ARID ZONE JOURNAL OF ENGINEERING, TECHNOLOGY & ENVIRONMENT
AZOJETE, September 2019. Vol. 15(3):488-509

Published by the Faculty of Engineering, University of Maiduguri, Maiduguri, Nigeria.
Print ISSN: 1596-2490, Electronic ISSN: 2545-5818

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488

ORIGINAL RESEARCH ARTICLE

ANALYTICAL, NUMERICAL AND COMPUTATIONAL MULTISCALE MODELLING
TECHNIQUES FOR HETEROGENOUS MATERIALS: A REVIEW

O. H. Aliyu
(Department of Building, Faculty of Environmental Sciences, The Federal University Birnin Kebbi,

P.M.B. 1157, Birnin Kebbi, Nigeria)
* Corresponding author’s email address: ohunene.hafsa@fubk.edu.ng

ARTICLE INFORMATION

Submitted 14 September, 2018
Revised 16 March, 2019
Accepted 18 March, 2019

Keywords:
Homogenization
heterogeneous material
representative volume
element (RVE)
effective properties
volume averages
computational
homogenization

ABSTRACT
This paper reviews the analytical, numerical as well as the
computational homogenization multiscale modelling schemes for
determining the effective material properties for heterogeneous
medium at the macroscopic level. It also looked at the limitations of
the analytical homogenization techniques in simulating the effective
non linear heterogeneous material behaviours (for example the rapid
localization of damage and so on) as well as the advancements of the
computational techniques in addressing these limitations. In addition,
the possible future trends for the computational technique such as the
development of a fully coupled micro-macro computational scheme
were also discussed. It was concluded that although, the analytical
technique was quite popular and straight forward, its inability to
capture rapid localization of damage limited its application and that
numerical and computational schemes were able to address these
limitations as they relied on the establishment of constitutive relations
for the macroscopic problems in a numerical form through which the
macroscopic problems were constructed and solved in a nested
manner.

© 2019 Faculty of Engineering, University of Maiduguri, Nigeria. All rights reserved.

1.0 Introduction
Most engineering materials (steel, concrete, polymer-blends, functional materials and so on) are
inherently heterogeneous in nature (Figure 1) when viewed at a particular scale and are mostly
referred to as composites (Nguyen et al., 2012; Matous et al., 2017).

Figure 1: Natural random heterogeneous material adapted from Torquato (2013).

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Studies (LLorca et al., 2007; Kouznetsova et al., 2012) have shown that, the macroscopic
behaviour of these composite materials are linked to their mechanical behaviour as well as the
details of their microstructure (that is volume fraction, spatial distribution of the different phases
and inter-phases, their morphological properties, sizes and shapes et cetera). These micro-
structural material details tend to evolve by undergoing phase transformations under the
application of thermo-mechanical loadings as well as advanced forming operations; thus,
making it essential to understand the relationship between the microstructure properties and
the macroscopic material behaviour (also known as the micro-macro structure property relation).
It is hoped that by taking advantage of this micro-macro structure-property relationship, it
should be possible to manipulate the material microstructure so as to influence the macroscopic
mechanical behaviour of the composite material to obtain desired results. For example,
“improved strength and toughness, high stiffness and high damping, improved thermal
conductivity and electrical permittivity, improved permeability” et cetera (Kanoute et al., 2009);
consequently, improving the materials performance in special industrial applications from an
engineering perspective, provides an economic alternative to the designing of new materials
with desirable macroscopic characteristics (Nguyen et al., 2012; LLorca et al., 2007).

Establishing this micro-macro structure-property relation has been a major challenge in
computational micromechanics as well as material science as it is not feasible to carry out an
upfront experimental measurement where all the micro-structural characteristics are explicitly
considered. It is the need to provide an efficient modelling strategy, to better understand the
relationship between material micro-structure and macroscopic properties (that is the micro-
macro structure property relation) leading to the determination of the general macroscopic
homogenised constant value for the heterogeneous material, that has lead to the development
of the multiscale models. This is because for the analysis of composite materials, it is the
effective or homogenised material property that is used as opposed to the properties of the
separate components that make up the material (Kanoute et al., 2009; Nguyen et al., 2012).
Multiscale modelling, is therefore, an area of study which involves resolving problems which
have significant characteristics at multiple scales of resolution. It usually entails modelling from
the smallest scale of the material to a much larger scale (for example, from nanoscale that is
quantum level to the macroscale that is continuum level as shown in figure 2) This is usually
achieved through the use of various models at the different scales of resolution which are then
coupled either analytically, numerically or computationally (Engquist et al., 2007; Weinan and
Jianfeng, 2011).

Figure 2: Illustration of the multi-scale modeling hierarchy adapted from Weinan and Jianfeng
(2011).

