-
si
ui926an D
Article history:Received 11 August 2008Received in revised form
22 January 2009Accepted 9 February 2009Available online 28 February
2009
Keywords:Nonlinear nite element analysis
Finite element (FE) response sensitivity analysis is an
essential tool for gradient-based optimization
Several methods are available for response sensitivity
computa-tion, including the nite difference method (FDM), the
adjointmethod (AM), the perturbation method (PM), and the direct
differ-entiation method (DDM). These methods are described by
Zhangand Der Kiureghian [3], Kleiber et al. [2], Conte et al. [46],
Guand Conte [7], Scott et al. [8], and Haukaas and Der
Kiureghian[9]. The FDM is the simplest method for response
sensitivity com-putation, but is computationally expensive and can
be negatively
Based on DDM, this paper presents a derivation of response
sen-sitivities with respect to material parameters of an existing
mate-rial model, the multi-yield-surface J2 plasticity model. This
modelwas rst developed by Iwan [10] and Mroz [11], then applied
byPrevost [1214] to soil mechanics. It was later modied and
imple-mented in OpenSees [1517] by Yang [18] and Elgamal et al.
[19].OpenSees is an open source software framework for
advancedmodeling and simulation of structural and geotechnical
systemsdeveloped under the auspice of the Pacic Earthquake
EngineeringResearch (PEER) Center. In contrast to the classical J2
(or VonMises) elasto-plastic behavior with a single yield
surface,
* Corresponding author. Tel.: +1 858 822 4545; fax: +1 858 822
2260.
Comput. Methods Appl. Mech. Engrg. 198 (2009) 22722285
Contents lists availab
A
.eE-mail address: [email protected] (J.P. Conte).1.
Introduction
Finite element (FE) response sensitivities represent an
essentialingredient for gradient-based optimization methods
required invarious sub-elds of structural and geotechnical
engineering suchas structural optimization, reliability analysis,
system identica-tion, and FE model updating [1,2]. In addition, FE
response sensitiv-ities are invaluable for gaining insight into the
effects and relativeimportance of system and loading parameters in
regards to systemresponse.
affected by numerical noise (i.e., truncation and round-off
errors).The AM is extremely efcient for linear and nonlinear
elastic sys-tems, but is not a competitive method for
path-dependent (i.e.,inelastic) problems. The PM is computationally
efcient, but gener-ally not very accurate. The DDM, on the other
hand, is general,accurate and efcient and is applicable to any
material constitutivemodel (both path-independent and
path-dependent). The compu-tation of FE response sensitivities to
system and loading parame-ters based on the DDM requires extension
of the FE algorithmsfor response-only computation [5].Response
sensitivity analysisMulti-yield-surface plasticity modelDirect
differentiation methodSoil material model0045-7825/$ - see front
matter 2009 Elsevier B.V. Adoi:10.1016/j.cma.2009.02.030methods
used in various sub-elds of civil engineering such as structural
optimization, reliability analy-sis, system identication, and nite
element model updating. Furthermore, stand-alone sensitivity
anal-ysis is invaluable for gaining insight into the effects and
relative importance of various system andloading parameters on
system response. The direct differentiation method (DDM) is a
general, accurateand efcient method to compute FE response
sensitivities to FE model parameters. In this paper, theDDM-based
response sensitivity analysis methodology is applied to a pressure
independent multi-yield-surface J2 plasticity material model, which
has been used extensively to simulate the nonlinearundrained shear
behavior of cohesive soils subjected to static and dynamic loading
conditions. Thecomplete derivation of the DDM-based response
sensitivity algorithm is presented. This algorithm isimplemented in
a general-purpose nonlinear nite element analysis program. The work
presented in thispaper extends signicantly the framework of
DDM-based response sensitivity analysis, since it enablesnumerous
applications involving the use of the multi-yield-surface J2
plasticity material model. Thenew algorithm and its software
implementation are validated through two application examples,
inwhich DDM-based response sensitivities are compared with their
counterparts obtained using forwardnite difference (FFD) analysis.
The normalized response sensitivity analysis results are then used
tomeasure the relative importance of the soil constitutive
parameters on the system response.
2009 Elsevier B.V. All rights reserved.a r t i c l e i n f o a b
s t r a c tFinite element response sensitivity analymodel by direct
differentiation method
Quan Gu a, Joel P. Conte b,*, Ahmed Elgamal b, ZhaohaAMEC
Geomatrix Consultants Inc., 510 Superior Avenue, Suite 200, Newport
Beach, CAbDepartment of Structural Engineering, University of
California at San Diego, 9500 GilmcURS Corporation, 1333 Broadway,
Suite 800, Oakland, CA, USA
Comput. Methods
journal homepage: wwwll rights reserved.s of multi-yield-surface
J2 plasticity
Yang c
63, USArive, La Jolla, CA 92093, USA
le at ScienceDirect
ppl. Mech. Engrg.
l sevier .com/locate /cma
-
Two application examples are provided to validate the new
re-sponse sensitivity algorithm and its implementation using the
-nite difference method (FDM). As an application of
responsesensitivity analysis, the response sensitivity results are
used tomeasure the relative importance of the soil material
parametersof different soil layers on the displacement response of
the soil.
2. Constitutive formulation of multi-yield-surface J2
plasticitymodel and numerical integration
2.1. Multi-yield surfaces
Each yield surface of this multi-yield-surface J2 plasticity
modelis dened in the deviatoric stress space as [23]
f 32s a : s a
12
K 0; 1
where s denotes the deviatoric stress tensor and a, referred to
asback-stress tensor, denotes the center of the yield surface {f =
0}in the deviatoric stress space. Parameter K represents the
size(3=2
ptimes the radius) of the yield surface which denes the re-
gion of constant plastic shear modulus. The dyadic tensor
productof tensors A and B is dened as A:B = AijBij (i, j = 1, 2,
3). The back-stress a is initialized to zero at the start of
loading.
In geotechnical engineering, the nonlinear shear behavior of
soilmaterials is described by a shear stressstrain backbone
curve[18,19] as shown in Fig. 1a. The experimentally determined
back-bone curve can be approximated by the hyperbolic formula [31]
as
Gc
1 2 3
l. Mech. Engrg. 198 (2009) 22722285 2273multi-yield-surface J2
plasticity employs the concept of a eld ofplastic moduli [1214] to
achieve a piecewise linear elasto-plasticbehavior under cyclic
loading conditions. This eld is dened by acollection of nested
yield surfaces of constant size (i.e., no isotropichardening) in
the stress space, which dene the regions of constantplastic shear
moduli (and therefore constant tangent shear mod-uli). The stress
sensitivity to material parameters is computed bydifferentiating
consistently the constitutive law integration algo-rithm, adding
the contributions from all yield surfaces that affectthe stress
computation at the current time step.
The existing implementation in OpenSees [15] of the
multi-yield-surface J2 plasticity model [18,19] considered is then
ex-tended to enable response sensitivity computation using
theDDM-based algorithm developed in this paper. The
DDM-basedalgorithmwas implemented in OpenSees by extending the
existingframework for sensitivity and reliability analysis
developed by DerKiureghian et al. [20], Haukaas and Der Kiureghian
[21], and Scottand Haukaas [22].
