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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
Int. J. Numer. Meth. Engng (2017)
Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.5476
Mixed Arlequin method for multiscale poromechanics problems
WaiChing Sun*,†, Zhijun Cai and Jinhyun Choo
Department of Civil Engineering and Engineering Mechanics, Columbia University in the City of New York,
614 SW Mudd, Mail Code: 4709 New York, NY, 10027, USA
SUMMARY
An Arlequin poromechanics model is introduced to simulate the hydro-mechanical coupling effects of uid-
inltrated porous media across different spatial scales within a concurrent computational framework. A two-
eld poromechanics problem is rst recast as the twofold saddle point of an incremental energy functional. We
then introduce Lagrange multipliers and compatibility energy functionals to enforce the weak compatibility
of hydro-mechanical responses in the overlapped domain. To examine the numerical stability of this hydro-
mechanical Arlequin model, we derive a necessary condition for stability, the twofold inf –sup condition for
multi-eld problems, and establish a modied inf–sup test formulated in the product space of the solution
eld. We verify the implementation of the Arlequin poromechanics model through benchmark problems
covering the entire range of drainage conditions. Through these numerical examples, we demonstrate the
performance, robustness, and numerical stability of the Arlequin poromechanics model. Copyright © 2016
John Wiley & Sons, Ltd.
Received 1 November 2015; Revised 9 November 2016; Accepted 10 November 2016
KEY WORDS: poromechanics; multiscale simulations; domain coupling; Arlequin method; inf–sup tests
1. INTRODUCTION
The mechanical behavior of a uid-inltrated porous solid is signicantly inuenced by the presence
and ow of uid in the pores. While the ow of the pore uid may introduce rate dependence to the
mechanical responses of a porous medium, the deformation of solid skeleton may also displace pore
uid and lead to build up of excess pore pressure. This hydro-mechanical coupling appears in a vari-
ety of natural and engineered materials, ranging from rocks, soils, and concretes to bones and soft
tissues. Reliable and efcient modeling of coupled hydro-mechanical processes in porous materials
is thus crucial to address many engineering problems, such as unconventional energy recovery, haz-
ards mitigation, and biomedical treatment [1–3]. Oftentimes, a major challenge in modeling these
coupled hydro-mechanical processes is the demand to capture physical phenomena occurring at mul-
tiple length scales that span several orders of magnitude. As an example, let us consider hydraulic
fractures in shale reservoirs. While these fractures are driven by uid inltration in nanometer-scale
pores, the information crucial for engineering application is the impact of these fractures on the
behavior and performance of kilometer-scale reservoirs. Nevertheless, due to its enormous compu-
tational costs, an explicit simulation of every grain-scale solid– uid interaction in such a large-scale
problem is impractical.
One feasible approach to incorporate small-scale dynamics into large-scale modeling is to make
use of multiscale coupling techniques [4–6]. A variety of multiscale modeling methods have been
developed and advanced over the past few decades in order to address interactions between micro-
scopic and macroscopic responses. According to Aubertin et al. [7], these multiscale methods can be
*Correspondence to: WaiChing Sun, Assistant Professor, Department of Civil Engineering and Engineering Mechanics,
Columbia University , 614 SW Mudd, Mail Code: 4709, New York, NY 10027.
†E-mail: wsun@columbia.edu
Copyright © 2016 John Wiley & Sons, Ltd.
W. SUN, Z. CAI AND J. CHOO
classied into three categories. The rst category pertains to methods that model inherent multiscale
characteristics by introducing a length scale through phenomenological laws. For instance, Fleck
and Hutchinson [8] incorporate a strain gradient term into constitutive models such that multiple
material length parameters can be dened for the eld equations corresponding to different domi-
nant mechanisms. Also, for capturing deformation bands much thinner than feasible mesh sizes, one
can insert enhanced basis functions or localization elements to embed strong or weak discontinuous
displacement elds [9–12].
The second category of multiscale methods is a class of hierarchical methods that incorporate
micro-structural information from unit cells to compute effective (homogenized) properties of coarse
(macroscopic) domains [13–18]. Kouznetsova et al. [19] present a gradient-enhanced homogeniza-
tion scheme which obtains macroscopic stress, strain measure, and their gradients from solutions of
boundary-value problems applied on representative volume elements. Ehlers et al. [20] describe a
homogenization procedure that upscales higher-order kinematics of particle ensembles to both con-
vectional stress measures and higher-order couple stresses. Liu et al. [15] apply a staggered nonlocal
scheme to introduce a physical length scale for hierarchically coupled discrete elements–nite ele-
ments modeling of granular materials in a corotational framework. Another related application of
homogenization-based multiscale methods has also been developed by several studies [4,5,21 – 23]
to conduct large-scale ow simulations using tomographic images. The performance of these hier-
archical multiscale methods relies on the existence of a representative elementary volume and the
design of sequential coupling schemes to establish an exchange of information across scales.
Multiscale methods in the third category are concurrent methods that apply ne-scale, computa-
tionally demanding models to local region(s) of high interest (e.g., crack tip), while use coarse-scale,
cost-efcient models elsewhere. Then, these methods extract high-resolution information in critical
region(s) performing concurrent simulations of physical processes across length scales. In these con-
current simulations, the ne, critical domain is connected to the coarse, non-critical domain either via
non-overlapped mortar interfaces (e.g., [24,25]) or overlapped handshaking domains (e.g., [26– 30]).
Previous work has successfully developed concurrent methods for coupling discrete and continuum
models (e.g., [7,28,31]), classical local and non-local elastic continua (e.g., [32,33]), and structural
elements with various mesh renements (e.g., [27]).
The major upshot of concurrent multiscale approach is that it enables computational resources
to be concentrated on region(s) of interest where important and complicated processes take place,
without neglecting the far-eld inuence. Such efcient allocation of computational resources can
signicantly improve our modeling capabilities for a wide spectrum of problems whereby cou-
pled hydro-mechanical processes occur across multiple spatial scales. Examples of these problems
range from needle insertion into biological tissues [34], bone fractures [2], to hydraulic fracturing
in unconventional reservoirs [35], and injection-induced seismic events [36]. Engineering designs
and predictions for those applications can be substantially improved if numerical models can ef-
ciently allocate computational resources to resolve the ne-scale information in the localized region
of interest without neglecting the far-eld inuences.
In this work, we extend a concurrent multiscale method to simulate the coupled hydro-mechanical
processes in uid-inltrated porous media across spatial scales. The proposed numerical model is
based on the Arlequin framework, which was rst proposed by Ben Dhia [26] as a general framework
for coupling different domains and models. The Arlequin framework has been successfully applied
to multiscale, multimodel simulations of a wide spectrum of multiscale solid mechanics problems
(e.g., [27,29,33,37,38]). The theoretical basis and numerical tools to predict the spatial stability have
been proposed in a few studies [28,33,38]. Each of these studies has proposed inf–sup tests for the
corresponding boundary value problems. However, to the best of our knowledge, there is not yet
any contribution dedicated to the extension of the Arlequin method to coupled hydro-mechanical
problems. The new contributions of this work include a compatibility energy functional for spatial
coupling of multiphase porous materials, a stability condition in the presence of both pore pressure
and domain coupling constraints, a combined inf–sup test derived to examine the spatial stability
using the product spaces of the displacement and pore pressure eld, and numerical experiments that
verify and demonstrate the robustness and accuracy of the Arlequin poromechanics model.
Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2017)
DOI: 10.1002/nme
MIXED ARLEQUIN METHOD FOR MULTISCALE POROMECHANICS PROBLEMS
The rest of this paper is organized as follows. In Section 2, we formulate the Arlequin poromechan-
ics model, introducing a time-discrete variational statement and Lagrange multipliers for domain
coupling. Subsequently, in Section 3, we derive a necessary condition for spatial stability of the
developed method as a twofold inf– sup condition, and propose a new type of inf– sup test to evaluate
the stability. In Section 4, we verify the developed Arlequin method in the entire range of drainage
condition through two numerical examples. Then, we present a more complex example that show-
cases the performance of the method for coupling different models—particularly an isogeometric
extended nite element model and a standard nite element model.
As for notations and symbols, bold-faced letters denote tensors; the symbol ‘·’ denotes a single
contraction of adjacent indices of two tensors (e.g., a·b=aibior c·d=cijdjk ); the symbol ‘:’ denotes
a double contraction of adjacent indices of tensor of rank two or higher (e.g., C∶𝝐e=Cijkle
kl);
the symbol ‘’ denotes a juxtaposition of two vectors (e.g., ab=aibj) or two symmetric second
order tensors (e.g., (𝜶𝜷)=ijkl ). Following the standard sign convention in mechanics, stress is
positive in tension.
2. MIXED ARLEQUIN FORMULATION FOR POROMECHANICS
This section presents a mixed Arlequin formulation for poromechanics problems. We begin by intro-
ducing the notion of domain partitioning that allows for the use of the Arlequin method. Governing
equations of a coupled poromechanics problem are then reviewed briey. To enforce the compat-
ibility of solid and uid motions between the sub-domains, we recast the coupled poromechanics
problem as a saddle point problem of an incremental energy functional, and then apply partition of
unity to the incremental energy density functional. Lastly, we derive the weak form and the matrix
form of the Arlequin formulation.
2.1. Domain partitioning
Consider a two-phase porous medium that occupies a domain B∈R3. To model critical and non-
critical regions differently, we partition the domain Binto a coarse sub-domain Band a sub-domain
Bsuch that B=B∪
B, see Figure 1. Note that the two domains are overlapped in Bc=B∩
B.
The coarse domain Binvolves non-critical processes that can be well simulated by relatively simple
models. Conversely, the ne domain
Bcontains critical regions and it is usually (much) smaller than
the coarse domain B. Therefore, we will concentrate more expensive, but more accurate models
(e.g., a more advanced constitutive model, ner/higher-order discretization) on the ne domain. It
is noted that even though the two models in the coarse and ne domains simulate the same physical
processes in the overlapped domain, they do not need to be identical. For notational consistency, we
shall denote quantities pertaining to Band
Bby (·) and
(·), respectively (see Sun and Mota [33] for
the same notation).
Figure 1. Partitioning of the domain Binto a coarse domain Band a ne domain
B. The coarse and ne
domains are overlapped in Bc.
Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2017)
DOI: 10.1002/nme
W. SUN, Z. CAI AND J. CHOO
The boundary of the entire porous medium Bis also partitioned as
B=B∪
B,(2.1)
where Bdenotes the boundary of the coarse domain with the unit normal vector n,and
B
the boundary of ne domain with the unit normal vector
n. Depending on the type of boundary
conditions, the coarse domain boundary Bis further decomposed as:
−uB: solid displacement boundary (Dirichlet).
−tB: solid traction boundary (Neumann).
−pB: uid pressure boundary (Dirichlet).
−qB: uid ux boundary (Neumann).
Similarly, the ne domain boundary
Bis also decomposed as
−
u
B: solid displacement boundary (Dirichlet).
−
t
B: solid traction boundary (Neumann).
−p
B: uid pressure boundary (Dirichlet).
−
q
B: uid ux boundary (Neumann).
The decompositions of Band
Bare subjected to the following restrictions
B=uB∪tB=pB∪qB,(2.2)
B=
u
B∪
t
B=p
B∪
q
B,(2.3)
∅=uB∩tB=pB∩qB,(2.4)
∅=
u
B∩
t
B=p
B∩
q
B.(2.5)
The solid displacement boundary of the entire domain is the union of the solid displacement bound-
aries of the coarse and ne domains, that is, uB=uB∪
u
B. Similarly, the union of the uid
pressure boundaries of the coarse and ne domains forms the entire uid pressure boundary, that is,
pB=pB∪p
B. Note that in both cases, the coarse and ne domains can have overlapped Dirichlet
boundaries, that is, uBc=uB∩
u
B≠∅and pBc=pB∩p
B≠∅. Similarly, the solid trac-
tion boundary of the entire domain is partitioned as tB=tB∪
t
Bwith an overlapped boundary
tBc=tB∩
t
B, and the uid ux boundary as qB=qB∪
q
Bwith an overlapped boundary
qBc=qB∩
q
B. To enforce the compatibility between the uid and solid motions of the coarse
and ne domains, we will introduce Lagrange multipliers to the sub-domains B⧵uB(for the solid
displacement) and B⧵pB(for the uid pressure).
Finally, the initial conditions at t=0aregivenby{uo,po}for the coarse domain and {
uo,po}for
the ne domain.
2.2. Field equations for poromechanics problems
In this work, our goal is to model coupled uid-diffusion–solid-deformation processes in uid-
saturated porous media. The formulation of the poromechanics model for uid-saturated porous
media has been well established in the literature. Readers interested in the details of the porome-
chanics theory are referred to standard texts such as [1,3,39]. Here, we provide a brief overview of
the subject for completeness.
In a nutshell, the continuum poromechanics model conceptualizes the uid-inltrated porous
media as a continuum mixture of the solid and uid constituents, each occupies a fraction of volume
in the macroscopic body. Provided that a representative elementary volume exists, the behavior of
porous media can be predicted by introducing proper constitutive laws for the solid skeleton and pore
uid ow as well as imposing conservation laws for linear momentum and mass as eld equations.
For simplicity, here we consider the case in which the pore space is lled up with a single-phase
uid (e.g., water). Therefore, in what follows, we consider a two-phase continuum composed of one
Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2017)
DOI: 10.1002/nme
MIXED ARLEQUIN METHOD FOR MULTISCALE POROMECHANICS PROBLEMS
solid and one uid constituents. In addition, the following assumptions are made to further simplify
the problem:
1. The isothermal condition holds.
2. The inertial effect is negligible.
3. There is no mass exchange between the solid and uid constituents.
4. Reynold’s number of the ow in the pore space is sufciently low such that convection is
negligible.
5. Darcy’s law is valid.
6. The deformation of the solid skeleton is elastic, isotropic, and innitesimal.
7. The effective stress principle is valid.
8. The pore space is fully saturated by a single type of uid.
The balance of linear momentum of the two-phase porous media then reads
·(𝝈′−Bp)+𝜸=𝟎,(2.6)
where 𝝈′is the effective stress, pis the pore pressure, and 𝜸is the body force vector. The second-
order tensor Bis a generalization of Biot’s coefcient in the isotropic elasticity case. For the linear
elasticity case, Bmay read (cf. [40]),
B=I−1
3Ks
Csk ∶I,(2.7)
where Ksis the intrinsic bulk modulus of the solid constituent, Csk is the fourth-order elasticity tensor,
and Iis the second-order identity tensor. Note that in the isotropic case, this expression simplies to
the classical expression of Biots coefcient B=BI=(1 −K∕Ks)Iwhere Kis the bulk modulus of
the solid matrix (cf. [3,41–43]). Thus, in this work, we consider the effective stress of the form
𝝈′=𝝈+BpI,(2.8)
where 𝝈is the total stress in the porous medium. When the solid deformation is innitesimal and
hyperelastic, constitutive law for the solid skeleton can be expressed as
𝝈′=W(𝝐)
𝝐,(2.9)
where W(𝝐) is the stored energy function of the solid skeleton. The innitesimal strain tensor of the
solid skeleton 𝝐reads,
𝝐=symu=1
2(u+Tu).(2.10)
The corresponding incremental form of the solid constitutive relation in (2.9) is given by
Δ𝝈′=D∶Δ𝝐(2.11)
where Dis a fourth-order tensor of tangent stiffness tensor. If the solid skeleton exhibits linear
isotropic elastic behavior, then D=Csk. For soils and soft rocks Biot’s coefcient B≈1, whereas
for hard rocks or biological materials Bis usually less than one. Finally, the body force 𝜸acting on
the mixture of the solid matrix and the pore uid is given by
𝜸=𝜸s+𝜸f=(1−f)sg+ffg,(2.12)
where fis the porosity (the volume fraction of the pore uid), sand fare the intrinsic densities
of the solid and uid constituents, respectively, and gis the gravitational acceleration. The balance
of mass with compressible uid and solid constituents reads,
1
Mp+B·
u+·q=0,(2.13)
Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2017)
DOI: 10.1002/nme
W. SUN, Z. CAI AND J. CHOO
where Mis Biot’s modulus and qis the seepage velocity. For fully saturated, isotropic porous
materials, Mmay be related to the intrinsic bulk moduli and porosity as follows [40,42,43]:
M=KsKf
Kf(B−f)+Ksf.(2.14)
If the ow in the pore space remains laminar and the uid movement is dominated by viscous forces,
then the seepage velocity and the gradient of the pore pressure can be related by Darcy’s law,
q=−
1
μk·(p−fg),(2.15)
where kis the effective permeability and μis the dynamic viscosity of the pore uid. In this
work, we shall assume isotropic permeability, that is, k=kI. Substituting this equation into (2.13)
leads to
1
Mp+B·
u−·1
μk(p−fg)=0.(2.16)
Observe that the two governing Eqs. (2.6) and (2.16) have two unknowns—the solid displacement
uand the pore pressure p. For this reason, this formulation is often referred to as a u∕pformulation
(cf. [1,16,44,45]). Alternatively, it is also possible to formulate a poromechanics problem in terms of
uid mass ux vector and solid displacement unknowns (cf. [46,47]) or in terms of Darcy’s velocity,
solid displacement, and pore pressure (cf. [1,48,49]). The application of the Arlequin framework for
the latter two formulations (i.e., u∕vand u∕v∕p) is not considered herein, and will be considered in
future studies.
2.3. Variational statement for poromechanics
The goal of this section is to derive a time-discrete weak form of the poromechanics problem, which
will then serve as the basis for the derivation of the Arlequin poromechanics model presented in
Section 2.4. In previous works, such as [1,45,50– 52], the nite element model for the poromechan-
ics problem is often formulated via a weight-residual argument. An alternative approach has been
explored by Biot [53] and Armero and Callari [10] who consider the porous medium an open sys-
tem in which eld equations can be recast as the Euler–Lagrange equation from a generalization
of d’Alembert’s principle. While both approaches may lead to the same set of eld equations, here
we adopt the latter approach in order to obtain an equivalent static problem between two incremen-
tal time steps before discretizing the spatial domain. An upshot of this approach is that it makes the
solutions of the equivalent static problem corresponding to an incremental saddle point energy func-
tional, hence allowing us to formulate the Arlequin poromechanics model via the energy blending
approach. Nevertheless, it should be noted that the Arlequin formulation may also be introduced by
using virtual work principle as the starting point, as performed in the pioneering work by Ben Dhia
[26] which rst introduced the Arlequin method. Our goal here is to derive a time-discrete energy
functional of which the corresponding Euler–Lagrange equation is the time-discrete weak form
of the eld equation presented in (2.6) and (2.16). In Section 2.4, we will show that the Arlequin
poromechanics problem can be formulated by applying partition of unity to the incremental energy
functional and introducing an additional energy functional to impose compatibility constraints in the
overlapped domain(s).
To derive a variational statement, we rst introduce two spaces for the trial functions for the
displacement and pore pressure elds, that is,
Vu={u∶B→R3u∈[H1(B)]3,uuB=upred},(2.17)
Vp={p∶B→Rp∈[H1(B)],ppB=ppred},(2.18)
Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2017)
DOI: 10.1002/nme
MIXED ARLEQUIN METHOD FOR MULTISCALE POROMECHANICS PROBLEMS
where H1denotes the Sobolev space of degree one, upred and ppred represent the prescribed bound-
ary values of the solid displacement and pore pressure, respectively. Accordingly, we also dene
admissible variations of displacement uand pore pressure pas
Vu={u∶B→R3u∈[H1(B)]3,uuB=0},(2.19)
Vp={p∶B→Rp∈[H1(B)],ppB=0}.(2.20)
To obtain an incremental energy functional of the poromechanics problem, we adopt an approach
used in other variational frameworks [54– 57] that performs temporal discretization rst and then
spatial discretization. In fact, this discretization sequence is the opposite to the typical sequence in
poromechanics formulations (e.g., [16,50,58–60]) whereby the spatial domain is discretized rst.
Yet, this typical sequence does not allow us to derive an incremental energy functional necessary for
the Arlequin formulation. Therefore, we rst discretize the time domain as [0,T]=∪
N
n=0[tn,tn+1]∈
R+. Then, our objective is the following: Given variables at the previous time step tn, obtain an
incremental update for the solid displacement and uid pressure at time step tn+1. Hereafter, we will
use the backward Euler method and assume that Biot’s coefcient B, Biot’s modulus M, and the
permeability kare constant.
