1

Multiscale Analysis of Large Scale Nonlinear Structures and Materials

Jacob Fish and Kamlun ShekCivil, Mechanical and Aerospace Engineering

Rensselaer Polytechnic Institute, Troy, NY 12180

1.0 Introduction

A typical structure (an airplane or a car) is of the order of magnitude of meters, whilethe size of the small features in the interconnect parts is of the order of millimeters. Inorder to determine the force redistribution between various structural components, fasten-ers, lugs and other interconnecting parts must be accurately modeled at a scale which isseveral orders of magnitude smaller than the entire structure. The useful life and cost ofmaintenance of the structure depends on the quality of modeling at each level and theability of a reliable transfer of the appropriate information between various modeling lev-els.

In analyzing a typical large scale structure the current practice is to carry out asequence of uncoupled analyses corresponding to different length scales:

(i) the structural or component level, and

(ii) the level of interconnects

(iii) the scale of material heterogeneity

On the structural (or global) level, structural components are treated discretely, whiledetails of interconnects are not represented, except perhaps for their overall contribution.It is only on the local level that the details of subcomponents are explicitly accounted for.For analysis on the scale of material heterogeneity individual phases (matrix and fibers incomposites, aggregates and cement in concrete, cords and rubber in tires [53], rods intruss-like materials [51], wires and vias in multichip modules [52]) are treated discretely,while the atomic scale is not recognized. Note that the three analysis steps are currentlyperformed independently in the sense that output from one is used as an input to the sec-ond.

The obvious question arises as to the validity of such an uncoupled three scale proce-dure. Is there a need for coupled approach that will consider phenomena simultaneously atseveral different scales? If the answer is positive, is it feasible to devise a coupledapproach that could be successfully utilized by structural designers in day-to-day practice?

In order to answer the first question it is important to recall that most of the structuresare often designed for temporary overloads in which case inelastic deformation is permit-

2

ted. The key question then becomes: Can the appropriate information extracted from theinelastic analysis of the interconnect be incorporated into the structural (global) analysisprocedures where the details of interconnects are excluded in order to reduce the size ofthe global problem? For linear problems this seems to be a viable approach, since it is fea-sible to idealize the interconnect as an elastic hinge or spring. Unfortunately, for nonlinearhistory dependent systems such an idealization is highly questionable since instantaneousmaterial properties in the vicinity of the interconnect undergoing plastic deformation mayvary in space and time due to their history dependence.

In what follows we briefly outline various research activities in the area of multiscalecomputational techniques which took place in the past 10 years. An excellent review ofearlier works in this field has been given by Dong [6]. Various multiscale computationaltechniques in the modern era can be classified into the following three categories: (a) mul-tiple scale expansion methods, and (b) superposition based methods.

Various forms of homogenization methods aimed at bridging material and structuralscales have been employed by Ghosh [8], [9], Herakovich [10], Kikuchi [11], [12], Fish[7], [14], [15] and others. Homogenization methods hinge on the following three assump-tions:

(i) The macrostructure is formed by a spatial periodicity of the Representative VolumeElement (RVE).

(ii) The solution is locally periodic in the statistical sense.

(iii) Macroscopic fields are constant within the Representative Volume Element.

Typically, these assumptions remain valid as long as the material heterogeneity is sig-nificantly smaller than the typical dimension of the macrostructure.

Superposition based methods asdvocated by Belytscko [16], Fish [17], [18], [19], [20],[21], Rank [22], Stanley [23], Hughes [24], [25], Reddy [27], [28], Noor [29] and others

hinge on the hierarchical decomposition of the solution space into global and local

effects, , and enforcement of solution compatibility by prescribing homo-

geneous boundary conditions on at the global-local interface , . Various

superposition based methods differ in the following three respects:

(i) Approximation of and ,

(ii) Selection of the interface ,

(iii) Solution of the global-local system of equations.

Hughes [24], [25] views and as resolvable and unresolvable parts of the solu-

tion, respectively. The fine-scale Green’s functions representation is employed for and

u uG

uL

u uL

uG+=

uL ΓGL u

L0=

uG

uL

ΓGL

uG

uL

uL

3

finite element discretization for . Belytschko et al [16] uses spectral approximation for

and finite element discretization for . Rank [22] reverses the roles of and formodeling of interconnects. A multilevel solution scheme, termed as the s-version of thefinite element method, has been developed by Fish [17], [18], [19], [20], [21] where eachlevel is discretized using a finite element mesh of arbitrary element size and polynomialorder. Fish [14], [21] Stanley [23] Reddy [27], [28] and Noor [29] applied the method ofsuperposition for modeling laminated plates and shells using a different mathematical

model for and .

