Sub-structure Coupling for Dynamic Analysis: Application to Complex Simulation-Based Problems Involving Uncertainty

231 Pages·2019·9.816 MB·English

by  Hector Jensen, Costas Papadimitriou

Lecture Notes in Applied and Computational Mechanics 89
Hector Jensen
Costas Papadimitriou
Sub-structure
Coupling
for Dynamic
Analysis
Application to Complex Simulation-Based
Problems Involving Uncertainty
Lecture Notes in Applied and Computational
Mechanics
Volume 89
Series Editors
Peter Wriggers, Institut für Baumechanik und Numerische Mechanik, Leibniz
Universität Hannover, Hannover, Niedersachsen, Germany
Peter Eberhard, InstituteofEngineering andComputational Mechanics, University
of Stuttgart, Stuttgart, Germany
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Hector Jensen Costas Papadimitriou
(cid:129)
Sub-structure Coupling
for Dynamic Analysis
Application to Complex Simulation-Based
Problems Involving Uncertainty
123
Hector Jensen Costas Papadimitriou
Federico SantaMaríaTechnical University University of Thessaly
Valparaiso, Chile Volos, Greece
ISSN 1613-7736 ISSN 1860-0816 (electronic)
Lecture Notesin AppliedandComputational Mechanics
ISBN978-3-030-12818-0 ISBN978-3-030-12819-7 (eBook)
https://doi.org/10.1007/978-3-030-12819-7
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Preface
The solution tomany complexsimulation-basedproblems involving finite element
models requires a large number of dynamic re-analyses. This class of problems
includesBayesian uncertainty quantification, structural dynamicsimulation,model
updating, reliability analysis, sensitivity analysis, uncertainty management in
structuraldynamics,reliability-baseddesignoptimization,globaloptimization,and
so on. These problems have a wide range of important applications in several
engineering fields. The corresponding computational demand depends on the
number offinite element analyses and on the time taken to perform an individual
analysis. For problems involving finite element models with a medium/large
numberofdegreesoffreedom,thecomputationaleffortmaybeexcessive.Tocope
with this difficulty, model reduction techniques are used to generate reduced-order
modelsinordertocarryoutthedifferentanalysesinasignificantlyreducedspaceof
generalizedcoordinates.Inparticular,amethodbasedonatypeofmodelreduction
techniques, known as substructure coupling for dynamic analysis, is used in the
context of this monograph. The method involves dividing the structure into a
number of substructures, obtaining reduced-order models of the substructures and
then assembling a reduced-order model of the entire structure. While the use of
reduced-order models alleviates part of the computational effort, their repetitive
generation during simulation processes can be computationally expensive, due to
the substantial computational overhead that arises at the substructure level. In this
regard, an efficient finite element model parametrization scheme is implemented.
When the division of the structural model is guided by such a parametrization
scheme, a drastic reduction in computational effort is achieved without compro-
misingtheaccuracyoftheresults.Thecapabilitiesofthedevelopedproceduresare
demonstrated in a number of simulation-based problems involving uncertainty.
These include reliability analysis, reliability sensitivity analysis, reliability-based
design optimization, and Bayesian model updating.
Valparaiso, Chile Hector Jensen
Volos, Greece Costas Papadimitriou
v
Acknowledgements
The assistance of our students from the Department of Civil Engineering at the
FedericoSantaMariaTechnicalUniversityandfromtheDepartmentofMechanical
Engineering at the University of Thessaly is gratefully acknowledged.
vii
Contents
Part I Reduced-Order Models
1 Model Reduction Techniques for Structural Dynamic Analyses . . . . 3
1.1 Structural Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Substructure Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Fixed-Interface Normal Modes . . . . . . . . . . . . . . . . . . . 5
1.2.2 Interface Constraint Modes. . . . . . . . . . . . . . . . . . . . . . 5
1.3 Reduced-Order Model: Standard Formulation. . . . . . . . . . . . . . . 7
1.3.1 Transformation Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.2 Reduced-Order Matrices. . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Reduced-Order Model: Improved Formulation . . . . . . . . . . . . . . 11
1.4.1 Static Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4.2 Improved Transformation Matrix. . . . . . . . . . . . . . . . . . 13
1.4.3 Enhanced Reduced-Order Matrices . . . . . . . . . . . . . . . . 15
1.4.4 Remarks on the Use of Residual Modes . . . . . . . . . . . . 15
1.5 Numerical Implementation: Pseudo-Code No. 1 . . . . . . . . . . . . . 16
1.6 Global Interface Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.6.1 Interface Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.6.2 Reduced-Order Matrices Based on Dominant
Fixed-Interface Modes . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.6.3 Reduced-Order Matrices Based on Residual
Fixed-Interface Modes . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.7 Numerical Implementation: Pseudo-Code No. 2 . . . . . . . . . . . . . 22
1.8 Local Interface Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.9 Numerical Implementation: Pseudo-Code No. 3 . . . . . . . . . . . . . 26
1.10 Reduced-Order Model Response . . . . . . . . . . . . . . . . . . . . . . . . 28