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Ohunene Hafsa Aliyu: Analytical, Numerical and Computational Multiscale Modelling Techniques for Heterogenous
Materials: A Review. AZOJETE, 15(3):488-509. ISSN 1596-2490; e-ISSN 2545-5818, www.azojete.com.ng

Corresponding author’s e-mail address: ohunene.hafsa@fubk.edu.ng 490

Figure 3: A seamless integration of models of the different length scales into one coherent
multiscale modelling frame work. Adapted from ICAMS webpage Ruhr University Germany

Although, most engineering problems are solved at the macroscopic scale, the results obtained
from such analysis are not sufficiently precise while the direct micro-scale modelling of
engineering problems are not cost effective and offer too much information; hence the need for
multiscale modelling (Weinan and Jianfeng, 2011; Kanoute et al., 2009). The driving force behind
carrying out multiscale modelling is to achieve a balance between accuracy and efficiency by
coupling together both the microscopic and macroscopic models; thus, providing an accurate
predictive tool which is considered crucial in engineering and material science (Engquist et al.,
2007).
This paper reviews the advancements in multiscale modelling of composite material (such as
steel, concrete, polymer-blends, and so on) which are used for various industrial applications to
determine their macroscopic material properties such as stress and stain fields in addition to
their macroscopic material behaviour such as their energy absorption at high strain rates,
cracking, creep, et cetera by using the various homogenization methods (Nguyen et al., 2012;
Aliyu, 2019).

2. Homogenisation techniques
As previously mentioned, the significance of modelling across multiple-scales is that the
predictions obtained from such modelling techniques are more precise and accurate (Weinan
and Jianfeng, 2011). The conventional multi-scale modelling of composites is either carried out
within the framework of homogenisation methods or the concurrent method depending on
whether the scales are distinctively separated or coupled (Nguyen et al., 2012).
The fundamental aim of homogenisation techniques is to determine the effective properties for
heterogeneous materials from the mechanical behaviour of the materials constituents’
microstructure; thus, providing a means for replacing the heterogeneous material with a
homogenous one.
For many decades now, quite a number of micromechanical modelling techniques dealing with
heterogeneous materials have been developed (Kanoute et al., 2009). The micromechanical
modelling technique to be adopted in achieving a homogenised constant material value for a
heterogeneous or composite material falls into one of three main categories. In the first category,
schemes employed used high resolutions at certain sections of the domain where fine
(microscopic) scale detail of the microstructure morphology (i.e. at cracked tips, and interfaces)
needs to be determined. For the second category, the schemes rely on the description of the

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macrostructure from the analysis of the underlying fine (microscopic) scale problem while the
third category uses schemes which allows for some sections of the domain to be completely
resolved at the fine microscopic scale and the other sections resolved by using the effective
macroscopic response (Nemat-Nasser and Hori 1999; Sluis et al., 2000; Kouznetsova et al., 2012;
Torquatom, 2013; Matous et al., 2017).
Schemes under the above three categories will be discussed in the following sections
Analytical/mathematical homogenisation, Numerical homogenisation as well as Computational
homogenisation.

2.1 Analytical/mathematical homogenisation
Initial research on homogenization techniques can be attributed to the works of Voigt (1889)
and Reuss (1929). They were both able to develop models which gave rough estimates of the
upper and lower bound values of the effective moduli of heterogeneous materials. Both models
only took into account the volume fraction. This limited its application as the influence of other
characteristic properties of the microstructure was not taken into consideration.

2.1.1 General procedure
As the principle objective of homogenization is to describe the effective properties of the
composite material from the mechanical behaviour of the materials constituents’ microstructure
as shown in Figure 4.

Figure 4: Schematic representation of homogenization of material (Temizer and Zohid 2006;
cited in Nyguyen et al. 2012).

The bridging of length scales is necessary and this is carried out by volume averaging and is
denominated homogenization (LLorca et al., 2007).

∈� =
1
Ω Ω

∈� (x)dΩ and σ� =
1
Ω Ω

σ� (x)dΩ (1)

where: Ω is the microscopic sample; �(�) microscopic stress; �(�) microscopic strain;
position vector; (�� )macroscopic strain and (��)macroscopic stress.
While the expression which gives the denominated localization is given by:

∈ (x) = A(x):∈� and σ(x) = B(x):σ� (2)
where: �(�) and �(�) are the stain and stress concentration tensors.

The increasing complexity of the microscopic mechanical behaviour and geometries hinders the
underlying objective of the above technique. Hence, the simplifications of the above

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Ohunene Hafsa Aliyu: Analytical, Numerical and Computational Multiscale Modelling Techniques for Heterogenous
Materials: A Review. AZOJETE, 15(3):488-509. ISSN 1596-2490; e-ISSN 2545-5818, www.azojete.com.ng

Corresponding author’s e-mail address: ohunene.hafsa@fubk.edu.ng 492

assumptions are necessary which leads to a number of homogenization models with the most
significant simplification being the “mean-field approach which approximates the strain and
stress fields by phase-wise constant fields” �� and �� given in equation 3.