Selection of the interface is one the critical issues in the superposition based

methods. Hughes [24], [25] assumes that the unresolvable scale, , vanishes at the ele-ment boundaries [24]. This choice provides a theoretical framework for stabilizationmethods. A rigorous mathematical analysis aimed at quantifying the “pollution” effects ofthe localized phenomena on the global behavior and subsequently identifying the optimallocation of the interface have been carried out by Babuska [30] and Fish [31]. In [31] an

adaptive strategy has been devised to construct an optimal discretization of and sothat the local and global phenomena of interest are resolved within the user-specifiedaccuracy.

The discrete system of equations resulting from the superposition of local and globalfields can be solved either exactly or approximately. An iterative global-local schemedeviced by Whitcomb [32] and Sun [33] can be interpreted as a block Gauss-Seidel itera-tive solution scheme. A single iteration of such a scheme coincides with a classical engi-neering submodeling procedure. The rate of convergence of various global-local iterativeschemes has been studied by McCormick [34], Bramble [35] and Fish [36]. Generaliza-tion to hybrid systems has been given by Fish [37].

The manuscript presents a nonlinear multiscale computational procedure based on thephilosophy of multilevel methods and which exploits the similarity between the classicalengineering global-local design practice and multilevel methods. Both approaches sepa-rate the system response into the global and local effects. While classical substructuring-based schemes separate the global and local effects on the basis of domain decompositionprinciples often resorting to intuition and understanding the physics of the problem, multi-level approach described here, carries out de facto a spectral decomposition automaticallyas a part of the solution process.

2.0 Problem Statement

We consider the strong form of the boundary value problem on the smallest scale of inter-est:

uG

uL

uG

uL

uG

uL

uG

ΓGL

uL

uL

uG

4

(1)

where and are the components of the stress and strain tensors, respectively; and

represent the body forces and prescribed boundary tractions on the boundary ,

respectively; are the components of the displacement vector; are the prescribed dis-

placements on the boundary ; represent the consistent (or algorithmic) instante-

neous components of the fourth order constitutive tensor; is a problem domain.

Symmetric gradient is denoted by . Standard tensorial notation

with summation over the repeated indexes is employed. The material time derivative is

denoted by a superposed dot. For example, is the velocity component; and the

components of the rate of deformation are denoted as . For simplicity, attention is

restricted to small deformation theory. We will refer to equation (1) as the exact mathe-matical model.

Our ultimate goal is to solve the system of nonlinear equations arising from the finiteelement discretization of (1) which yields:

(2)

where is set of continuous shape functions, such that and

; the upper case subscripts denote the degree-of-freedom and

the summation convention over repeated subscripts is employed for both the degrees-of-freedom and spatial dimensions; denotes the components of nodal displacement vec-

tor; and and are components of the internal and external force vectors, respec-

tively.

The Newton method for the discrete problem (2) is given as:

, (3)

where the tangent stiffness matrix

σij j, bi+ 0= on Ω

σ· ij Lijkl ε·kl= on Ω

εij u i j,( )= on Ω

ui ui= on Γu

σij nj ti= on Γt

σij εij bi

ti Γt

ui ui

Γu Lijkl

Ωϕ i j,( ) ϕi j, ϕj i,+( ) 2⁄=

vi ui·

=

ε·ij

rA fAint dA( ) fA

ext– 0= = fAint BijAσij Ωd

Ω∫=

NkA C0

uk NkAdA=

εkl N kA l,( )dA BklAdA≡=

dA

fAint fA

ext

di 1+

A di

A ∆dA+= Ki

AB∆dB ri

A=

5

(4)

is obtained via consistent linearization of the discrete equilibrium equations (2). For largescale structures with detailed representation of structural components the nonlinear systemof equation (2) may often contain over million unknowns. Moreover, if the discretizationemployed is at the level of material heterogeneity the number of unknowns could oftenexceed 10 digits. Since the brute force approach of solving (2) may not be feasible we willrestate our goal as follows. First (Section 3), we will attempt to construct simpler auxiliarymathematical model, that will provide a reasonable approximation of the solution. We will

denote such an approximation as . Then (in Section 4) we will develop a multiscaleiterative process for obtaining a discrete solution of equation (1) by utilizing the solutionof the auxiliary mathematical model.

3.0 Uncoupled Multiscale Mathematical Model

A typical large scale structure can be modeled on three or more scales: (i) the globalscale (the structural level), (ii) the local scale (the component level), and (iii) the materialscale (the level of microconstituents). There could be additional scales. For example, thecomponent scale can subdivided into two scales since the thickness of the shell is typicallymuch smaller than its in-plane dimension. Furthermore, the scale of material heterogeneityin each microphase, such as dislocations and grain boundaries could be considered as an-other scale, not to mention the atomic and electronic scales.

