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Comput. Methods Appl. Mech. Engrg. 198 (2009) 2272–2285

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Comput. Methods Appl. Mech. Engrg.

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Finite element response sensitivity analysis of multi-yield-surface J2 plasticitymodel by direct differentiation method

Quan Gu a, Joel P. Conte b,*, Ahmed Elgamal b, Zhaohui Yang c

a AMEC Geomatrix Consultants Inc., 510 Superior Avenue, Suite 200, Newport Beach, CA 92663, USAb Department of Structural Engineering, University of California at San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USAc URS Corporation, 1333 Broadway, Suite 800, Oakland, CA, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 11 August 2008Received in revised form 22 January 2009Accepted 9 February 2009Available online 28 February 2009

Keywords:Nonlinear finite element analysisResponse sensitivity analysisMulti-yield-surface plasticity modelDirect differentiation methodSoil material model

0045-7825/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.cma.2009.02.030

* Corresponding author. Tel.: +1 858 822 4545; faxE-mail address: [email protected] (J.P. Conte).

Finite element (FE) response sensitivity analysis is an essential tool for gradient-based optimizationmethods used in various sub-fields of civil engineering such as structural optimization, reliability analy-sis, system identification, and finite 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 efficient 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 finite element analysis program. The work presented in thispaper extends significantly 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 forwardfinite 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.

1. Introduction

Finite element (FE) response sensitivities represent an essentialingredient for gradient-based optimization methods required invarious sub-fields of structural and geotechnical engineering suchas structural optimization, reliability analysis, system identifica-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.

Several methods are available for response sensitivity computa-tion, including the finite 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. [4–6], 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

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: +1 858 822 2260.

affected by numerical noise (i.e., truncation and round-off errors).The AM is extremely efficient for linear and nonlinear elastic sys-tems, but is not a competitive method for path-dependent (i.e.,inelastic) problems. The PM is computationally efficient, but gener-ally not very accurate. The DDM, on the other hand, is general,accurate and efficient 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].

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 first developed by Iwan [10] and Mroz [11], then applied byPrevost [12–14] to soil mechanics. It was later modified and imple-mented in OpenSees [15–17] 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 Pacific Earthquake EngineeringResearch (PEER) Center. In contrast to the classical J2 (or VonMises) elasto-plastic behavior with a single yield surface,

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τmax

τm

G

γm γmax

τ

γ

Hyperbolic backbone curve

1H

m( )2⁄

1

13---τ3

13---τ1

13---τ2

Deviatoric plane

(a) Octahedral shear stress-strain (after [12])

fNYS

fm

fm 1+

τ3

τ1

τ2

(b) Von Mises multi-yield surfaces

τ1 τ2 τ3= =

f1

Fig. 1. Yield surfaces of multi-yield-surface J2 plasticity model in principaldeviatoric stress space.

Q. Gu et al. / Comput. Methods Appl. Mech. Engrg. 198 (2009) 2272–2285 2273

multi-yield-surface J2 plasticity employs the concept of a field ofplastic moduli [12–14] to achieve a piecewise linear elasto-plasticbehavior under cyclic loading conditions. This field is defined by acollection of nested yield surfaces of constant size (i.e., no isotropichardening) in the stress space, which define 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-basedalgorithm was 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 significantly 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 [12–14,16,18,19,23–26]. Also, it isoperational in OpenSees [18] through which soil–structure-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 significantly the computational effi-ciency of such applications to a wide class of geotechnical systems[27–29] 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 DDM methodology 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 defining the initial config-uration of the multi-yield surfaces are required at the initiationof the response and response sensitivity computation. Further-more, the sensitivities of the kinematic hardening parametersdefining the active and inner yield surfaces must be updated ateach load/time step.

Two application examples are provided to validate the new re-sponse sensitivity algorithm and its implementation using the fi-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 defined in the deviatoric stress space as [23]

f ¼ 32ððs� aÞ : ðs� aÞÞ

� �12

� K ¼ 0; ð1Þ

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(ffiffiffiffiffiffiffiffi3=2

ptimes the radius) of the yield surface which defines the re-

gion of constant plastic shear modulus. The dyadic tensor productof tensors A and B is defined 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 stress–strain backbone curve[18,19] as shown in Fig. 1a. The experimentally determined back-bone curve can be approximated by the hyperbolic formula [31] as

s ¼ Gc1þ ðc=crÞ

; ð2Þ

--

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 flow rule of multi-yield-surface J2 plasticity model.

