Development of a Nonperiodic Homogenization Method 

for One-Dimensional Continua 

by Michael J. Leamy, Peter W. Chung, and Raju Namburu 

ARL-TR-3197 May 2004 

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Army Research Laboratory 

Aberdeen Proving Ground, MD 21005-5067 

ARL-TR-3197 May 2004 

Development of a Nonperiodic Homogenization Method 

for One-Dimensional Continua 

Michael J. Leamy 

U.S. Military Academy 

Peter W. Chung and Raju Namburu 

Computational and Information Sciences Directorate, ARL 

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Development of a Nonperiodic Homogenization Method for One-Dimensional 

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Michael J. Leamy,* Peter W. Chung, and Raju Namburu 

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U.S. Military Academy 

Department of Mechanical Engineering 

West Point, NY 10996 

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*Assistant Professor, U.S. Military Academy, Department of Mechanical Engineering, West Point, NY 10996. 

ARL/USMA Faculty Summer Program. 

14. ABSTRACT 

This work describes a one-dimensional homogenization method that relaxes the usual assumption of periodicity of microscale 

displacements. As such, this method is appropriate for studying materials with periodic as well as nonperiodic microstructures. 

Expected applications for the new method include composite rods with manufacturing variability (where material periodicity is 

destroyed) and rods composed of functionally graded materials. The method is validated through direct comparison with a 

known exact solution, that being a rod with linearly varying axial rigidity. The method for this case is shown to be exact. 

Further application of the developed method to two different periodic materials is used to validate the usual periodic assumption 

employed in traditional homogenization methods. 

15. SUBJECT TERMS

homogenization, composite materials, multiscale 

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Peter W. Chung 

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iii

Contents 

1. Introduction 1

2. On a New Homogenization Procedure 3

2.1 Formulation .....................................................................................................................3 

2.2 Extension of Procedure to Include Higher-Order Terms in u ( x ) ε ε ...............................6 

2.3 Application to an Example Rod—Linearly Varying Axial Rigidity EA.........................8 

2.4 Application to an Example Rod—Periodically Varying Axial Rigidity EA...................8 

3. Conclusions 10

4. References 11

Distribution List 13 
iv

INTENTIONALLY LEFT BLANK. 
1

1. Introduction 

The traditional asymptotic homogenization method has found wide acceptance in the composites 

modeling community (Bendsoe and Kikuchi, 1988; Fish et al., 1999; Francfort and Murat, 1986; 

Ghosh et al., 1996; Hollister et al., 1994; Lene, 1986; Moulinec and Suquet, 1998; SanchezPalencia, 1980; Shkoller and Hegemier, 1995), for its ability to incorporate and estimate 

effective material properties from a subscale representative unit cell (Babuska, 1976). Central to 

the method is the principal of convergence, which has been studied in the mathematical literature 

for H-, G-, Γ-, and two-scale convergence. In simple terms, H-convergence, for example, states 

that over an open bounded domain Ω in RN, a sequence of bounded and finite matrices ε A is said 

to H-converge to a sequence of bounded and finite matrices A if, for all ∈ ( ) Ω −1 f H , the 

solution ε

u of ( ) , (Ω) 1u H0 − div A ∇u = f ∈ ε ε ε converges, as the scaling parameter ε goes to zero, 

in weak- ( ) Ω1 H0 to the solution u of ( ) , (Ω) 1

u H0 − div A∇u = f ∈ and if the sequence ε ε A ∇u

converges in weak- ( )N L Ω2 to A∇u . In general, the H-limit of ε A , which is A, is not explicit 

except under certain assumptions. The works of Bensoussan et al. (1978) and Sanchez-Palencia 

(1980) are commonly cited to show that the explicit H-limit is known for periodic systems. 

Loosely speaking, for the convergence of ε A in the appropriate space of linear operators, or the 

convergence of the Green’s operator, the literature refers to G-convergence (Spagnolo, 1976). 