Multiple scale expansion method (see for example [46]) provides a theoretical frame-work for the uncoupled mathematical model developed in this section. It is based on the

first order triple scale asymtotic expansion, which seeks to approximate solution, , of theboundary value problem (1) in the form

(5)

where , and denote position vectors in the global,local and material scales, respectively. From the mathematical point of view the asymptoticexpansion is considered uniformly valid if the terms in the expansion are rapidly decreas-ing. This in turn means that should me sufficiently small. From the physical point of viewfor the triple scale asymtotic expansion (5) to be valid, structural response quantities

( ) defined on should have a slow (global) variation from point to

point, an intermediate (local) variation within a small neighborhood of , and fast (ma-

terial) variation in the vicinity of . Since there is only a single spatial independent

variable, , any response quantitity can be represented as . The

indirect spatial derivatives of can be calculated by the chain rule as

KAB dB∂∂

rA≡ BijALijkl BklB ΩdΩ

∫=

uζ

uζ

uζ x( ) uo x( ) ζu1 x y,( ) ζ2u2 x y z, ,( )+ +=

x Ω⊂ y ζx= y ΩL⊂, z ζy= z ΩM⊂,

ζ

uζ εζ σζ, , Ω ΩL× ΩM×ζ x

ζ2y

x fζ

fς x( ) f x y x( ) z x( ),,( )≡fς

6

(6)

where the comma followed by a subscript variable denotes a partial derivative with re-

spect to the subscript variable (i.e. ). Summation convention for repeated sub-

scripts is employed, except for subscripts , and .

The triple scale asymtotic expansion (5) can be interpreted as a superposition of global,

local and material scale fields. We assume that vanishes on the boundary of ,

denoted by . Material scale solution is assumed to be periodic, i.e.,

or to vanish on the boundary of for nonperiodic

media; is a basic period of the periodic microstructure and is an arbitrary diag-onal integer matrix. In the case of periodic microstructures the instanteneous consistentstiffness is periodic, i.e., . We will refer to the aferomen-tioned Global-Local-Material scale interface conditions simply as the interface conditions.For simplicity we assume that .

Strain expansions on the composite domain are obtained by substituting(5) into (1) with consideration of the indirect differentiation rule (6)

(7)

where various order strain components are given as

(8)

and

. (9)

For convenience we define an overall local strain, , as an average strain over

(10)

and the overall global strain as an average strain over

(11)

f,xi

ς x( ) f,xix y z, ,( ) 1

ς--- f,yi

x y z, ,( ) 1

ς2----- f,zi

x y z, ,( )+ +=

xi

f,xi∂f ∂xi⁄≡

x y z

u1 x y,( ) ΩL

ΓLu2 x y z, ,( )

u2 x y z, ,( ) u2 x y z kZ+, ,( )= ΓM ΩM

Z k 3 3×

L x y z, ,( ) L x y z kZ+, ,( )=

b b x( )=

Ω ΩL× ΩM×

εijζ εij

0 x y z, ,( ) ςεi j1 x y z, ,( ) …+ +=

εijs εxij us( ) εyij us 1+( ) εzij us 2+( ), s+ + 0 1 …, ,= =

εxij us( ) u i ,xj( )s , εyij us( ) u i ,yj( )

s , εzij us( ) u i ,zj( )s===

εL ΩM

εijL 1

ΩM----------- εij

ζ ΩMd

ΩM∫ εxij u0( ) εyij u1( )+= =

ΩL

εijG 1

ΩL---------- εij

L ΩLd

ΩL∫ εxij u0( )= =

7

where the integrals of over and of over vanish due to the interface

conditions; and denote the volumes of and , respectively.

Substituting the asymtotic expansion of the strain field (7) into the constitutive equa-tion yields the asymptotic expansion of stress field

(12)

where

(13)

Inserting the stress expansion (12) into the equilibrium equation (1) yields the follow-ing equilibrium equations for various orders:

(14)

(15)

(16)

Equations (14), (15) and (16) represent the equilibrium condition on the material, localstructural and global structural scales, respectively.

