Multi-Scale Structural Optimization in the Presence of Uncertainty for Very Large

Composite Structures

Robert LiptonLSU

http://math.lsu.edu/~lipton

IMA WorkshopMathematical Modeling, Numerical Approximation

&Large Scale

Optimization of Complex Systems With Uncertainty

Oct 17-22, 2010

B. Alali (University Utah)I. Babuska (UT Austin, ICES) T. Breitzman (Wright Patterson AFB) Y. Chen (LSU)E. Iarve (Wright Patterson AFB)S. Jimenez (WPI)T. Mengesha (LSU)M. Stuebner (NCSU)

Supported by AFOSR Grant FA9550-08-1-0095NSF Grant DMS-0807265

Talk based on collaborations with

Robert Lipton

Outline of presentation

Overview of problemMultiple scales & sources of uncertaintyCertification of composite structures

Modeling and optimization with multiple scales1. Scale separation2. No scale separation

Robert Lipton

Wind Turbine Blade

Robert Lipton

Composite Wing

Robert Lipton

Ultimate Load Test: Composite Wing Attached to

Fuselage

Robert Lipton

Stress/strain transfer across multiple length scales I.

Structural Length Scale: Composite Stabilizer -777

Stack of laminates

Fiber length scale ~10-6m

Ply length Scale

~10-4m

Laminate length scale

~10 m

Robert Lipton

Stress/strain transfer across multiple length scales II: Joints connecting structures

Connections between stringers to center wing box

Robert Lipton

J. Aboudi, M. J. Pindera, and S. M. Arnold, Microstructural optimization of functionally graded composites subjected to thermal gradient via the coupled higher-order theory. Composites Part B, 28B (1997), pp. 93108.

C. Albert and G. Fernlund, Spring-in and warpage in curved laminates. Composites Science and Technology, 62 (2002), pp. 1895-1912.

I. Babuska, B. Andersson, P. J. Smith, and K. Levin, Damage analysis of fibrer composites, Part I: Statistical analysis on fiber scale, Comput. Methods Appl. Engrg., 172 (1999), pp. 27-77.

T. Breitzman, R. Lipton, & E. Iarve, Local field assessment inside multiscale composite architectures. SIAM MultiscaleModeling and Simulation, 2007.

Babuska, T. Strouboulis, and K. Copps, The design and analysis of the generalized finite element method, Comput. Meth. Appl. Mech. Engng., 181 (2000), pp.43-69.

B. Cox and Q. Yang, In quest of virtual tests for structural composites, Science,314 (2006), pp. 1102-1106.

I. M. Daniel and O. Ishai, Engineering Mechanics of Composite Materials. New York, Oxford University Press, 1994.

G. Dvorak and A. P. Suvorov, The effect of fiber pre-stress on residual stresses and the onset of damage in symmetric laminates, Composites Science and Technology, 60 (2000), pp. 1129-1139.

W. E, and B. Engquist, The heterogeneous multiscale methods, Commun. Math. Sci., 1 (2003), pp. 87-132.

Y. R. Effendiev, T. Y. Hou and X. Wu, Convergence of a nonconforming multiscale finite element method ,J Comput. Phys. 37 (2000), pp. 888-910.

Multi-scale AnalysisNumerical Methods, Applied Math, Mechanics

Robert Lipton

J. Fish and Z. Yuan, Multiscale enrichment based on partition of unity Int. J. Num. Meth. Engr., 62 (2005), pp. 1341-1359.

J. H. Gosse and S. Christensen, Strain invariant failure criteria for polymers in composite materials, AIAA, 1184 (2001), p. 11.

V. H. Hoang and C. Schwab, High-dimensional finite elements for elliptic problems with multiple scales, MultiscaleModel. Simul., 3 (2005), pp. 168-194.

T. Y. Hou and X. H. Wu, A multiscale finite element for elliptic problems in composite materials and porous media, J. Comput. Phys., 134 (1997), pp. 169-189.

