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: jpconte@ucsd.edu (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