Asymptotic Analysis of Fields in Multi-Structures

, by Kozlov, Vladimir; Maz'ya, Vladimir; Movchan, Alexander

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ISBN: 9780198514954 | 0198514956
Cover: Hardcover
Copyright: 10/28/1999

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The asymptotic analysis of boundary value problems in parameter-dependent domains is a rapidly developing field of research in the theory of partial differential equations, with important applications in electrostatics, elasticity, hydrodynamics and fracture mechanics. Building on the work of Ciarlet and Destuynder, this book provides a systematic coverage of these methods in multi-structures, i.e. domains which are dependent on a small parameter e in such a way that the limit region consists of subsets of different space dimensions. An undergraduate knowledge of partial differential equations and functional analysis is assumed.

List of symbols

xiv

Introduction to compound asymptotic expansions

(55)

Elementary examples of perturbation problems for ordinary differential equations

(4)

A one-dimensional singularly perturbed problem

(5)

Neumann boundary value problem in a domain with small cavity

(12)

Formulation of the problem

(2)

The leading order approximation

(1)

Remainder estimate

(3)

Complete asymptotic expansion

(4)

Asymptotic formula for the energy

(2)

Dirichlet boundary value problem in a domain with small inclusion

(8)

The leading order approximation

(2)

The next approximation

(1)

The complete asymptotic expansion

(5)

Mixed boundary value problem for the Laplacian in a thin rectangle

(8)

Formulation of the boundary value problem

(1)

Two-term approximation

(3)

The next approximation

(2)

Higher-order approximation

(2)

Problem of junction between thin bodies

(18)

Model problems

(7)

The leading order approximation

(2)

The next-order approximation

(2)

The complete asymptotic expansion

(2)

The remainder estimate

(3)

A boundary value problem for the Laplacian in a multistructure

(59)

Formulation of the problem

(1)

Model problems

(18)

Limit domains

(1)

Model problem in Ω

(3)

Model problem for the junction region

(7)

Junction layer

(1)

Model problem for the bottom region

(4)

Two model problems for a thin cylinder

(1)

Algebraic system for the skeleton

(1)

Right-hand sides

(4)

Local coordinates and limit domains

(1)

Cut-off functions

(1)

Asymptotic representations of the right-hand sides

(2)

The leading term of the asymptotic solution

(8)

Domain Ω

(2)

Junction layer

(2)

Thin cylinder

(2)

Bottom layer

(1)

Evaluation of W0(j)

(1)

Evaluation of the constants T0(j) and C0

(1)

Concluding remarks on formal algorithm

(1)

Complete asymptotic expansion

(3)

Structure of the asymptotic expansion

(2)

The asymptotic algorithm

(1)

Justification of the asymptotic expansion

(6)

Auxiliary estimates for functions in H1(ΩE)

(2)

Estimate for solutions

(3)

Estimate for the remainder term

(1)

A constant right-hand side

(2)

Application to the asymptotics of the energy integral

100

(5)

The case of the right-hand sides concentrated in Ω

101

(2)

The case of the Dirichlet data at the bases of thin cylinders

103

(2)

On a general 1D--3D multi-structure

105

(3)

A multi-structure with a thin-walled tube

108

(7)

Auxiliary facts from mathematical elasticity

115

(40)

Basic formulae of linear elasticity

115

(3)

Stress and strain

115

(1)

Equations of equilibrium and boundary conditions

116

(2)

Two-dimensional problems of linear elasticity

118

(2)

Plane strain

118

(1)

Anti-plane shear

119

(1)

Differential equations for engineering models of elastic rods

120

(1)

Classical solutions of linear elasticity for a half-space

121

(5)

Boussinesq--Cerruti's solution

121

(2)

Mindlin's solution

123

(1)

Connection between the Boussinesq--Cerruti and Mindlin solutions

124

(2)

Special solutions for a bounded two-dimensional domain

126

(3)

The torsion potential

127

(1)

The bending potentials

128

(1)

Example

128

(1)

Special solutions of linear elasticity for an infinite cylinder

129

(12)

Representation of differential operators

129

(1)

The spectral problem

130

(2)

