Calibration and Reduction of Large-Scale Dynamic Models - Application to Wind Turbine Blades

This thesis investigates the validity of structural dynamics models of wind turbine blades. An outlook on methods for model calibration to make models valid for their intended use is presented in the thesis. The intention is to make the models valid for robust predictions. The model validity is here assessed to be of hierarchical dual level. On one hand, a detailed structural dynamics model needs to be substantiated by good correlation between experimental results of wind turbine testing and theoretical simulation results using that model. On the other hand, after that detailed model has been validated, a model of significantly low order based on the detailed model has been validated by a good model-to-model correlation. With the connection between models, this implies that also the low order model is implicitly validated by testing. The development of a highly detailed structural dynamics model provides real physical insights to observation made during testing. This model is often developed using finite element analysis. A model verification and validation activity is done to create a three dimensional finite element model that is capable to predict the dynamics of wind turbine blade with sufficient accuracy. Integration of such large-scale models of wind turbine blades in aeroelastic simulations places an untenable demand on computational resources and, hence, means of speed-up become necessary. The common practice is to develop, calibrate and validate an industry-standard beam model against the simulated data obtained from the validated highly detailed rotor blade model. However, the validated beam model cannot well capture the coupling features of the highly detailed model because of its inherent limitations. Our scientific hypothesis is that it is possible to create low-order rotor blade models which preserve the vibrational pattern of the baseline model at its eigenfrequecies and also closely mimic its input-output behavior. Toward this end, a quasi optimal modal truncation algorithm is developed to yield reduced models which have the eigenmodes with highest contribution to the input-output map of the large-scale model. The predictive capability of the created reduced model is compared with that of the validated beam model

Calibration and Reduction of Large-Scale Dynamic Models - Application to Wind Turbine Blades

This thesis investigates the validity of structural dynamics models of wind turbine blades. An outlook on methods for model calibration to make models valid for their intended use is presented in the thesis. The intention is to make the models valid for robust predictions. The model validity is here assessed to be of hierarchical dual level. On one hand, a detailed structural dynamics model needs to be substantiated by good correlation between experimental results of wind turbine testing and theoretical simulation results using that model. On the other hand, after that detailed model has been validated, a model of significantly low order based on the detailed model has been validated by a good model-to-model correlation. With the connection between models, this implies that also the low order model is implicitly validated by testing. The development of a highly detailed structural dynamics model provides real physical insights to observation made during testing. This model is often developed using finite element analysis. A model verification and validation activity is done to create a three dimensional finite element model that is capable to predict the dynamics of wind turbine blade with sufficient accuracy. Integration of such large-scale models of wind turbine blades in aeroelastic simulations places an untenable demand on computational resources and, hence, means of speed-up become necessary. The common practice is to develop, calibrate and validate an industry-standard beam model against the simulated data obtained from the validated highly detailed rotor blade model. However, the validated beam model cannot well capture the coupling features of the highly detailed model because of its inherent limitations. Our scientific hypothesis is that it is possible to create low-order rotor blade models which preserve the vibrational pattern of the baseline model at its eigenfrequecies and also closely mimic its input-output behavior. Toward this end, a quasi optimal modal truncation algorithm is developed to yield reduced models which have the eigenmodes with highest contribution to the input-output map of the large-scale model. The predictive capability of the created reduced model is compared with that of the validated beam model

Stochastic model updating and model selection with application to structural dynamics

