# Integrated Digital Image Correlation for Micro-Mechanical Parameter
Identification in Multiscale Experiments111The post-print version of this
article is published in _Int. J. Solids Struct._ ,
10.1016/j.ijsolstr.2023.112130. This manuscript version is made available
under the CC-BY 4.0 license.

O. Rokoš<EMAIL_ADDRESS>R.H.J. Peerlings<EMAIL_ADDRESS>J.P.M.
Hoefnagels<EMAIL_ADDRESS>M.G.D. Geers<EMAIL_ADDRESS>Mechanics
of Materials, Department of Mechanical Engineering, Eindhoven University of
Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

###### Abstract

Micromechanical constitutive parameters are important for many engineering
materials, typically in microelectronic applications and material design.
Their accurate identification poses a three-fold experimental challenge: (i)
deformation of the microstructure is observable only at small scales,
requiring SEM or other microscopy techniques; (ii) external loadings are
applied at a (larger) engineering or device scale; and (iii) material
parameters typically depend on the applied manufacturing process,
necessitating measurements on material produced with the same process. In this
paper, micromechanical parameter identification in heterogeneous solids is
addressed through multiscale experiments combined with Integrated Digital
Image Correlation (IDIC) in conjunction with various possible computational
homogenization schemes. To this end, some basic concepts underlying multiscale
approaches available in the literature are first reviewed, discussing their
respective advantages and disadvantages from the computational as well as
experimental point of view. A link is made with recently introduced uncoupled
methods, which allow for identification of material parameter ratios at the
microscale, still lacking a proper normalization. Two multiscale methods are
analysed, allowing to bridge the gap between microstructural kinematics and
macroscopically measured forces, providing the required normalization. It is
shown that an integrated experimental–computational scheme provides relaxed
requirements on scale separation. The accuracy and performance of the
discussed techniques are analysed by means of virtual experimentation under
plane strain and large strain assumptions for unidirectional fibre-reinforced
composites. The robustness against image noise is also assessed. The obtained
results demonstrate that the expected accuracy is typically within $10\,\%$
RMS error for all multiscale methods, but decreasing to $1\,\%$ RMS error for
the optimal method.

###### keywords:

Integrated Digital Image Correlation , parameter identification , multiscale
methods , micromechanics , inverse methods , tensile test

††journal: Int. J. Solids Struct.

##  1 Introduction

In many engineering applications, accurate knowledge of micro-mechanical
parameters is of high relevance, especially for predicting the mechanical
response and the lifespan of advanced multi-phase materials or micro-electro-
mechanical devices. Through micro-structural numerical simulations, these
material parameters furthermore help to better understand individual
mechanical processes in solids occurring across the scales such as damage,
ductile fracture, or crack initiation and propagation, and thus contribute to
the design of new materials, see Hoc et al. [2003], Rupil et al. [2011],
Blaysat et al. [2015], Garoz et al. [2017], or Buljac et al. [2017].

Multiple aspects complicate the accurate identification of individual micro-
structural constituents, such as: (i) inherently small dimensions,
complicating manufacturing of microscopic specimens for micro-tensile tests
and subsequent measurements; (ii) in numerous situations, such as 3D printing,
material parameters depend on the manufacturing process and spatial position
in the structure (i.e., final dimensions, shape, and morphology of the entire
specimen are relevant and cannot be neglected); and (iii) external loadings
are applied at a larger engineering or device scale, the effect of which needs
to be assessed at the microscale. The last point (iii) requires a multi-scale
setting, in which the specimen length scale $L$ is significantly larger
compared to the length scale of the typical micro-structural features $\ell$,
i.e., the scale ratio $L/\ell\gg 1$. When mechanical properties of the micro-
structural constituents are of interest, their deformation needs to be
captured with a sufficiently high resolution using optical or scanning
electron microscopy, which implies inability to observe mechanical behaviour
of the entire specimen at the same time due to limitations in the Field Of
View (FOV). On the other hand, when deformation of the entire specimen is
captured, microstructural features are not resolved with a sufficiently high
accuracy due to limitations on image resolution.

One of the options available for non-destructive in-situ testing of
heterogeneous microstructures is nano-indentation, cf. Fröhlich et al. [1977]
or Griswold et al. [2005]. However, inaccuracies stemming from sample surface
preparation, surface roughness, precise indenter shape, unknown contact and
friction behaviour, and unresolved details on nano-plasticity, etc.,
significantly affect the resulting accuracy. Typically, this leads to
systematic overestimation of elastic stiffness properties, as discussed by
Hardiman et al. [2017]. As an alternative, direct measurements on micro-
mechanical specimens using micro-tensile testing machines can be used, as
reported in [Son et al., 2005, Du et al., 2017]. Such an approach brings,
nevertheless, significant challenges in conducting a well-defined uniaxial
micro-tensile test [Kang and Saif, 2010, Kang et al., 2010, Du et al., 2018],
and may result in significant changes in mechanical properties due to specimen
preparation [Greer and De Hosson, 2011]. Such shortcomings can be circumvented
by considering larger specimens, as discussed by Héripré et al. [2007] or
Bertin et al. [2016], where material properties in polycrystals have been
identified within larger samples. Yet, such an approach requires bridging the
gap between the microstructure and macroscopically applied loads. Such
bridging introduces normalization of the micromechanical constitutive
parameters, since when deformation of the microstructure is observed, only
material parameter ratios between individual constituents can be identified.
Knowledge of acting forces then allows for their normalization to proper
levels of magnitude. This is equivalent to a mechanical system subjected
solely to Dirichlet boundary conditions, for which its deformation state
remains the same as long as the ratios of the material parameters do not
change (although the magnitude of the reaction forces changes significantly).

This can be achieved, for instance, through an estimated underlying stress
field, as discussed by Schmidt et al. [2012], or by minimizing differences
between macroscopically measured and microstructurally estimated homogenized
stiffness properties, e.g., Matzenmiller and Gerlach [2005] or Blaheta et al.
[2015]. Typically, however, micro-mechanical parameter identification still
relies on macroscopic measurements (of displacements and/or forces), that need
to be linked to micro-mechanical properties using, e.g., computational
homogenization (FE2). This has been theoretically studied by Burczyński and
Kuś [2009], Klinge [2012a, b], Klinge and Steinmann [2015], Schmidt et al.
[2015, 2016], or Beluch and Hatlas [2017]. Typically, virtual tests were
employed, generating macroscopic measurements by forward evaluation of the
multiscale model, assuming validity of (i) the multiscale model (i.e., the
homogenization technique) used, and (ii) separation of scales. Problems with
uniqueness of micro-mechanical parameters leading to the same homogenized
macroscopic response is often mentioned. To remedy this, multiple tests have
to be performed or more robust optimization techniques have to be used,
seeking the global minimum, cf. Zohdi [2003], Chaparro et al. [2008], or Unger
and Könke [2017]. This issue becomes more relevant for a larger number of
microstructural constituents to be identified at the microscale, as upper
bounds for the maximum number of micromechanical parameters that can be
uniquely identified from macroscopic experimental observations exist, see
Schmidt et al. [2016]. Identification of micromechanical parameters within the
multiscale regime has been experimentally performed by Fedele et al. [2006],
where differences between the experimentally observed homogenized stress field
and predictions from numerical simulations carried out on a representative
volume were minimized, assuming certain micromechanical parameters to be known
a priori. Finally, parameter identification within the probabilistic realm,
e.g., Rappel et al. [2020], Hu et al. [2016], Oskay and Fish [2008], and Rosić
et al. [2013], and machine learning realm, e.g., Flaschel et al. [2021, 2022],
Joshi et al. [2022], have been discussed as well.

The starting point in this contribution is Digital Image Correlation (DIC),
which is a well-established non-intrusive full-field measurement technique. It
is especially suitable for micromechanical identification because it can be
readily used in combination with optical microscopy, or with some precautions
also with Scanning Electron Microscopy (SEM), cf. Sutton et al. [2007] and
Maraghechi et al. [2018, 2019]. Its integrated variant, called Integrated DIC
(IDIC), uses prior information on the experimentally measured specimen,
typically introduced through an underlying mechanical model, making the method
even more versatile. The prior information comprises constitutive laws,
morphology of the microstructure, and applied boundary conditions. In effect,
this allows for direct identification of all relevant material parameters of
the model, as well as high overall accuracy and robustness with respect to
measurement uncertainty, as proven in many publications, cf., e.g., Roux and
Hild [2006], Leclerc et al. [2009], Réthoré et al. [2009], Réthoré et al.
[2013], Neggers et al. [2015], Blaysat et al. [2015], or Ruybalid et al.
[2015]. Other approaches to inverse identification methods can be found in,
e.g., Avril et al. [2008], or Cameron and Tasan [2021] by direct stress
identification.

The integration of DIC with a numerical model introduces, nevertheless,
several complications because not all of the data required are readily
available, or the data is available at different scales. As an example,
consider a macroscopic specimen subjected to a uni-axial displacement-
controlled tensile test with measured reaction forces, as shown in Fig. 1. A
microscope used for the observation of microstructural features has a limited
FOV, capturing only a small portion of the entire specimen with sufficient
resolution and accuracy (note that scanning of the entire specimen at each
deformation increment, although theoretically possible, is practically
unfeasible). As a consequence, applied boundary conditions (prescribed
displacements and corresponding reaction forces) are observable far outside of
the FOV, even at a different scale (i.e., at the scale of the entire
specimen). As the underlying mechanical model requires combined information
from both scales, the gap between the two scales needs to be bridged by a
proper multiscale method.

$\color[rgb]{0,0.44921875,0.7421875}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.44921875,0.7421875}{\vec{F}_{\mathrm{exp}},\
\vec{u}_{\mathrm{D}}}$$\color[rgb]{0,0.44921875,0.7421875}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.44921875,0.7421875}{\vec{F}_{\mathrm{exp}},\
\vec{u}_{\mathrm{D}}}$matrix inclusions

Figure 1: Sketch of a multiscale experimental set-up in which a macroscopic
specimen is subjected to a uni-axial tensile test. Microscopic observations
are carried out using a microstructural field of view,
$\Omega_{\mathrm{fov}}^{\mathrm{m}}$, an underlying mechanical model inside a
micro-structural volume element, $\Omega_{\mathrm{mve}}^{\mathrm{m}}$, and a
microscopic region of interest where the experimental and numerical fields are
compared, $\Omega_{\mathrm{roi}}^{\mathrm{m}}$. At the macroscale, the
deformation field of the entire specimen $\Omega$, applied boundary
conditions, and measured reaction forces are observed through a macroscopic
field of view, $\Omega_{\mathrm{fov}}^{\mathrm{M}}$, and a macroscopic region
of interest, $\Omega_{\mathrm{roi}}^{\mathrm{M}}$.

The aim of this paper is to systematically address these challenges for
micromechanical parameter identification. To this purpose, experimental
measurements will be carried out at two scales, employing IDIC, thereby
assuming _all_ micromechanical parameters as unknown. An overview of several
methods available in the literature is provided in the first step. The basic
ideas behind these approaches are briefly described, and their respective
advantages and disadvantages are discussed from an experimental as well as a
computational perspective. In the second step, two methodologies are proposed
to provide the missing normalization of identified material parameter ratios
at the microscale by means of uncoupled methods. To a large extent, uncoupled
methods can eliminate problems with uniqueness in the identification of
mechanical parameters through residual images (indicating global minimum), and
reduce effects for a small scale separation by sampling true boundary
conditions. However, uncoupled methods only yield material parameter ratios
requiring, therefore, a suitable normalization. This is achieved in a first
novel method proposed herein by direct stress integration over a number of
experimentally observed Microstructural Volume Elements (MVE), relying on the
principle of virtual work. This methodology results in an integrated
experimental–computational scheme. Although satisfactory accuracy is reached,
substantial experimental effort is still required. A second novel method is
therefore proposed and elaborated, which reduces experimental efforts at the
expense of computational resources by making use of FE2. All tested IDIC
methodologies are summarized for convenience in Fig. 2.

(a) Direct Numerical Simulation (DNS), Section 4

(b) Concurrent Multiscale Modelling (CMM), Section 5

(c) Bridging Scale Method (BSM), Section 6

(d) Stress Integration (SI), Section 7.2

(e) Computational homogenization (FE2), Section 7.3

Figure 2: An overview of the different methodologies considered in this
contribution for micromechanical parameter identification. A more detailed
description on each methodology and its associated accuracy can be found in
the individual sections (listed in the sub-captions).

The remainder of this paper is divided into five parts. After the
introduction, the first part describes the basics of DIC (Section 2). The
reference virtual experiment, used as a test example for comparison of
individual methods, is introduced in the second part (Section 3). The existing
multiscale approaches along with the two newly proposed methodologies based on
stress integration and computational homogenization are elaborated in the
third part (Sections 4–7). In each section of the third part, the theory
underlying each method is first briefly outlined, followed by an analysis of
its results (such as convergence, typical accuracy, or performance). Mutual
comparison and robustness with respect to image noise is summarized in the
fourth part (Section 8). The paper finally closes with a summary and
conclusions in the fifth part (Section 9).

Throughout this paper, the following notation is adopted, using Cartesian
tensors:

  * 1.

scalars $a$,

  * 2.

vectors $\vec{a}$,

  * 3.

second-order tensors ${\boldsymbol{A}}$,

  * 4.

fourth-order tensors $\mathbb{C}$,

  * 5.

matrices ${\boldsymbol{\mathsf{A}}}$ and column matrices $\underline{a}$,

  * 6.

$\vec{a}\cdot\vec{b}=a_{i}b_{i}$,

  * 7.

${\boldsymbol{A}}\cdot\vec{b}=A_{ij}b_{j}\vec{e}_{i}$,

  * 8.

${\boldsymbol{A}}\cdot{\boldsymbol{B}}=A_{ik}B_{kj}\vec{e}_{i}\vec{e}_{j}$,

  * 9.