We first consider the rate of equilibrium equation on the material scale. After substitu-

tion of (10), (11) and (13) into (14) the equilibrium equation can be restated as

(17)

To solve for (17) up to a constant we introduce the following separation of variables

(18)

which yields the strong form of the linearized boundary value problem on

u i ,yj( ) ΩLu i ,zj( ) ΩM

ΩM ΩL ΩM ΩL

σijζ σij

0= x y z, ,( ) ςσij1 x y z, ,( ) …+ +

σ· ijs

Lijkl ε·kls

, s 0 1 …, ,==

O ς 2–( ): σij ,zj

0 0=

O ς 1–( ): σij ,yj

0 σij ,zj

1+ 0=

O ς0( ): σij ,xj

0 σij ,yj

1 σij ,zj

2 bi+ + + 0=

O ς 2–( )

Lijkl ε·klL

ε·zkl u2( )+( )

,zj

0 on ΩM=

u·i2 x y z, ,( ) Hikl

M z( )ε·klL

x y,( )=

ΩM

8

(19)

where . The weak form of the corresponding boundary value

problem states: Find the third order tensor, , such that for all the following weak statement holds

(20)

Integration by parts of (20) yields

(21)

To construct the finite element approximation of the space , we subdivide

into finite element subdomains, , such that ; and interpolate the

solution as

(22)

Substituting interpolants (22) into the weak form (21) yields a system of linear equa-

tions for the unknowns

(23)

Due to symmetry with respect to indices (i,j) and (m,n) it can be seen that the linearsystem (23) must be solved for 3 right hand side vector in 2D and 6 in 3D.

Lijkl Iklmn ΨklmnM+( )[ ] zj, 0=

ΨklmnM

H k zl,( )mnM= HM WM∈

WM HM z( ) HMH

1 ΩM( )( )3

Z-periodic,∈[ ], or

HM z( ) HMH

1 ΩM( )( )3

HM z( ),∈ 0 on ΓM=[ ]

=

Iklmn12--- δkmδnl δnkδml+( )=

HM WM∈ δHM WM∈

δHstiM

Lijkl Iklmn ΨklmnM+( )[ ] zj, ΩM

dΩM∫ 0=

δΨstijM

Lijkl ΨklmnM ΩM

dΩM∫ δΨstij

MLijkl ΩM

dΩM∫–=

WM

WM

ΩM ΩeM Ωe

M

e∪ ΩM=

HmnkM

NkAM

qmnAM= Ψmnkl

MBklA

MqmnA

M= BklAM

N kA l,( )M=

qmnAM

KABM

qmnBM

FmnAM= KAB

MBijA

MLijkl BklB

M ΩMd

ΩM∫= FmnA

MBijA

MLijmn ΩM

dΩM∫–=

9

We next proceed to the rate form of equilibrium equation. Substituting consti-

tutive equation (13) into the rate form of (15), integrating it over and exploiting theinterface conditions yields

(24)

where the overall local stress, , is defined as an average stress over . Substituting (13) and (18) into (24) yields

(25)

where represents the average instanteneous material properties on . It is a triv-

ial to show (see for example [7], [12]) that is symmetric provided that is sym-

metric. To solve for the local structural equilibrium equation (25) we introduce the

following approximation for :

(26)

which yields the strong form of the linearized boundary value problem on

(27)

Similarly to the weak form on , the weak form for the boundary value problem

(27) on seeks for the , such that for all the followingweak statement (after integration by parts) holds

(28)

To construct the finite element approximation of the space , we subdivide

into finite element subdomains, , such that , and interpolate as

O ζ 1–( )

ΩM

σ· ij y j,L

0= σijL 1

ΩM----------- σi j

ζ ΩMd

ΩM∫=

σL ΩM

LijklL ε·kl

L( ) yj, 0= Lijmn

L 1

ΩM----------- Lijkl Iklmn Ψklmn

M+( ) ΩM

dΩM∫=

LijmnL ΩM

LijklL

Lijkl

u·1

u·i1 x y,( ) Hikl

L y( )ε·klG

x( )=

ΩL

LijklL

Iklmn ΨklmnL+( )[ ] yj, 0=

ΨklmnL

H k zl,( )mnL= HL WL∈

WLHL y( ) HL

H1 ΩL( )( )

3 HL y( ),∈ 0 on ΓL=[ ]=

ΩM

ΩLH

L WL∈ δHL WL∈

δΨstijL

LijklL Ψklmn

L ΩLd

ΩL∫ δΨstij

LLijkl

L ΩLd

ΩL∫–=

WL

WL

ΩL ΩeL Ωe

L

e∪ ΩL

=

10

(29)

Substituting interpolants (29) into the weak form (28) yields a system of linear equa-

tions for the unknowns

(30)

Finally, the global equilibrium equations, , can be obtained by substituting

(13) into the rate form of (16), integrating over and ; and exploiting interface con-ditions, which yields:

(31)

where the overall global stress, , is defined as an average stress over .