We formulate a discrete Lagrangian such that the updated solid displacement and uid pressure
(un+1,pn+1 ) satisfying the time-discrete versions of (2.6) and (2.16) are the saddle point of the the
discrete energy functional. The total (discrete) free energy functional of the solid matrix Πsat time
tn+1 is the total free energy of the porous media subtracted by the total free energy contributed by the
pore uid of the same control volume [61],
[un+1,pn+1]n+1=s[un+1,pn+1]n+1−f[un+1,pn+1]n+1+ext[un+1,pn+1]n+1.(2.21)
Under the quasi-static and isothermal conditions, the internal energy of the solid matrix at time
t=tn+1 is given by [62,63]
s[un+1,pn+1]n+1=1
2∫B
(𝝈′
n+1)∶symun+1dV ,(2.22)
where Δt=tn+1 −tnis the time increment. The energy contribution of the pore uid at time t=tn+1
is given by
f[un+1,pn+1]n+1=f[un,pn]n+∫B
(pn+1−pn)2
2M+Bpn+1I∶(symun+1−sym un)dV,(2.23)
and the external work at t=tn+1 is
ext[un+1,pn+1]n+1=−
∫qB
q·npn+1d−∫tB
t·un+1d−∫B
𝜸·un+1dV,(2.24)
where
qand
tare the prescribed uid ux and traction, respectively. The dissipation due to seepage
of the pore uid at time t=tn+1 is approximated by the backward Euler method, that is,
[pn+1]n+1
n≈∫B
Δt
2μ(pn+1−fg)·k·(pn+1−fg)dV ,(2.25)
where the effective permeability tensor kis assumed to be symmetric and positive semi-denite. The
discrete Lagrangian H[un+1,pn+1 ]n+1 is therefore given by
H[un+1,pn+1]n+1=[un+1,pn+1]n+1−[un,pn]n−[pn+1]n+1
n.(2.26)
Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2017)
DOI: 10.1002/nme
W. SUN, Z. CAI AND J. CHOO
The rst variation that corresponds to the saddle point of the incremental energy functional with
respect to the incremental solution eld (un+1,pn+1 ) reads,
DH[u]n+1=∫B
(𝝈′
n+1∶symu−Bpn+1·u−𝜸u)dV −∫tB
t·ud, (2.27)
DH[p]n+1=∫B1
M(pn+1−pn)p+B·(un+1−un)p−Δt·1
μk·(pn+1+g)pdV
−∫qB
q·npd=0,
(2.28)
where the operator D(·) denotes the Gateaux derivative, that is, DH(u)= d
dH(u+u)=0.Note
that the system of equations listed in (2.27)–(2.28) may also be obtained by the weight-residual
method. In this case, the starting point is to use (2.6) and (2.16) as the strong form. Then, a weak form
of the linear momentum and mass conservation equations can be obtained using a weight-residual
argument and integration by parts with testing functions (𝜹u,p), followed by a discretization in the
temporal domain typically performed by a one-step or linear multi-step method (cf. [64]). For brevity,
this derivation is not repeated in this work. Interested readers may refer to [1,2,45,51,52,64,65] for
details.
2.4. Variational statement for Arlequin poromechanics
We now enforce the compatibility of the mechanical and hydrological states in the overlapped
domain. The Arlequin framework enforces the compatibility by suitably partitioning the energy of
the overlapped domain into the associated sub-domains. Following this approach, here we formu-
late a variational statement for the Arlequin poromechanics. For simplicity, we restrict our focus on
a sub-class of the Arlequin poromechanics problem where the overlapped domain consists of only
two models, which are referred to as coarse and ne models. In this case, the Arlequin model for the
poromechanics problem can be established by: (1) applying partition of unity to the coarse and ne
incremental energy functionals, and (2) introducing constraints and Lagrange multipliers to enforce
compatibility.
Recall the notation (·) for the coarse domain and
(·) for the ne domain. We dene the trial
spaces as
Vu={u∶B→R3u∈[H1(B)]3,uuB=upred}(2.29)
Vp={p∶B→Rp∈[H1(B)],ppB=ppred}(2.30)
V
u={
u∶B→R3
u∈[H1(B)]3,
u
uB=
upred}(2.31)
Vp={p∶B→Rp∈[H1(B)],ppB=ppred }.(2.32)
The admissible variations of displacement and pore pressure for the coarse and ne domains are
dened as
Vu={u∶B→R3u∈[H1(B)]3,uuB=0}(2.33)
Vp={p∶B→Rp∈[H1(B)],ppB=0}(2.34)
V
u={
u∶B→R3
u∈[H1(B)]3,
u
uB=0}(2.35)
Vp={p∶B→Rp∈[H1(B)],ppB=0}.(2.36)
Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2017)
DOI: 10.1002/nme
MIXED ARLEQUIN METHOD FOR MULTISCALE POROMECHANICS PROBLEMS
In the Arlequin framework, multiple models are used to simulate the same physical processes in
the overlapped domain. Accordingly, the energy of the system in the overlapped domain is partitioned
into the sub-domains by a weighting function ∶B→[0,1]. This weighting function can be chosen
as discontinuous, piece-wise constant or polynomials of various orders of completeness in space,
provided that partition of unity is satised [27,33]. For instance, in a one-dimensional problem, if
Bc=[a,b]∈Band B⧵Bc=[0,a[∈ B, then the weighting function can be dened as:
(x)=
0,for x∈B⧵Bc,
x
b−a−a
b−a,for x∈Bc,
1,for x∈
B⧵Bc.
(2.37)
For notational brevity, we dene 𝝋u={u,
u}and 𝝋p={p,p}. Then, the total free energy and viscous
dissipation are the linear combination of the coarse and ne energy functionals that are partitioned
via the weighting function ,thatis,
s
Ar[𝝋u,n+1,𝝋p,n+1]n+1=(1−)s[un+1,pn+1]n+1+
s[
un+1,pn+1]n+1,(2.38)
f
Ar[𝝋u,n+1,𝝋p,n+1]n+1=(1−)f[un+1,pn+1]n+1+
f[
un+1,pn+1]n+1,(2.39)
ext
Ar [𝝋u,n+1,𝝋p,n+1]n+1=(1−)ext [un+1,pn+1]n+1+
ext[
un+1,pn+1]n+1,(2.40)
[pn+1]n+1=(1−)D[pn+1]n+1+()
D[pn+1]n+1.(2.41)
Note that in previous work such as [27,66], the weighting functions of the external work and
the internal work are not necessarily identical. Nevertheless, we choose to use the same weighting
function to simplify the formulation. Therefore, we can write the expression for the total energy
of the porous medium in the context of the Arlequin framework as
Ar[𝝋u,n+1,𝝋p,n+1]n+1=s
Ar −f
Ar +ext
Ar
=Ar [un+1,pn+1]n+1+(1−)
Ar[
un+1,pn+1]n+1.(2.42)
To enforce the compatibility between the coarse and ne solutions in the overlapped domain,
the Arlequin incremental energy functionals must be augmented by constraints that minimize
the discrepancies of the coarse and ne solutions. The pioneering work by Ben Dhia [67] and
follow-up analyses such as [28,33,66] have analyzed various ways to enforce the compatibility for
displacement-based nite element models. These studies have commonly found that a constraint that
minimizes the displacement discrepancy measured by the H1norm seems to yield the most stable
and accurate formulation. In this work, we extend this H1coupling scheme to multiphase porous
media in which solid and uid motions at the same material point are not identical.
To enforce weak compatibility in the overlapped domain, we introduce Lagrange multipliers for
the solid displacement 𝝀u∈V𝝀uanduidpressurep∈Vpdened in the spaces of
V𝝀u={𝝀u∶B→R3𝝀u∈[H1(B)]3},(2.43)
Vp={p∶B→Rp∈[H1(B)]}.(2.44)
The admissible variations of the Lagrange multipliers belong to 𝝀u∈V𝝀uand p∈Vp. Then,
constraints that minimize the discrepancies between the solid displacement elds as well as those
between the uid pressure elds lead to the following compatibility energy functionals
[𝝋u,n+1,𝝋p,n+1,𝝀u,n+1,
p,n+1]n+1=u[𝝋u,n+1,𝝀u,n+1]n+1+p[𝝋p,n+1,
p,n+1]n+1(2.45)
Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2017)
DOI: 10.1002/nme
W. SUN, Z. CAI AND J. CHOO
where
u[𝝋u,n+1,𝝀u,n+1]n+1=∫Bc𝝀u,n+1·(un+1−
un+1)+l2sym𝝀u∶(sym un+1−sym
un+1)dV (2.46)
p[𝝋p,n+1,
p,n+1]n+1=∫Bcp(pn+1−pn+1)+l2p,n+1·(pn+1−pn+1)dV.(2.47)
Here, ∈[0,1] and ∈[0,1] are non-dimensional parameters and lis a scaling factor that has the
dimension of length. When both and are zero, the compatibility energy functionals minimize the
L2norm of the discrepancies between the coarse and ne solution elds. Otherwise, they minimize
the H1norm of the discrepancies.
Augmenting these compatibility functionals, we now state the incremental energy functional for
the Arlequin poromechanics as follows:
HAr[𝝋u,n+1,𝝋p,n+1,𝝀u,n+1,
p,n+1]n+1=(1−)H[un+1,pn+1]n+1+
H[
un+1,pn+1]n+1
−(1−)[pn+1]n+1
n+
[pn+1]n+1
n
+[𝝋u,n+1,𝝋p,n+1,𝝀u,n+1,
p,n+1]n+1.
(2.48)
The saddle point of the incremental Arlequin energy functional (2.48) results in the following set of
time-discrete governing equations that give incremental updates of the displacement, pore pressure,
and Lagrange multipliers:
DHAr[u]=∫B
𝝈′
n+1∶symu−Bpn+1·u−𝜸udV −∫tB
t·ud
+∫Bc𝝀u,n+1·(u)+l2sym 𝝀u,n+1∶(symu)dV =0,
(2.49)
DHAr[
u]=(1−)∫
B
𝝈′
n+1∶sym
u−Bpn+1·
u−
𝜸
udV −∫t
B
t·
ud
−∫Bc𝝀u,n+1·(
u)+l2sym𝝀u,n+1∶(sym
u)dV =0,
(2.50)
DHAr[p]=∫B
1
M(pn+1−pn)p+B·(un+1−un)pdV
−Δt∫B
·1
μk·(pn+1−fg)pdV−∫qB
q·npd
+∫Bcp,n+1p+l2p,n+1·pdV =0,
(2.51)
DHAr[p]=(1−)∫
B
1
M(pn+1−pn)p+B·(
un+1−
un)pdV
−Δt∫
B
·1
μk·(pn+1−fg)pdV−∫qB
q·
npd
+∫Bcp,n+1p+l2p,n+1·( p)dV =0,
(2.52)
DHAr[𝝀u]=∫Bc𝝀u·(un+1−
un+1)+l2sym𝝀u∶(sym un+1−sym
un+1)dV =0,(2.53)
DHAr[p]=∫Bcp(pn+1−pn+1)+l2p,n+1·(pn+1−pn+1)dV =0,(2.54)
Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2017)
DOI: 10.1002/nme
MIXED ARLEQUIN METHOD FOR MULTISCALE POROMECHANICS PROBLEMS
where the variations of solid displacement, (u,
u), uid pressure (p,p), and Lagrange multipliers
(𝝀u,
p)are arbitrary elds.