T. Y. Hou, X. H.Wu and Z. Cai, Convergence of a multiscale finite element method for elliptic problems with rapidly varying coecients, Math. Comp., 68 (1999), pp. 913-943.

T. Hughes, G. Feijoo, L. Mazzei, and J. Quincy, The variational multiscale method: A paradigm for computational mechanics, Comput. Meth. Appl. Mech. Engng., 166 (1999), pp.913-943.

P. Hutapea, F. G. Yuan, and N. J. Pagano, Micro-stress prediction in composite laminates with high stress gradients, Int. J. Solids Structures, 40 (2003), pp. 2215-2248.

J. T. Oden, K. Vemaganti, and N. Moes, Hierarchical modeling of heterogeneous solids, Comput. Methods Appl. Mech. Engng., 172 (1999), pp. 3-25.

J. T. Oden and T. I. Zohdi, Analysis and adaptive modeling of highly heterogeneous elastic structures, Comput. Methods Appl. Mech. Engng., 148 (1997), pp. 367-391.

N. J Pagano, G. A. Schoeppner, R. Kim, and F. L. Abrams, , Steady-state cracking and edge effects in thermo-mechanical transverse cracking of cross-ply lami-nates, Comp. Sci. and Tech., 58 (1998), pp. 18111825.

Multi-scale AnalysisNumerical Methods, Applied Math, Mechanics

Robert Lipton

Design Objectives

Overall Structural Performance

Associated Optimization problems:

Minimizing WeightMaximizing Lift

Reliability

Associated optimization problems:

Maximizing Service Lifetime

Robert Lipton

Forward problem: Finite Element Analysis supported with Experimental Testing of Components

FE Analysis1. FE Analysis extensively used to calculate stresses

at component level & structural level2. The cost of experiment increases dramatically

when going from components to the full structure

From Fawcett, Trostle, & Ward,11th International Conference on Composite Materials (ICCM-11),Australia, July 14-18, 1997

Robert Lipton

Dimension ReductionGlobal – Local Modeling & Aspects of

Uncertainty I.

Apply separation of length scales hypotheses.Overall structural modeling via effective stiffnessproperties-iterated homogenization. Introduction of uncertainties at smaller length scales.

1. Effective stiffness of plies. (Uncertainty in fiber configuration, fiber-matrix bonds and elastic properties of epoxy resin matrix)

2. Effective stiffness of laminates of plies. (Uncertainty of uniformity of epoxy resin rich zones between plies).

Global local analysis for optimizing overallstructural performance

Robert Lipton

Dimension Reduction Global – Local Modeling&

Aspects of Uncertainty II. Service lifetime related to fatigue resistance.• Highly dependent on local stresses and strains in epoxy

resin matrix near stress risers such as joints.

Introduction of uncertainties related to the physical state of the epoxy matrix and fiber to matrix bonding. (Moisture & temperature dependent properties.)

Scale separation hypothesis: spatially coarse components of stress and strain drive the oscillations & concentrations of local stress and strain over small volume elements containing fine structural length scales

1. Field concentrations inside plies. (Uncertainty in fiber configuration, adhesive bond to fibers and elastic properties of epoxy resin matrix)

2. Field concentrations between plies. (Uncertainty of uniformity of epoxy resin rich zones between plies).

Robert Lipton

Techniques for assessment of stress transfer in stochastic composite materials

Problem description:

Uncertainty: on the actual configuration of the microstructure.

Data: Partial statistical information on microstructure,such as local volume fraction and two-point correlation information, (obtained from image processing);and (or) effective properties obtained from measurements.

Goal: Use this statistical data to assess extent of high stress zones inside heterogeneous structures.

Assumptions: Length scale of heterogeneities much smaller thancharacteristic length scale of domain.

Mathematics: Configurations represented by coefficients of PDE.Here the coefficients interpreted as random fields.