${\boldsymbol{A}}:{\boldsymbol{B}}=A_{ij}B_{ji}$,

  * 10.

conjugate ${\boldsymbol{A}}^{c}={\boldsymbol{A}}^{\mathsf{T}}$,
$A_{ij}^{\mathsf{T}}=A_{ji}$,

  * 11.

gradient operator $\displaystyle\vec{\nabla}\vec{a}=\frac{\partial
a_{j}}{\partial X_{i}}\vec{e}_{i}\vec{e}_{j}$,

  * 12.

divergence operator $\displaystyle\vec{\nabla}\cdot\vec{a}=\frac{\partial
a_{i}}{\partial X_{i}}$,

  * 13.

derivatives of scalar functions with respect to second-order tensors
$\displaystyle\delta\Psi({\boldsymbol{F}};\delta{\boldsymbol{F}})=\left.\frac{\mathrm{d}}{\mathrm{d}h}\Psi({\boldsymbol{F}}+h\delta{\boldsymbol{F}})\right|_{h=0}=\frac{\partial\Psi({\boldsymbol{F}})}{\partial{\boldsymbol{F}}}:\delta{\boldsymbol{F}}$.

##  2 Digital Image Correlation

The guiding method underpinning micromechanical parameter identification is
Digital Image Correlation (DIC). DIC follows the usual scheme of inverse
problems, i.e., it minimizes the least squares difference

$\underline{\lambda}^{\star}\in\underset{\underline{\lambda}\in\mathbb{R}^{n_{\lambda}}}{\text{arg
min}}\
\frac{1}{2}\|f(\vec{X})-g(\vec{X}+\vec{u}(\vec{X},\underline{\lambda}))\|^{2}_{\Omega_{\mathrm{roi}}}$
(1)

between observation and model prediction. In Eq. (1),
$\|\bullet\|_{\Omega_{\mathrm{roi}}}$ denotes the $L^{2}$-norm defined over a
Region Of Interest (ROI), $\Omega_{\mathrm{roi}}\subset\mathbb{R}^{2}$, while
$\vec{X}=X_{1}\vec{e}_{1}+X_{2}\vec{e}_{2}$ is the position vector associated
with the reference configuration. $f$ and $g$ are observation data, more
precisely grey scale images capturing undeformed and deformed configurations
of the experimentally tested specimen. The image $g$ is mapped through a
displacement field
$\vec{u}(\vec{X},\underline{\lambda})=u_{1}(\vec{X},\underline{\lambda})\vec{e}_{1}+u_{2}(\vec{X},\underline{\lambda})\vec{e}_{2}$,
which is a function of a set of admissible model parameters, stored in a
column
$\underline{\lambda}=[\lambda_{1},\dots,\lambda_{n_{\lambda}}]^{\mathsf{T}}\in\mathbb{R}^{n_{\lambda}}$.
Through this mapping, a back-deformed image is constructed (through the model
prediction), which is compared against the reference image $f$.

Choosing the field $\vec{u}$ as a linear combination of user-selected basis
functions $\vec{\psi}_{i}(\vec{X})\in\mathbb{R}^{2}$, $i=1,\dots,n_{\lambda}$,
i.e.,

$\vec{u}(\vec{X},\underline{\lambda})=\sum_{i=1}^{n_{\lambda}}\vec{\psi}_{i}(\vec{X})\lambda_{i},\quad\vec{X}\in\Omega_{\mathrm{roi}},$
(2)

results in a so-called Global DIC scheme. In order to avoid ill-posedness or
local minima, the supports of the interpolation functions, $\vec{\psi}_{i}$,
need to span multiple pixels [cf., e.g., Horn and Schunck, 1981], be
regularized through an additional mechanical system and equilibrium gap method
[see, e.g., Tomičević et al., 2013] or by Tikhonov regularization [Leclerc et
al., 2011, Yang, 2014]. Note that the definition of GDIC adopted herein is not
unique, since some integrated approaches fulfil the definition of Eq. (2) as
well, cf., e.g., [Roux and Hild, 2006, Section 5].

On the contrary, in Integrated DIC, $\vec{u}$ is based on the solution of an
underlying mechanical model of the boundary value problem at hand, expressed
in abstract form as

$\vec{\mathcal{M}}(\vec{u}(\vec{X},\underline{\lambda}),\underline{\lambda},\vec{X})=\vec{0},\quad\vec{X}\in\Omega_{\mathcal{M}}.$
(3)

Obtaining $\vec{u}$ from Eq. (3) usually entails the solution of an associated
(system of) partial differential equation(s), carried out typically by means
of the Finite Element (FE) method. The model domain $\Omega_{\mathcal{M}}$
differs for the individual methods shown in Fig. 2, spanning either an inner
region of the specimen domain $\Omega$ or the entire specimen. Dependence on
the time-resolved nature of the problem has been omitted for simplicity in
Eqs. (1)–(3), but can easily be taken into account [Neggers et al., 2015].

The mechanical solution regularizes the minimization problem of Eq. (1) and
delivers accurate parameters when an appropriate model $\vec{\mathcal{M}}$
with no model error is used. As hinted to in both Eqs. (2) and (3),
$\underline{\lambda}$ may consist of kinematic Degrees Of Freedom (DOFs), such
as kinematic boundary conditions [Rokoš et al., 2018b], material parameters,
or topological properties associated with the underlying system. Accuracy of
IDIC can further be improved by including the force measurement in the
optimization scheme, i.e., by considering

$\frac{1}{2}\|\vec{F}_{\mathrm{exp}}-\vec{F}_{\mathrm{sim}}(\underline{\lambda})\|^{2}_{\mathrm{E}}$
(4)

as an additional objective function. The expression in Eq. (4) measures the
Euclidean norm $\|\bullet\|_{\mathrm{E}}$ of the difference between a vector
of experimentally observed reaction forces $\vec{F}_{\mathrm{exp}}$ and forces
predicted by the underlying mechanical model
$\vec{F}_{\mathrm{sim}}(\underline{\lambda})$. Formally, multi-objective
optimization technique should be used for the two objectives, which is
nevertheless typically avoided by using weighted sum scalarization, i.e.,
minimizing a weighted sum of the two objectives (see, e.g., Deb and Kalyanmoy
2001 and Neggers et al. 2015). In this contribution, a slightly different
approach will be adopted, as detailed below in Section 4.1.

The solution of the minimization problem of Eq. (1) is usually obtained by a
Gauss–Newton algorithm, convergence of which to the global minimum can be
readily verified by means of the image residual
$|f(\vec{X})-g(\vec{X}+\vec{u}(\vec{X},\underline{\lambda}))|$, where
$|\bullet|$ denotes the absolute value of a function $\bullet$. For further
details on minimization problem of Eq. (1) and its consistent linearisation
see Neggers et al. [2016]. Solution of the underlying mechanical system in Eq.
(3) follows standard procedures, discussed in, e.g., Zienkiewicz and Taylor
[1989], Crisfield [1997], or Jirásek and Bažant [2002].

A two-step Finite Element Model Updating (FEMU) procedure can be used as an
alternative to the presented (one-step) IDIC approach. In FEMU, first the
displacement fields are measured through standard DIC, typically Local DIC,
which are subsequently used in material parameter identification, carried out
in the second step, cf., e.g., Avril et al. [2008]. As shown by Ruybalid et
al. [2015], IDIC outperforms FEMU in terms of accuracy and robustness with
respect to image noise (especially when displacements are small and/or
sensitivity to material parameters is low). Only IDIC is therefore elaborated
further in this work.

##  3 Reference Model

The performance and accuracy of the different methodologies will be tested by
means of virtual experiments, i.e., observations are generated through Direct
Numerical Simulations (DNS) of the full sample with a finely discretized
microstructure. To this end, first the underlying model $\vec{\mathcal{M}}$ of
Eq. (3), governing the mechanics of the system, is specified. A standard
hyper-elastic mechanical model in absence of body forces, with equilibrium
equation

$\vec{\nabla}\cdot{\boldsymbol{P}}^{\mathsf{T}}(\vec{X})=\vec{0},\quad\vec{X}\in\Omega,$
(5)

under plane strain and large deformation assumptions is adopted. The
assumption on plane strain is used to remove any inaccuracies stemming from
out-of-plane motion, i.e., subsurface and 3D effects, since our primary
objective in this contribution lies in multiscale identification of
micromechanical parameters. Although 3D effects are crucial for accurate
identification, they are difficult to capture, and are capable of introducing
large errors when not treated properly. Methodologies accounting for out-of-
plane motion effects should be considered in such cases, including digital
height correlation [Kleinendorst et al., 2016, Uzun and Korsunsky, 2019],
digital volume correlation [Leclerc et al., 2011], and full 3D models. In Eq.
(5),

${\boldsymbol{P}}(\vec{X})=\frac{\partial
W(\vec{X},{\boldsymbol{F}}(\vec{u}(\vec{X})))}{\partial{\boldsymbol{F}}^{\mathsf{T}}},\quad\vec{X}\in\Omega,$
(6)

is the first Piola–Kirchhoff stress tensor, $\vec{\nabla}$ denotes the
gradient with respect to the reference configuration, ${\boldsymbol{F}}$ is
the deformation gradient, and $\vec{u}(\vec{X})$ the displacement field. The
elastic energy density $W$ is decomposed as

$W(\vec{X},{\boldsymbol{F}}(\vec{u}(\vec{X})))=[1-\chi(\vec{X})]W_{1}({\boldsymbol{F}}(\vec{u}(\vec{X})))+\chi(\vec{X})W_{2}({\boldsymbol{F}}(\vec{u}(\vec{X})))$
(7)

according to phases $\beta=1,\,2$, which are geometrically characterized by a
phase indicator function of the inclusions

$\chi(\vec{X})=\left\\{\begin{array}[]{ll}1,&\mbox{if~{}$\vec{X}$ is inside an
inclusion},\\\ 0,&\mbox{otherwise}.\end{array}\right.$ (8)

Individual energy densities are further assumed to follow a compressible
hyper-elastic Neo–Hookean constitutive law

$W_{\beta}({\boldsymbol{F}})=\frac{1}{2}G_{\beta}(\overline{I}_{1}-3)+\frac{1}{2}K_{\beta}(\ln(J))^{2},$
(9)

which corresponds to the matrix (for $\beta=1$) and inclusions (for
$\beta=2$). In Eq. (9), $J=\det{{\boldsymbol{F}}}$, and
$\overline{I}_{1}=J^{-2/3}\,\mathrm{tr\,{(}}{\boldsymbol{C}})$ is the first
isochoric invariant of the right Cauchy–Green deformation tensor
${\boldsymbol{C}}={\boldsymbol{F}}^{\mathsf{T}}\cdot{\boldsymbol{F}}$. For the
micro-mechanical parameter identification, the unknown constants are the
matrix shear and bulk moduli $G_{1}$ and $K_{1}$, and the inclusions’ shear
and bulk moduli $G_{2}$ and $K_{2}$, for convenience all normalized by the
shear modulus of the matrix $G_{\mathrm{m}}$. The reference normalized
material parameters are summarized in Tab. 1 as a function of the material
contrast ratio $\rho$, fixed to a value $\rho=4$. Throughout all examples, the
micro-structural morphology is assumed to be known exactly for all scans. In
practice, however, it should be identified from experimental data, thus
introducing additional measurement uncertainty.

For simplicity, the mechanical test adopted is a non-uniform tensile test,
sketched in Fig. 3, with a rectangular specimen domain
$\Omega=[-L/2,L/2]\times[-W/2,W/2]=[-54,54]d\times[-18,18]d$, having randomly
distributed inclusions of unit diameter $d=1$ with volume fraction
$\phi_{\mathrm{dns}}=42.9\,\%$. This means that the typical length scale of
the microstructural features, discussed earlier in the introduction, is
$\ell=d=1$. Because the employed constitutive models are size independent,
recall Eq. (9), without any influence on the mechanical behaviour all
geometric properties are normalized by the inclusions’ diameter $d=1$, which
can be thought of to be $50\,\mu\mathrm{m}$, for instance. Macroscopically
applied boundary conditions are prescribed to the two vertical edges, cf. Fig.
3, whereas the two horizontal edges are free surfaces. The induced tension
corresponds to $5\,\%$ of overall strain.

Table 1: Micromechanical parameters adopted in virtual experiment. The shear and bulk moduli $G_{\beta}$ and $K_{\beta}$ are normalized by the shear modulus of the matrix $G_{\mathrm{m}}$. Adopted contrast ratio is fixed to a value $\rho=4$. Constitutive parameters |  | matrix
---
($\beta=1$)
| inclusions
---
($\beta=2$)
Normalized shear modulus, $G_{\beta}$ | 1 |  | $\rho$ |
Normalized bulk modulus, $K_{\beta}$ | 3 |  | $3\rho$ |
Poisson’s ratio, $\nu_{\beta}=\frac{3K_{\beta}-2G_{\beta}}{2(3K_{\beta}+G_{\beta})}$ | 0 | .35 | 0 | .35
$\Omega$
---
a
---
b
---
d
---
c
---
$\xi$
---
$L=108d$
---

$W=36d$
---

Figure 3: A sketch of the specimen’s geometry with a random microstructure
subjected to $5\,\%$ overall non-uniform tensile strain. The microstructure
(randomly distributed circular inclusions of normalized diameter $d=1$) is
scanned only in certain parts, depending on the identification method used
(cf. Fig. 2). As an example, two micro ROIs with domains
$\Omega_{\mathrm{roi}}^{\mathrm{m}}$ of sizes $\ell_{\mathrm{roi}}=4d$ and
$10d$ are indicated at the specimen’s centre.

