Content uploaded by Abhilash Sreekumar
Author content
All content in this area was uploaded by Abhilash Sreekumar on Dec 13, 2020
Content may be subject to copyright.
Multiscale VEM for the Biot consolidation analysis of
complex and highly heterogeneous domains
Abhilash Sreekumara, Savvas P. Triantafylloub,∗, Fran¸cois-Xavier B´ecotc,
Fabien Chevillottec
aCentre for Structural Engineering and Informatics, Faculty of Engineering, The University
of Nottingham, UK
bInstitute for Structural Analysis and Aseismic Research, School of Civil Engineering,
National Technical University of Athens, Greece
cMatelys - Research Lab 7 rue des Maraˆıchers Bˆatiment B 69120 Vaulx-en-Velin France
Abstract
We introduce a novel heterogeneous multi-scale method for the consolidation
analysis of two-dimensional porous domains with a complex micro-structure. A
two-scale strategy is implemented wherein an arbitrary polygonal domain can
be discretised into clusters of polygonal elements, each with its own set of fine
scale discretization. The method harnesses the advantages of the Virtual El-
ement Method into accurately capturing fine scale heterogeneities of arbitrary
polygonal shapes. The upscaling is performed through a set of numerically
evaluated multi-scale basis functions. The solution of the coupled governing
equations is performed at the coarse-scale at a reduced computational cost. We
discuss the computation of the multi-scale basis functions and corresponding
virtual projection operators. The performance of the method in terms of ac-
curacy and computational efficiency is evaluated through a set of numerical
examples for poro-elastic materials with heterogeneities of various shapes.
Keywords: consolidation, porous materials, multiscale finite
element method, virtual element method
∗Corresponding author
Email address: savtri@mail.ntua.gr (Savvas P. Triantafyllou )
Preprint submitted to Journal of L
A
T
E
X Templates December 13, 2020
1. Introduction
It is often in nature that physics evolve across highly heterogeneous, geo-
metrically complex, and multiscale deformable domains; examples pertain to
sound absorption and transmission in foams and fibrous materials [1, 2, 3, 4]
and fracture analysis of composite components [5]. Focal point of this work is5
the mechanical response of highly heterogeneous saturated poro-elastic domains
where material variability spans several length scales, within the context of the
Biot consolidation theory [6]. Pertinent applications involve large scale geome-
chanics [7, 8], reservoir modelling and subsurface flows [9], and tissue modelling
[10].10
Achieving numerical solutions for the Biot consolidation problem in highly
heterogeneous domains with standard discretization methods, .e.g, the FEM,
BEM [11], etc. necessitates the explicit resolution of all underlying hetero-
geneities so that i) material distributions are accurately accounted for and ii)
geometrical interfaces are accurately resolved. In computational terms, this can15
be prohibitively expensive. Multiscale modelling methods have been developed
over the years to accurately treat heterogeneous material distributions across
scales while reducing computational costs using the robust mathematical frame-
work of homogenisation [12]. These include volume averaging [see, e.g., 13],
analytical homogenization [14] and computational homogenization approaches20
[see, e.g., 15, 16], see, also, FE2methods [17].
Homogenization theory relies on the assumptions of scale separation and
periodicity. However such assumptions do not necessarily hold for the case of
highly heterogeneous domains. Hence, alternative methods that do not rely
on this assumption have been developed, such as multiscale finite volume [18]25
and multiscale finite element methods (MsFEM) [19, 20]. A comparison between
different multiscale approaches in the context of elliptic problems was performed
in [21].
The MsFEM relies on the notion of nested computational domains and the
evaluation of a numerical basis that maps quantities, i.e., displacements, from30
2
the one to the other. Contrary to FE2(see, e.g., in [17]) methods where a micro-
scale finite element mesh is attached to each coarse scale integration point, in
the MsFEM the coarse scale is fully spanned by the fine scale. Hence, the
MsFEM is more suited to highly heterogeneous domains where scales cannot be
fully separated. The MsFEM was based on the pioneering work of [22] and was35
further developed by, e.g., [23] to resolve flows in highly heterogeneous media.
The Coupling Multiscale Finite Element Method (CMsFEM) [24] was de-
veloped to resolve the coupled field fully saturated porous media consolidation
problem using a two-scale (meso-macro) approach. Meso-scale heterogeneities
are mapped to the macroscopic scale using numerically computed multiscale ba-40
sis functions. A thorough discussion on the computational gains of the MsFEM
and the CMsFEM is provided in [23, 24, 25]. A more specific comparison of
different multiscale finite element approaches for composites and porous media
flows was done in [26].
In all the aforementioned multiscale finite element methods, scales are meshed45
with classical quadrilateral elements. Unfortunately, accounting for complex
meso-scale morphologies using such elements necessitates quite fine discretiza-
tions, rendering the computation of these multiscale basis functions expensive.
Optimization of the meshes involved could significantly improve the performance
of the method. To do this, one requires numerical methods that can handle more50
flexible element geometries.
Polygonal finite elements (PFEM) [see, e.g., 27, 28, 29] are used in fluid me-
chanics [30], contact mechanics [31], computational fracture and damage mod-
elling [32, 33, 34] and topology optimization [35], where one encounters complex
inclusion and interface geometries. Analytical basis functions [36] are employed55
over simplex polyhedra. Numerical approaches are necessary for computing
basis functions for non-convex domains such as maximum-entropy [37] and har-
monic [38] shape functions. This can significantly drive up computational costs
especially in the case of non-linear problems [39, 40]. The Virtual Element
Method (VEM) [41, 42, 43, 44, 45, 46, 47] is a recently developed numerical60
method that specifically addresses these limitations.
