mixed isogeometric collocation methods for the simulation
TRANSCRIPT
NOVEL COMPUTATIONAL APPROACHES TO OLD AND NEW PROBLEMS IN MECHANICS
Mixed isogeometric collocation methods for the simulationof poromechanics problems in 1D
S. Morganti . C. Callari . F. Auricchio . A. Reali
Received: 20 May 2017 / Accepted: 11 January 2018 / Published online: 30 January 2018
� Springer Science+Business Media B.V., part of Springer Nature 2018
Abstract Isogeometric collocation is for the first
time considered as a simulation tool for fluid-saturated
porous media. Accordingly, with a focus on one-
dimensional problems, a mixed collocation approach
is proposed and tested in demanding situations, on
both quasi-static and dynamic benchmarks. The
developed method is proven to be very effective in
terms of both stability and accuracy. In fact, the
peculiar properties of the spline shape functions
typical of isogeometric methods, along with the ease
of implementation and low computational cost
guaranteed by the collocation framework, make the
proposed approach very attractive as a viable alterna-
tive to Galerkin-based approaches classically adopted
in computational poromechanics.
Keywords Consolidation � Isogeometric
collocation � Mixed methods � Poromechanics
1 Introduction
Porous media theories play an important role in many
branches of engineering, including material science,
oil industry, soil mechanics, and biomechanics.
Modeling milestones in this field have been proposed
by Biot more than 50 years ago [1–3], based on the
work by Terzaghi [4]. These works also represent the
starting point of the present paper.
As in the majority of boundary value problems,
solutions to time dependent poromechanics equations
in closed form are unknown except for few particular
cases. Therefore, numerical methods are needed to
obtain a solution. Main contributions in this field
propose finite element approaches to obtain predictive
approximate solutions of quasi-static and dynamic
poromechanics problems. In particular, for poro-
dynamic problems, a representation of Biot’s theory
using the unknowns solid displacement and pore
pressure is usually considered [5–10], even if other
S. Morganti (&)
Department of Electrical, Computer and Biomedical
Engineering, University of Pavia, Pavia, Italy
e-mail: [email protected]
C. Callari
Department of Biosciences and Territory (DiBT) -
Engineering Division, University of Molise, Campobasso,
Italy
F. Auricchio � A. RealiDepartment of Civil Engineering and Architecture
(DICAr), University of Pavia, Pavia, Italy
A. Reali
Institute for Advanced Study (IAS), Technical University
of Munich, Garching, Germany
F. Auricchio � A. RealiInstitute for Applied Mathematics and Information
Technology (IMATI), National Research Council (CNR),
Pavia, Italy
123
Meccanica (2018) 53:1441–1454
https://doi.org/10.1007/s11012-018-0820-8
two-field or three-field formulations are available for
the finite element solution of these problems [11, 12].
More recently, also four-field finite element formula-
tions have been proposed to simulate the thermo-
hydro-mechanical coupled response of multiphase
porous media subjected to dynamic loading [13].
Several researchers have observed spurious spatial
oscillations of finite element solutions in terms of pore
pressures, also in simple one-dimensional quasi-static
problems (e.g., see [14, 15] and references therein). As
a consequence, it is typically recommended a mini-
mum threshold value for the time step size. Alterna-
tively, to limit the temporal discretization errors due to
large time steps, it is suggested a mesh refinement in
proximity of the boundaries where free-drainage
conditions are instantaneously imposed.
In 2013 a numerical formulation, alternative to
standard finite elements, for the prediction of the
behavior of deformable fluid-saturated porous media
has been proposed based on isogeometric analysis
(IGA) [16]. Introduced in 2005 [17], IGA aims at
integrating design and analysis by employing typical
functions from Computer Aided Design (such as, e.g.,
B-splines or NURBS) for describing both geometry
and field variables based on the isoparametric
paradigm. Besides a cost-saving simplification of the
typically expensive mesh generation and refinement
processes required by finite element analysis, thanks to
the high-regularity properties of its basis functions,
IGA showed a better accuracy per degree of freedom
and an enhanced robustness with respect to standard
finite elements [18].
In [16] it has been shown that, taking advantage of
the higher order continuity of spline basis functions,
such an approach typically leads to superior perfor-
mance with respect to finite elements in terms of
computational efficiency and cost. However, a well-
known important issue of IGA is related to the
development of efficient integration rules when
higher-order approximations are employed. In fact,
element-wise Gauss quadrature, typically used for
finite elements and originally adopted for Galerkin-
based IGA, does not properly take into account inter-
element higher continuity leading to sub-optimal array
forming costs, significantly affecting the performance
of IGA methods.
In an attempt to address this issue taking full
advantage of the special possibilities offered by IGA,
isogeometric collocation schemes have been proposed
in [19]. The aim was to optimize the computational
cost still relying on IGA geometrical flexibility and
accuracy. In contrast to Galerkin-type formulations,
collocation is based on the direct discretization of the
strong form of partial differential equations, evaluated
at suitable points. Consequently, isogeometric collo-
cation does not require integral computation, resulting
in a very fast method providing superior performance
with respect to Galerkin formulations in terms of both
assembling operations and order of convergence [20].
Since its introduction, many promising significant
works on isogeometric collocation methods have been
published in different fields, including phase-field
modeling [21], structural dynamics [22], contact
[23, 24], nonlinear elasticity [24], as well as several
interesting studies in the context of structural elements
[25–30], where isogeometric collocation has proven to
be particularly stable in the context of mixed methods.
