GeoELSEvp: a spectral element approach for dynamic elasto-viscoplastic problems
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GeoELSEvp: a spectral element approach for
dynamic elasto-viscoplastic problems
Marco Stupazzini,* Clara Zambelli**
Summary
A spectral element method for the numerical analysis of 2D and 3D wave propagation problems is equipped with a
viscoplastic constitutive model in order to provide a tool able to grant the flexibility of a finite element technique in dealing
with non linear constitutive models, together with the accuracy of spectral methods for propagating waves. The proposed
tool is both fast and behaves like a black-box requiring minimum interaction to the user. The implementation of the meth-
od is illustrated and applied to a set of 2D reference cases.
1. Introduction ear field and finally (iii) the scarce evidence of non
linear effects in the observed motion, other than liq-
Wave propagation phenomena can be easily uefied sites.
studied nowadays thanks to powerful numerical However in the last decades, the second and the
techniques and probably the most overwhelming third issues have changed completely their perspec-
advances have been recently achieved in computa- tives: the viscoplastic constitutive model has grown
tional seismology at global or regional scale significantly in importance among the solid dy-
[KOMATITSCH et al., 2003], [KOMATITSCH et al., 2004], namic scientific community, thanks to the effort de-
by employing visco-elastic constitutive models. Any- voted by many authors [PASTOR et al., 1995], [DI PR-
way, a large set of numerical techniques like finite ISCO et al., 2002], [MABSSOUT et al., 2003], and seems
differences (FD), finite elements (FE), spectral ele- to be one of the most promising models in order to
ments (SE) or integral boundary elements (IBE) study accurately non linear propagation phenom-
spurred by the computational power made available ena with numerical scheme based on explicit time
by parallel computers, have embraced the area of advancing discretization. Concerning the third is-
three-dimensional wave propagation and very inter- sue, a number of accelerograms have been recorded
esting results have been achieved particularly in the during strong earthquakes that have made possible
field of seismic wave propagation [M OCZO et al., to infer non linear response of soil, like reduction of
2003], [BIELAK et al., 1999], [BIELAK et al., 2003], shear wave velocity and the increase of soil damping
[STUPAZZINI, 2004], [SANCHEZ-SESMA et al., 1995]. Nev- with increasing load [HARDIN et al., 1972]. Reliable
ertheless, nearly all models used until recently in records of events available for the study of non lin-
seismology for predicting ground motion induced ear site effects have been provided in a particular
by earthquakes have been based on the assumption seismic region in Taiwan (SMART, Strong Motion
of linear elastic behavior of soil. Only few authors, in Accelerograph Array in Taiwan) [ABRAHAMSON et al.,
recent times, have shown the capabilities of taking 1987], [CHANG et al., 1989], [WEN, 1994], [BERESNEV,
into account non linear soil behavior in parallel fi- 1995]. Furthermore, the nonlinear behavior of soil
nite element code for large-scale applications [XU et during strong earthquakes, like Hygoken-Nambu
al., 2003]. 1995 [SATO et al., 1996] or Northridge 1994 [BARDET
This lack of numerical tools capable to solve et al., 1996], [TRIFUNAC et al., 1996], have been dem-
solid dynamic problems in non-linear field is basi- onstrated by different motion recorded. Finally, in
some strong ground motions accelerograms, ampli-
cally due to three main factors: (i) for years non-lin-
tude deamplification as well as changes in reso-
ear soil amplification has been routinely taken into
nance frequency have been identified [ARCHULETA,
account in geotechnical engineering practice by em-
1998], [ARCHULETA et al., 2000].
ploying simple non-linear elasticity [S EED et al.,
1969], (ii) the lack of satisfying constitutive model In this study a parallel spectral element method-
able to capture time rate dependence in the non lin- ology for modeling 2D and 3D wave propagation
problems is equipped with a viscoplastic constitutive
model in order to grant the flexibility of a finite el-
* Department für Geo- und Umweltwissenschaften Sektion ement technique in dealing with non linear consti-
Geophysik Ludwig-Maximilians Universität, Munich tutive model, together with the accuracy of spectral
** Department of Structural Engineering, Politecnico di Milano methods for propagating waves in heterogeneous
RIVISTA ITALIANA DI GEOTECNICA 4/2005GEOELSEVP: A SPECTRAL ELEMENT APPROACH FOR DYNAMIC ELASTOVISCOPLASTIC PROBLEMS 71
earth media. Due to the innovative aspects of a SE are finite. Summation on repeated indices will be
technique for dynamic analyses (for the static case understood, unless otherwise specified.
see, for instance, Cividini [CIVIDINI et al., 1997]) The previous equation must be supplemented
dealing with non linear constitutive model, a quite by suitable conditions prescribing the value of both
large dataset of comparisons with a very simple 2D u and ∂u/∂t for all x ∈ Ω at the initial time t=0. Con-
study case already developed in literature with FEM cerning the boundary conditions, for simplicity, but
based code [MABSSOUT et al., 2003] has been per- without loss of generality, displacements have been
formed and illustrated in the following. assumed to vanish on ΓD.