Robert Lipton

Example: A two phase material & a partial statistical description of the microstructure

)(1 yχVolume/Area fraction of each phase is known

x

r

Q(x,r))(2 yχ

dyyrxQ

xrxQ

r )(|),(|

1)(),(

2,2 ∫= χϑ

dyyrxQ

xrxQ

r )(|),(|

1)(),(

1,1 ∫= χϑ

=

=

1, y in material 10, otherwise

1, y in material 20, otherwise

Robert Lipton

Theme/Challenge: A composite made of two materials

Input: Plot of local area fraction θ1,r(x)

Output: Upper/Lower bounds oniso-stress level lines over all

composites associated with a given θ1,r(x)

Local average by area of material one over a cube centered at x of radius r given by θ1,r(x)

Robert Lipton

Formulate the problem subject to scale separation hypothesis and

``Look for the best microstructure’’ fora prescribed θ1,r

For this discussion ``best’’= mininum stress/strain

microstructure:

Extreme points in the set of coefficient

random fields

For given data θ1,r we discuss the problem offinding a lower bound on local stresses

Robert Lipton

Approach: 1. Review of homogenization theory. 2. Then see what other ingredients need to be added

Mathematical context: No assumptions placed on microstructure so will work in the general homogenization setting given by G-convergence (Spagnolo 1960’s).

The scale separation hypothesis suggests that some aspects of homogenization theory should be considered

What is a good mathematical context?

Robert Lipton

G-convergence

)(xεϕ

)()( xRx εε ϕσ ∇=

)(1 xεχ

10 <<< εLength scale of microstructure

Stress potential

Stress tensor

Local shear compliance )(xS ε

Sε is the shear compliance of a two phase material, and takes the two values, S1 and S2

Equation of elastic equilibrium

Boundary conditions

fSdiv =∇− )( εε ϕ0=εϕ ε

dxxdxxS

iS

i )()( ∫∫ → ϑχ ε

Indicator function of material 1

Separation of scales expressed by the weak convergence

For any subset S

Robert Lipton

Homogenization theory; convergence of averages

dxxdxxS

M

S∫∫ → )()( ϕϕε

dxxdxxS

M

S

)()( ∫∫ ∇→∇ ϕϕε

fSdiv ME =∇− )( ϕBoundary Condition

dxxSdxxSS

ME

S∫∫ ∇→∇ )()( ϕϕεε

0=Mϕ

And for any subset S:

Homogenized equation

Sε G-converges to SE if and only if for every load case f in H-1 :

Robert Lipton

Assessment of local field fluctuations

)(2 xεχ)(1 xεχ

|)(|)(,, 1 txxwhereSx >∇∈ εε ϕχS1ε(t) =

For any subset S examine the high stress zonesinside each material

|)(|)(,, 2 txxwhereSx >∇∈ εε ϕχS2ε(t) =

Denote the indicator functions of each phase by

Here the goal is to characterize the limits of the setsS1ε and S2

ε as ε goes to zero in terms of partialKnowledge of )(2 xεχ)(1 xεχ and

Separation of scales dxxdxxS

iS

i )()( ∫∫ → ϑχ ε for any subset S

Robert Lipton

Beyond average fields

Require new quantities in addition to G-limits that allow for the characterization of the sets Si

ε, for 0<ε<<1.

Robert Lipton

Localization step

Consider a square Q(x,r) centered at x of side length 2r.We impose a constantfield on the sample ofmicrostructure insideand see how it responds

x

r

Q(x,r)

Robert Lipton

Multi-scale quantities for local field assessment

Imposed constant field ei on Q(x,r) Here e1=(1,0,0); e2=(0,1,0); e3=(0,0,1)

Local periodic fluctuation wi,r,ε(x,y) solvesdiv(Sε(y)(∇wi,r,ε(x,y)+ei ))=0, y in Q(x,r).

Here x appears as parameter and y is the independent variable denoting a point insideQ(x,r).