Moreover, the combination with different spline
spaces, like hierarchical splines, generalized
B-splines, and T-splines, has been successfully tested
in [20, 31, 32], while alternative effective selection
strategies for collocation points have been proposed in
[33–35].
Given the good results provided by IGA in [16] and
the many advantages granted by a collocation
approach, in particular in the framework of dynamics
and mixed methods, in this work, we present a mixed
displacement-pressure isogeometric collocation
method aiming at demonstrating that this can represent
a very efficient technology for solving problems
coming from the poromechanics field. To this end,
we focus on one-dimensional problems and success-
fully test the proposed approach in demanding situa-
tions, on both quasi-static and dynamic benchmarks.
2 Governing equations of linear poroelasticity
We follow the Biot’s theory [1, 2] to model the linear
elastic response of a fully saturated porous medium.
The relevant governing equations for the case of a
porous solid X � R3 are briefly presented in this
section, together with the simplified version consid-
ered in this work. The linear momentum balance
reads:
divrþ qfm ¼ q€uþ qwn €u� €wð Þ ð1Þ
1442 Meccanica (2018) 53:1441–1454
123
where u and r are the fields of displacement and (total)
stress in the porous medium, respectively, w is the
displacement of the interstitial fluid, fm is the external
body load per unit of mass, q and qw are the densities
of the porous medium and of the fluid, respectively,
and n is the porosity, defined as the current void
volume per unit volume of the deformed porous solid.
We assume as positive the compressive fluid pressure
p and the tensile normal components of the stress
tensor r.
Denoting by M the fluid mass content for unit
reference volume of the porous solid and by qw the
mass flow of the fluid relative to the solid skeleton, the
fluid mass balance reads:
_M ¼ �divqw ð2Þ
Denoting by e :¼ rsu the infinitesimal strain field and
assuming a linear elastic response, the constitutive
equations of the porous medium reads (e.g., see [36]):
r ¼ Cske� bpM
qw¼ Cwpþ b : e ð3Þ
for the pore pressure p, the Biot’s tensor b, the storage
modulus Cw and the elastic tensor Csk of the solid
skeleton (the so-called ‘‘drained’’ elastic tensor).
A Darcy’s law is used to model the fluid flow:
qw ¼ �qwk rp� qwfm þ qw €uþ qa þ nqwn
€u� €wð Þh i
ð4Þ
where k is the permeability tensor and qa is the so-
called ‘‘apparent density’’ introduced by Biot [2] to
describe the dynamic interaction between fluid and
solid skeleton.
Under the assumption of negligible relative accel-
erations between the pore fluid and solid skeleton (i.e.,
€u� €w � 0) and neglecting also the solid-skeleton
acceleration term in (4), the following simplified
versions of linear momentum balance (1) and Darcy’s
law (4) are obtained, respectively:
divrþ qfm ¼ q€u qw ¼ �qwk rp� qwfmð Þ ð5Þ
The applicability of the corresponding simplified
formulation in the analysis of low-frequency problems
is demonstrated by Zienkiewicz et al. [5] and more
recently by Schanz and Struckmeier [10].
It is known that the finite element formulation of the
system of governing Eqs. (2), (3), (5) often exhibits
spurious spatial oscillations in the pore pressure
solution where its gradient is supposed to instanta-
neously localize, e.g., near free-drainage boundaries.
This issue is fully evident also in one-dimensional
quasi-static settings [37–41], such as the classic
problem of oedometric consolidation considered by
Terzaghi [42].
Hence, in view of the considerations above, we
examine in this paper the 1D problem of an infinite
poroelastic layer of thickness h in the direction y,
perpendicular to the layer, so it is X ¼ ½0; h� in this
case, with the boundary denoted by C ¼ f0g [ fhg.The layer is subjected to the pore pressure p and to the
normal strain u0 for the solid skeleton displacement u
oriented as y. We assume that the boundary C admits
decompositions C ¼ Cu [ Cu0 and C ¼ Cp0 [ Cp, with
Cu \ Cu0 ¼ £ and Cp0 \ Cp ¼ £ (see for example
Fig. 1 for an illustration of a particular case of this
setting).
Assuming an isotropic response of the layer mate-
rial, it is also b ¼ b1 for the Biot’s coefficient b and
k ¼ k1 for the permeability coefficient k. For zero
body forces, the corresponding 1D mixed problem for
the unknown displacements u and pore pressures p
describing longitudinal waves in poroelastic media,
can be stated as:
Fig. 1 1D infinite poro-elastic layer of thickness h ¼ 1:0m:
configuration of the quasi-static problem in terms of geometry
and boundary conditions on displacements u and pore pressures
p. A constant and uniform compressive load q ¼ 100 kPa is
applied at the pervious boundary
Meccanica (2018) 53:1441–1454 1443
123
Eoeu00 � q€u� bp0 ¼ 0
�b _u0 þ kp00 � Cw _p ¼ 0
�ð6Þ
where Eoe is the oedometric modulus. Equations (6)
are complemented by the following boundary
conditions:
u ¼ �uC on Cu
u0 ¼ �u0C on Cu0
�and
p ¼ �pC on Cp
p0 ¼ �p0C on Cp0
�
ð7Þ
for the given functions �uC, �u0C, �pC and �p0C. A quasi-
static formulation can be easily derived from Eq. (6)
setting the inertia term equal to zero:
Eoeu00 � bp0 ¼ 0;
�b _u0 þ kp00 � Cw _p ¼ 0;
�ð8Þ
complemented by the same boundary conditions (7).