ϵi = Ai :ϵ� and σi = Bi :σ� (3)
where:
�� � strain concentration tensor for the phase i
�� - stress concentration tensor for the phase i (LLorca et al., 2007)
In adopting this principle, Eshelby (1957), came up with an effective medium approximation in
obtaining the overall effective properties of the heterogeneous medium where the
microstructure is composed of a matrix and a spherical or ellipsoidal inclusion. He also noted
that the strain field within the inclusion was constant.
So that for a linear constitutive equation:

σi = Li:εi; (4)L =
i

N

fiLi:� Ai (5)

I =
i

N

fi� Ai =
i

N

fi� Bi (6)

ϵi = Aidil:ϵ� (7)
where Aidil = I + Si:Lm�1: Li � Lm

�1
(8)

The elastic stiffness tensor of the effective material can then be evaluated from equations (1),
(6) and (8) as:

L = Lm + fi Li � Lm :Aidil (9)
where: ��- Stiffness tensor for the matrix

��- Elastic stiffness tensor for the inclusion
�- Effective elastic stiffness tensor
N- Number of phases
I-Identity tensor
��- Eshelby’s tensor for the inclusion.
����� - Strain concentration tensor for the dilute condition
���- dilute
�� - Volume fraction

This Eshelby effective medium approximation was further extended by Hill (1965)
Along the lines of this mean field approximations, Kr�� ner (1958) came up with a self consistent
approach for obtaining the effective material property for a matrix and spherical or ellipsoidal
inclusions in the elastic regime. The main idea behind this approach is that for a composite
composed of scattered particles, it is possible to assume that the composite is composed of a
particle of one phase embedded in an effective medium. In his work, the strain concentration
tensor Aisc was obtained from the Eshelby dilute solution in Eqn. (8) above by replacing the
matrix elastic constant (Lm) in Eqn. (8) with those of the effective medium ( ) to give the
resulting expression in equation (10)

���� = � + ��:���: �� � � �� (1�)
The effective material property (�) was then be evaluated by substituting Eqns. (10) in (5) to
obtain Eqn. (11).

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493

L =
i

N

fiLi:� I + Si:L�1: Li � L �1 (11)

This self consistent approach was further extended to non-linear cases in the studies carried out
by Hill (1965), Hutchinson (1976) and Bereviller (1979).
The effective properties of the heterogeneous medium in an elastic regime can also be
evaluated using the variation principles. Employing this technique gives an upper and a lower
bound value for the effective property. The simplest works on this variation principle were those
of Voigit and Ruess (Hill, 1963). In their models the strain energy density function (x, ) and the
complementary energy density function (x, ) were expressed in terms of a homogenous phase
potential as:

ω(x,ε) =
r=1

N

χ(r)(x) ω(r)(ε)� (12)

u(x,σ) =
r=1

N

χ(r)(x) u(r)(σ)� (13)

where:
ω(r)(ε) and u(r)(σ) – are the homogeneous phase potentials and χ(r)(x)=1 if x � Ω(㌠) or zero.

Firstly, the effective strain and complementary potentials (W� (ε�) andU�(σ�)) for the heterogeneous
medium is achieved by averaging their minimum potential energies given as:

W� (ε�) = min ω(x,ε) (14)
�(x)∈k

U�(σ�) = min u(x,σ) (15)
�(x)∈s (Kanoute et al., 2009)

Where:
��(��) – Average minimum complimentary potential
(�� ) - Average of the actual strain field �
��(��) -Average minimum complimentary strain
(��) - Average of the actual strain field �
While the strain energy and complementary density functions as given in the McCauley brackets
(that is equation (14) and (15)) gives the volume averages in the domain ٠.; �� and �� are the
average strain and stress fields while k is the set of kinematically admissible strains and S is the
set of statically admissible stresses. The effective strain and stress potential for the elastic
heterogeneous medium at the macroscale is therefore given by the Voigt and Ruess model as
shown in Eqns. (18) and (19):

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(a) Voigt model W� (ε�) = 1
2
ε� : L�: ε� (16)

L� =
r=1

N

c(r) L(r) (17)�

To arrive at the above expressions, Voigt assumed that the strain field within the heterogeneous
medium was constant.

U�(σ�) = 1
2
σ� : M� : σ� (18)

(b) Ruess model L��1 �1 = r=1
N c(r) (L(r))�1 (19)�

The assumption in Ruess model is that a uniform stress was applied to the polycrystals with
resulting comparable stresses in the constituent crystallite.
where:

L- Is the effective elastic tensor
M- Compliance tensor

- The volume fraction for the phase r
�(㌠) - The elasticity tensor for the phase r

Hill (1952), noted that because of the assumptions of a constant strain field in Eqn. (14) and a
constant stress field in Eqn. (15), Voigt model gives an upper bound estimate as shown in Eqn.
(16) while Ruess model gives a lower bound estimate as shown in Eqn. (17)

W� (ε�) < 1
2
ε� : L�: ε� (2�)

U�(σ�) ≤ 1
2
σ� : M� : σ� (21)

In adopting the variation principle, Voigt and Ruess have only taken into account the volume
fraction of each phase (one-point limit) which limited the application of their model as the
influence of other characteristic properties of the microstructure on the effective properties of
the heterogeneous medium have not been taken into consideration.
Hashin and Shtrikman (1962a, 1962b and 1963), also employed the variation principles but took
into account the polarization fields and were able to derive much tighter bounds for the
effective properties of heterogeneous materials (two-point bounds) in comparison to that
obtained from the Voigt and Ruess model (one-point limit).