Now introduce the two scale quantity

)),((

,, |)()),((|),(rxQL

Mx

jrjii xexwrxF

j ∞∂+⋅∇= ϕχ εεε

Robert Lipton

Bounds on regions of high local stress I

0|),()(|suplim 0 =≥∇→CM

ii txftS ϕεε

)),( txf Mi ≥∇ϕ

txxf Mi ≥∇ ))(,( ϕ

L. (2004) Journal Mech. Phys. Solids. 52 pp. 1053-1069(2003) Phil. Trans. R. Soc. Lond. A 361 pp. 921-946.

Set ),(suplim))(,( 00 rxFxxf irM

iε

εϕ →→=∇

Theorem. Characterization of Siε for 0<ε<<1

Robert Lipton

Step two

In order to proceed we need to be able to characterize Si

ε for 0<ε<<1in terms of the statistics of the microgeometry.

We show how to get rigorous lower bounds on the set )),( txf M

i ≥∇ϕ

Robert Lipton

Partial statistical descriptions of the microstructure

dyyrxQ

xrxQ

r )(|),(|

1suplim)(),(

10,01 ∫→→= εε χϑ

Volume fractions of each phase

x

r

Q(x,r)dyyrxQ

xrxQ

r )(|),(|

1suplim)(),(

20,02 ∫→→= εε χϑ

Robert Lipton

Two point correlation function T

Two point correlation function in Q(x,r)

Probability that both ends of a segment with length and orientation v lies in yellow phase when thrown into composite randomly. Can be obtained using image processing methods (J.G. Berryman, J. Computational Physics (1988)).

v

∫ +=),(

11,

1 )()(|),(|

1)(rxQ

r dyvyyrxQ

vT εεε χχ

Robert Lipton

Tensor of geometric parameters

)||

)(,1(suplim)( 20

^

210,0 k

kkkrxMk

r T⊗

= ∑≠

→→ε

ϑϑε

J.R. Willis 1977, Tartar (H-Measures) 1990’s,

We make use of the invariants of M. The eigenvalues of M are denoted by:

10 321 ≤≤≤≤ ddd

Robert Lipton

Lower bounds

=∇ ))(),(( 11 xxL Mϕϑ0)(|,)(|

)))((1)(()(

121132

2 >∇

−−+

xxSSxxdS

S M ϑϕϑ

0)(,0 1 =xϑ

0)(|,)(|))(()(

)(2

12111

1 >∇

−+

xxSSxxdS

S M ϑϕϑ

0)(,0 2 =xϑ=∇ ))(),(( 22 xxL Mϕϑ

L. (2004) Journal of Applied Physics 96, 2821-2827

≥∇ ))(,(1 xxf Mϕ

))(),(())(,( 222 xxLxxf MM ϕϑϕ ∇≥∇

))(),(( 11 xxL Mϕϑ ∇

Robert Lipton

Worst case scenario: Break the ``best’’ design built

from the ensemble of configurations

),(),( txftL Mi

Mii ≥∇⊆≥∇ ϕϕϑ

The lower bounds give

Robert Lipton

),( tL Mii ≥∇ϕϑ ),( txf M

i ≥∇ϕ

Structural Domain

Optimality of Lower Bounds – Minimum Stress Microstructure

The lower bound on f1(x,∇φM(x)) is attained by Q(x,r) filled with the coated ellipsoid assemblage with core of material one and coating of material two with major axis aligned with ∇φM(x).

∇φM

(Lipton J.Appl.Phys.2004(96) 2821-2827)

Robert Lipton

Optimality of Lower Bounds II – Minimum Stress Microstructure

The lower bound on f2(x,∇φM(x)) is attained by the coated ellipsoid assemblage with core of material two and coating of material one with minor axis aligned with ∇φM.