The stress and strain tensors in (1) are related to
the displacement by Hooke’s law
2. GeoELSE
(2)
The computational code employed to perform
and
the numerical analyses is named GeoELSE (Geo
ELasticity by Spectral Elements [M AGGIO et al.,
2001], [STUPAZZINI, 2004], [MAGGIO et al., 2005]). , (3)
This is a Spectral Element [FACCIOLI et al., 1997],
[KOMATITSCH, 1998] code for the study of wave prop- where λ and µ are the Lamé elastic coefficient and
agation phenomena in 2D or 3D complex domains. δij is the Kronecker delta, such that δij =1, if i=j,
The recently presented version of the code includes: and δij =0, otherwise.
(i) the capability of dealing with fully unstructured In addition to the “classical” solution of the
computational domains and (ii) the parallel architecture. standard differential formulation of the elastody-
While the former feature allows to treat problems namic problem, the variational approach can pro-
involving complex geometries, the second is the vide other significant solutions, like those with dis-
natural approach for large scale applications. Be- continuous gradient arising in heterogeneous me-
fore describing the numerical analyses performed dia where ρ, λ and µ are bounded functions, not nec-
and geotechnical problem taken into consideration, essarily continuous (for instance, they can be piece-
wise constant). The Galerkin method stemming
it seems worth to briefly introduce the key features
from (1) is the basis for the spectral element method
of the spectral element method adopted and then to
advocated in finite element methods, as well as in
outline the constitutive viscoplastic model intro-
this paper.
duced.
3.1. Space discretization with spectral elements
3. Formulation of the problem
An appropriate numerical solution of (1) can be
We consider an elastic medium occupying the fi- achieved with discretization in space and time do-
nite region Ω⊂ℜd (d=2 or 3 number of space di- main. Herein, the former is done via spectral ele-
mension) with boundary Γ=ΓN ∪ ΓD, where ΓD is ments and the latter via finite differences.
the portion where the displacement vector u is con- The spectral element method (SEM) is usually
trolled and ΓN the portion subject to external forces regarded as a generalization of the finite element
(tractions). Through the principle of virtual work, method (FEM) based on the use of high order piece-
the dynamic equilibrium problem for the medium wise polynomial functions. The crucial aspect of the
method is the capability to provide an arbitrary in-
can be stated in the following weak, or variational
crease of accuracy simply enhancing the algebraic
form [ZIENKIEWICZ et al., 1989]:
degree of these functions (the spectral degree). On
find u=u(x,t) such that ∀t ∈ (0,T)
practical ground, this operation is completely trans-
parent for the users, who limit themselves to choose
the spectral degree at runtime, leaving to the com-
(1) putational code the task to build up suitable quadra-
ture points and new degrees of freedom. Obviously,
increasing the spectral degree has also the effect to
where t is the time, ρ=ρ(x) the material density, σij raise the computational effort of the problem.
the stress tensor, εij the small-strain tensor, f=f(x,t) On the other hand, one can also play on the
a known body force distribution, t=t (x,t) the vector grid refinement to improve the accuracy of the nu-
of external traction prescribed on ΓN, and v=v(x) is merical solution, thus following the standard finite
a generic function (candidate to represent admissi- element approach. Spectral elements are therefore
ble displacements) chosen so that all integrals in (1) a so-called “h-p” method [F ACCIOLI et al., 1996],
OTTOBRE - DICEMBRE 200572 STUPAZZINI - ZAMBELLI
where “h” refers to the grid size and “p” to the de- Boundary integrals are evaluated in similar
gree of polynomials. manner.
In detail, the SEM adopted in GeoELSE is es- In order to illustrate the algorithmic aspects of
sentially based on the formulation proposed by Fac- the method, it is important to clarify well the de-
cioli [FACCIOLI et al., 1997]. The approximate nu- scription of LGL points and Lagrange polynomials: the
merical solution, uN, as well as the admissible dis- former are the points where the Eq. (4) is actually
placement vN, are chosen as piecewise polynomial computed and where numerical integration and dif-
functions of degree N, continuous in the whole do- ferentiation are performed, while the latter are the
main. To simplify the notation, the set of such poly- interpolating polynomials which approximate, the
nomials is denoted by VN, and the subset of those solution uN through LGL points.
vanishing on ΓD by V 0N. Then Eq. (1) is approxi- The distribution of LGL nodes in 2D, for differ-
mated by the following “semi-discrete” formulation: ent values of N, is shown in Figure 1.