K1 +
v2

1
k2 � k1

+
3 1� v2
3k1 + 4μ1

≤ k∗ ≤ k2 +
1� v2

1
k1 � k2

+
3v2

3k2 + 4μ2

(22)

μ1 +
v2

1
μ2 � μ1

+
6 1 � v2 (k1 + 2μ1)
5μ1(3k1 + 4μ1)

≤ μ∗ ≤ μ2 +
1� v2

1
μ1 � μ2

+
6v2(k2 + 2μ2)
5μ2(3k2 + 4μ2)

(23)

where:
�1,�1– Bulk moduli in the first phase of the microstructure
�2,�2– Shear moduli in the second phase of the microstructure
�2 – Volume fraction in the second phase (Konoute et al., 2009)

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495

This technique was later expanded by Torquato (2001) by including the details of the phase
arrangement in his model (three-point bounds).
The variation technique was further extended to non-linear cases in the work carried out by
Talbot and Willis 1985 and Willis 1986. Further advancement in variation principles has lead to
the Ponte Castaneda (1991, 1992) and Talbot and Willis (1992) variation principles in estimating
the effective properties of non-linear composites (such as, the rapid localization of damage,
energy absorption at high strain rates that is large plastic deformations and so on) as well as the
Suquet variation principles for power law composites. As an improvement on Voigt upper
bound estimate of the effective macroscopic properties for rigid plastic material, Olson (1994)
came up with his own variation principles for perfectly plastic composites.
Lately, homogenization schemes employed in the analysis of elasto-plastic composite materials
has been reviewed by scholars such as: Ponte Castaneda and Suquet (1998) as well as Chaboche
et al. (2005). In their opinion the schemes can be categorised into two groups, namely, those
based on the use of secant stiffness tensor and those based on the use of tangent stiffness
tensor.
Schemes based on the secant formulations dealt with plastic deformations within the frame work
of non-linear elasticity, where the effective secant stiffness tensor is used to assess the
relationship between the stress and strains at the elasto-plastic stages. This approach is best
adopted when the composite material is subjected to monotonic and proportional loading
(LLorca et al., 2007). The elasto-plastic tangent modulus approach for evaluating the mechanical
response of composites was presented by Hill (1965) where he linearized the local constitutive
equation by bringing it back in an incremental form:

�(�) = �(�):�(�) (24)
where: L is the tangent modulus (Konoute et al., 2009).
The initial models based on this approach gave very stiff predictions of the flow stress. The
source of the inaccuracy was attributed to the anisotropic characteristic of the tangent stiffness
tensor. More accurate predictions of the flow stress were obtained by adopting an isotropic
estimation of the tangent operator (Chaboche et al., 2005).
Another homogenizational approach is the mathematical asymptotic homogenization theory
which employs an asymptotic expansion of the displacement and stress fields based on a ratio of
the length scales ( � ). This technique gives both the effective overall properties such as the
effective stiffness as well as the local stress and strain values (Kouznetsova et al., 2012).

2.2 Numerical Homogenization Schemes
In computational micromechanics, the macroscopic or effective behaviour of composites is
obtained through the solution of the boundary value problem (BVP) for a representative volume
element (RVE) of the microstructure by using numerical techniques. Within this frame work, it is
possible to obtain the local values of the field variable which facilitates the accurate simulation
of crack nucleation and growth of damage as well as the localization of strains leading to
damage which is a major advantage of the technique over analytical homogenizational schemes
(LLorca et al., 2007; Zohdi and Wriggers 2005).

2.2.1 Numerical simulation schemes
The current advancements in computing power and simulation tools, facilitating the solution of
boundary value problems (BVP) of representative volume element (RVE) of composites or

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Ohunene Hafsa Aliyu: Analytical, Numerical and Computational Multiscale Modelling Techniques for Heterogenous
Materials: A Review. AZOJETE, 15(3):488-509. ISSN 1596-2490; e-ISSN 2545-5818, www.azojete.com.ng

Corresponding author’s e-mail address: ohunene.hafsa@fubk.edu.ng 496

heterogeneous materials, of a size statistically sufficient enough to represent the microstructure
without introducing undesired or non-existent properties is a welcome development in the
computational micromechanics community. These numerical strategies (fast fourier transform
algorithms, boundary element method as well as the finite element method FEM) adopted in the
solution of BVP are not only computationally efficient but also facilitates the solution of
problems involving complex geometries. Although the FEM is the most preferred in the
micromechanics community because of its mesh generation capabilities as well as its ability to
implement complex constitutive equations; it tends to produce large models when simulating
micro fields in regions with large strain gradient (LLorca et al., 2007). The Voronoi cell finite
element method (VCFEM) was introduced as an innovation to the FEM in solving non-linear
problems in composite materials. This technique is designed to simulate the behaviour of the
composite material microstructure having different morphologies. In order to capture the
variation in morphology, the finite element discretization was obtained by using Voronoi
tessellations where every Voronoi cell contains an inclusion embedded in the matrix
representing the basic hybrid finite element (Moorthy and Ghosh 1994; LLorca et al., 2007).
This formulation lowers the cost of computation in determining the micro fields in
heterogeneous materials as the hybrid finite element reduces the number of degrees of freedom
(d.o.f) in the analysis (Moorthy and Ghosh 1996, 2000). A three-dimensional VCFEM (Figure 5) for
analysing composite material with randomly distributed composites was developed by Ghosh
and Moorth (2004).

