Optimization in Elliptic Problems With Applications to Mechanics of Deformable Bodies and Fluid Mechanics

by Litvinov, William G.

• ISBN13: 9783764361990
• ISBN10: 3764361999
• eBook ISBN(s): 9783034883870
• Format: Hardcover
• Copyright: 2000-06-01
• Publisher: Birkhauser
• More Book Details

Summary

This book is unique in that it presents a profound mathematical analysis of general optimization problems for elliptic systems, which are then applied to a great number of optimization problems in mechanics and technology. After the setting of a problem, attention is focused on existence theorems that lead to the construction of approximate solutions. The coefficients of the equations, the shape of the domain, and the right-hand sides of the equations are considered to be controls. Applications include optimization problems arising in mechanics of elastic solids, plates, shells, composite materials and structures fabricated with them, as well as fluid mechanics.The monograph is written in an accessible and self-contained manner. It will be of interest to research mathematicians and science engineers working in solid and fluid mechanics, and in optimization theory of partial differential equations. Moreover, it is suitable as a textbook for graduate courses in optimization of elliptic systems.

Table of Contents

Preface	xv
Introduction	xix
Basic Definitions and Auxiliary Statements
Sets, functions, real numbers	1	(3)
Notations and definitions	1	(1)
Real numbers	2	(2)
Topological, metric, and normed spaces	4	(6)
General notions	4	(1)
Metric spaces	5	(1)
Normed vector spaces	6	(4)
Continuous functions and compact spaces	10	(4)
Continuous and semicontinuous mappings	10	(2)
Compact spaces	12	(1)
Continuous functions on compact spaces	13	(1)
Maximum function and its properties	14	(3)
Discrete maximum function	14	(2)
General maximum function	16	(1)
Hilbert space	17	(19)
Basic definitions and properties	17	(3)
Compact and selfadjoint operators in a Hilbert space	20	(5)
Theorem on continuity of a spectrum	25	(6)
Embedding of a Hilbert space in its dual	31	(2)
Scales of Hilbert spaces and compact embedding	33	(3)
Functional spaces that are used in the investigation of boundary value and optimal control problems	36	(8)
Spaces of continuously differentiable functions	36	(1)
Spaces of integrable functions	37	(1)
Test and generalized functions	37	(2)
Sobolev spaces	39	(5)
Inequalities of coerciveness	44	(6)
Coercive systems of operators	44	(4)
Korn's inequality	48	(2)
Theorem on the continuity of solutions of functional equations	50	(1)
Differentiation in Banach spaces and the implicit function theorem	51	(3)
Frechet derivative and its properties	51	(1)
Implicit function	52	(1)
The Gateaux derivative and its connection with the Frechet derivative	53	(1)
Differentiation of the norm in the space Wmp(Omega;)	54	(4)
Auxiliary statement	54	(1)
Theorem on differentiability	55	(3)
Differentiation of eigenvalues	58	(12)
The eigenvalue problem	58	(2)
Differentiation of an operator-valued function	60	(1)
Eigenspaces and projections	61	(3)
Differentiation of eigenvalues	64	(6)
The Lagrange principle in smooth extremum problems	70	(2)
G-convergence and G-closedness of linear operators	72	(1)
Diffeomorphisms and invariance of Sobolev spaces with respect to diffeomorphisms	73	(8)
Diffeomorphisms and the relations between the derivatives	73	(2)
Sequential Frechet derivatives and partial derivatives of a composite function	75	(1)
Theorem on the invariance of Sobolev spaces	76	(2)
Transformation of derivatives under the change of variables	78	(3)
Optimal Control by Coefficients in Elliptic Systems
Direct problem	81	(5)
Coercive forms and operators	81	(1)
Boundary value problem	82	(4)
Optimal control problem	86	(17)
Nonregular control	86	(2)
Regular control	88	(2)
Regular problem and necessary conditions of optimality	90	(7)
Nonsmooth (discontinuous) control	97	(5)
Some remarks on the use of regular and discontinuous controls	102	(1)
The finite-dimensional problem	103	(2)
The finite-dimensional problem (another approach)	105	(12)
The set U(t)	105	(2)
Approximate solution of the problem (2.2.22)	107	(2)
Approximate solution of the optimal control problem when the set U°ad is empty	109	(1)
On the computation of the functional h → Ψk(h,uh)	110	(3)
Calculation and use of the Frechet derivative of the functional h → Ψk(h,uh)	113	(4)
Spectral problem	117	(3)
Eigenvalue problem	117	(1)
On the continuity of the spectrum	118	(2)
Optimization of the spectrum	120	(9)
Formulation of the problem and the existence theorem	120	(2)
