Table of Contents
Preface vii
Introduction 1
Part I Flutter of plates
1 Statement of the problem 5
2 Determination of aerodynamic pressure 6
3 Mathematical statement of problems 11
4 Reduction to a problem on a disk 14
5 Test problems 20
6 Rectangular plate 36
6.1 Problem statement and analytical solution 36
6.2 Numerical-analytical solution 38
6.3 Results 41
6.4 Bubnov-Galerkin (B-G) method 42
6.5 Dependence of critical flutter velocity on plate thickness 46
6.6 Dependence of critical flutter velocity on altitude 46
7 Flutter of a rectangular plate of variable stiffness or thickness 48
7.1 Strip with variable cross section -48
7.2 Rectangular plates 52
8 Viscoelastic plates 57
Part II Flutter of shallow shells
9 General formulation 63
10 Determination of aerodynamic pressure 66
11 The shallow shell as part of an airfoil 71
12 The shallow shell of revolution 74
13 The conical shell: external flow 78
14 The conical shell: internal flow 82
14.1 Statement of the problem 82
14.2 Determination of dynamic pressure 87
15 Example calculations 91
Part III Numerical methods for non-self-Adjoint eigenvalue problems
16 Discretization of the Laplace operator 99
16.1 The Sturm-Liouville problem 99
16.2 Interpolation formula for a function of two variables on a disk, and its properties 104
16.3 Discretization of the Laplace operator 108
16.4 Theorem of h-matrices 109
16.5 Construction of h-matrix cells by discretization of Bessel equations 112
16.6 Fast multiplication of h-matrices by vectors using the fast Fourier transform 114
16.7 Symmetrization of the ft-matrix 116
17 Discretization of linear equations in mathematical physics with separable variables 118
17.1 General form of equations with separable variables 118
17.2 Further generalization 119
18 Eigenvalues and eigenfunctions of the Laplace operator 122
18.1 The Dirichlet problem 123
18.2 Mixed problem 135
18.3 The Neumann problem 136
18.4 Numerical experiments 140
19 Eigenvalues and Eigenfunctions of a Biharmonic Operator 142
19.1 Boundary-value problem of the first kind 145
19.2 Boundary-value problem of the second kind 145
19.3 Numerical experiments 148
20 Eigenvalues and Eigenfunctions of the Laplace Operator on an Arbitrary Domain -151
20.1 Eigenvalues and eigenvectors of the Laplace operator 151
20.1.1 The Dirichlet problem 158
20.1.2 Mixed problem 158
20.1.3 The Neumann problem 159
20.1.4 Description of the program LAP123C 159
20.2 Program forconformal mapping 164
20.3 Numerical Experiments 166
21 Eigenvalues and Eigenfunctions of a Biharmonic Operator on an Arbitrary Domain 168
21.1 Eigenvalues and eigenfunctions of a biharmonic operator 167
21.1.1 Boundary-value problem of the first kind 173
21.1.2 Boundary-value problem of the second kind 173
21.1.3 Description of the program BIG12AG 173
21.2 Program for conformal mapping 177
21.3 Numerical experiments 179
22 Error Estimates for Eigenvalue Problems 180
22.1 Localization theorems 180
22.2 A priori error estimate in eigenvalue problems 183
22.3 A posteriori error estimate for eigenvalue problems 185
22.4 Generalization for operator pencil 185
Conclusion 187
Bibliography 189