The LGL nodes {ξp} correspond, in a normal-
ized 1D situation, to the zeroes of L’N, the first space
such that: derivatives of the Legendre polynomial LN of de-
gree N [ABRAMOVITZ et al., 1966] along with the ex-
(4) tremes of the interval:
. (6)
for all . The previous piecewise poly- For a degree N of the Legendre polynomial,
nomial function can be defined more precisely one has N-1 zeroes of L’N, plus the two extremes, i.e.
thanks to the following assumptions: the computa- a total of (N+1) LGL nodes in each space direction.
tional domain Ω is decomposed into a family of SH For 2D and 3D problems LGL points are (N+1)d
non overlapping quadrilaterals Ωk (hexahedra in (d=2,3).
3D), with typical linear size H and such that the in- Each element Ωk is obtained by a mapping of
tersection between two elements is either empty, or the master element Ωref through a suitable transfor-
– –
a vertex, or an edge, and Ω=∪k Ωk. A fully non- mation x=Fk , where {x} and are the cartesian
structured domain decomposition was chosen in or- coordinates in Ωk and Ωref, respectively (Figure 2).
der to deal with complex computational domains.
Thanks to transformation Fk , all computa-
Each element Ωk is mapped into a reference el-
tions are performed on the master element Ωref and
ement Ωref (the square [–1,+1]2, in 2D, and the cube
then brought back into each subdomain Ωk this is
[–1,+1]3, in 3D) which is used to evaluate derivatives the reason why LGL points are shown only in Ωref.
and integrals in (4).
1 The restriction of uN and vN to each quadrilate- Fk is characterized by the Jacobian Jk . Ac-
ral in 2D (or hexahedra in 3D) is the mapping ceptable elements are generated when the sign of
of polynomials of degree N with respect to each the Jacobian does not change.
space variables defined in Ωref; uN and vN are Once the LGL points have been defined, to im-
continuous across the element interface. plement the method we need a set of independent
2 Each derivative function in (4) is evaluated polynomials {pj}⊂ V 0N ; this enables to express the i-
through a linear combination of function values th component of the numerical solution as
at the so-called Legendre-Gauss-Lobatto (LGL) , where the ’s are the unknowns
nodes: the linear combination is expressed by
means of the spectral differentiation matrix. of the problem. Consequently, the Eq. (4) needs to
Details on the LGL nodes and the spectral diffe-
rentiation matrix are provided in the sequel.
3 Each volume integral in (4) is evaluated numeri-
cally by a high accuracy approximation, specifi-
cally the Legendre-Gauss-Lobatto quadrature
formula:
(5)
Fig. 1 – LGL points within SEM elements with different
where xj(k) is the vector of the coordinates of the
spectral degree N.
LGL nodes in Ωk and the ωj(k) are the corresponding Fig. 1 – Punti LGL all’interno di elementi spettrali aventi grado
weights [CANUTO et al., 1988], [DAVIS et al., 1984]. spettrale N differente.
RIVISTA ITALIANA DI GEOTECNICAGEOELSEVP: A SPECTRAL ELEMENT APPROACH FOR DYNAMIC ELASTOVISCOPLASTIC PROBLEMS 73
(7)
taking value 1 in ξi and 0 in ξj, for i≠j, i.e.:
(8)
For example, for N=6 one has 7 LGL points
and 7 Lagrange polynomials: some of them, in the
1D case, are shown in Figure 3. In this figure the
normalized basis [0,1] was used instead of [–1,1].
In 2D or 3D, the interpolation functions are ob-
tained by tensor product of 1D functions of Eq. (7).
For example in 2D, the generic spectral element shape
function, is:
(9)
In Figure 4 are depicted three Lagrange poly-
Fig. 2 – Computational domain Ω is decomposed into a nomials for a simple 2D case.
family of non overlapping quadrilaterals Ωk, obtained by The spectral domain decomposition formula-
a mapping of the master element Ωref =[–1,1]2 through a tion can now be written:
suitable transformation x=Fk [CASADEI et al., 2000].
Fig. 2 – Il dominio di calcolo Ω viene suddiviso in un insieme
di quadrilateri Ωk, che non si sovrappongono, come risultato di such that:
una mappatura dell’elemento di riferimento Ωref =[–1,1]2, tra-
mite un’opportuna trasformazione x=Fk [CASADEI et al.,
2000].
(10)
be satisfied only by such {pj}, i.e. taking vN = pjei, for
all j, where ei is the i-th base unit vector.
It turns out that the right choice for the space
components of {v} is the Lagrange polynomials of
degree N. In fact, this choice allows on one hand to where and f (i) denote the scalar
preserve the exponential accuracy of the spectral components of the vectors uN, vN, t and f, respecti-
method and, on the other hand, to minimize the
vely; denotes the part of the boun-
computational effort. Note that with the choice of
dary of Ωk shared with ΓN. In (10) the continuity of
Lagrange polynomials, the unknown coefficients
the displacements across the interface is ensured by
coincide with the values of the i-th component assumption. Further, it can be shown that the equa-
of uN(t) at the j-th grid point. tions of motion are satisfied by uN at all internal no-
In 1D, the i-th Lagrange polynomial can be ex- des, and that tractions at the interfaces can be di-
pressed by the classical product formula as: scontinuous but their “jump” vanishes as N → ∞.