The work presented in this paper extends signicantly
theframework of DDM-based response sensitivity analysis, since it
en-ables numerous applications involving the use of the
multi-yield-surface J2 plasticity material model. Although this
material modelis a rather old model, it remains an effective and
robust model tosimulate the undrained response of cohesive
materials under cyclicand seismic loading conditions
[1214,16,18,19,2326]. Also, it isoperational in OpenSees [18]
through which soilstructure-inter-action studies may be conducted
by a large user community. Thus,an area of application of the
present DDM-based FE response sen-sitivity analysis scheme is in
earthquake loading (undrained) forgeotechnical cohesive soils, with
applications to soil-foundation-structure interaction scenarios
[16,17]. Response sensitivity analy-sis results are needed as input
for reliability, optimization, and FEmodel updating applications.
Therefore, the contribution of thispaper potentially improves
signicantly the computational ef-ciency of such applications to a
wide class of geotechnical systems[2729] and
soil-foundation-structure interaction systems involv-ing the
dynamic undrained shear response of cohesive soils.
The developments presented in this paper include new
imple-mentation details of the DDM that can carry over to other
ad-vanced constitutive models. (1) To the authors knowledge, inpast
work, the DDM-based response sensitivity analysis methodol-ogy has
been implemented for uniaxial material constitutive mod-els
[2,5,6,8] and three-dimensional (3D) single surface J2
plasticitymodels [2,3] with implicit constitutive law integration
schemes. Inthis paper, the DDMmethodology is extended to a general
3D elas-to-plastic material constitutive model, in which the
multi-yield-surface J2 plasticity approach is utilized. (2) In this
plasticity model,the stress state at the current load/time step is
obtained through anexplicit corrective iteration scheme, which
accumulates contribu-tions from all yield surfaces involved, the
number of which variesfrom load/time step to load/ time step [30].
The DDM-based re-sponse sensitivity algorithm follows exactly the
corrective itera-tion process for stress computation. (3) The
computation of theDDM-based FE response sensitivity requires the
consistent andnot the continuum tangent material moduli [5]. The
consistent tan-gent moduli consist of an unsymmetrical fourth-order
tensor(exhibiting only minor symmetries, Dijkl = Djikl = Djilk =
Djilk, butDijkl Dklij). They are computed by differentiating the
stress tensorwith respect to the strain tensor by following exactly
the stresscomputation algorithm as presented in [30]. (4) The
sensitivitiesof the kinematic hardening parameters dening the
initial cong-uration of the multi-yield surfaces are required at
the initiationof the response and response sensitivity computation.
Further-
Q. Gu et al. / Comput. Methods Appmore, the sensitivities of the
kinematic hardening parametersdening the active and inner yield
surfaces must be updated ateach load/time step.3
2(b) Von Mises multi-yield surfacess 1 c=cr
; 2
max
m
G
m max
Hyperbolic backbone curve
1H m( ) 2
1
13---3
13---1
13---2
Deviatoric plane
(a) Octahedral shear stress-strain (after [12])
fNYSfm
fm 1+
1
= =
f1Fig. 1. Yield surfaces of multi-yield-surface J2 plasticity
model in principaldeviatoric stress space.
-
Q 1Q
ofor
12
in which Q ofor : oforn o1
2, represents the plastic ow direction normal
to the yield surface face {f = 0} at the current stress point.
ParameterL in Eq. (11), referred to as the plastic loading
function, is dened asthe projection of the stress increment vector
ds onto the directionnormal to the yield surface, i.e.,
L Q : ds: 13The symbol h i in Eq. (11) denotes the MacCauleys
brackets denedsuch that hLi = max(L, 0). The magnitude of the
plastic strain incre-ment, hLiH0 , is a non-negative function which
obeys the Kuhn-Tuckercomplementarity conditions expressed as hLiH0
f s; a 0, such thatthe plastic strain increment is zero in the
elastic case (i.e., whenf(s, a) < 0).
The ow rule dened above in differential (continuum) form
isintegrated numerically over a trial time step (or load step)
using anelastic predictor-plastic corrector procedure illustrated
in Fig. 2,which shows, as an illustration, two corrective
iterations beforeconvergence is achieved. In this gure, the rst
subscript n (orn + 1) attached to a material response parameter
denotes the last(or current) converged load/time step, while the
second subscripti indicates the ith corrective iteration (not to be
confused withthe iteration number of the NewtonRaphson scheme used
to solvethe nonlinear equilibrium equations at each time step).
Assumingthat the current active yield surface is themth surface {fm
= 0} withits center at amn , the elastic trial (deviatoric) stress
strn1;0 is ob-tained as
strn1;0 sn 2GDen1; 14
l. Mech. Engrg. 198 (2009) 22722285where s and c denote the
octahedral shear stress and shear strain,respectively, and G is the
low-strain shear modulus. Parameter cris a reference shear strain
dened as
cr cmaxsmax
Gcmax smax; 3
where smax, called shear strength, is the shear stress
correspondingto the shear strain c = cmax (selected sufciently
large so thatsmax s(c =1)) (see Fig. 1).
Within the framework of multi-yield-surface plasticity,
thehyperbolic backbone curve in Eq. (2) is replaced by a piecewise
lin-ear approximation as shown in Fig. 1a. Each line segment
repre-sents the domain of a yield surface {fi = 0} of size K(i)
characterized by an elasto-plastic shear modulus H(i) for i =
1,2, . . . , NYS, where NYS denotes the total number of yield
surfaces[1214]. Parameter H(i) (see Fig. 1a) is conveniently dened
asHi 2 si1sici1ci
. A constant plastic shear modulus H0i dened as
[32]
1
H0i 1
Hi 12G
4
is associated with each yield surface {fi = 0}.The stressstrain
points (sj, cj) (subscript j denotes the point
number) used to dene the piecewise linear approximation ofthe
shear stressstrain (s c) backbone curve are dened suchthat their
projections on the s axis are uniformly spaced (seeFig. 1).
Thus,
sj smax jNYS and cj sjcr
Gcr sjj 1;2; . . . ;NYS 1: 5
The j-th yield surface {fj = 0} is dened by the two points (sj,
cj) and(sj+1, cj+1) (see Fig. 1). For each yield surface j with 1 6
j 6 NYS 1,Dcj cj1 cj; 6Dsj sj1 sj; 7
Kj 32
p sj; 8
Hj 2DsjDcj; 9
H0j 2GHj
2G Hj: 10
For the outermost yield surface (failure surface), setH(NYS) =
H0NYS = 0. The yield surfaces in their initial positions (atthe
start of loading) represent a set of concentric cylindrical
sur-faces whose axes coincide with the hydrostatic axis in the
deviator-ic stress space as shown in Fig. 1b. The outermost yield
surface{fNYS = 0} represents a failure surface and therefore denes
a geo-metrical boundary in the deviatoric stress space.
It is worth mentioning that the yield surfaces may initially
becongured any way suggested by experimental data and
typicallywould not be concentric if calibration is based on
triaxial test datafor instance.
2.2. Flow rule
An associative ow rule is used to compute the plastic
strainincrements. In the deviatoric stress space, the plastic
strain incre-ment vector lies along the exterior normal to the
yield surface atthe stress point. In tensor notation, the plastic
strain increment isexpressed as
dp hLiQ ; 11
2274 Q. Gu et al. / Comput. Methods AppH0
where the second-order unit tensor Q dened aswhere sn is the
converged deviatoric stress at the last (nth) timestep, and Den+1
denotes the total (from last converged step) devia-
--
n 1 0,+tr
n 1 1,+tr
n 1 2,+tr
2 3 Kn 1 1,+
Pn 1 2,+Qn 1 2,+
nm( )
nm 1+( )
2 3 K m( )
n
fm 0=
f m 1+ 0=
Pn 1 1,+
P n 1 2,+
A
Pn 1 1,+
2 3 Kn 1 2,+
n 1 2,+*
n 1 1,+*
Qn 1 1,+Fig. 2. Schematic of ow rule of multi-yield-surface J2
plasticity model.