2274 Q. Gu et al. / Comput. Methods Appl. Mech. Engrg. 198 (2009) 2272–2285

where s and c denote the octahedral shear stress and shear strain,respectively, and G is the low-strain shear modulus. Parameter cr

is a reference shear strain defined as

cr ¼cmaxsmax

Gcmax � smax; ð3Þ

where smax, called shear strength, is the shear stress correspondingto the shear strain c = cmax (selected sufficiently 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[12–14]. Parameter H(i) (see Fig. 1a) is conveniently defined asHðiÞ ¼ 2 siþ1�si

ciþ1�ci

� �. A constant plastic shear modulus H0ðiÞ defined as

[32]

1

H0ðiÞ¼ 1

HðiÞ� 1

2Gð4Þ

is associated with each yield surface {fi = 0}.The stress–strain points (sj, cj) (subscript j denotes the point

number) used to define the piecewise linear approximation ofthe shear stress–strain (s � c) backbone curve are defined suchthat their projections on the s axis are uniformly spaced (seeFig. 1). Thus,

sj ¼ smaxj

NYSand cj ¼

sjcr

Gcr � sjðj ¼ 1;2; . . . ;NYS� 1Þ: ð5Þ

The j-th yield surface {fj = 0} is defined 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,

ðDcÞj ¼ cjþ1 � cj; ð6ÞðDsÞj ¼ sjþ1 � sj; ð7Þ

KðjÞ ¼ 3ffiffiffi2p sj; ð8Þ

HðjÞ ¼2ðDsÞjðDcÞj

; ð9Þ

H0ðjÞ ¼ 2GHðjÞ

2G� HðjÞ: ð10Þ

For the outermost yield surface (failure surface), setH(NYS) = H0ðNYSÞ = 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 defines a geo-metrical boundary in the deviatoric stress space.

It is worth mentioning that the yield surfaces may initially beconfigured 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 flow 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

d�p ¼ hLiH0

Q ; ð11Þ

where the second-order unit tensor Q defined as

Q ¼ 1Q

ofor

ð12Þ

in which Q ¼ ofor

: ofor

n o12, represents the plastic flow 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 defined asthe projection of the stress increment vector ds onto the directionnormal to the yield surface, i.e.,

L ¼ Q : ds: ð13Þ

The symbol h i in Eq. (11) denotes the MacCauley’s brackets definedsuch 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 hLi

H0 f ðs; aÞ ¼ 0, such thatthe plastic strain increment is zero in the elastic case (i.e., whenf(s, a) < 0).

The flow rule defined 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 figure, the first 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 Newton–Raphson scheme used to solvethe nonlinear equilibrium equations at each time step). Assumingthat the current active yield surface is the mth surface {fm = 0} withits center at a

ðmÞn , the elastic trial (deviatoric) stress str

nþ1;0 is ob-tained as

strnþ1;0 ¼ sn þ 2GDenþ1; ð14Þ

where sn is the converged deviatoric stress at the last (nth) timestep, and Den+1 denotes the total (from last converged step) devia-

T

T

n

nm( )

fm 0=

fm 1+ 0=

O

n 1+m( )

nm 1+( )

n 1+

n 1+


toric strain increment in the current time step. If trial the trial stressstr

nþ1;0 falls inside the current yield surface {fm = 0}, then the iterationprocess for the integration of the material constitutive law is con-verged, 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 defined as (see Fig. 2 fori = 1 and 2)

Pnþ1;i ¼ strnþ1;i�1 � str

nþ1;i ði ¼ 1;2;3 . . .Þ: ð15Þ

An important stress quantity called the contact stress s*n+1,i isdefined as the intersection point of vector str

nþ1;i�1 � aðmÞn and the

current active yield surface {fm = 0}, and can be computed as (seeFig. 2 for i = 1 and 2)

s�nþ1;i ¼KðmÞ

Knþ1;iðstr

nþ1;i�1 � aðmÞn Þ þ aðmÞn ; ð16Þ

where Kn+1,i (different from K(i) =ffiffiffiffiffiffiffiffi3=2

ptimes the radius of the i-th

yield surface) is defined as

Knþ1;i ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi32ðstr

nþ1;i�1 � aðmÞn Þ : ðstr

nþ1;i�1 � aðmÞn Þ

r; ð17Þ

which isffiffiffiffiffiffiffiffi3=2

ptimes the distance from str

nþ1;i�1 to aðmÞn . The unit ten-

sor normal to the current active yield surface {fm = 0} at s�nþ1;i is de-rived from Eq. (12) or Fig. 2 as

