multiscale analytical sensitivity analysis for composite materials

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2001; 50:1501–1520

Multiscale analytical sensitivity analysis forcomposite materials

Jacob Fish∗;† and Amine GhoualiDepartments of Civil; Mechanical and Aerospace Engineering; Rensselaer Polytechnic Institute; Troy;

New York 12180; U.S.A.

SUMMARY

We describe a methodology aimed at determining the sensitivity of the global structural behaviour, suchdeformation or vibration modes with respect to the local characteristics such as material constants of micro-constituents. An analytical gradient computation, which involves the direct di�erentiation of the multiplescale strong forms with respect to the design parameters is developed. Comparison of the multiple scalesensitivity analysis (MASA) to the central �nite di�erence (CFD) approximation in terms of accuracy andcomputation e�ciency is carried out. We demonstrate the robustness of the MASA approach compared tothe CFD approximation, which has been found to be highly sensitive to the choice of the step size, whoseoptimal value is problem dependent. Copyright ? 2001 John Wiley & Sons, Ltd.

KEY WORDS: analytical sensitivity design; composite materials; multiple scales

1. INTRODUCTION

Composite materials play an important role in many �elds of modern engineering due to lowweight=sti�ness ratio that makes their application in high-performance structures very desirable [1].One of the greatest challenges facing the composites designers today is in selecting nearly optimalmaterial architectures including micro-geometry and properties of micro-phases [2]. The ability toperform micro-structural optimization depends on the accuracy and computational e�ciency of thedesign sensitivity analysis [3; 4].Design sensitivity analysis deals with a gradient computation of an objective function and

constraints with respect to geometrical and=or material characteristics [5; 6]. Typically, this com-putation is performed numerically by means of the central �nite di�erence approximation [7; 8]. Inthis case the discrete thermo-mechanical problem is solved for at least two di�erent, but nearby,values of each design parameter. To ensure robustness a convergence study involving severaldiscrete analyses for each design parameter is often performed to determine the required step

∗Correspondence to: Jacob Fish, Department of Civil Engineering, Rensselaer Polytechnic Institute, 110 8th Street, Troy,NY 12180-3590, U.S.A.

†E-mail: �[email protected]

Contract=grant sponsor: O�ce of Naval Research; contract=grant number: N00014-97-1-0687

Received 28 December 1999Copyright ? 2001 John Wiley & Sons, Ltd. Revised 18 May 2000

1502 J. FISH AND A. GHOUALI

size. When the total number of independent design parameters becomes large this approach mayconsume more than two-thirds of the total computational cost of the optimization process [9].Furthermore, the accuracy of the approximations obtained by the �nite di�erence method may notbe adequate, in particular for non-linear history-dependent problems [10; 11], or when the actualsensitivities are of the order of numerical errors [12]. One of the alternatives, is the semi-analyticalmethod [13; 14], which is based on the di�erentiation of the discrete equations with respect to thedesign parameters and subsequent �nite di�erence approximation. Finally, the analytical approachis �nite di�erence-free, despite its high complexity [11; 15–18].The present manuscript is largely motivated by a desire to develop a multiscale analytical sensi-

tivity analysis (MASA), with the intent of calibrating microscale material properties to observableexperimental data. Multiscale calibration consists of solving the inverse problem leading to a uniqueand reliable determination of constitutive laws at multiple scales. The goal of the multiple scalesensitivity analysis is to determine the sensitivity of the global behaviour, such deformation orvibration modes with respect to material constants of micro-phases and vise versa.The manuscript is organized as follows. In Section 2, the multiple scale inverse problem is

formulated as a minimization of the di�erence between the measured and computed data of interest.The analytical gradient computation involves a direct di�erentiation of the multiple scale strongforms with respect to the design parameters. In Section 3, the mathematical homogenization theoryis given for the metal matrix composites. The use of a double-scale asymptotic expansion introducestwo uncoupled (micro- and macro-) direct problems. The weak form and the corresponding �niteelement discretization of the two problems are then stated. In Section 4, the multiscale analyticalsensitivity analysis (MASA) is developed using the double-scale asymptotic expansion for thesensitivities. In Section 5, the �nite element discretization and the non-linear solution of the discretemacro- and micro-sensitivity equations is presented. Finally, two examples are used to validate thepresent formulation: the �rst for a linear isotropic elastic composite, and the latter for the linearisotropic elastic reinforcement embedded in the elastoplastic matrix. Comparison of the MASAapproach to the CFD approximation in terms of accuracy and computation e�ciency completesthe manuscript. We show that the CFD approximation is highly sensitive to the choice of the stepsize, whose optimal value is problem dependent.

2. INDIRECT CALIBRATION OF MULTISCALE CONSTITUTIVE EQUATIONS

In this section, we focus on the mathematical formulation of the problem associated with the opti-mal design of material microstructure. The ultimate goal is to control the geometrical and materialcharacteristics of the micro-constituents by means of solving the inverse problem which calibratesthe computed data to the available experimental measurements at multiple scales. The experimen-tal data on the microscale can be obtained, for example, using the Moire interferometry technique[19; 20], which produces contour maps of the displacement and strain �elds. For an alternativetechnique based on the topological micro-structural optimization we refer to Reference [24].

