Variational problems in multi-domains. Modeling of junctions and applications. (Problèmes variationnels dans les multi-domaines. Modélisation des jonctions et applications.) (French) Zbl 0744.73027

Le Dret, Hervé

Recheres en Mathématiques Appliquées 19. Paris etc.: MASSON (ISBN 2-225-82543-2). x, 198 p. (1991).

This book is a successful attempt to explain an important new mathematical method to applied mathematicians and engineers interested in research and recent developments in the modelling of multi-structures. The multi-structures can comprise substructures and different dimensions of their components (three dimensional structures, plates, beams). The coupled and multidimensional boundary value problems of heat conduction and elasticity are considered. Asymptotic expansion methods and variational formulations are used to derive the essential equations.
Chapter 1 deals with the fundamental results of functional analysis (distributions, Sobolev spaces, variational problems). These mathematical backgrounds have been mainly used in the next chapters. In Chapter 2 junctions between slender structures in the statical case are treated in connection with the Laplace operator. The heat equation in \(L\)-shaped domains and three dimensional – two-dimensional domains in the dynamical case and the wave equation in \(L\)-shaped domains are considered in Chapter 3. The Chapter 4 deals with the junctions of elastic plates in detailed form. The statical and dynamical problems of multi-plate structures are studied by asymptotic analysis. The multi-rod structures and some generalizations are presented in the Chapter 5.
The main results of this book are as follows: (a) a general technique for modelling of junction; (b) the reducing of three (two)-dimensional problems to two (one)-dimensional problems.

Reviewer: István Ecsedi (Miskolc-Egyetemváros)

MSC:

74K20	Plates
49-02	Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
74-02	Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids
74S30	Other numerical methods in solid mechanics (MSC2010)
74P10	Optimization of other properties in solid mechanics
74E30	Composite and mixture properties
35C20	Asymptotic expansions of solutions to PDEs
74K10	Rods (beams, columns, shafts, arches, rings, etc.)

Keywords:

modelling of multi-structures; multi-dimensional boundary value problems; heat conduction; elasticity; slender structures; statical case; \(L\)- shaped domains; three dimensional; two dimensional domains; dynamical case; wave equation; junctions of elastic plates; multi-plate structure; multi–rod structures