Fig. 3 – Lagrange polynomials of degree 6 centered on different LGL nodes [CASADEI et al., 1997].
Fig. 3 – Polinomi di Lagrange di grado 6 posizionati su diversi nodi LGL [CASADEI et al., 1997].
OTTOBRE - DICEMBRE 200574 STUPAZZINI - ZAMBELLI
Fig. 4 – Lagrange polynomials for a 2D case (top), associated with different nodal points shown by solid dots, in a simple
grid with four elements Ωk (bottom). The plots refer to an internal point (left), an interface point (centre), and a cross-point
(right), respectively [FACCIOLI et al., 1997].
Fig. 4 – Polinomi di Lagrange in caso bidimensionale, associati a diversi punti nodali (punto in grossetto) in una maglia semplice formata
da 4 elementi Ωk. La figura fa riferimento ad un punto interno (sinistra), un punto di interfaccia (centro) e un cross-point (destra)
[FACCIOLI et al., 1997].
The traction condition on ΓN is satisfied in a similar
(12)
manner. A detailed analytical derivation for the 1D
cases is given elsewhere [FACCIOLI et al., 1996]. It
must be stressed that the actual implementation of where are the LGL points and the are the
the method here illustrated makes direct use of Eq. corresponding weights. From Eq. (10) a system of
(10), by picking the Lagrange functions associated algebraic equations is obtained by first introducing
with each nodal point. (11) for the unknown uN. Then, for each scalar com-
It is well known that when the “exact” solution u ponent of displacement, as many equations are gen-
is sufficiently smooth, the numerical solution uN erated as the number of nodal points. Each equa-
provided by the proposed method is expected to tion is obtained from Eq. (10) by choosing
converge u to as u–uN 1 ≤CHN exp(–N), where C is (i=1…d), where Ψr is the Lagrange
a constant independent from H and N. Therefore, polynomial associated with r-th LGL node. In this
spectral elements converge more rapidly to the way Eq. (10) can be written as an ordinary differen-
exact solution than finite elements, a property tial equations system with respect to time:
known in literature as spectral accuracy.
Assembling the grids of the single elements (13)
yields the global LGL grid, that is denoted by {xq} in
where matrices [M] and [K], respectively the mass
the sequel. As stated earlier, to the q-th node is asso-
and the stiffness matrix, arise from the terms on the
ciated a continuous piecewise polynomial function
left-hand side of (10), vectors F and T are due to the
of degree N that takes value 1 at the node consid-
contributions of external forces and tractions condi-
ered and vanishes at the others; this function is de-
tions, respectively, and U are the displacement val-
noted by Ψq(x). Owing to this choice, the i-th dis-
ues uN(xq,t) at LGL nodes. The advancement of nu-
placement component is expressed as:
merical solution in time will be discussed in the fol-
(11) lowing section.
where . Numerical differenti- 3.2. Time discretization
ation and integration make also use of LGL nodes.
Both are performed in the master element through The full numerical treatment of the problem at
the transformation Fk. hand requires also the time-discretization of
Integration is performed going back to the mas- Eq. (13). Several finite difference schemes have
ter element Ωref as follows; let the integrals appear- been tested for the scalar 2D wave equation [MAG-
ing in the reference element be approximated with GIO et al., 1994], both implicit and explicit. While
the quadrature formula: the former are unconditionally stable, the latter
RIVISTA ITALIANA DI GEOTECNICAGEOELSEVP: A SPECTRAL ELEMENT APPROACH FOR DYNAMIC ELASTOVISCOPLASTIC PROBLEMS 75
must satisfy the well known Courant-Friedrichs- capable of capturing all the phenomena associated
Levy (CFL) condition: to the coupling between water and soil skeleton, in
which soil liquefaction is included. Solely in the last
(14) decade non-linear numerical analyses of dynamic
problems have been performed. In this field the
where ∆xmin is the shortest spectral grid spacing, time factor plays a fundamental role, for this reason
cmax is an upper bound for the wave propagation ve- an elastoplastic approach is not suitable because the
locity, and γ is a positive constant strictly less than 1. dependency of the mechanical response on the load
Since ∆xmin corresponds to the LGL points closest to frequency, during cyclic tests, or on the load period
the boundary, where the spectral grid size is propor- when impulsive tests are analysed, is completely
tional to N–2, the condition for numerical stability provided by the inertial term. Conversely, the
can also be written as: elasto-viscoplastic constitutive models take actually
into consideration the dependency of the mechani-
cal behaviour of soils on the time factor, but such a
(15)
dependency is very well known as far as cohesive
soils are concerned but less when soft rocks and
where L is the typical linear dimension of Ωk. For granular soils [DI PRISCO et al., 1996] are considered.
