modelling ply failure with continuum damage mechanics and delamination with cohesive elements. Explicit modelling of discrete cracks in composites is state-of-the-art in damage modelling. Various methods are applied for crack modelling such as the extended finite element method (XFEM) (Moës et al. 1999), phantom node method (van der Meer and Sluys 2009) and floating node method (Chen et al. 2014; Hu et al. 2018b; Zhi and Tay 2019; Zhi et al. 2019). Adaptive methods such as adaptive discrete-smeared crack model (Lu et al. 2019b) have also been developed for efficient modelling of damage and cracks. Despite the high computational cost to model structures with the assumption of independent behaviour of the plies, this level of modelling is the most widely used since both interply and intraply damage can be accurately captured.

2.3 Macroscale models

Macroscale models (Fig. 2(c)) are currently the only viable method for modelling large structures. These models are designed to simulate the structure’s overall response in terms of the laminate as a homogenized equivalent medium. The macroscale models are incapable of predicting damage in individual plies. Numerical efficiency and acceptable overall structural response prediction are the main objectives of such macroscale models.

At a level between the ply and laminate levels, the sub-laminate based approach introduced by Williams et al. (2003), regards the sub-laminate as the basic building block of the laminated composite structure. The objective of the sub-laminate approach is to predict the laminated composite structure’s overall non-linear response (such as stiffness, stability, load-carrying capacity and post-peak response) in a homogenized manner due to progressive damage caused by a loading condition rather than predicting the damage of every single ply. Since the sub-laminate is the basic building block, each variant of the stacking sequence must be treated as a separate material for which the material properties need to be recalibrated. However, this approach assumes no damage within each individual ply of the sub-laminate block.

Since non-linear behaviour in heterogeneous materials occurs across multiple length scales (microns to meters), multi-scale methods for modelling composites are reviewed in the following section.

3 Multiscale methods

Multiscale methods will help optimize the macroscopic structural design and material microstructure design. A multiscale simulation encompasses the modelling of material/structure at different scales. These simulations are required not just to understand the response but also for studying the nature of failure or damage of the composites under various loads. The formation and propagation of the damage in heterogeneous materials like composites occur at multiple scales such as fibre-matrix debonding at the ply-constituent level and delamination at the laminate level in laminated fibre reinforced composites. The multiscale nature of damage and failure in these materials has been a major driving force for developing multiscale models (Forghani et al. 2015). As stated earlier, an impetus for multiscale model development is virtual testing to reduce composite component development cost and time.

In general, multi-scale approaches fall into two categories—information passing (also known as hierarchical) and concurrent. Hierarchical methods are characterized by the presence of different levels(scales) of computation. The term “concurrent” refers to solution methods where different scales/methodologies are used in different subvolumes of the global domain in the literature (Fish 2006). Concurrent methods handle two length scales simultaneously in time and may contain a hand-shake region where the two scales meet. On the other hand, hierarchical models, at any given point of time, deal with a single length scale simulation and can pass information from one scale to another in one (sequential homogenization) or both directions(computational homogenization-FE²).First generation of hierarchical models is also termed as sequential methods since they pass information in only one direction. This is the reason sequential methods lose information about the interaction between different constituent phases when passing information from the constituent micro level to the homogenized macro level.

Hierarchical models (Chamis et al. 2000; Fish and Yu 2001; Ladeveze 2004; Mayes and Hansen 2004; Gonzalez and LLorca 2006; Wu et al. 2012) are the first generation of multiscale models. These models involve calculating effective macroscale properties from the composite unit cell through homogenization. The behaviour of a heterogeneous material is averaged out to be used as a fictitious anisotropic (mostly orthotropic) homogeneous material at a larger structural length scale. This averaging or homogenization (also referred to as upscaling) is the process of replacing an equation with highly oscillatory periodic coefficients with one having a homogenous (uniform) coefficient (Fish 2013). Homogenization is a widely used form of hierarchical modelling. The motivation for homogenization of continua is that the governing equations of individual phases, including their geometry and constitutive equations, are better understood at the fine-scale phases than at the coarse-scale phases (Fish 2013). The objective of homogenization is to obtain macroscale constitutive equations that incorporate the role of nonlinearities at the microscale.

Early research on homogenization techniques focused on calculating bounds for mechanical properties (André 2002). The use of computational techniques within homogenization theory has gained considerable attention since the early 1990′s (Guedes and Kikuchi 1990) and shifted the objective of homogenization research from calculating bounds to modelling macroscale behaviour from heterogeneous microstructures. Computational homogenization methods connect the micro and macro scales using an iterative (i.e. nested) coupled algorithm such as FE², which is defined and described in Sect. 4. The computational intensity arises as it is required to perform analysis of models at two scales through a control code to handle the information transfer between the two scales. The main advantage of such an approach is that it is possible to take into account precisely the interaction between the different phases in heterogeneous materials like composites (Feyel 2003).

While hierarchical models can predict the elastic response of materials with heterogeneous microstructures, they are faced with the issue of homogenization being invalidated due to strain localization when it comes to damage or failure prediction. As a result, classical hierarchical models are incapable of predicting the size of Fracture Process Zones (FPZs), size effect and fracture energy. The characteristic length linked to the constitutive model at the macroscale cannot be derived from lower scales hierarchically(Forghani et al. 2015). Concurrent models avoid this by incorporating both scales in a single model. In undamaged areas, macroscale models with homogenized behaviour are employed while in damaged areas (where homogenization cannot be performed for reasons discussed in Sect. 2.1), the model resolution is increased by accounting for meso/micro features explicitly.

Concurrent methods (Broughton et al. 1999; Oden et al. 1999; Ghosh et al. 2001, 2007; Ladevèze et al. 2001; Fish and Yuan 2005; Hund and Ramm 2007; Elmekati and El Shamy 2010; Temizer and Wriggers 2011; Wellmann and Wriggers 2012; Lloberas-Valls et al. 2012) contain information regarding both the macroscale and microscale in the same model and solutions at both scales are solved simultaneously. The concurrent modelling framework has been applied to model progressive damage in 3D braided composites (Šmilauer et al. 2011).