The convergence of ( ) ε ε ε ε A ∇u ,∇u , u is referred to as Γ-convergence (Tartar, 1978), and a more 

specific case of H-convergence, which takes into account the periodicity of the solution, is 

referred to as two-scale convergence (Allaire, 1992). 

The engineering and computational mechanics literature have made such extensive use of these 

fundamental mathematical developments that it is difficult to provide a truly exhaustive 

bibliography. We note among many, however, the work on structural optimization (Bendsoe and 

Kikuchi, 1988), biomechanics (Hollister et al., 1994), and multiphysics solid mechanics (Fish et 

al., 1999; Fish and Yu, 2001; Ghosh et al., 1996, 2001; Lene, 1986; Moulinec and Suquet, 1998; 

Shkoller and Hegemier, 1995; Terada et al., 2000; Terada and Kikuchi, 2001). Until now, 

however, the engineering developments in homogenization have mainly considered specialized 

cases of two-scale convergence where the periodicity is built into the computational procedure. 

The application of the so-called y-periodicity generally precludes the ability to consider 

nonperiodic systems. In this regard, developments in the engineering and mathematical literature 

have been on divergent paths. 

We have found that nonperiodicity can occur under two scenarios. The first is when the finite 

boundary has a direct influence on the solution, or the reduction-of-dimension situation, which 

can occur typically in laminated plates or stratified rods. The second is when the material 

exhibits finite microstructure but in a nonperiodic sense. This can commonly be encountered in 

the form of localized behavior, such as that found in damage mechanics or fracture, or in 
2

smoothly varying microscopic material dependence on macroscopic coordinates, such as that 

found in functionally graded composites. 

Limited works by the engineering community have attempted to modify the homogenization 

method for systems involving finitely thick plates (Rostam-Abadi et al., 2000). Still others have 

considered the presence of strong anharmonic local fields that disrupt the periodicity and require 

increased resolution solutions (Oden and Vemaganti, 1999; Ghosh et al., 2001; Oden et al., 

2001). In these works, the traditional periodic homogenization method has been modified at a 

schematic level using the laminated plate theory (Rostam-Abadi et al., 2000) or h and p finite 

element adaptivity techniques (Ghosh et al., 2001; Oden et al., 2001) so that directions in 

Cartesian space or regions within a nonperiodically deforming body are specifically accounted 

for by small changes in the periodic homogenization procedure. On the other hand, 

developments in the mathematical literature have been tied to special cases where material 

dependence on spatial coordinates is strictly demonstrable (Briane, 1994; Shkoller, 1997; Fabre 

and Mossino, 1998). Namely, they consider material types where key heterogeneous features are 

within a diffeomorphic mapping and linear mapping away from a periodic arrangement. 

Briane (1994) developed a nonperiodic homogenization method in which a highly oscillatory but 

nonperiodic matrix ε A is compared to a periodic matrix ε B as the microscale feature size goes 

(slower) to zero as ε goes to zero. The limit of ε B is a function of every point in the material. 

Recognizing the computationally infeasible nature of this approach, Shkoller (1997) showed that 

the problem could be simplified to a unit cell approach, where the microscale feature size is kept 

finite and the error in the approximation is of the order of the unit cell size. In this sense, the 

asymptotic nature of homogenization is not being fully assumed. Fabre and Mossino (1998) 

completed a similar study, looking at H-convergent multiplicable matrices that may be used to 

perform a linear mapping that takes a periodic material into a nonperiodic one thereby allowing 

periodic homogenization of a mapped nonperiodic material. Gustafsson and Mossino (2003) 

developed a nonperiodic homogenization method for diffusion equations based on the Hconvergent multiplicable matrices of Fabre and Mossino (1998) to write explicit expressions of 

homogenization for a plate and less explicit expressions for a thin cylinder. 

Developments in nonperiodic homogenization stem from the general H-convergence properties 

of the homogenization theory combined with mapping principles and dimension reduction. In its 

most general and primitive form, homogenization makes no presumption of periodicity, which 

indicates that its application to nonperiodic systems can be made forthwith. Bensoussan et al. 