In the following section, we present the computa-
tional framework adopted to numerically solve the
equations above in both a quasi-static and dynamic
regime. The results of representative numerical exam-
ples are discussed in Sect. 4.
3 Isogeometric collocation for poromechanics
In this paper we adopt an isogeometric collocation
approach where B-splines [43] are in general used to
represent both geometry and problem variables in an
isoparametric fashion [17, 18] to numerically solve
partial differential equations (PDEs) in strong form.
The basic ingredient for the construction of B-
splines basis functions is the knot vector, i.e., a set of
non-decreasing coordinates in the parameter space:
N ¼ fn1 ¼ 0; . . .; nnþqþ1 ¼ 1g, where q is the degree
of the B-spline and n is the number of basis functions.
In the present work, we always employ so-called open
knot vectors, i.e., the first and the last knots have
multiplicity qþ 1. Basis functions formed from open
knot vectors are interpolatory at the ends of the
parametric interval [0; 1] but are not, in general,
interpolatory at interior knots.
3.1 Mixed u–p formulation
To construct the 1D isogeometric collocation method
we follow [22] by seeking the approximations uM and
pN to the unknown displacement field u and pressure
field p, respectively, that can be expressed in the
following form:
uM ¼XMi¼1
uiuðxÞui; ð9Þ
pN ¼XNj¼1
ujpðxÞpj; ð10Þ
where the unknown scalar values ui 2 R and pj 2 R
are referred to as displacement coefficients (or control
variables) and pressure coefficients (or control vari-
ables), respectively, uiu and uj
p are the corresponding
B-spline basis functions, and M and N are the number
of displacement and pressure collocation points,
respectively. Following the isogeometric collocation
paradigm, expressions (9) and (10) are then substituted
into Eq. (6) (or Eq. (8) in the simplified quasi-static
case) evaluated at a proper set of collocation points
within the problem domain, as well as substituted into
the corresponding boundary conditions (7).
Following [22], we now choose M collocation
points suk , k ¼ 1; . . .;Mf g located at the images of the
Greville abscissae of the knot vector for the displace-
ment field and, similarly, we choose N collocation
points spl , l ¼ 1; . . .;Nf g located at the images of the
Greville abscissae of the knot vector for the pressure
field.
The Greville abscissae [44] related to a spline space
of degree q and knot vector N ¼ fn1; . . .; nnþqþ1g are
points of the parametric space defined as:
�ni ¼niþ1 þ niþ2 þ � � � þ niþq
q: ð11Þ
Accordingly, we can collocate domain equations on
the indicatedM? N collocation points, obtainingM?
N equations to be then solved for the displacement and
pressure coefficients.
Focusing for example on the more general dynamic
case, we obtain ðM � 2Þ � ðN � 2Þ scalar equations bycollocating Eq. (6) at the points suk and spl ,k ¼ 2; . . .;M � 1f g, l ¼ 2; . . .;N � 1f g in the domain
interior X:
Eu00M � q€uM � bp0N� �
ðsukÞ ¼ 0;
�b _u0M þ kp00N � Cw _pN� �
ðspl Þ ¼ 0;
(ð12Þ
complemented by the Dirichlet boundary conditions
1444 Meccanica (2018) 53:1441–1454
123
uM ¼ �uC on Cu
pN ¼ �pC on Cp;ð13Þ
and by the Neumann boundary conditions
u0M ¼ �u0C on Cu0
p0N ¼ �p0C on Cp0 :ð14Þ
For the quasi-static case described by Eq. (8), the
collocated equations are the same, we simply neglect
the inertial term q€uM:
Eu00M � bp0N� �
ðsukÞ ¼ 0;
�b _u0M þ kp00N � Cw _pN� �
ðspl Þ ¼ 0;
(ð15Þ
We note that in the incompressible limit p corresponds
to the hydrostatic pressure. Otherwise, p is simply a
scalar field defined by Eq. (6)2 (see [45] for elabora-
tion). Despite this, in the remainder of the paper, we
refer to p simply as the ‘‘pressure’’.
3.2 Time integration
The semidiscrete equations obtained after discretiza-
tion in space via isogeometric collocation need then to
be integrated in time, as explained in this section for
both the quasi-static and the dynamic case.