3
The VEM has found extensive application in linear elasticity [48, 49, 50, 51],
topology optimization [52], modelling of plates and shells [53, 54, 55], linear
and ductile fracture mechanics [56, 57, 58], contact mechanics [59], homogeniza-
tion of fiber reinforced composites [60], geomechanics and porous media flows65
[61, 62] and mixed VEM finite-volume discretization of Biot poromechanics
[63]. The scheme has recently been extended to account for curved geometries
[64, 65, 66]. The method naturally emerged from advances in Mimetic Finite
Difference (MFD) methods [see, e.g., 67, 68, 69, 70, 71, 72]. MFDs, when used
in conjunction with the Finite Element Method, seek to model trial and test70
functions spaces without resorting to explicit representations of basis functions
over the element interior. When extended to non-standard element geometries,
the accuracy of the method is improved by enriching the function spaces with
possibly non-polynomial expressions. The basis functions, which are allowed
to assume complex non-polynomial forms, are implicitly defined through care-75
fully chosen degrees of freedom. This implicit representation does away with
the problem of analytically or numerically deriving basis functions over complex
element domains. A significant point of departure of the VEM from MFDs lies
in VEM’s attempt to preserve polynomial accuracy over element boundaries
[41]. This allows for extension to more generalized inter-element continuity and80
conformity requirements [73]. The authors have introduced a multiscale VEM
formulation for elasto-statics, where the VEM has been introduced within a
multiscale setting considering the case of regular coarse element domains only
[74]. Very recently, the VEM has been employed within a mixed-formulation
setting to address elliptic problems [43, 75, 76]. Furthermore, a three field VEM85
formulation for the Biot consolidation equations has been presented in [77].
In this work, we harness the merits of the VEM in accurately resolving com-
plex material interfaces and develop a novel Coupled Multiscale Virtual Element
Method (CMsVEM) for the consolidation analysis of highly heterogeneous de-
formable domains across multiple length scales. To achieve this, we recast the90
three field VEM formulation for the Biot consolidation equations [77] within an
engineering context and originally employ it to compute fine scale Representa-
4
tive Volume Element (RVE) state matrices. Contrary to the work of [63], we
employ the VEM to resolve both the solid and pore-pressure governing equa-
tions. Further to the methodology provided in [74], the proposed CMsVEM95
is specifically designed to treat the generic case of arbitrary polygonal coarse
element geometries. Using this novel approach, we derive multiscale basis func-
tions to upscale highly heteregoneous porous domains and perform the solution
procedure in the time domain at the macroscopic scale at a reduced computa-
tional cost. We investigate the potential merits and bottlenecks of the proposed100
scheme in terms of solution accuracy and computational merits. The influence
of the type of boundary conditions used to evaluate the multiscale basis is also
examined.
The rest of this manuscript is structured as follows. In Section 2, the gov-
erning equations and the VEM formulation for the Biot consolidation problem105
for fully saturated poroelastic media is presented. The upscaling procedure and
associated kinematical constraints used in deriving a CMsVEM is discussed in
Section 3. Numerical examples are provided in Section 4 to verify the method
and assess its efficiency when comparted to the standard FEM, VEM, and the
CMsFEM. Concluding remarks and future outlooks are provided in Section 5.110
2. Preliminaries
2.1. Problem Statement
The case of an arbitrary continuous two-dimensional porous domain Ω ⊂R2
with a domain boundary Γ is considered as shown in Fig. 1a. The domain is
subjected to a set of prescribed displacements ¯
uon Γu, enforced pressures ¯p115
on Γp, applied tractions ¯
ton Γtand applied volume fluxes ¯
qon Γqsuch that
Γu∩Γt=∅and Γp∩Γq=∅. The domain is also subjected to body forces b
and a source/sink term Q.
Considering the case of a linear elastic material, small strains, isothermal
conditions, and neglecting inertial effects, the governing equations of the con-
5
Figure 1: Schematic representation of a two dimensional domain Ω with boundary ∂Ω. (a)
Essential and natural boundaries for the solid phase ¯
uand ¯
tare prescribed on Γuand Γt
respectively. Similarly, the relevant boundaries for the fluid phase ¯p and ¯
qare applied on Γp
and Γqrespectively. (b) The discretized domain This decomposed into polygonal elements.
solidation problem assume the following form [24]
div(Dε(u)) = div(αBmp) −b(1a)
αB˙
εvol + Sε˙p = divk
γf
∇p + Q,(1b)
where uand p are the vectors of the solid skeleton displacements and the
pore-fluid pressures, respectively.120
In Eq. (1a), ε(·) is the strain operator that under the assumption of small
strains assumes the following form
ε(u) = 1
2(∇u+ (∇u)T),(2)
and mis the identity tensor in Voigt notation, i.e., m=h110iT. The
quantity ˙
εvol represents the rate of volumetric strain, i.e., ˙
εvol =mTε(u). Fur-
thermore, Dis the 2-D elastic constitutive matrix and αBis the Biot’s coefficient.
