Multi-scale modelling for structures and composites

409 Pages·2005·10.309 MB·English

by  G P Panasenko

Multi-scale Modelling for Structures
and Composites
by
G. PANASENKO
AC.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 1-4020-2981-0 (HB)
ISBN 1-4020-2982-9 (e-book)
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Contents
Preface ix
Notations xii
Chapter 1. Introduction: Basic Notions and Methods 1
1.1. What is an inhomogeneous rod? 1
1.2. What are effective coefficients? 3
1.3. A scheme for calculating effective coefficients 5
1.4. Microscopic structure of a field 9
1.5. What is the homogenization method? 9
1.6. What is a finite rod structure? 10
1.7. What is a lattice structure? 13
1.8. Advantages and disadvantages of the asymptotic approach 16
1.9. Appendices 17
1.A1. Appendix 1: What is the
Poincar´e-Friedrichs-Korn inequalities? 17
Chapter 2. Heterogeneous Rod 21
2.1. Homogenization (N.Bakhvalov’s ansatz and the boundary
layer technique) 22
2.1.1. Bakhvalov’s ansatz 22
2.1.2. An example of formal asymptotic solution 23
2.1.3. The boundary conditions corrector 26
2.1.4. Introduction to the boundary layer technique 27
2.1.5. Homogenization in IRs 29
2.1.6. Boundary layer correctors to homogenization
in IRs 32
2.2. Steady-state conductivity of a rod 36
2.2.1. Statement of the problem 36
2.2.2. Inner expansion 38
2.2.3. Boundary layer corrector 43
2.2.4. The justification of the asymptotic expansion 48
2.3. Steady state elasticity equation in a rod 56
2.3.1. Formulation of the problem 57
2.3.2. Inner expansion 59
2.3.3. Boundary layer corrector 65
2.3.4. The boundary layer corrector when
the left end of the bar is free 69
2.3.5. The boundary layer corrector for
the two bar contact problem 73
2.3.6. Homogenized problem of zero order 79
2.3.7. The justification of the asymptotic expansion 82
2.4. Non steady-state conductivity of a rod 98
2.4.1. Statement of the problem 98
2.4.2. Inner expansion 98
VI
2.4.3 Boundary layer corrector 100
2.4.4. Justification 100
2.5. Non steady-state elasticity of a rod 103
2.5.1. Statement of the problem 103
2.5.2. Inner expansion 104
2.5.3. Boundary layer corrector 106
2.5.4. Justification 106
2.6. Contrasting coefficients (Multi-component homogenization) 110
2.7. EFMODUL: a code for cell problems 120
2.8. Bibliographical Remark 126
Chapter 3. Heterogeneous Plate 129
3.1. Conductivity of a plate 130
3.1.1. Statement of the problem 130
3.1.2. Inner expansion 131
3.1.3. Boundary layer corrector 134
3.1.4. Algorithm for calculating the
effective conductivity of a plate 136
3.1.5. Justification of the asymptotic expansion 138
3.2. Elasticity of a plate 146
3.2.1. Statement of the problem 146
3.2.2. Inner expansion 146
3.2.3. Boundary layer corrector 149
3.2.4. Proof of Theorem 3.2.1. 151
3.2.5. Algorithm for calculating
the effective stiffness of a plate 154
3.3. Equivalent homogeneous plate problem 156
3.4. Time dependent elasticity problem for a plate 158
3.5. Bibliographical Remark 160
Chapter 4. Finite Rod Structures 161
4.1. Definitions. L-convergence 161
4.1.1. Finite rod structure 162
4.1.2. L-convergence method for a finite rod structure 165
4.2. Shape optimization of a finite rod structure 173
4.2.1. Stored energy as the cost 174
4.2.2. Simplification of the set of
finite rod structures. Initial configuration 176
4.2.3. An iterative algorithm for the optimal
design problem 177
4.2.4. Some results of numerical experiment 180
4.3. Conductivity: an asymptotic expansion 183
4.3.1. Construction of asymptotic expansion 183
4.3.2. The leading term of the asymptotic expansion 192
4.4. Elasticity: an asymptotic expansion 194
4.4.1. Construction of asymptotic expansion 194
4.4.2. The leading term of asymptotic expansion 212
4.5. Flows in tube structures 215