Figure 5: A two-dimensional Voronoi cell finite-element mesh and detail of one Voronoi finite
element adapted from Llorca et al. (2007)

This technique was further extended by Li and Ghosh (2006). They came up with the extended
Voronoi cell finite element method (X-VCFEM) used to model interfacial debonding in
heterogeneous materials. In computational micromechanics, the use of efficient simulation tools
(that is the domain decomposition method, iterative solvers such as: the conjugate gradient
method) capable of solving BVP for RVE is key as most micromechanics simulations require very
large models with complex non-linear behaviour (Zohdi and Wriggers, 2005).
The numerical homogenization discussed, gives emphasis on multiscale techniques based on
coupled homogenization techniques and numerical analysis of the RVE for both periodic and
non periodic (random) microstructure morphological compositions as shown in Figure 4.

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497

Figure 6: Schematic representation of periodic and non periodic RVE from Nguyen et al. (2012).

2.2.2 Asymptotic homogenization
One such multiscale modelling technique is the coupling of asymptotic homogenization and
finite elements. In this method the mathematical asymptotic concept is adopted in solving
multiple scale problems by coupling finite element problems across length scales. This
homogenization technique adopts an asymptotic expansion of the displacement and stress field
on natural scale parameters see Eqn. (25) (Kanoute et al., 2009; Nguyen et al., 2012). As such, it is
important to distinguish both length scales, as the ratio of the length scales � � describes the
small parameter (
�㇀ඟ� �◠ � � = � (25)
where: R and r are the position vectors on the macro and microscale respectively (Figure 6).

u R,r =
R
� = u� R + �u1 R,r + O2 (26)

Eqn. (26) gives the asymptotic expansion of the displacement field
where:
�� � – Macroscopic displacement field
�1 �,㌠ – Periodic first order fluctuations or perturbations due to the microstructure
referred to as fast and slow variables respectively.
� - A small positive number linking the length scales at the macro and micro level
u- Displacement field

Figure 7: Schematic strategy for the multi-scale simulation based on the finite element method
and asymptotic homogenization adapted from LLorca et al. (2007)

This asymptotic expansion is regarded as an approximate solution of the boundary value
problem (BVP) especially when � is reasonably small.
The stress field is given as:

ϵ R,r = grad� ���� s u = Gradsu + 1
�
gradsu

= Gradsu� + gradsu1 + � gradsu�1 (27)

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Ohunene Hafsa Aliyu: Analytical, Numerical and Computational Multiscale Modelling Techniques for Heterogenous
Materials: A Review. AZOJETE, 15(3):488-509. ISSN 1596-2490; e-ISSN 2545-5818, www.azojete.com.ng

Corresponding author’s e-mail address: ohunene.hafsa@fubk.edu.ng 498

While the asymptotic expansion of the stress field assuming a linear elastic behaviour is given in
Eqn. (28) as:

σ R,r = L :ϵ R,r = L :grad� ���� s u

= σ1 R,r + �σ2 R,r + … (28)

The homogenized stiffness tensor L� according to Llorca et al. (2007) is computed from mean
strain value ��−1 as:

σ� = 1
Ω Ωσ

1� dΩ = 1
Ω Ω L : I + ∂χ r

∂r
d� :Gradsu� = L� :Gradsu� (29)

Although, this technique can be used for composites with either a local periodic microstructure
(that is microstructure having different morphologies) or global periodicity (that is having
repetitive cell units) as shown in Figure 7, it is however, mostly restricted to simple microscopic
geometries at very small strains ( ) as well as simple material models (mostly at small strains)
where the macroscopic fields are virtually stable at the microscopic length scale.

Figure 8: A representation of a microstructure (a) with local periodicity and (b) with global
periodicity from Kouznetsova et al. cited in Galvanetto and Aliabadi (2010)

This homogenizational approach offers not only the effective material properties, but also gives
information on the local micro fields (LLorca et al., 2007; Kanoute et al., 2009; Kouznetsova et al.,
2001).

2.2.3 Mesh Superposition Method
This multiscale technique was introduced by Fish (1992), where he superimposed an additional
mesh on the macroscopic (or global) mesh especially in the regions of high stress and strain
gradients with the aim of obtaining detailed information on the heterogeneous material
microstructure at critical regions (crack tips, interfaces et cetera).The less critical benign regions
of the heterogeneous materials, were modelled using coupled finite element method with
asymptotic homogenization technique. In summary, numerical homogenization is used in the
field having benign deformations while concurrent method is used in the critical regions of high
stress and strain gradients where there is a substantial variation in the macroscopic field as
shown in Figure 9 (Nguyen et al., 2012).
The grading arrangement of the displacement field (u) in Figure 9 is of a hierarchical nature as
the displacement field has been separated into global and local regions respectively that is

(30)
(31)

uL � displacement field in the local mesh with respect to the global mesh

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499

.
Figure 9: Schematic representation of the mesh superposition method from LLorca et al. (2007).
This hierarchical nature of the displacement field facilitates the modelling of the exact behaviour
of the heterogeneous material microstructure in areas of high strain gradient. To ensure
homogenous displacement compatibility at the boundary points between the local and global
meshes, the local displacement field must equal zero (that is uL = � at ∂Ω) while stiffness matrix
( �GL ) that links the displacement degrees of freedom of the global and local meshes are
obtained from a system of equilibrium equations as given in Eqn. (32)