Finite-dimensional approximation of the optimal control problem	122	(5)
Computation of eigenvalues	127	(2)
Control under restrictions on the spectrum	129	(9)
Optimal control problem	129	(2)
Approximate solution of the problem (2.7.7)	131	(1)
Second method of approximate solution of the problem (2.7.7)	132	(3)
Differentiation of the functionals h → Aiμ(h) and necessary conditions of optimality	135	(3)
The basic optimal control problem	138	(4)
Setting of the problem. Existence theorem	138	(2)
Approximate solution of the problem (2.8.6)	140	(2)
The combined problem	142	(3)
Optimal control problem for the case when the state of the system is characterized by a set of functions	145	(4)
Setting of the problem	145	(1)
The existence theorem	146	(3)
The general control problem	149	(10)
Bilinear form aq and the corresponding equation	150	(3)
Bilinear form br and the spectral problem	153	(1)
Basic control problem	154	(3)
Application of the basic control problem (combined problem)	157	(2)
Optimization by the shape of domain and by operators	159	(9)
Domains and bilinear forms	159	(1)
Optimization problem connected with solution of an operator equation	160	(2)
Eigenvalue optimization problem	162	(2)
Some realizations of the spaces Ml and Nl	164	(4)
Optimization problems with smooth solutions of state equations	168	(9)
Systems of elliptic equations	168	(2)
Elliptic problems in domains and in a fixed domain	170	(3)
The problem of domain shape optimization	173	(1)
Approximate solution of the direct problem ensuring convergence in the norm of a space of smooth functions	174	(3)
Control by the Right-hand Sides in Elliptic Problems
On the minimum of nonlinear functionals	177	(6)
Setting of the problem. Auxiliary statements	177	(2)
The existence theorem	179	(2)
Characterization of a minimizing element	181	(1)
Functionals continuous in the weak topology	182	(1)
Approximate solution of the minimization problem	183	(8)
Inner point lemma	183	(2)
Finite-dimensional problem	185	(6)
Control by the right-hand side in elliptic problems provided the goal functional is quadratic	191	(7)
Setting of the problem	191	(1)
Existence of a solution. Optimality conditions	192	(2)
An example of a system described by the Dirichlet problem	194	(4)
Minimax control problems	198	(3)
Control of systems whose state is described by variational inequalities	201	(8)
Setting of the problem	201	(2)
The existence theorem	203	(2)
An example of control of a system described by a variational inequality	205	(4)
Direct Problems for Plates and Shells
Bending and free oscillations of thin plates	209	(14)
Basic relations of the theory of bending of thin plates	209	(2)
Orthotropic plates	211	(1)
Bilinear form corresponding to the strain energy of the plate	212	(3)
Problem of bending of a plate	215	(6)
Problem of free oscillations of a plate	221	(2)
Problem of stability of a thin plate	223	(19)
Stored energy of a plate	223	(3)
Conditions of stationarity	226	(2)
Auxiliary statements	228	(3)
Transformation of the problem (4.2.27), (4.2.28)	231	(4)
Stability of a plate and bifurcation	235	(4)
An example of nonexistence of stable solutions	239	(3)
Model of the three-layered plate ignoring shears in the middle layer	242	(4)
Basic relations	242	(2)
Problems of the bending and of the free flexural oscillations	244	(2)
Model of the three-layered plate accounting for shears in the middle layer	246	(11)
Basic relations	246	(4)
Bilinear form corresponding to the three-layered plate	250	(3)
Bending of the three-layered plate	253	(2)
Natural oscillations of three-layered plate	255	(2)
Basic relations of the shell theory	257	(3)
Shells of revolution	260	(5)
Deformations and functional spaces	260	(2)
The bilinear form ah	262	(2)
The subspace of functions with zero-point strain energy	264	(1)
Shallow shells	265	(2)
Problems of statics of shells	267	(1)
Free oscillations of a shell	268	(2)
Problem of shell stability	270	(4)
On some approaches to stability problems	270	(1)
Reducing of the stability problem to the eigenvalue problem	271	(1)
Spectral problem (4.10.12)	272	(2)
Finite shear model of a shell	274	(8)
Strain energy of an elastic shell	274	(2)
Shallow shell	276	(2)
A relation between the Kirchhoff and Timoshenko models of shell	278	(4)
Laminated shells	282	(5)
The strain energy of a laminated shell	282	(2)
Shell of revolution	284	(2)
Shallow shells	286	(1)
Optimization of Deformable Solids
Settings of optimization problems for plates and shells	287	(4)
Goal functional and a function of control	287	(2)
Restrictions	289	(2)