∇φM

(Lipton J.Appl.Phys.2004(96) 2821-2827)

Robert Lipton

Mapping from volume fraction information to local stress information I

),(),( txftL Mi

Mii ≥∇⊆≥∇ ϕϕϑ

Suppose we know the local area fractions θi(x). Use local effective compliance SE(x) associated with minimum stress microstructure then on solving the homogenized problem

We get ∇φM and the lower bounds give

fSdiv ME =∇− )( ϕ

Robert Lipton

An example-2d torsion

)(1 xϑ

+++−++

=)()()(

21221

122211 SSSS

SSSSSxS E

ϑϑ

2/121 ==dd

Data: We are given local area fractions of each material

and The microstructure is statistically isotropic

hence

)(2 xϑ

1)( =∇− MESdiv ϕ

The minimum stress microstructure Has local effective property SE given by

For torsion φM=0 on the boundary and

And we can compute the level sets of ))(),(( 11 xxL Mϕϑ ∇

Robert Lipton

Location of high stress zones

))(),(( 11 xxL Mϕϑ ∇Plot of area fraction θ1 Level lines of

S

),(),( txftL Mi

Mii ≥∇⊆≥∇ ϕϕϑ

Robert Lipton

Industrial work: Aircraft Industry Boeing

Strain Invariant Failure Criteria-with micromechanical enhancement.

J. H. Gosse and S. Christensen, Strain invariant failure criteria for polymers in composite materials, AIAA, 1184 (2001), p. 11.

Robert Lipton

Failure in epoxy resin

Design of graded composite structures in an uncertain environment assuming scale

separationGiven a local volume fraction use minimum stress microstructuresSuppose we know the stiff material fails under large shear stressFor a prescribed macro shear load choose a micro-geometry with minimum shear stress in the stiff material. Recall we assume scale separation so the homogenized stress is constant over the length scale of the microstructure.For this case the minimum stress microstructure isgiven by a laminate (B. Alali & L. 2009).

Robert Lipton

Minimum Stress Microstructures for Pure Shear Stress

)( abbasD

⊗+⊗=σ

Let material one have larger shear modulus and consider the pure shear stress

In both 2 and 3-d the bound is attained by stress fieldsInside a simple laminate with layering direction n=a or

n=b.

|||)(|DD x σσ ≥Then in material 1

n=aB. Alali and R.L. SIAM J. Applied Math 2009

Robert Lipton

Design of graded composite structures –using minimum stress micro geometries

Coarse graining and multi-scalinggive good parameterizations for design of graded composite structures.(Joint work with M. Stuebner NCSU)

Example: The design a locally layered structure first for overall stiffness, then for strength.

Robert Lipton

Library of minimum stress microstructures

Locally layered materials made from twoelastic components: E1=240, E2=30; nu1=nu2=1/3.

Robert Lipton

Use coarse graining and multi-scaling to parameterize the library

Effective elastic property a function AE(θ1,n)of local proportion θ1(x) of material one and layerdirection n(x).

Local stress amplification in material one M1(θ1 ,n)is a function of θ1 and local layer direction n(x)

Design variable θ1(x) can be thought of as phase field.

Here we constrain θ1(x) to take values between0.1 and 0.9 and the total amount of Material 1 in design domain = 30%

Robert Lipton

Inverse homogenization theorem

There is a locally layered microstructure with elastic tensor Aε and stress σε such that:

Aε → AE(θ1,n) in homogenization and σε→σH weakly

Denote the set of points in material 1 such that |σε|2 >t by Sε(t)

Denote the set of points such that M1(θ1 ,n) σH · σH >t by SH(t)

Then measure(Sε(t)⋂ CSH(t))→0 as ε→0.

LS 2005

Robert Lipton

Design problemsFind the best distribution θ1(x) and n(x)inside design domain to: 1)Mininize work done against the load. 2) Minimize the size of the sets SH(t) where t is greater than an acceptable value. We enforce a resource constraint on ʃ θ1(x) dx.

?? ?