2.5. Galerkin and matrix forms
We now apply the standard Galerkin method to discretize (2.49) –(2.54) in space. In doing so, we
employ equal-order spatial discretization for the solid displacement and uid pressure variables for
both the ne and coarse domains [52]. The rationale behind employing equal-order mixed nite
elements instead of inf–sup stable higher-order elements is to reduce the minimum number of
nite-dimensional spaces required for the Arlequin method from 4 to 2. This reduction signicantly
simplies the formulation.
Let the Lagrange multiplier for solid displacement 𝝀u(x) be spanned by the same set of basis
functions that interpolate the coarse displacement eld. Similarly, let the Lagrange multiplier for the
uid pressure p(x) be also spanned by the same set of basis functions that interpolate by the coarse
uid pressure eld. This arrangement is motivated by the nding of Guidault and Belytschko [66]
that the use of the same basis functions for interpolating the Lagrange multiplier and ne displace-
ment eld would lead to severe locking. This treatment also simplies the setup of the boundary
value problem, because there is no need to generate extra meshes for the Lagrange multiplier. Thus,
we have
uh(x)=NA(x)UA∈Vh
u, uh(x)=NB(x)UB∈Vh
u,
uh(x)=
Na(x)
Ua∈Vh
u,
uh(x)=
Nb(x)
Ub∈Vh
u,
ph(x)=NA(x)PA∈Vh
p, ph(x)=NB(x)PB∈Vh
p,
ph(x)=
Na(x)
Pb∈Vh
p,
ph(x)=
Nb(x)
Pb∈Vh
p,
𝝀h
u(x)=NA(x)𝜦A∈Vh
𝝀u,𝝀h
u(x)=NB(x)𝜦B∈Vh
𝝀u,
h
p(x)=NA(x)A∈Vh
p,
h
p(x)=NB(x)B∈Vh
p,
(2.55)
where (NA,NB)are the basis functions for the interpolated solutions and admissible variations of the
coarse displacement and pore pressure, whereas (
Na,
Nb)are the basis functions for the interpolated
solutions and admissible variations of the ne displacement and pore pressure. Also, (Na,Nb)are the
basis functions for the Lagrange multipliers, which are the same as those for the coarse displacement
and pore pressure except that they are conned in the overlapped domain. In addition, Vh
uVu,
Vh
uV
u,Vh
pVp,Vh
pVp,Vh
𝝀u
V𝝀u,andVh
𝝀p
V𝝀pare nite-dimensional subspaces spanned by
the corresponding interpolation functions.
We now develop the matrix form of the problem. Following the standard nite element proce-
dure, we rst insert the interpolated solutions and variations in (2.55) into (2.49)– (2.54), and then
eliminate the nodal arbitrary variables for the trial functions. This leads to the following matrix form,
K0GT0CT
u0
0
K0
GT−
CT
u0
G0𝜣00CT
p
0
G0
𝜣0−
CT
p
Cu−
Cu00 0 0
00
Cp−
Cp00
U
U
P
P
𝜦U
𝜦P
=−
Ru
R
u
Rp
Rp
R𝝀u
Rp
,(2.56)
where U,
U,P,
P,𝚲U,𝚲Pare the solution increments during iterations. The residual vectors
in the right hand side are given by:
Ru=∫B
BT·(𝝈′
n+1−BIpn+1)−NT𝜸dV −∫TB
NT
td
+∫BcNT𝝀u,n+1+l2BT𝝀u,n+1dV =0,
(2.57)
Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2017)
DOI: 10.1002/nme
W. SUN, Z. CAI AND J. CHOO
R
u=(1−)∫
B
BT·(
𝝈′
n+1−BIpn+1)−
NT𝜸dV −∫T
B
NT
td
−∫Bc
NT𝝀u,n+1+l2
BT𝝀u,n+1dV =0,
(2.58)
Rp=∫B
NT1
M(pn+1−pn)+NTB·(un+1−un)dV
−Δt∫B
NT·1
μk(pn+1−fg)dV −Δt∫qB
NTqd
+∫BcNT𝝀p,n+1+NTl2k𝝀p,n+1dV =0,
(2.59)
Rp=(1−)∫
B
NT1
M(pn+1−pn)+
NTB·(
un+1−
un)dV
−Δt∫
B
NT·1
μk(pn+1−fg)dV −Δt∫qB
NTqd
−∫Bc
NT𝝀p,n+1+
NTl2𝝀p,n+1dV =0,
(2.60)
R𝝀u=∫BcNT·(un+1−
un+1)+l2BT·(sym un+1−sym
un+1)dV =0,(2.61)
Rp=∫BcNT(pn+1−pn+1)+l2NT(pn+1−pn+1)dV =0,(2.62)
where N=[N1,N2,…,Nnu ]and
N=[
N1,
N2,…,
Nnu]are the Nmatrices for the coarse and ne
domains where the basis function are stored in the matrix form (nuand nuare the numbers of nodes
in the coarse and ne domains). Also, B=[B1,B2,…,Bnu]and
B=[
B1,
B2,…,
Bnu]are the B
matrices for the coarse and ne domains, given by
BA=
NA,100
0NA,20
00
NA,3
NA,2NA,10
0NA,3NA,2
NA,10NA,3
;
Ba=
Na,100
0
Na,20
00
Na,3
Na,2
Na,10
0
Na,3
Na,2
Na,10
Na,3
(2.63)
The submatrices of the tangent (Jacobian) matrix in (2.56) are obtained as follows. First, the
tangential stiffness matrices of the solid skeleton in the coarse and ne domains are given by
K=Ru
U
=∫B
BTDBdV,
K=R
u
U=∫
B
(1−)
BT
D
BdV,(2.64)
where Dand
Dare the tangential stiffness tensors dened in (2.11), corresponding to the models
used in the coarse and ne domains. Once again, we emphasize that while these two models capture
the same physical processes in the overlapped domain, but the choices of the mesh size, constitutive
laws and basis functions used to interpolate the solution can be different.
Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2017)
DOI: 10.1002/nme
MIXED ARLEQUIN METHOD FOR MULTISCALE POROMECHANICS PROBLEMS
The two non-zero components of the 2-by-2 block at the center of the Jacobian matrix are the
hydraulic components that consist of two terms—one for the uid mass changes due to local changes
of pore pressure and another for the uid diffusion. For instance, if the effective permeability and
Biot’s modulus are constant, then the two matrices read,
𝜣=Rp
P
=−tE−𝜱,
𝜣=Rp
P=−t
E−
𝜱,(2.65)
where
E=∫B
NT1
μkNdV,
E=∫
B
(1−)
NTk
μ
NdV,(2.66)
𝜱=∫B
NT1
MNdV,
𝜱=∫
B
(1−)
NT1
M
NdV.(2.67)
The coupling matrices that represent the hydro-mechanical coupling are dened as
G=Rp
U
=∫B
NTbdV,
G=Rp
U=∫
B
NT
bdV,(2.68)
where b=B·𝟏and
b=
B·𝟏with 1=[111000]
TB, again, is Biot’s coefcient. The coupling
matrices that enforce the compatibility of the solid displacement are dened as
Cu=R𝝀u
U
=∫Bc
NTN+l2BTBdV,
Cu=R𝝀u
U=∫Bc
NT
N+l2BT
BdV.(2.69)
Lastly, the coupling matrices that enforce the compatibility of the uid pressure are dened as
Cp=Rp
P=∫Bc
NTN+l2NTNdV,
Cp=Rp
P=∫Bc
NT
N+l2NT
NdV.(2.70)
Because we use the basis function of the coarse solution eld to interpolate the Lagrange multipliers,
the numerical integration of all block matrices—except
Cuand
Cp—can be performed via standard
Gauss quadrature rules. The calculation of
Cuand
Cpcan be more complicated, as the
Cuand
Cp
matrices consist of expressions in terms of both the coarse and ne basis functions. Here, we numer-
ically integrate the integrand expressed in the isoparametric (natural) coordinates of the ne mesh
and perform the standard full integration over the ne mesh elements. This approach requires one
to evaluate the coarse basis function at non-standard locations in the isoparametric coordinates of
the ne mesh. and are dimensionless parameters that control the weight of the gradient term in
the norms used to measure the discrepancies. If ==0, then the L2error of displacement and
pore pressure is minimized. lis a parameter which makes the constraint dimensionless. Our analysis
indicates that introducing the gradient term in the compatibility functionals may improve numerical
stability. This result is explained further in the next section.
We note that this matrix system may be ill-conditioned for two reasons. One reason is the intro-
duction of Lagrange multipliers for domain coupling [33], and another is the coupling between solid
deformation and uid ow [68,69]. Therefore, making the use of an iterative solver for this linear sys-
tem requires a preconditioning strategy that effectively handles these two sources of ill-conditioning.
White et al. [70] have recently developed a framework for block-partitioned solvers for coupled
poromechanics problems, whereas Sun and Mota [33] present a block solver for overlapped domain
problems in solid mechanics problem. In the former case, the condition number of the matrix system
of the poromechanics problem may increase signicantly when a smaller time step is used. While
we use direct solvers in this work and hence the usage of preconditioner is less critical, the usage of
direct solvers is rarely a practical option for large-scale problems due to the relatively high compu-
tational demands and lower speed of the direct solver. As a result, a proper design for preconditioner
Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2017)
DOI: 10.1002/nme
W. SUN, Z. CAI AND J. CHOO
for the Arlequin poromechanics problems is an important future work, especially for solving large-
scale problems that require an efcient iterative solver. The design of such a preconditioner is outside
the scope of this work but will be considered in future studies.