2.1. Multiscale inverse problem

We consider a periodic composite structure de�ned by two scales: the macroscale with a positionvector x(x1; x2; x3) de�ned on the domain and the microscale with a position vector y(y1; y2; y3)de�ned on the representative volume element (RVE) domain � as shown in Figure 1. The

Copyright ? 2001 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2001; 50:1501–1520

COMPOSITE MATERIALS 1503

Figure 1. (a) The macro-domain ; and (b) the representative volume element domain �.

position vectors on the micro- and macro-scales are related by the small parameter ∈ (y= x=∈),which is the ratio of the two scale characteristic lengths. The micro-constituents are assumed topossess homogeneous properties and to satisfy equilibrium, constitutive, kinematics and compati-bility equations, as well as jump conditions at the interface between the micro-constituents.The two-scale inverse problem consists of minimizing the error between the measured and

computed data. Assuming that the experimental data, such as displacements u and strain �eld areavailable, the minimization problem can be stated as follows:

Find the control �eld P(p�) such that

MinP∈�

(�(P)=$1

∫∗

‖u− u‖2 d + $2

∫∗

[1

|�∗|∫�∗

‖�(u)− �‖2 d�]d

)(1)

where � is the space of admissible design variables; �∗ and ∗ are the portions of the micro-and macro-domains, respectively, where the experimental data is available; P(p�) is the set ofparameters de�ning constitutive laws on the microscale, such as Young modulus, Poisson ratio,hardening modulus and yield stress of micro-phases; $1 and $2 are weighting parameters; u isthe computed displacement �eld on ∗; and �(u) is the computed local strain �eld on �∗.Various sensitivity analysis approaches required for solving the constrained minimization prob-

lem (1) are brie y summarized in the next section.

2.2. Design sensitivity analysis

One approach by which the gradient computation can be obtained is the central �nite di�erence(CFD) approximation

d�dp�

∼= ��p�

≡ �(p� + �p�)− �(p� − �p�)2�p�

(2)

where �p� denotes a small step size.Typically, the gradient computation using the �nite di�erence approximation requires several

solutions of the problem for each �nite di�erence increment. In order to ensure su�ciently accuratesensitivities a convergence study is often performed [21], which involves several analyses for eachdesign parameter. For large-scale problems, the computational cost might be prohibitive as it mayentail more than two-thirds of the optimization process [12]. Furthermore, the accuracy of the



approximations obtained by the �nite di�erence approximation may not be adequate in particularfor non-linear history-dependent problems [10; 11] or when the actual sensitivities are of the orderof numerical errors [16]. An alternative, which is adopted in the present manuscript, is based onthe analytical di�erentiation [3; 18]. In the present context it involves a direct di�erentiation ofthe multiple scale strong forms with respect to the design parameters.

3. REVIEW OF THE MATHEMATICAL HOMOGENIZATION THEORY

3.1. Macro-scale direct problem

Due to the periodic nature of the micro-structure any response function g∈(x) = g(x; y(x)), suchas displacements u∈

i , stresses �∈ij , and strains �ij(u∈), de�ned on the composite domain (x; y), are

also y-periodic. The boundary value problem is governed by the following equations:

�∈ij; xj + bi(x) = 0 in (x; y) (3)

�∈ij = Lijkh�kh(u∈) in (x; y) (4)

�ij(u∈) = u∈(i; xj) in (x; y) (5)

u∈i = �ui(x) in �t (6)

�∈ij nj = �ti(x) in �� (7)

where the superimposed dot represents the material time derivative; u∈i and �∈ij are the velocity

vector and the rate of deformation tensor, respectively; the subscript pair with parentheses denotesthe symmetric gradient de�ned as u∈

(i; xj) = (u∈i; xj+ u∈

j; xi)=2; bi(x); �ui(x) and �ti(x) are the body forceson , the prescribed displacements on �u and the prescribed boundary tractions on �t, respectively;Lijkh is the instantaneous constitutive tensor and nj is the normal to �t. Attention is restricted tosmall deformation theory.The multiple scale analysis of the strong form (3)–(7) is obtained by approximating the velocity

�eld as

u∈i (x) = ui(x; y) = u0i (x; y) + ∈ u1i (x; y) + · · · (8)

Using the chain rule, the macroscopic spatial derivative of the velocity �eld (8) is given by

u∈i; xj (x) =

@u∈i (x)@xj

=(

@@xj

+ ∈−1 @@yj

)ui(x; y) = ui; xj (x; y) + ∈−1ui; yj (x; y) (9)

From (7) and (8), we obtain the following expansion of the rate of strain �eld

�ij(u∈(x))≡ �ij(u(x; y))≈∈−1�−1ij (u(x; y)) + �0ij(u(x; y)) + ∈ �1ij(u(x; y)) + · · · (10)

where

�−1ij (u(x; y)) = u0(i; yj)(x; y) (11)