KG KGL
KGL KL

uG
uL

= f
� (32)

which can be arrived at by deriving the Navier equations from the overall displacement field
given in Eqn. (30)
where:
KG and KL – Stiffness matrix obtained from local and global meshes
KGL - Stiffness matrix that combines the displacement d.o.f of the global and local meshes
f - Nodal force vector

2.2.4 Embedded Cell Method
For this technique the model adopted describes the characteristics of the microstructure in detail
at critical core regions which are then embedded within a homogenous medium. The finite
element mesh of the core region is much finer than the embedding medium hence; it is capable
of capturing fast variations in field which ensues as a result of damage. The mesh of the
embedding medium is coarse, it is able to relay the macroscopic far field to the core region by
implementing displacement continuity which links the core and the embedding region (LLorca et
al., 2007; Nguyen et al., 2012). This technique has proven to be quite invaluable in accurately
simulating the fracture process in localized regions. The technique as shown in Figure 10 was
adopted Gonzalez and LLorca (2006), in simulating fracture behaviour in fibre reinforced
composites. They were able to accurately show from the multiscale computational model the
main deformation and failure micromechanism.

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Figure 10: Schematic representation of the embedded cell method from LLorca et al. (2007).

2.3 Modern Multiscale Modelling
As stated earlier the essence of multi-scale modelling is to take advantage of simplicity and
efficiency that exists by coupling both microscopic and macroscopic models. It can therefore be
stated that the task of multiscale modelling is to devise a coupled macroscopic-microscopic
computational method which is not only efficient but also capable of giving the required
information with the desired accuracy as opposed to solving the full microscopic model (Weinan
and Jianfeng, 2011). While the traditional multiscale solvers as given in Table 1 are general
purpose microslovers which give solution of the microscale model (that is it focuses on just on
scale), the modern multiscale method also listed in Table 1 gives macroscopic behaviour from
data obtained from the microstructure either sequentially or concurrently at a cost much less
than the general microsolvers such that:

Cost of multiscale method
cost of microscale slover on the full domain

≪ 1 (33)

Table 1: Classical and modern multiscale technique
Traditional Multiscale Methods New Multiscale Techniques
Multigrid Method Car- Parrinello Method
Domain Decomposition Method Quasi-continuum Method
Wavelet-based Methods Optimal Prediction
Adaptive Mesh Refinement Heterogeneous Multiscale Method
Fast Multipole Method Gap-tooth Scheme
Conjugate Gradient Method Adaptive Mode Refinement
Adapted from Weinan et al. (2007).
The objective of multiscale modelling, is striking a balance between accuracy and feasibility
hence, the focus here will be on the heterogeneous multiscale method (HMM) with a close
methodology to the computational homogenization method used in computational mechanics
(Matous et al., 2017). The idea behind this technique is to work out the incomplete macroscopic
model by obtaining the required data from the microstructural model. There are two major
components to the heterogenous multiscale methodology namely:
The overall macroscopic scheme for approximating the macroscopic variable U

U = Qu (34)
Also known as the compression operator, where:

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501

U- is the local averages of the microscopic variables (�) over the macro cells
The scheme for evaluating the missing macroscopic data from the microscopic model u
The overall macroscopic scheme for approximating the missing macroscopic data (for example
the stiffness matrix, transition rates and so on) from the microscopic model is achieved by
solving the micro models locally subject to the following constraint:

Q�u = U (35)
Where:
-is an estimate of and in the present state of the macrostructure

The above two processes (1 and 2) as well as the macro state variable U are linked to each other
by the compression (U) as well as the reconstruction (R) operators given in Eqns. (36) and (37)
in that order; having the property known as the gradient operator given in equation (38).

U = Qu (36)
RU = u (37)

In summary, The heterogeneous multiscale methodology, essentially entails reconstructing the
micro state from U then evolving the micro state using the microscopic model subject to certain
constraint ( ) to evaluate the missing data from u. This way the coupling between both
the macro and micro models is done in such a way that the constraints for setting up the micro
models are obtained from the macro models while the micro model in turn, provides the much
needed constitutive data for the macro model (Weinan and Jianfeng, 2011;Weinan et al., 2007;
Weinan et al., 2003 ).

2.3.1 Computational Homogenization
The last few decades have seen the development of an alternative multiscale method for micro-
macro coupling adopted in the homogenization of complex heterogeneous materials know as
the computational homogenizational technique also called global-local analysis or finite element
analysis in a more particular form (Kouznetsova et al., 2012). With computational
homogenization the details of the macroscale models are computed on-the-fly from the
microscale models as simulations are ongoing (Weinan and Jianfeng, 2011). This characteristic
nature of the computational homogenizational method makes it suitable in obtaining the
effective mechanical behaviour of materials with evolving microstructural geometries, phase
transformation as well as nonlinear mechanical problems (large deformations, initiation and
propagation of cracks, dislocation mechanics and defects, creep and stress relaxation) (Matous
et al., 2017). In this method, materials are generally examined using characteristic volume (that is
RV�◠) as shown in Figure 11.

Figure 11: A two dimensional representative volume element from Kouznetsova et al. (2001).