An early example of concurrent models proposed by Ghosh et al. is an adaptive multi-level computational model (Lee et al. 1999; Ghosh et al. 2001; Raghavan and Ghosh 2004a, b) for damage prediction due to matrix cracking in a composite ply. Their concurrent modelling strategy involves three levels of computation carried out simultaneously as shown in Fig. 3.—(i) Level 0: macro in Fig. 3(a), (ii) Level 1: macro/micro (RVE), which is an FE² implementation (Fig. 3(b)) and (iii) Level 2: purely microscopic domain when RVEs cease to exist as can be seen in Fig. 3(b). Level switching is controlled by employing different criteria (Lee et al. 1999). The level-0 to level-l transition is governed by a plastic work gradient-based criterion, while the level 1- to level 2 transition is guided by a microstructurally based damage criterion. Transition elements (Fig. 3(b)) are used between level 2 and level 1/0 elements to “smooth” the stress transfer between these dissimilar elements. This hierarchy allows for differentiating between critical and non-critical regions and increases the computational efficiency through selective ‘zoom in’ regions.

Fig. 3

A two-scale concurrent computational model for the analysis of a structure: a macroscopic model i.e., level 0; b zoomed in image of microscale model i.e., level 2. (Lee et al. 1999)

The limitations of the concurrent models are their massive computational costs along with the necessity to characterize small-scale models, which can be challenging to implement. For these reasons, concurrent models are currently not yet widely adopted.

Another way to classify multiscale methods is to view them as either (i) mean field or (ii) full field approaches. Mean field approaches (describing the macroscale response using microstructure with the Mori–Tanaka method Mori and Tanaka 1973; Doghri and Ouaar 2003) or self-consistent method (Hill 1965; Walpole 1966; Budiansky 1983; Milton and Kohn 1988)) are sufficient for linear problems but are inadequate and inaccurate for problems with non-linear constitutive behaviour. Moreover, mean field approaches do not provide the local response and its interaction within the material but gives the overall macroscale behaviour. This has motivated the development of full field multiscale methods to model heterogeneous material structures with non-linear two-scale analysis (Fish et al. 1997; Feyel and Chaboche 2000; Miehe et al. 2002; Asada and Ohno 2007). FE² is one of the non-linear two-scale methods that models and describes macroscale structural behaviour derived from the material microstructure.

4 FE² method

FE² refers to a subclass of computational homogenization schemes which handles both the influence of microstructure and the coupling with macroscale response as presented by Smit et al. (1998), Feyel and Chaboche (2000) and Kouznetsova et al. (2001). It is understood that the singular noun is used throughout this paper to refer to the generic class of methods known as “the FE² method”. Computational homogenization essentially involves the construction of a microscale boundary value problem, that is used to determine the local governing behaviour at the macroscale (Geers et al. 2010). When FEM is employed in the computation for both scales in the form of a nested boundary value problem, it is sometimes referred to as an FE² approach. When it is stated that the two scales in FE² are calculated or analysed concurrently, it is the FE² nested solution scheme that is generally being referred to in the literature (Yvonnet et al. 2013; Papadopoulos and Tavlaki 2016; Tan et al. 2020). However, the FE² method solves a single scale at a time and passes information from one scale to another, which is why the FE² method or computational homogenization in general is classified as an information passing hierarchical multiscale method (Fish 2013; Matouš et al. 2017) as opposed to a concurrent method, in which both scales(present in different regions of the same model) are solved simultaneously. In literature, the terms FE² and multi-level FEM mostly refer to nested FE² implementations depending on the context.

In FE², FE calculations are performed at two scales, namely, the macroscale (homogenized material structural model) and microscale (that specifies the heterogeneous micro-structure details). An FE analysis of a RVE of the microstructure is carried out for every macroscale FE integration point calculation as shown in Fig. 4. FE² is traditionally implemented in such a way that two finite-element simulations are run in a nested manner. The passing of information between the two scales is bi-directional to link the macroscale FE analysis to one or many, in the case parallelized computing, microscale FE analyses at any point in time (Smit et al. 1998; Feyel and Chaboche 2000; Feyel 2003; Yuan and Fish 2007; Tchalla et al. 2013; Tikarrouchine et al. 2018).

Fig. 4

A classical FE² implementation (Smit et al. called the method ‘multi-level FEM’) (Smit et al. 1998), where F_macro is the macroscopic deformation, gradient, σ_macro is the local macroscopic stress at macroscopic integration point, S_macro is the local macroscopic tangential stiffness matrix at macroscopic integration point, u_i is the nodal displacement of RVE (microscopic) nodes, y_0i are the positions of RVE nodes in the initial undeformed reference configuration, ${\tilde{\sigma }}_{RVE}$ is the RVE averaged stress, ${\stackrel{-}{S}}_{\mathrm{RVE}}$ is the RVE averaged stiffness matrix

The heterogeneous material or structure is discretized into homogenized continuum finite elements at the macroscale. The microscale FE simulations are carried out on multiple computational RVE models, in which the various phases of the heterogeneous material are explicitly modelled. Each integration point of the macroscale finite elements is associated with its own microscale RVE. The FE² method consists of three main components according to Feyel (Feyel 1999, 2003), which are:

• 1. A geometrical description and a FE model of the unit cell, which is the RVE.

• 2. The local constitutive laws describing the response of each component of the composite within the unit cell.

• 3. Scale transition relationships that define the connection between the microscopic and the macroscopic fields (stress and strain). These are: (i) a downscaling rule that determines the microscale boundary problem for the RVE given the macroscopic deformation measures; (ii) an upscaling rule for the macroscopic stress given the micromechanical stress state.

FE² is often employed for problems with a clear separation of length scales between the size of the RVEs and the macroscale finite elements, i.e., “the microscopic length scale is assumed to be much smaller than the characteristic length over which the macroscopic loading varies in space” (Geers et al. 2010). This implies that the macroscale deformation gradient remains constant over the RVE length scale. It can be stated that the scale of microscopic fluctuations $l$ _µ is smaller than the scale of the RVE, $l$ _m, which in turn is smaller than the scale of macroscopic fluctuations, $l$ _M as shown in Eq. (1):

$${l}_{\mu }<{l}_{m}\ll {l}_{M}$$

(1)

The principle of separation of length scales is well satisfied for many practical problems but is violated in situations when the macro fluctuation scale tends to be small (e.g. high-frequency waves, high gradients, deformation localization) or when the microstructural length scale tends to be large (e.g. the presence of long-range correlations, large microstructural features or percolation phenomena) (Matouš et al. 2017).