�0ij(u(x; y)) = u0(i; xj)(x; y) + u1(i; yj)(x; y) (12)

�1ij(u(x; y)) = u1(i; xj)(x; y) + u2(i; yj)(x; y) (13)



The expansion of the stress rate is obtained by substituting the strain rate approximation (10) intothe constitutive equation (3) which yields

�∈ij (x)≡ �ij(x; y)≈∈−1�−1

ij (x; y) + �0ij(x; y) + ∈ �1ij(x; y) + · · · (14a)

where

�sij = Lijkh�skh(u(x; y)); s= − 1; 0; 1; : : : (14b)

Similarly, the expansion for the stress �eld is given by

�∈ij (x)≡ �ij(x; y)≈∈−1�−1

ij (x; y) + �0ij(x; y) + ∈ �1ij(x; y) + · · · (15)

Inserting the stress asymptotic expansion (14a) into equilibrium equations (3) and making use ofthe chain rule yields various order equilibrium equations:

O(∈−2): �−1ij; yj = 0 (16)

O(∈−1): �−1ij; xj + �0ij; yj = 0 (17)

O(∈0): �0ij; xj + �1ij; yj + bi = 0 (18)

O(∈s): �sij; xj + �s+1

ij; yj = 0; s=1; 2; : : : (19)

To solve for the O(∈−2) equilibrium equation we pre-multiply the rate form of (16) by u0i andintegrate it over RVE domain, �, which yields∫

�u0i �

−1ij; yj d�=0 (20)

and subsequent integration by parts results in∫��

u0i �−1ij nj d�� −

∫�u0(i; yj)Lijkhu0(h;yk ) d�=0 (21)

where the �rst term in the above equation vanishes due to periodicity of the boundary conditionson ��. Assuming that the instantaneous sti�ness tensor Lijkh(y) is positive de�nite the uniquesolution of the equilibrium equation (21) is

u(i;yj)0 = 0 ⇒ ui0 = ui0(x) (22)

The strong form of the O(∈−1) equilibrium follows from (15) and the rate form of (17) whichyields (

Lijkh(u0(k; xh) + u1(k;yh)

));yj=0 (23)

To solve for (23) up to a constant we introduce the following separation of variables:

ui1(x; y)=<mn

i (y) u(m; xn)0 (x) (24)

in which <mni represents the instantaneous y-periodic function with symmetry with respect to

indexes m and n.We further denote

mnij (y)=<mn

(i;yj)(y) (25)



and the equilibrium equation (23) can be expressed as(Lijkh

(Ikhmn +kh

mn(y))u0(m; xn)(x)

);yj=0 (26)

where

Ikhmn= 12(�km�hn + �kn�hm) (27)

and �km is the Kronecker delta. For an arbitrary macroscopic velocity �eld u0i , Equation (26) yields(Lijkh

(Ikhmn +kh

mn(y)))

;yj=0 (28)

To solve for the O(∈0) equilibrium equation we integrate the rate form of (18) over RVE domainwhich yields∫

�

(Lijkh(y)u1(k; xh)

);yjd� +

∫�Lijkh(y)

(u1(k;yh) + u0(k; xh)

); xjd� +

∫�bi d�=0 (29)

where the �rst term vanishes due to the periodicity resulting in([∫�Lijkh(y) (Ikhmn +mn

kh ) d�]u0(m; xn)

); xj

+ |�|bi=0 (30)

where |�| is the volume of the RVE. The rate of the local stress �eld,�ij =Lijkh (y) (Ikhmn +mn

kh ) ˙�mn(u) (31)

the macroscopic strain rate �eld, ˙�mn(u)= u0(m; xn)(x), and the overall stress �eld

˙�ij =1|�|

∫��ij d�=

1|�|

∫�Lijkh(y) (Ikhmn +mn

kh ) d� ˙�mn(u) (32)

are de�ned in the usual manner [1; 22; 23].From (29) the macroscopic equilibrium equation can be cast into the familiar form

�ij; xj + bi=0 (33)

Further denoting the homogenized sti�ness tensor components as

Lijkh=1|�|

∫�Lijmn(y)(Ikhmn +mn

kh ) d� (34)

the strong form of the macro-scale direct global problem (3)–(7) can be re-written as follows:

�ij; xj = −bi in ˙�ij = Lijkh(<) ˙�kh(u) in

˙�ij(u) = u0(i; xj) in

ui = �ui on �u�ijnj = �ti on �t

(35)

3.2. Micro-scale direct problem

The solution of the boundary value problem (35) is a function of the characteristic function <mni .

Further expressing the stress concentration factors on � as

skhij =Lijmn(y)(Ikhmn + kh

mn

)(36)



the characteristic functions <mni and skh

ij are the solutions of the following micro-scale directproblem:

<mni is y-periodic

skhij;yj = 0 in �

skhij = Lijmn(y)(Ikhmn +kh

mn

)in �

[|<mni |]= 0 on �

[|skhij nj|]= 0 on �

skhij nj is y-antiperiodic

(37)

where[| • |] denotes the jump operator at the interface between the micro-constituents �.