Two speckle patterns, $f_{\mathrm{m}}$ (approx. $900\times 900$ pixels) and
$f_{\mathrm{M}}$ (approx. $2500\times 1000$ pixels), are applied at the
microscale (inside $\Omega_{\mathrm{fov}}^{\mathrm{m}}$) and at the macroscale
(inside $\Omega_{\mathrm{fov}}^{\mathrm{M}}$). Rectangular micro ROIs are
considered with edge length $\ell_{\mathrm{roi}}$, i.e.,
$\Omega_{\mathrm{roi}}^{\mathrm{m}}$ has the size
$\ell_{\mathrm{roi}}\times\ell_{\mathrm{roi}}$. To mimic a fixed camera
resolution, $f_{\mathrm{m}}$ is mapped onto
$\Omega_{\mathrm{fov}}^{\mathrm{m}}$ (which is slightly larger than
$\Omega_{\mathrm{roi}}^{\mathrm{m}}$), meaning that the micro-pixel size
depends on $\ell_{\mathrm{roi}}$. Typical pixel densities corresponding to the
two patterns thus are $20\times 20$ pixels per unit area $d^{2}$ for the
macro-image, and range from $50\times 50$ pixels (corresponding to
$\ell_{\mathrm{roi}}=20d$ and scale ratio between the micro ROI size and the
specimen size equal to $5$) to $200\times 200$ pixels (corresponding to
$\ell_{\mathrm{roi}}=4d$ and scale ratio $25$) per unit area $d^{2}$ for the
micro-images depending on magnification of the microscopic observation. Figure
4 shows both speckle patterns along with their corresponding histograms and
Auto-Correlation Functions (ACF). Several quality parameters are listed in
Tab. 2, definitions of which can be found in Pan et al. [2008] or Dong and Pan
[2017]. Note that $\ell_{\mathrm{c}}$ is defined such that ACF equals $1/2$,
and that in order to closely approximate realistic conditions, physical
speckle patterns have been employed, adopted from [Hoeberichts, 2014, Section
4.2.5] for $f_{\mathrm{m}}$, and [DIC challenge SEM, 2017, Reu et al., 2018,
Sample 17] for $f_{\mathrm{M}}$. A multiscale pattern is furthermore assumed,
i.e., the micro-image $f_{\mathrm{m}}$ is applied inside
$\Omega_{\mathrm{fov}}^{\mathrm{m}}$ only, whereas the macro-image
$f_{\mathrm{M}}$ is considered in the remaining part of the domain, i.e., in
$\Omega_{\mathrm{fov}}^{\mathrm{M}}\subseteq\Omega\backslash\Omega_{\mathrm{fov}}^{\mathrm{m}}$.
More details on the application of multiscale patterns for _in situ_ FE-SEM
observations of carbon fibre-reinforced polymer composites can be found, e.g.,
in Tanaka et al. [2011].

(a) micro speckle pattern $f_{\mathrm{m}}$

(b) histogram of $f_{\mathrm{m}}$

(c) ACF of $f_{\mathrm{m}}$, $\ell_{\mathrm{c}}=1.8$ pix

(d) macro speckle pattern $f_{\mathrm{M}}$

(e) histogram of $f_{\mathrm{M}}$

(f) ACF of $f_{\mathrm{M}}$, $\ell_{\mathrm{c}}=2.8$ pix

Figure 4: Employed speckle patterns (a) $f_{\mathrm{m}}$ [Section 4.2.5 of Hoeberichts, 2014, totally approx. $900\times 900$ pixels], and (d) $f_{\mathrm{M}}$ [DIC challenge SEM, 2017, Reu et al., 2018, Sample 17, totally approx. $2500\times 1000$ pixels], along with the corresponding grey value histograms (b) and (e), and auto-correlation functions (c) and (f). Table 2: Effective parameters of the speckle patterns used at the macroscale $f_{\mathrm{M}}$ and microscale $f_{\mathrm{m}}$. Speckle pattern quality parameters | image $f_{\mathrm{M}}$ | image $f_{\mathrm{m}}$
---|---|---
Root-mean-square value, | $\mathrm{RMS}$ | 123 | .083 | 119 | .943
Mean intensity gradient, | $\delta_{f}$ | 30 | .047 | 25 | .327
Correlation length, | $\ell_{\mathrm{c}}$ | 1 | .796 pix | 2 | .745 pix
Quality factor, | $Q=\delta_{f}/\ell_{\mathrm{c}}$ | 16 | .728 | 9 | .226

Experimental observations $g_{\mathrm{m}}$, $g_{\mathrm{M}}$, and
$\vec{F}_{\mathrm{exp}}$, are generated through DNS. A Total Lagrangian
formulation is used to model the underlying physics of Eq. (5), employing
quadratic isoparametric triangular elements with three-point Gauss integration
rule. The mesh generator _gmsh_ by Geuzaine and Remacle [2009] has been used
for the triangulation, as shown in Fig. 6a. Only two images, i.e., the
reference $f$ and deformed $g$, will be used, although all inverse methods
presented below can easily be extended to simultaneous correlation of multiple
images in a time-integrated scheme as would be done in real tests to improve
accuracy, see Neggers et al. [2015]. The synthetic deformed images are
generated such that the reference image $f(\vec{X})$ is pixelized, with
discrete pixel positions $\vec{X}_{i}$. Since isoparametric elements are used,
natural coordinates $\vec{\eta}_{i}$ of all pixels are found by inverting the
isoparametric mapping. The displacement field is then computed exactly for
each of the pixels as $\vec{u}_{i}=\vec{u}(\vec{X}_{i})$, based on the assumed
shape of the interpolation functions, and each of the pixels is translated
accordingly, i.e., $g(\vec{X}_{i})=f(\vec{X}_{i}-\vec{u}_{i})$. Since computed
displacements are of non-integer pixel values, the resulting images
$g_{\mathrm{m}}$ and $g_{\mathrm{M}}$ are interpolated at the reference
(integer) pixel locations of the underlying grid $\vec{X}_{i}$ using cubic
interpolation.

For further analyses, effective material parameters $\widetilde{G}$ and
$\widetilde{K}$ obtained for a MVE positioned at the specimen’s centre
$\vec{X}=\vec{0}$ (as shown in Fig. 3), are computed to provide a measure for
the representativeness of individual MVEs as a function of their size
$\ell_{\mathrm{mve}}$. To this end, first the homogenized anisotropic
constitutive tangent of each MVE, $\overline{\mathbb{C}}_{\mathrm{m}}$, is
computed, considering periodic boundary conditions with exact constitutive
models for both constituents in the reference configuration, i.e.,

$\overline{{\boldsymbol{\mathsf{C}}}}_{\mathrm{m}}=\frac{1}{|\Omega_{\mathrm{mve}}^{\mathrm{m}}|}{\boldsymbol{\mathsf{B}}}^{\mathsf{T}}({\boldsymbol{\mathsf{A}}}{\boldsymbol{\mathsf{K}}}_{\partial}^{-1}{\boldsymbol{\mathsf{A}}}^{\mathsf{T}})^{-1}{\boldsymbol{\mathsf{B}}}.$
(10)

In Eq. (10), $\overline{{\boldsymbol{\mathsf{C}}}}_{\mathrm{m}}$ is a matrix
representation of $\overline{\mathbb{C}}_{\mathrm{m}}$,
${\boldsymbol{\mathsf{K}}}_{\partial}$ is a stiffness matrix condensed to the
nodes of the MVE boundary $\partial\Omega_{\mathrm{mve}}^{\mathrm{m}}$, and
$|\Omega_{\mathrm{mve}}^{\mathrm{m}}|$ is the size of the micro-model domain
$\Omega_{\mathrm{mve}}^{\mathrm{m}}$. The coefficient matrices
${\boldsymbol{\mathsf{A}}}$ and ${\boldsymbol{\mathsf{B}}}$ correspond to the
discretized version of periodic boundary conditions, expressed as a set of
equality constraints

${\boldsymbol{\mathsf{A}}}\underline{u}_{\partial}={\boldsymbol{\mathsf{B}}}(\underline{\overline{F}}_{\mathrm{M}}-\underline{I}),$
(11)

where $\underline{u}_{\partial}$ is a column of MVE boundary displacements,
and $\underline{\overline{F}}_{\mathrm{M}}$ and $\underline{I}$ are column
representations of the macroscopically prescribed deformation gradient
$\overline{{\boldsymbol{F}}}_{\mathrm{M}}$ and the identity tensor
${\boldsymbol{I}}$; for further details see [Miehe, 2003, Miehe and
Bayreuther, 2007, Larsson and Runesson, 2007]. Next, $\widetilde{G}$ and
$\widetilde{K}$ are derived by fitting an isotropic stiffness tensor
$\mathbb{C}_{\mathrm{iso}}$ to the homogenized anisotropic constitutive
tangent $\overline{\mathbb{C}}_{\mathrm{m}}$,

$\\{\widetilde{G},\widetilde{K}\\}=\underset{G,K}{\text{arg min}}\
\|\mathbb{C}_{\mathrm{iso}}(G,K)-\overline{\mathbb{C}}_{\mathrm{m}}\|_{\mathrm{T}}^{2},$
(12)

where $\|\mathbb{C}\|^{2}_{\mathrm{T}}=C_{ijkl}C_{ijkl}$ denotes the tensor
norm of fourth-order tensors. More details on constitutive tangent fitting can
be found in Moakher and Norris [2006]. The resulting effective parameters
(normalized by their limit values) are shown in Fig. 5a. As expected, for
small MVE sizes, the homogenized shear and bulk moduli are not representative,
whereas the error slowly decreases with increasing MVE size (especially for
$\ell_{\mathrm{mve}}\geq 16d$). The limit values, corresponding to the entire
specimen, $\ell_{\mathrm{mve}}=L$, which are used for normalization, read as
$\widetilde{G}_{\mathrm{dns}}=1.656$, $\widetilde{K}_{\mathrm{dns}}=4.656$,
see Tab. 3. The representativeness of individual MVEs in terms of homogenized
bulk and shear moduli correlates well with the volume fraction of inclusions
shown in Fig. 5b. Overall, all effective quantities fluctuate approximately
within $5\,\%$ error relative to their limit values. The small fluctuation
observed in both graphs of Fig. 5 towards the limit $\ell_{\mathrm{mve}}=L$ is
explained by the fact that all inclusions are not allowed to intersect
specimen’s boundary, cf. Fig. 3. For more details on the representativeness of
random microstructures see, e.g., Torquato [2002].

(a) normalized effective material parameters

(b) normalized volume fraction

Figure 5: (a) Convergence of the normalized effective material parameters $\widetilde{K}$ and $\widetilde{G}$ based on MVE simulations and Eqs. (10) and (12) for various sizes of $\Omega_{\mathrm{mve}}^{\mathrm{m}}$ ($\ell_{\mathrm{mve}}\times\ell_{\mathrm{mve}}$). (b) Convergence of the normalized volume fraction $\phi$ as a function of $\ell_{\mathrm{mve}}$. Normalization limit values corresponding to $\ell_{\mathrm{mve}}=L$ are listed in Tab. 3. Table 3: Effective parameters corresponding to the entire specimen $\Omega$. Effective parameter | Value
---|---
Volume fraction, $\phi_{\mathrm{dns}}$ | 0 | .429
Shear modulus, $\widetilde{G}_{\mathrm{dns}}$ | 1 | .656
Bulk modulus, $\widetilde{K}_{\mathrm{dns}}$ | 4 | .656

##  4 Full-Scale Direct Numerical Simulations

###  4.1 Methodology

Full-Scale Direct Numerical Simulations (DNS), cf. Fig. 2a and Proc. 1,
consider a fully resolved microstructure throughout the entire specimen
domain, thus requiring considerable experimental as well as computational
efforts. The IDIC model $\vec{\mathcal{M}}$ spans the entire domain (i.e.,
$\Omega_{\mathcal{M}}\equiv\Omega$), for which experimentally applied
kinematic boundary conditions and reaction forces are known. For the region of
interest, typically microscopic images $\Omega_{\mathrm{roi}}^{\mathrm{m}}$
are used, cf. Fig. 1. In principle, DNS constitutes a single-scale approach
operating at the fully expanded microscale only, and hence is considered as
the reference solution.

The normalization procedure through experimentally measured forces is
implemented as follows. Instead of scalarization, which requires prior
knowledge of appropriate weights, one can minimize the objectives of Eqs. (1)
and (4) sequentially, making use of the fact that the deformation state of the
considered (elastic) specimen under the applied load does not depend on any
particular material parameter normalization (recall that during the tensile
experiment adopted, displacement control with force measurement is used). The
numerically computed force, $\vec{F}$, is therefore known only up to a
multiplicative constant $\alpha$, i.e., the simulated force reads
$\vec{F}_{\mathrm{sim}}=\alpha\vec{F}$. The multiplicative constant can be
eliminated from the quadratic objective of Eq. (4), providing optimal value
denoted by the asterisk

$\alpha^{\star}=\frac{\vec{F}_{\mathrm{exp}}\cdot\vec{F}(\underline{\lambda})}{\vec{F}(\underline{\lambda})\cdot\vec{F}(\underline{\lambda})}.$
(13)

Although normalization through the measured force in Eq. (13) is used for
simplicity, we note that other approaches can be used, following for instance
the line of reasoning proposed in Rahmani et al. [2014], where homogenized
stiffness properties of the microstructure have been fitted to the
experimentally observed stiffness, or [Réthoré, 2010, Réthoré et al., 2013],
where equilibrium gap method and the balance between external and internal
forces has been adopted to extend the least-squares objective of Eq. (1).

(a) DNS mesh (image generation)

(b) DNS mesh (IDIC)

Figure 6: Triangulations associated with two DNS systems used for (a)
generating experimental observations, and (b) for micro-mechanical
identification. List of procedures 1 Full-Scale Direct Numerical Simulation
(DNS) used for micromechanical parameter identification. 1. Experimental
observations: (a) before the test, record a high-resolution image of the
microstructure of the entire specimen $\Omega$ in undeformed configuration
(located inside $\Omega_{\mathrm{fov}}^{\mathrm{M}}$) which typically requires
multiple microscopic scans, (b) at each load step, record a (high-resolution)
scan of the microscopic region of interest
$\Omega_{\mathrm{roi}}^{\mathrm{m}}$ (not necessarily spanning the entire
specimen to save experimental efforts), the applied macroscopic kinematic
boundary conditions $\vec{u}_{\mathrm{D}}$, and the resulting reaction force
$\vec{F}_{\mathrm{exp}}$. 2. Construct an underlying IDIC model, taking into
account the entire microstructure (i.e., in Eq. (3),
$\Omega_{\mathcal{M}}\equiv\Omega$), and apply the measured kinematic boundary
conditions $\vec{u}_{\mathrm{D}}$ directly to the model. 3. Perform IDIC, by
correlating the scans of the micro ROI (i.e., in Eq. (1),
$\Omega_{\mathrm{roi}}\equiv\Omega_{\mathrm{roi}}^{\mathrm{m}}$), followed by
normalization through Eqs. (4) and (13) to obtain the resulting set of
micromechanical parameters $\underline{\lambda}$.