3. NUMERICAL STABILITY
The objective of this section is to derive a numerical test that examines the spatial stability of the
Arlequin poromechanics formulation. A necessary (but not sufcient) condition for the convergence
of an implicit solution method for nonlinear problems is the stability of the linearized system of
equations such as (2.56). For single-eld nite elements, well-posedness and numerical stability of
an incremental update can be guaranteed if the tangent operator is both coercive and consistent. How-
ever, a tangent operator in mixed nite elements, such as those for solving multiphysical problems
like poromechanics, may lack coercivity. In such cases, one must ensure that at least one diagonal
block of the block-partitioned Jacobian matrix is positive denite [71,72].
Mixed nite elements for constrained multi-eld problems should satisfy the inf–sup condition to
maintain the spatial stability of numerical solutions [71–74]. When the inf–sup condition is not sat-
ised, the kernel space of the coupling operator(s) in the governing equation is spanned by non-trivial
basis, which in turn gives rise to spurious (physically meaningless) oscillations in the numerical
solutions. In poromechanics, it is well known that the inf–sup condition arises when the pore uid
imposes an incompressibility constraint in the solid deformation—which is common especially in
the early stages of loading. As such, mixed nite elements that employ equal-order interpolations
for the displacement and pore pressure elds may result in spurious oscillations in the pore pressure
eld, see [1,51,52,75–78] for example.
The introduction of the Arlequin method for poromechanics problems complicates the analysis of
spatial stability because it adds additional type of constraints. Hence, multiple Lagrange multipliers
are used in the overlapping domains to enforce compatibility for both the solid and uid constituents.
As shown in (2.56), the Arlequin method embeds the hydro-mechanical coupling blocks into another
block system that incorporates constraints enforcing domain compatibility. Previous studies have
analyzed the inf–sup condition of the Arlequin method for single-physics solid mechanics problems
[28,33]. Furthermore, Jamond and Ben Dhia [38] have analyzed, within the Arlequin framework,
critical zones for an incompressible medium modeled via a two-eld mixed formulation. To avoid
redundancy in the overlapped domain that may lead to pathological results, they weakly enforce the
incompressibility for only one of the two super-imposed models. Interestingly, the poromechanics
theory may also lead to a two-eld u∕pmixed nite element formulation and a similar block matrix
system that resembles the incompressible elasticity counterpart at the undrained limit. However,
because the ow of the pore uid is related to the gradient of pore pressure, the pore pressure eld
must exist in all sub-domains to capture the hydro-mechanical coupling effects. As a result, the anal-
ysis of spatial stability of the Arlequin method for poromechanics requires a further endeavor and
examination. In particular, this analysis requires us to tackle the spatial stability of the Arlequin cou-
pling model whereby two primary elds are both constrained, which has not yet been attempted to
the best of the authors’ knowledge. Our new contribution in this work is to address the inf – sup con-
dition that arises from the multiscale coupling of the mixed Arlequin poromechanics model. In doing
so, we ensure the spatial stability in non-overlapped domains by using the polynomial projection
stabilization procedure, which has been successfully applied to various constrained problems includ-
ing poromechanics [45,51,52,75,77]. In essence, the stabilization procedure augments the following
additional term to the incremental energy functional HAr[𝝋u,n+1 ,𝝋p,n+1,𝝀u,n+1,𝝀p,n+1 ]n+1 in (2.48)
Wstab(Δph,Δph)= 1
2∫B
(Δp−Δp)2dV +1
2∫B
(1−)(Δp−Δp)2dV ,(3.1)
where (·) denotes a projection operator that project the interpolated pore pressure eld onto an
element-wise constant, and is the stabilization parameter whose optimal value depends on both the
effective diffusivity and the element size.
Lastly, we emphasize that it is also possible to couple nite elements that are individually inf – sup
stable within the Arlequin framework. However, this approach requires selecting multiple nite
Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2017)
DOI: 10.1002/nme
MIXED ARLEQUIN METHOD FOR MULTISCALE POROMECHANICS PROBLEMS
dimensional spaces for the coarse and ne displacements, pore pressures, and Lagrangian multipliers.
Consequently, more elaborated efforts must be spent for the implementation, without any guaran-
tee that the coupling of two inf–sup stable mixed nite elements through Lagrangian multipliers
remains stable in the overlapped domain.
3.1. Twofold inf – sup condition for Arlequin poromechanics problems
Here, we show that one can analyze the spatial stability of the Arlequin poromechanics problem by
checking the twofold inf–sup condition, the inf–sup condition arising from a twofold saddle point
problem. Howell and Walkington [79] derived this condition, and Sun [52] and Choo and Borja [77]
developed stabilization methods for this condition in the contexts of thermo-poro-mechanics and
two-scale poromechanics, respectively.
The key idea of this work is to analyze the multi-eld Arlequin problem with the product spaces
equipped with weighted product norms. For instance, one may group the solution of the system of
equations in (2.56) in the following three product spaces: Vh
u×Vh
u,Vh
p×Vh
p,andVh
𝝀u×Vh
p.The
corresponding matrix form therefore reads
KG
TCT
u
GDC
T
p
CuCp0
Δ
U
Δ
P
Δ𝜦
=−
Ru
Rp
R
,(3.2)
where the block matrices in (3.2) are given by
K=K0
0
K,D=𝜣0
0
𝜣,G=G0
0
G,Cu=Cu−
Cu
00
,Cp=00
Cp−
Cp,(3.3)
and Δ
U,Δ
P,and𝜦are the column vectors of the solution elds, that is,
Δ
U=ΔU
Δ
U,Δ
P=ΔP
Δ
P,Δ𝜦=Δ𝜦U
Δ𝜦P.(3.4)
The product spaces of the solutions for the combined solid displacement, pore pressure, and Lagrange
multipliers therefore read
u∈Vh
u=Vh
u×Vh
u,
p∈Vh
p=Vh
p×Vh
p,𝝀∈V𝝀=Vh
𝝀u×Vh
p,(3.5)
which are equipped with the following product norms,
uV
u=u2+
u2;
pV
P
=p2+
p2;𝝀W=wu𝝀u2+wp𝝀p2,(3.6)
where ·2denotes the L2norm, and wuand wpare the weighting functions for Lagrange multipliers.
Following Auricchio et al. [80], we further condense the system of equations as a problem coupled
by a composite coupling operator B,thatis,
AB
T
B0x
y=f
g(3.7)
where:
A=KG
T
GD
,B=CuCp,x=Δ
U
Δ
P,y=Δ𝜦.(3.8)
Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2017)
DOI: 10.1002/nme
W. SUN, Z. CAI AND J. CHOO
The kernel of coupling operator is made by (Δ
U,Δ
P) such that:
CuΔ
U+CpΔ
P=0.(3.9)
The key step to simplify the analysis is to take advantage of the fact that the spaces Vh
u,Vh
p,and
V𝝀are all nite dimensional and spanned by the basis function (cf. [74,80]). This fact allows
us to construct a bijective map from the nite dimensional spaces Vh
u,Vh
p,andV𝝀onto the
Euclidean space of the nodal solution. Let Nu,Np,andNbe the dimensions of the nodal solu-
tion
U∈RNu,
P∈RNp,and𝚲∈RN. Then, we can express the twofold inf – sup condition
as follows:
Twofold inf–sup condition. There exists a positive constant s,independent of the mesh size,
such that:
sup
(
U∈RNu⧵{0},
P∈RNp⧵{0},)
𝜦TCu
U+𝜦TCp
P
PVP+
UVU
⩾s∀𝜦V𝜦>0,∀𝜦∈V𝜦,(3.10)
where ·VU,·VP,and·V𝚲are the norms of the product spaces of the nodal displace-
ment, pore pressure, and Lagrange multipliers, respectively. Notice that if sdoes not exist, then
the kernel space of Bmay contain non-trivial basis. This non-trivial basis may cause spurious
oscillations in the solution elds.
3.2. Discrete twofold inf– sup test
Having formulated the stability requirement as the twofold inf–sup condition (3.10), we now intro-
duce a simple numerical procedure that can determine whether this condition is satised. The
numerical procedure we propose here is an extension of the inf–sup test used in [73,74,81]. We
emphasize that this numerical test by no means supersedes the analytical proof of the stability of
nonlinear boundary value problems. However, because the analytical proof is notoriously difcult,
here we aim at developing a practical means for investigating the existence of spurious modes at a
given incremental step.
Let ISdenote the matrix associated with the the scalar product of a nite element space S[38].
Following the standard procedure to establish an inf– sup test [33,74], we rewrite the twofold inf– sup
condition (3.10) as
𝜳u𝜳pCuCpIVU0
0IVPCT
u
CT
p𝜳u
𝜳p⩾s𝜳u𝜳p𝜳u
𝜳p,(3.11)
where 𝜳u∈RNu⧵{0}and 𝜳p∈RNp⧵{0}. The existence of the positive constant sis there-
fore guaranteed if the smallest eigenvalue of the following generalized eigenvalue problem is larger
than zero:
CuCpIVU0
0IVPCT
u
CT
p𝜳u
𝜳p=𝜳u
𝜳p.(3.12)
The square root of the minimum eigenvalue value of (3.12) is commonly referred to as the inf – sup
value [33,73,74,81]. The inf–sup value is crucial and widely used in computational mechanics
problems, as it enables one to easily detect the onset of spurious modes by checking whether the
inf–sup value becomes zero. This procedure—checking the existence of a positive non-zero inf –sup
value—is called the inf–sup test. This test involves the calculation of inf–sup values correspond-
ing to the same boundary value problems discretized by typically at least four successively rened
meshes. In the next section, we will conduct inf–sup tests for multiscale coupling operators [CuCp]
in various numerical examples to examine whether spurious oscillation mode(s) may occur in the
overlapped domain(s).
Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2017)
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4. NUMERICAL EXAMPLES
This section presents numerical examples to verify the proposed multiscale hydro-mechanical model
and demonstrate its performance. For verication of the model in terms of hydro-mechanical cou-
pling, we rst use the classical Terzaghi’s consolidation problem of which an analytical solution
exists. Subsequently, to verify the model under incompressible/compressible deformations in the
undrained/drained conditions, we extend Cook’s membrane problem to poroelasticity and compare
the results with benchmark values of incompressible and compressible cases. Lastly, to demonstrate
the performance of the Arlequin model whereby different models are coupled, we simulate uid
injection into a pre-existing crack of a porous medium coupling an isogeometric extend nite ele-
ment model with a polynomial-based standard nite element model. In this example, the uid–solid
interactions in both the near-eld and far-eld of a pre-existing crack are concurrently simulated. To
investigate the spatial stability of the model, in every example, we perform the inf–sup test described
in the previous section.