3.2.1 Quasi-static case
We employ a backward Euler scheme for the time
integration of the quasi-static case. As a consequence,
the time derivatives of u and p appearing in Eq. (15)
are approximated as follows:
_unþ1 ¼unþ1 � un
Dtand _pnþ1 ¼
pnþ1 � pn
Dt; ð16Þ
where Dt ¼ tnþ1 � tn indicates the time increment,
while subscripts nþ 1 and n indicate evaluations at
t ¼ tnþ1 and t ¼ tn, respectively. The quasi-static
counterpart of (12) thus reads
Eoeu00Mjnþ1 � bp0N jnþ1
� �ðsukÞ ¼ 0;
�b _u0Mjnþ1 þ kp00N jnþ1 � Cw _pN jnþ1
� �ðspl Þ
¼ �b _u0Mjn � Cw _pN jn� �
ðspl Þ:
8><>:
ð17Þ
or, in an explicit format,
Eoeu00uðsukÞ � bu0
pðsukÞ�bu0
uðspl Þ kDtu00
pðspl Þ � Cwupðs
pl Þ
" #unþ1
pnþ1
� �
¼0
�bu0nðspl Þ � Cwpnðspl Þ
� �
ð18Þ
where suk and spl represent the displacement and
pressure collocation points, respectively, unþ1 and
pnþ1 collect the unknown displacement and pressure
coefficients at time t ¼ tnþ1, respectively, while uu
and up contain row vectors collecting the shape
functions evaluations at each collocation point and are
defined as follows
uu ¼
u1uðsu1Þ u2
uðsu1Þ � � � � � �u1uðsu2Þ u2
uðsu2Þ � � � � � �� � � � � � � � � � � �
u1uðsuMÞ u2
uðsuMÞ � � � � � �
26664
37775;
up ¼
u1pðs
p1Þ u2
pðsp1Þ � � � � � �
u1pðs
p2Þ u2
pðsp2Þ � � � � � �
� � � � � � � � � � � �u1pðs
pMÞ u2
pðspMÞ � � � � � �
266664
377775
ð19Þ
and unðspl Þ and pnðspl Þ are displacements and pressures
evaluated at each pressure collocation point at time
t ¼ tn, i.e.,
unðspl Þ ¼
uMðsp1ÞjnuMðsp2Þjn
� � �uMðspNÞjn
26664
37775; pnðs
pl Þ ¼
pNðsp1ÞjnpNðsp2Þjn
� � �pNðspNÞjn
26664
37775:
ð20Þ
3.2.2 Dynamic case
For the dynamic case, generalized Newmark methods
are used to integrate in time Eq. (12). In particular,
GN22 is used for displacements and GN11 for
pressures [7]. As a consequence of these time
integration schemes, the first time derivatives of u
and p as well as the second time derivative of u are
approximated as follows:
Meccanica (2018) 53:1441–1454 1445
123
_unþ1 ¼cbunþ1 � un
Dt� c
b� 1
� �_un �
c2b
� 1
� �€unDt
_pnþ1 ¼cbpnþ1 � pn
Dt� c
b� 1
� �_pn
€unþ1 ¼1
bunþ1 � un
Dt2� 1
b_unDt
� 1
2b� 1
� �€un:
ð21Þ
After consideration of expressions (21), Eq. (12) read:
leading to the following matrix representation
Eoe
qu00uðsukÞ �
1
bDt2uuðsukÞ � b
qu0pðsukÞ
� bcbDt
u0uðs
pl Þ ku00
pðspl Þ �
cCw
bDtupðs
pl Þ
2664
3775
�unþ1
pnþ1
� �
ð23Þ
where
_unðsukÞ ¼
_uMðsu1Þjn_uMðsu2Þjn� � �
_uMðsuMÞjn
26664
37775; €unðsukÞ ¼
€uMðsu1Þjn€uMðsu2Þjn
� � �€uMðsuMÞjn
26664
37775; etc:
ð25Þ
4 Representative numerical examples
The performance of the mixed collocation method for
1D poroelasticity presented above is assessed in the
next sections by means of some simple but very
representative problems assuming both quasi-static
(Sect. 4.1) and dynamic (Sect. 4.2) conditions. To this
end, the response computed with our method is
compared with finite element results and analytical
solutions. In particular we employ a 2-D finite element
formulation for porous media, which uses backward
Euler and generalized Newmark schemes (GN22,
GN11) for time integration of the quasi-static and
dynamic solutions, respectively. The resulting coupled
finite element equations are solved through a mono-
lithic approach [46, 47]. All the finite element
solutions reported in this section are obtained with
mixed triangular elements. In particular, for the solid-
skeleton problem, we consider both the two mixed
B-bar formulations P1P0 (3-noded triangle with linear
interpolation of the displacements and constant inter-
polation of the mean stresses) and P2P1 (6-noded
triangle with quadratic displacements and discontin-
uous linear mean stresses) [45]. Both these formula-
tions are combined with a linear interpolation of the
pore pressure, thus obtaining the two mixed triangles
denoted in the following as P1P0/P1 and P2P1/P1,
respectively.
Eoe
qu00Mjnþ1 �
1
bDt2uMjnþ1 �
b
qp0N jnþ1 ¼
1
bDt2uM jn �
1
bDt_uM jn �
1
2b� 1
� �€uMjn
� bcbDt
u0M jnþ1 þ kp00N jnþ1 �CwcbDt
pN jnþ1 ¼ � bcbDt
u0M jn � bcb� 1
� �_u0Mjn
�bc2b
� 1
� �Dt€u0Mjn �
cCw
bDtpN jn � Cw
cb� 1
� �_pN jn;
8>>>>>>><>>>>>>>:
ð22Þ
¼� 1
bDt2unðsukÞ �
1
bDt_unðsukÞ �
1
2b� 1
� �€unðsukÞ
� bcbDt
u0nðspl Þ � b
cb� 1
� �_u0nðs
pl Þ � b
c2b
� 1
� �Dt€u0nðs
pl Þ �
cCw
bDtpnðs
pl Þ � Cw
cb� 1
� �_pnðs
pl Þ
26664
37775; ð24Þ
1446 Meccanica (2018) 53:1441–1454
123
For both the numerical examples presented in
Sects. 4.1 and 4.2, we consider the Biot’s extension
[1] of the classical one-dimensional problem origi-
nally formulated by Terzaghi [4]. The conditions
illustrated in Fig. 1 for the quasi-static and in Fig. 7
for the dynamic problems are imposed at the bound-
aries of an infinite poroelastic layer of thickness
h ¼ 1 m. A compressive and uniform load q ¼100 kPa is constantly applied at one surface of the
layer.