Robert Lipton

The objective functions

For optimizing compliance the objectivefunction is

To minimize SH(t) the objective function is

dxug H∫ •

dxnM HH σσθ •∫ ),( 11

Robert Lipton

Local StressSH(t)Compliance

Robert Lipton

Convergence history of objective function for

minimizing local stress

Robert Lipton

Comparison of level setsSH(t)

Optimized to minimize compliance

Optimized to minimize

dxnM HH σσθ •∫ ),( 11

Robert Lipton

Dimension reduction: no scale separation-forward problem

(Collaboration with I Babuska)

Motivates use of a Multiscale GFEM in Global-local analysis.Best 2 scale basis for representing macro and micro length scales when they are not separated``Best’’ = minimum approximation error for a fixed number of degrees of freedom

Robert Lipton

Boundary value problem for two-phase materials

Solve: div (A(x) grad u)=0, with Neumann BC: n·(A grad u)=g,

and A(x) given by:

Robert Lipton

Discussion in the context of the Partition of Unity (PUM) and Generalized FEM methods

(a partial list)Babuska Caloz Osborn (1994)Babuska and Melenek (1997),

Babuska Osborn (2000)Stroboulis, Zhang, & Babuska (2003), Babuska, Stroboulis, & Copps (2000)

Robert Lipton

Partition of unity method: set up

Given a partition of unity φi with support sets ωiΣφi=1 on Ω

Introduce local approximation space for each ωi

Vip (ωi)=w in H1(ωi): w in span (vi1,vi2,…,vip)

Xi

ωi

Robert Lipton

GFEM-- A Galerkin methodTrial and test spaces for GFEM constructed from local approximation spaces: GFEM trial and test space S(Ω)⊂Η1(Ω)

The GFEM approximation is given by the minimizer of

Vip (ωi)

S(Ω)=ϕ: ϕ=Σi φi wi ; with wi taken from Vi

p (ωi)

)(;min Ω∈+∇•∇ ∫∫Ω∂Ω

SdsgdxA ϕϕϕϕ

Robert Lipton

Motivation: Local Error Estimates for PUM Method and GFEM for Multiscale problems

An implementation of the PUM method in 2d. Take uniform mesh with vertices Xi, i=1,…N

Xi

ωiIdea 1. Use local information

on the solution operatordiv(A(x) grad v)=0 on ωi

Ideas of Hou (1997)E., Engquist (2007)

Babuska, Caloz, Osborn(1994)

Local Enhancement on ωi:Vi

p=spanwi in H1(ωi): div(A(x)grad wi )=0For partition of unity φi subordinate to ωi,

The PUM approximation is given by uApp=ΣiNφiwi, wi in Vi

p, i=1,…,N.

Robert Lipton

Multiscale GFEM Motivation: Dimension Reduction

GFEM trial and test space S(Ω)⊂Η1(Ω)

Reduces the size of the stiffness matrixby an order of magnitude.

S(Ω)=ϕ: ϕ=Σi φi wi ; with wi taken from Vi

p (ωi)

Robert Lipton

Local approximation controls global error estimate

12 ||||),(|||| 1)( HLi upiwui

εω

≤−∩Ω

12 ||||),(|||| 2)( HLi upiwui

εω

≤∇−∇∩Ω

12 ||||)),((max|||| 1,..1)( HNiLApp upiCuu ε=Ω

∗≤−

Application of the triangle inequality shows thatif have ε1 and ε2 such that for each ωi

there is a wi in Vip

such that:

12 ||||),()(

),(|||| 221

1,...,1

)( max HiNi

LApp upiC

diampiCCuu

+

∩Ω∗≤∇−∇

=Ω

εω

ε

and

Then there is a uApp satisfying:

Robert Lipton

Framework

2/1)( )(|||| dxA

SSE ∫ ∇•∇= ϕϕϕ

We discuss the optimal local basis to discover nearlyexponential convergence for ε2(i,p) that holds for

all L∞ coefficients: A(x) with aI<A(x)<bI

dxAS

SE ∫ ∇•∇= δϕδϕ )(),(

Robert Lipton

Motivation: More on choice of local basis functions in the PUM Method and GFEM

Implementation of the PUM method in 2d. Take uniform mesh with vertices Xi, i=1,…N

Xi

Ωi

ωi

Use this extra information to conduct a betterlocal approximating basis Vi

p over ωi

For partition of unity φi subordinate to ωi,The PUM approximation is given by uApp=Σi

Nφiwi, wi in Vip, i=1,…,N.