Q nþ1;i ¼ðs�nþ1;i � a

ðmÞn Þ

½ðs�nþ1;i � aðmÞn Þ : ðs�nþ1;i � a

ðmÞn Þ�

12: ð18Þ

The plastic stress correction tensor Pn+1, i (i = 1, 2, 3, . . .) can be de-rived as [32,33]

Pnþ1;1 ¼ 2GQ nþ1;1 : ðstr

nþ1;0 � s�nþ1;1ÞðH0ðmÞ þ 2GÞ

Q nþ1;1 ð19Þ

and

Pnþ1;i ¼ 2G �Q nþ1;i : ðstr

nþ1;i�1 � s�nþ1;iÞðH0ðmÞ þ 2 � GÞ

� ðH0ðm�1Þ � H0ðmÞÞ

H0ðm�1Þ � Q nþ1;i

ði ¼ 2;3;4; . . .Þ: ð20Þ

The trial stress after the plastic correction for the current activeyield surface is obtained using Eq. (15) as

strnþ1;i ¼ str

nþ1;i�1 � Pnþ1;i ði ¼ 1;2;3; . . .Þ: ð21Þ

If the trial stress strnþ1;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 str

nþ1;i

falls inside the next outer yield surface. After ‘‘convergence” ofthe deviatoric stress str

nþ1;i to strnþ1 is achieved following the above

iterative algorithm, the volumetric stress rvolnþ1 is updated to

rvolnþ1 ¼ rvol

n þ BðD�nþ1 : IÞ; ð22Þ

where B = elastic bulk modulus, D�n+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

rnþ1 ¼ snþ1 þ rvolnþ1 � I: ð23Þ

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]).

2.3. Hardening law

A 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, all

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 defined after [19] as

lnþ1 ¼ ½sT � aðmÞn � �KðmÞ

Kðmþ1Þ ½sT � aðmþ1Þn �; ð24Þ

where sT is the deviatoric stress tensor defining 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 a

ðmÞn of the current yield surface and the

current stress state (sn+1) at the end of the trial time/load step.The hardening rule defined 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 ¼ aðmÞn þ nðsnþ1 � aðmÞn Þ; ð25Þ

where 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

Deviatoric Plane

fNYS

fm 1– f1

n 1+fm

13---τ3

13---τ1

13---τ2

Fig. 4. Inner yield surface movements.


ðsT � aðmþ1Þn Þ : ðsT � aðmþ1Þ

n Þ � 23ðKðmþ1ÞÞ2 ¼ 0: ð26Þ

Substituting Eqs. (25) into (26) yields the following scalar quadraticequation to be solved for parameter n:

An2 þ Bnþ C ¼ 0; ð27Þ

where the coefficients A, B, and C are given by

A ¼ ðsnþ1 � aðmÞn Þ : ðsnþ1 � aðmÞn Þ; ð28ÞB ¼ 2ðaðmÞn � aðmþ1Þ

n Þ : ðsnþ1 � aðmÞn Þ; ð29Þ

C ¼ ðaðmÞn � aðmþ1Þn Þ : ðaðmÞn � aðmþ1Þ

n Þ � 23ðKðmþ1ÞÞ2: ð30Þ

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,

½snþ1 � ðaðmÞn þ flnþ1Þ� : ½snþ1 � ðaðmÞn þ flnþ1Þ� �23ðKðmÞÞ2 ¼ 0; ð31Þ

which reduces to the following quadratic equation:

A0f2 þ B0fþ C 0 ¼ 0; ð32Þ

where the coefficients A0, B0, and C0 are given by

A0 ¼ lnþ1 : lnþ1; ð33ÞB0 ¼ �2lnþ1 : ðsnþ1 � aðmÞn Þ; ð34Þ

C0 ¼ ðsnþ1 � aðmÞn Þ : ðsnþ1 � aðmÞn Þ �23ðKðmÞÞ2: ð35Þ

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

amnþ1 ¼ am

n þ flnþ1: ð36Þ

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 {fm�1 = 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]

snþ1 � aðmÞnþ1

KðmÞ¼

snþ1 � aðm�1Þnþ1

Kðm�1Þ ¼ � � � ¼snþ1 � a

ð1Þnþ1

Kð1Þ: ð37Þ

From Eq. (37), the updated center of each of the inner yield surfacesis obtained as

aðiÞnþ1 ¼ snþ1 �

KðiÞðsnþ1 � aðmÞnþ1Þ

KðmÞð1 6 i 6 m� 1Þ: ð38Þ

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 definition, the sensitivity of r with re-spect to the (material or loading) parameter h is expressedmathematically as the absolute partial derivative of r with respectto the variable h; or

oh

��h¼h0

, where h0 denotes the nominal value taken

by the sensitivity parameter h for the finite element responseanalysis.