In Eq. (1b), Sεis the storage coefficient and k,γfare the specific permeabil-
ity and pore-fluid specific viscosity, respectively. Finally, div(·) and ∇(·) denote
the divergence and gradient operators, respectively and ˙
(·) denotes differentia-
tion with respect to time. The storage coefficient Sεis evaluated through the
6
following expression,
Sε= nβ+ (αB−n)Cs,(3)
where β, Csare the pore-fluid and solid-grain compressibility, respectively.
The linear elastic constitutive equations for the solid phase are expressed as
σ=Dε,(4)
where σdenotes the Cauchy stress tensor. The constitutive relation for the fluid
phase is expressed through the static Darcy law for a single phase flow through
a porous medium
q=k
γf
∇p,(5)
where qdenotes the specific discharge, i.e., the specific volume of pore-fluid
exiting a control volume; this is expressed as
q= n(vf−vs),(6)
with vfand vsbeing the velocities of the pore-fluid and the solid-skeleton,125
respectively.
The coupled system of Eqs. (1) is supplemented by the following set of initial
and boundary conditions
u=u0,p = p0,in Ω −initial conditions (7a)
u=¯
uon Γu,p = ¯p,on Γp−enforced boundary conditions (7b)
t=¯
ton Γt,q=¯
q,on Γq−natural boundary conditions,(7c)
where u0and p0denote the initial displacement and pressure distributions over
the domain at time t= 0. The Dirichlet boundary values for solid and fluid
phases are represented by ¯
uand ¯p, respectively. The Neumann traction and
flux boundary values are contained in ¯
tand ¯
q, respectively.130
7
2.2. Virtual Element discretization
The weak form of the governing Eqs. (1) is derived accordingly as:
Find (u,p) ∈ V1× V2:= [H1(Ω)]2×[H1(Ω)] such that :
aε(u,v)D−a(ε,0)(p,v)αBm= fu(v)∀v∈ V1
a(ε,0)(˙
u,w)αBmT+ a0( ˙p,w)Sε+ a∇(p,w)k/γf= fp(w) ∀w∈ V2,
(8)
where a(·), f(·)are bilinear and linear functional operators and vand w are
appropriate test functions such that u,v∈ V1and p,w∈ V2. The spaces V1
and V2denote the spaces of admissible displacements and pressures, respectively.
These assume standard two-dimensional [H1(Ω)]2and one-dimensional H1(Ω)135
Hilbert Spaces, respectively.
In this work, the coupled weak form of Eq. (8) is discretized using the Virtual
Element Method to account for non-simplex and non-convex element domains.
Within this setting, the displacement field is approximated through the following
finite dimensional approximation, i.e.,
uh,vh∈ Vh1 ⊂ V1,(9)
where uhand vhare the discretized trial and test functions, respectively; these
are defined over a finite-dimensional subspace Vh1.
Similarly, the discretized pressure field trial phand test whfunctions are
defined accordingly as
ph,wh∈ Vh2 ⊂ V2,(10)
over the finite-dimensional subspace Vh2.
Using the discrete approximations introduced in Eqs. (9) and (10), the ab-
stract formulation of the discretized weak form is expressed
Find (uh,ph)∈ Vh1 × Vh2 such that :
aε(uh,vh)D−a(ε,0)(ph,vh)αBm= fu(vh)∀vh∈ Vh1
a(ε,0)(˙
uh,wh)αBmT+ a0( ˙ph,wh)Sε+ a∇(ph,wh)k/γf= fp(wh)∀wh∈ Vh2 ,
(11)
8
where a(·), f(·)are bilinear and linear functional operators. The individual op-
erators are defined as follows:
aε(uh,vh)D=Z
Ω
ε(uh)Dε(vh) dΩ (12a)
aε,0(ph,vh)αBm=Z
Ω
ε(vh)αBmphdΩ (12b)
aε,0(˙
uh,wh)αBmT=d
dtZ
Ω
ε(uh)αBmTwhdΩ (12c)
a0( ˙ph,wh)Sε=d
dtZ
Ω
phSεwhdΩ (12d)
a∇(ph,wh)k/γf=Z
Ω
∇(ph)k
γf
∇(wh) dΩ (12e)
fu(vh) = Z
Γt
¯
t·vhdΓ + Z
Ω
b·vhdΩ,(12f)
fp(wh) = −Z
Γq
¯
q·whdΓ + Z
Ω
QwhdΩ.(12g)
The nature of the finite-dimensional subspaces Vh1 and Vh2 chosen for the140
discrete problem varies slightly between the classical FEM and VEM. The FEM
approach allows for solutions over simplex elements where the basis functions
are explicitly expressed. To extend this approach to account for non-simplex,
non-convex element domains, certain conditions on the approximating subspace
need to be relaxed. In particular, one allows for a more flexible discretization of145
Ω into nel non-intersecting polygonal elements Ki,i = 1,...,nel with arbitrarily
defined edges and convexities. The same condition on completeness is required,
i.e. Ω ≈ Th1=S
Ki∈Th
Kias illustrated in Fig. 1b.
To accommodate for such arbitrary elements, the necessary finite-dimensional
space needs to be enlarged, i.e., a weaker definition that allows for non-polynomial
function definitions over the element interior, is required. The global virtual
1The parameter h is interpreted as the maximum diameter of all elements contained in Th.