(1978) have shown a one-dimensional result for homogenization that is implicit in its absence of 

periodicity constraints on the microstructure. 

In this report, we develop a one-dimensional nonperiodic computational homogenization method 

with explicit equations and show that it recovers the periodic case. This is done by relaxing the 

periodicity assumption of the microscale displacement field, integrating analytically, and 

applying a boundedness argument to the result. The procedure is entirely different from that of 
3

Bensoussan et al. (1978) yet is shown to yield the same result. This enables the development of 

the new computational homogenization procedure for the one-dimensional continua (i.e., for the 

elastic rod). Generalization to the three-dimensional continua is expected to follow in a later 

paper. When compared to the engineering literature, the procedure described differs in two 

ways: (1) the approximation space is introduced early in the procedure, and (2) no periodicity 

condition is imposed on the microscale displacement field. This second difference is expected to 

make the new procedure relevant to a wider class of problems than the preceding 

homogenization procedures. Using the example of an end-displaced rod with linearly varying 

axial rigidity EA, a comparison is made between the displacement field predicted by the new 

homogenization procedure and the displacement field of the exact solution. A second example 

of a rod with periodically varying axial rigidity is used to discuss the appropriateness of the 

microscale periodicity condition imposed by preceding homogenization procedures. 

2. On a New Homogenization Procedure 

2.1 Formulation 

Consider a straight rod characterized by an axial rigidity EA, whose deformation is described by 

a displacement field ) (

ε ε u x governed by the equilibrium equation 

f

x

u EA

x = − 















∂

∂

∂

∂ ε

ε

ε , (1) 

where for the simplicity of discussion, the external forcing f will be considered to be absent. 

Consider further the rod to be fixed rigidly at its initiating end with a prescribed displacement at 

its terminating end: 

 , ( = 0) = 0 ε ε u x (2) 

and 

ul u (x = l) = ε ε . (3) 

This choice of end conditions is motivated by an eventual finite element implementation of the 

described procedure, in which the end conditions take the form of equations 2 and 3, and a rigid 

body translation. The axial rigidity EA of the described rod is considered to be nonuniform. It 

can be decomposed into a constant macroscale axial rigidity and a nonconstant microscale axial 

rigidity 

 ) ( 0 1 EA = EA + EA y , (4) 
4

where dependence on the macroscale implies ε x = x dependence and dependence on the 

microscale implies ε ε y = x dependence, ε being a small parameter. The equilibrium equation 

can be updated to reflect dependence on the two scales by using the derivative expression 

x x ∂y

∂

+

∂

∂ = ∂∂

ε ε

1 . (5) 

As in previous homogenization procedures, the displacement field is expanded using ε: 

uε (xε ) = u

0

(x, y) + εu

1

(x, y) +", (6) 

where ) ( , 0

u x y and ) ( , 1u x y denote macro- and microscale displacement fields, respectively. 

Interchangeably, ) ( , 0

u x y will be termed the O(ε

0

) displacement, where O( ) denotes the order 

operator, and ) ( , 1

u x y will be termed the O(ε

1

) displacement. Introducing equations 5 and 6 into 

equation 1 while neglecting quantities of O(ε

2

) in equation 6, followed by a separation of orders 

of ε, yields the following governing equations: 

for ε

–2, 

0

0

=















∂

∂

∂

∂

y

u EA

y

; (7) 

for ε

–1, 

2 0 2

0 1 2 0 2 1

= ∂

∂

+

∂ ∂

∂

+















∂

∂

+

∂

∂

∂

∂

y

u EA

x y

u EA

y

u

x

u

y

EA ; (8) 

for ε

0

, 

2 0

2 1

2

1 2 0

= ∂ ∂

∂

+

∂

∂

+

∂

∂

∂

∂

x y

u EA

x

u EA

x

u

y

EA ; (9) 

and for ε

1

, 

0 2

2 1

= ∂

∂

x

u EA . (10) 

Without loss of generality, the boundary inhomogeneity ul is not expanded and the boundary 

conditions can be restated as 

 , (0,0) 0 0

u = (11) 