As the macroscale FE model contains multiple integration points, there are multiple FE simulations of microscale RVEs corresponding to each macroscale integration point for every increment. An FE² simulation requires large computing memory and processing resources to parallelize these multiple microscale FE simulations (as shown in Fig. 5 (Tikarrouchine et al. 2018)) to ensure it is viable and efficient computationally.

Fig. 5

Parallelization of the macro and micro calculations in ABAQUS/Standard for non-linear problems (Tikarrouchine et al. 2018)

Several approaches to improve the robustness and reduce the computational costs of FE² have been carried out—adaptive sub-incremental strategies to ensure the convergence of the multiscale solution in the presence of several sources of non-linearity (Somer et al. 2009; Reis and Pires 2013), adaptive strategies to solve the microscale problem for only the minimum number of times necessary (Unger 2013; Otero et al. 2015), high-performance non-linear solvers such as the asymptotic numerical method (Nezamabadi et al. 2009, 2015; Xu et al. 2019) and model order reduction techniques (Yvonnet and He 2007; Monteiro et al. 2008; Hernández et al. 2014) which employ methods like proper orthogonal decomposition, empirical cubature (Hernández et al. 2017) or proper generalized decomposition (Lamari et al. 2010; El Halabi et al. 2013).

4.1 Fundamentals of the FE² method

A brief introduction to the fundamentals of the FE² method is given in the following to provide a framework and context for further discussion in subsequent sections. This is also referred to as the first-order computational homogenization. The FE² method as solved in the context of a mechanics problem is reviewed in this section. The variables with subscript M refer to the macroscale and variables with a subscript m refer to the microscale.

The governing equation at the macroscale is the mechanical equilibrium equation

$${\overrightarrow{\nabla }}_{M}.{P}_{M}=\overrightarrow{0 }$$

(2)

where ${P}_{M}$ is the macroscale Piola–Kirchoff stress. To solve (2), constitutive relations between the stress field and deformation gradient field ${F}_{M}$ are required, given by Eq. (3)

$${P}_{M}={P}_{M}\left({F}_{M}\right)$$

(3)

In iterative implicit methods like Newton Raphson scheme used for solving the non-linear macroscale problem, a constitutive tangent ${C}_{M}$ from the linearized form of the constitutive relation is needed.

$$\delta {P}_{M}={C}_{M} :{\delta F}_{M}$$

(4)

Computational homogenization uses the knowledge (and models) of the behaviour of single phases and constituents instead of identifying effective overall material parameters of heterogeneous multi-phase microstructures. The main objective in FE² is to derive the constitutive relations for the macroscale problem numerically by constructing and solving a microscale problem. The macroscale and microscale problems are solved iteratively in a nested loop (Fig. 14(b)). All important microstructural features are taken into account without any assumptions on their influence of macroscale behaviour.

The microscale boundary value problem (BVP) is formulated based on a set of consistent scale transition relations. The microscale is described by a RVE. The primary field in the microscale is denoted by ${a}_{m}$ with deformation gradient ${F}_{m}={\nabla }_{m}{a}_{m}$. The mechanical equilibrium at the microscale is given by

$${\overrightarrow{\nabla }}_{m}.{P}_{m}=\overrightarrow{0 }$$

(5)

where ${P}_{m}$ is the microscale Piola–Kirchoff stress. The constitutive behaviour necessary to solve the problem is assumed to be known for every microstructural constituent $l$

$${P}_{m}^{(l)}={P}_{m}^{(l)}\left({F}_{m}^{(l)};{\varphi }_{m}^{(l)}\right)$$

(6)

where ${\varphi }_{m}^{(l)}$ represents a set of internal variables, that describe and capture all relevant microscale processes in the micro constituent $l$. Note that in FE², internal variables do not appear in the macroscale constitutive relation Eq. (3) explicitly since all internal variables are contained within the microscale problem.

To complete the microscale problem, i.e., Eqs. (5) and (6), boundary conditions need to be specified. These are defined through scale transition relations. To this objective, the microscale field ${a}_{m}$ is written as a superposition of the macroscale homogeneous part, given by the macroscale deformation gradient field ${F}_{M}$ and microscale fluctuation field ${w}_{m}$. ${F}_{M}$ is the deformation gradient at the macroscale point for which the microscale problem is being constructed. ${w}_{m}$ is the local variation of the microscale field due to the heterogeneous microstructure.

$$\Delta {a}_{m}={F}_{M}.\Delta {X}_{m}+ {w}_{m}$$

(7)

where ${X}_{m}$ is the position vector of the points in the reference state of the RVE (microscale); $\Delta$ denotes the relative value of a field with respect to an arbitrary reference point.

The microscale gradient computed from Eq. (7) gives

$${F}_{m}={\nabla }_{m}{a}_{m}={F}_{M}+{\nabla }_{m}{w}_{m}$$

(8)

The first scale transition relationship is postulated requiring the RVE volume average of the microscale deformation gradient to be equal to its macroscale counterpart

$${F}_{M}=\frac{1}{{V}_{m}}{\int }_{{V}_{m}}{F}_{m}d{V}_{m}$$

(9)

where ${V}_{m}$ denotes the volume of the RVE(microscale). Upon substituting Eq. (8) into (9), it can be observed that the scale transition relationship is fulfilled if the volume-averaged gradient of the microscale fluctuation field vanishes

$$\frac{1}{{V}_{m}}{\int }_{{V}_{m}}{\nabla }_{m}{w}_{m}d{V}_{m}=\frac{1}{{V}_{m}}{\int }_{{S}_{m}}{n}_{m}\otimes {w}_{m}d{S}_{m}=0$$

(10)

where ${S}_{m}$ is the RVE boundary and ${n}_{m}$ is the outward normal and the gauss divergence theorem has been applied. Equation (10) can be fulfiled a priori by prescribing appropriate boundary conditions on the micro-fluctuation field ${w}_{m}$.The following microscale boundary conditions are available.