9
space of kth-order VVEM
his defined as
VVEM
h={v∈[H1(Ω) ∩C0(Th)] : v|K∈ VK
h(K),∀ K ∈ Th},(13)
where the local virtual space VK
his defined over an element Kas
VK
h(K) = {v∈[H1(K)∩C0(K)] : v,i|e∈Pk(e) ∀e∈∂K;
∆v,i|K∈Pk−2(K),for i= 1,2}.(14)
The virtual element spaces Vh1 and Vh2 now assume the following form:
Vh1 = [VVEM
h]2;Vh2 =VVEM
h.(15)
Based on the aforementioned, we consider a polygonal element Kof Nv
number of edges and vertices, and area |K|, with arbitrarily chosen polynomial150
order k ≥1, as shown in Fig. 2; the corner nodes are represented by νj,j =
1,...,Nv. Each edge ej,j = 1,...,Nvconnects nodes νjand νj+1 and contains
k−1 internal nodes per edge, which are denoted by νe.
Figure 2: Conventions adopted for computing barycentric coordinates over a polygonal ele-
ment. The nodes are shown for a k = 2 pentagonal virtual element
The bilinear and linear functional forms used in Eq. (11) can now be com-
puted through assembling local element-wise operators as shown in Eqs. (16).
10
aε(·,·) = X
K∈Th
aε
K(·,·),(16a)
a∇(·,·) = X
K∈Th
a∇
K(·,·),(16b)
a0(·,·) = X
K∈Th
a0
K(·,·),(16c)
a(ε,0)(·,·) = X
K∈Th
a(ε,0)
K(·,·),(16d)
fu(·) = X
K∈Th
fu
K(·),(16e)
fp(·) = X
K∈Th
fp
K(·).(16f)
The arguments of these functionals belong to either VK
h1(K)⊂ Vh1 (Th) or
VK
h2(K)⊂ Vh2 (Th). The functions belonging to VK
h1(K) and VK
h2(K) are not155
explicitly defined, They are defined implicitly through carefully chosen degrees
of freedom (DoFs)2. These DoFs are defined in Table 1, where Mk−2(K) and
[Mk−2(K)]2denote spaces containing scalar and vector valued monomials of
order k−2, respectively.
There are three primary operations performed on the discrete functions as160
detailed in Eq. (11), i.e., ε(·) contained in aε(·,·), ∇(·) contained in a∇(·,·), and
(·) contained in a0(·,·).
The operators ε(·) and ∇(·) cannot directly act upon functions belonging
to the virtual spaces VK
h1 and VK
h2 as they are not explicitly defined. Further,
performing numerical integration for all three cases can be computationally165
expensive. This is because high order quadrature rules are necessary to obtain
accurate results for non polynomial integrands.
To avoid this, three operation specific projectors, i.e., Πε
k, Π∇
kand Π0
kare in-
troduced to replace the ε(·), ∇(·) and (·) operators respectively. These operators
2The member functions of the virtual element spaces VK
h1 and VK
h2 are often referred to
in VEM literature as a canonical basis Φ. These basis functions are defined implicitly in a
barycentric fashion, i.e., Φi(xj) = δij where δij is the Kronecker Delta function.
11
DoF Type Location Vh1 Vh2
Number
of DoFs
Description Number
of DoFs
Description
Corner vertices of K2Nv
uh(νj),
j = 1,...,Nv
Nv
ph(νj),
j = 1,...,Nv
Edge
internal boundary
points on each
edge of K
2Nv(k −1)
uh(νe
j),
j = 1,...,k−1
for each edge
Nv(k −1)
ph(νe
j),
j = 1,...,k−1
for each edge
Area
Moment
point lying in
interior of
domain K
2k(k−1)
2
1
|K| R
K
uh·mdK
∀m∈[Mk−2(K)]2
k(k−1)
2
1
|K| R
K
ph·mdK
∀m∈Mk−2(K)
Table 1: Degrees of Freedom for VK
h1 and VK
h2. For Area moment, the monomials belong to
[Mk−2]2and Mk−2spaces, respectively.
project the virtual functions onto an appropriate scalar, or vector polynomial170
space, denoted by [Pk(K)]d,d = 1,2, respectively .
This approximation induces additional error into the formalism. Minimiz-
ing the influence of this error is critical to the performance of the VEM. The
projectors are defined to this end using the following optimality criteria:
Πε
k:VK
h1(K)→[Pk(K)]2:= aε
K(uh−Πε
kuh,m) = 0,∀uh∈ VK
h1(K),m∈[Pk(K)]2,
(17a)
Π∇
k:VK
h2(K)→Pk(K) := a∇
K(ph−Π∇
kph,m) = 0,∀ph∈ VK
h2(K),m∈Pk(K),
(17b)
Π0
k:VK
h2(K)→Pk(K) := a0
K(ph−Π0
kph,m) = 0,∀ph∈ VK
h2(K),m∈Pk(K).
(17c)
The criteria enforced in these definitions ensure that the errors arising from
these projections, i.e., uh−Πε
kuh,ph−Π∇
kphand ph−Π0
kphare energet-
ically orthogonal to the approximating subspaces, [Pk(K)]2and Pk(K), respec-
12
tively. It follows that the energies associate with these bilinear functionals are175
computed exactly, despite the simplifying assumptions introduced by the poly-
nomial approximations. This property is referred to in standard VEM literature
as polynomial k-consistency [41].