 0 0 0 1

u ( , ) = , (12) 

( , ) , 0 0ul u l l ε = (13) 

and 
5

( , ) 0 1u l l ε = . (14) 

Integrating equation 7 twice with respect to y yields 

 ) ( , ) ( ) ( ) ( 1 2

0

u x y = C x α y + C x , (15) 

where ∫ = dy EA y

1

α( ) . Assuming ( ) 1 C x is not proportional to ε, so as to preserve the relative 

ordering of all terms in (equation 6) for all y, the presence of α( y) in equation 15 results in 

( , ) 0

u x y being unbounded as y → ∞ , or equivalently then, as ε → 0 . For example, retaining 

only the macroscale term in the axial rigidity for ease of explanation, at the terminal end 

α( y = l ε ) evaluates to 0

1

EA

l

ε

, which is unbounded as ε → 0 . We require that the solution 

procedure be bounded, or appealing to a physical argument, be applicable to the case of a 

homogenous rod where ε is arbitrarily small, in which case, the need for a bounded ) ( , 0

u x y

requires that ( ) 1 C x be zero. 

To further facilitate a solution, an approximating space is introduced for ( ) 2 C x . Specifically, we 

search for an approximation to ) ( , 0

u x y such that ( ) 2 C x ( ) 1 ∈ H x (i.e., the Sobolev space of 

degree 1) (Hughes, 1987). For simplicity herein, a single element that spans the rod’s length l 

will be considered, but, more generally, the use of the 1 H Sobolev approximation space admits 

usage of multiple elements while insuring displacement continuity across the element 

boundaries. In accordance with this discussion, a linear polynomial approximation for ( ) 2 C x is 

introduced: 

( ) , 2 2 2 C x = a x + b (16) 

where 2 a and 2 b are unknown constants. Applying the boundary conditions (equations 11 and 

13) yields 

x

l

u

u x y u x l

0

0 0 ( , ) = ( ) = . (17) 

With the determination of the O(ε

0

) displacement completed, the analysis of the O(ε

1

) 

displacement follows. Substituting equation 17 into equations 8, 9, and 10 yields 

0 2

0 1 2 1

= ∂

∂

+















∂

∂

+

∂

∂

y

u EA

y

u

l

u

y

EA l , (18) 

2 0

1 2 1

= ∂ ∂

∂

+

∂

∂

∂

∂

x y

u EA

x

u

y

EA , (19) 

and 

0 2

2 1

= ∂

∂

x

u EA . (20) 
6

Equation 18 can be rewritten as 





 





∂

∂ = − 















∂

∂

∂

∂ 0

1

ul l

EA

y y

u EA

y

, (21) 

which can be integrated once and rearranged to yield 

( ) 3

0

1

u C x

l

EA

y

u EA + l = ∂∂ . (22) 

Note that equation 22 states that a function of x and y (say g(x,y) = 0

1

ul l

EA

y

u EA +

∂

∂ ) is equal to a 

function of x only (say f(x) = ) (3 C x ) for all x,y—this can imply that either g(x, y) is a function of 

x only or both g(x, y) and f(x) are constant. The former does not lead to an immediate solution,*

while the latter leads to the solution 

( ) , ( , ) ( ) ( ) 4

0

3

1

3 3 u C x

l

y C x C u x y C y = = α − l + , (23) 

where C3 is a pure constant to be determined by the boundary conditions. The substitution of 

equation 23 into equation 19 and the enforcement of the resulting equation for all y determine 

that 4 4 C (x) = C (i.e., that C4 is also a pure constant to be determined by the boundary 

conditions).†

 Note that the resulting solution for the microscale displacement 1

u is independent 

of x and satisfies equation 20 identically. Following application of equations 12 and 14, the 

microscale displacement is given by 

0

0

1 ( ) ( ) ( ) l

l u

l

y y l

u

u y = α − εα ε. (24) 

Due to the presence of the multiplier ε in equation 6, ) (

1 ε u y is bounded for all y asε → 0 . Note 

that the microscale displacement is not necessarily periodic in y even if the axial rigidity EA is. 

