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Optimal Combination of Reduction Methods in Structural Mechanics and Selection of a Suitable Intermediate Dimension: Optimal Combination of Reduction Methods in Structural Mechanics and Selection of a Suitable Intermediate Dimension
A two-step model order reduction method is investigated in order to overcome problems of certain one-step methods. Not only optimal combinations of one-step reductions are considered but also the selection of a suitable intermediate dimension (ID) is described. Several automated selection methods are presented and their application tested on a gear box model. The implementation is realized using a Matlab-based Software MORPACK. Several recommendations are given towards the selection of a suitable ID, and problems in Model Order Reduction (MOR) combinations are pointed out. A pseudo two-step is suggested to reduce the full system without any modal information. A new node selection approach is proposed to enhance the SEREP approximation of the system’s response for small reduced representations.:Contents
Kurzfassung..........................................................................................iv
Abstract.................................................................................................iv
Nomenclature........................................................................................ix
1 Introduction........................................................................................1
1.1 Motivation........................................................................................1
1.2 Objectives........................................................................................1
1.3 Outline of the Thesis........................................................................2
2 Theoretical Background.......................................................................3
2.1 Finite Element Method......................................................................3
2.1.1 Modal Analysis...............................................................................4
2.1.2 Frequency Response Function.......................................................4
2.2 Model Order Reduction.....................................................................5
2.3 Physical Subspace Reduction Methods.............................................7
2.3.1 Guyan Reduction...........................................................................7
2.3.2 Improved Reduced System Method...............................................8
2.4 Modal Subspace Reduction Methods...............................................10
2.4.1 Modal Reduction...........................................................................11
2.4.2 Exact Modal Reduction..................................................................11
2.4.3 System Equivalent Reduction Expansion Process.........................13
2.5 Krylov Subspace Reduction Methods...............................................14
2.6 Hybrid Subspace Reduction Methods..............................................15
2.6.1 Component Mode Synthesis........................................................16
2.6.2 Hybrid Exact Modal Reduction......................................................19
2.7 Model Correlation Methods.............................................................21
2.7.1 Normalized Relative Frequency Difference...................................21
2.7.2 Modified Modal Assurance Criterion.............................................22
2.7.3 Pseudo-Orthogonality Check.......................................................22
2.7.4 Comparison of Frequency Response Function.............................23
3 Selection of Active Degrees of Freedom............................................25
3.1 Non-Iterative Methods...................................................................26
3.1.1 Modal Kinetic Energy and Variants..............................................26
3.1.2 Driving Point Residue and Variants..............................................27
3.1.3 Eigenvector Component Product..................................................28
3.2 Iterative Reduction Methods...........................................................29
3.2.1 Effective Independence Distribution.............................................29
3.2.2 Mass-Weighted Effective Independence.......................................32
3.2.3 Variance Based Selection Method.................................................33
3.2.4 Singular Value Decomposition Based Selection Method................34
3.2.5 Stiffness-to-Mass Ratio Selection Method.....................................34
3.3 Iterative Expansion Methods...........................................................35
3.3.1 Modal-Geometrical Selection Criterion...........................................36
3.3.2 Triaxial Effective Independence Expansion...................................36
3.4 Measure of Goodness for Selected Active Set..................................39
3.4.1 Determinant and Rank of the Fisher Information Matrix................39
3.4.2 Condition Number of the Partitioned Modal Matrix........................40
3.4.3 Measured Energy per Mode..........................................................40
3.4.4 Root Mean Square Error of Pseudo-Orthogonality Check.............41
3.4.5 Eigenvalue Comparison................................................................41
4 Two-Step Reduction in MORPACK.......................................................42
4.1 Structure of MORPACK.....................................................................42
4.2 Selection of an Intermediate Dimension.........................................43
4.2.1 Intermediate Dimension Requirements........................................44
4.2.2 Implemented Selection Methods..................................................45
4.2.3 Recommended Selection of an Intermediate Dimension...............48
4.3 Combination of Reduction Methods.................................................49
4.3.1 Overview of All Candidates..........................................................50
4.3.2 Combinations with Modal Information.........................................54
4.3.3 Combinations without Modal Information....................................54
5 Applications........................................................................................57
5.1 Gear Box Model...............................................................................57
5.2 Selection of Additional Active Nodes................................................58
5.3 Optimal Intermediate Dimension......................................................64
5.4 Two-Step Model Order Reduction Results........................................66
5.5 Comparison to One-Step Model Order Reduction Methods..............70
5.6 Comparison to One-Step Hybrid Model Order Reduction Methods...72
5.7 Proposal of a New Approach for Additional Node Selection..............73
6 Summary and Conclusions...................................................................77
7 Zusammenfassung und Ausblick..........................................................79
Bibliography............................................................................................81