VII
4.5.1. Definitions. One bundle structure 215
4.5.2. Tube structure with m bundles of tubes 227
4.6. Bibliographical Remark 230
4.7. Appendices 230
4.A1. Appendix 1: estimates for traces in the
pre-nodal domain 230
4.A2. Appendix 2: the Poincar´e
and the Friedrichs inequalities for a
finite rod structure 233
4.A3. Appendix 3: the Korn inequality for the
finite rod structures 242
Chapter 5. Lattice Structures 247
5.1. Definition of lattice structure 247
5.2. L-convergence homogenization of lattices 249
5.2.1. L-convergence for the simplest lattice 249
5.2.2. Some auxiliary inequalities 251
5.2.3. FL-convergence. Relation to the L-convergence 254
5.2.4. Proof of Theorem 5.2.1 255
5.3. Non-stationary problems 258
5.3.1. Rectangular lattice 258
5.3.2. Lattices: general case 259
5.3.3. Proof of Theorem 5.3.2 262
5.4. L- and FL-Convergence in elasticity 269
5.5. Conductivity of a net 270
5.6. Elasticity of a net 273
5.7. Conductivity of a lattice: an expansion 278
5.8. High order homogenization of elastic lattices 291
5.9. Random coefficients on a lattice 307
5.9.1. The Simplest Lattice. The main result 307
5.9.2. Proof of theorem 5.9.1 309
5.10. Bibliographical Remark 314
5.11. Appendices 318
5.A1. Appendix 1: the Poincar´e
and the Friedrichs inequalities for
lattices 318
5.A2. Appendix 2: the Korn inequality for
lattices 321
Chapter 6. The Multi-Scale Domain Decomposition 337
6.1. Differential version 341
6.1.1. General description of the differential version 341
6.1.2. Model example 343
6.1.3. Poisson equation in a rod structure 349
6.2. Variational version 354
6.2.1. General description of the variational version 354
6.2.2. Model example 357
6.2.3. Elasticity equations 358
VIII
6.3. Decomposition of a flow in a tube structure 364
6.4. The partial homogenization 373
6.5. Bibliographical Remark 382
Bibliography 385
Subject index 397
PREFACE.
Rod structures are widely used in modern engineering. These are bars,
beams, frames and trusses of structures, gridwork, network, framework and
otherconstructions. AvarietyoftheoriesbasedontheKirchhoff-Love,Kirchhoff-
Clebschandotherhypothesesareappliedfortheiranalysis. Structuralmechan-
icssoftwarebasedonmaterialstrengththeorymethodsalsoexists. Atthesame
time the questions concerning the limits of applicability of these hypotheses
and theories and the possibilities of their refinement are very important. In
this connection we develop the multi-scale asymptotic analysis of equations of
mathematical physics, and in particular the elasticity equations of set in the
rod structures (without these hypotheses and simplifying assumptions being
imposed) . Problems with one small parameter (the ratio of bar diameter to
its length) as well as problems with two and more small parameters (periodic
frameworksystems,wherethesecondparameterrepresentstheratioofaperiod
tothecharacteristicspacedimensionoftheproblem,weaklycompressiblebars,
etc.) are studied.
The homogenization technique for partial differential equations described
in the book by N.Bakhvalov and G.Panasenko [16] and the boundary layer
techniquesareusedasamaintoolintheseinvestigations. Thephysicalprocesses
are simulated by partial differential equations set in ”thin” domains containing
asmallparameter. Theasymptoticanalysisisappliedforinvestigationofthese
partial differential equations.
The multi-scale models are developed according to two main schemes:
- the up-scaling procedure of asymptotic derivation of macroscopic models
from the microscopic ones (ie., the homogenization approach) and
- the hybrid multi-scale models, combining two scales inside one model,
making a microscopic zoom inside the macroscopic model (ie., the asymptotic
partial domain decomposition, partial homogenization).