The oedometric modulus Eoe ¼ 2:4� 104 kPa is
obtained assuming E ¼ 2:0� 104 kPa and m ¼ 0:25
for the Young’s modulus and the Poisson’s coefficient,
respectively. Further settings are q ¼ 2:0Mg=m3 and
qw ¼ 1:0Mg=m3 for the densities of the porous
medium and of the fluid, respectively, b ¼ 1 for the
Biot’s coefficient and k ¼ 1� 10�8m2=ðkPa sÞ for thepermeability (thus, the corresponding hydraulic per-
meability is kh ¼ qwg k ¼ 1� 10�7m=s). Finally, the
storage modulus value Cw ¼ 3� 10�7kPa�1 is
obtained from the known relation (see, e.g., [7, 47]):
Cw ¼ n
jwþ b� n
jsð26Þ
for the porosity n ¼ 0:5, the bulk modulus jw ¼2� 106kPa of the pore fluid and the bulk modulus
js ¼ 1� 107kPa of the solid phase.
4.1 1D consolidation in quasi-static conditions
The evolution is assumed to be quasi-static, and the
problem is therefore governed by Eq. (8), comple-
mented by the following boundary conditions (Fig. 1):
u0ð0; tÞ ¼ �q=Eoe
pð0; tÞ ¼ 0
uðh; tÞ ¼ 0
p0ðh; tÞ ¼ 0ð27Þ
and by the initial conditions:
uðx; 0Þ ¼ 0 pðx; 0Þ ¼ q=b: ð28Þ
We remark again that the use of a small initial time
step for time integration of the problem at hand often
induces spurious spatial oscillations in the finite
element pore pressure solution evaluated near the
free-drainage boundary. In view of this issue, several
researchers have presented theoretical estimates of the
so-called ‘‘critical’’ time step DtFEMc , i.e., of minimum
time step size which ensures a stable finite element
solution [37–41].
Accordingly, we consider as a reference solution
the pore-pressure profiles obtained for the problem at
hand with the 2-D finite element formulations for
porous media mentioned in Sect. 4. In particular, we
employ two regular meshes consisting of the same
number (2� 5� 10, with 10 subdivisions along the
layer thickness) of P1P0/P1 and of P2P1/P1 triangular
elements, respectively. Since the magnitude of pres-
sure oscillations tends to reduce over time, our
0 0.2 0.4 0.6 0.8 1y/h
0
0.2
0.4
0.6
0.8
1
1.2
1.4
p/q
FEM p1 (Δt = 0.1ΔtcFEM)
FEM p1 (Δt = 0.5ΔtcFEM)
FEM p1 (Δt = ΔtcFEM)
0 0.2 0.4 0.6 0.8 1y/h
0
0.2
0.4
0.6
0.8
1
1.2
1.4
p/q
FEM p2 ( t = 0.1 tcFEM)
FEM p2 ( t = 0.5 tcFEM)
FEM p2 ( t = tcFEM)
(a) (b)
Fig. 2 Finite element solutions of 1D quasi-static consolidation
in terms of pore pressure distributions along the layer at t ¼ Dt,for different choices of Dt ¼ DtFEMc =n (for n ¼ 1; 2; 10)
obtained using 2� 5� 10 triangular mixed B-bar P1P0/P1
(a) and P2P1/P1 (b) elements
Meccanica (2018) 53:1441–1454 1447
123
attention is focused on the results at the first time step,
as detailed in the following. In Fig. 2, we plot the
distributions of pore pressure calculated at the end of
the first integration time step (i.e., for t ¼ Dt), fordifferent choices of its size: Dt ¼ DtFEMc =n for
n ¼ 1; 2; 10; 30, where the critical time step size
DtFEMc ¼ 21 s is calculated as suggested in [40] for
the considered setting of porous solid parameters, size
of finite elements along y, interpolation order for pore
pressures and time-integration scheme.
Figure 2a shows that the solution obtained with
P1P0/P1 elements is affected by high spurious oscil-
lations for the sizeDtFEMc =10 of the first time step. Also
the solution obtained with P2P1/P1 elements exhibits
spurious oscillations in Fig. 2b for the same size of the
first time step. The magnitude of these oscillations is
less than that obtained with P1P0/P1 elements and this
result is consistent with the improved performance of
triangular Taylor-Hood elements (P2/P1) in terms of
satisfaction of the inf-sup (or LBB) condition, with
respect to equal order approximations (see, e.g., [14]
and references therein). However, the oscillations
observed in Fig. 2b are not negligible and also this
evidence is consistent with the already observed
performances of one-dimensional [14] and quadrilat-
eral Q2/Q1 [15] Taylor-Hood elements in the solution
of the classical 1-D consolidation problem at hand.
The solutions computed with the herein proposed
1D mixed collocation method are reported in Fig. 3.