The global problem is discretized using finite element method on the coarse mesh,

with displacement and strain interpolants denoted as: and .

The resulting nonlinear algebraic system equations is solved by the Newton or relatedmethod. The steps carried out in computation of the consistent global tangent stiffnessmatrix

(32)

are as follows: (i) Solution of the material scale problem (23) and evaluation of ,

(ii) evaluation of the overall instanteneous material properties (25), (iii) solution of the

local structural problem (30) and computation of , (vi) evaluation of (31), and

finally, computation of from (32). The global stress update procedure denoted as

, consists of (i) integration of , (ii) integration of

, (iii) stress integration on material scale , evaluation of

by averaging on (24), evaluation of by averaging on . For moredetails on computational algorithms for two scale problems see [15].

HmnkL

NkAL

qmnAL

= ΨmnklL

BklAL

qmnAL

= BklAL

N kA l,( )L

=

qmnAL

KABL

qmnBL

FmnAL= KAB

LBijA

LLijkl

LBklB

L ΩLd

ΩL∫= FmnA

LBijA

LLijmn

L ΩLd

ΩL∫–=

O ζ 1–( )

ΩM ΩL

σij x j,G

bi+ 0= σijG 1

ΩL---------- σij

L ΩLd

ΩL∫=

σ· ijG

LijklG ε·kl

G= Lijmn

G 1

ΩL---------- Lijkl

LIklmn Ψklmn

L+( ) ΩLd

ΩL∫=

σG ΩL

ukG

NkAG

dAG= εkl

GBklA

GdA

G=

KABG

BijAG

LijklG

BklAG ΩG

dΩG∫=

HM GM,

LL

HL

GL, L

G

KABG

σi G ∆εεG,( ) σσi 1+ G→ εε·L

∆εεL→

ε·M

∆εεM→ σσi M ∆εεM,( ) σσi 1+ M→

σi 1+ L ΩM σi 1+ G ΩL

11

4.0 Coupled multiscale analysis

In the limit as the solution of the source problem (1) approaches the solution ofthe uncoupled multiscale model provided that the error introduced due to the approxima-tion of the interface conditions is negligible. Unfortunately, in many practical situationswhen the value of is finite and/or the local solution is neither periodic nor vanishes atthe interface, the uncoupled multiscale model may err badly in comparison with the exactsolution of the source problem (1). The most significant errors are encountered in the por-tions of the problem domain where the solution has high gradients. Ironically, these areprecisely the regions of major interest from the practical standpoint.

In what follows we describe a novel approach which takes advantage of the specialnature of differential equations with multiple spatial scales in order to develop fast itera-tive solvers for system of linearized equations arising from such differential operators.This is accomplished using a multilevel approach with special intergrid transfer operator.

The classical multigrid approach with standard linear interpolation operators is notwell suited to approximate the lower frequency response of the source system, mainlybecause the lower frequency modes are not smooth on the local structural and materialscales. On the other hand, the solution of the uncoupled multiscale mathematical model isin good agreement with the lower frequency response of the source problem (1) as evi-denced by the studies conducted in [46],[47]. The basic idea of the coupled multiscaleapproach presented here is to utilize the (lower frequency) solution obtained from theuncoupled multiscale model within the framework of the multilevel method.

The remainder of this section is organized as follows. In Section 4.1 we review thebasic principles of the multilevel methods. Dedicated interscale transfer operators arederived in section 4.2. Regularization of the interscale transfer operators is given in Sec-tion 4.3. Finally, Section 4.4 presents the framework for the coupled multiscale analysis.

4.1 Newton-Multilevel method

As a prelude to subsequent derivations we briefly outline the basic principles of multi-level algorithm for solving a linearized system of equations (3)

(33)

The multilevel method is an iterative process which resolves the higher frequencyresponse of the linearized system by means of relaxation methods, while the remainingsmooth components of the solution (in the sense of energy) are captured on the auxiliarycoarse model. The multilevel algorithm is outlined below:

• Starting from the fine scale approximation , which is obtained at the end of cycle

(inner iteration) , perform several pre-smoothing iterations to smooth out high fre-

quency components of the error and evaluate the fine scale residual . The left sub-

script denotes the cycle count.

ζ 0→

ζ

Ki

AB∆dB ri

A=

∆dj B

j

rij A

12

• Restrict the residual from the fine scale model to the auxiliary coarse model

(34)

where lower case subscripts denote the degrees-of-freedom in the auxiliary coarse

model and is the restriction operator

• Compute the coarse model correction

(35)

where is the coarse model stiffness matrix obtained either directly on the aux-

iliary coarse model, or variationally, by restricting the fine grid stiffness matrix

(36)

The solution of (35) can be carried out by a direct method, or by introducing anothercoarser auxiliary model and using one or more cycles of the two-level algorithm.

