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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

X

i=1

pα T

m(i)Sel,α

m(i)pα

m(i) (47d)

Equating Eqs. (46a) and (47a) the following expression is derived

uT

M(α)Kel

M(α)uM(α)=uT

M(α)

nmel

X

i=1 NuT

m(i)Kel,α

m(i)Nu

m(i)uM(α),(48)

that holds if and only if

Kel

M(α)=

nmel

X

i=1

Kel

M(α),m(i) (49)

24