Idea 2. Use fact that theactual solution ``u’’ solvesdiv(A(x) grad u)=0 on a slightly larger domain Ωi

that contains ωi

Examples: Oversampling: Hou, Wu, Zhang (2004)

Buffers:Strouboulis, Zhang Babuska

(2004).

Robert Lipton

On the optimal local basis Notion of Kolmogorov n-width

Set of all non constant φ in H1(Ω) that solve: div(A gradφ)=0 on Ω is denoted by HA(Ω)/R.

P: HA(Ω)/R-->HA(ω) is the restriction of φ to ω.

ω

∂Ω

1

1/2

−

=ΩΩ∈Ω∈ )(

)(

)( ||||

||)(||];[ infsup

E

E

OwHp

wPd

pA

ϕ

ϕω ω

ϕ

−

=ΩΩ∈Ω∈⊂ )(

)(

)()( ||||

||)(||];[ infsupinf

E

E

WwHHWp

wPd

pAA

p ϕ

ϕω ω

ϕω

From the Caccioppoli inequality P is compact operator, i.e.,energy bounded sequences in HA(Ω)/R are precompact in HA(ω)

with respect to the H1(ω) norm. The optimal approx space Op satisfies

Robert Lipton

Optimal local approximation OP

iiiPP ϕλϕ =*

ii Pϕψ =,,, 21 p

p spanO ψψψ =

Op follows from general considerationscontrolled by the singular values of P.

Introduce the adjoint P*: HA(ω)HA(Ω)/R

1];[ +=Ω ppd λω

)()( ),(),( Ω= EiiEi δϕλδϕ ω For all δ in HA(Ω)/R

Robert Lipton

Nearly exponential decay

For ε>0 there is an Nε>0 such that for all p>Nε

−≤Ω

−

+ε

ω 11

exp];[ dp pd

Theorem 1.

I. Babuska & R. Lipton 2010

Robert Lipton

B. Alali & R. Lipton, Optimal lower bounds on local stress inside random media. SIAM J. On Applied Mathematics 70 (2009) 1260-1282.

I. Babuska, I. & R. Lipton, Optimal local approximation spaces for generalized finite element methods with application to multiscaleproblems. arXiv:1004.3041 [math.NA] 18 Apr. 2010. Submitted to SIAM Multiscale Modeling and Simulation.

I. Babuska, R. Lipton, & M. Stuebner, The penetration function and its application to microscale problems. BIT Numerical Mathematics. 48 (2008) 167-187.

T. Breitzman, R. Lipton, & E. Iarve, Local field assessment inside multiscale composite architectures. SIAM Multiscale Modeling and Simulation, 2007.

Y. Chen & R. Lipton, Optimal lower bounds on the local stress inside random thermoelastic composites. Acta Mechanica. 213 (2010) 97-109.

S. Jimenez & R. Lipton, Correctors and field fluctuations for the pε(x)-Laplacian with rough exponents. Journal of Mathematical Analysis and Applications. 372 (2010) 448-469.

R. Lipton & M. Stuebner, Optimization of composite structures subject to local stress constraints. Computer Methods in Applied Mechanics and Engineering 196 (2006) 66 -75.

R. Lipton & M. Stuebner. A new method for design of composite structures for strength and stiffness. American Institute of Aeronautics and Astronautics Paper AIAA 2008-5986.

R. Lipton, Homogenization and field concentrations in heterogeneous media.SIAM J. Math. Analysis 38 (2006) 1048 -1059.

R. Lipton. Assessment of the local stress state through macroscopic variables. Philosophical Transactions of the Royal Society of London A. 361 (2003) 921 - 946.

R. Lipton. Bounds on the distribution of extreme values for the stress in composite materials. Journal of the Mechanics and Physics of Solids. 52 (2004) 1053 – 1069.

R. Lipton. Homogenization theory and the assessment of extreme field values in composites with random microstructure. SIAM J. Applied Math. 65 (2004) 475 – 493.

Related PublicationsRobert Lipton