example if N = 5, –γ ≅ 4.3γ. The condition (15) A standard Perzyna [PERZYNA, 1963] type elasto-
shows that the stability requirement on ∆t becomes
viscoplastic constitutive model for the ideal cohe-
prohibitive for small values of L/N2 (which corre-
sive-frictional material has been implemented in
sponds to the p-version), for which an implicit time
this spectral element numerical algorithm.
discretization is strongly recommended. For the h
The material mechanical response is assumed to
and h-p version, the best tradeoff in terms of accu-
be characterized by the superimposition of an in-
racy, stability and computational complexity (also in
stantaneous elastic strain increment and a delayed
view of parallel computing), is provided by the ex-
plastic strain:
plicit 2nd order leap-frog scheme (LF2-LF2) [MAG-
GIO et al., 1994].
(17)
where is the elastic strain rate tensor and is
(16)
the viscoplastic one.
The stress rate is given by
Finally, it is worth summarizing the crucial as-
pects of the here adopted SEM technique, com- (18)
pared with a classical FEM: (i) spectral elements
converge more rapidly to the exact solution or,
equivalently, it is more accurate for a given number where Deijhk is the elastic constitutive tensor (Eqs. 2
of nodal points (spectral accuracy), when the exact and 3) and εhk the strain tensor. The viscoplastic
solution is sufficiently smooth; (ii) the polynomial strain increment [PERZYNA, 1963] may be defined
order of the method can be set at run-time, without in the following way:
modifying the computational grid like in FEM; (iii)
the mass matrix is diagonal for construction.
Thanks to this latter fact the inversion needed into (19)
an explicit time discretisation come up to be trivial
in terms of computational effort. where f is the yield function, g the plastic potential,
φ( f ) the viscous nucleus and γ is a constitutive para-
meter, that influences the strain rate and conse-
4. GeoELSEvp quently the rapidity with which the asymptotic
strain value is reached. In above, ⟨.⟩ are the Macau-
The kernel described up to now follows the most lay brackets,
simple way of interpreting the dynamic response of
soils, that consists in defining an elasto-viscous
model in which the elastic stresses are added to the
viscous ones and both contributes are obtained by
assuming the linearity of the constitutive relation- The plastic potential g defines the direction of
ship. By following such an approach, the coupling the viscoplastic strain increment tensor and the
between the shear and the volumetric components yield function f influences its modulus by means of
is missing, and, consequently, these models are not the viscous nucleus φ( f ).
OTTOBRE - DICEMBRE 200576 STUPAZZINI - ZAMBELLI
By using the present flow rule, it is not necessary 5.1. Biaxial compression test
to introduce the consistency rule, as the viscoplastic
strain rate modulus depends directly on the func- The aim of this section is the analysis of a bidi-
tion describing the viscous nucleus and not on the mensional specimen subjected to a rapid wave load-
plastic multiplier. The yield function may be posi- ing on its upper face. The study case is sketched in
tive or negative, without any constraint, i.e. the Figure 5 and it consists of a square 1 m wide, but for
stress state may be external or internal to the yield the symmetry only one half has been considered in
locus. It is important to underline that from a nu- the numerical analyses. The imposed boundary
merical point of view, this approach allows to avoid conditions are the following:
the loading-unloading criterion definition; as a con- (i) on Γ1 ux =uy =0
sequence the numerical implementation of this kind (ii) on Γ2 the symmetry results on ux =0
of constitutive models into FEM and SEM codes is
(iii) Γ3 is a stress free boundary (σxx =0 and σxy =0)
more simple.
(iv) finally, displacement at Γ4 is given by (Figure 6a):
Among the several alternative choices for the
function φ( f ), it has been used the following ux =0
uy =u0(t),
where u0 (t) is given by:
(20) u0 (t)=0.06667 · t 0≤ t≤tf,
=0.01 t>tf,
where N is a model parameter, and f is the yield fun- where tf has been taken as 0.15 s.
ction. The value f0 characterizes the stress level be- The numerical simulations (Fig. 7) show clearly
low which no flow occurs. In the case of a Von Mises
that the strain localization is captured by GeoEL-
material, f is taken as the effective stress σ–
SEvp both playing on the grid size (“h”) or on the
spectral degree (“p”). This feature must be consid-
(20a) ered like a fundamental one into a SE technique
and it is preserved even if a non-linear constitutive
where sij is the deviatoric stress tensor. The size of
model is introduced. The analysis has been per-
the yield surface varies according to the following
formed with two different structured meshes: the
suitable hardening/softening law:
“fine” mesh is characterized by a large number of el-
ements (200 el.) and a spectral degree (SD) equal to
(21) 2, that can be considered as a low interpolation de-
gree for a SE technique. The second mesh, “coarse”,
where H is the hardening modulus.