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while the homogenization process relies on the principles of separation of scales where the
molecular length scale is smaller than microscopic length scale of the RVE which in turn is
much smaller than the typical size of the macroscopic sample that is as shown
in Figure 12 . The information obtained from this length scales are volumetrically averaged in a
hierarchical order( that is ) to obtain the homogenous macroscopic properties
such as stiffness tensor, the gradient tensor (FM) , the stress tensor (PM) and so on
(Kouznetsova et al., 2012); Geers et al., 2010; Matous et al., 2017).

Figure 12: Length scales relevant for the separation of scales principles adapted from Matous et
al. (2017).

2.3.2 Basic Theory of Micro-Macro Modelling
In computational homogenization, the macroscopic or effective behaviour of composites is
obtained by the following basic procedures listed below and schematically shown in Figure 13.
Firstly, define the RVE and the local constitutive equations to obtain input variables. Next carry
out a construction of boundary conditions for the RV from the above input variables as well as
the solution of the boundary value problem (BVP) for the (RV ) of the composite using
numerical techniques. This is also known as down scaling or macro-to-micro transition. The
internal fields (output variables) obtained from the solution of the BVP of the RV are then
volumetrically averaged (or homogenized) using averaging theorems (i.e. coupling between
micro and macro levels) to obtain the regularized or effective material properties used in the
macroscopic analysis at component level. This is also known as up scaling or micro-to-macro
transition (Kouznetsova et al., 2001; Grytz and Mechke 2007; Nguyen et al., 2012; Petracca et al.,
2016).
This regularization process is therefore referred to as “homogenization” (Zohdi and Wriggers
2008). An advantage of this computational homogenization over other homogenizational
schemes is that the microscopic constitutive behaviour is defined as the simulations are ongoing
thus, making it suitable for predicting the mechanical behaviour of composites with complex
microstructure as well as adequately accounting for microstructure evolution under complex
loading (Feyel and Chaboche 2000; Kouznetsova et al., 2001; Ozedmir et al., 2008; Weinan and
Jianfeng, 2011; Weinan et al., 2007; Nguyen et al., 2012).

Figure 13: A schematic representation of the computational homogenizational scheme adapted
from Kouznetsova et al., cited in Galvanetto and Aliabadi (2010).

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In the computational homogenizational scheme shown, is the gradient tensor for any point
on the macro solid which is used to set up the boundary condition imposed on the RVE
boundary (Figures 11 and 12) with a microscopic domain of size . A solution of the
BVP for the RVE should give the macroscopic stress tensor which is achieved by the volume
averaging of the microstructural stresses over the RVE (Kouznetsova et al., cited in Galvanetto
and Aliabadi 2010). Implementing the above process in a finite element frame work gives the
finite element method for computational homogenization which has been used in two-
dimensional problems. The results from two-dimensional scheme showed the importance of
fibre/matrix interface strength on the macroscopic behaviour (damage and failure phases) of the
composite materials (Feyel 1999; Kouznetsova et al., 2001). This scheme has been further
extended to treat three dimensional problems by facilitating large deformations and rotations
making it appropriate for dynamic problems on a finer scale. It can also be used on
homogenous materials (on a finer-scale) with various shapes of RVE (Wiechert and Wall 2010).

2.3.3 Averaging theorems
Based on the principles of separation of scales, the RVE is much smaller than the characteristic
macroscopic sample that is therefore; the internal body forces at the macroscale are
generally overlooked so that the static equilibrium state of the RVE is mathematically expressed
as:

∇m. σm = � in Ωm (39)
(Kouznetsova et al., cited in Galvanetto and Aliabadi 2010)
where:

– Gradient operator
– Microscopic stresses

2.3.3.1 Deformation averaging theorem
As the actual coupling of the micro-macro kinematic quantities are based on averaging
theorems (Kouznetsova et al., cited in Galvanetto and Aliabadi 2010), the deformation gradient
tensor at any point on the macroscopic solid (Figure 13) is given by volume averages of the
microstructural deformation gradient tensor.

FM =
1
Vo Vo

Fm� dVo =
1
Vo Γo

x N� dΓo (4�)

The above expression for was obtained after transforming the integral of the un-deformed
volume of the RVE to a surface integral using the divergent theorem (Kouznetsova et al., cited in
Galvanetto and Aliabadi 2010).
where:

��- Macroscopic deformation gradient tensor
�� - Microscopic deformation gradient tensor
��– Reference domain
� - for every �x in the domain
� – Un-deformed position
x - Deformed position (Kouznetsova et al., cited in Galvanetto and Aliabadi 2010).

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2.3.3.2 Energy averaging theorem
Generally referred to as the Hill-Mandel macro-homogeneity principle, ascertains that the
macroscopic volume averages on the RVE (given by the expression in Eqn. (41) which is
expressed in terms of work conjugate set as given in Eqn. (41) can also be stated in terms of RVE
surface quantities given in Eqn. (42).

1
�� ��

��: ���
�� ��� = ��: ���

� (41)

δWOM =
1
Vo Vo

Pm: δFmC� dVo =
1
Vo Γo

P.� δxdΓo (42)

where: �� is the first Piola – Kirchhoff stress tensor
(Grytz and Mechke 2007; Kouznetsova et al., cited in Galvanetto and Aliabadi 2010).

