Maximum and Minimum Principles: A Unified Approach with Applications

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S Title


Saddle function


Maximum and minimum principles: A unified approach, with applications


Copyright

     Cambridge University Press 1987

     Re-issued in this digitally printed version 2007

     ISBN 978-0-521-33244-6 hardback

     ISBN 978-0-521-34876-8 paperback

     QA316.535 1987 511'.66

     LCCN 86-20791


Contents


Preface



1  Saddle function problems

     1.1. The basic idea

          (i) A simple saddle function

          (ii) A convex function of one variable

          (iii) A general saddle function

          (iv) Equivalence problems

          Exercises 1.1

          (v) Quadratic example

          Exercises 1.2

          (vi) A saddle quantity

     1.2. Inequality constraints

          (i) A single variable example

          Exercises 1.3

          (ii) An inequality problem generated by a saddle function

          (iii) A graphical illustration of equivalence

          Exercises 1.4

     1.3. Transition to higher dimensions

          (i) Notation

          (ii) Basic equivalence problems

          (iii) Definition of a saddle function in higher dimensions

          (iv) Second derivative hypotheses

          (v) Quadratic example

          (vi) Invariance in saddle definitions

          Exercises 1.5

     1.4. Upper and lower bounds and uniqueness

          (i) Introduction

          (ii) Upper and lower bounds

          (iii) Example

          (iv) Uniqueness of solution

          (v) Embedding in concave linear L [x, u]

          Exercises 1.6

     1.5. Converse theorems: extremum principles

          (i) Introduction

          (ii) Extremum principles for nonlinear L [x, ii]

          (iii) Maximum principle by embedding

          (iv) A stationary value problem under constraint

          Exercises 1.7

     1.6. Examples of links with other viewpoints

          (i) Linear programming

          (ii) Quadratic programming

          (iii) Decomposition of a nonhomogeneous linear problem

          (iv) Hypercircle

          Exercises l.8

     1.7. Initial motion problems

          (i) Mechanical background

          (ii) Initial motion problem

          (iii) Generating saddle function

          (iv) Unilateral constraints

          (v) Examples

          (vi) Simultaneous extremum principles

          (vii) Cavitation

     1.8. Geometric programming

          (i) Introduction

          (ii) Primal problem

          (iii) Saddle function

          (iv) Governing conditions

          (v) Dual problem

          (vi) Alternative version of the dual problem

          (vii) Inequality constraints

     1.9. Allocation problem

          (i) Introduction

          (ii) Saddle function and governing conditions

          (iii) Simultaneous extremum principles



2  Duality and Legendre transformations

     2.1. Introduction

     2.2. Legendre transformation

          (i) Introduction

          (ii) Duality between a point and a plane

          (iii) Polar reciprocation

          (iv) Plane locus of pole dual to point envelope of polar

          (v) Duality between regular branches

          (vi) Classification of singularities

          (vii) Stability of singularities

          (viii) Bifurcation set

          Exercises 2.1

     2.3. Legendre transformations in one active variable, with singularities

          (i) Inflexion and cusp as dual isolated singularities

          (H) Dual distributed and accumulated singularities

          (iii) Some general theorems

          Exercises 2.2

          (iv) Ladder for the cuspoids

          (v) Half-line dual of a pole at infinity

          (vi) Response curves and complementary areas

          (vii) Nondecreasing response characteristics

     2.4. Legendre transformations in two active variables, with singularities

          (i) Umbilics, illustrating dual isolated singularities

          (ii) Accumulated duals of nonplanar distributed singularities

          Exercises 2.3

     2.5. Closed chain of Legendre transformations

          (i) Indicial notation

          (ii) Inner product space notation

          (iii) Convex and saddle-shaped branches

          Exercises 2.4

     2.6. Examples of quartets of Legendre transformations

          (i) Introductory examples

          (ii) Strain energy and complementary energy density

          (iii) Incremental elasticlplastic constitutive equations

          Exercises 2.5

          (iv) Other physical examples of Legendre transformations

          (v) Stable singularities in a constrained plane mapping

     2.7. The structure of maximum and minimum principles

          (i) Bounds and Legendre transformations

          (ii) Bifurcation theory

          Exercises 2.6

          (iii) Generalized Hamiltonian and Lagrangian aspects

          (iv) Supplementary constraints

          Exercises 2.7

     2.8. Network theory

          (i) Introduction

          (ii) A simple electrical network

          (iii) Node-branch incidence matrix

          (iv) Electrical branch characteristics

          (v) Saddle function and governing equations

          (vi) Bounds and extremum principles

          (vii) Equivalent underdetermined systems

          (viii) Loop-branch formulation

          (ix) Branch characteristics with inequalities



3  Upper and lower bounds via saddle functionals

     3.1. Introduction

     3.2. Inner product spaces

          (i) Linear spaces

          (ii) Inner product spaces

     3.3. Linear operators and adjointness

          (i) Operators

          (ii) A djoin to ess of linear operators

          (iii) Examples of adjoint operators

               1. Transposition of a matrix

               2. Integration by parts

               3. Green's Gauss'Idivergence theorem

               4. A vector identity

               5. Virtual work transformation

               6. Integral operators

               7. Sequences and integrable functions

          (iv) The operator T * T

          Exercises 3.1

     3.4. Gradients of functionals

          (i) Derivatives of general operators

          (ii) Gradients of functionals

          (iii) Partial gradients of functionals

     3.5. Saddle functional

          (i) Saddle quantity

          (ii) Definition of a saddle functional

     3.6. Upper and lower bound

          (i) Introduction

          (ii) A central theorem in infinite dimensions

          (iii) Quadratic generating functional

     3.7. An ordinary differential equation

          (i) Introduction

          (ii) Intermediate variable

          (iii) Inner product spaces and adjoins operators

          (iv) Hamiltonian functional

          (v) Generating functional

          (vi) Identification of A and B

          (vii) Stationary principle

          (viii) Saddle functional L[x, r ]