In this paper, following the notation proposed in Ref. [2], thescalar response quantity r(#) = r(f(#), #) depends on the parametervector # (defined 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, dr

d# denotesthe gradient or total derivative of r with respect to #; dr

dhirepresents

the 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 or

ohi

��z

denotes the partial derivative of r

with respect to parameter hi when the vector of variables z is keptconstant (fixed). In the particular and important case in which

z = f(#), the expression orohi

��z

reduces 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, dueto the uncoupled nature of the sensitivity equations with respectto different sensitivity parameters.

In the context of nonlinear finite 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 finite elements can be found in anumber of Refs. [4,6–9,20–22,33,34].

3.2. Displacement-based FE response sensitivity analysis using DDM

After spatial discretization using the finite element method, theequations of motion of a materially-nonlinear-only model of astructural system take the form of the following nonlinear matrixdifferential equation:


MðhÞ€uðt; hÞ þ CðhÞ€uðt; hÞ þ Rðuðt; hÞ; hÞ ¼ Fðt; hÞ; ð39Þ

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):

Wðunþ1Þ ¼ eFnþ1 �1

bðDtÞ2Munþ1 þ

abðDtÞ2

Cunþ1 þ Rðunþ1Þ" #

¼ 0; ð40Þ

where

eFnþ1 ¼ Fnþ1 þM1

bðDtÞ2un þ

1bðDtÞ

_un � 1� 12b

� €un

" #

þ Ca

bðDtÞun � 1� ab

� _un � ðDtÞ 1� a

2b

� €un

�; ð41Þ

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 (finite 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.,

Rðunþ1Þ ¼ ANel

e¼1fRðeÞðuðeÞnþ1Þg; ð42Þ

where ANele¼1f. . .g denotes the direct stiffness assembly operator from

the element level (in local elements coordinates) to the structure le-vel in global reference coordinates, Nel represents the number of fi-nite elements in the FE model, R(e) and uðeÞnþ1 denote the internalresisting force vector and nodal displacement vector, respectively,of element e.

Typically, Eq. (40) is solved using the Newton–Raphson 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 specified 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

bðDtÞ2Mþ a

bðDtÞCþ ðKstatT Þnþ1

" #dunþ1

dh

¼ � 1

bðDtÞ2dMdhþ a

bðDtÞdCdh

!unþ1 �

oRðunþ1ðhÞ; hÞoh

��unþ1

þ deFnþ1

dh;

ð43Þ

where

deFnþ1

dh¼ dFnþ1

dhþ dM

dh1

bðDtÞ2un þ

1bðDtÞ

_un � 1� 12b

� €un

!

þM1

bðDtÞ2dun

dhþ 1

bðDtÞd _un

dh� 1� 1

2b

� d €un

dh

" #

þ dCdh

abðDtÞun � 1� a

b

� _un � ðDtÞ 1� a

2b

� €un

�þ C

abðDtÞ

dun

dh� 1� a

b

� d _un

dh� ðDtÞ 1� a

2b

� d€un

dh

�:

ð44Þ

In Eq. (43), ðKstatT Þnþ1 denotes the consistent (or algorithmic) tangent

(static) stiffness matrix of the structure/system, which is definedas the assembly of the element consistent tangent stiffness matricesas

ðKstatT Þnþ1 ¼

oRðunþ1Þounþ1

¼ ANel

e¼1

oRðeÞðunþ1ÞouðeÞnþ1

( )¼ A

Nel

e¼1

ZXðeÞ

BTDnþ1BdXðeÞ� �

;

ð45Þ

where B is the strain–displacement transformation matrix, and Dn+1

denotes 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.,

Dnþ1 ¼ornþ1ðrn; �n; �� n; . . .Þ

o�nþ1: ð46Þ

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 fixed. It is com-puted through direct stiffness assembly of the element resistingforce derivatives as

oRðunþ1ðhÞ; hÞoh

��unþ1

¼ ANel

e¼1

ZXðeÞ

BTorð�nþ1ðhÞ; hÞoh

��nþ1

dXðeÞ( )

: ð47Þ

In the following section, computation of the unconditional (i.e., un+1

is not fixed) stress sensitivities to parameters of the multi-yield-surface material plasticity model is presented in detail. The condi-tional (i.e., un+1 is fixed) 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 configuration of multi-yield-surface plasticity model