Uncertainty induced by our incomplete state of knowledge about engineering systems and their surrounding environment give rise to challenging problems in the process of building predictive models for the system behavior. One such challenge is the model selection problem, which arises due to the existence of invariably multiple candidate models with different mathematical forms to represent the system behavior, and so there is a need to assess their plausibility based on experimental data. However, model selection is a non-trivial problem since it involves a trade-off between predictive power and simplicity. Another challenge is the model updating problem, which refers to the process of inference of the unknown parameters of a specific model structure based on experimental data so that it makes more accurate predictions of the system behavior. However, the existence of modeling errors and uncertainties, e.g., the measurement noise and variability in material properties, along with sparsity of data regarding the parameters often make model updating an ill-conditioned problem. In this thesis, probabilistic tools and methodologies are established for model updating and selection of structural dynamic systems that can deal with the uncertainty arising from missing information, with special attention given to systems which can have high-dimensional uncertain parameter vector. The model updating problem is first formulated in the \textit{Frequentist} school of statistical inference. A framework for stochastic updating of linear finite element models and the uncertainty associated to their parameters is developed. It uses the techniques of damping equalization to eliminate the need for mode matching and bootstrapping to construct uncertainty bounds on the parameters. A combination of ideas from bootstrapping and unsupervised machine learning algorithms lead to an automated modal updating algorithm suitable for identification of large-scale systems with many inputs and outputs. The model updating problem is then formulated in the \textit{Bayesian} school of statistical inference. A recently appeared multi-level Markov chain Monte Carlo algorithm, ABC-SubSim, for approximate Bayesian computation is used to solve Bayesian model updating for dynamic systems. ABC-SubSim exploits the Subset Simulation method to efficiently draw samples from posterior distributions with high-dimensional parameter spaces. Formulating a dynamic system in form of a general hierarchical state-space model opens up the possibility of using ABC-SubSim for Bayesian model selection. Finally, to perform the exact Bayesian updating for dynamic models with high-dimensional uncertainties, a new multi-level Markov chain Monte Carlo algorithm called Sequential Gauss-Newton algorithm is proposed. The key to success for this algorithm is the construction of a proposal distribution which locally approximates the posterior distribution while it can be readily sampled

Stochastic model updating and model selection with application to structural dynamics

Uncertainty induced by our incomplete state of knowledge about engineering systems and their surrounding environment give rise to challenging problems in the process of building predictive models for the system behavior. One such challenge is the model selection problem, which arises due to the existence of invariably multiple candidate models with different mathematical forms to represent the system behavior, and so there is a need to assess their plausibility based on experimental data. However, model selection is a non-trivial problem since it involves a trade-off between predictive power and simplicity. Another challenge is the model updating problem, which refers to the process of inference of the unknown parameters of a specific model structure based on experimental data so that it makes more accurate predictions of the system behavior. However, the existence of modeling errors and uncertainties, e.g., the measurement noise and variability in material properties, along with sparsity of data regarding the parameters often make model updating an ill-conditioned problem. In this thesis, probabilistic tools and methodologies are established for model updating and selection of structural dynamic systems that can deal with the uncertainty arising from missing information, with special attention given to systems which can have high-dimensional uncertain parameter vector. The model updating problem is first formulated in the \textit{Frequentist} school of statistical inference. A framework for stochastic updating of linear finite element models and the uncertainty associated to their parameters is developed. It uses the techniques of damping equalization to eliminate the need for mode matching and bootstrapping to construct uncertainty bounds on the parameters. A combination of ideas from bootstrapping and unsupervised machine learning algorithms lead to an automated modal updating algorithm suitable for identification of large-scale systems with many inputs and outputs. The model updating problem is then formulated in the \textit{Bayesian} school of statistical inference. A recently appeared multi-level Markov chain Monte Carlo algorithm, ABC-SubSim, for approximate Bayesian computation is used to solve Bayesian model updating for dynamic systems. ABC-SubSim exploits the Subset Simulation method to efficiently draw samples from posterior distributions with high-dimensional parameter spaces. Formulating a dynamic system in form of a general hierarchical state-space model opens up the possibility of using ABC-SubSim for Bayesian model selection. Finally, to perform the exact Bayesian updating for dynamic models with high-dimensional uncertainties, a new multi-level Markov chain Monte Carlo algorithm called Sequential Gauss-Newton algorithm is proposed. The key to success for this algorithm is the construction of a proposal distribution which locally approximates the posterior distribution while it can be readily sampled