Approximate solution of direct and optimization problems for plates and shells	291	(9)
Direct problems and spline functions	291	(1)
The spaces Vm for plates	292	(2)
The spaces Vm for shells	294	(3)
Direct problems for nonfastened plates and shells	297	(1)
Solution of optimization problems	298	(2)
Optimization problems for plates (control by the function of the thickness)	300	(12)
Optimization under restrictions on strength	300	(5)
Stability optimization problem	305	(6)
Optimization of frequencies of free oscillations	311	(1)
Combined optimization problem and optimization for a class of loads	312	(1)
Optimization problems for shells (control by functions of midsurface and thickness)	312	(7)
Problem of optimization of a shell of revolution with respect to strength	313	(3)
Optimization according to the stability of a cylindrical shell subject to a hydrostatic compressive load	316	(3)
Control by the shape of a hole and by the function of thickness for a shallow shell	319	(7)
Problem of optimization according to strength	319	(3)
Approximate solution of the optimization and direct problems	322	(2)
Problem of optimization of eigenvalues	324	(1)
Approximate solution of the eigenvalue problem	325	(1)
Control by the load for plates and shells	326	(7)
General problem of control by the load	326	(1)
Optimization problems for plates	327	(6)
Optimization of structures of composite materials	333	(40)
Concept of a composite material	333	(1)
Homogenization (averaging) of a periodical structure based on G-convergence	334	(9)
Effective elasticity characteristics of granule and fiber reinforced composites	343	(5)
Optimization of the effective elasticity constants of a composite	348	(6)
Optimization of a granule reinforced composite	354	(3)
Optimization of composite laminate shells	357	(10)
Optimization of the composite structure	367	(6)
Optimization of laminate composite covers according to mechanical and radio engineering characteristics	373	(10)
Propagation of electromagnetic waves through a laminated medium	373	(7)
Optimization problems	380	(3)
Shape optimization of a two-dimensional elastic body	383	(5)
Sets of controls and domains in the optimization problem	383	(1)
Problems of elasticity in domains	384	(2)
The optimization problem	386	(2)
Optimization of the internal boundary of a two-dimensional elastic body	388	(3)
Optimization problems on manifolds and shape optimization of elastic solids	391	(18)
Optimization problem for an elastic solid	392	(6)
Spaces and operators on R/2πZ, auxiliary statements	398	(7)
Optimization problem on R/2πZ	405	(4)
Optimization of the residual stresses in an elastoplastic body	409	(22)
Force and thermal loading of a nonlinear elastoplastic body	410	(11)
Residual stresses and deformations	421	(3)
Temperature pattern in a medium	424	(2)
Optimization problem	426	(5)
Optimization Problems for Steady Flows of Viscous and Nonlinear Viscous Fluids
Problem of steady flow of a nonlinear viscous fluid	431	(12)
Basic equations and assumptions	431	(3)
Formulation of the problem	434	(5)
Existence theorem	439	(4)
Theorem on continuity	443	(3)
Continuity with respect to the shape of the domain	446	(8)
Formulation of the problem	446	(2)
Lemmas on operators Lq and Bq	448	(3)
Theorem on continuity	451	(3)
Control of fluid flows by perforated walls and computation of the function of filtration	454	(6)
The problem of flow in a circular cylinder and the function of filtration	455	(3)
The passage factor for the power model	458	(1)
Control of the surface forces at the inlet by the perforated wall	459	(1)
The flow in a canal with a perforated wall placed inside	460	(3)
Basic equations	460	(2)
Generalized solution of the problem	462	(1)
Optimization by the functions of surface forces and filtration	463	(8)
Formulation of the problem and the existence theorem	463	(2)
On the differentiability of the function T → (v(T),p(T))	465	(5)
Differentiability of the functionals Φi and necessary optimality conditions	470	(1)
Problems of the optimal shape of a canal	471	(7)
Set of controls and diffeomorphisms	472	(2)
Optimization problems	474	(4)
A problem of the optimal shape of a hydrofoil	478	(17)
State equation for a moving hydrofoil	478	(7)
Fixed-domain problem and Frechet differntiability of the functionals	485	(8)
Optimization problem	493	(2)
Direct and optimization problems with consideration for the inertia forces	495	(8)
Setting and solution of the direct problem	495	(5)
Approximation of the problem (6.9.10)-(6.9.12)	500	(1)
Some remarks on models, optimization problems, and existence results	501	(2)
Bibliography	503	(16)
Index	519