In this section, the sensitivities of the parameters (see Eqs. (3)–(10)) defining the initial configuration of the material model to thesensitivity parameter h are derived. Differentiating Eq. (3) with re-spect to h yields


dcr

dh¼ 1

ðGcmax � smaxÞ2� cmax �

dsmax

dh� ðGcmax � smaxÞ

� cmax � smax �

dGdh� cmax �

dsmax

dh

� �: ð48Þ

For each yield surface {fj = 0} (1 6 j 6 NYS � 1), Eq. (5) is differenti-ated with respect to h as

dsj

dh¼ dsmax

dh� jNYS

; ð49Þ

dcj

dh¼ 1

ðGcr � sjÞ2dsj

dh� cr þ sj �

dcr

dh

� � ðGcr � sjÞ

�sjcr �

dGdh� cr þ G � dcr

dh� dsj

dh

� �: ð50Þ

Differentiating Eqs. (6)–(10) with respect to h yields

dðDcÞjdh

¼dcjþ1

dh�

dcj

dh; ð51Þ

dðDsÞjdh

¼ dsjþ1

dh� dsj

dh; ð52Þ

dKðjÞ

dh¼ 3ffiffiffi

2p dsj

dh; ð53Þ

dHðjÞ

dh¼ 2

ðDcÞ2j

dðDsÞjdh

� ðDcÞj � ðDsÞj �dðDcÞj

dh

�; ð54Þ

dH0ðjÞ

dh¼ 2

ð2G� HðjÞÞ2dGdh

HðjÞ þ GdHðjÞ

dh

" #(

� ð2G� HðjÞÞ � GHðjÞ 2dGdh� dHðjÞ

dh

! !): ð55Þ

The outermost yield surface (failure surface) is characterized byH(NYS) = H0ðNYSÞ = 0. from which it follows that dHðNYSÞ

dh ¼ 0 anddH0ðNYSÞ

dh ¼ 0. Furthermore, for each yield surface, the sensitivity to hof the initial back-stress a(j) = 0 (initial center of the yield surface)is zero, i.e., daðjÞ

dh ¼ 0 ð1 6 j 6 NYSÞ.

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 termoRðunþ1ðhÞ;hÞ

oh

��unþ1

. The latter requires computation of the conditional

(i.e., un+1 is fixed case sensitives, orð�nþ1ðhÞ;hÞoh j�nþ1

, at each integration

(Gauss) point. Eq. (43) is solved for the displacement response sen-sitivity dunþ1

dh . From the relationship between nodal displacementsand strains at the element level, the strain and deviatoric strain re-sponse sensitivities (d�nþ1

dh and denþ1dh , respectively) can be readily ob-

tained. Then, the unconditional (i.e., un+1 is not fixed) 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 fixed) stress response sensitivities are evaluated usingthe same equations as the unconditional ones, but imposing theconstraint that un+1 is fixed, which implies that the strain �n+1 isfixed for displacement-based finite 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

dstrnþ1;0

dh¼ dsn

dhþ 2

dGdh

Denþ1 þ 2GdDenþ1

dhð56Þ

where Den+1 = en+1 � en and thus dDenþ1dh ¼ denþ1

dh �dendh . When comput-

ing the conditional stress sensitivity (see Eq. (47)) for which the

strain �n+1 and therefore the deviatoric strain en+1 remain fixed,denþ1

dh ¼ 0 and dDenþ1dh ¼ � den

dh . 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 s�nþ1;i to parameter h is ob-tained by differentiating Eq. (16) with respect to h as

ds�nþ1;i

dh¼ 1

K2nþ1;i

dKðmÞ

dhðstr

nþ1;i�1 � aðmÞn Þ þ KðmÞdstr

nþ1;i�1

dh� da

ðmÞn

dh

!" #(

� Knþ1;i � KðmÞðstrnþ1;i�1 � aðmÞn Þ

dKnþ1;i

dh

�þ da

ðmÞn

dh; ð57Þ

where the sensitivity of Kn+1, i to h is obtained from Eq. (17) as

dKnþ1;i

dh¼ 3

2Knþ1;i

dstrnþ1;i�1

dh� da

ðmÞn

dh

!: ðstr

nþ1;i�1 � aðmÞn Þ: ð58Þ

Eq. (18) yields the following sensitivity to h of the unit tensor nor-mal to the yield surface at s�nþ1;i:

dQ nþ1;i

dh¼

ds�nþ1;idh �

daðmÞn

dh

� �½ðs�nþ1;i � a

ðmÞn Þ : ðs�nþ1;i � a

ðmÞn Þ�1=2

�ds�

nþ1;idh �

daðmÞn

dh

� �: ðs�nþ1;i � a

ðmÞn Þ

ððs�nþ1;i � aðmÞn Þ : ðs�nþ1;i � a

ðmÞn ÞÞ3=2 � ðs

�nþ1;i � aðmÞn Þ: ð59Þ

Then, the sensitivity of the plastic stress correction tensor is derivedfrom Eq. (19) as