4.1. Terzaghi’s 1D consolidation problem
Terzaghi’s 1D consolidation problem is one of the few poromechanics problems of which analytical
solutions exist [82]. As such, this problem has been widely used as benchmark to verify porome-
chanical models in the literature [1,3,16,65]. In this study, we use this problem as the rst benchmark
problem to verify the proposed Arlequin formulation for coupled hydro-mechanical problems.
Figure 2 illustrates the domain conguration and boundary conditions of the boundary value prob-
lem. As for the boundary conditions for uid ow, the top boundary is a zero pressure boundary
where drainage is allowed. On the other hand, the bottom and lateral boundaries are impermeable,
that is, no ux boundaries. The solid boundary conditions are imposed such that the material deforms
in the vertical direction only: the top boundary is subjected to compressive stress of 90 kPa while
the bottom boundary is xed and the lateral boundaries are constrained horizontally. As a result, the
problem is essentially one-dimensional, allowing for an analytical solution.
Figure 2. Domain conguration and boundary conditions of the 1D consolidation problem.
Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2017)
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We model the problem by overlapping two domains, as shown in Figure 2. Specically, we dis-
cretize upper 45%of the domain by a ner mesh (ne domain) while discretize lower 65%by a
coarser mesh (coarse domain). Both the coarse and ne domains are modeled by stabilized nite
elements that use linear polynomials to interpolate the displacement and pore pressure elds and pro-
jection based strategy to stabilize the pore pressure eld [45,51,52,75,77]. In the overlapped domain,
each coarse-scale nite element is coupled with eight ne-scale nite elements. The H1coupling
scheme is used throughout the verication simulations. The material parameters used in this problem
are summarized in Table I.
The simulations are performed using two types of weighting functions: (1) constant weighting
function and (2) linear weighting function. Figure 3 presents the results of the two cases in terms of
the normalized pore pressure p∕p0along the normalized location z∕hwhere p0=90 kPa. Here, we
observe that the numerical solutions agree with the analytical solutions irrespective of the weighting
functions.
Additional numerical experiments are conducted to investigate whether it is necessary to introduce
constraints for both the displacement and pore pressure elds, not just for one of them. Figure 4(a)
and 4(b) show the results when a constraint is applied to either the pore pressure or the displacement,
respectively. As shown in Figure 4(a), the use of the pore pressure constraint alone is able to enforce
the compatibility of pore pressures in the two domains, but it produces incorrect results. On the other
hand, Figure 4(b) indicates that if the pore pressure constraint is not enforced, the coarse and ne
pore pressure elds are not compatible with each other. These results demonstrate that both of the
pore pressure and displacement constraints are essential to ensure the compatibility between the two
domains.
Table I. Parameters for the 1D consolidation problem.
Parameter Value Unit
Young’s modulus E70 MPa
Poisson’s ratio 0.4 −
Biot coefcient B1.0 −
Porosity n0.3 −
Permeability k1.57 ×10−13 m2
Viscosity of pore uid μ1.0×10−3Pa ·s
Bulk modulus of pore uid Kf2.2 GPa
Figure 3. Pore pressure proles of the 1D consolidation problem at 250, 500, 750, and 1000 s (from right
to left in each gure), from the simulations employing: (a) constant weighting function and (b) linear
weighting function.
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Figure 4. Results of the consolidation problem (at time =250 s) with different constraints: (a) Pressure
constraint only and (b) Displacement constraint only.
Figure 5. Inf–sup test results of the 1D consolidation problem.
4.1.1. Numerical inf–sup test for the 1D consolidation problem. To investigate the spatial stabil-
ity of the proposed Arlequin method, we perform a numerical inf–sup test for this consolidation
problem and present the results in Figure 5. We test both H1and L2coupling systems with a linear
weighting function, and obtain the inf–sup values rening both the ne and coarse meshes. In this
gure, Ncstands for the number of nite elements in the coarse domain. The results indicate that
the H1coupling system always yields higher inf–sup values than the L2counterpart. Note that this
observation is consistent with previous nding from a single-physics displacement-based Arlequin
model [33]. The inf–sup values decrease with mesh renement in both simulations. However, it is
clear that the inf–sup value of the H1coupling performs better. As mentioned in [81], the inf – sup
test is only considered successful when a lower bound of inf–sup values is likely to be found from
underneath. However, in practice, the effectiveness of the inf– sup test, as explained and demon-
strated through numerical examples in Bathe et al. [83], is highly sensitive to the norm one used
to constructed in the inf–sup test. This is particularly true for problems that may exhibit boundary
layer effect, such as Terzaghi’s consolidation problem and convection – diffusion problems [84]. The
difculty of these problems is that at near time when the time step is small, the solution in the inte-
rior domain is smooth, while the solution near the boundary may exhibit sharp gradient. Hence, as
mentioned in Bathe et al. [83], an ideal norm must be able to measure equally well in the smooth
and non-smooth parts of the solution. This effective norm has not yet been found, according to the
best knowledge of the authors, but we may consider it for future research. As a result, we may only
Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2017)
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draw the conclusion that the H1coupling strategy seems to generate more stable results than the L2
counterpart, but it is not clear whether the inf–sup values will be bounded in either cases.
4.2. Modied Cook’s membrane problem for poroelasticity
In the second example, we investigate the accuracy of the proposed Arlequin method when it is used
to capture the transition from undrained to drained behaviors of a poroelastic medium. According to
poroelasticity theory, a uid-saturated porous solid may deform isochorically even if the solid matrix
is compressible. This is possible if: (1) an incompressible uid is trapped in the pore space because
the loading rate is much higher than the hydraulic diffusivity, and (2) the solid and uid constituents
are both incompressible [51,85]. In other words, incompressibility can arise when the undrained bulk
modulus approaches innity despite the fact that the drained bulk modulus remains nite [86]. In the
poromechanics and geotechnical engineering literature, this condition is usually referred to as the
undrained condition, as opposed to the drained condition in which the pore uid is free to migrate
and the excess pore pressure is dissipated such that the presence of uid in the pore space does not
lead to an incompressibility constraint. These two distinctive types of isochoric or nearly isochoric
deformation of porous media have been extensively studied and compared in Levenston et al. [87]
and Sun et al. [51]. In the context of mixed nite elements for poromechanics, this incompressibility
can trigger volumetric locking as well as spurious oscillations in the pore pressure eld which was
discussed in Section 3. However, how this incompressibility in the undrained condition can affect
the convergence and stability of the proposed Arlequin coupling model is elusive, as the Arlequin
framework has not yet been applied to a poromecahanics problem.
The specic purpose of this example is to investigate whether any convergence or stability problem
arises during the transition from the undrained condition to the drained condition. For this purpose,
we modify Cook’s membrane problem [88], which has served as a benchmark problem in many
studies to assess a numerical technique designed to prevent volumetric locking in solid mechanics
problems. Figure 6 depicts the mesh and boundary conditions of the modied Cook’s problem which
is modeled by two overlapped domains as in the previous example. For our purpose, we model
the membrane as a porous solid fully saturated by a (nearly) incompressible uid. This allows the
Figure 6. Domain conguration and boundary conditions of the modied Cook’s membrane problem.
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membrane to be incompressible at the undrained limit without setting the Poisson ratio approaching
1/2. Meanwhile, the membrane may exhibit volumetric deformation as the excess pore pressure
dissipates in time and eventually reaches the drained condition under which the membrane again
behaves like a compressible material. The boundary conditions for the solid are identical to those of
the original problem, and the traction applied to the right edge is 1 N. However, for the pore uid, we
introduce additional boundary conditions: the left edge is a zero pressure boundary while all other
edges are no-ux boundaries.
The parameters for this problem are summarized in Table II. The material parameters for the solid
are the same as those used in the classical Cook’s membrane benchmark. On the other hand, the
material parameters for the uid (e.g., permeability) are chosen such that the material is subjected to
the undrained condition after the rst time step. We set the Poisson ratio of the solid matrix =0.3
so as to compare (1) the tip displacement in the undrained condition with the benchmark value in
the incompressible case, and (2) the tip displacement in the drained condition with the benchmark
value in the compressible case with =0.3.
Figure 7 presents the time evolution of the tip displacement after the loading. We observe that the
displacement is about 16.5 m at the rst time step and it converges to about 21.5 m with time. Note
that the rst time step is in the undrained condition while the nal step is in the drained condition.
The benchmark values of the tip displacement, which can be computed by Richardson’s extrapola-
tion [88], are found to be 16.43258437 m for the incompressible case (=0.5) and 21.5234479 m
for the compressible case (=0.3). Figure 8 shows that the tip displacement converges to the bench-
mark values for both the incompressible (undrained) and compressible (drained) cases. In Figure 9,
we also plot pore pressure elds in three time steps during the simulation. These contours indicate
that the pore pressure developed by the tip loading is dissipated as time increases. The fact that
both the deformation and pressure match well in the overlapped domain again demonstrates that the
constraints used in the Arlequin model are appropriate.
Table II. Parameters for the modied Cook’s membrane problem.
Parameter Value Unit
Young’s modulus E1Pa
Poisson’s ratio 0.3 −
Biot coefcient B1.0 −
Porosity n0.3 −
Permeability k1.57 ×10−10 m2
Viscosity of pore uid μ1.0×10−3Pa ·s
Bulk modulus of pore uid Kf2.2 GPa
Figure 7. Tip displacement with time in the modied Cook’s membrane problem.
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Figure 8. Normalized error of the tip displacement with mesh renement: (a) Initial time step (incompress-
ible/undrained) and (b) Final time step (compressible/drained).
Figure 9. Pore pressure elds in the deformed membrane (magnication: 1x) at various time steps: (a) Step
#100, (b) Step #200, and (c) Step #750.
4.2.1. Numerical inf–sup test for the modied Cook’s membrane problem. We conduct two inf –sup
tests for the modied Cook’s membrane to investigate the effect of mesh ratio on the inf–sup values.
In the rst inf–sup test, we rene both the coarse and ne meshes three times and compute the
inf–sup values for both the L2and H1coupling models. In all four simulations, the coarse-to-ne
mesh ratio remains the same. Figure 10 shows the results of this constant mesh ratio inf–sup test.
In this test, the inf–sup value decreases upon mesh renement, similar to what we observed in the
previous inf–sup test conducted for Terzaghi’s problem.