          (ix) Upper and lower bounds

          (x) Associated inequality problems

          (xi) Solution of individual constraints

          (xii) Evaluation of simultaneous bounds

          (xiii) Specific example

          (xiv) The fundamental lemma of the calculus of variations

          (xv) Complementary stationary principles

          (xvi) Weighting function with isolated zeros

          Exercises 3.2

     3.8. Solution of linear constraints

          (i) Consistency conditions

          (ii) General formulae for J[ua] and K [x,]

          (iii) Separate upper and lower bounds, not simultaneous

          (iv) Example of a single bound

     3.9. A procedure for the derivation of bounds

          (i) Introduction

          (ii) Steps in the procedure

          (iii) Hierarchy of smoothness in admissible fields

     3.10. A catalogue of examples

          (i) Introduction

          (ii} Obstacle problem

          (iii) Euler equation and Hamilton's principle

          (iv) Foppl-Hencky equation

          (v) A partial differential equation

          (vi) A free boundary problem

          Exercises 3.3

     3.11. Variational inequalities

          (i) Introduction

          (ii) A general definition

          (iii) Examples

     3.12. Nonnegative operator equations

          (i) Introduction

          (ii} Examples

          (iii) General results

          (iv) Laplacian problems

          (v) A comparison of equivalent differential and integral equations

          (vi) Alternative bounds

          (vii) Wave scattering at a submerged weir



4  Extensions of the general approach

     4.1. Introduction

     4.2. Bounds on linear functionals

          (i) Introduction

          (ii) Nonnegative operator problems

          (iii) Other saddle-generated problems

          Exercises 4.1

          (iv) Comparison problems for a cantilever beam

               (a) Given problem

               (b) Saddle functional

               (c) First comparison problem

               (d) Second comparison problem

          (v) Other examples of point wise bounds

               (a) A unit spike function

               (b) Example of T * Tx + E [x] = O

               (c) A two dimensional problem

               (d) Further examples

          (vi) Boundedness hypotheses

          (vii) Embedding method

          (viii) Examples

               1. Bounds on a solution of an algebraic equation

               2. A nonlinear ordinary ififferential equation

               3. A nonlinear integral equation

          Exercises 4.2

     4.3. Initial value problems

          (i) Introduction

          (ii) The role of the adjoint problem

          (iii) A simple example

          (iv) Change of variable

          (v) A self-adjoins representation

          (vi) A particular change of variable

          (vii) Example

          (viii) A general first order system

          (ix) A second order equation

          (x) A heat conduction problem

          (xi) Alternative approaches

          Exercise 4.3

     4.4. Diffusion of liquid through a porous medium

          (i) Governing equations

          (ii) Adjointness

          (iii) Governing equations as Ti + h = 0

          (iv) Nonnegative T

          (v) Generating saddle functional

          (vi) Upper and lower bounds

     4.5. Comparison methods

          (i) The basic idea

          (ii) Example of the general setting

          (iii) Easy and hard problems

          (iv) Difference variables

          (v) Stationary and saddle properties

          (vi) Hashin---Shtrikman functional



5  Mechanics of solids and fluids

     5.1. Introduction

     5.2. Thermodynamics

          (i) Thermodynamic potential functions

          (ii) Second derivatives of thermodynamic potentials

          (iii) Simple fluids

          Exercises 5.1

     5.3. Compressible inviscid flow

          (i) Thermodynamics

          (ii) Another Legendre transformation

          (iii) Constitutive surfaces

          (iv) Balance of mass, momentum and energy

          (v) Simultaneous upper and lower bounds for steady irrotational flow

          (vi) General flows

          Exercises 5.2

     5.4. Magnetohydrodynamic pipe flow

          (i) Governing equations

          (ii) Decomposition method

          (iii) Embedding method

     5.5. Inner product spaces and adjoint operators in continuum mechanics

          (i) Inner product spaces

          (ii) A djoin tress

          (iii) Virtual work

          (iv) Discontinuities of stress and displacement

     5.6. Classical elasticity

          (i) Governing equations

          (ii) Generalized Hamiltonian representation

          (iii) Generating saddle functional

          (lv) Simultaneous upper and lower bounds

     5.7. Slow viscous flow

          (i) Governing equations

          (ii) Generalized Hamiltonian representation

          (iii) Generating saddle functional

          (iv) Historical remarks and applications

          Exercises 5.3

     5.8. Rigid/plastic yield point problem

          (i) Governing equations

          (ii) Generalized Hamiltonian representation

          (iii) Generating saddle functional

          (iv) Uniqueness of stress

          (v) Simultaneous upper and lower bounds

          Exercises 5.4

     5.9. Finite elasticity

          (i) Strain measures

          (ii) Stress measures

          (iii) Hyperelastic solid

          (iv) Governing equations

          (v) Stationary principles

          Exercises 5.5

     5.10. Incremental elastic/plastic distortion

          (i) Governing equations

          (ii) Strain-rate

          (iii) Stress-rate

          (iv) Constitutive rate equations

          (v) Continuing equilibrium

          (vi) Adjointness

          (vii) Generating functional

          (viii) Upper and lower bounds

          (ix) Weaker hypotheses

          (x) Kinematical constraints

          (xi) Equilibrium constraints

          (xii) Application of supplementary constraints

          Exercises 5.6


References


Subject index


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