These results are obtained adopting approximation
degrees pu ¼ 4 and pp ¼ 3 for displacements and
pressures, respectively, and M ¼ N ¼ ncp ¼ 20 con-
trol points for both variables. We investigate the
stability of the solution for the same choices of time
steps as in the finite element case. Figure 3 clearly
shows that the proposed collocation approach combi-
nes extremely good accuracy and stability properties,
constituting a viable and inexpensive alternative to
standard finite element methods. With the adopted
setting (see below for some studies on the effect of
mesh refinement and degree elevation), oscillations
are observed only for Dt ¼ DtFEMc =30 which are
however very limited in amplitude compared to the
ones reported in Fig. 2 for the finite element solutions.
We also have to remark here that the cost of one
collocation degree of freedom is much lower than that
of one degree of freedom computed in the Galerkin
framework (both in the case of isogeometric or finite
element formulations), since in the collocation
approach integrals are not to be computed and only
one evaluation per degree of freedom is required.
Accurate studies showing the cost-effectiveness of
isogeometric collocation with respect to Galerkin
approaches can be found in [20] and are beyond the
scope of the present work.
In Fig. 4, isogeometric collocation results are
reported in terms of distributions of pore pressure
and of its gradient (which is proportional to the fluid
flow qw, see Eq. 52) computed along the layer
thickness at instants t ¼ 10Dt, t ¼ 20Dt, t ¼ 30Dt,t ¼ 40Dt. Such results are computed again using
degrees pu ¼ 4 and pp ¼ 3 for, respectively, displace-
ments and pressures, M ¼ N ¼ ncp ¼ 20 control
points for both variables, and an integration time step
Dt ¼ DtFEMc . In the same figure, the comparison of
IGA-C results with the corresponding analytical
solutions confirms the good behavior of the proposed
method in representing not only primary variables like
the pore pressure, but also their gradients. In partic-
ular, the higher regularity of splines guarantees a
continuous pressure gradient (and, therefore, fluid
flow) profile, something that cannot be achieved
within the standard finite element framework, charac-
terized by C0-continuous inter-element regularity. In
addition, it is also to be remarked that, in correspon-
dence of the pervious boundary (i.e., at y ¼ 0), the
0 0.2 0.4 0.6 0.8 1y/h
0
0.2
0.4
0.6
0.8
1
1.2
1.4p/
q
IGA-C (Δt = ΔtcFEM)
IGA-C (Δt = 1/2ΔtcFEM)
IGA-C (Δt = 1/10ΔtcFEM)
IGA-C (Δt = 1/15ΔtcFEM)
IGA-C (Δt = 1/30ΔtcFEM)
Fig. 3 Isogeometric collocation (IGA-C) solutions of 1D quasi-
static consolidation in terms of pore pressure distributions along
the layer at t ¼ Dt, for different choices of Dt ¼ DtFEMc =n (for
n ¼ 1; 2; 10; 15; 30). Results obtained using
M ¼ N ¼ ncp ¼ 20, pu ¼ 4, pp ¼ 3
1448 Meccanica (2018) 53:1441–1454
123
proposed method is able to accurately reproduce the
flat profile (i.e., p00jy¼0 ¼ 0) of the analytical solution.
We finally study the method performance with
respect to the choices of number of collocation points
and of polynomial degrees. Accordingly, in Figs. 5
and 6 several combinations of approximations are
studied for the demanding situation
t ¼ Dt ¼ DtFEMc =10. It is possible to observe that
good and stable results are obtained whenever not too
coarse meshes are adopted.
It is interesting to remark that, for the problem at
hand, even equal order approximations perform well.
However, selecting a displacement approximation one
order higher than that for the pressure grants better
results in terms of stability, and, therefore, should be in
general the preferred choice.
4.2 1D dynamic coupled problem
The developed mixed isogeometric collocation for-
mulation is also adopted to solve a demanding one-
0 0.2 0.4 0.6 0.8 1y/h
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1p/
q
IGA-C (t = 10Δt)IGA-C (t = 20Δt)IGA-C (t = 30Δt)IGA-C (t = 40Δt)analytical sol.
0 0.2 0.4 0.6 0.8 1y/h
0
50
100
150
200
250
300
-p
[kP
a/m
]
IGA-C (t = 10Δt)IGA-C (t = 20Δt)IGA-C (t = 30Δt)IGA-C (t = 40Δt)analytical sol.