• ${w}_{m}=0$ for all points on ${S}_{m}$ the RVE boundary (Voight/Taylor model)

• Periodic boundary conditions: for a RVE of initially periodic shape, the normals are opposite on the ‘ + ’ and ‘−’ sides of the RVE boundary. ${n}_{m}^{+}=-{n}_{m}^{-},$ require the periodicity of the microfluctuation field ${w}_{m}^{+}={w}_{m}^{-}$

• Minimum essential boundary condition: impose Eq. (10) directly requiring the integral over the entire RVE boundary to vanish. (Mesarovic and Padbidri 2005; Inglis et al. 2008)

The RVE BVP can be readily solved with any of the above-listed boundary conditions. After solving the microscale problem, another scale transition relationship is required to upscale the microscale constitutive response to the macroscale. The scale transition relationship is obtained by equating local macroscale energy (per unit volume) and the volume-averaged energy of the corresponding microscale RVE,

$${P}_{M}:\delta {F}_{M}=\frac{1}{{V}_{m}}{\int }_{{V}_{m}}{P}_{m}:\delta {F}_{m}d{V}_{m}$$

(11)

Equation (11) is known as the Hill–Mandel Lemma or macrohomogeneity condition. This principle simply expresses the volumetric consistency of the macroscale virtual work and that of the underlying microscale volume. Since $F$ is the deformation gradient, applying the chain rule and the Gauss theorem to Eq. (11), leads to the relation between the microscale and macroscale stresses

$${P}_{M}=\frac{1}{{V}_{m}}{\int }_{{V}_{m}}{P}_{m} d{V}_{m}$$

(12)

Therefore computing ${P}_{M}$ from the microscale simulations at each increment provides the macroscale constitutive relation (Eq. (3)) in a numerical form. Numerical differentiation and techniques based on the Schur complement of the microscale linearized system of equations can be used to obtain the constitutive tangent ${C}_{M}$ defined in Eq. (4) (Kouznetsova et al. 2001; Miehe 2002).

4.2 FE² literature review

The research work carried out in FE² method can be broadly classified as development and implementation of the FE² method (Smit et al. 1998; Feyel and Chaboche 2000; Feyel 2003; Yuan and Fish 2007; El Halabi et al. 2013; Tchalla et al. 2013; Tikarrouchine et al. 2018) and applications of the FE² method which are discussed in Sect. 4.3.

FE² method development can be subcategorized further as (a) RVE design, which focusses on the effects of the RVE model parameters like size, volume fraction, and inclusion/reinforcement architecture on the macroscopic response (Ohser and Mücklich 2000; Kanit et al. 2003; Stroeven et al. 2004; Temizer and Zohdi 2007; Pelissou et al. 2009) and (b)scale transition relations, which is the exploration of how various measures (e.g. stress, strain, heat flux) in each scale interact and affect one and another.

Terada et al. showed that although converging trends are observed in both macroscale and microscale variables as the size of RVE becomes large, the error due to the size and morphology of RVE regions is much more severe in microscale variables than in macroscale ones in their study of multi-scale convergence. The error seems to be related to the simplification and idealization of the RVE geometry. The RVE size should be as large as possible since the accurate evaluation of the microscopic stress is necessary in non-linear computational homogenization analyses (Terada et al. 2000).

The effects of shape and volumetric distribution of reinforcements and heterogeneities on the composite materials’ macroscale response have been investigated in literature through numerical and analytical methods for linear elastic and thermal properties of real and virtual computer-generated fibrous composites (Altendorf et al. 2014), the tensile behaviour of particle reinforced composites (Ayyar et al. 2008), elastic moduli, thermal expansion coefficient, stress concentration factor, viscoelastic relaxation modulus and creep compliance of particle-filled polymers at finite concentrations (Chow 1980), elastic properties of ellipsoidal and spherical particle reinforced composites (El Moumen et al. 2015), analysis of the reinforcing particle shape and interface strength effects on the deformation and fracture behavior of an Al/Al₂O₃ composite (Romanova et al. 2009), effective properties of the random media using XFEM coupled with Monte Carlo Simulations (Savvas et al. 2014), particle/matrix interface debonding in SiC particle reinforced Al alloy matrix composites (Williams et al. 2012) and effect of particle clustering on the mechanical properties of composites (Segurado et al. 2003).

Scale transition research can be further classified as upscale transition (Bensoussan et al. 1979; Willis 1981; Müller 1987; Suquet 1987; Nemat-Nasser and Hori 1993) and downscale transition. Upscale transition research involves the averaging of the microscale measures through various homogenization techniques, while downscale transition deals with determining the appropriate micro-scale boundary problem for the given macroscale deformation.

The upscale transition carried out using homogenization theory, involves volume averaging of the microscale quantities (stresses, strains) of the RVE unit cell to obtain the macroscale quantities. The constitutive model for the macro scale FE calculation is obtained from microscale FE analysis and an algorithm for information interchange between the scales is necessary (Feyel and Chaboche 2001; Feyel 2003; Schröder et al. 2016; Herwig and Wagner 2018); i.e. macroscopic constitutive calculations through an appropriate homogenization scheme of the local microscale RVEs (Bensoussan et al. 1979; Willis 1981; Müller 1987; Suquet 1987; Nemat-Nasser and Hori 1993).

The downscaling transition problem involves the determination of the appropriate microscale boundary conditions to be applied at the RVEs associated with each macroscale integration point. Miehe (2002) investigated the three classical boundary conditions—(a)linear displacements, (b) constant tractions and (c) periodic displacements and anti-periodic tractions to be applied on deformation-driven microstructures. Nezamabadi (Nezamabadi et al. 2015) used classical periodic conditions citing that they have the least drawbacks such as overestimated overall stiffness for linear displacement boundary conditions and underestimated overall stiffness for constant traction boundary conditions. Kaczmarczyk et al. (2008) studied scale transition and enforcement of generalized RVE boundary conditions for second-order homogenization. The boundary conditions are discussed in detail by several researchers (Terada et al. 2000; van der Sluis et al. 2000; Kouznetsova et al. 2001; Miehe and Bayreuther 2007; Perić et al. 2011). Microscale boundary conditions to model-specific problems such as strain localization have also been proposed (Larsson et al. 2011; Coenen et al. 2012b). Larsson et al (2011) proposed a novel variational formulation of the RVE-problem, based on the assumption of weak micro-periodicity of the displacement fluctuation field. A parameterized transition between the conventional “strong” periodicity and Neumann boundary conditions is allowed due to an independent FE-discretization of boundary tractions (using Lagrange multipliers). Percolation‐path‐aligned boundary conditions (Coenen et al. 2012b) are able to capture the constraining effect of the surrounding material upon developing localization bands. The highly strained band can cross the RVE and fully develop with negligible interference of the applied boundary conditions.