-
process for the integration of the material constitutive law is
con-
yield surfaces may translate in the deviatoric stress space to
thecurrent stress point without changing in size (i.e., no
isotropichardening). In the context of multi-surface plasticity,
translationof the current active yield surface {fm = 0} is
generally governedby the consideration that no overlapping is
allowed between thecurrent and next yield surfaces [11]. On this
basis, the translationdirection ln+1 as shown in Fig. 3 is dened
after [19] as
ln1 sT amn Km
Km1sT am1n ; 24
where sT is the deviatoric stress tensor dening the position
ofstress point T, see Fig. 3, as the intersection of {fm+1 = 0}
(the outeryield surface next to the current active yield surface)
with the vec-tor connecting the center amn of the current yield
surface and thecurrent stress state (sn+1) at the end of the trial
time/load step.The hardening rule dened in Eq. (24) is also based
on Mroz conju-gate-points concept [11], and guarantees no
overlapping of yieldsurfaces [19]. Once the translation direction
ln+1 is computed fromEq. (24), the current active yield surface {fm
= 0} is translated (or up-dated) in the direction ln+1 until it
touches the current stress pointsn+1. After the active yield
surface ({fm = 0}) is updated, all the inneryield surfaces are
updated based on the current active yield surface.For the detailed
updating process of the active and inner yield sur-faces, the
following two steps are performed.
2.3.1. Active yield surface updateCompute the deviatoric stress
sT (see Fig. 3) as
sT amn nsn1 amn ; 25where the unknown scalar parameter n is
obtained from the condi-tion that sT lies on the yield surface
{fm+1 = 0}, i.e., sT has to satisfy
l. Mech. Engrg. 198 (2009) 22722285 2275verged, otherwise a
plastic correction (or corrective iteration) is ap-plied as
follows. The plastic stress correction tensor Pn+1,i for thecurrent
active yield surface ({fm = 0}) is dened as (see Fig. 2 fori = 1
and 2)
Pn1;i strn1;i1 strn1;i i 1;2;3 . . .: 15
An important stress quantity called the contact stress s*n+1,i
isdened as the intersection point of vector strn1;i1 amn and
thecurrent active yield surface {fm = 0}, and can be computed as
(seeFig. 2 for i = 1 and 2)
sn1;i Km
Kn1;istrn1;i1 amn amn ; 16
where Kn+1,i (different from K(i) =3=2
ptimes the radius of the i-th
yield surface) is dened as
Kn1;i 32strn1;i1 amn : strn1;i1 amn
r; 17
which is3=2
ptimes the distance from strn1;i1 to a
mn . The unit ten-
sor normal to the current active yield surface {fm = 0} at sn1;i
is de-rived from Eq. (12) or Fig. 2 as
Q n1;i sn1;i amn
sn1;i amn : sn1;i amn 12: 18
The plastic stress correction tensor Pn+1, i (i = 1, 2, 3, . .
.) can be de-rived as [32,33]
Pn1;1 2GQ n1;1 : strn1;0 sn1;1
H0m 2G Q n1;1 19
and
Pn1;i 2G Q n1;i : strn1;i1 sn1;i
H0m 2 G H0m1 H0m
H0m1 Q n1;i
i 2;3;4; . . .: 20The trial stress after the plastic correction
for the current activeyield surface is obtained using Eq. (15)
as
strn1;i strn1;i1 Pn1;i i 1;2;3; . . .: 21If the trial stress
strn1;i lies outside the next yield surface
{fm+1 = 0}, the active yield surface index is set to m =m + 1,
thecorrective iteration number is set to i = i + 1 and the plastic
correc-tion process (Eqs. (16)(21)) is repeated until the trial
stress strn1;ifalls inside the next outer yield surface. After
convergence ofthe deviatoric stress strn1;i to s
trn1 is achieved following the above
iterative algorithm, the volumetric stress rvoln1 is updated
to
rvoln1 rvoln BDn1 : I; 22where B = elastic bulk modulus, Dn+1 =
total strain tensorincrement, and I = second order unit tensor.
Then, the new totalstress (at the end of the integration of the
material constitutivelaw over a trial time/load step) referred to
as the current stresspoint is given by
rn1 sn1 rvoln1 I: 23
2.3. Hardening lawtoric strain increment in the current time
step. If trial the trial stressstrn1;0 falls inside the current
yield surface {fm = 0}, then the iteration
Q. Gu et al. / Comput. Methods AppA pure deviatoric kinematic
hardening rule is employed to cap-ture the Masing-type hysteretic
cyclic response behavior of claysunder undrained shear loading
conditions [19]. Accordingly, allT
T
n
nm( )
fm 0=fm 1+ 0=
O
n 1+m( )
nm 1+( )
n 1+
n 1+
Fig. 3. Hardening rule of multi-yield-surface J2 plasticity
model where {fm = 0}
represents the current active yield surface, sn is the converged
deviatoric stress atthe last time step, and sn+1 is the current
stress at the end of the trial time/load step(after [19]).
-
A s am : s am; 28
to the uncoupled nature of the sensitivity equations with
respectto different sensitivity parameters.
In the context of nonlinear nite element (FE) response
analysis,the consistent FE response sensitivities based on the
direct differ-entiation method (DDM) are computed at each time or
load step,after convergence is achieved for the response
computation. Thisrequires consistent differentiation of the FE
algorithm for the re-sponse-only computation (including the
numerical integrationschemes for the various material constitutive
laws used in the FEmodel) with respect to each sensitivity
parameter h. Consequently,the response sensitivity computation
algorithm involves the vari-ous hierarchical layers of FE response
analysis, namely the: (1)structure/system level, (2) element level,
(3) Gauss point level (orsection level), and (4) material level.
Details on the derivation ofthe DDM-based sensitivity equations for
classical displacement-based, force-based and mixed nite elements
can be found in anumber of Refs. [4,69,2022,33,34].
3.2. Displacement-based FE response sensitivity analysis using
DDM
After spatial discretization using the nite element method,
the
l. Mewhich reduces to the following quadratic equation:
A0f2 B0f C 0 0; 32where the coefcients A0, B0, and C0 are given
by
A0 ln1 : ln1; 33B0 2ln1 : sn1 amn ; 34
C0 sn1 amn : sn1 amn 23Km2: 35
From the geometric interpretation of Eq. (31) (see Fig. 3), it
followsthat Eq. (32) has two real positive roots. The smaller root
is the solu-tion to the problem. After parameter f is obtained, the
center of theactive yield surface {fm = 0} is updated to
amn1 amn fln1: 36
2.3.2. Inner yield surface updateAfter the current active yield
surface {fm = 0} is updated, all the
inner yield surfaces, {f1 = 0},{f2 = 0}, . . . , and {fm1 = 0},
are updatedsuch that all yield surfaces {f1 = 0} to {fm = 0} are
tangent to eachother at the current stress point s as shown in Fig.