The instantaneous strain concentration function =ijkh(y) is denoted as

=ijkh(y)= Iijkh +

ijkh(y) (38)

so that the local strain rate �eld �ij(u(x; y)) is given by

�ij(u(x; y))==ijmn(y) ˙�mn(u) (39)

3.3. The weak and the discrete forms

The weak form of (35) is given as

�nd u0i ∈Uad such that:∫�ij(u0)�ij(’) d=

∫bi’i d +

∫�t

�ti’i d�; ∀’i ∈U∗ad (40)

where Uad is the admissible space de�ned by

Uad ={ui ∈

[H 1()

]3=ui= �ui on �u

}(41)

H 1() is the Hilbert space and U∗ad is the vector space de�ned as

U∗ad =

{ui ∈ [H 1()]3=ui=0 on �u

}(42)

The system of non-linear equations arising from the �nite element discretization of (40) yields

rA=fextA − fintA =0 (43)

where rA is the residual vector and

fintA (ui)=∫�kh(ui)BkhA d; fextA =

∫bjN

jA d +∫�t

�tjNjA d� (44)

are the components of the internal and external force vectors, respectively; NjA are the shape

functions de�ned on ; BkhA are the corresponding spatial derivatives (BjiA=N

( jA; i)).Let dA (ui=N

iA dA) be the nodal components of the displacement �eld. The discrete equation(43) is solved using the Newton method such that

(k+1)dA= (k)dA +�dA (45)



where the displacement increment, �dA, for each iteration (k) is obtained from

(k)KAB�dA = (k)rA (46)

KAB =@rA@dB

=∫BmnALmnkhBkhB d (47)

The instantaneous characteristic function <rsi is a solution of the following micro-scale weak form

Find <rsi ∈ [

H 1(�)]3such that

∫�<rs

i; jLijkh’k; h d�=−∫�Lijrs’i; j d�; ∀’i ∈

[H 1(�)

]3(48)

Let 6rsA (<rsi =N�

iA6rsA ) be the nodal components of the characteristic functions, then the corre-sponding discrete form of (48) yields

K�AB 6rsB =(∫

�B�mnALmnkh(y)B�khB d�

)6rsB =−

∫�Lkhrs(y)B�khA d� (49)

where K�AB is the RVE sti�ness matrix; Lmnkh(y) the instantaneous constitutive tensor of the micro-phases; N�

iA the shape functions on � and B�khA the corresponding symmetric spatial derivatives.The above two-scale non-linear problem is solved incrementally (see References [22; 23]). The

stress update procedure is outlined below.

3.4. Stress update procedure

The constitutive relation for the elasto-plastic microstructure at a typical Gauss point y� ∈� is

��=L��e�(u)=L�(��(u)− �p� (u)

)(50)

where L� is the elastic sti�ness matrix; �e�(u) is the elastic part of the microscopic strain rate (39).Following the associative ow rule, the plastic strain rate �p� (u) is given by

� p� (u)= ��(u)N�; N�=@��(�)@��

(51)

where �(u) is a plastic ow parameter; ��(�) is VonMises yield function at a Gauss point, given by

��(�)= 12�T� P�� − 1

3 �2� =0 (52)

For Von Mises plasticity, P is de�ned as

P= PTTP; P=13

2 −1 −1 0 0 0

−1 2 −1 0 0 0

−1 −1 2 0 0 0

0 0 0 3 0 0

0 0 0 0 3 0

0 0 0 0 0 3

(53)

in which T is 6× 6 diagonal matrix with 1 in the �rst three diagonal locations and 2 in theremaining three diagonal entries; P is a projection operator satisfying P= PP and P= PP=PP,



which transforms a vector to deviatoric space; �� is the yield stress which evolves according tothe hardening law as

˙��= 23 ��(u)H��; (k+1)��= (k)�� + 2

3��(u)H (k+1)�� (54)

where the backward Euler integration scheme is exercised; H is a hardening parameter de�ned asthe ratio between e�ective stress rate and e�ective plastic strain rate.Inserting (39) and (51) into (50) yields

�� = L�

(=� ˙��(u)− �(u)P��

)(55)

and applying the backward Euler scheme gives(k+1)�� = (I +��(u)L�P)−1�trial� (56)

where �trial� is the trial stress given by

�trial� = (k)�� + L�=��(u) (57)

Note that the instantaneous strain concentration factor =� is obtained from the solution of (48)which in turn depends on the instantaneous properties at each Gauss point. For simplicity ofnumerical implementation we assume

(k+1)=� = (k)=� (58)

Substituting (56) into the yield function (52) yields the non-linear equation with unknown plasticparameter increment ��(u) at each Gauss point which can be solved using Newton method

(k+1)��(u) = (k)��(u)−(

@��

@��

)−1�|(k)�� (59)

where

@��

@��=−�T� P((L�P)−1 + ��(u)H)−1�� − 4

9H�2� (60)

Once the incremental plastic parameter ��(u) is obtained, the local stress �eld follows from theintegration of (56).

