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

This book is intended to be both a thorough introduction to contemporary research in optimization theory for elliptic systems with its numerous applications and a textbook at the undergraduate and graduate level for courses in pure or applied mathematics or in continuum mechanics. Various processes of modern technology and production are described by el­ liptic partial differential equations. Optimization of these processes reduces to op­ timization problems for elliptic systems. The numerical solution of such problems is associated with the solution of the following questions. 1. The setting of the optimization problem ensuring the existence of a solution on a set of admissible controls, which is a subset of some infinite-dimensional vector space. 2. Reduction of the infinite-dimensional optimization problem to a sequence of finite-dimensional problems such that the solutions of the finite-dimensional problems converge, in a sense, to the solution of the infinite-dimensional problem.3. Numerical solution of the finite-dimensional problems.

1 Basic Definitions and Auxiliary Statements.- 1.1 Sets, functions, real numbers.- 1.1.1 Notations and definitions.- 1.1.2 Real numbers.- 1.2 Topological, metric, and normed spaces.- 1.2.1 General notions.- 1.2.2 Metric spaces.- 1.2.3 Normed vector spaces.- 1.3 Continuous functions and compact spaces.- 1.3.1 Continuous and semicontinuous mappings.- 1.3.2 Compact spaces.- 1.3.3 Continuous functions on compact spaces.- 1.4 Maximum function and its properties.- 1.4.1 Discrete maximum function.- 1.4.2 General maximum function.- 1.5 Hilbert space.- 1.5.1 Basic definitions and properties.- 1.5.2 Compact and selfadjoint operators in a Hilbert space.- 1.5.3 Theorem on continuity of a spectrum.- 1.5.4 Embedding of a Hilbert space in its dual.- 1.5.5 Scales of Hilbert spaces and compact embedding.- 1.6 Functional spaces that are used in the investigation of boundary value and optimal control problems.- 1.6.1 Spaces of continuously differentiable functions.- 1.6.2 Spaces of integrable functions.- 1.6.3 Test and generalized functions.- 1.6.4 Sobolev spaces.- 1.7 Inequalities of coerciveness.- 1.7.1 Coercive systems of operators.- 1.7.2 Korn’s inequality.- 1.8 Theorem on the continuity of solutions of functional equations.- 1.9 Differentiation in Banach spaces and the implicit function theorem.- 1.9.1 Fréchet derivative and its properties.- 1.9.2 Implicit function.- 1.9.3 The Gâteaux derivative and its connection with the Fréchet derivative.- 1.10 Differentiation of the norm in the space Wpm(?).- 1.10.1 Auxiliary statement.- 1.10.2 Theorem on differentiability.- 1.11 Differentiation of eigenvalues.- 1.11.1 The eigenvalue problem.- 1.11.2 Differentiation of an operator-valued function.- 1.11.3 Eigenspaces and projections.- 1.11.4 Differentiation of eigenvalues.- 1.12 The Lagrange principle in smooth extremum problems.- 1.13 G-convergence and G-closedness of linear operators.- 1.14 Diffeomorphisms and invariance of Sobolev spaces with respect to diffeomorphisms.- 1.14.1 Diffeomorphisms and the relations between the derivatives.- 1.14.2 Sequential Fréchet derivatives and partial derivatives of a composite function.- 1.14.3 Theorem on the invariance of Sobolev spaces.- 1.14.4 Transformation of derivatives under the change of variables.- 2 Optimal Control by Coefficients in Elliptic Systems.- 2.1 Direct problem.- 2.1.1 Coercive forms and operators.- 2.1.2 Boundary value problem.- 2.2 Optimal control problem.- 2.2.1 Nonregular control.- 2.2.2 Regular control.- 2.2.3 Regular problem and necessary conditions of optimality.- 2.2.4 Nonsmooth (discontinuous) control.- 2.2.5 Some remarks on the use of regular and discontinuous controls.- 2.3 The finite-dimensional problem.- 2.4 The finite-dimensional problem (another approach).- 2.4.1 The set U(t).- 2.4.2 Approximate solution of the problem (2.2.22).- 2.4.3 Approximate solution of the optimal control problem when the set ?ad is empty.- 2.4.4 On the computation of the functional h ? ?k(h,uh).- 2.4.5 Calculation and use of the Fréchet derivative of the functional h ? ?ma(h,uh).- 2.5 Spectral problem.- 2.5.1 Eigenvalue problem.- 2.5.2 On the continuity of the spectrum.- 2.6 Optimization of the spectrum.- 2.6.1 Formulation of the problem and the existence theorem.- 2.6.2 Finite-dimensional approximation of the optimal control problem.- 2.6.3 Computation of eigenvalues.