4. The updatingof the inner yield surfaces is achieved through
similarity as [12]
sn1 amn1Km
sn1 am1n1
Km1 sn1 a
1n1
K1: 37
From Eq. (37), the updated center of each of the inner yield
surfacesis obtained as
ain1 sn1
Kisn1 amn1Km
1 6 i 6 m 1: 38
3. Derivation of response sensitivity algorithm for
multi-yield-surface J2 plasticity model
3.1. Introduction
If r denotes a generic scalar response quantity (e.g.,
displace-ment, strain, stress), then by denition, the sensitivity
of r with re-n1 n n1 nB 2amn am1n : sn1 amn ; 29
C amn am1n : amn am1n 23Km12: 30
From the geometric interpretation of Eq. (25) (see Fig. 3), it
followsthat Eq. (27) has two real roots of opposite signs. The
positive root isthe solution retained for n. Once sT is known, the
translation direc-tion ln+1 (not necessarily a unit vector) can be
computed from Eq.(24). After ln+1 is obtained, the magnitude of the
translation (i.e.,fln+1 where f is a positive scalar to be
determined) is computedfrom the condition that the current stress
sn+1 lies on the mth yieldsurface {fm = 0} after it is translated.
Thus,
sn1 amn fln1 : sn1 amn fln1 23Km2 0; 31sT am1n : sT am1n 23Km12
0: 26
Substituting Eqs. (25) into (26) yields the following scalar
quadraticequation to be solved for parameter n:
An2 Bn C 0; 27where the coefcients A, B, and C are given by
2276 Q. Gu et al. / Comput. Methods Appspect to the (material or
loading) parameter h is expressedmathematically as the absolute
partial derivative of r with respectto the variable h; oroh
hh0 , where h0 denotes the nominal value takenby the sensitivity
parameter h for the nite element responseanalysis.
In this paper, following the notation proposed in Ref. [2],
thescalar response quantity r(#) = r(f(#), #) depends on the
parametervector # (dened by n time-independent sensitivity
parameters,i.e., # = [h1 hn]T) both explicitly and implicitly
through the vectorfunction f(#). According to the notation adopted
herein, drd# denotesthe gradient or total derivative of r with
respect to #; drdhi representsthe absolute partial derivative of
the response quantity r with re-spect to the scalar variable hi, i
= 1, . . . ,n, (i.e., the derivative of rwith respect to parameter
hi considering both explicit and implicitdependencies of r on hi),
and orohi
zdenotes the partial derivative of r
with respect to parameter hi when the vector of variables z is
keptconstant (xed). In the particular and important case in
which
z = f(#), the expression orohi
zreduces to the partial derivative of r
considering only the explicit dependency of r on parameter
hi.For # = h = h1 (case of a single sensitivity parameter), the
adoptednotation reduces to the usual elementary calculus notation.
Thederivations below consider the case of a single (scalar)
sensitivity(material or loading) parameter h without loss of
generality, due
Deviatoric Plane
fNYSfm 1 f1
n 1+fm
13---3
13---1
13---2
Fig. 4. Inner yield surface movements.
ch. Engrg. 198 (2009) 22722285equations of motion of a
materially-nonlinear-only model of astructural system take the form
of the following nonlinear matrixdifferential equation:
-
l. MeMhut; h Chut; h Rut; h; h Ft; h; 39
where t = time, h = scalar sensitivity parameter (material
orloading variable), u(t) = vector of nodal displacements, M =
massmatrix, C = damping matrix, R(u, t) = history dependent
internal(inelastic) resisting force vector, F(t) = applied dynamic
load vec-tor, and a superposed dot denotes one differentiation with
respectto time.
We assume without loss of generality that the time
continuousspatially discrete equation of motion (39) is integrated
numericallyin time using the well-known Newmark-b time-stepping
methodof structural dynamics [35], yielding the following nonlinear
ma-trix algebraic equation in the unknowns un+1 = u(tn+1):
Wun1 eFn1 1bDt2
Mun1 abDt2
Cun1 Run1" #
0; 40where
eFn1 Fn1 M 1bDt2
un 1bDt _un 112b
un
" #
C abDtun 1
ab
_un Dt 1 a2b
un
; 41
a and b are parameters controlling the accuracy and stability of
thenumerical integration algorithm and Dt is the time integration
stepassumed to be constant.
Eq. (40) represents the set of nonlinear algebraic equations
thathave to be solved at each time step [tn, tn+1] for the unknown
re-sponse quantities un+1. In general, the subscript (. . .)n+1
indicatesthat the quantity to which it is attached is evaluated at
discretetime tn+1. In the direct stiffness (nite element) method,
the vectorof internal resisting forces R(un+1) in Eq. (40) is
obtained by assem-bling, at the structure level, the elemental
internal resisting forcevectors, i.e.,
Run1 ANel
e1fReuen1g; 42
where ANele1f. . .g denotes the direct stiffness assembly
operator fromthe element level (in local elements coordinates) to
the structure le-vel in global reference coordinates, Nel
represents the number of -nite elements in the FE model, R(e) and
uen1 denote the internalresisting force vector and nodal
displacement vector, respectively,of element e.
Typically, Eq. (40) is solved using the NewtonRaphson itera-tion
procedure, which consists of solving a linearized system
ofequations at each iteration. Assuming that un+1 is the
convergedsolution (up to some iteration residuals satisfying a
specied toler-ance usually taken in the vicinity of the machine
precision) for thecurrent time step [tn, tn+1], and differentiating
Eq. (40) with respectto h using the chain rule, recognizing that
R(un+1) = R(un+1(h), h)(i.e., the structure inelastic resisting
force vector depends on h bothimplicitly, through un+1, and
explicitly), we obtain the following re-sponse sensitivity equation
at the structure level:
1
bDt2M a
bDtC KstatT n1
" #dun1dh
1bDt2
dMdh
abDt
dCdh
!un1 oRun1h; hoh
un1
deFn1dh
;
Q. Gu et al. / Comput. Methods App43
wheredeFn1dh
dFn1dh
dMdh
1
bDt2un 1bDt _un 1
12b
un
!
M 1bDt2
dundh
1bDt
d _undh
1 12b
d undh
" #
dCdh
abDtun 1
ab
_un Dt 1 a2b
un
C a
bDtdundh
1 ab
d _undh
Dt 1 a2b
dundh
:
44
In Eq. (43), KstatT n1 denotes the consistent (or algorithmic)
tangent(static) stiffness matrix of the structure/system, which is
denedas the assembly of the element consistent tangent stiffness
matricesas
KstatT n1 oRun1oun1
ANel
e1oReun1
ouen1
( ) A
Nel
e1
ZXe
BTDn1BdXe
;
45
where B is the straindisplacement transformation matrix, and
Dn+1denotes the point-wise matrix of material consistent (or
algorith-mic) tangent moduli obtained through consistent
linearization ofthe constitutive law integration scheme [5,33,36],
i.e.,
Dn1 orn1rn; n; n; . . .on1 : 46
The consistent tangent modulus for the presented
multi-yield-sur-face J2 plasticity material model is computed by
differentiatingthe stress tensor with strain tensor, following the
explicit correctiveiteration process of the stress computation.
This modulus is anunsymmetrical fourth-order tensor (i.e., it
exhibits only minor sym-metry, kijkl = kjikl = kijlk = kjilk, but
kijkl kklij). The derivation andimplementation of the consistent
tangent modulus was docu-mented elsewhere in [30,33].