Δ
(a) (b)
Fig. 4 1D quasi-static consolidation. Isogeometric collocation
(IGA-C) results at different time instants for
M ¼ N ¼ ncp ¼ 20, pu ¼ 4, pp ¼ 3, and Dt ¼ DtFEMc ¼ 21 s,
compared with relevant analytical solutions: distributions along
the layer thickness of pore pressure (a) and of pore pressure
gradient (b)
0 0.2 0.4 0.6 0.8 1y/h
0
0.2
0.4
0.6
0.8
1
1.2
p/q
IGA-C (ncp = 10)IGA-C (ncp = 15)IGA-C (ncp = 20)IGA-C (ncp = 25)
0.04 0.06 0.08 0.1 0.12
0.94
0.96
0.98
1
1.02
1.04
1.06
1.08
0 0.2 0.4 0.6 0.8 1y/h
0
0.2
0.4
0.6
0.8
1
1.2
p/q
IGA-C (ncp = 10)IGA-C (ncp = 15)IGA-C (ncp = 20)IGA-C (ncp = 25)
0.02 0.04 0.06 0.08 0.1 0.12 0.14
0.9
0.95
1
1.05
(a) (b)
Fig. 5 1D quasi-static consolidation. Pore pressure distributions at the instant t ¼ Dt ¼ DtFEMc =10 computed with isogeometric
collocation (IGA-C) using different numbers of collocation points: a case with pu ¼ 4, pp ¼ 3; b case with pu ¼ pp ¼ 4
Meccanica (2018) 53:1441–1454 1449
123
dimensional dynamic coupled problem of stress and
pore pressure wave propagation. We consider the
configuration of the poroelastic layer problem illus-
trated in Fig. 7 characterized by the following bound-
ary conditions:
uð0; tÞ ¼ 0
p0ð0; tÞ ¼ 0
u0ðh; tÞ ¼ �q=Eoe
pðh; tÞ ¼ 0ð29Þ
and by the initial conditions:
uðx; 0Þ ¼ 0 pðx; 0Þ ¼ 0: ð30Þ
The employed values of all the parameters are again
those listed in Sect. 4. The evolution is assumed to be
fully dynamic, and the problem is therefore governed
by Eq. (6) complemented by appropriate boundary
and initial conditions.
This benchmark is numerically solved with the
proposed isogeometric collocation approach, consid-
ering two combinations of approximation degrees,
namely pu ¼ pp þ 1 ¼ 4 and pu ¼ pp þ 1 ¼ 5, and
three different meshes consisting of
M ¼ N ¼ ncp ¼ 100, M ¼ N ¼ ncp ¼ 150, and M ¼N ¼ ncp ¼ 200 control points. The obtained results are
compared with the analytical solution of the uncou-
pled undrained limit and with the numerical solution
computed using a regular mesh of 2� 4� 40 (with 40
subdivisions along the layer thickness) triangular
linear mixed B-bar P1P0/P1 finite elements. It can
be noted that, with respect to the quasi-static case
presented in the previous section, we employ herein a
finer finite element mesh and a higher number of
collocation points in order to obtain an improved
spatial resolution able to capture dynamic phenomena.
In all the simulations presented in this section, time
0 0.2 0.4 0.6 0.8 1y/h
0
0.2
0.4
0.6
0.8
1
1.2p/
q
IGA-C (pu = 3, pp = 2)IGA-C (pu = 4, pp = 3)IGA-C (pu = 5, pp = 4)IGA-C (pu = 6, pp = 5)
0.02 0.04 0.06 0.08 0.1 0.120.88
0.9
0.92
0.94
0.96
0.98
1
1.02
0 0.2 0.4 0.6 0.8 1y/h
0
0.2
0.4
0.6
0.8
1
1.2
p/q
IGA-C (pu = pp = 2)IGA-C (pu = pp = 3)IGA-C (pu = pp = 4)IGA-C (pu = pp = 5)IGA-C (pu = pp = 6)
0.05 0.1 0.150.9
0.95
1
1.05
1.1
(a) (b)
Fig. 6 1D quasi-static consolidation. Pore pressure distribu-
tions at the instant t ¼ Dt ¼ DtFEMc =10 computed with isogeo-
metric collocation (IGA-C) using M ¼ N ¼ ncp ¼ 20 and
different approximation degrees of displacement and pressure:
a Case: pu ¼ pp þ 1; b Case: pu ¼ pp
Fig. 7 1D infinite poro-elastic layer of thickness h ¼ 1:0m:
configuration of the dynamic problem in terms of geometry and
boundary conditions on displacements u and pore pressures p. A
constant and uniform compressive load q ¼ 100 kPa is instan-
taneously applied at the pervious boundary of the layer
1450 Meccanica (2018) 53:1441–1454
123
integration is performed using a time step
Dt ¼ 2� 10�7 s.
Figures 8 and 9 report the obtained results in terms
of time evolution of the pore pressure at y ¼ h=2 for
different durations (Tf ¼ 0:8� 10�3 s and
Tf ¼ 3:2� 10�3 s, respectively), showing the
expected response due to the stress and pore-pressure
wave propagation and reflection at the layer
boundaries.
In particular, in Fig. 8, we can appreciate that, as
the load is applied, a pore pressure wave of magnitude
p ¼ 100 kPa (p=q ¼ 1) starts propagating in the
poroelastic layer. With proper combinations of the
numbers of collocation points ncp and polynomial
degrees for the shape functions of the two fields (e.g.,
for ncp ¼ 150 and pu ¼ pp þ 1 ¼ 4, or for ncp ¼ 100
and pu ¼ pp þ 1 ¼ 5), the isogeometric collocation
scheme is able to provide a pore pressure response
without the significant spurious oscillations that are
observed using mixed P1P0/P1 finite elements.
Figure 9 shows the pore pressure evolution in the
case of a longer duration, which allows the develop-
ment of the expected effects of wave propagation and
reflection. In particular, we report the solutions
obtained with the proposed isogeometric collocation
method using ncp ¼ 150 and approximation degrees
pu ¼ pp þ 1 ¼ 4 or pu ¼ pp þ 1 ¼ 5, as well as the
finite element and the analytical solutions. The
stability and accuracy properties granted by the use
in this context of an isogeometric collocation approach
can be clearly appreciated.
Figure 10 reports the same results proposed in
Fig. 8, but this time in terms of spatial evolution at a
given instant t ¼ 1:2� 10�3 s. The same considera-
tions above hold true also for this setting.