- 2.7 Control under restrictions on the spectrum.- 2.7.1 Optimal control problem.- 2.7.2 Approximate solution of the problem (2.7.7).- 2.7.3 Second method of approximate solution of the problem (2.7.7).- 2.7.4 Differentiation of the functionals h ? Aiµ(h) and necessary conditions of optimality.- 2.8 The basic optimal control problem.- 2.8.1 Setting of the problem. Existence theorem.- 2.8.2 Approximate solution of the problem (2.8.6).- 2.9 The combined problem.- 2.10 Optimal control problem for the case when the state of the system is characterized by a set of functions.- 2.10.1 Setting of the problem.- 2.10.2 The existence theorem.- 2.11 The general control problem.- 2.11.1 Bilinear form aq and the corresponding equation.- 2.11.2 Bilinear form br and the spectral problem.- 2.11.3 Basic control problem.- 2.11.4 Application of the basic control problem (combined problem).- 2.12 Optimization by the shape of domain and by operators.- 2.12.1 Domains and bilinear forms.- 2.12.2 Optimization problem connected with solution of an operator equation.- 2.12.3 Eigenvalue optimization problem.- 2.12.4 Some realizations of the spaces Ml and Nl.- 2.13 Optimization problems with smooth solutions of state equations.- 2.13.1 Systems of elliptic equations.- 2.13.2 Elliptic problems in domains and in a fixed domain.- 2.13.3 The problem of domain shape optimization.- 2.13.4 Approximate solution of the direct problem ensuring convergence in the norm of a space of smooth functions.- 3 Control by the Right-hand Sides in Elliptic Problems.- 3.1 On the minimum of nonlinear functionals.- 3.1.1 Setting of the problem. Auxiliary statements.- 3.1.2 The existence theorem.- 3.1.3 Characterization of a minimizing element.- 3.1.4 Functionals continuous in the weak topology.- 3.2 Approximate solution of the minimization problem.- 3.2.1 Inner point lemma.- 3.2.2 Finite-dimensional problem.- 3.3 Control by the right-hand side in elliptic problems provided the goal functional is quadratic.- 3.3.1 Setting of the problem.- 3.3.2 Existence of a solution. Optimality conditions.- 3.3.3 An example of a system described by the Dirichlet problem.- 3.4 Minimax control problems.- 3.5 Control of systems whose state is described by variational inequalities.- 3.5.1 Setting of the problem.- 3.5.2 The existence theorem.- 3.5.3 An example of control of a system described by a variational inequality.- 4 Direct Problems for Plates and Shells.- 4.1 Bending and free oscillations of thin plates.- 4.1.1 Basic relations of the theory of bending of thin plates.- 4.1.2 Orthotropic plates.- 4.1.3 Bilinear form corresponding to the strain energy of the plate.- 4.1.4 Problem of bending of a plate.- 4.1.5 Problem of free oscillations of a plate.- 4.2 Problem of stability of a thin plate.- 4.2.1 Stored energy of a plate.- 4.2.2 Conditions of stationarity.- 4.2.3 Auxiliary statements.- 4.2.4 Transformation of the problem (4.2.27), (4.2.28).- 4.2.5 Stability of a plate and bifurcation.- 4.2.6 An example of nonexistence of stable solutions.- 4.3 Model of the three-layered plate ignoring shears in the middle layer.- 4.3.1 Basic relations.- 4.3.2 Problems of the bending and of the free flexural oscillations.- 4.4 Model of the three-layered plate accounting for shears in the middle layer.- 4.4.1 Basic relations.- 4.4.2 Bilinear form corresponding to the three-layered plate.- 4.4.3 Bending of the three-layered plate.- 4.4.4 Natural oscillations of three-layered plate.- 4.5 Basic relations of the shell theory.- 4.6 Shells of revolution.- 4.6.1 Deformations and functional spaces.- 4.6.2 The bilinear form ah.- 4.6.3 The subspace of functions with zero-point strain energy.- 4.7 Shallow shells.- 4.8 Problems of statics of shells.- 4.9 Free oscillations of a shell.- 4.10 Problem of shell stability.- 4.10.1 On some approaches to stability problems.- 4.10.2 Reducing of the stability problem to the eigenvalue problem.- 4.10.3 Spectral problem (4.10.12).- 4.11 Finite shear model of a shell.- 4.11.1 Strain energy of an elastic shell.- 4.11.2 Shallow shell.- 4.11.3 A relation between the Kirchhoff and Timoshenko models of shell.- 4.12 Laminated shells.- 4.12.1 The strain energy of a laminated shell.- 4.12.2 Shell of revolution.- 4.12.3 Shallow shells.- 5 Optimization of Deformable Solids.- 5.1 Settings of optimization problems for plates and shells.- 5.1.1 Goal functional and a function of control.- 5.1.2 Restrictions.- 5.2 Approximate solution of direct and optimization problems for plates and shells.