The second term on the right-hand-side (RHS) of Eq. (43)
repre-sents the partial derivative of the internal resisting force
vector,R(un+1), with respect to sensitivity parameter h under the
conditionthat the nodal displacement vector un+1 remains xed. It is
com-puted through direct stiffness assembly of the element
resistingforce derivatives as
oRun1h; hoh
un1
ANel
e1
ZXe
BTorn1h; h
oh
n1
dXe( )
: 47
In the following section, computation of the unconditional
(i.e., un+1is not xed) stress sensitivities to parameters of the
multi-yield-surface material plasticity model is presented in
detail. The condi-tional (i.e., un+1 is xed) stress sensitivities
are obtained as a specialcase of the unconditional ones.
3.3. Stress sensitivity for multi-yield-surface J2 material
plasticitymodel
Without loss of generality, the material sensitivity
parametersthat are considered in this paper are: (1) the low-strain
shear mod-ulus G, (2) the bulk modulus B, and (3) the shear
strength smax (seeFig. 1).
3.3.1. Parameter sensitivity of initial conguration of
multi-yield-surface plasticity model
In this section, the sensitivities of the parameters (see Eqs.
(3)
ch. Engrg. 198 (2009) 22722285 2277(10)) dening the initial
conguration of the material model to thesensitivity parameter h are
derived. Differentiating Eq. (3) with re-spect to h yields
-
l. Medcrdh
1Gcmax smax2
cmax dsmaxdh
Gcmax smax
cmax smax dGdh
cmax dsmaxdh
: 48
For each yield surface {fj = 0} (1 6 j 6 NYS 1), Eq. (5) is
differenti-ated with respect to h as
dsjdh
dsmaxdh
jNYS
; 49dcjdh
1Gcr sj2
dsjdh
cr sj dcrdh
Gcr sj
sjcr
dGdh
cr G dcrdh
dsjdh
: 50
Differentiating Eqs. (6)(10) with respect to h yields
dDcjdh
dcj1dh
dcjdh
; 51dDsjdh
dsj1dh
dsjdh
; 52dKj
dh 3
2p dsj
dh; 53
dHj
dh 2Dc2j
dDsjdh
Dcj Dsj dDcjdh
; 54
dH0j
dh 22G Hj2
dGdh
Hj GdHj
dh
" #(
2G Hj GHj 2dGdh
dHj
dh
! !): 55
The outermost yield surface (failure surface) is characterized
byH(NYS) = H0NYS = 0. from which it follows that dH
NYSdh 0 and
dH0NYSdh 0. Furthermore, for each yield surface, the sensitivity
to h
of the initial back-stress a(j) = 0 (initial center of the yield
surface)is zero, i.e., dajdh 0 1 6 j 6 NYS.
3.3.2. Stress response sensitivityAt each time or load step,
once convergence is achieved for the
response computation (at the structure level), the RHS of the
re-sponse sensitivity equation (at the structure level), Eq. (43),
isformed which includes the computation of the termoRun1h;h
oh
un1
. The latter requires computation of the conditional
(i.e., un+1 is xed case sensitives,orn1h;h
oh jn1 , at each integration(Gauss) point. Eq. (43) is solved
for the displacement response sen-sitivity dun1dh . From the
relationship between nodal displacementsand strains at the element
level, the strain and deviatoric strain re-sponse sensitivities
(dn1dh and
den1dh , respectively) can be readily ob-
tained. Then, the unconditional (i.e., un+1 is not xed)
stressresponse sensitivities at each integration (Gauss) point are
com-puted as they are needed to form the RHS of the structural
re-sponse sensitivity equation at the next time step. The
conditional(i.e., un+1 is xed) stress response sensitivities are
evaluated usingthe same equations as the unconditional ones, but
imposing theconstraint that un+1 is xed, which implies that the
strain n+1 isxed for displacement-based nite elements.
Next, the stress response sensitivity to material
constitutiveparameters (G, B, and smax) is derived through
consistent differen-tiation of the algorithm for stress response
computation presentedin Section 2. From Eq. (14), it follows
that
dstrn1;0dh
dsndh
2dGdhDen1 2GdDen1dh 56
2278 Q. Gu et al. / Comput. Methods Appwhere Den+1 = en+1 en and
thus dDen1dh den1dh dendh . When comput-ing the conditional stress
sensitivity (see Eq. (47)) for which thestrain n+1 and therefore
the deviatoric strain en+1 remain xed,den1dh 0 and dDen1dh dendh .
This represents the only difference be-tween conditional and
unconditional stress response sensitivitycomputations. All the
formulas for unconditional sensitivity compu-tations derived in the
sequel of Section 3.3 apply equally to condi-tional stress
sensitivity computation.
The sensitivity of the contact stress sn1;i to parameter h is
ob-tained by differentiating Eq. (16) with respect to h as
dsn1;idh
1K2n1;i
dKm
dhstrn1;i1 amn Km
dstrn1;i1dh
damn
dh
!" #(
Kn1;i Kmstrn1;i1 amn dKn1;idh
da
mn
dh; 57
where the sensitivity of Kn+1, i to h is obtained from Eq. (17)
as
dKn1;idh
32Kn1;i
dstrn1;i1dh
damn
dh
!: strn1;i1 amn : 58
Eq. (18) yields the following sensitivity to h of the unit
tensor nor-mal to the yield surface at sn1;i:
dQ n1;idh
ds
n1;idh da
mndh
sn1;i amn : sn1;i amn 1=2
ds
n1;idh da
mndh
: sn1;i amn
sn1;i amn : sn1;i amn 3=2 sn1;i amn : 59
Then, the sensitivity of the plastic stress correction tensor is
derivedfrom Eq. (19) as
dPn1;1dh
2dGdh
Q n1;1 : strn1;0 sn1;1H0m 2G Q n1;1
2GQ n1;1 : strn1;0 sn1;1
H0m 2G dQ n1;1
dh
2GH0m 2G2
dQ n1;1dh
: strn1;0 sn1;1
Q n1;1 :dstrn1;0dh
dsn1;1dh
H0m 2G
Q n1;1 : strn1;0 sn1;1:dH0m
dh 2dG
dh
!)Q n1;1 60
for the rst corrective iteration (subscript i = 1) and from Eq.
(20) as
dPn1;idh
2dGdh
Q n1;i : strn1;i1 sn1;iH0m 2G
Q n1;i
"
2GQ n1;i : strn1;i1 sn1;i
H0m 2GdQ n1;idh
#H0m1 H0m
H0m1
2GH0m1 H0m
H0m 2G2 H0m1dQ n1;idh
: strn1;i1 sn1;i
Q n1;i :dstrn1;i1
dh ds
n1;idh
H0m 2G
Q n1;i : strn1;i1 sn1;idH0m
dh 2dG
dh
!)Q n1;i
2GQ n1;i : strn1;i1 sn1;i
H0m 2G" #
ch. Engrg. 198 (2009) 22722285 H0m
H0m12dH0m1
dh 1H0
m1dH0m
dhQ n1;i 61
-
l. Mefor the second and subsequent corrective iterations (i = 2,
3, 4, . . .).The sensitivity to h of the trial stress tensor after
the ith plastic cor-rection is obtained from Eq. (21) as
dstrn1;idh
dstrn1;i1dh
dPn1;idh
i 1;2; . . .: 62
If the trial stress strn1;i lies outside the next outer yield
surface{fm+1 = 0}, the active yield surface index is set to m =m +
1, the cor-rective iteration number is set to i = i + 1, the
plastic correction pro-cess and its sensitivity (Eqs. (57)(62)) is
repeated until the trialstress falls inside the next outer yield
surface.