4. MULTISCALE ANALYTICAL SENSITIVITY ANALYSIS

The multiscale analytical sensitivity analysis is based on the the direct di�erentiation of the directproblem (3)–(7) and subsequent multiple scale asymptotic analysis. In the present manuscript,only two-scale sensitivity analysis is considered, even though the methodology can be gener-alized to multiple scales. Attention is restricted to the sensitivities with respect to materialconstants.

4.1. Macro-scale sensitivity problem

We assume that traction and displacement boundary conditions as well as the body force areindependent of the design variables. The direct di�erentiation of the strong form (3)–(7) with



respect to the design variables P(p�) yields

�∈ij; xj = 0 in (x; y) (61)

�∈ij = Lijkh�kh(w∈) + L′

ijkh�kh(u∈) in (x; y) (62)

�ij(w∈) = w∈(i; xj) in (x; y) (63)

w∈i = 0 in �u (64)

�∈ijnj = 0 in �t (65)

where

L′ijkh =

@Lijkh

@P(66)

is the sensitivity of the constitutive tensor and

w∈i =

@u∈i

@P(67)

�∈ij =

@�∈ij

@P(68)

are the sensitivities of the displacement and the stress �elds, respectively; w∈i and �

∈ij denote the

corresponding time derivatives.The double-scale asymptotic analysis of the sensitivity problem (61)–(65) is carried out by

approximating the solution sensitivity w∈i (x) as follows:

w∈i (x) = wi(x; y) = w0i (x; y)+ ∈ w1i (x; y) + · · · (69)

Applying the chain rule to (69) yields the following expansion of the strain rate sensitivity

�ij(w∈(x)) ≡ �ij(w(x; y))≈ ∈ −1�−1ij (w(x; y)) + �0ij(w(x; y))+ ∈ �1ij(w(x; y)) + · · · (70)

where

�−1ij (w(x; y)) = w0(i;yj)(x; y) (71)

�1ij(w(x; y)) = w0(i; xj)(x; y) + w1(i;yj)(x; y) (72)

�1ij(w(x; y)) = w1(i; xj)(x; y) + w2(i;yj)(x; y) (73)

The approximation of the stress rate sensitivity �eld is obtained by inserting (70) into the consti-tutive equation (62), which gives

�∈ij(x) ≡ �ij(x; y)≈ ∈−1 �−1

ij (x; y) + �0ij(x; y)+ ∈ �1ij(x; y) + · · · (74)

where

�sij = Lijkh(y)�skh(w(x; y)) + L′

ijkh(y)�skh(u(x; y)) s= − 1; 0; 1; : : : (75)



The analogous expression for the stress sensitivity is

�∈ij(x) ≡ �ij(x; y)≈ ∈−1 �−1

ij (x; y) + �0ij(x; y)+ ∈ �1ij(x; y) + · · · (76)

Inserting the asymptotic expansion (76) into (61) and using the chain rule yields various ordersof sensitivity equations

O(∈−2) : �−1ij;yj = 0 (77)

O(∈−1) : �−1ij; xj +�

0ij;yj = 0 (78)

O(∈0) : �0ij; xj +�1ij;yj = 0 (79)

O(∈s) : �sij; xj +�

s+1ij;yj = 0; s = 1; 2; : : : (80)

To solve for the O(∈−2) sensitivity equation we pre-multiply the time derivative of (77) by w0i ,integrate it over the � and then perform integration by parts which yields∫

��

w0i �−1ij nj d�� −

∫�w0(i;yj)L

′ijkhu

0(h;yh) d�−

∫�w0(i;yj)Lijkh w0(h;yh) d� = 0 (81)

where the �rst and the second term vanish due to the periodicity of the boundary conditions on�� and (22), respectively. Assuming that L′

ijkh is positive de�nite yields the unique solution

w0(i;yj) = 0 ⇒ w0i = w0i (x) (82)

The strong form of the O(∈−1) equation is obtained from (75) and the rate form of (78) whichyields (

Lijkh(w0(k; xh) + w1(k;yh)

));yj+(L′ijkh

(u0(k; xh) + u1(k;yh)

));yj= 0 (83)

Using (24), the above equation can be re-written as follows:(Lijkh

(w0(k; xh) + w1(k;yh)

));yj+(L′ijkh

(Ikhmn +kh

mn

));yj

u0(m; xn) = 0 (84)

The solution of (84) for w1i (x; y) up to a constant is constructed in the form of

w1i (x; y) = �mni (y)u

0(m; xn)(x) + <mn

i (y) w0(m; xn)(x) (85)

in which �mni (y) is the y-periodic sensitivity of the characteristic function <mn

i (y)

�mni (y) =

@<mni (y)@P

(86)