- 5.2.1 Direct problems and spline functions.- 5.2.2 The spaces Vm for plates.- 5.2.3 The spaces Vm for shells.- 5.2.4 Direct problems for nonfastened plates and shells.- 5.2.5 Solution of optimization problems.- 5.3 Optimization problems for plates (control by the function of the thickness).- 5.3.1 Optimization under restrictions on strength.- 5.3.2 Stability optimization problem.- 5.3.3 Optimization of frequencies of free oscillations.- 5.3.4 Combined optimization problem and optimization for a class of loads.- 5.4 Optimization problems for shells (control by functions of midsurface and thickness).- 5.4.1 Problem of optimization of a shell of revolution with respect to strength.- 5.4.2 Optimization according to the stability of a cylindrical shell subject to a hydrostatic compressive load.- 5.5 Control by the shape of a hole and by the function of thickness for a shallow shell.- 5.5.1 Problem of optimization according to strength.- 5.5.2 Approximate solution of the optimization and direct problems.- 5.5.3 Problem of optimization of eigenvalues.- 5.5.4 Approximate solution of the eigenvalue problem.- 5.6 Control by the load for plates and shells.- 5.6.1 General problem of control by the load.- 5.6.2 Optimization problems for plates.- 5.7 Optimization of structures of composite materials.- 5.7.1 Concept of a composite material.- 5.7.2 Homogenization (averaging) of a periodical structure based on G-convergence.- 5.7.3 Effective elasticity characteristics of granule and fiber reinforced composites.- 5.7.4 Optimization of the effective elasticity constants of a composite.- 5.7.5 Optimization of a granule reinforced composite.- 5.7.6 Optimization of composite laminate shells.- 5.7.7 Optimization of the composite structure.- 5.8 Optimization of laminate composite covers according to mechanical and radio engineering characteristics.- 5.8.1 Propagation of electromagnetic waves through a laminated medium.- 5.8.2 Optimization problems.- 5.9 Shape optimization of a two-dimensional elastic body.- 5.9.1 Sets of controls and domains in the optimization problem.- 5.9.2 Problems of elasticity in domains.- 5.9.3 The optimization problem.- 5.10 Optimization of the internal boundary of a two-dimensional elastic body.- 5.11 Optimization problems on manifolds and shape optimization of elastic solids.- 5.11.1 Optimization problem for an elastic solid.- 5.11.2 Spaces and operators on ?/2??, auxiliary statements.- 5.11.3 Optimization problem on ?/2??.- 5.12 Optimization of the residual stresses in an elastoplastic body.- 5.12.1 Force and thermal loading of a nonlinear elastoplastic body.- 5.12.2 Residual stresses and deformations.- 5.12.3 Temperature pattern in a medium.- 5.12.4 Optimization problem.- 6 Optimization Problems for Steady Flows of Viscous and Nonlinear Viscous Fluids.- 6.1 Problem of steady flow of a nonlinear viscous fluid.- 6.1.1 Basic equations and assumptions.- 6.1.2 Formulation of the problem.- 6.1.3 Existence theorem.- 6.2 Theorem on continuity.- 6.3 Continuity with respect to the shape of the domain.- 6.3.1 Formulation of the problem.- 6.3.2 Lemmas on operators $$ {\tilde L_q}and{\tilde B_q}$$.- 6.3.3 Theorem on continuity.- 6.4 Control of fluid flows by perforated walls and computation of the function of filtration.- 6.4.1 The problem of flow in a circular cylinder and the function of filtration.- 6.4.2 The passage factor for the power model.- 6.4.3 Control of the surface forces at the inlet by the perforated wall.- 6.5 The flow in a canal with a perforated wall placed inside.- 6.5.1 Basic equations.- 6.5.2 Generalized solution of the problem.- 6.6 Optimization by the functions of surface forces and filtration.- 6.6.1 Formulation of the problem and the existence theorem.- 6.6.2 On the differentiability of the function T?(v(T), p(T)).- 6.6.3 Differentiability of the functionals ?iand necessary optimality conditions.- 6.7 Problems of the optimal shape of a canal.- 6.7.1 Set of controls and diffeomorphisms.- 6.7.2 Optimization problems.- 6.8 A problem of the optimal shape of a hydrofoil.- 6.8.1 State equation for a moving hydrofoil.- 6.8.2 Fixed-domain problem and Fréchet differentiability of the functionals.- 6.8.3 Optimization problem.- 6.9 Direct and optimization problems with consideration for the inertia forces.- 6.9.1 Setting and solution of the direct problem.- 6.9.2 Approximation of the problem (6.9.10)–(6.9.12).- 6.9.3 Some remarks on models, optimization problems, and existence results.