###  4.2 Results

The numerical models used for identification of micromechanical parameters
employ the discretization shown in Fig. 6b, which is $1.5$ times coarser
compared to the forward-model mesh used for generating the virtual
measurements (shown in Fig. 6a). Correlation is performed for microscopic
images $f_{\mathrm{m}}$ and $g_{\mathrm{m}}$, using
$\Omega_{\mathrm{roi}}^{\mathrm{m}}$ positioned at the specimen’s centre. In
order to quantify the accuracy of all four identified micromechanical
parameters ($G_{\beta}$ and $K_{\beta}$, $\beta\in\\{1,2\\}$) through only one
simple quantity, the following Root-Mean-Square (RMS) norm is introduced

$\eta_{\mathrm{rms}}=\sqrt{\frac{1}{n_{\lambda_{\mathrm{mat}}}}\sum_{i=1}^{n_{\lambda_{\mathrm{mat}}}}\left(\frac{\lambda_{i}}{\lambda_{\mathrm{ex},i}}-1\right)^{2}},$
(14)

where $\lambda_{\mathrm{ex},i}$ is the exact value of corresponding $i$-th
material parameter $\lambda_{i}$. The achieved typical accuracy of DNS is
$\eta_{\mathrm{rms}}=0.03\,\%$, as reported in Tab. 4 of Section 8.1, where an
overall comparison of all methods is presented.

For consistency with other methods, the Rigid Body Motion (RBM) has been
corrected for in IDIC to eliminate related systematic errors by adding the
corresponding DOFs to $\underline{\lambda}$. As expected, the RBM
displacements vanish, namely $\|\vec{u}_{\mathrm{rbm}}\|_{\mathrm{E}}\leq
1\times 10^{-4}$. Furthermore, various sizes of
$\Omega_{\mathrm{roi}}^{\mathrm{m}}$ positioned at the specimen’s centre have
been considered to test the influence of the micro-image resolution in the
range $\ell_{\mathrm{roi}}\in[4,20]d$, used later for Stress Integration (SI)
and computational homogenization (FE2) methods described in Sections 7.2 and
7.3. No significant influence on the accuracy of identified parameters has
been observed, suggesting that the resolution $\ell_{\mathrm{roi}}=20d$ (i.e.,
density of $50\times 50$ pixels per $d^{2}$) still suffices for accurate
micromechanical identification (obtained accuracy $\eta_{\mathrm{rms}}$
corresponds to $0.08\,\%$).

##  5 Concurrent Multiscale Modelling

###  5.1 Methodology

In order to reduce computational and experimental efforts, Concurrent
Multiscale Modelling (CMM) [see Tian et al., 2010, Gracie and Belytschko,
2011, Hütter et al., 2014] takes into account only a small portion of the
microstructure (cf. Fig. 2b), whereas a homogeneous material with a suitable
(effective) constitutive law is assumed elsewhere. The two regions, with
either the fully resolved microstructure
($\Omega_{\mathrm{full}}^{\mathrm{m}}$) or the homogeneous material (outside
$\Omega_{\mathrm{full}}^{\mathrm{m}}$), are seamlessly bridged by a gradually
refined mesh. The homogenized material transmits forces applied at the
specimen’s boundary to the fully resolved microstructural region, inside which
the images are correlated (i.e., inside $\Omega_{\mathrm{roi}}^{\mathrm{m}}$).

The identification itself proceeds in two steps and requires observations at
two scales, cf. Proc. 2. First, macroscopic IDIC is performed using the
applied kinematic boundary conditions, measured reaction forces,
macroscopically observed images inside $\Omega_{\mathrm{roi}}^{\mathrm{M}}$,
and a macroscopic homogeneous model $\vec{\mathcal{M}}^{\mathrm{hom}}$. This
model spans the entire domain (i.e.,
$\Omega_{\mathcal{M}}^{\mathrm{hom}}\equiv\Omega$), in which homogeneous
effective material properties are adopted. In the subsequent identification
stage, the actual microstructural material parameters are sought for, which
are considered only inside the fully resolved region of which
$\Omega_{\mathrm{roi}}^{\mathrm{m}}$ is part. To this end, a multiscale
heterogeneous model $\vec{\mathcal{M}}^{\mathrm{het}}$ is constructed, which
uses the same domain (i.e., $\Omega_{\mathcal{M}}^{\mathrm{het}}\equiv\Omega$)
and boundary conditions as the homogeneous model
$\vec{\mathcal{M}}^{\mathrm{hom}}$. The only difference is that the multiscale
model $\vec{\mathcal{M}}^{\mathrm{het}}$ considers fully refined
microstructure in certain part of the domain $\Omega$ (having there an
independent set of micromechanical parameters) and denoted as
$\Omega_{\mathrm{full}}^{\mathrm{m}}$ (of size
$\ell_{\mathrm{full}}\times\ell_{\mathrm{full}}$), whereas outside this region
the previously identified homogeneous material is used.

List of procedures 2 Concurrent Multiscale Modelling (CMM) used for
micromechanical parameter identification. 1. Experimental observations: (a) at
each load step, record a macroscopic image spanning
$\Omega_{\mathrm{fov}}^{\mathrm{M}}$ with low spatial resolution (no
microstructure needed), the applied displacements $\vec{u}_{\mathrm{D}}$, and
the reaction forces $\vec{F}_{\mathrm{exp}}$, (b) at each load step, also
record a microscopic image inside $\Omega_{\mathrm{fov}}^{\mathrm{m}}$ with a
high spatial resolution (capturing micro-structural morphology with sufficient
accuracy). 2. Construct two models: (a) a homogeneous model,
$\mathcal{M}^{\mathrm{hom}}$,
$\Omega_{\mathcal{M}}^{\mathrm{hom}}\equiv\Omega$, in which constant effective
material parameters are adopted everywhere, (b) and a multiscale heterogeneous
model, $\mathcal{M}^{\mathrm{het}}$,
$\Omega_{\mathcal{M}}^{\mathrm{het}}\equiv\Omega$, in which effective material
parameters are used outside the fully resolved region
$\Omega_{\mathrm{full}}^{\mathrm{m}}$, whereas inside this region an
independent set of micro-mechanical parameters is used. 3. Perform IDIC using
the homogeneous model $\mathcal{M}^{\mathrm{hom}}$ to identify effective
material parameter ratios by correlating inside
$\Omega_{\mathrm{roi}}^{\mathrm{M}}$. Normalize obtained parameter ratios by
the measured forces $\vec{F}_{\mathrm{exp}}$ through Eq. (13). 4. Perform IDIC
using the heterogeneous model $\mathcal{M}^{\mathrm{het}}$, in which the
normalized effective material parameters (Step $3$) are used and kept fixed
outside the fully resolved region. Inside the fully resolved region (of which
$\Omega_{\mathrm{roi}}^{\mathrm{m}}$ is part of), an independent set of micro-
mechanical parameters is identified by correlating the microscopic images
inside $\Omega_{\mathrm{roi}}^{\mathrm{m}}$. (Note that no additional
normalization is required, because homogeneous material parameters have
already been normalized in Step $3$.)

###  5.2 Results

_Step 1:_ Using the triangulation of Fig. 7a in the homogeneous model
$\vec{\mathcal{M}}^{\mathrm{hom}}$ of Step 3, Proc. 2, the _effective_
constitutive parameters are obtained through IDIC by correlating images inside
the macroscopic ROI, $\Omega_{\mathrm{roi}}^{\mathrm{M}}$, assuming the
constitutive law of Eq. (9). Obtained values are $\overline{G}=1.649$ and
$\overline{K}=4.584$, which are within $2\,\%$ of relative error compared to
the reference homogenized values $\widetilde{G}_{\mathrm{dns}}$ and
$\widetilde{K}_{\mathrm{dns}}$ (obtained for $\ell_{\mathrm{mve}}=L=108d$)
reported in Tab. 3. The discrepancy between limit values and IDIC values
$\overline{G}$ and $\overline{K}$ are explained by different state of the
material and its non-linearities. In particular, reference configuration has
been used for the identification of $\widetilde{G}_{\mathrm{dns}}$ and
$\widetilde{K}_{\mathrm{dns}}$, which means that the constitutive law
effectively corresponds to the Hooke’s law. On the contrary, deformed
configuration was used for the identification of $\overline{G}$ and
$\overline{K}$, for which constitutive law does not fulfil Hooke’s law anymore
due to large strains and hyperelasticity, recall Eq. (9). The differences in
the deformed state then give rise to observed discrepancy.

_Step 2:_ Micromechanical parameter identification using a multiscale
heterogeneous model $\vec{\mathcal{M}}^{\mathrm{het}}$ is performed, cf. Step
4, Proc. 2, using $\overline{G}$ and $\overline{K}$ outside the fully resolved
region of the heterogeneous microstructure. Typical triangulation is shown in
Fig. 7b for the size of the fully resolved region
$\ell_{\mathrm{full}}=\ell_{\mathrm{roi}}=4d$. Rigid body motion is corrected
for, which now does not vanish due to neglected heterogeneities outside
$\Omega_{\mathrm{full}}^{\mathrm{m}}$, achieving values as large as
$\|\vec{u}_{\mathrm{rbm}}\|_{\mathrm{E}}\approx 1.2\times 10^{-2}$ (cf. also
Fig. 11, where the systematic errors in MVE BCs are compared for the different
methods). When $\vec{u}_{\mathrm{rbm}}$ is neglected, however, large errors in
the identified material parameters result.

In order to evaluate the influence of the size of the fully resolved region on
the accuracy of the micromechanical parameter identification, the following
convergence study is performed. The microscopic region of interest
$\Omega_{\mathrm{roi}}^{\mathrm{m}}$ (of constant size
$\ell_{\mathrm{roi}}=4d$) is fixed at the specimen’s centre, whereas the size
of the fully resolved region is varied as $\ell_{\mathrm{full}}\in[4,108]d$.
With increasing $\ell_{\mathrm{full}}$, the effect of the homogeneous material
fades out (i.e., inaccurate displacements at the interface between the
homogeneous and heterogeneous regions, cf. Fig. 11). The effect of the
homogeneous region can be observed in Fig. 9a, where the residual
corresponding to $\ell_{\mathrm{full}}=5.6d$ and $\ell_{\mathrm{roi}}=4d$ is
shown. In terms of $\eta_{\mathrm{rms}}$, shown in Fig. 8a, two regions can be
distinguished: (i) $\ell_{\mathrm{full}}$ is smaller than the specimen’s width
$W$ (solid line); and (ii) $\ell_{\mathrm{full}}$ is larger than the
specimen’s width (dashed line). For smaller sizes of the fully resolved
regions, the achieved accuracy decreases due to neglected heterogeneities
outside the fully resolved region, and achieves on average
$\overline{\eta}_{\mathrm{rms}}\approx 7\,\%$ of RMS for
$4d\leq\ell_{\mathrm{full}}\leq 20d$. In the second case, the method is no
longer in a multiscale regime and hence, these results should be interpreted
with caution as they are not representative for cases aimed for in this
contribution (for $\ell_{\mathrm{full}}=48d$, $\eta_{\mathrm{rms}}=0.2\,\%$,
which is an accurate result). For very large fully resolved regions
($\ell_{\mathrm{full}}>80d$), the accuracy decreases again, which is caused by
too narrow homogenized regions positioned between the fully resolved region
and the two fixed vertical boundaries. Note that the limit in which the entire
specimen is fully resolved cannot be reached as no homogenized material
outside $\Omega^{\mathrm{m}}_{\mathrm{full}}$, needed for normalization, is
included. In this case, one would obviously opt for the full DNS model
instead.

(a) triangulation of the homogeneous model $\vec{\mathcal{M}}^{\mathrm{hom}}$

(b) triangulation of the heterogeneous model
$\vec{\mathcal{M}}^{\mathrm{het}}$

Figure 7: Two triangulations used for the Concurrent Multiscale Modelling
(CMM). (a) A triangulation with a homogeneous constitutive model used in the
first step for identification of the homogenized properties $\overline{G}$ and
$\overline{K}$. (b) A seamless triangulation bridging two scales with
heterogeneous constitutive properties used for identification of the
micromechanical parameters. A quarter of the entire domain $\Omega$ is shown
in both cases. For this particular case, the size of the fully resolved
region, in which the heterogeneous microstructure is considered, corresponds
to $\ell_{\mathrm{full}}=\ell_{\mathrm{roi}}=4d$.

(a) accuracy of CMM

(b) accuracy of BSM

Figure 8: (a) Convergence of $\eta_{\mathrm{rms}}$ as a function of the size
of the fully resolved region $\ell_{\mathrm{full}}$ for the Concurrent
Multiscale Modelling (CMM). (b) Convergence of $\eta_{\mathrm{rms}}$ as a
function of the MVE size $\ell_{\mathrm{mve}}$ for the Bridging Scale Method
(BSM). Micro ROI size is fixed in both cases to $\ell_{\mathrm{roi}}=4d$.

##  6 Bridging Scale Method

###  6.1 Methodology

The Bridging Scale Method (BSM) [see, e.g., Tang et al., 2005, Kaye et al.,
2013], similarly to the CMM, fully resolves the microstructure inside a small
region of interest, as indicated in Fig. 2c. It differs, however, in the fact
that there is no seamless transition between the micro- and macro-scale by a
gradually refined mesh, but the microscale model is coupled to the macroscale
one. Although multiple options exist, a kinematic coupling is adopted in what
follows.

Two separate macro- and microscopic models are introduced in this approach,
which are solved sequentially in two steps, cf. Proc. 3. In the first step,
IDIC correlating on $\Omega_{\mathrm{roi}}^{\mathrm{M}}$ is performed through
a macroscopic model $\vec{\mathcal{M}}^{\mathrm{M}}$, containing no
microstructure and spanning the entire specimen
($\Omega_{\mathcal{M}}^{\mathrm{M}}\equiv\Omega$). This step provides
homogenized material properties, and displacement and stress fields inside
$\Omega_{\mathcal{M}}^{\mathrm{M}}$. Next, boundary conditions are sampled
from the macroscopic model and prescribed to a MVE, represented by a
microscopic model $\vec{\mathcal{M}}^{\mathrm{m}}$ (in which
$\Omega_{\mathcal{M}}^{\mathrm{m}}\equiv\Omega_{\mathrm{mve}}^{\mathrm{m}}$),
where the full microstructure is considered. Using this model, a second IDIC
is performed, by correlating the images inside
$\Omega_{\mathrm{roi}}^{\mathrm{m}}\subseteq\Omega_{\mathrm{mve}}^{\mathrm{m}}$.
Because Dirichlet BCs are applied along the entire boundary
$\partial\Omega_{\mathrm{mve}}^{\mathrm{m}}$, only material parameter ratios
can be extracted. In order to normalize them, the difference between the
macroscopic ${\boldsymbol{P}}_{\mathrm{M}}$ and effective MVE stress
$\overline{{\boldsymbol{P}}}_{\mathrm{m}}$ tensors is minimized in analogy to
Eq. (4), i.e.