For the sake of simplicity, in the following an as-
sociated flow rule has been assumed and the accu-
mulated viscoplastic strain is defined as follows:
(22)
.
5. Numerical examples
Two different “ideal tests” will be discussed and
numerically analysed. Both the first and the second
are biaxial tests in which the specimen is subjected
to an imposed upper bound displacement charac-
terized, respectively, by a (i) rapid compression
wave and (ii) a tensile shock wave. All study cases are
characterized by an ideal material with the following
properties: Young’s modulus E=8×10–7 Pa, Pois-
–
son’s ratio ν=0.3, an initial yield stress σ=5×10 5
Fig. 5 – Bidimensional specimen subjected to a displace-
Pa and a softening modulus of H=–E/10. Parame- ment loading.
ters of Perzyna’s model are γ=20s–1 and N=1. Den- Fig. 5 – Campione bidimensionale soggetto ad uno spostamento
sity is ρ=2000 kg/m3. imposto.
RIVISTA ITALIANA DI GEOTECNICAGEOELSEVP: A SPECTRAL ELEMENT APPROACH FOR DYNAMIC ELASTOVISCOPLASTIC PROBLEMS 77
discretizes the domain with only 8 elements but the
spectral degree is chosen equal to 10. It is evident
from Figure 7c and Figure 7d that almost the same
results in terms of accumulated viscoplastic strain
are obtained in these two different analyses (snap-
shot taken at t = 0.15 s. of Fig. 6).
Thanks to the capabilities of GeoELSE kernel to
handle unstructured domain it is possible to study
the same problem with the two very different un-
structured meshes depicted in Figure 8a and b. Dif-
ferences between the two meshes lay on (i) the
Fig. 6 – Imposed vertical displacement on the bidimensional
number of elements adopted, (ii) the spectral degree
specimen upper bound as (a) rapid wave or (b) shock wave. and (iii) the “orientation” of the general pattern.
Fig. 6 – Spostamento verticale imposto sul bordo superiore del It can be stated that even with an unstructured
campione bidimensionale come (a) onda veloce o (b) onda di “shock”. mesh (Fig. 8) GeoELSEvp is able to reproduce the
same viscoplastic strain pattern obtained with the
Fig. 8 – Influence of the mesh size and spectral degree in
Fig. 7 – Influence of the mesh size and spectral degree in strain softening regime: deformed mesh (a) 14 elements
strain softening regime: deformed mesh (a) “fine” with with SD=7 and (b) 34 elements with SD=4. Accumulated
SD=2 and (b) “coarse” with SD=10. Accumulated viscopla- viscoplastic strain respectively for mesh (c) with 14 el. and
stic strain respectively for mesh (c) “fine” and (d) “coarse”. (d) with 34 el.
Fig. 7 – Influenza della dimensione della griglia e del grado Fig. 8 – Influenza della dimensione della griglia e del grado
spettrale per materiale avente comportamento fragile: griglia spettrale per materiale avente comportamento fragile: griglia
deformata (a) “fine” con grado spettrale 2 e (b) “grossolana” con deformata (a) 14 elementi con grado spettrale 7 e (b) 34 elementi
grado spettrale 10. Deformazione viscoplastica accumulata, con grado spettrale 4. Deformazione viscoplastica accumulata,
rispettivamente per la griglia (c) “fine” e (d) “grossolana”. rispettivamente per la griglia (c) 14 el. e (d) 34 el.
OTTOBRE - DICEMBRE 200578 STUPAZZINI - ZAMBELLI
structured meshes (Figure 7). Obviously, due to the PRISCO et al., 2002], [DI PRISCO et al., 2003]. It is im-
mesh dependence phenomena, the stencil of the portant to underline that the viscoplastic constitu-
spectral mesh triggers the shear band in different tive model is not capable of avoiding numerical
ways (see Figs. 8c and d). solution mesh dependency due to the particular
This kind of problem is observed when strain viscous nucleus definition as well as the viscous
localisation is numerically analysed. To avoid constitutive parameter value. In fact, the constitu-
mesh dependence of solution, viscoplastic consti- tive parameter γ represents the characteristic time
tutive models can be employed. Unfortunately, as which defines the evolution of the material micro-
already shown by Sluys [SLUYS, 1992] and by di Pri- structure on the material mechanical response,
sco [DI PRISCO et al., 2003], this approach is not al- this is experimentally calibrated by means of
ways sufficient to avoid mesh dependence when creep tests and not on the localization profile and,
the strain softening regime is considered. One as a consequence, not to regularize the numerical
possible solution will be the implementation of solution.
nonlocal constitutive model into GeoELSEvp ker- Figure 10 shows the load-displacement curves
nel as it has already been successfully introduced obtained by means of the four meshes previously
in finite difference and finite element codes [ DI considered (Figs. 7 and 8). Up to 0.007 m of dis-
Fig. 10 – Load-displacement curves obtained during a
strain softening regime; data are filtered with a low-pass
300 Hz filter.
Fig. 10: Curve carico-spostamento per materiale avente
comportamento fragile; i dati sono stati filtrati passa-basso a
300Hz.
