Design-Oriented Analysis of Structures (eBook)

Table of Contents 7
Preface 11
Acknowledgements 15
PART ONE CONCEPTS AND METHODS 16
1. Introduction 18
1.1 ANALYSIS AND REANALYSIS 18
1.1.1 Structural Analysis 18
1.1.2 Design Variables 20
1.1.3 Changes in the Structural Model 22
1.1.4 Reanalysis of Structures 26
1.2 SCOPE OF TEXT 27
1.3 REFERENCES 30
2 Structural Analysis 32
2.1 LINEAR ANALYSIS OF FRAMED STRUCTURES 33
2.1.1 Basic Relations 33
2.1.2 Solution by the Displacement Method 35
2.2 CONTINUUM STRUCTURES 38
2.3 NONLINEAR ANALYSIS 45
2.3.1 Geometrical Non-linearity 46
2.3.2 Material Non-linearity 48
2.4 DYNAMIC ANALYSIS 48
2.4.1 The Eigenproblem 49
2.5 COLLAPSE AND BUCKLING ANALYSIS 53
2.6 REFERENCES 55
3 Reanalsis of Structures 56
3.1 FORMULATION OF REANALYSIS PROBLEMS 56
3.1.1 Linear Reanalysis 56
3.1.2 Nonlinear Reanalysis 57
3.1.3 Vibration Reanalysis 59
3.2 REANALYSIS METHODS 61
3.2.1 Direct Methods 62
3.2.2 Approximate Methods 62
3.3 REFERENCES 66
4 Direct Methods 70
4.1 A SINGLE RANK-ONE CHANGE 70
4.2 MULTIPLE RANK-ONE CHANGES 72
4.3 GENERAL PROCEDURE 75
4.4 REFERENCES 78
5 Local Approximations 80
5.1 SERIES EXPANSION 80
5.1.1 The Taylor Series 80
5.1.2 The Binomial Series 81
5.1.3 Homogeneous Functions 83
5.2 INTERMEDIATE VARIABLES 85
5.2.1 Conservative and Convex Approximations 86
5.2.2 Intermediate Response Functions 88
5.3 IMPROVED SERIES APPROXIMATIONS 90
5.3.1 Scaling of the Initial Design 91
5.3.2 Scaling of Displacements 97
5.4 REFERENCES 99
6 Global Approximations 100
6.1 POLYNOMIAL FITTING AND RESPONSE SURFACE 101
6.1.1 Polynomial Fitting 101
6.1.2 Least-Square Solutions 105
6.2 REDUCED BASIS 107
6.2.1 Static Analysis 108
6.2.2 Dynamic Analysis 110
6.3 THE CONJUGATE GRADIENT METHOD 116
6.3.1 Solution Procedure 116
6.3.2 Preconditioned Conjugate Gradient 118
6.4 REFERENCES 120
PART TWO A UNIFIED APPROACH 122
7 Combined Approximations (CA) 124
7.1 COUPLED BASIS VECTORS 125
7.1.1 Determining the Basis Vectors 125
7.1.2 Solution Procedure 127
7.2 UNCOUPLED BASIS VECTORS 130
7.2.1 Determining the Basis Vectors 130
7.2.2 Solution Procedure 133
7.3 ACCURATE SOLUTIONS 134
7.3.1 Linearly Dependent Basis Vectors 134
7.3.2 Equivalence of the CA Method and the PCG Method 135
7.3.3 Error Evaluation 137
7.3.4 Scaled and Nearly Scaled Designs 139
7.3.5 High-Order Approximations 147
7.4 REFERENCES 150
8 Simplified Solution Procedures 152
8.1 LOW-ORDER APPROXIMATIONS 152
8.1.1 Structural Optimization 154
8.1.2 Reanalysis of Damaged Structures 159
8.1.3 Efficiency of the Calculations 163
8.1.4 Limitations on Design Changes 165
8.2 EXACT SOLUTIONS 166
8.2.1 Multiple Rank-One Changes 167
8.2.2 Equivalence of the CA Method and the S-M Formula 168
8.2.3 Equivalence of the CA Method and the Woodbury Formula 169
8.2.4 Solution Procedure 169
8.3 REFERENCES 173
9 Topological and Geometrical Changes 176
9.1 TOPOLOGICAL CHANGES 177
9.1.1 Number of DOF is Unchanged 178
9.1.2 Number of DOF is Decreased 184
9.1.3 Number of DOF is Increased 187
9.2 GEOMETRICAL CHANGES 191
9.2.1 Accurate Solutions 192
9.2.2 Exact solutions 199
9.3 REFERENCES 200
10 Design Sensitivity Analysis 202
10.1 EXACT ANALYTICAL DERIVATIVES 203
10.1.1 Direct Method 204
10.2 APPROXIMATE FIRST-ORDER DERIVATIVES 205
10.2.1 Direct Approximations (DA) 206
10.2.2 Adjoint-Variable Approximations (AVA) 207
10.2.3 Finite Difference Approximations (FDA) 208
10.3 COMPARISON OF RESULTS 211
10.3.1 Accuracy of the Calculations 211
10.3.2 Computational Efficiency 211
10.3.3 Ease-of-Implementation 212
10.4 SECOND-ORDER DERIVATIVES 212
10.5 COMPUTATIONAL PROCEDURE 213
10.6 REFERENCES 219
11 Nonlinear Reanalysis 222
11.1 GEOMETRIC NONLINEAR ANALYSIS 222
11.2 NONLINEAR ANALYSIS BY THE CA METHOD 223
11.3 NONLINEAR REANALYSIS BY THE CA METHOD 225
11.4 REFERENCES 235
12 Vibration Reanalysis 236
12.1 VIBRATIONANALYSIS 236
12.2 FORMULATION OF EIGENPROBLEM REANALYSIS 238
12.3 REANALYSIS BY THE CA METHOD 239
12.4 EVALUATION OF MODIFIED EIGENVALUES 245
12.5 REFERENCES 246
Subject Index 248
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6 Global Approximations (p. 85-86)