dPnþ1;1

dh¼ 2

dGdh

Q nþ1;1 : ðstrnþ1;0 � s�nþ1;1Þ

ðH0ðmÞ þ 2GÞQ nþ1;1

þ 2GQ nþ1;1 : ðstr

nþ1;0 � s�nþ1;1ÞðH0ðmÞ þ 2GÞ

�dQ nþ1;1

dh

þ 2G

ðH0ðmÞ þ 2GÞ2dQ nþ1;1

dh: ðstr

nþ1;0 � s�nþ1;1Þ�

þ Q nþ1;1 :dstr

nþ1;0

dh�

ds�nþ1;1

dh

� �ðH0ðmÞ þ 2GÞ

� Q nþ1;1 : ðstrnþ1;0 � s�nþ1;1Þ:

dH0ðmÞ

dhþ 2

dGdh

!)Q nþ1;1 ð60Þ

for the first corrective iteration (subscript i = 1) and from Eq. (20) as

dPnþ1;i

dh¼ 2

dGdh

Q nþ1;i : ðstrnþ1;i�1 � s�nþ1;iÞ

ðH0ðmÞ þ 2GÞQ nþ1;i

"

þ 2GQ nþ1;i : ðstr

nþ1;i�1 � s�nþ1;iÞðH0ðmÞ þ 2GÞ

dQ nþ1;i

dh

#ðH0ðm�1Þ � H0ðmÞÞ

H0ðm�1Þ

þ 2GðH0ðm�1Þ � H0ðmÞÞðH0ðmÞ þ 2GÞ2 � H0ðm�1Þ

dQ nþ1;i

dh: ðstr

nþ1;i�1 � s�nþ1;iÞ�

þ Q nþ1;i :dstr

nþ1;i�1

dh�

ds�nþ1;i

dh

� �ðH0ðmÞ þ 2GÞ

� Q nþ1;i : ðstrnþ1;i�1 � s�nþ1;iÞ

dH0ðmÞ

dhþ 2

dGdh

!)Q nþ1;i

þ 2GQ nþ1;i : ðstr

nþ1;i�1 � s�nþ1;iÞðH0ðmÞ þ 2GÞ

� H0ðmÞ

½H0ðm�1Þ�2dH0ðm�1Þ

dh� 1

H0ðm�1Þ

dH0ðmÞ

dh

" #Q nþ1;i ð61Þ


for 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

dstrnþ1;i

dh¼

dstrnþ1;i�1

dhþ dPnþ1;i

dhði ¼ 1;2; . . .Þ: ð62Þ

If the trial stress strnþ1;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 strnþ1;i is achieved,

the sensitivity to h of the updated volumetric stress rvolnþ1 is com-

puted from Eq. (22) as

drvolnþ1

dh¼ drvol

n

dhþ dB

dh� ðD�nþ1Þii þ B � dðD�nþ1Þii

dh; ð63Þ

where dðD�nþ1Þiidh ¼ dD�nþ1;11

dh þ dD�nþ1;22dh þ dD�nþ1;33

dh . It is worth noting that inthe process of computing the conditional stress sensitivities, thequantity dðD�nþ1Þii

dh ¼ 0, since un+1 is considered fixed. Finally, the sen-sitivity to h of the total stress is obtained from Eq. (23) as

drnþ1

dh¼ dsnþ1

dhþ

drvolnþ1

dh� I: ð64Þ

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 ðaðiÞn ;1 6 i 6 mÞ, 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

dsT

dh¼ da

ðmÞn

dhþ dn

dhðsnþ1 � aðmÞn Þ þ n

dsnþ1

dh� da

ðmÞn

dh

!; ð65Þ

where dndh is obtained from Eq. (27) as

dndh¼� dA

dh n2 � dBdh n� dC

dh

2Anþ Bð66Þ

and where dAdh,

dBdh, and dC

dh are obtained in turn from Eqs. (28)–(30) as

dAdh¼ 2

dsnþ1

dh� da

ðmÞn

dh

!: ðsnþ1 � aðmÞn Þ; ð67Þ

dBdh¼ 2

daðmÞn

dh� da

ðmþ1Þn

dh

!: ðsnþ1 � aðmÞn Þ

þ 2ðaðmÞn � aðmþ1Þn Þ :