Fig. 9 – Influence of the mesh size and spectral degree in
strain hardening regime: deformed mesh (a) “fine” with
SD=2 and (b) 34 elements with SD=4. Accumulated vi-
scoplastic strain respectively for mesh (c) with “fine” and
(d) with 34 el. Fig. 11 – Load-displacement curves obtained during a
Fig. 9: Influenza della dimensione della griglia e del grado strain hardening regime; data are filtered with a low-pass
spettrale per materiale avente comportamento incrudente: griglia 300 Hz filter.
deformata (a) “fine” con grado spettrale 2 e (b) 34 elementi con Fig. 11 – Curve carico-spostamento per materiale avente
grado spettrale 4. Deformazione viscoplastica accumulata, comportamento incrudente; i dati sono stati filtrati passa-basso a
rispettivamente per la griglia (c) “fine” e (d) 34 el. 300Hz.
RIVISTA ITALIANA DI GEOTECNICAGEOELSEVP: A SPECTRAL ELEMENT APPROACH FOR DYNAMIC ELASTOVISCOPLASTIC PROBLEMS 79
placement, all results seem to coincide; subse- based on a velocity-stress formulation, by Mabssout
quently the curves suddenly differ because of the and Pastor [MABSSOUT et al., 2003].
mesh dependence. The bidimensional specimen of Figure 5 is sub-
On the other hand, the results obtained with the jected to a shock wave on its upper boundary.
same analysis and hardening behavior are depicted The imposed displacement at Γ 4 is given by
in Figure 11; it is evident that all the curves coin- (Figure 6b):
cide, and therefore, mesh dependence disappears ux =0
in this case. Thanks to the assumption of associated uy =–u0(t),
flow rule, the strain pattern obtained in hardening where u0 (t)is given by:
regime does not show a clear localization direction u0 (t)=1·t 0≤t≤tf,
(Figure 9). It is worth noting that the snapshot scale = 5×10-3 t>tf,
adopted in softening or hardening regime does not
where tf has been taken as 5×10–3s.
coincide.
The shear band starts from point A (Fig. 5)
when the wave is reflected at the base.
5.2. Strain localization in 2D specimen The amplitude of the wave has been chosen
such that the stresses do not reach the yield surface
The second example is taken from literature until the wave is reflected at the bottom, which
and has been already analysed with a FEM code, happens at t = 4.5 × 10 –3 s. Hereafter, the stress
Fig. 12 – Influence of the mesh size and spectral degree: (a) fine mesh with SD=1, (b) 8 elements structured mesh with
SD=7, (c) 2 elements mesh with SD = 13 and (d) 34 elements unstructured mesh with SD =3.
Fig. 12 – Influenza della dimensione della griglia e del grado spettrale: (a) maglia fine con grado spettrale pari a 1, (b) maglia strutturata
ad 8 elementi con grado spettrale pari a 7, (c) maglia strutturata di 2 elementi con grado spettrale pari a 13 e (d) maglia non strutturata
di 34 elementi e grado spettrale pari a 3.
OTTOBRE - DICEMBRE 200580 STUPAZZINI - ZAMBELLI
doubles, the stress path crosses the yield surface, captured at 9 ms. Therefore if the simulation went
and the strain localizes in the form of a shear band on, shear band will increase and once again mesh
which is incepted at point A (Fig. 5). Inclination of dependence could be observed.
the shear band is 58°, which agrees well with ana- Finally, it is crucial to underline, once again,
lytical solution. the capability of GeoELSEvp to provide an arbi-
The numerical results obtained in terms of de- trary increase of accuracy, achieved just increasing
formed mesh are shown in Figure 12, in particular: the spectral degree at run time and preserving the
(a) fine mesh with SD equal to 1, (b) 8 elements user from any remeshing operation. The code is
structured mesh with SD equal to 7, (c) 2 elements encharged to fulfill the task to build up a suitable
mesh with SD equal to 13 and (d) 34 elements un- quadrature points and new degree of freedom.
structured mesh with SD equal to 3. While the cor- This aspect is clearly shown in Figure 13, where the
responding contours of the accumulated viscoplas- results obtained analyzing only the 8 elements
tic strain are given respectively in Figure 12 (e), (f), mesh with different spectral degree are reported.