4.3 Applications and extensions

At present, the FE² method has been successfully applied for studying different types of materials, quasi-static and transient mechanical loading, coupled mechanical problems and multi-physics phenomena and structural elements such as beams, plates and shells. These applications are listed in Sect. 4.3.1 and some are discussed in Sect. 4.3.2. The extensions of FE² to overcome the limitations of the first-order method are discussed in Sect. 4.3.3.

4.3.1 Applications

The FE² method has been used to study technical textiles (Fillep et al. 2015), reinforced elastomers (Matouš and Geubelle 2006), granular materials (Li et al. 2014), trabecular bone (Wierszycki et al. 2014), biological tissue (Breuls et al. 2002), and Li-ion battery cells (Salvadori et al. 2014). Some other materials to which computational homogenization has been applied includes: polycrystalline metals (Matous and Maniatty 2009), cellular materials (Nguyen and Noels 2014) and porous media (Su et al. 2011; Gao et al. 2015; Jänicke et al. 2015; Zhuang et al. 2015). Kohlhaas and Klinkel (Kohlhaas and Klinkel 2015) proposed a FE² approach for modelling the memory effect of randomly oriented and distributed shape memory alloy (SMA) fibre-reinforced composites. A 3D modeling of SMA fibre-reinforced composites has also been carried out using the FE² method (Xu et al. 2018).

A FE² approach to model adhesive layers and heterogeneous interfaces couples a (zero thickness) cohesive zone type description at the macroscale to an RVE at the microscale. The microstructure of the layer (example shown in Fig. 6) as well as damage within is fully resolved by this technique and has been studied by some researchers (Matouš et al. 2008; Hirschberger et al. 2009; Verhoosel et al. 2010; Nguyen et al. 2012; Mosby and Matouš 2015, 2016).

Fig. 6

Computational homogenization between the interface integration point (IP) of the interface element on ${\bar{\Gamma }}_{0}^{e}$ at the macro scale and the underlying discretized representative volume element ${B}_{0}^{h}$ with boundary ${\partial B}_{0}^{h}$ (Hirschberger et al. 2009)

The FE² method has been applied to problems involving purely mechanical non-linear quasi-static loading problems (Suquet et al. 1985; Ghosh et al. 1996, 2001; Smit et al. 1998; Miehe et al. 1999; Feyel and Chaboche 2000; Terada et al. 2000; Kouznetsova et al. 2001; Terada and Kikuchi 2001; Miehe 2002), the interaction of microscopic and macroscopic mechanical instabilities (Miehe et al. 2002; Nezamabadi et al. 2009), transient large deformation (Fish and Fan 2008), impact loading (Souza et al. 2008) and transient response of acoustic metamaterials (Pham et al. 2013; Sridhar et al. 2016).

The FE² approach has also been applied to problems in fields, such as thermal loading (Özdemir et al. 2008a; Larsson et al., 2010), thermo-mechanical behaviour (Özdemir et al. 2008b; Regener et al. 2011; Fleischhauer et al. 2016, 2020), electro-magneto-mechanical behaviour (Javili et al. 2013; Zäh and Miehe 2013; Keip et al. 2014; Niyonzima et al. 2014; Schröder et al. 2016), diffusion (Nilenius et al. 2014) and liquid-phase sintering (Öhman et al. 2013). Naturally multi-scale non-linear problems, such as contact and friction have also been modelled with FE²(De Lorenzis and Wriggers 2013; Temizer 2014a, b).

Beams, plates, and shells have been modelled with the FE² approach using microscale models with a thickness equal to the physical thickness of the shell. In-plane homogenization is carried out, along with through thickness integration (Geers et al. 2007; Coenen et al. 2010; Helfen and Diebels 2012, 2014; Gruttmann and Wagner 2013). A new paradigm for Carrera’s Unified Formulation (CUF) based on multiscale structural modelling has been established by bridging micromechanics and the advanced CUF one-dimensional/beam structural theories through a FE² framework (Hui et al. 2019).

4.3.2 Application of the FE² method to composite structures

A few applications of the FE² method for modelling various behaviour in composites are highlighted in this section. Feyel and Chaboche (Feyel and Chaboche 2000) have applied the FE² method for SiC fibre-titanium metal matrix composite in four-point bending tests and rotor blades subjected to centrifugal loading. It is noted that the FE² solution gives a satisfactory approximation of the reference solution (obtained from direct numerical simulation with FE of the heterogenous structure) for a RVE volume (fibre diameter) of the same magnitude as the volume around a Gauss point of the FE² macroscale mesh.

A toolbox for implementing FE² in ABAQUS using a FORTRAN subroutine and python scripts for linear and non-linear problems is demonstrated with numerical examples (Yuan and Fish 2007; Tchalla et al. 2013). Using a similar approach, Tikarrouchine et al. developed a 3D FE² implementation in ABAQUS (see Fig. 5) and modelled short glass fibre composites for pure mechanical (Tikarrouchine et al. 2018) and coupled thermomechanical problems (Tikarrouchine et al. 2019) as shown in Figs. 7, 8, and 9.

Fig. 7

Applied thermo-mechanical boundary conditions and the mechanical loading path on the macroscopic structure (Tikarrouchine et al. 2019)

Fig. 8

Macroscopic spatio-temporal temperature distribution (in K) in the composite structure for analysis time of a t = 100 s, b t = 200 s and c t = 300 s. The heterogeneous temperature field tends to be uniform by the end of the analysis (Tikarrouchine et al. 2019)

Fig. 9

Local dissipation distribution field in the microstructure corresponding to the characteristic critical point A in Fig. 8 of the structure for analysis time of a t = 100 s and b t = 300 s (in) (Tikarrouchine et al. 2019)

The fully coupled 3D thermo-mechanical homogenization framework (Tikarrouchine et al. 2019), was used to model the thermoelastic-viscoplastic regime and to capture the rate-dependency in the structural behavior and the thermo-mechanical coupling in polymer composites under complex thermo-mechanical loading. The boundary conditions and the macroscopic temperature distribution at different times are shown in Fig. 7 and 8, respectively for one of the example problems. The interest in the FE² approach resides also in the estimation of the local and global dissipation of energy and can be very useful in studying thermal performance. It is not possible to obtain local dissipation at the microscale (such as shown in Fig. 9) with single scale homogenized macroscale models.