• Prolongate the displacement correction from the coarse model to the fine grid andupdate

(37)

where is a prolongation operator related to the restriction operator by .

• Perform post-smoothing operations starting with the updated solution on the fine grid

to obtain a new approximation

• Accelerate the multigrid cycle using CG method for positive definite systems or QMRand GMRES for indefinite systems [48],[49].

• Check convergence. If necessary, start a new cycle.

In Section 4.2 we show that the tangent stiffness matrix obtained by restriction of the finescale model (37) coincides with the tangent stiffness matrix of the detailed structural

model . Likewise, we show that restriction of tangent stiffness matrix corresponding

to the detail structural model coincides with the tangent stiffness matrix of the global

model .

4.2 The interscale transfer operators

In this section we focus on the central issue of constructing the interscale tranfer oper-

ators for the three scale model. As a prelude we make the following definitions. Let ,

ri coarsej B QBA r

ij A=

QBA

∆dB

j coarse

Ki coarse

AB ∆dB

j coarse ri coarsej B=

Ki coarse

AB

Ki coarse

CD QCA Ki

ABQBD=

di

j 1+ B dij B QBA ∆d

Aj coarse+=

QBb QBb QbBT=

dBi 1+

Kˆ

ABL

Kˆ

ABG

Nˆ

iAG

13

and be the global shape functions obtained by assembly of local shape

functions , and , respectivelly; and let , and be the corre-

sponding discrete solution on . The goal is to relate , and

and ultimaly establish the relation between the tangent stiffness matrices of variousmodels.

We start from the discrete form of the asymptotic expansion of the solution

(38)

From (19)-(22) it can be easily seen that , i.e.,

is independent of the choice of . We further discretize

which yields

(39)

We assume that the shape functions in the detailed structural model can be

expressed as a linear combination of the shape functions in the model discretized on thescale material heterogeneity, i.e.,

(40)

To construct local-to-material instanteneous interscale transfer operator, , we

evaluate at the nodes of the material scale model, which after substitution of (40)

into (39) yields

(41)

where underlined subscripts indicate no summation over repeated indices. In (41) we

denoted velocities along the spatial coordinate at the node as a degree-of-freedom

in the material scale model, i.e., . Thus

, (42)

Nˆ

iAL

Nˆ

iAM

NiA≡

NiAG

NiAL

NiAM

dAG

dAL

dAM

dA≡

Ω ΩL

L∪ ΩM

M∪= = dA

GdA

L

dAM

u·iζ

u·iL

ζ2u·i

2+ N

ˆiAL

ζ2Hijk

M z( )Bˆ jkA

L+( )d

·AL

= =

ζ2Hijk

M z( ) HijkM ζ2z( ) H

ˆijkM

x( )= =

ζ2Hijk

Mz( ) ζ H

ˆijkM

x( ) Nˆ

iBM

qjkBM

=

u·iζ

Nˆ

iAL

Nˆ

iBM

Bˆ

jkAL

qjkBM

+( ) d·AL

=

Nˆ

iAL

Nˆ

iAL

Nˆ

iDM

GDAL

=

QCAL

u·iζ

xCM

u·iζ

xCM( ) d

·CM

Nˆ

iDM

xCM( ) GDA

LBˆ

jkAL

xCM( )qjkD

M+

d·AL

= =

i xCM

C

u·iζ

xCM( ) d

·CM

and Nˆ

iDM

xCM( ) δDC= =

d·CM

QCAL

d·AL

= QCAL

GCAL

Bˆ

jkAL

xCM( )qjkC

M+=

14

It remains to establish the relation between the tangent stiffness matrices in thedetailed structural and material scale models. For this purpose we construct the expres-

sion for the rate of the strain field, , in terms of the solution in the detailed structural

model and the and the sequence of uncoupled the solutions, , in the material

cells

(43)

The derivation of (43) follows from the the asymtotic expansion, ,interpolations at various scales and equation (40).

The tangent stiffness matrix of the detailed structural model is given as

(44)

Equation (44)c (line 3) follows from the assumption of infinitisimality of the material

cell, , i.e., the fields on the structural level, such as are taken to be constant in the

vicinity of a point in the material cell . Equation (44)e follows from the expression

of the rate of strain field (43), whereas (44)e (line 5) directly follows from the definition ofthe prolongation operator in (42).