The approximating subspaces, [Pk(K)]2and Pk(K) are spanned by scaled
monomials belonging to [Mk(K)]2and Mk(K), respectively. These monomial180
spaces also contain members that contribute zero energy to aε
K(·,·) and a∇
K(·,·),
e.g., ε([1,0]T) = [0,0,0]T,∇(1) = [0,0]T. These zero energy modes are specific
to the operator considered and are called the kernel of the operator. To avoid ill
conditioned matrices and consequent spurious results, these are excluded when
computing the projectors. The a0
K(·,·) has no zero energy modes.185
Following this reasoning, Eq. (17) can finally be established as follows:
Πε
k:= aε
K(uh−Πε
kuh,mj) = 0,∀uh∈ VK
h1(K),mj∈[Mk(K)]2\Kε(K),
(18a)
Π∇
k:= a∇
K(ph−Π∇
kph,mj) = 0,∀wh∈ VK
h2(K),mj∈Mk(K)\K∇(K),
(18b)
Π0
k:= a0
K(ph−Π0
kph,mj) = 0,∀ph∈ VK
h2(K),mj∈Mk(K),(18c)
where Kε(K) and K∇(K) belong to the kernels of zero energy modes of aε
K(·,·)
and a∇
K(·,·), respectively. The contents of these spaces can be derived using
kinematical decomposition relations mentioned in [76]. The monomials spaces
used for the VEM formulation are detailed in Appendix A. The procedure fol-
lowed in deriving the necessary virtual projectors Πε
k, Π∇
k, Π0
kis provided in190
Appendix B. Consequently the associated bilinear functionals aε
K(·,·), a∇
K(·,·)
and a0
K(·,·) are discussed within a multiscale context in the following section.
13
(a)
(b)
Figure 3: (a) Multiscale mesh with nM= 16 coarse nodes and nMel = 9 quadrilateral coarse
elements, each clustering its own fine quadrilateral mesh. (b) Multiscale mesh with nM= 22
coarse nodes and nMel = 9 polygonal coarse elements, each clustering its own fine polygonal
mesh.
3. Coupled Multiscale Virtual Element Methods for polygonal do-
mains
3.1. Overview195
The standard CMsFEM accounts for rectangular elements in the coarse scale
and quadrilaterals in the fine scale as shown in Fig. 3a; this limits the applica-
bility of the method especially for the case of inclusions or voids of an arbitrary
and typically non-convex geometry. In principle, one would be able to account
for such heterogeneities via a very fine finite element discretization; this would200
considerably increase the number of elements to be resolved at the micro-scale
14
Coarse Fine
Nodes Elements Nodes Elements
Fig. 3a nM= 16 nMel = 9 nm= 100 nmel = 81
Fig. 3b nM= 22 nMel = 9 nm= 100 nmel = 81
Table 2: Total number of coarse and fine-scale nodes and elements in the multiscale domains
illustrated in Fig. 3
hence countering the computational advantages of the multiscale procedure.
Our objective is to treat the most general case of arbitrarily shaped domains
at both the macro- and the micro-scale, see, also, Fig. 3b. To achieve this we
harness the flexibility of the the VEM to efficiently resolve non-simplex and205
non-convex geometries. The discretizations used at both scales in Fig. 3 are
summarized in Table 2.
In the proposed CMsVEM, each polygonal coarse-element KM(α), α =
1. . . nMel , clusters its own underlying fine-scale virtual element mesh comprising
micro-elements Km(i),i = 1 . . . nα
mel , where nMel and nα
mel denote the number of210
coarse-elements and micro-elements clustered in KM(α), respectively . This is
illustrated for the case of the coarse element M2M3M9M8M7M6in Fig. 4.
The resolved parameters at the fine scale are mapped to the coarse scale
where the solution of the governing equations is performed. The CMsVEM pro-
cedure is schematically depicted in Fig. 4. The upscaling procedure is achieved215
by numerically deriving appropriate multiscale basis functions to perform this
mapping. It is critical that these basis functions sufficiently capture all signifi-
cant static modes of the coarse element under consideration. In coupled porous
consolidation problems, this is equivalent to capturing deformation modes of
the solid skeleton and pressure gradient modes of the pore-fluid.220
Within this setting, two sets of multiscale basis functions are computed. One
set describes the solid skeleton displacements and the other captures the pore-
fluid pressure. For these evaluations, both phases are assumed to be decoupled
from each other. The basis functions for the displacement field are evaluated
15
through the solutions of the following homogeneous sub-problems
Find uh∈ Vh1(KM(α)) such that
aε(uh,vh)D= 0 ∀vh∈ Vh1 (KM(α)).
(19)
Similarly, the fluid phase multiscale basis functions are evaluated as
Find ph∈ Vh2(KM(α)) such that
a∇(ph,wh)k/γf= 0 ∀wh∈ Vh2 (KM(α)).
(20)
These equations are subjected to Dirichlet constraints, which are imposed
at the coarse element boundary. In CMsFEM the constraints are either linear
or periodic in character. Periodic boundaries are not possible in polygonal
RVEs. Alternately, a reduced version of the cell problems Eqs. (19) and (20)
are solved at the boundary called oscillatory boundary conditions [78]. The225
implementation of linear and oscillatory boundary conditions is discussed in
Section 3.3. The accuracy of the method depends heavily on the ability of
the constraints to satisfactorily reflect the physical behaviour of the problem.