$\alpha^{\star}=\underset{\alpha\in\mathbb{R}}{\mbox{arg min}}\
\|{\boldsymbol{P}}_{\mathrm{M}}-\alpha\overline{{\boldsymbol{P}}}_{\mathrm{m}}(\underline{\lambda})\|_{\mathrm{T}}^{2},\quad\mbox{providing}\quad\alpha^{\star}=\frac{{\boldsymbol{P}}_{\mathrm{M}}:\overline{{\boldsymbol{P}}}_{\mathrm{m}}^{\mathsf{T}}(\underline{\lambda})}{\overline{{\boldsymbol{P}}}_{\mathrm{m}}(\underline{\lambda}):\overline{{\boldsymbol{P}}}_{\mathrm{m}}^{\mathsf{T}}(\underline{\lambda})},$
(15)

where

$\overline{{\boldsymbol{P}}}_{\mathrm{m}}(\underline{\lambda})=\frac{1}{|\Omega_{\mathrm{mve}}^{\mathrm{m}}|}\int_{\Omega_{\mathrm{mve}}^{\mathrm{m}}}{\boldsymbol{P}}_{\mathrm{m}}(\vec{X}_{\mathrm{m}},\underline{\lambda})\,\mathrm{d}\vec{X}_{\mathrm{m}}$
(16)

denotes the volume average of the microscopic first Piola–Kirchhoff stress
tensor,
$\|{\boldsymbol{A}}\|_{\mathrm{T}}^{2}={\boldsymbol{A}}:{\boldsymbol{A}}^{\mathsf{T}}$
is the square of the tensor norm induced by the corresponding scalar product,
and $\alpha^{\star}$ the optimal normalization constant. In practice, the
average stress of Eq. (16) computes as

$\overline{\underline{P}}_{\mathrm{m}}=\frac{1}{|\Omega_{\mathrm{mve}}^{\mathrm{m}}|}{\boldsymbol{\mathsf{B}}}^{\mathsf{T}}\underline{\mu},$
(17)

where $\overline{\underline{P}}_{\mathrm{m}}$ is a column storing components
of the homogenized microscopic stress tensor
$\overline{{\boldsymbol{P}}}_{\mathrm{m}}$, $\underline{\mu}$ is the Lagrange
multiplier enforcing the periodicity of Eq. (11), and
${\boldsymbol{\mathsf{B}}}$ is the coefficient matrix, recall Eq. (10) and the
discussion therein.

A potential drawback of this approach is that boundary conditions prescribed
to the MVE model are sampled from the meso- or macroscopic model
$\Omega_{\mathcal{M}}^{\mathrm{M}}$, and hence they do not contain any micro-
structural features and fluctuations, which are known to be essential for
accurate microstructural IDIC material identification without stress
normalization, cf. Rokoš et al. [2018b].

List of procedures 3 Bridging Scale Method (BSM) used for micromechanical
parameter identification. 1. Experimental observations: (a) at each load step,
record a macroscopic image spanning $\Omega_{\mathrm{fov}}^{\mathrm{M}}$ with
low spatial resolution (no microstructure is needed), the applied
displacements $\vec{u}_{\mathrm{D}}$, and reaction forces
$\vec{F}_{\mathrm{exp}}$, (b) at each load step, also record a microscopic
image $\Omega_{\mathrm{fov}}^{\mathrm{m}}$ with a high spatial resolution
(capturing the microstructural morphology with sufficient accuracy). 2.
Construct a homogeneous macroscopic model $\vec{\mathcal{M}}^{\mathrm{M}}$ of
the entire specimen ($\Omega_{\mathcal{M}}^{\mathrm{M}}\equiv\Omega$); perform
IDIC inside $\Omega_{\mathrm{roi}}^{\mathrm{M}}$ to obtain the effective
homogeneous material parameters. 3. Construct a heterogeneous microscopic
model $\vec{\mathcal{M}}^{\mathrm{m}}$ inside the MVE (i.e.,
$\Omega_{\mathcal{M}}^{\mathrm{m}}\equiv\Omega_{\mathrm{mve}}^{\mathrm{m}}$),
taking into account the microstructural morphology. Sample the kinematic MVE
boundary conditions along $\partial\Omega_{\mathrm{mve}}^{\mathrm{m}}$ and the
corresponding average macroscopic stress ${\boldsymbol{P}}_{\mathrm{M}}$ from
the macroscopic model $\vec{\mathcal{M}}^{\mathrm{M}}$. Perform IDIC on the
MVE model, correlating the microscopic images inside
$\Omega_{\mathrm{roi}}^{\mathrm{m}}$, to obtain the micromechanical parameter
ratios, while simultaneously enforcing normalization according to Eq. (15).

###  6.2 Results

The first identification step towards homogenized material parameters in BSM
is identical to that of CMM, and hence provides the same results. The second
step consists in prescribing kinematic boundary conditions, sampled from the
homogeneous model, along the entire MVE boundary
$\partial\Omega_{\mathrm{mve}}^{\mathrm{m}}$. Finally, micromechanical
parameters are normalized by comparing homogenized stresses of the MVE model
to stresses of the homogeneous model. The MVE rigid body motion has been
corrected for ($\|\vec{u}_{\mathrm{rbm}}\|_{\mathrm{E}}\approx 1.1\times
10^{-2}$), which is again significant due to inaccuracies in the applied
boundary conditions.

A convergence study is carried out, which consists in enlarging the employed
MVE ($\Omega_{\mathrm{mve}}^{\mathrm{m}}$, with
$\ell_{\mathrm{mve}}\in[4,108]d$) while using a ROI with a fixed size
$\ell_{\mathrm{roi}}=4d$. The results are shown in Fig. 8b, where the best
accuracy, within multiscale regime (i.e., for $4d\leq\ell_{\mathrm{mve}}\leq
20d$), is achieved for
$\eta_{\mathrm{rms}}(\ell_{\mathrm{mve}}=16.6d)=1.8\,\%$. The average
performance corresponds approximately to
$\overline{\eta}_{\mathrm{rms}}\approx 8\,\%$ of RMS. The image residual
corresponding to $\ell_{\mathrm{mve}}=4.4d$ is shown in Fig. 9b, which
indicates a relatively poor correlation.

(a) CMM

(b) BSM

(c) GDIC-IDIC

(d) BE-IDIC $0.2$
---
$0.1$
---
$0.3$
---
$0.0$
---
$0.4$
---

Figure 9: Microscopic image residuals
$|f_{\mathrm{m}}-g_{\mathrm{m}}|(\vec{X}_{\mathrm{m}})/2^{8}$ (normalized by
dynamic contrast $2^{8}$) corresponding to (a) Concurrent Multiscale Method
(CMM) with $\ell_{\mathrm{full}}=5.6d$ (RMS grey value of $0.06$), (b)
Bridging Scale Method (BSM) with $\ell_{\mathrm{mve}}=4.4d$ (RMS grey value of
$0.09$), (c) uncoupled GDIC-IDIC with $\ell_{\mathrm{mve}}=4d$ (RMS grey value
of $0.02$), and (d) uncoupled BE-IDIC with $\ell_{\mathrm{mve}}=4d$ (RMS grey
value of $0.02$). In all cases, $\ell_{\mathrm{roi}}=4d$ has been used.

##  7 Normalization by Newly Proposed Methods

Two novel methodologies for the identification of micro-mechanical parameters
are described below in this section. They both utilize the same ingredient,
uncoupled microscale methods described first in Section 7.1, which serve to
identify micromechanical parameter ratios at the microscale. Proper
normalization constants are obtained through Stress Integration (SI) or
computational homogenization (FE2) methods, described subsequently in Sections
7.2 and 7.3.

Although both methodologies are presented in a simplified context, i.e.,
focusing on conservative hyperelastic systems and considering only two
configurations (reference and deformed), they can both be extended to time-
evolving and history-dependent problems. To this end, simultaneous
correlations of multiple images in a time-integrated scheme would need to be
considered, see Neggers et al. [2015]. In the case of SI (Section 7.2), the
displacement $\vec{u}_{\mathrm{M}}$ and strain ${\boldsymbol{B}}$ fields would
need to be considered incrementally at each of the time steps (to approximate
virtual test fields), effectively minimizing the difference between the
internal and external forces over all time steps tested with the incremental
displacement and strain fields, cf. also [Pierron and Grédiac, 2012, Chapter
4]. For FE2 (Section 7.3), time-integrated difference between the observed
$\vec{F}_{\mathrm{exp}}$ and simulated $\vec{F}_{\mathrm{sim}}$ forces would
need to be minimized. Since microstructural simulations are run only with
Dirichlet boundary conditions, the parameters of an elastic-plastic
constitutive law are expected to have a small influence on the displacement
field, see, e.g., Réthoré et al. [2013]. For that reason, low accuracy of
history-dependent variables might occur. Moreover, based on history evolution
of the resulting macroscopic forces, it might be difficult to distinguish
between non-linearities stemming from elastic and plastic deformations, cf.,
e.g., Rappel et al. [2020]. Additional investigations are thus needed to
obtain quantitative as well as qualitative performance of the newly proposed
methods for history-dependent materials. This is outside of the scope of the
current manuscript, and will be considered in future studies.

###  7.1 Uncoupled Methods

####  7.1.1 Methodology

Uncoupled methods rely on IDIC being carried out inside an MVE only, as
sketched in Fig. 10, requiring microstructural scans of
$\Omega_{\mathrm{roi}}^{\mathrm{m}}$ and a corresponding IDIC MVE model
$\vec{\mathcal{M}}^{\mathrm{m}}$. The MVE model necessitates kinematic
boundary conditions, applied along the entire MVE boundary
$\partial\Omega_{\mathrm{mve}}^{\mathrm{m}}$, which are generally unknown and
highly irregular due to material heterogeneities. Furthermore, because
Dirichlet BCs are prescribed, only material parameter ratios can be extracted.
The key question here is how sufficiently accurate microscopic boundary
conditions along the MVE can be obtained, as discussed below.

Two alternatives are considered, as summarized in Proc. 4 and 5: (i)
sequential identification of microscopic displacement fields by GDIC followed
by material parameter identification through IDIC (introduced by Hild et al.
2016 and Shakoor et al. 2017), abbreviated as GDIC-IDIC in what follows; and
(ii) simultaneous identification of boundary conditions and material
parameters, abbreviated as Boundary-Enriched IDIC (BE-IDIC), cf. Fedele and
Santoro [2012] and Rokoš et al. [2018b]. Because the GDIC-IDIC approach is
subject to the well-known GDIC compromise (i.e., lack of kinematic freedom for
coarse GDIC meshes and random errors due to image noise for fine meshes, cf.
Leclerc et al. 2009), the identified kinematic boundary conditions (and hence
identified micromechanical parameters) suffer from inaccuracies for highly
heterogeneous microstructures [Rokoš et al., 2018b].

The second alternative (i.e., BE-IDIC) is used to eliminate the GDIC step.
Material parameters as well as all kinematic displacements at the MVE boundary
are considered as DOFs in Eq. (1), i.e.,

$\underline{\lambda}=[\underline{\lambda}_{\mathrm{mat}}^{\mathsf{T}},\underline{\lambda}_{\mathrm{kin}}^{\mathsf{T}}]^{\mathsf{T}},$
(18)

where

$\displaystyle\underline{\lambda}_{\mathrm{mat}}$
$\displaystyle=[G_{1},K_{1},G_{2},K_{2}]^{\mathsf{T}},$ (19)
$\displaystyle\underline{\lambda}_{\mathrm{kin}}$
$\displaystyle=\underline{u}_{\mathrm{mve}}(\vec{X}),\quad\vec{X}\in\partial\Omega_{\mathrm{mve}}^{\mathrm{m}}.$

In Eq. (19), $\underline{u}_{\mathrm{mve}}(\vec{X})$,
$\vec{X}\in\partial\Omega_{\mathrm{mve}}^{\mathrm{m}}$, denotes a column of
all DOFs situated at the MVE boundary. A standard IDIC procedure with an
extended set of DOFs is then conducted.

Note that for both uncoupled methods the MVE chosen does not have to be
representative [Rokoš et al., 2018b], meaning that it should merely contain
all relevant microstructural constituents. This is in contrast with
normalization approaches which require representative effective stresses
(typically evaluated at macroscopic Gauss integration points). Accuracy of the
uncoupled methods can be further improved by considering multiple MVE
realizations, and averaged over the ensemble of identified material parameter
ratios to eliminate random scatter (cf. Rokoš et al. 2018b, where it was shown
that the identified micromechanical parameter ratios show a low systematic
bias for the BE-IDIC case).