2.3.4 Representative Volume Elements (RV�� )
The RVE is the smallest material volume statistically representative of the microstructure of the
heterogeneous material or the smallest volume of the heterogeneous material that exactly
replicates the macroscopic behaviour (Drugan and Willis 1996; LLorca et al., 2007; Kouznetsova
et al., cited in Galvanetto and Aliabadi 2010). Accurate simulations in continuum micromechanics
rely on simulating the behaviour of the RVE of the heterogeneous material. With traditional
computational homogenizational models, different RV�◠ have been used at various points on
the microstructure to ensure accuracy (LLorca et al., 2007).
Based on the model introduced by Grytz and Meschike (2007) shown in Figure 14 it is possible
to use the same RVE at all points on the microstructure.

Figure 14 Schematic representation of the micro-macro computational homogenization of finite
deformation in curvilinear coordinates adapted from Grytz and Mechke (2007).

Similarly, the computational schemes addressed so far are applicable in the standard Cartesian
coordinates (that is one physical space), but with this development, the scheme can be applied
to address problems formulated in curvilinear coordinates in the three-dimensional Euclidean
space both at the macro and micro levels. Grytz and Mechke 2007, employed this technique in
the multiscale analysis of a spherical shell subjected to internal pressure shown in figure 15
where the homogenization of the 3D microstructure was carried out numerically at the macro
level. Similarly, only two RVEs were considered all through the shell thickness.

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505

Figure 15: Spherical shell subjected to internal pressure-undeformed and deformed
configurations of the macro-structure and the underlying micro-structure with the distribution of

the Cauchy stresses ��11 adapted from Grytz and Mechke (2007).

From their study, they came to the conclusion that simplification of recommended
computational scheme for problems formulated in curvilinear coordinates (such as the use of the
same RVE and so on), greatly simplified the solution of the multiscale problem when assessed
against the traditional approaches using only a single physical space for both micro and macro
scales thus demonstrating the efficiency of the proposed scheme over the traditional schemes
particularly for problems in 3D Euclidean space.
From this study, the efficient application of computational homogenization in different physical
space at different scales was demonstrated (Grytz and Mechke 2007).

3. Conclusion
From the review of the existing mathematical, numerical and computational homogenizational
scheme for composite materials, it has been shown that:
Prior to the development of digital computers as well as the corresponding development in
simulation tools (that is the fast fourier transform, the boundary element method and more
recently the finite element method) as a means of analysing multi-scale models, the
Analytical/mathematical homogenization technique was considered the main workhorse to
simulate the macroscopic mechanical behaviour of multi-phase heterogeneous material from
the behaviour of the representative volume element (RVE) of the microstructure of the material.
The straight forwardness of the fundamental assumptions of the technique made it quite
popular. Despite its popularity (in computing internal stress in multiphase materials during
deformation) the inability of the method to capture rapid localization of damage has limited the
application of the method in computing mechanical behaviour of material where progressive
damage is involved. Although, the method describes quite accurately the effective macroscopic
properties of the heterogeneous material in the elastic system it is not as effective in the
nonlinear regime which has been the major drawback for this technique.
With the current technological developments in simulating technologies the above limitation of
the mathematical homogenization schemes in simulating the effective non linear heterogeneous
material behaviour were addressed by adopting computational homogenizational schemes
which rely on established constitutive relations for the macroscopic problems in a numerical
form through which the macroscopic problems are constructed and solved in a nested manner.
It can therefore be seen that:

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The homogenization schemes in general, have proven to be an invaluable tool for obtaining
heterogeneous macroscopic material properties from the mechanical properties of the material
microstructure.
The accuracy of the analytical schemes is limited to elastic systems which are mostly applicable
for accurately modelling microstructures with simple geometrical features. A limitation which has
largely been attributed to the simplifications adopted in the analytical homogenisation models
preventing it from capturing rapid localization of strains, nucleation, growth of damage as well
as complex loading paths which causes phase transformation .
Computational homogenization on the other hand,
Has been able to give accurate predictions of the macroscopic material behaviour in non-linear
systems.
It allows the pragmatic assumption of local periodicity in the material microstructure which
further improves on the accuracy of the results obtained.
The major drawback of this technique is that the completely coupled micro-macro technique is
quite expensive to simulate computationally.
This has however been compensated for by adopting numerical strategies where the benign
fields are modelled using coupled homogenization and finite elements and then carrying out
detailed micro-macro analysis in the core regions (crack tips, interfaces) where it is essential to
obtain a detailed information of the stress and strain fields at the core regions since it is the
information from these regions that govern failure mechanisms.

4. Future trends
A major hindrance of the computational homogenizational scheme is in the development of a
fully coupled micro-macro technique. A development of a fully coupled scheme is highly needed
in order to be able to validate computational homogenizational multiscale models against
experimental results without have any discrepancies at relatively low cost.
Furthermore, the application of this computational homogenizational scheme in designing as
well as enhancing existing engineering materials by manipulating the microstructure is still
lacking and needs more study. A significant improvement in this area will be highly invaluable in
applications of engineering material for particular industrial applications. For example, the
designing of novel lightweight materials such as: magnesium alloys, aluminium alloys,
amorphous polymers etcetera which are energy efficient and aid in facilitating the reduction of
greenhouse gas emissions from buildings, transportation and industrial sector.

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