Finally, starting from the discretization of the asymptotic expansion of the detailedstructural solution

(45)

ε·ijζ

d·AL

qmkDM

ΩM

ε·ijζ

Bˆ

mnAL

I imjn Hˆ

i j,( )mnM

+( )d·AL

Bˆ

ijDM

GDAL

Bˆ

mkAL

qmkDM

+( )d·AL

= =

ε·ijζ

ε·klL

ε·zkl u2( )+=

KABL

Bˆ

i jAL

LijklL

Bˆ

klBL

ΩdΩ∫ B

ˆijAL

LijklL

Bˆ

klBL

ΩLd

ΩL∫

L∑= =

Bˆ

ijAL

Bˆ

klBL

ΩM 1

ΩM----------- I imjn H

ˆi j,( )mn

M+( )Lijkl Ikplq H

ˆk l,( )pq

M+( ) ΩM

d

ΩM

∫dΩM∫

M∑=

Bˆ

mnAL

I imjn Hˆ

i j,( )mnM

+( )Lijkl Ikplq Hˆ

k l,( )pqM

+( )Bˆ pqBL

ΩMd

ΩM

∫M

∑=

Bˆ

ijDM

GDAL

Bˆ

mnAL

xDM( ) qmnD

M+

Lijkl GCBL

Bˆ

pqBL

xCM( ) qpqC

M+

Bˆ

klCM

ΩMd

ΩM

∫M

∑=

QADL

Bˆ

ijD

MLijkl B

ˆklCM

QCBL ΩM

d

ΩM

∫M

∑= QDAL( )

TKˆ

DC

MQCB

LKˆ

ABL

≡=

ΩMBˆ

ijDM

xDM ΩM

u·iL

u·iG

ζu·i1

+ Nˆ

iAG

ζHijkL

y( )Bˆ jkA

G+( )d

·AG

= =

15

and following derivations analogous to equations (38)-(44) it can be shown that the glo-

bal-to-local instanteneous interscale transfer operator, , is given by

, (46)

and the tangent stiffness matrix of the global structural model can be obtained by restrict-ing the tangent stiffness matrix of the detailed structural model, i.e.,

(47)

Remark. The prolongated velocities at any scale have to maintain continuity. Unfortu-

natelly, the interscale tranfer operators, , do

not uniquely determine the velocity field at the boundaries between local and material

subregions since and are continuous, i.e., they are discontinuous at the

boundaries. In Section 4.3 we develop a methodology which generates continuoussolutions at any scale.

4.3 Prolongation by aggregation In the previous section we have derived the interscale transfer operators assuming an

infinitesimality of the unit cell. In practice when the value of is finite, it is necessary to

reformulate the interscale transfer operator to maintain continuity of the prolongatedsolution. We ephasize that the direct application of the prolongation operators (42), (46)does not uniquely determine solution at the boundaries of the representative volume ele-

ments, since the global and local strain fields , are con-

tinuous functions. Consequently, the prolongated solution is also discontinuous at theinterface between the representative volume elements.

To develop a prolongation operator, that generates continuous displacements,we will employ aggregation approach [2], [36]. Alternative strategy based on the genera-

tion of strains has been presented in [47].

In the aggregation approach the coarse model is directly constructed from the sourcegrid by decomposing the whole set of nodes into non-intersecting groups to be referred toas aggregates, and then for each aggregate assigning a reduced number of degrees of free-dom. By doing so one reduces dimensionality of the source problem, while maintainingthe compatibility of the solution.

The non-intersecting groups of elements are formed on the basis of “stiff” and “soft”elements. The element is considered “stiff” if the spectral radius of its stiffness matrix isrelatively large compared to other elements and vice versa. It has been shown in [2], [36]that it is advantageous to place “soft” elements at the interface between the aggregates,

QCAG

d·CL

QCAG

d·AG

= QCAG

GCAG

Bˆ

jkAG

xCL( )qjkC

L+=

QDAG( )

TKˆ

DC

LQCB

GKˆ

ABG

≡

C0

GCAG

Bˆ

jkAG

xCL( )qjkC

L+ GCA

LBˆ

jkAL

xCM( )qjkC

M+

Bˆ

jkAG

Bˆ

jkAL

C1–

C0

ζ

C0

Bˆ

jkAG

xCL( )qjkC

LBˆ

jkAL

xCM( )qjkC

MC

1–

C0

C0

C0

16

and “stiff” elements within the aggregates. The approximation for the maximum eigen-value can be easily estimated using the Gerschgorin theorem. A typical aggregated modelis shown in Figure 1.