In the second inf–sup test, we rene the ne mesh consecutively three times while keeping the
coarse mesh size constant. As a result, upon the renement, the coarse-to-ne mesh ratio decreases
from 1:1 to 1:8. The inf–sup value of this changing-mesh-ratio inf– sup test is shown in Figure 11,
where Nfrepresents the number of ne elements of the model. Surprisingly, the result suggests that
the inf–sup value is not sensitive to the mesh ratio. This result suggests that the mesh ratio may have
little inuence on the spatial stability of the Arlequin poromechanics model. Further studies must
be conducted to check whether this observation can be applied to other problems, but the relatively
small changes in the inf–sup test indicate that coupling between models designed for signicantly
different length scales can be numerically stable.
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Figure 10. Inf–sup test results of the modied Cook’s membrane problem with a constant mesh ratio.
Figure 11. Inf–sup test results of the modied Cook’s membrane problem with various coarse-to-ne mesh
size ratios ranging from 1:1 to 1:8.
4.3. Fluid injection into a pre-existing crack of a porous medium
We consider the problem of uid injection into a pre-existing crack of a porous medium, which arises
in many applications including hydraulic fracturing, geological sequestration of uidized green-
house gases, and geological disposal of contaminants and hazardous wastes [89,90]. As pointed
out recently by Kim and Selvadurai [90], previous work on the uid injection problem often treats
injection activities as a static process where the geological material is assumed to be elastic and the
injection process is simulated by a distribution of dilatation acting inside the geological formation.
A more rigorous treatment of this important problem is to simulate the uid injection problem in
the framework of poroelasticity, such that transient effect induced by the pore uid diffusion inside
the pre-existing crack and the porous medium and the undrained and drained behaviors at t=0and
t→∞are properly captured.
Unlike the previous examples in which the solutions in the coarse and ne domains are both
interpolated by polynomial basis functions, in this example, we purposely assign different nite
dimensional spaces to interpolate solution eld in different regions. Having veried the Arlequin
poromechanics model through two numerical examples that couple different meshes together, this
setup provides us an opportunity to assess the robustness and stability of the Arlequin coupling
techniques when the coupling domains are represented by different numerical methods.
In this problem, complicated processes are expected to be concentrated on a local region around
the crack, whereas relatively simple processes may occur in regions far from the pre-existing crack.
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These treatments render the Arlequin method an efcient, attractive strategy for a uid injection
problem. For the same reasons, the Arlequin method is also an appealing choice for modeling other
problems where one can easily identify a domain of interest (e.g., existing fault and ow barrier).
Figure 12 illustrates the conguration of the uid injection problem. Here we inject a constant
ux of 0.1 mm/s into a 4.5 m long horizontal crack, which is located at the center of the left edge of
a poroelastic domain. While no ow is allowed through the rest of the left edge, drainage is allowed
through all other boundaries where pore pressure is zero. As for the solid boundary conditions, the
left edge is supported by horizontal rollers except the bottom node which is xed for stability. Using
symmetry, we model a half of the domain. Table III summarizes the material parameters.
We partition the domain into three sub-domains: one domain that accommodates the crack
(ne domain) while the other domains that are continuous porous media (intermediate and coarse
domains). As shown in Figure 12, the level of spatial discretization is highest in the cracked
ne domain and gradually decreases with the distance from the crack. There are two overlapped
Figure 12. Domain conguration and boundary conditions for the uid injection problem. The nest domain
(zoomed mesh) is modeled by the isogeometric extended nite element method. Symbols in the zoomed mesh
denote enriched nodes. Other domains are modeled by the polynomial-based standard nite element method.
Because of the axial symmetry, only half of the domain is discretized and a zero horizontal displacement
boundary condition is applied on the left side of the domain.
Table III. Parameters for the uid injection problem.
Parameter Value Unit
Young’s Modulus E144 MPa
Poisson’s ratio 0.2 -
Biot coefcient B1.0 -
Porosity n0.3 -
Permeability k2×10−11 m2
Viscosity of pore uid μ1.0×10−3Pa ·s
Bulk modulus of pore uid Kf3.0 GPa
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domains—one linking the ne and the intermediate domains, and the other linking the intermedi-
ate and coarse domains. In these two overlapped domains, the ner meshes are conformal with the
coarser meshes, and each coarser element is overlapped with nine ner elements in the gluing zone.
To take further advantage of the Arlequin method and to test the robustness of the Arlequin frame-
work for coupling models of different natures, we employ different numerical methods and meshes
of different sizes for the sub-domains in this last numerical example. For the cracked domain, we use
the extended nite element method (XFEM) [91]—which allows one to capture strong discontinuity
in the interior of nite elements without re-meshing—in conjunction with the isogeometric anal-
ysis (IGA) [92]—which interpolates solutions by non-uniform rational basis spline (NURBS) that
can lead to higher-order accuracy than polynomial basis functions. It is worth noting that De Luy-
cker et al. [93] and Verhoosel et al. [94] have pointed out IGA employing XFEM to represent strong
discontinuities lead to higher asymptotic convergence rate and accuracy with the same amount of
degree of freedoms, in comparison with XFEM with conventional nite elements of equal degree.
Because of these appealing features, we combine XFEM and IGA to model the cracked ne domain.
Meanwhile, we use only polynomial-based nite elements to model the far-eld domain, as the sharp
gradient is absent there. Therefore, we purposely use the standard, polynomial-based nite elements
for these non-cracked sub-domains and test whether the coupling in the overlapped domain is stable
during the simulation. Once again, we emphasize that the capability of applying different numerical
methods in different regions—which in turn allows the modelers to conne highly sophisticated, yet
cost-demanding models to the domains of interest—is a major advantage of the Arlequin method as
previously pointed out in [27,28,33,38]. As demonstrated in this numerical example, this advantage
is also extended to multi-physical problems if a proper coupling strategy is used.
Hydro-mechanical processes in the cavity are modeled following standard approaches in the liter-
ature (e.g., [95]). Using XFEM, we introduce additional global degrees of freedom to the elements
that accommodate the crack (these enriched nodes are denoted by red symbols in Figure 12). The
uid ow along the crack is modeled by the cubic law, which assumes that the transmissivity of
a ow channel is proportional to the cube of the hydraulic aperture in 3D and the square of the
hydraulic aperture in plane strain 2D problem [96–99]. In this numerical example, we assume that
the mechanical and hydraulic apertures are identical. Hence, the hydraulic aperture (denoted as h
herein) is dened as the separation distance between the upper and lower surfaces of the crack. In a
2D plane strain setting, the intrinsic permeability of the fracture along the crack reads,
kf=1
f
h2
3μ,(4.1)
where fis a coefcient, which is typically within the range of (1.04, 1.65), according to Moham-
madnejad and Khoei [98]. In this numerical example, we set f=1. For brevity, the details of the
formulation and implementation are included in Appendix A. Our main focus here is to demon-
strate the possibility of using the Arlequin method to couple domains modeled by the isogeometric
XFEM and the classical polynomial-based nite element. As a result, the boundary value problem we
intended to solve is simplied such that the pre-existing crack would not propagate and the deforma-
tion of the solid skeleton is innitesimal. More complex problems close to the setup of a eld-scale
operation problem will be considered in the future. In addition, we acknowledge that the Arlequin
coupling framework is sufciently exible to accommodate other numerical methods for modeling
localized phenomena in porous media, such as the strong discontinuity approach that enhances local
shape functions for strain and ow [10,12]. In other words, the use of XFEM in this example is just
one of the many possible ways to capture localized displacement jump and uid ow within the
multi-model Arlequin framework.
Figures 13 and 14 show pore pressure and Darcy velocity elds at t=2andt=10 s in deformed
domains (deformations are magnied by 6000 times). The overall pressure and ow patterns indicate
that the injection drives ow of the pore uid into the drainage boundaries of the porous media.
Because of the relatively high effective permeability of the the host matrix, the pore pressure eld
appears to be continuous and without sharp gradient. As the boundary layer of the pressure plume
penetrates into the bulk materials, the pore pressure jump has not been observed even though the
extended nite element formulation we adopted does include enhancement in the pore pressure eld.
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Figure 13. Pore pressure eld in the deformed domain (magnication: 6000x) modeled by the three-level
mesh.
Further discussion on various modeling techniques for the uid ow in embedded discontinuity can
be found in [10,12,94], and in the recent review in de Borst [99], but it is out of the scope of this study
which mainly focuses on Arlequin coupling models. We note that these injection-induced pressure
build-up and ow are signicant in near-elds of the crack but they are marginal in far-elds. In
Figure 15, we show engineering shear strains at the same time instants. We observe that shear strains
are developed in near-elds of the crack due to hydro-mechanical coupling between uid injection
and crack opening. Again, we see that shear strains are marginal in far-elds. These contrasting
complexities of physical processes in the near and far-elds clearly justify the use of the Arlequin
method for introducing different models to the near and far-elds.
To investigate the effects of hydro-mechanical interactions among near-elds and far-elds, we
repeat the problem employing three levels of meshes, namely one-level, two-level, and three-level
meshes. In the one-level mesh model, we simulate the uid injection simulation only in the nest
mesh. Then, we conduct the two-level mesh model by placing an additional (intermediate) mesh
Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2017)
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Figure 14. Darcy velocity eld in the deformed domain (magnication: 6000x) modeled by the three-level
mesh.
and bundling it with the ne mesh to enlarge the domain. Finally, in the three-level mesh model,
we continue to enlarge the domain by introducing the coarse mesh such that it coincides the model
we have considered so far. In other words, all three models share the ne domain with the extended
isogeometric nite element, and the two-level and three-level mesh models commonly employ the
intermediate domain of the three-level mesh. Note that while the Arlequin method is not essential to
place more degree of freedoms near the crack tip, it enables coupling different numerical methods
(in this case NURBS-based XFEM and polynomial-based standard FEM), which is very challenging
otherwise.
In Figure 16, we compare results of the three models in terms of opening displacements along
the crack in the ne domain. Generally, models with larger domains tend to predict larger crack
openings. This trend can be attributed to the fact that larger domains are less affected by bound-
ary effects and thus closer to a poroelastic half space. At t=2 s, the opening cracks of the
two-level and three-level mesh models are almost identical; however, they become different as
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Figure 15. Shear strain eld in the deformed domain (magnication: 6000x) modeled by the three-level mesh.
time proceeds to t=10 s. This observation indicates that the opening crack displacement
grows faster in the model with the far-eld than in the model with only the ne isogeometric
domain.