Inserting (85) into (84) yields(Lijkh

(Ikhmn +kh

mn

));yj

w0(m; xn) +(Lijkh˝kh

mn + L′ijkh

(Ikhmn +kh

mn

));yj

u0(m; xn) = 0 (87)

where ˝mnij (y) is the y-periodic sensitivity de�ned as

˝khij (y) = �kh

(i;yj)(y) =@kh

ij (y)

@P(88)



For arbitrary values of ˙�ij(w) = w0(i; xj) and˙�ij(u) = u0(i; xj), Equation (87) yields the following two

governing equations on the RVE domain(Lijkh(y)

(Ikhmn +kh

mn(y)))

;yj= 0 (89)

which is a direct RVE problem, and(Lijkh(y)˝kh

mn(y) + L′ijkh(y)

(Ikhmn +kh

mn(y)))

;yj= 0 (90)

which is the sensitivity RVE problem.To solve for the O(∈0) sensitivity equation we integrate the rate form of (73) over the RVE

domain which gives

∫�

(Lijkh(y)w1(k; xh)

);yjd� +

∫�

(L′ijkh(y)u

1(k; xh)

);yjd� +

∫�Lijkh(y)(w1(k;yh) + w0(k; xh)); xj d�

+∫�L′ijkh(y)

(u1(k;yh) + u0(k; xh)

); xjd� = 0 (91)

The �rst two terms vanish due to the y-periodicity. Consequently, inserting (24) and (85) into(91) yields (∫

�

{Lijkh(y)˝mn

kh +L′ijkh(y)(Ikhmn+mn

kh )}d�u0(m; xn)

); xj

+(∫

�{Lijkh(y) (Ikhmn+mn

kh )} d�w0(m; xn)

); xj

=0

(92)

Let us denote the sensitivity of the macro-stress rate �eld as

˙�ij =1|�|

∫|�|�ij d� = Lijkh ˙�kh(w) + L′

ijkh˙�kh(u) (93)

where L′ijkh is the sensitivity of the instantaneous macroscopic constitutive tensor given by

L′ijkh =

1|�|

∫�

{Lijmn(y)˝mn

kh + L′ijmn(y) (Ikhmn +mn

kh )}d� (94)

From (92) one can derive the following macroscopic sensitivity equation

�ij; xj = 0 (95)

In summary, the macro-scale sensitivity problem (61)–(65) has the following concise form:

�ij; xj = 0 in

˙�ij = Lijkh(<) ˙�kh(w) + L′ijkh(<; �) ˙�kh(u) in

˙�ij(w) = w0(i; xj) in

wi = 0 on �u

�ijnj = 0 on �t

(96)



4.2. Micro-scale sensitivity problem

Since the displacement ui and the characteristic functions <khi are known prior to the sensitivity

analysis (from (35) and (37)), the solution of the boundary value problem (96) is function of �khi ,

which can be calculated from the following micro-scale sensitivity problem:

�mni is y-periodic

qkhij; yj = 0 in �

qkhij = Lijmn(y)�kh

(m;yn) + L′ijmn(y)(�km�hn + <kh

(m;yn)) in �[|�mn

i |] = 0 on �[|qkhij nj|

]= 0 on �

qkhij nj is y-antiperiodic

(97)

where qkhij is the sensitivity of the stress concentration factors skhij

qkhij =

@skhij@P

(98)

The sensitivity �ij(w) of the local strain rate �eld is given by

�ij(w)==ijmn(y) ˙�mn(w) +˝ij

mn(y) ˙�mn(u) (99)

The stress rate sensitivity in � is given as

�ij =Lijrs=rsmn(y) ˙�mn(w) +

[L′ijrs=rs

mn(y) + Lijrs˝rsmn(y)

]˙�mn(u) (100)

4.3. The weak and discrete forms of the sensitivity problem

The weak formulation of the macro-scale sensitivity problem (96) is given as:

Find w0i ∈[H 1()

]3such that

∫�ij(w)�ij(’) d=0; ∀’i ∈ [H 1()]3 (101)

The discretization of (101) yields the following system of non-linear equations

RA=FextA − F intA =0 (102)

where RA is the residual vector, and

F intA =∫�khBkhA d; FextA =0 (103)

The nodal components of the displacement sensitivity W 0A (w0i =N

iA W 0A ) are obtained using Newton

method

(k+1)W 0A =

(k)W 0A +�W 0

A (104)

(k)KAB�W 0B =

(k)RA − (k)K ′AB�d0B (105)



where KAB is the macro-sti�ness matrix for the direct problem (47) and K ′AB the coupling term

given by

K ′AB=

∫BmnAL

′mnkh BkhB d (106)

Note that L′ijkh is a function of � rs

i , which is the solution of the following weak formulation onthe micro-scale:

Find �rsi ∈ [

H 1(�)]3such that∫

�� rsi; j Lijkh’k; h d�=−

∫�<rs

i; j L′ijkh’k; h d�−

∫�L′ijrs ’i; j d�; ∀’i ∈

[H 1(�)

]3(107)

Further discretizing �rsi =N�

iA ℵmnA , the discrete form of (107) yields

(∫�

B�khA Lkhrs(y) B�khB d�)

ℵrsB =−

(∫�B�khA L′

khrs(y) B�rsB d�

)6rsB −

∫�L′khrs(y) B

�khA d�

(108)

where ℵmnA are the nodal components of the characteristic functions on �; Note that the same

sti�ness matrix is used to solve for the micro-scale direct and sensitivity problems. Procedures forintegrating the stress sensitivities are outlined in Section 4.4.