Otero et al. (2015) developed an efficient non-linear FE² analysis for composites, where the computational efficiency was improved by defining a criterion for deciding which of the macroscopic elements will need an FE² analysis. When the criterion determines that an element deformation had exceeded the elastic limit, selective FE² analysis is performed. They highlighted that the method reduced the computational cost of analysing a composite engine stiffener plate as shown in Fig. 10 from 32 days and 14 h (required by a classical multi-scale method) to less than 12 h.

Fig. 10

An efficient FE² analysis of an engine stiffener plate (Otero et al. 2015)

Ghosh et al. proposed an adaptive multi-level computational model as mentioned in Sect. 3 in which FE² is employed at level 1 (Fig. 3)to calculate fibre-matrix interfacial stresses in composite laminates (Raghavan et al. 2001). Damage in composites has also been modelled using this approach (Lee et al. 1999; Ghosh et al. 2001). They then adopted the method to model microstructurally debonding materials in a double lap bonded composite to demonstrate the capability in handling coupon test length-scale problems(Ghosh et al. 2007).

Composite damage in compression is difficult to model since it involves instability due to fibre microbuckling that could lead to macroscale laminate instability (buckling). Asymptotic Numerical Method (ANM) in combination with FE² can be used to study instabilities across the micro and macro scales(Nezamabadi et al. 2009; Xu et al. 2019). ANM is a path following solution technique in which every step is defined by a Taylor series with respect to a path parameter. ANM transforms a non-linear microscopic problem into a sequence of linear problems. Compared to incremental-iterative algorithms, ANM can be considered as a high order predictor without any need of a corrector iteration, which is effective in handling instabilities. This is advantageous since fibre microbuckling is influenced by both material and geometric quantities. The elastoplastic shear behaviour of the matrix and the fibre misalignment at the local level is also affected by macrostructural quantities. The effects of fibre wavelength on the buckling and post-buckling of macro structure were investigated by Xu et al. through this approach (Xu et al. 2019) while Nezamabadi et al (2015) used the method to model compressive failure of composites as shown in Fig. 11.

Fig. 11

Compressive failure of the composite beam (Nezamabadi et al. 2015)

Another important mode of failure in composites, delamination, has been modelled with FE² (Herwig and Wagner 2018), in which the interply microstructure is not considered in a partial homogenization delamination model, as shown in Fig. 12. The macro-scale composite structure is discretized by a combined shell interface shell (SIFS) element for modelling the delamination kinematics correctly. The mesoscale is modelled with a coupled representative volume element (cRVE) that models the whole stacking sequence along with the interfacial layer. Numerical examples showed the applicability and improvements over single scale models by reducing the oscillations of the load–displacement curve during the softening process and the influence of the macroscopic mesh discretization under single mode (Double cantilever beam tests) and mixed mode (Mixed mode bending test) loading.

Fig. 12

FE² delamination model with the delamination process zone (shaded regions) under double cantilever beam loading (Herwig and Wagner 2018)

The influence of interfacial shear strength and weight fraction on carbon nanotube polymer composite cantilever beam is analyzed using a three-level multiscale method (Papadopoulos and Tavlaki 2016). The atomic level, modelled with a space frame structure using modified molecular structural mechanics is converted to microscopic level RVE with an equivalent beam element (EBE) approximation. The microscopic and macroscopic levels are modelled with an FE² approach as shown in Fig. 13.

Fig. 13

The influence of interfacial properties on the macroscopic performance of carbon nanotube composites (Papadopoulos and Tavlaki 2016)

4.3.3 Extensions

The classical FE² method described in previous sections is also referred to as first-order computational homogenization in which only the deformation gradient at the macroscale is used and not any higher order gradients. This poses limitations on the first-order method namely: (i) macroscale is limited to standard continuum mechanics theory since the method complies with the principle of local action and material point concept, (ii) large spatial gradients across the macroscale cannot be resolved, which implies that it is not suited to solve localization problems and (iii) the clear separation of length scales implies an infinitely small RVE size compared to the macroscale point, which does not allow the proper study of size effects on macroscale behaviour.

The above limitations of the first-order framework have been overcome by developing higher order homogenization schemes to account for higher order deformation gradients at the macroscale. The second order computational homogenization couples the higher order gradient continuum at the macroscale to first-order continuum microstructures for mechanical problems (Kouznetsova et al. 2002, 2004; Geers et al. 2003; Kaczmarczyk et al. 2008, 2010; Bacigalupo and Gambarotta 2011). The separation of scales assumption is relaxed in this approach, which allows moderate localization of the macroscale field and accounts for the absolute size effect of the microstructure in the homogenization scheme. Higher order computational homogenization based on the computational continua approach relaxes the constraints on higher-order continuity requirements and boundary conditions (Fish and Kuznetsov 2010; Fish et al. 2015).

Modelling strong localization of macroscale fields due to strain softening plasticity or material failure is a challenging topic in the FE² method (Loehnert and Belytschko 2007; Belytschko et al. 2008; Hettich et al. 2008). To model localization, strain gradients need to be captured accurately. This is not possible in the first-order methods since deformation gradients are assumed constant over the spatial length scale associated with the RVE size due to the clear separation of length scales. Continuous–discontinuous homogenization–localization approaches have addressed localization of macroscale fields for bulk homogenization. In such approaches, the microscale properties inside the localization bands are upscaled directly to the macroscale, that is enriched by either a weak discontinuity(discrete band) (Massart et al. 2007a, b) or a strong discontinuity (discontinuous jump) (Coenen et al. 2012a, c; Bosco et al. 2014, 2015) representing the propagating fracture at the macroscale. Another viable solution for modelling damage and fracture is the computational homogenization of interfacial volumes towards cohesive zones (Matouš et al. 2008).