$\Omega_{\mathrm{roi}}^{\mathrm{m}}\equiv\Omega_{\mathrm{mve}}^{\mathrm{m}}$
---
$\Omega_{\mathrm{roi,gdic}}^{\mathrm{m}}$
---
$\Omega_{\mathrm{fov}}^{\mathrm{m}}$
---
(a) GDIC-IDIC

$\Omega_{\mathrm{roi}}^{\mathrm{m}}\equiv\Omega_{\mathrm{mve}}^{\mathrm{m}}$
---
$\Omega_{\mathrm{fov}}^{\mathrm{m}}$
---
(b) BE-IDIC

Figure 10: A sketch of uncoupled methods. (a) The GDIC-IDIC method identifies
in the first step kinematic boundary conditions by means of a GDIC mesh
constructed over
$\Omega_{\mathrm{roi,gdic}}^{\mathrm{m}}\subset\Omega_{\mathrm{fov}}^{\mathrm{m}}$,
and subsequently performs IDIC on a smaller region,
$\Omega_{\mathrm{roi}}^{\mathrm{m}}$. (b) BE-IDIC uses material parameters as
well as kinematic displacements at the MVE boundary as IDIC DOFs
($\Omega_{\mathrm{roi}}^{\mathrm{m}}\equiv\Omega_{\mathrm{mve}}^{\mathrm{m}}\subset\Omega_{\mathrm{fov}}^{\mathrm{m}}$).
List of procedures 4 Uncoupled method GDIC-IDIC for identification of
micromechanical parameter ratios. 1. Experimental observations: (a) at each
load step, record a microscopic image inside
$\Omega_{\mathrm{fov}}^{\mathrm{m}}$ with a high spatial resolution. 2.
Construct a GDIC triangulation and perform GDIC on
$\Omega_{\mathrm{roi,gdic}}^{\mathrm{m}}$,
$\Omega_{\mathrm{roi}}^{\mathrm{m}}\subset\Omega_{\mathrm{roi,gdic}}^{\mathrm{m}}\subset\Omega_{\mathrm{fov}}^{\mathrm{m}}$
(cf. Fig. 10a). 3. Construct a microscopic MVE model
$\vec{\mathcal{M}}^{\mathrm{m}}$ over
$\Omega_{\mathrm{mve}}^{\mathrm{m}}\subseteq\Omega_{\mathrm{roi,gdic}}^{\mathrm{m}}$
(cf. Fig. 10a where
$\Omega_{\mathrm{mve}}^{\mathrm{m}}\equiv\Omega_{\mathrm{roi}}^{\mathrm{m}}$),
applying the MVE BCs along $\partial\Omega_{\mathrm{mve}}^{\mathrm{m}}$
sampled from the GDIC data of Step 2. 4. Perform IDIC to obtain
micromechanical parameter ratios, correlating on
$\Omega_{\mathrm{roi}}^{\mathrm{m}}\subseteq\Omega_{\mathrm{roi,gdic}}^{\mathrm{m}}$.
List of procedures 5 Uncoupled method BE-IDIC for identification of
micromechanical parameter ratios. 1. Experimental observations: (a) at each
time step, record a microscopic image inside
$\Omega_{\mathrm{fov}}^{\mathrm{m}}$ with a high spatial resolution. 2.
Construct a microscopic MVE model $\vec{\mathcal{M}}^{\mathrm{m}}$ over
$\Omega_{\mathrm{mve}}^{\mathrm{m}}\equiv\Omega_{\mathrm{roi}}^{\mathrm{m}}$
(cf. Fig. 10b). 3. Perform IDIC, correlating on
$\Omega_{\mathrm{roi}}^{\mathrm{m}}$ while optimizing all material and
kinematic DOFs (cf. Eqs. (18) and (19)).

####  7.1.2 Results

The uncoupled methods are designed to identify accurately material parameters
at the microscale, assuming that one of the material parameters is a priori
known to provide normalization. The micromechanical boundary conditions are
accounted for accurately, whereby BE-IDIC outperforms GDIC-IDIC [cf. Rokoš et
al., 2018b]. The achieved overall precision in terms of identified material
parameter ratios is typically $\eta_{\mathrm{rms}}\approx 0.3\,\%$ (results
not shown), which is one order of magnitude more accurate compared to CMM and
BSM in the multiscale regime, recall Fig. 8. This result is obtained by
normalizing all parameters $G_{1}$, $K_{1}$, and $G_{2}$, relative to a fixed
inclusions’ bulk modulus $K_{2}$. Corresponding micro-residual images are
shown in Figs. 9c and 9d, resulting in a RMS grey value of $0.02$.

An example of displacement components $u_{1}(\xi)$ and $u_{2}(\xi)$ along the
micro ROI boundary $\partial\Omega_{\mathrm{roi}}^{\mathrm{m}}$, parametrized
by $\xi\in[0,16]d$ for $\ell_{\mathrm{roi}}=4d$, is shown in Fig. 11. The
results show that, in contrast to CMM and BSM methods, uncoupled approaches
(especially BE-IDIC) achieve practically indistinguishable profiles compared
to the DNS results. Even the peak displacement $u_{1}\approx 0.1d$ of $20$
pixels is captured accurately, see zoom inset in Fig. 11a. Note that in
contrast to other methods, the rigid body motion is accounted for
automatically.

zoom
---
(a) horizontal component $u_{1}(\xi)$

(b) vertical component $u_{2}(\xi)$

Figure 11: Line traces of the displacement field
$\vec{u}(\xi)=u_{1}(\xi)\vec{e}_{1}+u_{2}(\xi)\vec{e}_{2}$, as obtained for
individual methods, compared against the exact DNS result. The parametric
coordinate $\xi$ sweeps along the ROI boundary
$\partial\Omega_{\mathrm{roi}}^{\mathrm{m}}$. The chosen ROI is positioned at
the specimen’s centre (cf. Fig. 3) and has a size $\ell_{\mathrm{roi}}=4d$.
The horizontal components $u_{1}(\xi)$ are shown in (a), whereas the vertical
components $u_{2}(\xi)$ in (b).

###  7.2 Stress Integration

####  7.2.1 Methodology

In sequential multiscale modelling, the microscale information, such as the
first Piola–Kirchhoff stress tensor obtained from the uncoupled methods of
Section 7.1, is computed and passed to the macroscale. By integrating
homogenized stresses at the macroscale over $\Omega$, the resulting
macroscopic reaction forces are assembled and compared against experimentally
observed data to provide proper normalization. In particular, the stress
Integration (SI) method, summarized in Proc. 6 and sketched in Fig. 2d,
employs experimental observations at both scales. At the microscale, images
with sufficiently high resolution capturing all morphological features are
observed (at specific locations, as detailed below). Microstructural IDIC
identification is subsequently performed on
$\Omega_{\mathrm{roi}}^{\mathrm{m}}$ for all MVEs simultaneously to yield
micromechanical parameter ratios by means of one of the uncoupled methods.
This step also provides homogenized stresses for each MVE. At the macroscale,
the reaction forces, applied displacements, and coarse images are exploited to
provide normalization and strain fields.

Normalization of micromechanical parameters is achieved through six steps, as
follows:

  1. 1.

The Principle of Virtual Work (PVW) is first employed, i.e.,

$\underbrace{\int_{\Omega}{\boldsymbol{P}}_{\mathrm{M}}(\vec{X}_{\mathrm{M}}):\delta{\boldsymbol{F}}^{\mathsf{T}}_{\mathrm{M}}(\vec{X}_{\mathrm{M}})\,\mathrm{d}\vec{X}_{\mathrm{M}}}_{\delta
W^{\mathrm{int}}}=\underbrace{\int_{\Gamma_{\mathrm{D}}}\vec{t}_{\mathrm{M}}(\vec{X}_{\mathrm{M}})\cdot\delta\vec{u}_{\mathrm{M}}(\vec{X}_{\mathrm{M}})\,\mathrm{d}\vec{X}_{\mathrm{M}}}_{\delta
W^{\mathrm{ext}}},$ (20)

where $\Gamma_{\mathrm{D}}\subseteq\partial\Omega$ is the fixed part of the
domain boundary along which the traction $\vec{t}_{\mathrm{M}}$ is determined
from the measured force (which can be done for a specific loading
configuration such as a uni-axial tensile test),
${\boldsymbol{P}}_{\mathrm{M}}$ is the macroscopic first Piola–Kirchhoff
stress tensor, $\delta\vec{u}_{\mathrm{M}}$ macroscopic virtual displacement
field, and
$\delta{\boldsymbol{F}}_{\mathrm{M}}^{\mathsf{T}}=\delta({\boldsymbol{I}}+\vec{\nabla}_{\mathrm{M}}\vec{u}_{\mathrm{M}})=\vec{\nabla}_{\mathrm{M}}\delta\vec{u}_{\mathrm{M}}$
its gradient; note that $\vec{\nabla}_{\mathrm{M}}=\partial/\partial
X_{\mathrm{M},i}$ denotes the gradient operator with respect to the
macroscopic coordinates $X_{\mathrm{M},i}$ in the reference configuration.

  2. 2.

The macroscopic stress field ${\boldsymbol{P}}_{\mathrm{M}}$, entering the
internal virtual work $\delta W^{\mathrm{int}}$, is next approximated by its
homogenized counterpart (available only point-wise, typically in Gauss
integration points, and computed at the microscale through one of the
uncoupled methods), i.e.,

${\boldsymbol{P}}_{\mathrm{M}}(\vec{X}_{\mathrm{M}})\approx\alpha\,\overline{{\boldsymbol{P}}}_{\mathrm{m}}(\vec{X}_{\mathrm{M}})=\frac{\alpha}{|\Omega_{\mathrm{mve}}^{\mathrm{m}}|}\int_{\Omega_{\mathrm{mve}}^{\mathrm{m}}}{\boldsymbol{P}}_{\mathrm{m}}(\vec{X}_{\mathrm{M}},\vec{X}_{\mathrm{m}},\underline{\lambda})\,\mathrm{d}\vec{X}_{\mathrm{m}},$
(21)

which is known only at the positions of the measured MVEs (denoted
$\vec{X}_{\mathrm{M}}$), and up to a yet to be identified normalization
constant $\alpha$.

  3. 3.

The macroscopic virtual displacement $\delta\vec{u}_{\mathrm{M}}$ and strain
$\delta{\boldsymbol{F}}_{\mathrm{M}}$ fields in $\delta W^{\mathrm{int}}$ are
approximated by the kinematics experimentally identified at the macroscale,
i.e.,

$\delta{\boldsymbol{F}}^{\mathsf{T}}_{\mathrm{M}}(\vec{X}_{\mathrm{M}})\approx\vec{\nabla}_{\mathrm{M}}\vec{u}_{\mathrm{M}}(\vec{X}_{\mathrm{M}}).$
(22)

The displacement field $\vec{u}_{\mathrm{M}}$ is typically obtained by GDIC
with globally or locally supported polynomials, or by IDIC with homogeneous
material in which only the strain field is of interest.

  4. 4.

The continuous integration over $\Omega$ in $\delta W^{\mathrm{int}}$ of Eq.
(20) is approximated numerically considering a set of macroscopic Gauss
integration points:

$\displaystyle\delta
W^{\mathrm{int}}=\int_{\Omega}{\boldsymbol{P}}_{\mathrm{M}}(\vec{X}_{\mathrm{M}}):\delta{\boldsymbol{F}}_{\mathrm{M}}^{\mathsf{T}}(\vec{X}_{\mathrm{M}})\,\mathrm{d}\vec{X}_{\mathrm{M}}$
$\displaystyle\approx\alpha\int_{\Omega}\overline{{\boldsymbol{P}}}_{\mathrm{m}}(\vec{X}_{\mathrm{M}}):\underbrace{\vec{\nabla}_{\mathrm{M}}\vec{u}_{\mathrm{M}}(\vec{X}_{\mathrm{M}})}_{{\boldsymbol{B}}(\vec{X}_{\mathrm{M}})}\,\mathrm{d}\vec{X}_{\mathrm{M}}$
(23)
$\displaystyle\approx\alpha\underbrace{\sum_{i=1}^{n_{\mathrm{g}}}w_{i}J_{i}\overline{\underline{P}}_{\mathrm{m},i}^{\mathsf{T}}\underline{B}_{i}}_{\delta\widetilde{W}^{\mathrm{int}}}.$

In Eq. (23), $\overline{\underline{P}}_{\mathrm{m},i}$ and $\underline{B}_{i}$
are columns storing components of
$\overline{{\boldsymbol{P}}}_{\mathrm{m}}(\vec{X}_{\mathrm{M},i})$ and
${\boldsymbol{B}}(\vec{X}_{\mathrm{M},i})$ at positions
$\vec{X}_{\mathrm{M},i}$ of $n_{\mathrm{g}}$ adopted integration points with
weights $w_{i}$ and Jacobians $J_{i}$. It is clear that the spatial positions
of individual MVEs need to match with the particular integration scheme
chosen, influencing the experimental set-up. Therefore, an integrated
experimental–computational scheme results. Note that the approximation steps
of Eqs. (21)–(23) resemble a Reduced Order Modelling (ROM) approach [see,
e.g., Yvonnet and He, 2007], in which ${\boldsymbol{B}}(\vec{X}_{\mathrm{M}})$
corresponds to a strain snapshot (or a mode) of the considered specimen with
domain $\Omega$. This also suggests that more efficient ROM integration
techniques (compared to the Gauss integration scheme suggested here) can be
employed [see, e.g., An et al., 2008, Ryckelynck, 2009, Hernández et al.,
2017, Yano and Patera, 2019]. Some affinity with the Virtual Fields Method
(VFM) can be observed as well (cf., e.g., Pierron and Grédiac 2012), in which
${\boldsymbol{B}}(\vec{X}_{\mathrm{M}})$ represents a single virtual field.
This means that the sensitivity of the VFM on the choice of multiple virtual
fields [as discussed, e.g., by Rahmani et al., 2014] is relaxed, as only one
parameter needs to be identified (i.e., the normalization parameter $\alpha$).

  5. 5.

In analogy to the internal virtual work $\delta W^{\mathrm{int}}$, the
external virtual work $\delta W^{\mathrm{ext}}$ is estimated based on
available data (that is, experimentally measured applied macroscopic
displacements and reaction forces), i.e.,

$\delta
W^{\mathrm{ext}}=\int_{\Gamma_{\mathrm{D}}}\vec{t}_{\mathrm{M}}(\vec{X}_{\mathrm{M}})\cdot\delta\vec{u}_{\mathrm{M}}(\vec{X}_{\mathrm{M}})\,\mathrm{d}\vec{X}_{\mathrm{M}}\approx\underbrace{\sum_{i=1}^{n_{\mathrm{f}}}\underline{F}_{\mathrm{exp},i}^{\mathsf{T}}\underline{u}_{\mathrm{D},i}}_{\delta\widetilde{W}^{\mathrm{ext}}}.$
(24)

Note that accurate experimental observations of tractions and displacements
are not straightforward, if even possible. The formulation is, nevertheless,
introduced here in the context of PVW for completeness in the form of the
left-hand-side of Eq. (24). In practice, force resultants are considered
instead, see, e.g., [Réthoré et al., 2013, Section 3.4] or [Pierron and
Grédiac, 2012, Section 1.2 and Chapter 2], whereas displacements are obtained
through DIC or by using boundary enrichments (i.e., using virtual boundaries
as described in Section 7.1, Eqs. (18)–(19)). Within an experimental setting
considered hereafter, the knowledge of $\vec{t}_{\mathrm{M}}$ and
$\Gamma_{\mathrm{D}}$ under specific considerations reduces to a set of
reaction force resultants $\underline{F}_{\mathrm{exp},i}$ along with their
corresponding displacements $\underline{u}_{\mathrm{D},i}$,
$i=1,\dots,n_{\mathrm{f}}$.