Traditional homogenization procedures summarily impose periodicity on the microscale 

displacement. As a final note, for the case of a homogenous rod (i.e., 0 ( ) 1 EA y = ), equation 24 

evaluates to zero, the macroscale displacement is given by equation 17, and the exact solution is 

recovered. 

2.2 Extension of Procedure to Include Higher-Order Terms in u ( x ) ε ε

The solutions presented in section 2.1 were found by truncating the asymptotic expansion 

(equation 6) up to O(ε

3

). If instead truncation is not performed, a new term involving ) ( , 2u x y

 *

Simplification of the equations does not occur for this choice, but the possibility of a second solution arising from this choice 

may exist. 

†

For the case of a nonzero external load f(x), C4 is replaced by ( ) C4 EA y

f x dx + ∂ ∂

∫ . 
7

appears in equations 9 and 19. Similarly, new terms involving ) ( , 2

u x y and ) ( , 3u x y appear in 

equations 10 and 20. Note that equation 10, and thus equation 20, were not used in developing 

the solution procedure but instead were checked for satisfaction. In the solution presented for 

this section, attention is therefore focused only on solving the new equation 19, namely 

2 0

1 2 1 2

=















∂

∂

∂

∂

+

∂ ∂

∂

+

∂

∂

∂

∂

y

u EA

x y y

u EA

x

u

y

EA , (25) 

with the still valid equations 22 and 23. In this manner, macro- and microscale displacements 

will again be developed without the need to first truncate the displacement asymptotic series 

given by equation 6. 

Substituting equation 23 into equation 25 and redistributing yields 

y

EA

y

u EA

y C

∂

∂

















∂

∂

∂

∂ −

′ =

2

4 , (26) 

where a prime denotes differentiation with respect to x. Note that equation 26 states that a function 

of x (say f(x) = C4

′) is equal to a second function of both x and y (say g(x,y) = y

EA

y

u EA

y

2

∂

∂

















∂

∂

∂

∂ − ). 

As in the previous section, a solution in which both f(x) and g(x,y) are the same constant is sought. 

This leads to the expression 4 5 C6 C = C x + and an updated expression for ) ( , 1

u x y , 

5 6

0

3

1

( , ) ( ) u C x C

l

y u x y C y = α − l + + , (27) 

with three undetermined true constants. Application of the first boundary condition 

(equation 12) yields C6 = 0, while the second boundary condition (equation 13) yields 

0 1

5

0

3  − + = 

 





u C l l C l

ε ε

α . (28) 

Separating equation 28 into O(ε

–1) and O(ε0

) terms yields two equations for the remaining 

constants: 

for ε

–1, 

0 1 0

3  − = 

 





ul

l C

ε ε

α ; (29) 

and for ε

0

, 

 0 C5l = , (30) 
8

which, when satisfied again, yields equation 24. 

2.3 Application to an Example Rod—Linearly Varying Axial Rigidity EA

The new homogenization procedure described in section 2.1 is applied to a rod of unit length 

with a linearly increasing axial rigidity given by 

EA = 3 + 0.01y , (31) 

and with an end displacement given by 0.1 0 ul = . The microscale y is chosen to be one onehundredth of the macroscale x such that ε = 0.01. This information, evaluated with the 

definition of α( y) , the truncated expansion (equation 6), and the ordered displacements 

(equations 17 and 24), yields a predicted solution for the displacement field 

 ) ( ) 0.382 0.348ln(3 ε ε ε u x = − + + x , (32) 

where all calculations have been carried out to three significant figures. For comparison, an 

exact solution exists for the linearly varying rod. Collecting the relevant information, the 

boundary value problem governing the rod’s exact displacement field is as follows: 

 = 0 

 





∂

∂

∂

∂

x

u EA

x , EA = 3 + 0.01y , u(0) = 0, u(1) = 0.1, (33) 

which has the solution 

ln(3) 2 ln(2)

ln(3 )

10

1

ln(3) 2 ln(2)

ln(3)

10

1 ( ) −

+ − − = x

u x . (34) 

Evaluating equation 34 to three significant figures yields the exact displacement 

 ) u(x) = −0.382 + 0.348ln(3 + x , (35) 

which matches the perturbation solution (equation 32). The same match occurs when the 

calculations are carried out to any number of significant figures, indicating that for the linearly 

varying rod, the newly described homogenization procedure recovers the exact solution. 