5 Alternative FE² approaches

An implementation of classical FE² mentioned previously involves extensive programming and parallelization techniques to run both the macro and micro FE calculations in a nested manner simultaneously. Apart from the implementation difficulties, the computation time and resources required for increased problem domain size and model complexity for real-world application increases as multiple instances of FE simulations are needed at different scales. Parallelizing of several FE simulations (up to the number of macro-scale integration points for a single increment) of micro-scale RVEs is also required for every macro-scale strain increment for the method to be computationally viable. Since each material constitutive model requires different internal variables like strain rate for rate-dependent constitutive model in addition to stress, strain—a separate algorithm will be needed for different combinations of macro and micro-level constitutive models to handle the data transfer between the scales. This requires considerable programming and constitutive modelling expertise. Although this leads to intensive computations, the cost is still less expensive than the full FE brute force approach in which all heterogeneities are modelled and meshed explicitly.

It would be advantageous if the number of FE simulations can be reduced or the microscale and macroscale FE simulations can be combined so that FE² simulation of structures become computationally attractive and can be made more accessible. This section briefly discusses other FE² implementations namely: FE² using model reduction techniques, boundary value-driven homogenization (BVDH) (Fleischhauer et al. 2016), Direct FE² (Tan et al. 2020), and data-driven FE² schemes (Le et al. 2015; Lu et al. 2019a; Xu et al. 2020; Nguyen-Thanh et al. 2020).

The novelty of FE² using model reduction techniques lies in using proper generalized decomposition (PGD) (Lamari et al. 2010; El Halabi et al. 2013) or proper orthogonal decomposition (POD) (Yvonnet and He 2007; Monteiro et al. 2008; Hernández et al. 2014) for solving the microscale RVE step to improve computational efficiency. Using these multi-dimensional model reduction techniques, computational costs are significantly reduced as the primary field such as displacement over the RVE is calculated by simple algebraic operations for any set of parameters (material properties, boundary condition, loads, etc.). The microscale components can be recovered for problems in which the detailed local displacement, strain and stress fields are required.

The boundary value driven homogenization (BVDH) (Fleischhauer et al. 2016) has been implemented and demonstrated for coupled thermomechanical problems. In this method, the boundaries of the microscale RVEs are constrained by the macroscale interpolation functions. The method was benchmarked against FE² implementations with linear displacement and periodic boundary conditions (Miehe 2003). The reliability of the BVDH method is demonstrated by comparing to reference solutions and its efficiency is shown to be better than standard computational homogenization. The method is demonstrated with examples involving 1D thermoelasticity at small strains, purely mechanical problems—3D uniaxial tension and 3D inhomogeneous compression and a 3D coupled thermomechanical analysis of uniaxial tensile test specimen of short-glass-fibre reinforced thermoplastic with fibre-orientations of 0°, 30° and 90°.

Neural networks have been used in FE² schemes to construct strain energy density, for the homogenized composite material, which acts as the macroscale constitutive equation (Le et al. 2015; Lu et al. 2019a; Nguyen-Thanh et al. 2020). These methods improve computational efficiency by performing offline microscale computations and using neural networks to construct macroscale strain energy density as a function of macroscale strains and microstructural parameters. Xu et al. proposed a novel data-driven multiscale FE method (data-driven FE²) for composite materials and structures (Xu et al. 2020). A material genome database (rather than constructing a macroscale constitutive law) is utilized in the macroscale data-driven analysis to benefit from the data-driven computing paradigm(Kirchdoerfer and Ortiz 2016). The microscale problems are calculated beforehand to create this offline material genome database. In the data-driven FE² method, the online computation of microscale problems in classical FE² method is substituted with a search of data points over the offline database. Thus, the online computational efficiency of structural analysis is improved. Microscale fibre instability was studied using this proposed approach.

The Direct FE² (Tan et al. 2020) combines the microscale and macroscale simulations into a single analysis through two key ideas—(i) defining a scale transition relationships between the micro and macroscale degrees of freedom through multi-point constraints (MPCs), which are equations linking various degrees of freedom in a model through algebraic equations and (ii)scaling the size of the RVEs in FE² such that the internal virtual work for the microscale FE analysis is made equal to that of the macroscale FE analysis.

The theory, validation and applications of Direct FE² implemented in ABAQUS™ FE software are given in Raju (2019) and Tan et al. (2020). The Direct FE² approach in Fig. 14(c) with its scale transition relationship with MPCs effectively solves for only the microscale degrees of freedom $\stackrel{\sim }{d}$ and thus removes the nested loop (Fig. 14(b)) in the classical FE² approach. The macroscale degrees of freedom d (as denoted in Fig. 14(c)) are obtained from the scale transition relations denoted as L in Fig. 14(c). These relations (the matrix L) can be defined at the pre-processing stage of the FE analyses based on the macroscale mesh at the reference state and the choice of interpolation functions. There is no need to work out the overall tangent moduli for Direct FE² at the macroscale (Kouznetsova et al. 2001; Miehe 2003), as required in classical FE². Computational cost comparison for a single macroscale element for Direct FE² and classical FE² implemented in ABAQUS is given in the appendix. The Direct FE² method solves for both scales simultaneously compared with the classical FE² implementation, which solves both scales iteratively in a nested manner (Fig. 14(b)). This implies that computational costs associated with data transfer between the nested computations in classical FE² is not incurred in Direct FE². The method is attractive since it removes the need for extensive programming expertise and it enables the use of various constitutive models already available in commercial FE codes, as shown with the following example in Sect. 5.1.

Fig. 14

Solution algorithms for a standard FE simulation, b classical FE² and c Direct FE². $\stackrel{\sim }{f}$ is the vector of microscale nodal forces (Tan et al. 2020)

5.1 Direct FE² modelling of a composite cantilever beam with viscoelastic matrix using inbuilt modelling capabilities of commercial code

Hysteresis effects in a composite cantilever beam with viscoelastic matrix were modelled with Direct FE² and a brute force full FE method (Tan et al. 2020). As a demonstration of Direct FE²′s capability to access the in-built features and capabilities of commercial code, the viscoelastic constitutive model available in the ABAQUS library was used to represent the behaviour of the matrix. No extra programming is required to incorporate viscoelasticity in the Direct FE² cantilever model. A Direct FE² model used for elastic analysis can be used for simulation with viscoelastic matrix material just by simply switching the material model (along with assigning appropriate properties) and solver (a separate viscoelastic solver is used which accounts for strain rate in stress calculations without considering mass) in the ABAQUS GUI. The Direct FE² model is solved with scale transition relations defined using either linear or quadratic interpolation functions (Fig. 15(a)). The residual stress contours for both the Direct FE² and brute force full FE model are shown in Fig. 15(b)–(d). The Direct FE² models are able to capture the hysteresis (shown in Fig. 15(a)) and the residual stresses exhibited by the more computationally expensive brute force reference full FE model.