  6. 6.

The optimal normalization constant is finally obtained by equating the
approximate virtual internal and external work, providing

$\alpha^{\star}=\frac{\delta\widetilde{W}^{\mathrm{ext}}}{\delta\widetilde{W}^{\mathrm{int}}}.$
(25)

Unlike the uncoupled methods introduced in Section 7.1, it may furthermore
become clear that individual MVEs should be representative in order to provide
accurate homogenized macroscopic properties
$\overline{{\boldsymbol{P}}}_{\mathrm{m}}$. This may entail an increase in
experimental and computational costs because larger portions of the
microstructure need to be accurately scanned and modelled.

List of procedures 6 Stress Integration (SI) method used for micromechanical
parameter identification. 1. Experimental observations: (a) at each load step,
record a macroscopic image spanning $\Omega_{\mathrm{fov}}^{\mathrm{M}}$ with
low spatial resolution (no microstructure is needed), applied displacements
$\vec{u}_{\mathrm{D}}$, and reaction forces $\vec{F}_{\mathrm{exp}}$, (b) at
each load step, also record a microscopic image inside
$\Omega_{\mathrm{fov}}^{\mathrm{m}}$ with high spatial resolution for all
MVEs; individual MVEs are positioned close to Gauss integration points of the
entire specimen $\Omega$. 2. Use one of the uncoupled methods of Section 7.1
to identify micromechanical parameter ratios (simultaneously for all MVEs to
achieve the best accuracy). 3. Determine all macroscopic quantities to be able
to compute $\delta\widetilde{W}^{\mathrm{int}}$ in Eq. (23) and
$\delta\widetilde{W}^{\mathrm{ext}}$ in Eq. (24). Compute the optimal
normalization constant $\alpha^{\star}$ from Eq. (25).

The above description suggests that the main advantages of the SI are as
follows:

  * 1.

No constraints are implied for MVE kinematic boundary conditions; all
fluctuations are captured accurately (up to image noise present and MVE
discretization used);

  * 2.

The homogenized stresses $\overline{{\boldsymbol{P}}}_{\mathrm{m}}$ correspond
to the true deformation state of the underlying microstructure, as observed
experimentally (no assumption on homogeneous deformation);

  * 3.

The macroscopic strain mode ${\boldsymbol{B}}$ results directly from the
underlying physics at the corresponding scale, as it has been obtained
experimentally.

On the contrary, several disadvantages need to be emphasized as well:

  * 1.

The method requires substantial experimental effort, because $n_{\mathrm{g}}$
MVEs need to be observed at a priori chosen positions (e.g., Gauss integration
points);

  * 2.

The experimentally observed strain ${\boldsymbol{B}}$ and displacement
$\vec{u}_{\mathrm{M}}$ fields may introduce excessive inaccuracies due to
image noise and systematic errors (as a result of, e.g., too coarse or fine
GDIC mesh);

  * 3.

The virtual displacement and strain fields should ideally be incremental to
maintain a high accuracy (i.e., infinitesimally small), which is not always
guaranteed, but can be improved using a time-incremental scheme.

Although the effect of noise and systematic error in GDIC can be mitigated,
the number of required microstructural observations $n_{\mathrm{g}}$ and force
measurements may be too large for complex geometries to be experimentally
feasible. To this end, an alternative methodology, described in Section 7.3
below, is proposed as well.

####  7.2.2 Results

For the adopted simple tensile test example shown in Fig. 3, the resulting
experimental forces have only horizontal components
$\vec{F}_{\mathrm{exp}}=\pm F_{\mathrm{exp}}\vec{e}_{1}$, induced through
prescribed displacements at both ends, $\vec{u}_{\mathrm{D}}=\pm
u_{\mathrm{D}}\vec{e}_{1}$, where the displacements are constant along the two
vertical edges. Substituting these assumptions into Eq. (24) yields for the
variation of the external virtual work $\delta
W^{\mathrm{ext}}=2F_{\mathrm{exp}}u_{\mathrm{D}}$, which eventually gives
through Eq. (25) the normalization constant

$\alpha^{\star}=\frac{2F_{\mathrm{exp}}u_{\mathrm{D}}}{\delta\widetilde{W}^{\mathrm{int}}},$
(26)

where Eq. (23) is used to compute the approximate variation of the internal
virtual work, i.e.,
$\delta\widetilde{W}^{\mathrm{int}}=\sum_{i=1}^{n_{\mathrm{g}}}w_{i}J_{i}\overline{\underline{P}}_{\mathrm{m},i}^{\mathsf{T}}\underline{B}_{i}$,
Eq. (17) to compute homogenized stresses
$\overline{{\boldsymbol{P}}}_{\mathrm{m},i}$, and $w_{i}$ and $J_{i}$ result
from the adopted integration rule.

The results are summarized in terms of $\eta_{\mathrm{rms}}$ in Fig. 12, where
three aspects of the SI methodology have been tested for the case of only a
single integration point positioned at the specimen’s centre, i.e., at
$\vec{X}_{\mathrm{M}}=\vec{0}$. First, the influence of the macroscopic strain
field ${\boldsymbol{B}}(\vec{X}_{\mathrm{M}})$ is investigated by considering
four options:

  1. 1.

Strain field obtained from macroscopic IDIC, which correlates inside
$\Omega_{\mathrm{roi}}^{\mathrm{M}}$ with the non-linear homogeneous hyper-
elastic Neo–Hookean constitutive law of Eq. (9) and the mesh of Fig. 7a,
abbreviated SIidic;

  2. 2.

Macroscopic GDIC with FE hat-shaped functions, employing quadratic triangular
elements and the mesh shown in Fig. 13b, referred to as SIfe;

  3. 3.

Macroscopic GDIC with globally-supported smooth polynomials of fifth order,
abbreviated as SIpoly;

  4. 4.

And a macroscopic homogeneous virtual strain field of the form

${\boldsymbol{B}}(\vec{X})=\frac{2u_{\mathrm{D}}}{L}\vec{e}_{1}\vec{e}_{1},$
(27)

which in this simple case effectively provides normalization by the
experimentally applied homogenized stress field
$\sigma_{\mathrm{exp}}=F_{\mathrm{exp}}/W$, i.e.,

$\alpha^{\star}=\frac{\sigma_{\mathrm{exp}}}{\overline{P}_{\mathrm{m},11}},$
(28)

referred to as SIvf (recall also the discussion of point (iv), Eq. (23), on
the relation of SI to VFM).

The results corresponding to the normalization through individual strain
fields (a)–(d) are presented in Figs. 12a–12d. The best normalization in terms
of $\eta_{\mathrm{rms}}$ is achieved by the virtual strain field, i.e., SIvf,
whereas the other options provide a lower accuracy. Note that results
presented for the strain fields (a)–(c) have been obtained for only one time
increment, which is not an optimal choice. A higher accuracy could be reached
with a time incrementation scheme.

The second aspect investigated is the influence of the MVE size
$\ell_{\mathrm{mve}}$, which has been assessed by considering
$\ell_{\mathrm{mve}}\in[4,20]d$. As can be observed in Fig. 12, for small MVE
sizes significant fluctuations in the achieved accuracy are present for all
four strain fields (due to the lack of MVE representativeness, cf. also Fig.
5), for larger MVE sizes ($\ell_{\mathrm{mve}}\geq 16$) fluctuations in
$\eta_{\mathrm{rms}}$ rapidly decay.

Finally, sensitivity of spatial positioning of the MVE is investigated by
considering five different positions depicted in Fig. 13a (with shifts
$\Delta\vec{X}=\pm 4d\vec{e}_{1}\pm 4d\vec{e}_{2}$), which might yield
different results due to mild inhomogeneity of the considered deformation
state of the specimen. Corresponding results are presented in Fig. 12 through
individual lines which, except for the SIfe case that is sensitive to local
GDIC fluctuations, reveal a typical error level with increasing
$\ell_{\mathrm{mve}}$.

In general we can conclude that the SI method provides a typical accuracy
below $1\,\%$ when the normalization through a virtual field is used,
approaching for large MVEs a lower bound emanating from the adopted uncoupled
methods (i.e., $0.3\,\%$ RMS). On the contrary, normalization through a
virtual field can only be performed reliably for simple geometries, in which
an appropriate virtual field can be established. This may not be possible for
complex geometries, for which inhomogeneous stress and strain fields result.
As an alternative, a normalization through a strain field obtained from the
macroscopic IDIC scheme can be used instead, providing a typical accuracy of
$\eta_{\mathrm{rms}}=3\,\%$.

Although the choice of a single integration point might seem overly
simplified, it minimizes the experimental efforts required to perform
identification. Multiple integration points could in principle be introduced
for more accurate predictions, which is especially important in cases in which
significantly varying macroscopic stress and strain fields are present. The
positions and weights of integration points should be chosen optimally with
respect to the integrand, i.e., the product
${\boldsymbol{P}}_{\mathrm{M}}(\vec{X}_{\mathrm{M}}):{\boldsymbol{B}}_{\mathrm{M}}(\vec{X}_{\mathrm{M}})$,
see Eq. (23). However, for the simple geometry chosen here for the
demonstration purposes (cf. Fig. 3), a single integration point positioned at
the centre of the specimen has already proven to be sufficient because of
approximately constant strain field. For completeness, multiple macroscopic
integration points have been considered (not reported here in full detail for
brevity), using Gauss integration rule with $n_{\mathrm{g}}\in\\{4,9\\}$
integration points for a rectangular domain. The results are in both cases of
higher accuracy compared to those of a single integration point, since in case
of multiple integration points the identified forces are a weighted average
over stresses evaluated at those integration points (or arithmetic average in
the case of SIvf because the strain field is constant). This also leads to
higher robustness of the identification in addition to higher accuracy. In
particular, the highest error for $n_{\mathrm{g}}=9$ and
$\ell_{\mathrm{mve}}=4d$ was $0.7\,\%$ RMS for SIpoly, whereas the lowest
error was $0.1\,\%$ RMS for SIvf. In both cases, this is a significant
improvement, cf. Fig. 12, although at the cost of additional experimental
effort.

(a) SIidic

(b) SIfe

(c) SIpoly

(d) SIvf

Figure 12: Accuracy of the Stress Integration (SI) method with respect to the
size of the MVE $\ell_{\mathrm{mve}}\in[4,20]d$. Four strain fields
${\boldsymbol{B}}$ for the normalization have been used, obtained through (a)
IDIC with homogeneous material properties, (b) GDIC with finite element
interpolation functions (cf. employed mesh in Fig. 13b), (c) GDIC with
polynomial functions of fifth order, and (d) spatially constant virtual strain
field of Eq. (27). Individual lines in each of the plots correspond to various
MVE spatial positioning, as indicated in Fig. 13a.

MVE 1
---
(a) $n_{\mathrm{g}}=1$

(b) FE-GDIC mesh

(c) FE2 mesh

Figure 13: (a) Position of Gauss integration point as employed in the Stress
Integration (SI) method, MVE size $\ell_{\mathrm{mve}}=4d$ (four shifted RVEs
are shown in grey). Macroscopic triangulation (b) used in FE-GDIC for
identification of the strain field ${\boldsymbol{B}}(\vec{X}_{\mathrm{M}})$,
using quadratic triangular elements, and (c) triangulation used in FE2 with
$n_{\mathrm{e}}=16$ elements.

###  7.3 Computational Homogenization

####  7.3.1 Methodology

Normalization by FE2 method, sketched in Fig. 2e and summarized in Proc. 7,
requires in contrast to the SI method only a single experimental MVE
observation. This is possible because deformation state of the microstructure
associated with each macroscopic quadrature point is computed numerically
rather than observed experimentally. This in turn allows through a macroscale
model to estimate macroscopic reaction forces to be compared against
experimental observations, and thus provides normalization of micromechanical
parameters. Overall, the methodology proceeds as follows. First, material
parameter ratios are extracted from a single MVE, using one of the uncoupled
methods. Subsequently, the same MVE is assigned to all macroscopic Gauss
integration points of a macroscopic model $\vec{\mathcal{M}}^{\mathrm{M}}$
(assuming, however, that microstructural properties do not significantly vary
in space). Periodic MVEs are adopted by using periodic MVE meshes, although
the microstructural morphology is not necessarily periodic. Next, an FE2
simulation is performed in order to obtain the macroscopic displacement field
and reaction forces, which provide normalization of the microstructural
material parameter ratios upon comparison with $\vec{F}_{\mathrm{exp}}$
through Eq. (13). Boundary conditions according to Fig. 3 are directly applied
to the macroscopic model, by prescribing horizontal displacements at the two
vertical edges to a given value while fixing all vertical displacements to
zero. Note that when normalization through Eq. (13) is not used, unscaled
material parameters are identified, which typically means that one of the
microstructural parameters is fixed to an arbitrary value.

Unlike the stress integration method, where the micro–macro loop is open and
no iterations take place (cf. Fig. 2d), computational homogenization requires
equilibration of the macroscale system, and hence knowledge of the homogenized
stresses $\overline{{\boldsymbol{P}}}_{\mathrm{m}}$ (recall Eq. (17)) as well
as homogenized constitutive tangents $\overline{\mathbb{C}}_{\mathrm{m}}$, if
a Newton solver is used (recall Eq. (10)). Due to the assumption on
periodicity, normalization through FE2 may suffer from a higher inaccuracy
compared to the stress integration method because of inexact displacements at
the MVE boundary. Assumption on separation of scales adopted in the first-
order FE2, i.e., periodicity along with homogeneity of the underlying
deformation ${\boldsymbol{F}}_{\mathrm{M}}$ over each MVE, may thus limit the
accuracy of the method. On the other hand, as the displacement field at the
macroscale is now available, the macroscopic image residual can be constructed
and the quality of the entire multiscale model can be assessed. When the
adopted MVE is not representative enough, or the microstructural constitutive
models are inadequate, one expects these deficiencies to emerge at the
macroscale level through the macroscopic image residuals.

Note that although the first-order computational homogenization scheme has
been adopted in this work as an example, any other multiscale approach (which
may be more relevant for a given microstructure) can be used as well,
including second-order computational homogenization or micro-morphic schemes
[see, e.g., Kouznetsova et al., 2004, Biswas and Poh, 2017, Rokoš et al.,
2018a].