4.4. Stress sensitivity update procedure

We start from the rate of stress sensitivity given by

��=L�(��(w)− �p�(w)

)+ L′

�

(��(u)− �p�(u)

)(109)

where L′� is the sensitivity of elastic sti�ness matrix, which is equal to zero if the sensitivity is

not with respect to one of the elastic constants.The sensitivity of the plastic strain rate, �p�(w), is given as follows:

�p�(w)= ��(w)N� + ��(u)N ′� ; N ′

� =@ �(�)@��

(110)

where ��(w) is the sensitivity of the plastic ow parameter. �(�) is the sensitivity of theVon Mises yield function at a Gauss point given as

�(�)= �T� P�� − 23 ��=0 (111)

�� is the sensitivity of the yield stress ��:

˙��= 23(��(w)H�� + ��(u)H ′�� + ��(u)H ��) (112)

which after the backward Euler integration gives

(k+1)��=(I +��(u)H)−1[(k)�� + 2

3

(��(w)H (k+1)�� +��(w)H ′ (k+1)��

)](113)



Figure 2. Unidirectional periodic composite: (a) macro domain ; and (b) RVE domain �.

where H ′ is equal to 1 if the sensitivity parameter is the hardening parameter and zero otherwise.Substituting Equations (39), (51), (99) and (110) into (109) yields

��=L�(=� ˙��(w) + � ˙��(u)− ��(w)P�� − ��(u)P��) + L′�(=� ˙��(u)− �(u)P��) (114)

Assuming that (k+1) = (k ) and applying the backward Euler scheme to (114) yields(k+1)��=(I +��( u)L�P)−1�trial� (115)

where �trial� is the trial stress sensitivity given at increment (k + 1) by

�trial� = (k)�� + L�(k)=��(w) + (L′

�(k)=� + L�

(k)�)��(u)− (k+1)��P(L′

��(u) + L��(w))(116)

The sensitivity ��(w) of the plastic parameter increment is obtained from the Newton method:

(k+1)��(w)= (k)��(w)−(

@ �

@��(w)

)−1 |(k)��(w) (117)

where

@ �

@��(w)=−(k+1)�T� P((L�P)−1 + ��(u) P)−1(k+1)�� − (I +��(u)H)−1

49H (k+1)�2� (118)

5. NUMERICAL EXAMPLES

In this section we investigate the accuracy and computational e�ciency of the proposed multi-scale analytical sensitivity analysis. For simplicity, we consider a composite specimen subjectedto uniform tension load. The state of uniform macro-stress is modelled using a single eight-nodehexahedral element as shown in Figure 2(a).



Table I(A). Sensitivity of macro-constitutive tensorcomponents with respect to E1.

MASA CFD

LE11111 (10

−1) 6.091605 6.091500

LE13333 1.038295 1.038350

LE11212 (10

−1) 3.602927 3.602950

LE12323 (10

−1) 2.083333 2.083350

Table I(B). Sensitivity of the displacement �eldwith respect to E1.

MASA CFD

wE11 (10−9) 4.753287 4.755000

wE13 (10−8) −4.777277 −4.77600

Table II(A). Sensitivity of macro-constitutive tensorcomponents with respect to E2.

MASA CFD

LE21111 (10

−1) 5.537794 5.538000

LE23333 (10

−1) 2.403461 2.403500

LE21212 (10

−2) 9.382623 9.383000

LE22323 (10

−1) 2.000000 2.000000

Table II(B). Sensitivity of the displacement �eldwith respect to E2.

MASA CFD

wE21 (10−9) 5.070172 5.065000

wE23 (10−8) −1.337427 −1.338000

Two RVE models have been considered. The �rst, with isotropic linear elastic micro-constituents.The multiscale analytical sensitivity analysis (MASA) is performed with respect to the Young’smodulus, E, and Poisson ratio, �. In the second numerical example, we consider an elastic rein-forcement embedded in the elasto-plastic matrix. For discretization details see References [22; 23].The sensitivity analysis is performed with respect to the hardening parameter, H . In both cases,the sensitivity results obtained by the MASA approach are compared to those obtained using CFDmethod.



Table III(A). Sensitivity of macro-constitutive ten-sor components with respect to �1.

MASA CFD

L�11111 (10

+4) 1.019670 1.020000

L�13333 (10

+4) 1.297869 1.295000

L�11212 (10

+3) −3.002439 −3.00000L�12323 (10

+3) −1.736111 −1.73500

Table III(B). Sensitivity of the displacement �eldwith respect to �1.