List of Tables..........................................................................................86
List of Figures.........................................................................................88
A Appendix.............................................................................................89
A.1 Results of Two-Step Model Order Reduction.....................................89
A.2 Data CD............................................................................................96Mehrschrittverfahren der Modellreduktion werden untersucht, um spezielle Probleme konventioneller Einschrittverfahren zu lösen. Eine optimale Kombination von strukturmechanischen Reduktionsverfahren und die Auswahl einer geeigneten Zwischendimension wird untersucht. Dafür werden automatische Verfahren in Matlab implementiert, in die Software MORPACK integriert und anhand des Finite Elemente Modells eines Getriebegehäuses ausgewertet. Zur Auswahl der Zwischendimension werden Empfehlungen genannt und auf Probleme bei der Kombinationen bestimmter Reduktionsverfahren hingewiesen. Ein Pseudo- Zweischrittverfahren wird vorgestellt, welches eine Reduktion ohne Kenntnis der modalen Größen bei ähnlicher Genauigkeit im Vergleich zu modalen Unterraumverfahren durchführt. Für kleine Reduktionsdimensionen wird ein Knotenauswahlverfahren vorgeschlagen, um die Approximation des Frequenzganges durch die System Equivalent Reduction Expansion Process (SEREP)-Reduktion zu verbessern.:Contents
Kurzfassung..........................................................................................iv
Abstract.................................................................................................iv
Nomenclature........................................................................................ix
1 Introduction........................................................................................1
1.1 Motivation........................................................................................1
1.2 Objectives........................................................................................1
1.3 Outline of the Thesis........................................................................2
2 Theoretical Background.......................................................................3
2.1 Finite Element Method......................................................................3
2.1.1 Modal Analysis...............................................................................4
2.1.2 Frequency Response Function.......................................................4
2.2 Model Order Reduction.....................................................................5
2.3 Physical Subspace Reduction Methods.............................................7
2.3.1 Guyan Reduction...........................................................................7
2.3.2 Improved Reduced System Method...............................................8
2.4 Modal Subspace Reduction Methods...............................................10
2.4.1 Modal Reduction...........................................................................11
2.4.2 Exact Modal Reduction..................................................................11
2.4.3 System Equivalent Reduction Expansion Process.........................13
2.5 Krylov Subspace Reduction Methods...............................................14
2.6 Hybrid Subspace Reduction Methods..............................................15
2.6.1 Component Mode Synthesis........................................................16
2.6.2 Hybrid Exact Modal Reduction......................................................19
2.7 Model Correlation Methods.............................................................21
2.7.1 Normalized Relative Frequency Difference...................................21
2.7.2 Modified Modal Assurance Criterion.............................................22
2.7.3 Pseudo-Orthogonality Check.......................................................22
2.7.4 Comparison of Frequency Response Function.............................23
3 Selection of Active Degrees of Freedom............................................25
3.1 Non-Iterative Methods...................................................................26
3.1.1 Modal Kinetic Energy and Variants..............................................26
3.1.2 Driving Point Residue and Variants..............................................27
3.1.3 Eigenvector Component Product..................................................28
3.2 Iterative Reduction Methods...........................................................29
3.2.1 Effective Independence Distribution.............................................29
3.2.2 Mass-Weighted Effective Independence.......................................32
3.2.3 Variance Based Selection Method.................................................33
3.2.4 Singular Value Decomposition Based Selection Method................34
3.2.5 Stiffness-to-Mass Ratio Selection Method.....................................34
3.3 Iterative Expansion Methods...........................................................35
3.3.1 Modal-Geometrical Selection Criterion...........................................36
3.3.2 Triaxial Effective Independence Expansion...................................36
3.4 Measure of Goodness for Selected Active Set..................................39
3.4.1 Determinant and Rank of the Fisher Information Matrix................39
3.4.2 Condition Number of the Partitioned Modal Matrix........................40
3.4.3 Measured Energy per Mode..........................................................40
3.4.4 Root Mean Square Error of Pseudo-Orthogonality Check.............41
3.4.5 Eigenvalue Comparison................................................................41
4 Two-Step Reduction in MORPACK.......................................................42
4.1 Structure of MORPACK.....................................................................42
4.2 Selection of an Intermediate Dimension.........................................43
4.2.1 Intermediate Dimension Requirements........................................44
4.2.2 Implemented Selection Methods..................................................45
4.2.3 Recommended Selection of an Intermediate Dimension...............48
4.3 Combination of Reduction Methods.................................................49
4.3.1 Overview of All Candidates..........................................................50
4.3.2 Combinations with Modal Information.........................................54
4.3.3 Combinations without Modal Information....................................54
5 Applications........................................................................................57
5.1 Gear Box Model...............................................................................57
5.2 Selection of Additional Active Nodes................................................58
5.3 Optimal Intermediate Dimension......................................................64
5.4 Two-Step Model Order Reduction Results........................................66
5.5 Comparison to One-Step Model Order Reduction Methods..............70