The present monograph consists of six chapters. The first chapter is intro-
ductory. It presents the main notions and methods of the book, advantages
and disadvantages of these methods. In the second chapter we consider the
three-dimensional conductivity problem as well as theory for elasticity prob-
lems and other equations of mathematical physics in a thin cylindrical domain
(bar)withnon-homogeneousstructure. Fullasymptoticexpansionsofsolutions
are constructed for the small parameter equal to the ratio of the bar diameter
to the length; boundary layers are investigated and one-dimensional equations
for bars are derived. Then we consider the problem of junction of two hetero-
geneous rods. The connection conditions for rods result from the analysis of
boundary layers arising in the neighborhood of the bounds of the rods. Time
dependent models are considered.
Thethirdchapterisdevotedtothesimilaranalysisofaheterogeneouslayer
(plate). Full asymptotic expansions of solutions are constructed for the small
parameter equal to the ratio of the plate thickness to the length of the plate;
boundary layers are studied and two-dimensional equations for plates are de-
IX
X
rived.
Similar questions for systems with finite number of bars are studied in the
Chapter 4. The asymptotic analysis of a structure with finite number of bars
is developed first for nonlinear equations in some weak norms (L-convergence
method),andthenthedetailedasymptoticanalysisisdone,theasymptoticex-
pansionsofsolutionsareconstructed. TheKorninequalityor,morespecifically,
the investigation of how the constant depends on the small parameters in this
inequalityisessentialforthejustification. Finallytheresultsobtainedbymeans
of L-convergence method are applied to shape design of finite rod structures.
ThecodeOPTIFORimplementsthisalgorithm. Wediscusssomemodelexam-
ples. Thesimilaranalysisisdeveloped forsomeproblemsofflowinasystemof
tubes described by Stokes and Navier-Stokes equations.
In Chapter 5, periodic framework structures (lattice-like domains) are con-
sideredandtheirhomogenizationiscarriedout,i.e.,systemswithgreatnumber
of bars are investigated. The question of the existence of homogenized models
andtheconvergenceoftheexactsolutionstothemisstudied. Insomecasesthe
firstapproximationoftheasymptotictheorymaybealsoobtainedbymeansof
classical structural mechanics. At the same time, the asymptotic theory gives
the opportunity to obtain corrections to the structural mechanics models. It is
possible to obtain analytical formulas for these corrections in some examples.
Chapter 6 is devoted to a new multi-scale method of solution of different
problemswithsmallparameter. Itisthemethodofasymptoticdomaindecom-
position. The direct numerical solution of partial derivative equations in finite
rod structures is very expensive because the complicated geometry demands a
large number of nodes in the grid. The complete asymptotic expansions are of-
ten cumbersome. So we propose a hybrid numerical-asymptotic method which
uses a combined 3D-1D models: it is three-dimensional in the boundary layer
domain and it is one-dimensional outside of the boundary layer domain. We
cuttherodsatsomedistancefromtheendsoftherods, wekeepthedimension
three in the neighborhood of the ends and we reduce dimension on the trun-
cated (main) part of rods. So the principal idea of the method is to extract
the subdomain of singular behavior of the solution and to reduce dimension of
the problem in the subdomain of regular behavior of the solution. Of course
the most important question is: what are the interface conditions between 3D
and 1D parts? We formulate two approaches of construction of such hybrid
models and justify the closeness of the partially decomposed model and initial
model. Weanalyzesuchhybridmodelsforconductivityandelasticityequations
statedinrodstructuresaswellasStokesandNavier-Stokesequationsstatedin
a system of thin tubes.
Weconsiderbelowtwoversionsofthemethodofasymptoticpartialdecom-
position of domain. The first version is ”differential”, i.e. we work with the
differential formulation of the initial problem , we obtain the 1D differential
equation in the reduced part of the rod structure and we add the differential
interface conditions on the boundary between 3D and 1D parts . Of course we
can pass to a variational formulation of the partially decomposed problem but
it is generated by a differential one.