MASA CFD

w�11 (10−3) −3.395203 −3.400000

w�13 (10−3) −1.317338 −1.300000

Table IV(A). Sensitivity of macro-constitutive ten-sor components with respect to �2.

MASA CFD

L�21111(10

+4) 1.432844 1.43500

L�23333(10

+3) 8.972921 8.95000

L�21212(10

+3) −1:501220 −1:50000L�22323(10

+3) −3:20000 −3:20000

Table IV(B). Sensitivity of the displacement �eldwith respect to �2.

MASA CFD

w�21 (10

−3) −3:186963 −3:185000w�23 (10

−4) 6.179274 6.200000

5.1. Linear elastic composite

The properties of the elastic composite considered are as follows:

Matrix (1): Young’s modulus E1 = 105 Mpa, Poisson Ratio �1 = 0:25.

Fibre (2): Young’s modulus E2 = 2:105 Mpa, Poisson Ratio �2 = 0:20.

In the following we denote Lp� = @L=@p� and wp� = @u=@p�. Tables I(A), II(A), III(A), IV(A),give the sensitivity of the macro-constitutive tensor components, L1111; L3333; L1212 and L2323,with respect to the elastic parameters E1; �1; E2; �2, respectively. The sensitivities obtained by the



Figure 3. Evolution of x-component of the displace-ment sensitivity versus load parameter for H=E= 1

5 .Figure 4. Evolution of z-component of the displace-ment sensitivity versus load parameter for H=E= 1

5 .

MASA approach are compared to those computed using CFD method for �pa=1 × 10−2×pa.The x-component w1 (w2 =w1) and the z-component w3 of the displacement sensitivity at node ©8are shown in Tables I(B), II(B), III(B), IV(B), respectively.In terms of accuracy, the sensitivities obtained by the MASA approach agree well with those

computed using CFD approach (0 to 0.4 per cent di�erence). In terms of computational e�ciency,CFD approximation requires two to three factorizations of the elastic macro-sti�ness K and micro-sti�ness K� for each design parameter totaling 16–24 factorizations. Using MASA approach, onthe other hand, the same sensitivities can be obtained by performing two factorizations only.

5.2. Non-linear elastoplasticity problems

The properties of micro-phases considered are as follows:

Matrix: Young’s modulus E1 = 103 Gpa; Poisson Ratio �1 = 0:3; yield stress �m=24Mpa;isotropic hardening modulus H =2× 102 Gpa:Fibre: Young’s modulus E2 = 8× 103 Gpa; Poisson Ratio �2 = 0:2:

The sensitivity analysis is performed with respect to plastic hardening parameter H . Figures3 and 4 show the evolution of displacement sensitivities, w1 and w3, versus the load parameter,respectively. Results of MASA approach are compared to those of CFD approach for two di�erentvalues of design parameter step size, �H =10−2H and �H =10−5H .It can be seen that the sensitivity results obtained by the CFD approximation with the step size,

�H =2, is in good agreement with the MASA approach (0–1 per cent relative error). On the otherhand, for the value of step size, �H =0:002, the CFD gives inaccurate resolution of sensitivities. InFigures 4 and 5 we considered the hardening parameter to Young’s modulus ratio of H=E= 1

5 . Wefurther study the sensitivities for the hardening parameter to Young’s modulus ratio of H=E= 1

50 .The results presented in Figures 5 and 6 show that the optimal value of the step size selected inthe previous case results in oscillatory response of the sensitivities. This suggests that the optimalvalue of the step size in CFD approximation is problem dependent.

6. SUMMARY AND FUTURE RESEARCH DIRECTIONS

Multiscale analytical sensitivity analysis approach has been developed for inelastic periodiccomposites. Comparison of the multiple scale sensitivity analysis to the central �nite di�erence



Figure 5. Evolution of the x-component of the dis-placement sensitivity (H =2×101 Gpa) versus load

parameter.

Figure 6. Evolution of the z-component of the dis-placement sensitivity (H =2×101 Gpa) versus load

parameter.

approximation revealed its advantage in terms of accuracy and computation e�ciency. The CFDapproximation is highly sensitive to the choice of the step size, whose optimal value is problemdependent. When the step size is increased the CFD sensitivity error is dominated by the trunca-tion error which depends not only on the step size value but also on the problem parameters. Atthe other extreme, when the step size is signi�cantly reduced, the CFD sensitivity error might bedominated by the rounding error [6].Despite its signi�cant advantage over the CFD approach, the proposed MASA approach still

requires signi�cant computational resources for large-scale macro-problems with detailed repre-sentative volume elements. For non-linear history-dependent problems the RVE problem has tobe solved at every increment and for each macroscopic Gauss point. To alleviate this di�culty,in the future work we consider multiple scale sensitivity analysis in the context of mathematicalhomogenization theory with eigenstrains [22; 23]. By this approach, the history data are updatedonly at two or three points within a micro-structure, i.e. one for each phase.

ACKNOWLEDGEMENTS

The support of the O�ce of Naval Research through grant number N00014-97-1-0687 is gratefully acknowl-edged.

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