List of procedures 7 Computational homogenization (FE2) used for
micromechanical parameter identification. 1. Obtain microscopic images inside
$\Omega_{\mathrm{fov}}^{\mathrm{m}}$ with high spatial resolution (for a
single MVE only). 2. Use one of the uncoupled methods to identify material
parameter ratios. 3. Build a mesh of the macroscopic specimen $\Omega$ and set
up a macroscale model $\vec{\mathcal{M}}^{\mathrm{M}}$. With each Gauss
integration point, $i=1,\dots,n_{\mathrm{g}}$, associate the experimentally
obtained MVE and perform FE2 simulation (using periodic BCs). 4. Minimize the
difference between macroscopic forces predicted by FE2 through Eq. (13) to
normalize material parameter ratios. 5. If macroscopic observations inside
$\Omega_{\mathrm{fov}}^{\mathrm{M}}$ are available in addition to Step $1$,
macroscopic residual images may help to reveal any shortcomings of the
multiscale model used.

####  7.3.2 Results

The performance of the computational homogenization scheme is tested for
various numbers of macroelements $n_{\mathrm{e}}\in\\{2,4,16,24,48\\}$, and
for two types of shape interpolation functions: linear and quadratic. In the
case of linear triangles, one integration point per element is used, whereas
in the case of quadratic triangles, three Gauss integration points are
considered. Regular triangulations with isoparametric elements are employed,
as shown in Fig. 13c for $n_{\mathrm{e}}=16$. Finer or coarser discretizations
use a similar mesh with right-angled triangles. The displacement field
$\vec{u}_{\mathrm{M}}(\vec{X}_{M})$ is obtained using these discretizations
which solve for the equilibrium of the macroscopic system, while all nodes on
the two vertical lateral edges are considered with known prescribed
displacements (for both linear as well as quadratic elements).

The resulting accuracy is shown in Figs. 14a and 14b in terms of
$\eta_{\mathrm{rms}}$, where a clear correlation exists between the non-
monotonic trend of $\eta_{\mathrm{rms}}$ data along the
$\ell_{\mathrm{mve}}$-axis and the MVE representativeness of Fig. 5a. The
relatively large errors indicate that the normalization step through the force
measurements is responsible for most of the errors observed, as the BE-IDIC
used at the microscale level is again within $0.3\,\%$ RMS error in terms of
material parameter ratios, even for only a single MVE. The results further
show that a relatively large kinematic freedom needs to be provided to the
macroscopic system (quadratic interpolation with relatively large number of
elements) in order to achieve good accuracy and to capture inhomogeneity of
the overall deformation. Average error for the finest macroscopic
triangulations correspond to $2$ and $3\,\%$ or RMS error for $48$ linear and
quadratic elements.

Because for FE2 the displacement field
$\vec{u}_{\mathrm{M}}(\vec{X}_{\mathrm{M}})$ is available at the macroscale,
the image residual inside $\Omega_{\mathrm{roi}}^{\mathrm{M}}$ can be
computed. This is shown for $n_{\mathrm{e}}=16$ and quadratic elements in Fig.
15, indicating that close to the corner points the residual is relatively poor
(values are as high as $40\,\%$). Finally, the macro-residual, i.e.,

$\frac{1}{2}\|f_{\mathrm{M}}(\vec{X}_{\mathrm{M}})-g_{\mathrm{M}}(\vec{X}_{\mathrm{M}}+\vec{u}_{\mathrm{M}}(\vec{X}_{\mathrm{M}}))\|^{2}_{\Omega^{\mathrm{M}}_{\mathrm{roi}}},$
(29)

is reported for each FE2 configuration in Figs. 14c and 14d, revealing a clear
decrease with increasing number of macro-elements $n_{\mathrm{e}}$. However,
except for this general trend, no correlation of the image residual (Figs. 14c
and 14d) with the accuracy of the FE2 method (Figs. 14a and 14b) can be
observed, meaning that exact reproduction of the macroscopic displacement
field does not necessarily guarantee accurate micromechanical material
parameter identification for finite separation of scales in random
heterogeneous microstructures.

(a) $\eta_{\mathrm{rms}}$, linear elements

(b) $\eta_{\mathrm{rms}}$, quadratic elements

(c) macroscopic grey residual, linear elements

(d) macroscopic grey residual, quadratic elements

Figure 14: Results for the computational homogenization (FE2) scheme in
connection with BE-IDIC for micromechanical parameter identification as a
function of the number of macro-elements $n_{\mathrm{e}}\in\\{2,4,16,24,48\\}$
and size of the MVE $\ell_{\mathrm{mve}}\in\\{4,6,8,10,12,14,16,18,20\\}d$.
RMS error $\eta_{\mathrm{rms}}$ for (a) linear, and (b) quadratic triangular
macro-elements. Macro residual of Eq. (29) for (c) linear, and (d) quadratic
triangular macro-elements.

$0.3$
---
$0.2$
---
$0.1$
---
$0.4$
---
$0.0$
---
$0.3$
---
$0.2$
---
$0.1$
---
$0.4$
---
$0.0$
---

Figure 15: Macroscopic image residual
$|f_{\mathrm{M}}-g_{\mathrm{M}}|(\vec{X}_{\mathrm{M}})/2^{8}$ (normalized by
dynamic contrast $2^{8}$) corresponding to computational homogenization (FE2)
with $n_{\mathrm{e}}=16$ quadratic triangular macro-elements and
$\ell_{\mathrm{mve}}=6d$ ($0.07$ RMS in grey values).

##  8 Comparison

###  8.1 Comparison of Individual Methods

A short overview of all methods is compiled in Tab. 4 along with parameters
such as size of the microscopic ROI ($\ell_{\mathrm{roi}}$), fully resolved
region ($\ell_{\mathrm{full}}$), MVE size used ($\ell_{\mathrm{mve}}$), and
the number of Gauss integration points ($n_{\mathrm{g}}$), or number of
elements ($n_{\mathrm{e}}$). These results have been chosen such that they
reflect the typical results of each of the multiscale methods, which are
compared to the DNS results serving as the reference. In three cases (DNS,
CMM, and BSM), the same size of the ROI region has been used
($\ell_{\mathrm{roi}}=4d$), whereas for SI and FE2 the size of the ROI is the
same as the MVE size, i.e. $\ell_{\mathrm{roi}}=\ell_{\mathrm{mve}}$.

The Concurrent Multiscale Method (CMM) provides a relatively stable accuracy
of approximately $7\,\%$ for the entire range of the fully resolved region
$4d\leq\ell_{\mathrm{full}}\leq 20d$, which is still in a multiscale regime.
Beyond this, once the entire height of the specimen is fully resolved (i.e.,
$\ell_{\mathrm{full}}\geq W=36d$), a much better accuracy of $0.2\,\%$ can be
achieved.

The Bridging Scale Method (BSM) has a comparable performance to CMM in spite
of the fact that all fluctuations at the MVE boundary
$\partial\Omega_{\mathrm{mve}}^{\mathrm{m}}$ are neglected (recall Fig. 11).
The overall accuracy corresponds to approximately
$\overline{\eta}_{\mathrm{rms}}\approx 8\,\%$ of RMS error for
$4d\leq\ell_{\mathrm{mve}}\leq 20d$, and the best result is
$\eta_{\mathrm{rms}}(\ell_{\mathrm{mve}}=16.6d)=1.8\,\%$ RMS. In contrast to
the CMM method, $\ell_{\mathrm{mve}}\geq 36d$ does not seem to yield a higher
accuracy of identification. BSM is generally highly sensitive to
$\ell_{\mathrm{mve}}$, which makes it impossible to choose an optimal
$\ell_{\mathrm{mve}}$, and hence errors as high as $20\,\%$ should be
expected. It is worth mentioning that BSM is quite similar to SIidic,
neglecting, however, heterogeneities at the MVE boundary
$\partial\Omega_{\mathrm{mve}}^{\mathrm{m}}$. This seems to be at the root of
its lower accuracy, as SIidic performs almost three times better.

The sequential multiscale method performs adequately, especially in
combination with a virtual strain field (SIvf), which achieves an accuracy as
low as $1\,\%$, although the material parameter ratios can be identified more
accurately through uncoupled methods (error as low as $0.3\,\%$ of RMS). The
normalization step is responsible for the increase in the error, mainly due to
the lack of MVE representativeness. The presented result is therefore
achievable under the condition that MVEs used are representative. The high
accuracy is explained by the fact that the maximum of experimentally observed
data is used (micro- as well as macro-images, and force and displacement
measurements at the macroscale), which reflect the actual underlying physics.
When the strain field obtained through a macroscopic IDIC is used, the overall
error increases to $3\,\%$ of RMS.

Computational homogenization (FE2) provides a relatively good accuracy while
saving experimental efforts compared to SI, as it only requires a single MVE
even for complex structures. On the contrary, relatively large computational
efforts (fine macro-discretization) need to be used to provide an accurate
normalization. Similarly to the SI method, high accuracy can be achieved only
with a representative MVE. Note that in that case (i.e.,
$\ell_{\mathrm{mve}}\geq 16d$ according to Fig. 16a), combined area of all
MVEs considered is comparable to the area of the entire specimen, when
considering $n_{\mathrm{e}}=16$ and linear triangular elements as shown in
Fig. 13c. Although computational effort scales directly with the number of
macroscopic Gauss integration points $n_{\mathrm{g}}$, this is still an
efficient approach, since only one MVE needs to be observed experimentally,
thus substantially saving experimental efforts in comparison to SI. Overall
result achieved corresponds to approximately $2\,\%$ of RMS.

Table 4: Typical RMS errors (corresponding to Eq. (14)) in the identified micromechanical parameters for all tested methodologies (as detailed in Sections 4–7) considered for $\ell_{\mathrm{mve}},\ell_{\mathrm{full}}\in[4,20]d$, and for two levels of image noise $\zeta$. Method | DNS | CMM | BSM | SIvf | FE2
---|---|---|---|---|---
$\zeta\ [\%]$ | $0$ | $2$ | $0$ | $2$ | $0$ | $2$ | $0$ | $2$ | $0$ | $2$
$\eta_{\mathrm{rms}}$ [%] | $0.03$ | $0.05$ | $7.0$ | $7.1$ | $8.0$ | $8.0$ | $1.0$ | $1.1$ | $2.0$ | $2.1$
Parameters | — | $\ell_{\mathrm{full}}$ | $=5d$ | $\ell_{\mathrm{mve}}$ | $=6.8d$ | $\ell_{\mathrm{mve}}$ | $=6.4d$ | $\ell_{\mathrm{mve}}$ | $=10d$
$\ell_{\mathrm{roi}}$ | $=4d$ | $\ell_{\mathrm{roi}}$ | $=4d$ | $\ell_{\mathrm{roi}}$ | $=4d$ | $\ell_{\mathrm{roi}}$ | $=6.4d$ | $\ell_{\mathrm{roi}}$ | $=10d$
— | — | — | $n_{\mathrm{g}}$ | $=1$ | $n_{\mathrm{e}}^{\mathrm{quad}}$ | $=48$

###  8.2 Image Noise Study

The robustness of individual multiscale methods is evaluated through an image
noise study by adding Gaussian white noise with a standard deviation
$\zeta\in[0,0.05]\,\%$ of the image dynamic range to all (virtual)
experimental observations. The force measurements are, however, assumed noise-
free. The obtained results are summarized in Fig. 16b for the configurations
reported in Tab. 4. The results show that whereas CMM and BSM methods are
practically insensitive to image noise (relative to their overall accuracy),
DNS, SIvf, and FE2 reveal a mild dependence on $\zeta$.

(a) comparison in multiscale regime

(b) image noise study

Figure 16: (a) Comparison of individual methods in the multiscale regime
($\ell_{\mathrm{mve}}\leq 20d$). (b) Accuracy in terms of the RMS error of Eq.
(14) achieved by the individual multiscale identification methods, expressed
as a function of added image noise $\zeta\in[0,0.05]\,\%$ of the dynamic image
contrast $2^{8}$ for the configurations of Tab. 4. The force measurements are
assumed noise-free.

##  9 Summary and Conclusions

In this paper, a comparison of various multiscale parameter identification
methodologies has been performed, aiming for micromechanical parameter
identification with Integrated Digital Image Correlation (IDIC). A theoretical
overview of the adopted techniques with their description was given, combined
with numerical assessments on the basis of virtual experiments. The
convergence and performance of individual methods was investigated, including
an image noise study to test the robustness. The main results of this
contribution can be summarized as follows:

  1. 1.

The Concurrent Multiscale Method is robust with respect to the size of the
fully resolved region and image noise. However, its accuracy is lower ($7\,\%$
RMS) than other multiscale techniques.

  2. 2.

The Bridging Scale Method suffers from the assumed homogeneous boundary
conditions along the adopted Microstructural Volume Element (MVE) boundary.
This results in an accuracy that slowly improves with increasing size of the
MVE reaching, nevertheless, a stable accuracy of $8\,\%$ RMS error.

  3. 3.

The uncoupled methods provide only micromechanical parameter ratios with an
accuracy that is within $0.3\,\%$ RMS error.

  4. 4.

The normalization of micromechanical parameters by Stress Integration provides
an overall RMS error of $1\,\%$. A high correlation with the
representativeness of the MVEs as well as a mild sensitivity to image noise
has been observed.

  5. 5.

Computational homogenization is of comparable accuracy to Stress Integration,
i.e., of $2\,\%$ RMS error. High requirements for the multiscale model in
terms of kinematic freedom need to be satisfied to achieve such a result
(i.e., a fine macroscopic mesh), although less experimental effort is
required. Some dependence on MVE representativeness and sensitivity with
respect to image noise is also observed.

Throughout this contribution, a 2D plane-strain configuration was considered,
neglecting any inaccuracies due to out-of-plane motion and subsurface effects.
Furthermore, the micro-structural morphology and constitutive model were
assumed to be known exactly, while in reality they may introduce significant
additional errors. Considerably lower accuracies of all tested methodologies
may, therefore, be expected in real in-situ experiments compared to the
results reported here for virtual experiments. We suggest that in such cases
the most accurate multiscale methodology would be used, although approaches
capable of addressing respective sources of inaccuracies should be considered
as well, including digital height correlation or digital volume correlation in
case of 3D or subsurface effects, for instance.

## Acknowledgements

The research leading to these results has received funding from the European
Research Council under the European Union’s Seventh Framework Programme
(FP7/2007-2013)/ERC grant agreement № [339392].

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