(g) and (h). Though the meshes considered are ex- The spectral degree 4 (Fig. 13a) is already capable
tremely different, among each others, both from of capturing the shear band occurence, anyway go-
the grid size point of view and from the spectral ing up to spectral degree 7 (b) and spectral degree
degree adopted, the localization of shear band 10 (c), the solution is improved. On the contrary,
shows a very similar shape. Even if this second ex- from spectral degree equal to 10 up to spectral de-
ample seems to highlight no mesh dependence, it gree 13 (Fig. 13d) no further solution refinement
is worth to stress out the fact that the snapshot are is recognizable.
Fig. 13 – Influence of spectral degree with an 8 elements structured mesh: (a) SD=4, (b) SD=7, (c) SD=10 and (d) SD=13.
Fig. 13 – Influenza del grado spettrale per una maglia strutturata di 8elementi: (a) grado spettrale pari a 4, (b) grado spettrale pari a
7, (c) grado spettrale pari a 10 e (d) grado spettrale pari a 13.
RIVISTA ITALIANA DI GEOTECNICAGEOELSEVP: A SPECTRAL ELEMENT APPROACH FOR DYNAMIC ELASTOVISCOPLASTIC PROBLEMS 81
Conclusions ARCHULETA R.J. (1998) – Direct observation of nonlinear
soil response in acceleration time histories. Seism. Res.
GeoELSE is a Spectral Elements code for the Lett., 69, 149.
study of wave propagation phenomena in 2D or 3D BARDET J.P., DAVIS C., (1996) – Engineering observa-
complex domains. One of the peculiarities of the tions on ground motion at the Van Norman complex af-
formulation of SEM adopted in such a numerical ter the 1994 Northridge earthquake. Bull. Seism. Soc.
code is the strong similarity with the FEM. This al- Am., 86, pp. 333-349.
lows the authors to introduce a viscoplastic constitu- BERESNEV I.A., WEN K.L., YEH Y.T. (1995) – Seismo-
tive model quite easily into the GeoELSE kernel and logical evidence for nonlinear elastic ground behavior
testing its capabilities to handle non linear behavior during large earthquakes. Soil Dyn. Earthquake
with time rate dependence. Eng., 14, pp. 103-114.
The very simple constitutive law introduced for BIELAK J., XU J., GHATTAS O. (1999) – Earthquake
the soil is justified by the goal of the research work ground motion and structural responsein alluvial val-
which does not consist in reproducing a real case leys. J. Geotech. Geoenviron. Eng., 125, pp. 413-
but in illustrating the capability of the approach. 423.
The “ideal test” numerically simulated concerns the B IELAK J., L OUKAKIS K., H ISADA Y., Y OSHIMURA C.
localization of a soil specimen characterized by a (2003) – Domain reduction method for three-dimen-
marked strain softening. The 2D examples pro- sional earthquake modeling in localized regions. Part I.
vided show that the SEM version adopted is able to Theory Bull. Seism. Soc. Am., 93, 2, pp. 817-824.
capture non linear phenomena with good accuracy. CANUTO C., FUNARO D. (1988) – The Schwarz algo-
In particular, it is worth noting that (i) the shear rithm for spectral methods. SIAM J. Numer. Anal.,
band formation (both in rapid wave and shock wave) 25, pp. 24-40.
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refinement of the computational domain. 3D coupled spectral element / finite element solver for
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lations. Part I – Models. EUR 17730 EN.
CASADEI F., FOTIA G., GABELLINI E., MAGGIO F., QUAR-
Acknowledgments TERONI A. (2000) – A mortar spectral/finite element
method for complex 2D and 3D elastodynamic prob-
The authors are grateful to Prof. C. di Prisco for
lems. Computer Methods in Applied Mechanics
his stimulating suggestions and capability to
and Engineering, 191, pp. 5119-5148.
strongly encourage this research. Eng. L. Frassine
C HANG C.Y., P OWER M.S., TANG Y.K., M OK C.M.
for proposing the “young researchers project”
(1989) – Evidence of nonlinear soil response during a
thanks to which the research had been partly sup-
moderate earthquake. In “Proc. 12th Int. Conf. on
ported. Dott. F. Maggio and Eng. L. Massidda of Soil Mechanics and Foundation Engineering”, 3,
CRS4 for allowing the user to adopt GeoELSE ker- pp. 1-4, Rio de Janeiro.
nel and their general support. Prof. H. Igel, coordi- CIVIDINI A., ZAMPIERI E., (1997) – Non linear stress anal-
nator of SPICE project, for his support. Prof. R. ysis problems by spectral collocation methods. Comp.
Paolucci and Prof. E. Faccioli for their suggestions. Meth. Appl. Mech. Eng., 145, pp. 185-201.
The research was partially supported by the Euro- DAVIS P., RABINOWITZ P. (1984) – Methods of numerical
pean Union (Project DIGA and Project SPICE). integration. 2nd edtn, Academic Press, Orlando.
DI PRISCO C., IMPOSIMATO S. (1996) – Time dependent
mechanical behaviour of loose sand. Mechanics of co-
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