Figure 17 shows the von Mises stress distribution behind the pre-existing crack (along the hori-
zontal line that separate the upper and lower domains evenly). Because there is no traction-separation
law employed at the interface, the von Mises stress is zero inside the crack. Unlike the pore pres-
sure and displacement elds of which the existence of nodal values allows one to interpolate via the
basis function, the stress tensor is only evaluated at the Gauss point. Hence, we use a L2projection
scheme to project the discrete data point to the interpolated eld spanned by the same basis functions
we used to interpolate displacement and pore pressure [100,101]. For brevity, we will not provide
details of the projection scheme. Interested readers are referred to Mota et al. [101] for details. As
expected, the von Mises stress is concentrated near the crack tip and gradually decreases.
Figures 18 present the results of the three models in terms of the pore pressure along the crack.
Note that as each coarse mesh is added, the numerical solution behaves closer to a poro-elastic half
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Figure 16. Opening displacement along the crack: (a) t=2sand(b)t=10 s.
Figure 17. Variation of von Mises stress behind the crack tip: (a) t=2sand(b)t=10 s
space. While the pore pressure and the von Mises stress are different in all three cases, they show
consistent trends with the increase of the level of meshes.
These ndings from the three-level mesh study indicate that incorporating far-eld effects can
enhance the resolution of near eld solutions. Yet in practical problems such incorporation of far-eld
can be prohibitively expensive without an efcient concurrent multiscale, multimodel approach like
the Arlequin method. In this example, we just added 778 polynomial-based standard nite elements
for incorporating far-eld effects to the one-level mesh model. The added computational cost is a
fraction of the pre-existing cost from the NURBS-based extended nite elements in the ne domain,
let alone the cost of a naive extension that uses the same NURBS-based nite elements for far-elds.
This minimized cost showcases the advantage of the Arlequin method for concurrent multiscale
modeling.
4.3.1. Numerical inf–sup test for the uid injection problem. For this uid injection problem, we
perform four inf–sup tests rening only the ne domain modeled by NURBS-based XFEM. During
the test, the ne-to-coarse mesh ratio varies from 16:1 to 1:1. Figure 19 shows the results of the
inf–sup tests for L2and H1coupling cases. We observe that H1coupling results in higher inf–sup
values, consistent with previous ndings [33,66]. More importantly, the absence of any signicant
drop in the inf–sup values conrms the stability of our Arlequin poromechanics model. This result
Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2017)
DOI: 10.1002/nme
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Figure 18. Variation of pore pressure from the injection point: (a) t=2sand(b)t=10 s.
Figure 19. Inf– sup test results of the uid injection problem.
is particularly encouraging, as it suggests that the numerical stability is insensitive to the mesh size
ratio (in this case from 16:1 to 1:1) as well as the numerical methods (in this case from NURBS-
based to polynomial-based nite elements) in the overlapped region. Generalizing this conclusion to
other hydro-mechanical problems via a more rigorous mathematical analysis deserves future work.
5. CONCLUSION
A concurrent multiscale model for coupled poromechanics has been developed. Built on the Arlequin
framework, this model enforces weak compatibility of displacement and pore pressure elds between
domains by distributing energies. We have derived a necessary condition for spatial stability as a
twofold inf–sup condition, and proposed a discrete twofold inf– sup test to check the numerical
stability. Through two benchmark problems, we have veried the developed model under various
renement levels, mesh ratios, and drainage conditions. We have also presented a numerical example
that couples NURBS-based and polynomial-based nite element models to simulate uid injection
into a cracked porous material. This example has demonstrated how the Arlequin model can cou-
ple different poromechanics models of multiple length scales in an efcient manner. This feature
can signicantly advance our modeling capabilities of emerging and complicated hydro-mechanical
problems whereby highly contrasting processes take place at multiple scales and regions, such as
those involving injection, shear banding, faulting, or fractures.
Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2017)
DOI: 10.1002/nme
MIXED ARLEQUIN METHOD FOR MULTISCALE POROMECHANICS PROBLEMS
APPENDIX A: FORMULATION OF THE ISOGEOMETRIC EXTENDED FINITE ELEMENTS
IN THE FLUID INJECTION PROBLEM
This appendix describes the formulation of the isogeometric extended nite element model used
in the uid injection problem (Section 4.3). For more details, we refer to Nguyen et al. [102] and the
open-source code base igafem with which we implement the Arlequin poromechanics model. In
the following, we adopt the notation in Nguyen et al. [102]. First, let us construct a B-spline with a
knot vector
={1,
2,…,
n+p+1},
n⩽n+1,(A1)
where is the i-th knot, nis the number of B-spline basis function and pis the polynomial order.
Then, the corresponding set of B-spline basis functions can be dened by the Cox-de-Boor formula,
for the zeroth order basis function (p=0):
Ni,0()=1,if i⩽⩽i+1,
0,otherwise,(A2)
and for a polynomial order p⩾1
Ni,p()= −i
i+p−i
Ni,p−1()+ i+p+1−
i+p+1−i+1
Ni+1,p−1(),(A3)
where the fractions with the form 0∕0 are dened as zero. Therefore, the NURBS basis function can
be dened as follows:
Ri,p()=Ni,p()i
W()=Ni,p()i
n
î=1î
,(A4)
where i>0 is the set of NURBS weights.
Employing these NURBS shape functions and the extended nite element method, we express the
two-dimensional displacement eld in the cracked elements as follows:
uh(x)=
I∈S
RI(x)uI+
J∈Sc
RJ(x)H(x)aJ+
K∈Sf
RK(x)4
=1
Bb
K,(A5)
where RI,J,Kare the NURBS basis functions dened previously, uIrepresents the standard degrees
of freedom for the displacement, aJthe enrichment degrees of freedom for the crack, and b
Kthe
enrichment degrees of freedom for the crack tip. The set Sincludes the standard points, while the set
Scincludes the control points/nodes whose supports are cut by the crack and the set Sfare control
points with the crack tip. H(x) is the Heaviside function given by:
H(x)=+1,if (x−x∗)·n⩾0,
−1,otherwise,(A6)
where nis the outward normal vector to the crack and x*denotes the projection of point xon the
crack. The branch functions B, which span the crack tip displacement eld, are dened as follows:
[B1,B2,B3,B4](r,)=rsin
2,rcos
2,rsin
2cos , rcos
2cos ,(A7)
Here rand are the polar coordinates in the local crack front. Accordingly, now the
strain–displacement matrix Baccommodates the enrichment degrees of freedom as follows:
B=[Bstd Benr],(A8)
where Bstd is the Bmatrix related to the standard degrees of freedom while Benr is that related to
the enrichment degree of freedoms. This enriched strain–displacement matrix is thus a function
Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2017)
DOI: 10.1002/nme
W. SUN, Z. CAI AND J. CHOO
of both the NURBS basis functions and the global enrichment introduced to represent the strong
discontinuity (crack). In two-dimensional case, this enriched strain–displacement matrix is given by
Benr
I=
(RI),xI+RI(I),x0
0(RI),yI+RI(I),y
(RI),yI+RI(I),y(RI),xI+RI(I),x
.(A9)
Depending on whether control point Inear the tip is enriched, Ican be either the branch functions
Bor the Heaviside function H. In Armero and Callari [10] and Réthoré et al. [96], the pore pressure
is assumed to remain continuous, while a jump may exist in the Darcy’s velocity component normal
to the crack faces. As recently explained in de Borst [99], this assumption can be relaxed such that
the pore pressure itself can be discontinuous within the content of extended nite elements [99,103].
This approach is adopted in this study. To accommodate the strong discontinuity in the pore pressure
eld, we also introduce global enrichment to the pore pressure solution eld. In the element where
the strong discontinuity exists, the discrete gradient operator Bpis replaced by an enriched discrete
gradient operator given by
BP=[Bstd,PBenr,P],(A10)
where Bstd,Pis the standard gradient matrix BPwhile Benr,Pis the enriched BPmatrix for the enriched
pore pressure degree of freedom, that is,
Benr,P
I=(RI),xI+RI(I),x
(RI),yI+RI(I),y.(A11)
The permeability inside the crack is modeled by the cubic law which relates the intrinsic permeability
along the crack to cubic of the crack aperture. In this work, we employ an implicit scheme and denes
the residual vector Riat the i-th iteration as follows:
Rn+1
i=
00
CT
u
GD
(1)CT
p
CuCp0
Un+1
i
Pn+1
i
𝜦n+1
i
i
+
KG
TCT
u
0ΔtD(2)CT
p
CuCp0
Un+1
i
Pn+1
i
𝜦n+1
i
i
+
Fn
inter
ΔtQn
inter
0i
−
Fn
ext
ΔtQn
ext
0i
,(A12)
where the subscript iin (A12) denotes the iteration, and D1and D2refer to the terms related to the
local change of pore pressure and the diffusion-induced change of pore pressure, respectively. Here,
the traction across the interior discontinuity is given by
Finter =∫d
[N]Ttdd−∫d
[N]TndNdP
+∫d
(1−)[
N]T
tdd−(1−)∫d
[
N]Tnd
Nd
P,
(A13)
along dwhere enriched nodes exist. Similarly, the interfacial ux vector is given by
Qinter =∫d
NTndqdd+∫d
(1−)
NTnd
qdd, (A14)
where nddenotes the unit normal vector of the interface d,tdand
tddenote the cohesive tractions
across the interface between the coarse and ne sub-domains, which vanish for a open crack, qdand
qdare the uid uxes along the interface in the coarse and ne sub-domains.
Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2017)
DOI: 10.1002/nme
MIXED ARLEQUIN METHOD FOR MULTISCALE POROMECHANICS PROBLEMS
ACKNOWLEDGMENTS
The authors thank graduate student SeonHong Na for reviewing this manuscript. This research is
nancially supported by the Earth Materials and Processes program at the US Army Research Ofce
under grant contract W911NF-14-1-0658 and W911NF-15-1-0581, the National Science Foundation
under grant contracts CMMI-1462760 and EAR-1516300, and Sandia National Laboratories under
grant contact PO-1557089. These supports are gratefully acknowledged. Sandia National Labora-
tories is a multi-mission laboratory managed and operated by Sandia Corporation, a wholly owned
subsidiary of Lockheed Martin Corporation, for the US Department of Energy’s National Nuclear
Security Administration under contract DE-AC04-94AL85000. The authors are also grateful to the
two anonymous reviewers for their detailed reviews and insightful suggestions.
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