4.4 Three-scale multilevel processThe three-scale multilevel procedure starts by performing several smoothing iterations

on the material-scale. Consequently, the higher frequency modes of error are damped outimmediately. The remaining part of the solution error is smooth, and hence, can be effec-tively eliminated on the auxiliary coarse mesh. It has been shown in [46] that the finite el-ement mesh on the local scale serves as a perfect mechanism for capturing the lowerfrequency response on the material scale. Therefore, the residual in the finite element meshon the material scale is restricted to the local scale, while the smooth part of the solution iscaptured by the finite element mesh on the local scale. The oscillatory part of the solutionon the local scale is again damped out by a smoothing procedure. The lower frequency re-sponse on the local scale is resolved on the global model. The resulting solution on the localscale is obtained by prolongating displacements from the global mesh back to the finite el-ement mesh on the local scale and by adding the oscillatory part of the solution previouslycaptured on the local scale. Likewise, the solution on the material scale is obtained by pro-longating the smooth part of the solution from the local scale and by adding the oscillatorypart that has been obtained by smoothing. This process is repeated until satisfactory accu-racy is obtained.

For periodic heterogeneous media the rate of convergence of the two-scale process hasbeen studied in [47]. It has been proved that for problems in a periodic 1-D heterogeneousmedium, the rate of convergence of the two-scale method with special inter-scale transferoperators is governed by:

Aggregates

Interfaceelements

Figure 1: Aggrgated Model

17

(48)

where is the norm of solution error in iteration , represent the stiffnesses of

microconstituents. Note that if the material is homogeneous and the mesh is uniform, then and we recover the classical two-grid estimate: = 1/3 . Otherwise

q < 1 resulting in: / . < 1/3.

On the other hand if the stiffness of the inclusion is significantly higher than that of amatrix, i.e. D1>>D 2, then the two-scale method converges in a single iteration. This behav-ior of the two-scale method for periodic heterogeneous media together with its linear de-pendence on the number of degrees-of-freedom, makes it possible to solve large coupledglobal-local problems with the same amount of computational effort, or faster, than wouldbe required to solve the corresponding uncoupled problem using direct solvers.

5.0 Numerical examples

Seven problems have been considered to validate the present formulation and to com-pare its performance in terms of iteration count, the CPU time and memory requirementsto the state-of-the-art direct method [50]. Geometries and finite element meshes for theseven problems considered are shown in Figure 2. The Composite Diffuser Casing ismade of woven composite material with bundle material behaving linearly elastic andmatrix obeying von Mises plasticity. The Composite Diffuser Casing problem is modeledat three scales: Global-Local-Material. Material properties considered were as follows:

Bundle: Young’s modulus = 379.2 GPa, Poisson’s ratio = 0.21Matrix: Young’s modulus = 68.9 GPa, Poisson’s ratio = 0.33, Yield stress = 24 MPa, isotropic hardening = 14 GPa

The remaining six problems were modeled at two scales, global and local, only. Thematerial is assumed to be homogeneous and obey von Mises plasticity with material con-stants identical to those of the matrix phase in the Composite Diffuser Casing problem.For all problems considered the loading was selected so that at least 25% of global ele-ments experience plastic deformation.

Results are summarized in Table 1. For all seven test problems considered, the cou-pled multiscale solver with interscale transfer operator derived from the multiple scaleanalysis outperformed the multifrontal solver by a factor ranging from 5 to 25.

Acknowledgment:The support of the National Science Foundation under Agreement number CMS-9713337and SANDIA National Laboratory under contract number AX-8516 are gratefullyacknowledged.

ei 1+q

4 q–------------ ei= q

D1D2

0.5 D1 D2+( )--------------------------------=

ei 1+ i Di

D1 D2= ei 1+ ei

ei 1+ ei

18

.*Tolerance for Global Iterations was selected to be .

Table 1: Comparison of Solvers

Problem ElementNumber of equat.

GlobalIter*

Multi-frontalCPU(s)

Multi-frontalMemory

(MB)

Multi-ScaleAv.

LocalIterations

Multi-Scale

CPU (s)

Multi-Scale

Memory (MB)

Casting Tet-10 158166 11 65494 1019 45 8423 199

Nozzle Tet-10 131565 7 36288 1003 37 4877 174

Ring-strt Tet-4 102642 7 4326 345 26 1112 87

SDRC1 Tetra-10 178536 8 75568 1409 16 3127 232

SDRC2 Tetra-10 145911 9 28656 992 21 3634 195

Comp Diffuer

Tetra-4 137391 5 9465 667 28 1516 147

Three scale

Figure 2: Geometry and FE meshes

10 5–

19

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Table 1: Comparison of Solvers

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