When encountering comparable coarse and fine length scales in heterogeneous
problems, one can artificially enlarge the sampling domain to control resonance230
errors through oversampling methodologies [25] .
3.2. Virtual fine-scale state matrices
The bilinear forms defined over the coarse-element domain KM(α)in Eqs. (19)
and (20), respectively can be assembled from individual fine element contribu-
tions as shown in Eq. (16):
aε(uh,vh) =
nα
mel
X
i=1
aε
Km(uh,vh),∀(uh,vh)∈ VK
h1(Km(i) )⊂ Vh1(KM(α)) (21a)
a∇(ph,wh) =
nα
mel
X
i=1
aε
Km(ph,vh),∀(ph,wh)∈ VK
h2(Km(i) )⊂ Vh2(KM(α)) (21b)
Employing the VEM, the solid phase bilinear form at the micro-scale is ex-
pressed as
aε
Km(uh,vh) = aε
Km(uh−Πεm
kuh) + Πεm
Kuh),(vh−Πεm
kvh) + Πεm
kvh)(22)
16
Figure 4: Schematic representation of the CMsVEM upscaling procedure.
and the fluid-phase bilinear form becomes
a∇
Km(ph,wh) = a∇
Km(ph−Π∇m
kph)+ Π∇m
Kph),(wh−Π∇m
Kwh)+ Π∇m
kwh),(23)
where Πεm
kand Π∇m
krepresent the projectors discussed in Section 2.2.
Expanding Eqs. (22) and (23), exploiting the symmetry properties of bilinear
functionals and the orthogonality conditions laid out in Eq. (18), one obtains
the following relations for the solid phase
aε
Km(uh,vh) = aε
Km(Πεm
kuh,Πεm
kvh)
|{z }
solid phase consistency term
+ aε
Km(uh−Πεm
kuh,vh−Πεm
kvh)
|{z }
solid phase stability term
,(24)
and the fluid-phase
a∇
Km(ph,wh) = a∇
Km(Π∇m
Kph,Π∇m
Kwh)
|{z }
fluid phase consistency term
+ a∇
Km(ph−Π∇m
Kph,wh−Π∇m
Kwh)
|{z }
fluid phase stability term
,
(25)
respectively. The corresponding fine scale VEM expressions for a0(ph,wh) and
a(ε,0)(uh,wh) are derived in a similar manner and are omitted for brevity.235
Remark 1. The consistency terms comprise entirely polynomial terms and hence
can be computed analytically. However, this is not coercive. The stability term is
17
introduced to overcome this rank-deficiency. Any bilinear form satisfying basic
coercivity and stability properties can be taken up as the stability term. It is de-
signed to reduce to zero over polynomial subspaces. The stability terms contain240
non-polynomial integrands uh,vh, phand whthat have no explicit expression
over the element domain, therefore an exact computation is impossible. Ob-
taining close numerical approximations require higher order numerical quadra-
ture rules, rendering the procedure computationally expensive. Conversely, the
stability terms can be estimated with easy to compute forms that approximate245
the energy contributed by higher order modes. The additional error introduced
through this approximation is chosen such that optimal error convergence rates
are still achieved.
Based on the approximation of the stability term, the expression of the solid-
phase element-wise bilinear operator at the micro-scale assumes the following
form
aε
Km(uh,vh)≈aε
Km(Πεm
kuh,Πεm
kvh) + Sε
Km(uh−Πεm
kuh,vh−Πεm
kvh),(26)
where Sε
Km(·,·) denotes the stability term approximation.
The corresponding element-wise approximations for the fluid phases are ex-
pressed as
a∇
Km(ph,wh)≈a∇
Km(Π∇m
Kph,Π∇m
Kwh) + S∇
Km(ph−Π∇m
Kph,wh−Π∇m
Kwh)
(27a)
a0
Km(ph,wh)≈a0
Km(Π0
kph,Π0
kwh) + S0
Km(ph−Π0
kph,wh−Π0
kwh),(27b)
where S∇
Km(·,·) and S0
Km(·,·) denote the stability term approximations for250
a∇
Km(·,·) and a0
Km(·,·), respectively.
Finally, the VEM approximation for the coupling term is expressed as
a(ε,0)
Km(uh,wh)≈a(ε,0)
Km(Π∇m
Kuh,Π∇m
Kwh) + S(ε,0)
Km(uh−Π∇m
Kuh,wh−Π∇m
Kwh),
(28)
where S(ε,0)
Kmis the corresponding stability term. The choice of these stability
terms are not unique. One is referred to [79, 80, 81] for an extensive discussion
on the properties of stability terms.
18
Expanding Eqs. (26)-(28) and re-writing in matrix form following Eq. (12),
the following set of state-matrices is eventually defined at the micro-scale, i.e.,
Kel,α
m(i) ≈aε
Km(uh,vh)D=Z
Km(i)
εΠεm
k(uh)TDεΠεm
k(vh)dK+SK
Km(uh,vh)
(29a)
Qel,α
m(i) ≈a(ε,0)
Km(uh,wh)αBm=Z
Km(i)
εΠεm
k(uh)TαmΠ0
k(wh)dK+SQ
Km(uh,wh),
(29b)
Hel,α
m(i) ≈a∇
Km(ph,wh)k/γf=Z
Km(i)
∇Π∇m
k(ph)Tk
γf
∇Π∇m
k(wh)dK+SH
Km(ph,wh)
(29c)
Sel,α
m(i) ≈a0
Km(ph,wh)Sε=Z
Km(i) Π0
k(ph)TSεΠ0
k(wh)dK
|{z }
consistency
+SS
Km(ph,wh)
|{z }
stability
.