Finally, in Fig. 11 we report the distribution of
displacements u obtained with the proposed
0 1 2 3 4 5 6 7 8
t [s] × 10-4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2p/
q
IGA-C (pu=4, pp=3, ncp=100)IGA-C (pu=4, pp=3, ncp=150)IGA-C (pu=4, pp=3, ncp=200)FEManalytical sol.
4 4.5 5× 10-4
0.9
0.95
1
1.05
1.1
1.15
0 1 2 3 4 5 6 7 8
t [s] × 10-4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
p/q
IGA-C (pu=5, pp=4, ncp=100)IGA-C (pu=5, pp=4, ncp=150)IGA-C (pu=5, pp=4, ncp=200)FEManalytical sol.
4 4.5 5× 10-4
0.9
0.95
1
1.05
1.1
(a) (b)
Fig. 8 Pore pressure evolution at y ¼ h=2 for a duration Tf ¼ 0:8� 10�3 s, compared with the finite element (P1P0/P1) and the
analytical solutions: a isogeometric collocation for pu ¼ pp þ 1 ¼ 4; b isogeometric collocation for pu ¼ pp þ 1 ¼ 5
0 0.5 1 1.5 2 2.5 3
t [s] × 10-3
-0.5
0
0.5
1
1.5
2
2.5
p/q
IGA-C (pu=4, pp=3, ncp=150)IGA-C (pu=5, pp=4, ncp=150)FEManalytical sol.
1.1 1.15 1.2 1.25 1.3 1.35× 10-3
1.85
1.9
1.95
2
2.05
2.1
2.15
Fig. 9 Pore pressure evolution at y ¼ h=2 for a duration
Tf ¼ 3:2� 10�3 s: isogeometric collocation results for different
approximation degrees and ncp ¼ 150 are compared with the
finite element (P1P0/P1) and the analytical solutions
Meccanica (2018) 53:1441–1454 1451
123
isogeometric collocation method using ncp ¼ 150 and
approximation degrees pu ¼ pp þ 1 ¼ 4 or
pu ¼ pp þ 1 ¼ 5, as well as the finite element and
the analytical solutions, all computed at a given instant
t ¼ 1:2� 10�3 s. It is easily seen that isogeometric
collocation is able to attain high accuracy also in terms
of displacements.
5 Conclusions
In the present paper, isogeometric collocation has been
used for the first time to simulate the behavior of fluid-
saturated porous media. Amixed collocation approach
has been developed in the one-dimensional framework
and tested in demanding situations on both quasi-static
and dynamic benchmarks. The proposed methodology
has proven to be very effective in terms of both
stability and accuracy, thanks to the peculiar proper-
ties of the spline shape functions typical of isogeo-
metric methods. These properties, along with the ease
of implementation and low computational cost guar-
anteed by the collocation framework, make the
proposed approach very attractive as a viable alterna-
tive to classical finite elements and other Galerkin-
based approaches. Accordingly, the potential of iso-
geometric collocation to successfully deal with the
simulation of poromechanics problems deserves to be
further and deeply explored, and upcoming studies
include the extension to higher space dimensions and
to the nonlinear framework.
Acknowledgements This work was partially funded by
Fondazione Cariplo-Regione Lombardia through the project
‘‘Verso nuovi strumenti di simulazione super veloci ed accurati
basati sull’analisi isogeometrica’’, within the program RST-
rafforzamento. C. Callari was partially supported by the DiBT
Department Project ‘‘Computational modeling of erosion effects
on dams, levees, and historic artefacts’’.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1y/h
0.8
1
1.2
1.4
1.6
1.8
2
2.2p/
q
IGA-C (pu=4, pp=3, ncp=100)IGA-C (pu=4, pp=3, ncp=150)IGA-C (pu=4, pp=3, ncp=200)FEManalytical sol.
0.51.9
1.95
2
2.05
2.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1y/h
0.8
1
1.2
1.4
1.6
1.8
2
2.2
p/q
IGA-C (pu=5, pp=4, ncp=100)IGA-C (pu=5, pp=4, ncp=150)IGA-C (pu=5, pp=4, ncp=200)FEManalytical sol.
0.5
1.85
1.9
1.95
2
2.05
2.1
(a) (b)
Fig. 10 Pore pressure wave profile at the instant t ¼ 1:2� 10�3 s, compared with the finite element (P1P0/P1) and the analytical
solutions: a isogeometric collocation for pu ¼ pp þ 1 ¼ 4; b isogeometric collocation for pu ¼ pp þ 1 ¼ 5
0 0.2 0.4 0.6 0.8 1y/h
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
u [m
]
× 10-5
IGA-C (pu=4, pp=3, ncp=150)IGA-C (pu=5, pp=4, ncp=150)FEManalytical sol.
0.52 0.54 0.56 0.58
-3.45-3.4
-3.35-3.3
-3.25-3.2
-3.15-3.1
-3.05× 10-5
Fig. 11 Distribution of displacements at instant
t ¼ 1:2� 10�3 s: isogeometric collocation results for different
approximation degrees and ncp ¼ 150 are compared with the
finite element (P1P0/P1) and the analytical solutions
1452 Meccanica (2018) 53:1441–1454
123
Compliance with ethical standards
Conflict of interest The authors declare that they have no
conflict of interest.
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