5.6 Comparison to One-Step Hybrid Model Order Reduction Methods...72
5.7 Proposal of a New Approach for Additional Node Selection..............73
6 Summary and Conclusions...................................................................77
7 Zusammenfassung und Ausblick..........................................................79
Bibliography............................................................................................81
List of Tables..........................................................................................86
List of Figures.........................................................................................88
A Appendix.............................................................................................89
A.1 Results of Two-Step Model Order Reduction.....................................89
A.2 Data CD............................................................................................9
Building Cox-Type Structured Hazard Regression Models with Time-Varying Effects
In recent years, flexible hazard regression models based on penalised splines have been developed that allow us to extend the classical Cox-model via the inclusion of time-varying and nonparametric effects. Despite their immediate appeal in terms of flexibility, these models introduce additional difficulties when a subset of covariates and the corresponding modelling alternatives have to be chosen. We present an analysis of data from a specific patient population with 90-day survival as the response variable. The aim is to determine a sensible prognostic model where some variables have to be included due to subject-matter knowledge while other variables are subject to model selection. Motivated by this application, we propose a twostage stepwise model building strategy to choose both the relevant covariates and the corresponding modelling alternatives within the choice set of possible covariates simultaneously. For categorical covariates, competing modelling approaches are linear effects and time-varying effects, whereas nonparametric modelling provides a further alternative in case of continuous covariates. In our data analysis, we identified a prognostic model containing both smooth and time-varying effects
Bayesian Cluster Enumeration Criterion for Unsupervised Learning
We derive a new Bayesian Information Criterion (BIC) by formulating the
problem of estimating the number of clusters in an observed data set as
maximization of the posterior probability of the candidate models. Given that
some mild assumptions are satisfied, we provide a general BIC expression for a
broad class of data distributions. This serves as a starting point when
deriving the BIC for specific distributions. Along this line, we provide a
closed-form BIC expression for multivariate Gaussian distributed variables. We
show that incorporating the data structure of the clustering problem into the
derivation of the BIC results in an expression whose penalty term is different
from that of the original BIC. We propose a two-step cluster enumeration
algorithm. First, a model-based unsupervised learning algorithm partitions the
data according to a given set of candidate models. Subsequently, the number of
clusters is determined as the one associated with the model for which the
proposed BIC is maximal. The performance of the proposed two-step algorithm is
tested using synthetic and real data sets.Comment: 14 pages, 7 figure
Performance test of QU-fitting in cosmic magnetism study
QU-fitting is a standard model-fitting method to reconstruct distribution of
magnetic fields and polarized intensity along a line of sight (LOS) from an
observed polarization spectrum. In this paper, we examine the performance of
QU-fitting by simulating observations of two polarized sources located along
the same LOS, varying the widths of the sources and the gap between them in
Faraday depth space, systematically. Markov Chain Monte Carlo (MCMC) approach
is used to obtain the best-fit parameters for a fitting model, and Akaike and
Bayesian Information Criteria (AIC and BIC, respectively) are adopted to select
the best model from four fitting models. We find that the combination of MCMC
and AIC/BIC works fairly well in model selection and estimation of model
parameters in the cases where two sources have relatively small widths and a
larger gap in Faraday depth space. On the other hand, when two sources have
large width in Faraday depth space, MCMC chain tends to be trapped in a local
maximum so that AIC/BIC cannot select a correct model. We discuss the causes
and the tendency of the failure of QU-fitting and suggest a way to improve it.Comment: 8 pages, 9 figures, submitted to MNRA
Optimal model-free prediction from multivariate time series
Forecasting a time series from multivariate predictors constitutes a
challenging problem, especially using model-free approaches. Most techniques,
such as nearest-neighbor prediction, quickly suffer from the curse of
dimensionality and overfitting for more than a few predictors which has limited
their application mostly to the univariate case. Therefore, selection
strategies are needed that harness the available information as efficiently as
possible. Since often the right combination of predictors matters, ideally all
subsets of possible predictors should be tested for their predictive power, but
the exponentially growing number of combinations makes such an approach
computationally prohibitive. Here a prediction scheme that overcomes this
strong limitation is introduced utilizing a causal pre-selection step which
drastically reduces the number of possible predictors to the most predictive
set of causal drivers making a globally optimal search scheme tractable. The
information-theoretic optimality is derived and practical selection criteria
are discussed. As demonstrated for multivariate nonlinear stochastic delay
processes, the optimal scheme can even be less computationally expensive than
commonly used sub-optimal schemes like forward selection. The method suggests a
general framework to apply the optimal model-free approach to select variables
and subsequently fit a model to further improve a prediction or learn
statistical dependencies. The performance of this framework is illustrated on a
climatological index of El Ni\~no Southern Oscillation.Comment: 14 pages, 9 figure
Knot selection by boosting techniques
A novel concept for estimating smooth functions by selection techniques based on boosting is developed. It is suggested to put radial basis functions with different spreads at each knot and to do selection and estimation simultaneously by a componentwise boosting algorithm. The methodology of various other smoothing and knot selection procedures (e.g. stepwise selection) is summarized. They are compared to the proposed approach by extensive simulations for various unidimensional settings, including varying spatial variation and heteroskedasticity, as well as on a real world data example. Finally, an extension of the proposed method to surface fitting is evaluated numerically on both, simulation and real data. The proposed knot selection technique is shown to be a strong competitor to existing methods for knot selection
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