(29d)
where the expressions for the consistency and stability state matrices are pro-255
vided in Appendix C and Appendix D, respectively.
These fine scale matrices can be assembled over the coarse element domain
to provide the RVE state matrices using a direct approach:
Kα
m=
nmel
A
i=1 Kel,α
m(i),Qα
m=
nmel
A
i=1 Qel,α
m(i),Hα
m=
nmel
A
i=1 Hel,α
m(i),Sα
m=
nmel
A
i=1 Sel,α
m(i).
(30)
The state-matrices defined in (30) are used to evaluate the multiscale basis
functions as discussed in Section 3.3.
Remark 2. A virtual element formulation is not necessary for the load vectors
for our upscaling purposes. This is so because while the state matrices are evalu-260
ated at the fine scale for an RVE, the load vectors, in the absence of source/sink
terms and body forces, can directly be evaluated over the boundary at the coarse
scale thus rendering a virtual element approach unnecessary.
19
3.3. Constructing multiscale basis functions
The micro-nodal field variables uα
mand pα
mare mapped to the associated
coarse-nodes using the following relations:
uα
mx,i =
nM
X
J=1
Nu
iJxxuMx,J +
nM
X
J=1
Nu
iJxyuMy,J
uα
my,i =
nM
X
J=1
Nu
iJyxuMx,J +
nM
X
J=1
Nu
iJyyuMy,J
pα
m,i =
nM
X
J=1
Np
iJpM,J ,
(31)
where umx,i, umx,i and pm,i , i = 1 . . . nα
mare the displacement and pressure
components of the ith micro-node, nα
mis the number of micro-nodes within the
coarse-element αand nMis the number of coarse-nodes belonging to the αth
coarse-element. The terms uMx,J, uMy,J and pM,J denote the displacement and
pressure components at the Jth macro-node of the αth coarse-element. The
multiscale basis functions Nu
iJxx, Nu
iJxy, Nu
iJyx, Nu
iJyy and Np
iJ interpolate the
fine-scale displacements and pressures, respectively. The relations in Eq. (31)
hold only if:
nM
P
J=1
Nu
IJxx = 1
nM
P
J=1
Nu
IJxy = 0
,I = 1 . . . nM,
nM
P
J=1
Nu
IJyx = 0
nM
P
J=1
Nu
IJyy = 1
nM
P
J=1
Np
IJ = 1
(32)
The RVE specific fine-element nodal displacements uα
m= [umx,umy]T, and
pressures pα
mare associated with the corresponding coarse-element field variables
through the following equations, i.e.,
uα
m(i) =Nu
m(i)uM(α)(33a)
pα
m(i) =Np
m(i)pM(α),(33b)
where uα
m(i) and pα
m(i) denote the displacement and pressure vectors for the265
ith fine-element in the αth element. The arrays Nu
m(i) and Np
m(i) represent the
multiscale basis functions mapping the αth coarse-element nodal displacements
uM(α)and pressures pM(α)to the fine-scale.
20
Collecting the contributions from each fine-element, Eq. (33) can be ex-
pressed over the entire RVE:
uα
m=Nu
muM(α)(34a)
pα
m=Np
mpM(α),(34b)
where Nu
mand Np
mcorrespond to the coarse element multi-scale basis functions
for the displacement and pressure field, respectively. Each column of these
arrays corresponds to a possible static displacement or pressure mode of the
RVE. To compute these snapshots of the system in a manner consistent with
Eqs. (31) and (32), the discretized matrix forms of the boundary value sub-
problems in Eqs. (19) and (20) are solved over the RVE domain:
Kα
muα
m={∅} , on KM(α)
uS=¯
uIJ , on ∂KM(α)
, I = 1 . . . nM, J = 1,2 (35)
Hα
mpα
m={∅} , on KM(α)
pS=¯
pIJ , on ∂KM(α)
, I = 1 . . . nM, J = 1 (36)
where Kα
mand Hα
mare the RVE specific state matrices, which are assembled
from fine-element contributions using Eq. (29) and (30), respectively.270
In the CMsFEM, the prescribed displacements uSand pressures pSat the
RVE boundary are assigned linear or periodic kinematical constraints ¯
uand ¯
p.
For generalized polygonal RVEs (Fig. 5b), assigning periodic constraints is not
possible. Alternatively, oscillatory boundaries are used, i.e., reduced versions of
Eq. (35) and Eq. (36) are solved over the required edges of the RVE.275
In comparison to prescribing linear constraints (Fig. 5a), oscillatory condi-
tions allow for a less rigid enforcement of displacement and pressure profiles
along the RVE boundaries (Fig. 5b). Furthermore, the effect of material het-
erogeneities along the boundaries naturally emerges in the evaluation of the of
the corresponding displacement profiles hence providing a physically rigorous280
approach to the evaluation of the multiscale basis functions. The procedure
followed in assigning these kinematical constraints is provided in Appendix E
for the sake of completeness.