dsnþ1

dh� da

ðmÞn

dh

!; ð68Þ

dCdh¼ 2

daðmÞn

dh� da

ðmþ1Þn

dh

!: ðaðmÞn � aðmþ1Þ

n Þ � 43

Kðmþ1Þ:dKðmþ1Þ

dh: ð69Þ

Then, the sensitivity to h of the translation direction ln+1 iscomputed by differentiating Eq. (24) with respect to h as

dlnþ1

dh¼ dsT

dh� da

ðmÞn

dh

" #�

dKðmÞ

dh Kðmþ1Þ � KðmÞ dKðmþ1Þ

dh

½Kðmþ1Þ�2

!½sT � aðmþ1Þ

n �

� KðmÞ

Kðmþ1ÞdsT

dh� da

ðmþ1Þn

dh

" #: ð70Þ

After dlnþ1dh is obtained, the sensitivity to h of the translation param-

eter f is derived from Eq. (32) as

dfdh¼� dA0

dh f2 � dB0

dh f� dC0

dh

2A0fþ B0; ð71Þ

where dA0

dh , dB0

dh , and dC0

dh are obtained from Eqs. (33)–(35) as

dA0

dh¼ 2

dlnþ1

dh: lnþ1; ð72Þ

dB0

dh¼ �2

dlnþ1

dh: ðsnþ1 � aðmÞn Þ � 2lnþ1 :

dsnþ1

dh� da

ðmÞn

dh

!; ð73Þ

dC0

dh¼ 2

dsnþ1

dh� da

ðmÞn

dh

!: ðsnþ1 � aðmÞn Þ �

43

KðmÞ � dKðmÞ

dh: ð74Þ

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

daðmÞnþ1

dh¼ da

ðmÞn

dhþ df

dhlnþ1 þ f

dlnþ1

dh: ð75Þ

The sensitivity to h of the updated back-stress of each of the inneryield surfaces are obtained from Eq. (38) as

daðiÞnþ1

dh¼dsnþ1

dh

�KðmÞ dKðiÞ

dh ðsnþ1�aðmÞnþ1ÞþKðmÞKðiÞ dsnþ1

dh �daðmÞnþ1

dh

� �dKðmÞ

dh KðiÞðsnþ1�aðmÞnþ1Þ

ðKðmÞÞ2;

ð76Þ

where 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 firstapplication and validation example. The block is discretized into 8brick elements defined as displacement-based eight-noded, trilin-ear isoparametric finite 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,() Poisson’s ratio = 0.38), and maximum shear stress smax = 30 k-Pa. The bottom nodes of the finite element (FE) model are fixedand top nodes {A, B, C} and {A, D, E} are subjected to five cyclesof harmonic, 90 degrees out-of-phase, concentrated horizontalforces Fx1 ðtÞ ¼ 2:0 sinð0:2ptÞ½kN� and Fx2 ðtÞ ¼ 2:0 sinð0:2ptþ0:5pÞ½kN�, 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 ðtÞ, while thehysteretic shear stress–strain response (r31 � e31) at Gauss pointG (see Fig. 5) is plotted in Fig. 7. These figures indicate that signif-icant yielding of the clay is observed during the cyclic loadingconsidered.

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5x 10−4

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

F x1

kN[]

ux1

[m]

Fig. 6. Force–displacement response at node A.

−3 −2 −1 0 1 2 3 4 5x 10−4

−3

−2

−1

0

1

2

3

31 [k

Pa]

31

Fig. 7. Shear stress–strain response at Gauss point G (see Fig. 5).

0 10 20 30 40 50−12

−10

−8

−6

−4

−2

0

2

4

6

x 10−9

DDMΔ θ /θ = 1e−1Δ θ /θ = 1e−3Δ θ /θ = 1e−5

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 finite 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 10−9


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 finite difference(zoom view of Fig. 8).

0.5 m 0.5 m

0.5 m

0.5

m0.

5 m

BA C

D

E

G

Fx1

Fx1

Fx1

Fx2Fx2

Fx2

x1

x2

x3

0.5 m

Fig. 5. Solid block of clay subjected to horizontal quasi-static cyclic loading underundrained condition.

0 10 20 30 40 50−4

−3

−2

−1

0

1

2 x 10−10


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 finite difference for threedifferent values of relative parameter perturbation Dh/h.


The 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

0 10 20 30 40 50−7

−6

−5

−4

−3

−2

−1

0

1

2 x 10−5


du dτm

ax-----

---------

m3

kN-------

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.

Ground surface

0.00.6–

8.0–

17.7–

45.1–

73.3–

5.0

u1u2

u3u4

u5

.C

u6.D

.