After convergence of the deviatoric stress strn1;i is
achieved,the sensitivity to h of the updated volumetric stress
rvoln1 is com-puted from Eq. (22) as
drvoln1dh
drvoln
dh dBdh
Dn1ii B dDn1ii
dh; 63
where dDn1iidh dDn1;11
dh dDn1;22
dh dDn1;33
dh . It is worth noting that inthe process of computing the
conditional stress sensitivities, thequantity dDn1iidh 0, since
un+1 is considered xed. Finally, the sen-sitivity to h of the total
stress is obtained from Eq. (23) as
drn1dh
dsn1dh
drvoln1dh
I: 64
After the sensitivity to h of the current stress rn+1 is
computed, thesensitivity of the hardening parameters, namely the
back-stress(center of the yield surface) (1 6 i 6m) for the current
active andeach of the inner yield surfaces ain ;1 6 i 6 m, must be
computedand updated (in the case of unconditional stress
sensitivity only),which is the object of the next section. The
sensitivities of thehardening parameters are needed for computing
the conditionaland unconditional stress sensitivities at the next
time step (i.e.,at tn+2).
3.3.3. Sensitivity of hardening parameters of active and inner
yieldsurfaces
In this section, the sensitivity to h of the kinematic
hardeningparameters of the active and inner yield surfaces are
derived bydifferentiating with respect to h the updating equations
for thesehardening parameters presented in Section 2.3 (Eqs.
(24)(38)).
From Eq. (25), it follows that the sensitivity to h of the
deviatoricstress tensor sT at point T of the m-th yield surface
(see Fig. 3), isgiven by
dsTdh
damn
dh dndh
sn1 amn ndsn1dh
damn
dh
!; 65
where dndh is obtained from Eq. (27) as
dndh
dAdh n
2 dBdh n dCdh2An B 66
and where dAdh,dBdh, and
dCdh are obtained in turn from Eqs. (28)(30) as
dAdh
2 dsn1dh
damn
dh
!: sn1 amn ; 67
dBdh
2 damn
dh da
m1n
dh
!: sn1 amn
2amn am1n :dsn1dh
damn
dh
!; 68
dCdh
2 damn
dh da
m1n
dh
!: amn am1n
43Km1:
dKm1
dh: 69
Q. Gu et al. / Comput. Methods AppThen, the sensitivity to h of
the translation direction ln+1 iscomputed by differentiating Eq.
(24) with respect to h asdln1dh
dsTdh
damn
dh
" #
dKmdh K
m1 Km dKm1dhKm12
!sT am1n
Km
Km1dsTdh
dam1n
dh
" #: 70
After dln1dh is obtained, the sensitivity to h of the
translation param-eter f is derived from Eq. (32) as
dfdh
dA0dh f
2 dB0dh f dC0
dh
2A0f B0 ; 71
where dA0
dh ,dB0dh , and
dC0dh are obtained from Eqs. (33)(35) as
dA0
dh 2dln1
dh: ln1; 72
dB0
dh 2dln1
dh: sn1 amn 2ln1 :
dsn1dh
damn
dh
!; 73
dC0
dh 2 dsn1
dh da
mn
dh
!: sn1 amn
43Km dK
m
dh: 74
Finally, from Eq. (36) the sensitivity to h of the updated
back-stress(center of yield surface) of the current active yield
surface is givenby
damn1dh
damn
dh dfdh
ln1 fdln1dh
: 75
The sensitivity to h of the updated back-stress of each of the
inneryield surfaces are obtained from Eq. (38) as
dain1dh
dsn1dh
Km dK
idh sn1amn1KmKi dsn1dh
damn1dh
dKmdh Kisn1amn1
Km2 ;
76where 1 6 i 6m 1.
4. Application examples
4.1. Three-dimensional block of clay subjected to quasi-static
cyclicloading
In this section, a three-dimensional (3D) solid block of
dimen-sions 1 m 1 m 1 m subjected to quasi-static cyclic loading
inboth horizontal directions simultaneously, see Fig. 5, is used as
rstapplication and validation example. The block is discretized
into 8brick elements dened as displacement-based eight-noded,
trilin-ear isoparametric nite elements with eight integration
pointseach. The block material consists of a medium clay with the
follow-ing material constitutive parameters [19]: low-strain shear
modu-lus G = 6.0 104 kPa, elastic bulk modulus B = 2.4 105 kPa,()
Poissons ratio = 0.38), and maximum shear stress smax = 30 k-Pa.
The bottom nodes of the nite element (FE) model are xedand top
nodes {A, B, C} and {A, D, E} are subjected to ve cyclesof
harmonic, 90 degrees out-of-phase, concentrated horizontalforces
Fx1 t 2:0 sin0:2ptkN and Fx2 t 2:0 sin0:2pt0:5pkN, respectively.
The number of yield surfaces is set to 20.A time increment of Dt =
0.01 s is used to integrate the equationsof quasi-static
equilibrium (i.e., without inertia and dampingeffects).
The displacement response of node A (see Fig. 5) in the
x1-direc-tion is shown in Fig. 6 as a function of the force Fx1 t,
while thehysteretic shear stressstrain response (r31 e31) at Gauss
point
ch. Engrg. 198 (2009) 22722285 2279G (see Fig. 5) is plotted in
Fig. 7. These gures indicate that signif-icant yielding of the clay
is observed during the cyclic loadingconsidered.
-
l. Mech. Engrg. 198 (2009) 227222852280 Q. Gu et al. / Comput.
Methods AppThe sensitivities of the displacement response u(t) of
node A inthe x1-direction to the shear modulus G, bulk modulus B
and shearstrength smax computed using the DDM are compared in Figs.
8, 10
2 1.5 1 0.5 0 0.5 1 1.5 2 2.5x 104
2
1.5
1
0.5
0
0.5
1
1.5
2
F x1
kN[]
ux1
[m]
Fig. 6. Forcedisplacement response at node A.
3 2 1 0 1 2 3 4 5x 104
3
2
1
0
1
2
3
31 [k
Pa]
31
Fig. 7. Shear stressstrain response at Gauss point G (see Fig.
5).
0 10 20 30 40 5012
10
8
6
4
2
0
2
4
6
x 109
DDM / = 1e1 / = 1e3 / = 1e5
du dG-------
m3
kN-------
Time [sec]
Fig. 8. Sensitivity of displacement response u(t) of node A in
the x1-direction to thelow-strain shear modulus G computed using
DDM and forward nite difference forthree different values of
relative parameter perturbation Dh/h.
42 42.5 43 43.5 44 44.5
9
8.8
8.6
8.4
8.2
8
7.8
7.6
7.4
7.2
7
x 109
DDM / = 1e1 / = 1e3 / = 1e5
du dG-------
m3
kN-------
Time [sec]
Fig. 9. Sensitivity of displacement response u(t) of node A in
the xr direction to thelow-strain shear modulus G computed using
DDM and forward nite difference(zoom view of Fig. 8).
0.5 m 0.5 m
0.5m
0.5
m0.
5 m
BA C
D
E
G
Fx1
Fx1
Fx1
Fx2Fx2Fx2
x1
x2
x3
0.5m
Fig. 5. Solid block of clay subjected to horizontal quasi-static
cyclic loading underundrained condition.