X3-polynomial solutions

132

(1)

Biorthogonality conditions

133

(2)

The normalised stiffness coefficients

135

(1)

Biorthogonality relations for eigenvectors and generalised eigenvectors

136

(2)

There are no other polynomials

138

(3)

Green's matrix in Ω

141

(2)

Definition

141

(1)

Asymptotics

141

(2)

Korn's inequalities

143

(8)

The case of bounded Lipschitz domains

143

(3)

Half-space and a cylinder

146

(5)

Asymptotics at infinity for solutions to the traction problem for a half-cylinder

151

(4)

Elastic multi-structure

155

(58)

Multi-structure and boundary value problem

156

(2)

Model problems

158

(22)

Limit domains

158

(1)

Model problem for the body Ω

158

(2)

Junction layer

160

(7)

Model problem for the bottom layer

167

(4)

Model problem for a bounded two-dimensional domain

171

(1)

Model problems on the axis of an elastic rod

172

(2)

Model matrices and the pile structure

174

(5)

Special cases

179

(1)

Asymptotic expansion of the solution

180

(18)

Asymptotic representation of the right-hand sides

180

(2)

Description of the asymptotic series for the solution

182

(2)

Auxiliary solutions of the Lame system in a thin elastic rod

184

(3)

Expansions for displacement in a thin rod

187

(2)

Junction layer

189

(1)

Displacement in Ω

190

(1)

Bottom layer

191

(2)

Functions uk(m,j)

193

(4)

The recurrent procedure for the asymptotic expansion

197

(1)

Justification of the asymptotic expansion

198

(5)

Korn's inequality in Ωε

198

(1)

An estimate for the solution

199

(4)

The leading order approximation

203

(5)

The term u(-1)

203

(1)

The term u(0)

204

(4)

Physical interpretation of the results

208

(5)

The case |M3| + |F1| + |F2| ≠ 0

208

(2)

The case F1 = F2 = M3 = 0

210

(3)

Non-degenerate elastic multi-structures

213

(35)

Pile structure model

214

(10)

Skeleton of the multi-structure

214

(2)

The pile structure

216

(1)

Mathematical model of the pile structure

216

(1)

Solution of the pile structure equations

217

(2)

Algebraic system for the pile structure model

219

(2)

Non-degenerate and degenerate pile structures

221

(1)

Examples

222

(2)

Multi-structure and the boundary value problem

224

(3)

Description of the multi-structure

224

(2)

Formulation of the boundary value problem

226

(1)

Model problems

227

(4)

Junction layer

227

(3)

Remaining model problems

230

(1)

Asymptotic expansion of the solution

231

(13)

Cut-off functions

231

(2)

Asymptotic representation of the right-hand sides for the case of a non-degenerate multi-structure

233

(1)

Structure of the asymptotic series for the displacement field in Ωε

233

(2)

The junction layer

235

(2)

Displacement in Ω

237

(1)

The bottom layer

238

(1)

Functions ui(m,j)

239

(2)

Evaluation of the lock forces and moments at junction points

241

(1)

Algebraic system for α(m), β(m)

242

(1)

The recurrent procedure for the asymptotic expansion

243

(1)

Estimate for the remainder of the asymptotic expansion

244

(1)

Analysis of the leading term

245

(1)

Physical interpretation

246

(2)

Spectral analysis for 3D--1D multi-structures

248

(26)

An abstract scheme for the asymptotics of eigenvalues

249

(4)

Spectral problem for the Laplacian

253

(4)

The first eigenvalue

253

(2)

The first eigenfunction

255

(2)

Asymptotics of first eigenvalues of the Lame operator

257

(10)

Spectral problem

258

(1)

Korn-type inequalities

258

(5)

Spaces X0 and H0

263

(1)

Asymptotic formula for the eigenvalues

263

(4)

Spectral problem for an inhomogeneous elastic multistructure

267

(7)

The spectral problem

267

(1)

Asymptotic formulae for the eigenvalues

268

(6)

Bibliographical remarks

274

(2)

Bibliography

276

(5)

Index

281

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Asymptotic Analysis of Fields in Multi-Structures

Summary

Table of Contents

Supplemental Materials