XI
The second version is a direct variational approach, when the 3D integral
identity for the original problem is restated for a special subspace of functions
having a form of the ansatz of the asymptotic solution in the regular thin part
of the rod structure. We give some examples of application of this method.
As a rule the results are presented in the form of theorems. The obtained
asymptotic approximations are justified: estimates of their closeness to exact
solutions are proved.
The book is destined to graduate and postgraduate students, specialists in
applied mathematics, specialists in mechanics, engineers, university professors.
ItispartiallybasedoncoursesoflecturesdeliveredbytheauthorattheDepart-
ment of Mechanics and Mathematics at Moscow State University Lomonossov
and at the Mathematical Department of the University Jean Monnet (Saint
Etienne, France). Knowledge acquired after the first three years of advanced
mathematicaltrainingatacollegeoruniversityissufficienttoreadthemostof
thebook. Presentationofthematerialisbasedontheprinciple”fromsimpleto
complicated,”everychapterbeginningwithanelementaryexampletoillustrate
the main idea of the method to be described.
XII
Notations
ε,µ,ω 1 -small parameters
−
G - a domain in IIIRs, i.e., an open connected set (usually bounded) in the
s dimensional space where s is equal to 2 or 3
−
∂G -boundary of the domain G
G¯ -closure of the domain G, i.e., G¯ =G ∂G
∪
B -finite rod structure i.e., a connected union of a finite number of thin
µ
cylinders (the diameter of the base is of order of µ and the height is of order of
1); e is a segment inside of a cylinder constituting B , concluded between the
µ
bases
B -latticestructurei.e.,aconnectedε periodicunionofaninfinitenum-
ε,µ
−
berofthincylinders(thediameterofthebaseisoforderofproductεµandthe
height is of order of ε)
x=(x ,...,x ) -point in IIIRs, slow (macroscopic) variable
1 s
ξ =(ξ ,...,ξ ) -fast (microscopic) variable, ξ =x/ε or ξ =x/µ
1 s
z = (z ,...,z ) -fast (microscopic) variable in a boundary layer, z = x/µ or
1 s
z =x/(εµ)
L2(G)(thesamethatL (G))spaceoffunctionswithboundednorm u =
2 (cid:2) (cid:2)L2(G)
u(x)dx
G
(cid:1)(cid:2)(cid:2)(cid:2)CC(nG(¯G¯))-s-pspacaeceofoffufunnctcitoinonssconnttiinmueosudsiffinerG¯en,t(cid:2)iaub(cid:2)lCe(Gi¯n)G=¯ supx∈G¯|u(x)|
H1(G), HK(G), HHH1(G) -are the classical Sobolev spaces
0
u -trace on ∂G
∂G
|
[u] -means the difference of the limit values of function u on the two sides
Σ
|
of surface Σ
f(x ,...,x ,ξ ,...,ξ ) is a mean over the period, i.e.,
1 s 1 s
(cid:3) (cid:4)
1 1
... f(x ,...,x ,ξ ,...,ξ )dξ ...dξ
1 s 1 s 1 s
(cid:3)(cid:3)(cid:3)0 (cid:3)(cid:3)(cid:3)0
(a,b) -inner product of the vectors a and b
div, div , div -divergence with respect to the variables x or ξ
x ξ
grad, grad , grad -gradient with respect to the variables x or ξ
x ξ
, , -the same
x ξ
∇ ∇ ∇
rot, rot , rot -curl with respect to the variables x or ξ
x ξ
∆, ∆ , ∆ -Laplacian with respect to the variables x or ξ
x ξ
˜(ξ) - matrix of rigid displacements (translations and rotations)
I
i=(i ,...,i) is a multi-index, i = l is its length, i 1,...,s
1 l j
| | ∈{ }
Di = ∂l (partial derivative)
∂xi1...∂xil
f(x,ε) ∼ ∞j=0εjgj(x,ε) is an asymptotic expansion (a.e.) of f(x,ε), i.e.,
for any real N there exists M such that, for all m M the relation holds:
(cid:4) ≥