0 10 20 30 40 504
3
2
1
0
1
2 x 1010
DDM / = 1e2 / = 1e3 / = 1e4
du dB-------
m3
kN-------
Time [sec]
Fig. 10. Sensitivity of displacement response u(t) of node A in
the x1-direction to thebulk modulus B computed using DDM and
forward nite difference for threedifferent values of relative
parameter perturbation Dh/h.
-
3
2
1
0
1
2 x 105
u max
--------
m3
kN-------
73.3 .Cx
Fig. 12. Layered soil column subjected to total base
acceleration with nite elementmesh in thin lines (unit: [m]). Soil
layers are numbered from top to bottom.
0 2 4 6 8 102
0
2
Time [sec]
Acce
l. [m
/s2 ]
t = 0.01 [sec]
Fig. 13. Total acceleration time history at the base of the soil
column.
Q. Gu et al. / Comput. Methods Appl. Meand 11, respectively,
with the corresponding sensitivities esti-mated using the forward
nite difference (FFD) method withincreasingly small perturbations
Dh of the sensitivity parameterh. Fig. 9 shows a zoom view from
Fig. 8 that allows to better appre-ciate the convergence trend of
the FFD results to the DDM one. Inall cases, it is observed that
the FFD results converge asympoticallyto the DDM results as Dh/h
becomes increasingly small. In eachcase, particular attention was
given to the choice of the lower valueof the parameter perturbation
so as to avoid numerical problemsrelated to round-off errors. A
perfect match between DDM andFFD results was achieved with relative
perturbation Dh/h of1.0e5, 1.0e5 and 1.0e4, respectively, for the
shear modulusG, bulk modulus B, and shear strength smax.
4.2. Multi-layered soil column subjected to earthquake base
excitation
The second benchmark problem consists of a multi-layered
soilcolumn subjected to earthquake base excitation. This soil
columnis representative of the local soil condition at the site of
the Hum-boldt Bay Middle Channel Bridge near Eureka in northern
Califor-
0 10 20 30 40 507
6
5
4 DDM / = 1e2 / = 1e4 / = 1e5
d d------
Time [sec]
Fig. 11. Sensitivity of displacement response u(t) of node A in
the x1-direction to theshear strength smax computed using DDM and
FFD for three different values ofrelative parameter perturbation
Dh/h.nia [16]. A Multi-yield-surface J2 plasticity model with 20
yieldsurfaces and different parameter sets given in Table 1 is used
torepresent the various soil layers. The soil column is discretized
intoa 2D plane-strain nite element model consisting of 28
four-nodequadratic bilinear isoparametric elements with four Gauss
pointseach as shown in Fig. 12. The soil column is assumed to be
undersimple shear condition, and the corresponding nodes at the
samedepth on the left and right boundaries are tied together for
bothhorizontal and vertical displacements. First, gravity is
applied qua-si-statically and then base excitation is applied
dynamically. Thetotal horizontal acceleration at the base of the
soil column, seeFig. 13, was obtained from another study [16]
through deconvolu-
Table 1Material properties of various layers of soil column
(from ground surface to base ofsoil column).
Material # G (kPa) B (kPa) smax (kPa)
1 54450 1.6 105 332 33800 1.0 105 263 96800 2.9 105 444 61250
1.8 105 355 180000 5.4 105 606 369800 1.1 106 86Ground
surface0.00.6
8.0
17.7
45.1
5.0
u1u2u3u4
u5
u6.D
.
.F7.1
E
z
Layer #1
Layer #6
Layer #5
Layer #2Layer #3Layer #4
ch. Engrg. 198 (2009) 22722285 2281tion of a ground surface free
eld motion. The Newmark-beta di-rect step-by-step integration
method with parameters b = 0.2756and c = 0.55 is used with a
constant time step Dt = 0.01 s for inte-grating the equations of
motion of the soil column. The horizontaldisplacement response of
the soil column (relative to the base) atthe top of each soil layer
is shown in Fig. 14. The shear stressstrain(rxz,exz) hysteretic
response at Gauss points C, D, E, F (see Fig. 12) ofthe soil column
is shown in Fig. 15. The response simulation resultsin Figs. 14 and
15 indicate that the soil material undergoes signif-icant nonlinear
behavior.
Response sensitivity analysis is performed with the
low-strainshear modulus Gi, the bulk modulus Bi, and the shear
strengthsmax, i of the various soil layers (subscript i denotes the
soil layernumber, see Fig. 12) selected as sensitivity parameters.
TheDDM-based response sensitivity results are veried using the
FFDmethod. Representative comparisons between response
sensitivi-ties obtained using DDM and FFD with increasingly smaller
pertur-bations of the sensitivity parameters are shown in Figs.
1621.From these gures and closeups, the FFD results are shown to
con-verge asymptotically to the corresponding DDM results as the
per-turbation of the sensitivity parameter becomes increasingly
small.
Finite element response sensitivity analysis can be used as
atool to investigate the relative importance of various
systemparameters in regards to the system response. Normalized
-
response sensitivities (obtained through scaling each response
sen-sitivity by the nominal value of the corresponding
sensitivityparameter) can be used to quantify the relative
importance of sys-tem parameters. Each normalized sensitivity can
be interpreted as100 times the change in the considered response
quantity due toone percent change in the corresponding sensitivity
parameter. Asa rst illustration, Fig. 22 shows the normalized
sensitivities of thehorizontal displacement response of the soil
column top (groundsurface) to the six most sensitive material
parameters based onthe peak (absolute) value of the normalized
response sensitivitytime history. These results indicate that the
ground surface hori-zontal displacement response is most sensitive
to, in order ofdecreasing importance: (1) smax, 4, (2) smax, 5, (3)
smax, 6, (4) G4,
soil layers are very important in controlling the ground surface
dis-placement response, since the soil undergoes signicant
nonlinear-ities when responding the earthquake excitation
considered. Thistype of information may be extremely useful to
geotechnical engi-neers seeking an optimum strategy (for example
among variousground improvement techniques) to reduce the maximum
groundsurface displacement response during an earthquake. FE
responsesensitivities to material parameters are also invaluable to
engi-neers when performing FE model updating, since they point
to
0 2 4 6 8 100.08
0.06
0.04
0.02
0
0.02
0.04
0.06
u6u5u4u3u2u1
Dis
plac
emen
t [m
]
Time [sec]
Fig. 14. Relative horizontal displacement time histories at the
top of each layer ofthe soil column (see Fig. 12).
15
10
0 2 4 6 8 101.5
1
0.5
0
0.5
1
1.5
2 x 1011
DDM / = 1e2 / = 1e3 / = 1e4
du6
dG1
----------
m3
kN-------
Time [sec]
Fig. 16. Sensitivity of displacement response u6 (see Fig. 12)
to shear modulus G1obtained using DDM and FFD with increasingly
small perturbations of sensitivityparameter.
2282 Q. Gu et al. / Comput. Methods Appl. Mech. Engrg. 198
(2009) 22722285(5) G5, (6) G6. Thus, the shear strength parameters
of the bottom
15
10
5
0
5
10
15Point F
xz [k
Pa]1 0.5 0 0.5 1x 103
20
2 1 0 1 2 3x 103
60
40
20
0
20
40
60
80Point D
xz [k
Pa]
xz [-]
xz [-]
Fig. 15. Shear stressstrain hysteric responsesthe most sensitive
parameters which should be adjusted or
5
0
5
10
15
20Point E
xz[k
Pa]8 6 4 2 0 2 4 6x 104
20
1 0.5 0 0.5 1 1.5 2x 103
80
60
40
20
0
20
40
60
80
100
xz[k
Pa]
xz [-]
Point C
xz [-]
at Gauss points C, D, E, and F (see Fig. 12).