.F7.1–

E

x

z

Layer #1

Layer #6

Layer #5

Layer #2Layer #3Layer #4

Fig. 12. Layered soil column subjected to total base acceleration with finite elementmesh in thin lines (unit: [m]). Soil layers are numbered from top to bottom.

0 2 4 6 8 10−2

0

2

Time [sec]

Acce

l. [m

/s2 ]

Δt = 0.01 [sec]

Fig. 13. Total acceleration time history at the base of the soil column.


and 11, respectively, with the corresponding sensitivities esti-mated using the forward finite 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 of1.0e�5, 1.0e�5 and 1.0e�4, 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-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 finite 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 86

tion of a ground surface free field 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 stress–strain(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 verified 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. 16–21.From these figures 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

0 2 4 6 8 10−0.08

−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). 0 2 4 6 8 10

−1.5

−1

−0.5

0

0.5

1

1.5

2 x 10−11


du6

dG1

----------

m3

kN-------

Time [sec]

Fig. 16. Sensitivity of displacement response u6 (see Fig. 12) to shear modulus G1

obtained using DDM and FFD with increasingly small perturbations of sensitivityparameter.


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 as‘‘100 times the change in the considered response quantity due toone percent change in the corresponding sensitivity parameter”. Asa first 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,(5) G5, (6) G6. Thus, the shear strength parameters of the bottom

−1 −0.5 0 0.5 1x 10−3

−20

−15

−10

−5

0

5

10

15

−2 −1 0 1 2 3x 10−3

−60

−40

−20

0

20

40

60

80

Point F

Point D

xz [k

Pa]

xz [-]

xz [k

Pa]

xz [-]

Fig. 15. Shear stress–strain hysteric responses

soil layers are very important in controlling the ground surface dis-placement response, since the soil undergoes significant 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 tothe most sensitive parameters which should be adjusted or

−8 −6 −4 −2 0 2 4 6x 10−4

−20

−15

−10

−5

0

5

10

15

20

−1 −0.5 0 0.5 1 1.5 2x 10−3

−80

−60

−40

−20

0

20

40

60

80

100

xz[k

Pa]

xz [-]

Point E

Point C

xz[k

Pa]

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 10−12


du6

dG1

----------

m3

kN-------

Time [sec]

Fig. 17. Sensitivity of displacement response u6 (see Fig. 12) to shear modulus G1

obtained using DDM and FFD with increasingly small perturbations of sensitivityparameter (zoom view of Fig. 16).

0 2 4 6 8 10−2

−1.5

−1

−0.5

0

0.5

1 x 10−9


du1

dB6

---------

m3

kN-------

Time [sec]

Fig. 18. Sensitivity of displacement response u1 (see Fig. 12) to bulk modulus B6

obtained 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 10−10


du1

dB6

---------

m3

kN-------

Time [sec]

Fig. 19. Sensitivity of displacement response uj (see Fig. 12) to bulk modulus B6

obtained using DDM and FFD with increasingly small perturbations of sensitivityparameter (zoom view of Fig. 18).

0 2 4 6 8 10−1

−0.5

0

0.5

1

1.5

2 x 10−4


du2

dτm

ax, 2

----------

---------

m3

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 10−5


du2

dτm

ax2

,-----

----------

----m

3

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 10−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

τmax,4

τmax,4

τmax,5τmax,6G4G5G6

du1

dθi

--------

θ im[

]

Importance ranking:

Time [sec]

> τmax,5 > τmax,6 > >G4 >G5 G6

Fig. 22. Relative importance of material parameters on horizontal displacementresponse of ground surface (top of soil column, see Fig. 12).


0 2 4 6 8 10−0.02

−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 influence of shear modulus G6 on horizontal displacementresponse of soil column at different soil layers (see Fig. 12).


calibrated 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 G6

of 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) u3

and 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)may 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 efficient method to compute finite 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 forfinite element analysis of structural and/or geotechnical systems.Two application examples are presented to validate, using forwardfinite difference analysis, the response sensitivity results obtainedfrom the proposed DDM-based algorithm. The second applicationexample is also employed to illustrate the use of finite element re-sponse sensitivity analysis to investigate the relative importance ofmaterial parameters on system response.

The algorithm developed herein for nonlinear finite element re-sponse sensitivity analysis of geotechnical systems modeled usingthe multi-yield-surface J2 plasticity model has direct applications

in optimization, reliability analysis, and nonlinear FE modelupdating.

Acknowledgements

Support of this research by the Pacific 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, findings, conclusions, or recommen-dations expressed in this publication are those of the authors anddo not necessarily reflect the views of the sponsors.

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