Fig. 15

a Force displacement of viscoelastic composite beam. Residual stresses (MPa) at the end of loading cycle within the RVEs of Direct FE² models with b linear and c quadratic macroscale elements and d full FE model (Tan et al. 2020)

6 Conclusions

The FE² method for modelling composites is discussed following a brief overview of traditional modelling and multi-scale modelling methods of composites. The numerous publications in developing and applying the FE² method for computational homogenization is an indication of the capability and potential of the method for predicting behaviour of heterogeneous composite materials and structures. High fidelity, physics-based modelling of damage in composites accounting for all damage mechanisms (tensile and compressive failure of the constituent, reinforcement debonding and delamination) at the lower scale is an ambitious and exciting challenge.

Damage and fracture are currently being modelled with second order FE² formulation (with higher order deformation gradients)) for moderate strain localization and other approaches (mentioned in Sect. 4.3.3) for the transition to macroscale fracture and handling severe strain localization. The current challenges of FE² method development and application are tackled in attempts to model fully coupled multi-physics material phenomena, damage and fracture while trying to improve computational efficiency. Improving the computational efficiency of the FE² method with model reduction schemes such as using PGD (El Halabi et al. 2013), POD (Yvonnet and He 2007) and data-driven methods (Xu et al. 2020) have tremendous potential for future research. As the method is made more computationally efficient, adaptive modelling of composite components and larger structures (e.g. Otero et al. 2015) with FE² will become increasingly feasible in future.

Although computational homogenization methods such as FE² are well developed to model heterogeneous materials ranging from mechanical quasi-static non-linear loads to coupled thermo-mechanical and multi-physics phenomena, the learning curve to implement this method is steep and involves a fair amount of code development to handle the data transfer between the simulations of the two scales and to parallelize the computation to make the method more viable. Some work to implement these methods on commercial software have been carried out to make the method more available to FE users and researchers (Yuan and Fish 2007; Tchalla et al. 2013; Tikarrouchine et al. 2018). These research works discuss the implementation of the FE² method in ABAQUS using FORTRAN user subroutines and have been used to solve mechanical and coupled thermomechanical problems.

Alternative implementations to the classical nested FE² implementation have been proposed. One such alternative, the Direct FE² method, is highlighted for combining both micro and macro scales using MPCs, thereby effectively removing the computationally expensive nested calculations in classical FE², thus also reducing data transfer costs between the nested calculations. The method has been demonstrated for quasi-static non-linear mechanical loading for composites (Tan et al. 2020) and the ability to use inbuilt modelling features such as non-linear constitutive models from commercial code libraries is highlighted. Adaptive remeshing of macroscale elements to model post localization behaviour will be very interesting and challenging to explore. Developing Direct FE² models with multiple realizations of random RVEs in a single model, accounting for variations in microstructures, is a possible future direction of research. Application of the Direct FE² method by the authors to model transient mechanical loading and coupled thermomechanical problems are ongoing.

Appendix: Comparison of computational costs of Direct FE2 and classical FE2

This section compares the computational costs of Direct FE² and classical FE² implemented in ABAQUS. The various computational operations in a FE² simulation can be categorised as macroscale stress computations from RVE simulations, macroscale tangent moduli calculations and data transfer operations. Data transfer costs refer to the computational operations incurred for the data transfer between the scales(strain or deformation gradient to the microscale and computed macroscale tangent moduli and stress to the macroscale) as well as the I/O overheads that incur when we start microscale simulation and postprocess the microscale RVE simulation output. The data transfer costs exclude the operations needed to compute the macroscale tangent which is factored in the macroscale tangent moduli calculation costs. There are no macroscale tangent moduli calculations and data transfer costs associated with the Direct FE² method.

A single macroscale element which is equivalent to 20 × 20 unit cells is simulated considering geometric non-linearity with the boundary conditions shown in Fig. 16(b) and RVE shown in Figure 16(a). The properties of the fibre are Ef = 74GPa, υf = 0.2 and those of the matrix are Em = 3.35GPa, υm = 0.35.

Fig. 16

a Representative Volume Element(RVEs) with 0.6 fibre volume fraction and 5 µm diameter fibre (grey) and matrix(green); b macroscale element which is 20 × 20 RVEs in size and fixed horizontally and vertically at the left and given a displacement of 11.44 µm at the right

Fig 17

a Force–Displacement; b Computational cost of Direct FE² and classical FE²

The force–displacement behaviour of the various models is shown in Fig. 17(a). The full FE reference is a model with all the fibres meshed and is basically constructed with 400 RVEs (stacking 20 rows and 20 columns of the RVE shown in Fig. 16(a)) such that the overall size is identical to the single macroscale element model in Fig. 16(b).

The number of floating-point operations (flops) for Direct FE² and classical FE² are tabulated accounting for all iterations in Table 1. For calculation and comparison of floating-point operations, the simulation was carried out in a single increment at the macroscale for both Direct FE² and classical FE². The number of floating- point operations listed here are obtained from the ABAQUS simulation summary.

Table 1 Computational cost comparison for Direct FE² vs Classical FE²

The floating-point operations are estimated as below.

Flops in Direct FE² = (no. of iterations) × (flops/iteration).

Flops in classical FE² = (no. of macroscale iterations) × (no. of RVEs) x (no. of microscale iterations) × (flops/microscale iteration).

It should be noted that the data transfer costs (which includes the I/O overheads), and macroscale moduli calculation costs are not included in the classical FE² flops calculation. The macroscale flops for classical FE² is also excluded from the above estimate since it is negligible (204 flops/iteration).

A single Direct FE² iteration is 15% more expensive compared to a single classical FE² iteration, however, there are no separate iterations at the microscale and macroscale for Direct FE². A comparison of the total number of floating-point operations is given in Fig. 17(b).