-
5.3 5.32 5.34 5.36 5.38 5.4
9
9.2
9.4
9.6
9.8
10
10.2
10.4
10.6x 1012
DDM / = 1e2 / = 1e3 / = 1e4
du6
dG1
----------
m3
kN-------
Time [sec]
Fig. 17. Sensitivity of displacement response u6 (see Fig. 12)
to shear modulus G1obtained using DDM and FFD with increasingly
small perturbations of sensitivityparameter (zoom view of Fig.
16).
0 2 4 6 8 102
1.5
1
0.5
0
0.5
1 x 109
DDM / = 1e2 / = 1e3 / = 1e4
du1
dB6
---------
m3
kN-------
Time [sec]
Fig. 18. Sensitivity of displacement response u1 (see Fig. 12)
to bulk modulus B6obtained using DDM and FFD with increasingly
small perturbations of sensitivityparameter.
3.3 3.35 3.4 3.45 3.5 3.55 3.6 3.65
10.5
10
9.5
9
8.5
x 1010
DDM / = 1e2 / = 1e3 / = 1e4
du1
dB6
---------
m3
kN-------
Time [sec]
Fig. 19. Sensitivity of displacement response uj (see Fig. 12)
to bulk modulus B6obtained using DDM and FFD with increasingly
small perturbations of sensitivityparameter (zoom view of Fig.
18).
0 2 4 6 8 101
0.5
0
0.5
1
1.5
2 x 104
DDM / = 1e2 / = 1e4 / = 1e6
du2
dm
ax,
2-----
----------
----m
3
kN-------
Time [sec]
Fig. 20. Sensitivity of displacement response u2 (see Fig. 12)
to shear strengthsmax,2 obtained using DDM and FFD with
increasingly small perturbations ofsensitivity parameter.
6.64 6.645 6.65 6.655 6.668
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
x 105
DDM / = 1e2 / = 1e4 / = 1e6
du2
dm
ax2
,
----------
---------
m3
kN-------
Time [sec]
Fig. 21. Sensitivity of displacement response u2 (see Fig. 12)
to shear strengthsmax,2 obtained using DDM and FFD with
increasingly small perturbations ofsensitivity parameter (zoom view
of Fig. 20).
0 2 4 6 8 100.06
0.04
0.02
0
0.02
0.04
0.06
0.08
max,4
max,4
max,5max,6G4G5G6
du1
di
--------
im[
]
Importance ranking:
Time [sec]
> max,5 > max,6 >>G4 >G5 G6
Fig. 22. Relative importance of material parameters on
horizontal displacementresponse of ground surface (top of soil
column, see Fig. 12).
Q. Gu et al. / Comput. Methods Appl. Mech. Engrg. 198 (2009)
22722285 2283
-
l. Mecalibrated with the highest priority. In a second
illustration of theuse of response sensitivity analysis, the
effects of the low-strainshear modulus G6 (of bottom soil layer) on
the relative horizontaldisplacement response of the soil column at
various soil layersare investigated. Fig. 23 shows the normalized
sensitivities to G6of the horizontal displacement response of the
soil column at var-ious soil layers (u1 through u6). From these
results, it follows thatthe ranking of the displacement responses
u1 through u6, indecreasing order of their sensitivity to G6, is:
(1) u1 and u2, (2) u3and u4, (3) u5, (4) u6. These results indicate
that the low-strainshear stiffness of the deepest soil layer
affects most the soil re-sponse at the ground surface level.
For the two application examples presented above, convergenceof
the FFD response sensitivity results to the ones based on
DDMvalidates the DDM-based algorithms for response sensitivity
com-putation presented in this paper as well as their computer
imple-mentation. Regarding response sensitivity computation using
theFFD, it is worth mentioning the step-size dilemma [7,37]: if
theparameter perturbation Dh is selected to be small, so as to
reducethe truncation error, the condition error (due to round-off
errors)
0 2 4 6 8 100.02
0.015
0.01
0.005
0
0.005
0.01
0.015
0.02
0.025
u1u2u3u4u5u6
dui
dG6
----------
G6
m[]
Relative influence of G6: u1 u2 > u3 u4 > u5 > u6
Time [sec]
Fig. 23. Relative inuence of shear modulus G6 on horizontal
displacementresponse of soil column at different soil layers (see
Fig. 12).
2284 Q. Gu et al. / Comput. Methods Appmay be excessive. In some
cases, there may not be any value ofthe parameter perturbation Dh
which yields an acceptable error.
5. Conclusions
The direct differentiation method (DDM) is a general,
accurateand efcient method to compute nite element (FE)
responsesensitivities to FE model parameters, especially in the
case onnonlinear materials. This paper applies the DDM-based
responsesensitivity analysis methodology to a pressure
independentmulti-yield-surface J2 plasticity material model, which
has beenused extensively to simulate the behavior of nonlinear clay
soilmaterial subjected to static and dynamic loading conditions.
Thealgorithm developed is implemented in a software framework
fornite element analysis of structural and/or geotechnical
systems.Two application examples are presented to validate, using
forwardnite difference analysis, the response sensitivity results
obtainedfrom the proposed DDM-based algorithm. The second
applicationexample is also employed to illustrate the use of nite
element re-sponse sensitivity analysis to investigate the relative
importance ofmaterial parameters on system response.
The algorithm developed herein for nonlinear nite element
re-sponse sensitivity analysis of geotechnical systems modeled
usingthe multi-yield-surface J2 plasticity model has direct
applicationsin optimization, reliability analysis, and nonlinear FE
modelupdating.
Acknowledgements
Support of this research by the Pacic Earthquake
EngineeringResearch (PEER) Center through the Earthquake
Engineering Re-search Centers Program of the National Science
Foundation underAward No. EEC-9701568 and by Lawrence Livermore
National Lab-oratory with Dr. David McCallen as Program Leader is
gratefullyacknowledged. Any opinions, ndings, conclusions, or
recommen-dations expressed in this publication are those of the
authors anddo not necessarily reect the views of the sponsors.
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Q. Gu et al. / Comput. Methods Appl. Mech. Engrg. 198 (2009)
22722285 2285
Finite element response sensitivity analysis of
multi-yield-surface J2 plasticity model by direct differentiation
methodIntroductionConstitutive formulation of multi-yield-surface
J2 plasticity model and numerical integrationMulti-yield
surfacesFlow ruleHardening lawActive yield surface updateInner
yield surface update
Derivation of response sensitivity algorithm for
multi-yield-surface J2 plasticity
modelIntroductionDisplacement-based FE response sensitivity
analysis using DDMStress sensitivity for multi-yield-surface J2
material plasticity modelParameter sensitivity of initial
configuration of multi-yield-surface plasticity modelStress
response sensitivitySensitivity of hardening parameters of active
and inner yield surfaces
Application examplesThree-dimensional block of clay subjected to
quasi-static cyclic loadingMulti-layered soil column subjected to
earthquake base excitation
ConclusionsAcknowledgementsReferences