Global (multipoint) approximations are obtained by analyzing the structure at a number of design points, and they are valid for the whole design space (or, at least, large regions of it). This type of approximation may require much computational effort, particularly in problems with large numbers of design variables. This difficulty can be alleviated by the approach presented in Chapter 7.

Polynomial fitting and response surface methods are introduced in Section
6.1. In response surface methods, the response functions are replaced by simple functions (polynomials), which are fitted to data computed at a set of selected design points. So far in practice, the use of these methods has been limited to problems with a few design variables.

Reduced-basis methods are presented in Section 6.2. Using this approach, we approximate the response of a large system, which is originally described by many degrees of freedom, by a linear combination of a few pre-selected basis vectors. The problem is then stated in terms of a small number of unknown coefficients of the basis vectors. This approach is most effective in cases where highly accurate approximations can be achieved by the reduced system of equations. A basic question in using reduced basis methods relates to the choice of an appropriate set of the basis vectors. Response vectors of previously analyzed designs could be used, but an ad hoc or intuitive choice of these vectors may not lead to satisfactory approximations.

In addition, calculation of the basis vectors requires several exact analyses of the structure for the basis design points, which might involve extensive computational effort. A method for selecting the basis vectors that provides efficient and accurate results is presented in Section 7.1.1.

The conjugate gradient method described in Section 6.3 is an iterative method for solving a set of linear equations. The problem can be stated equivalently as the minimization of a quadratic function. The method generates a set of conjugate vectors such that the solution requires little storage and computation. If the quadratic function is minimized sequentially in an n dimensional space, once along each of a set of n conjugate directions, the minimum will be found at or before the nth step, regardless of the starting point.

A preconditioned conjugate gradient method, intended to accelerate convergence of ill-conditioned problems by transformation of the set of linear equations, is then developed.

6.1 POLYNOMIAL FITTING AND RESPONSE SURFACE

When the number of design variables is small it might be practical to analyze the structure at a number of design points, and use the response at those points to construct a polynomial approximation to the response at other points. Polynomial approximations obtained by analyzing the structure at a number of design points are global approximations. Obtaining such approximations can be quite expensive for problems with large numbers of design variables. For example, if the object is to fit the structural response by a quadratic polynomial, it is necessary to analyze the structure for at least n(n+1)/2 design points (typically, many more are required to ensure a robust approximation), where n is the number of design variables [1]. The most common global approximation is the response surface approach. Using this approach, we compute the response functions at a number of points, and then fit an analytical response surface, such as a polynomial, to the data.