21
(a) (b)
Figure 5: An example 6-noded coarse element (α= 2) clustering nmel =9 fine elements and
nm= 21 fine nodes. (a) Linear kinematical constraints are prescribed over edges ΓM2M3and
ΓM3M9, (b) Oscillatory kinematical constraints are prescribed over edges ΓM7M6and ΓM6M2.
Remark 3. The terminologies ”Coarse-Element” and ”Representative Volume
Element (RVE)” are used interchangeably here. This is to remain consistent285
with the literature. A coarse-element, as employed in this work, is not truly
representative of the entire domain, and should therefore not be confused with
the classical notion of the RVE found in homogenization theory with scale sep-
aration.
3.4. Governing multiscale equilibrium equations290
The element-wise governing equations introduced in Eq. (11) are expressed
in matrix form as
Kel,α
m(i) −Qel,α
m(i)
0 Hel,α
m(i)
uα
m(i)
pα
m(i)
+
0 0
Qel,α T
m(i) Sel,α
m(i)
˙
uα
m(i)
˙
pα
m(i)
=
fu el,α
m(i)
fp el,α
m(i)
.(37)
where the state matrices Kel,α
m(i),Qel,α
m(i),Hel,α
m(i),Sel,α
m(i) are evaluated using the
VEM according to Eq. (29). The vectors fu el,α
m(i) and fp el,α
m(i) correspond to the
nodal forces and outflows, respectively at the ith micro-element.
Substituting the micro to macro mapping Eqs. (33) into Eq. (37) and pre-
multiplying the first row-set of equations by NuT
mand the second row-set by
22
NpT
mthe following equation is obtained
Kel
M(α),m(i) −Qel
M(α),m(i)
0 Hel
M(α),m(i)
uM(α)
pM(α)
+
0 0
Qel T
M(α),m(i) Sel
M(α),m(i)
˙
uM(α)
˙
pM(α)
=
fu el
M(α),m(i)
fp el
M(α),m(i)
,
(38)
where the stiffness and coupling matrices of the ith micro-element mapped at
the coarse element nodes are expressed as
Kel
M(α),m(i) =NuT
m(i)Kel,α
m(i)Nu
m(i) (39)
Qel
M(α),m(i) =NuT
m(i)Qel,α
m(i)Np
m(i).(40)
Furthermore, permeability and compressibility matrices are expressed as
Hel
M(α),m(i) =NpT
m(i)Hel,α
m(i)Np
m(i) (41)
Sel
M(α),m(i) =NpT
m(i)Sel,α
m(i)Np
m(i).(42)
Finally, the forcing terms assume the following form
fu el
M(α),m(i) =NuT
m(i)fu el,α
m(i) (43)
fp el
M(α),m(i) =NpT
m(i)fp el,α
m(i) (44)
for the nodal forces and outflows, respectively.
In principle, the coarse-element equilibrium equations could be expressed in
a form analogous to Eq. (37), i.e.,
Kel
M(α)−Qel
M(α)
0 Hel
M(α)
uM(α)
pM(α)
+
0 0
Qel,T
M(α)Sel
M(α)
˙
uM(α)
˙
pM(α)
=
fu el
M(α)
fp el
M(α)
,(45)
where Kel
M(α),Qel
M(α),Hel
M(α),Sel
M(α)denote the coarse-element state matrices295
and fu el
M(α),fp el
M(α)denote the coarse-element load vectors, respectively. Due to
the heterogeneous material distribution at the fine scale, explicit expressions for
these matrices do not exist. Yet, these can be evaluated on the basis of energy
equivalence between the coarse element domain Eqs. (45) and the upscaled fine-
element components Eqs. (39)-(44).300
23
Considering Eq. (45), the internal energy associated with each operator is
defined as
EK
int =Z
KM(α)
εT
MDεMdK=uT
M(α)Kel
M(α)uM(α)(46a)
EQ
int =Z
KM(α)
pT
MαBmεMdK=pT
M(α)Qel
M(α)uM(α)(46b)
EH
int =Z
KM(α)
∇pT
M
k
γf
∇pMdK=pT
M(α)Hel
M(α)pM(α)(46c)
ES
int =Z
KM(α)
pT
MSεpMdK=pT
M(α)Sel
M(α)pM(α)(46d)
where εMand pMcorrespond to the strain and pressure fields defined over the
coarse element domain.
The internal energy of the RVE is also additively decomposed into the con-
tributions of its underlying fine-elements, i.e.,
EK
int =
nmel
X
i=1 Z
Km(i)
εT
m(i) Dεm(i) dK=
nmel
X
i=1
uα T
m(i)Kel,α
m(i)uα
m(i) (47a)
EQ
int =
nmel
X
i=1 Z
Km(i)
pT
m(i) αBmεm(i) dK=
nmel
X
i=1
pα T
m(i)Qel,α
m(i)uα
m(i) (47b)
EH
int =
nmel
X
i=1 Z
Km(i)
∇pT
m(i)
k
γf
∇pm(i) dK=
nmel
X
i=1
pα T
m(i)Hel,α
m(i)pα
m(i) (47c)
ES
int =
nmel
X
i=1 Z
Km(i)
pT
m(i) Sεpm(i) dK=
nmel