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
ix
x Contents
2 Parametrization of Reduced-Order Models Based on Normal
Modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 Parametrization Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2.1 Substructure Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2.2 Normal Modes and Interface Constraint Modes . . . . . . . 35
2.3 Parametrization of Reduced-Order Matrices . . . . . . . . . . . . . . . . 36
2.3.1 Unreduced Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3.2 Transformation Matrix TD . . . . . . . . . . . . . . . . . . . . . . 37
^ ^
2.3.3 Reduced-Order Matrices MD and KD . . . . . . . . . . . . . . 38
2.3.4 Transformation Matrix TR . . . . . . . . . . . . . . . . . . . . . . 39
^ ^
2.3.5 Reduced-Order Matrices MR and KR . . . . . . . . . . . . . . 41
^ ^
2.3.6 Expansion of MR and KR Under Partial Invariant
Conditions of TR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.4 Numerical Implementation: Pseudo-Code No. 4 . . . . . . . . . . . . . 43
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3 Parametrization of Reduced-Order Models Based on Global
Interface Reduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.1 Meta-Model for Global Interface Modes . . . . . . . . . . . . . . . . . . 49
3.1.1 Baseline Information . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.1.2 Approximation of Interface Modes . . . . . . . . . . . . . . . . 50
3.1.3 Determination of Interpolation Coefficients . . . . . . . . . . 52
3.1.4 Higher-Order Approximations. . . . . . . . . . . . . . . . . . . . 53
3.1.5 Support Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2 Numerical Implementation: Pseudo-Code No. 5 . . . . . . . . . . . . . 54
3.3 Reduced-Order Matrices Based on Global Interface
Reduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3.1 Transformation Matrix TDI . . . . . . . . . . . . . . . . . . . . . . 57
^ ^
3.3.2 Reduced-Order Matrices MDI and KDI . . . . . . . . . . . . . 58
3.3.3 Transformation Matrix TRI . . . . . . . . . . . . . . . . . . . . . . 59
^ ^
3.3.4 Reduced-Order Matrices MRI and KRI . . . . . . . . . . . . . 59
^ ^
3.3.5 Expansion of MRI and KRI Under Global Invariant
Conditions of TRI . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.4 Numerical Implementation: Pseudo-Code No. 6 . . . . . . . . . . . . . 61
3.5 Treatment of Local Interface Modes . . . . . . . . . . . . . . . . . . . . . 63
3.6 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Contents xi
Part II Application to Reliability Problems
4 Reliability Analysis of Dynamical Systems . . . . . . . . . . . . . . . . . . . . 69
4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2 Reliability Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . 70
4.3 Reliability Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.3.1 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.3.2 Basic Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.3.3 Failure Probability Estimator. . . . . . . . . . . . . . . . . . . . . 73
4.4 Numerical Implementation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.4.1 Basic Implementation. . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.4.2 Implementation Issues . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.5 Stochastic Model for Excitation. . . . . . . . . . . . . . . . . . . . . . . . . 75
4.5.1 General Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.5.2 High-Frequency Components . . . . . . . . . . . . . . . . . . . . 76
4.5.3 Pulse Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.5.4 Synthesis of Near-Field Ground Motions. . . . . . . . . . . . 77
4.5.5 Seismicity Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.6 Application Problem No. 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.6.1 Model Description and Substructures
Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.6.2 Reduced-Order Model Based on Dominant
Fixed-Interface Normal Modes . . . . . . . . . . . . . . . . . . . 80
4.6.3 Reduced-Order Model Based on Dominant
and Residual Fixed-Interface Normal Modes . . . . . . . . . 84
4.6.4 Reduced-Order Model Based on Interface Reduction . . . 86
4.6.5 Reliability Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.6.6 Remarks on the Use of Reduced-Order Models . . . . . . . 91
4.6.7 Support Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.6.8 Reliability Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.6.9 Computational Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.7 Application Problem No. 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.7.1 Structural Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.7.2 Definition of Substructures . . . . . . . . . . . . . . . . . . . . . . 98
4.7.3 System Reliability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.7.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.7.5 Computational Effort . . . . . . . . . . . . . . . . . . . . . . . . . . 107
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5 Reliability Sensitivity Analysis of Dynamical Systems. . . . . . . . . . . . 113
5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.2 Reliability Sensitivity Analysis Formulation . . . . . . . . . . . . . . . . 114
5.3 Sensitivity Measure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.4 Failure Probability Function Representation . . . . . . . . . . . . . . . . 115