2.4 Application to an Example Rod—Periodically Varying Axial Rigidity EA

As a second example, the new homogenization procedure is applied to a rod of unit length with a 

periodically varying axial rigidity given by 

) 53 sin( y EA π = + , (36) 

and with an end displacement given by 1 0. 0 ul = . The microscale y is chosen to be one onehundredth of the macroscale x such that ε = 0.01 (note that this gives a y-period of 10 for the 

axial rigidity). This information, evaluated with the definition of α( y) , the truncated expansion 

(equation 6), and the ordered displacements (equations 17 and 24), yields a predicted solution for 
9

Figure 1. (a) The function α(y), (b) the microscale displacement field εu1(y), and (c) the predicted displacement 

field uε

(x

ε

) for the example rod with periodically varying axial rigidity (i) EA = 3 + sin(π y/5), 

(ii) EA = 3 + 2.9sin(π y/5), and (iii) EA = 3 + 2.9signum(sin(π y/5)). 

the displacement field. The top portion of figure 1 shows the results for the function ) α( y , the 

predicted microscale displacement field εu

1

(y), and the predicted displacement field uε(x

ε

) for 

this example. 

It is interesting to note from this example that the microscale displacement is indeed y-periodic, 

as imposed a priori in the traditional homogenization procedure. The effects of further 

increasing the variability of the axial rigidity are shown in the middle portion of the figure. 

Finally, the results for a rod with nonsinusoidal, but still periodic, axial rigidity are given in the 

bottom portion of the figure. This last example gives further evidence supporting the imposition 

of microscale displacement periodicity in the standard homogenization method (note that, 

indeed, the microscale displacement is periodic). 

(b)

(i) 

(iii) 

(ii) 

(a) (c) 
10

3. Conclusions 

Using the well-established homogenization theory, a modification has been proposed for 

engineering applications in which the microscale displacement field is nonperiodic. The result is 

a novel nonperiodic homogenization procedure for functionally graded and reduced dimension 

problems where y-periodicity breaks down because of material or dimensional considerations. A 

specialized one-dimensional formulation was presented, verified against an exact solution, and 

subsequently demonstrated. A fully three-dimensional formulation is currently underway and 

will be presented in a sequel paper. 
11

4. References 

Allaire, G. Homogenization and Two-Scale Convergence. SIAM J. Math. Anal. 1992, 23, 1482–

1518. 

Babuska, I. Homogenization and Its Applications. Mathematical and Computational Problems. 

In Numerical Solution of Partial Differential Equations – III Synspade 1975; Hubbard, B., 

Ed.; Academic Press, Inc.: New York, 1976; pp 89–116. 

Bendsoe, M. P.; Kikuchi, N. Generating Optimal Topologies in Structural Design Using a 

Homogenization Method. Comp. Meth. Appl. Mech. and Eng. 1988, 71, 197–224. 

Bensoussan, A.; Lions, J.-L.; Papanicolaou, G. Asymptotic Analysis for Periodic Structures; 

North-Holland Publishing Company: Amsterdam, 1978. 

Briane, M. Homogenization of a Nonperiodic Material. J. Math. Pures Appl. 1994, 73, 47–66. 

Fabre, S.; Mossino, J. H-Convergence of Multiplicable Matrices. Calc. Var. Partial Dif. 1998, 

7, 125–139. 

Fish, J.; Yu, Q. Multiscale Damage Modeling for Composite Materials: Theory and 

Computational Framework. Int. J. Numer. Methods Eng. 2001, 52, 161–192. 

Fish, J.; Yu, Q.; Shek, K. L. Computational Damage Mechanics for Composite Materials Based 

on Mathematical Homogenization. Int. J. Numer. Methods Eng. 1999, 45, 1657–1679. 

Francfort, G. A.; Murat, F. Homogenization and Optimal Bounds in Linear Elasticity. Arch. 

Rat. Mech. Anal. 1986, 94, 307–334. 

Ghosh, S.; Lee, K.; Moorthy, S. Two-Scale Analysis of Heterogeneous Elastic-Plastic Materials 

With Asymptotic Homogenization and Voronoi Cell Finite Element Model. Comp. Meth. 

Appl. Mech. Eng. 1996, 132, 63–116. 

Ghosh, S.; Lee, K.; Raghavan, P. A Multilevel Computational Model for Multiscale Damage 

Analysis in Composite and Porous Materials. Int. J. Solids Struct. 2001, 38, 2335–2385. 

Gustafsson, B.; Mossino, J. Nonperiodic Explicit Homogenization and Reduction of Dimension: 

The Linear Case. IMA J. Appl. Math. 2003, 68, 269–298. 

Hollister, S. J.; Brennan, J. M.; Kikuchi, N. A Homogenization Sampling Procedure for 

Calculating Trabecular Bone Effective Stiffness and Tissue-Level Stress. J. Biomech. 1994, 

27, 433–444. 

Hughes, T. J. R. The Finite Element Method: Linear Static and Dynamic Finite Element 

Analysis; Prentice-Hall: New Jersey, 1987. 
12

Lene, F. Damage Constitutive Relations for Composite Materials. Eng. Fract. Mech. 1986, 25, 

713–728. 

Moulinec, H.; Suquet, P. A Numerical Method for Computing the Overall Response of 

Nonlinear Composites With Complex Microstructure. Comput. Method Appl. M. 1998, 157, 

69–94. 

Oden, J. T.; Vemaganti, K. Adaptive Hierarchical Modeling of Heterogeneous Structures. 

Physica D 1999, 133, 404–415. 

Oden, J. T.; Prudhomme, S.; Hammerand, D. C.; Kuczma, M. S. Modeling Error and Adaptivity 

in Nonlinear Continuum Mechanics. Comput. Method Appl. M. 2001, 190, 6663–6684. 

Rostam-Abadi, F.; Chen, C. M.; Kikuchi, N. Design Analysis of Composite Laminate Structures 

for Light-Weight Armored Vehicle by Homogenization Method. Comput. Struct. 2000, 76, 

319–335. 

Sanchez-Palencia, E. Nonhomogeneous Media and Vibration Theory. In Lecture Notes in 

Physics 127; Springer-Verlag: Berlin, 1980. 

Shkoller, S. An Approximate Homogenization Scheme for Nonperiodic Materials. Comput. 

Math. Appl. 1997, 33, 15–34. 

Shkoller, S.; Hegemier, G. Homogenization of Plain-Weave Composites Using Two-Scale 

Convergence. Int. J. Solids Struct. 1995, 32, 783–794. 

Spagnolo, S. Convergence in Energy for Elliptic Operators. In Numerical Solution of Partial 

Differential Equations – III Synspade 1975; Hubbard, B., Ed.; Academic Press, Inc.: New 

York, 1976; pp 469–498. 

Tartar, L. Nonlinear Constitutive Relations and Homogenization. In Contemporary 

Developments in Continuum Mechanics and Partial Differential Equations; de La Penha, 

G. M., Medeiros, L. A., Eds.; North-Holland Publishing Company: Amsterdam, 1978; 

pp 472–484. 

Terada, K.; Kikuchi, N. A Class of General Algorithms for Multiscale Analyses of 

Heterogeneous Media. Comput. Method Appl. M. 2001, 190, 5427–5464. 

Terada, K.; Hori, M.; Kyoya, T.; Kikuchi, N. Simulation of the Multiscale Convergence in 

Computational Homogenization Approaches. Int. J. Solids Struct. 2000, 37, 2285–2311.
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