Modal Reduction Based on Accurate Input-Output Relation Preservation

An eigenmode based model reduction technique is proposed to obtain low-order models which contain the dominanteigenvalue subspace of the full system. A frequency-limited interval dominancy is introduced to this technique to measure the output deviation caused by deflation of eigenvalues from the original system in the frequency range of interest. Thus, the dominant eigensolutions with effective contribution can be identified and retained in the reduced-order model. This metric is an explicit formula in terms of the corresponding eigensolution. Hence, the reduction can be made at a low computational cost. In addition, the retained low-order model does not contain any uncontrollable and unobservable eigensolutions. The performance of the created reduced-order models, in regard to the approximation error, is examined by applying three different input signals; unit-impulse, unit-step and linear chirp

Modal Reduction Based on Accurate Input-Output Relation Preservation

An eigenmode based model reduction technique is proposed to obtain low-order models which contain the dominanteigenvalue subspace of the full system. A frequency-limited interval dominancy is introduced to this technique to measure the output deviation caused by deflation of eigenvalues from the original system in the frequency range of interest. Thus, the dominant eigensolutions with effective contribution can be identified and retained in the reduced-order model. This metric is an explicit formula in terms of the corresponding eigensolution. Hence, the reduction can be made at a low computational cost. In addition, the retained low-order model does not contain any uncontrollable and unobservable eigensolutions. The performance of the created reduced-order models, in regard to the approximation error, is examined by applying three different input signals; unit-impulse, unit-step and linear chirp

On Gramian-Based Techniques for Minimal Realization of Large-Scale Mechanical Systems

Abstract In this paper, a review of Gramian-based minimal realization algorithms is presented, several comments regarding their properties are given and the ill-condition and efficiency that arise in balancing of large-scale realizations is being addressed. A new algorithm to treat non-minimal realization of very large second-order systems with dense clusters of close eigenvalues is proposed. The method benefits the effectiveness of balancing techniques in treating of non-minimal realizations in combination with the computational efficiency of modal techniques to treat large-scale problems

A Metric for Modal Truncation in Model Reduction Problems Part 1: Performance and Error Analysis

The strength of the modal based reduction approach resides in its simplicity, applicability to treat moderate-size systems and also in the fact that it preserves the original system pole locations. However, the main restriction has been in the lack of reliable techniques for identifying the modes that dominate the input-output relationship. To address this problem, an enhanced modal dominancy approach for reduction of second-order systems is presented. Briefly stated, a modal reduction approach is combined with optimality considerations such that the difference between the frequency response function of the full and reduced modal model is minimized in H2 sense. A modal ranking process is performed without solving Lyapunov equations. In the first part of this study, a literature survey on different model reduction approaches and a theoretical investigation of the modified modal approach is presented. The error analysis of the proposed dominancymetric is carried out. Furthermore, the performance of the method is validated for a lightly damped structure and the results are compared with other dominancy metrics. Finally the optimality of the obtained reduced model is discussed and the results are compared with the optimum solution

On Grammian-based reduction methods for moderate size systems

Over the last decades, there has been a constantly increasing interest in the compact reduced dynamical models. The central idea of model reduction is to systematically capture the main input-output properties by a much simpler model than needed for describing the entire states of the system. Among the most popular model reduction approaches, particularly in systems in the order of a couple of thousands, singular value decomposition based are most common model reduction schemes. In this note, a survey of Grammian-based model reduction techniques for moderate size systems is presented. Comments regarding their properties and discussion about their computational issues are given. Computational efforts needed in reduction methods based on Sylvester and Lyapunov equation are being compared. This investigation is followed by a numerical moderate-size example with dense clusters of close eigenvalues. Finally, results of the competing reduction approaches are compared with respect to computational cost and approximation error for same size approximants

Modal Dominancy Analysis Based on Modal Contribution to Frequency Response Function ℋ2-Norm

A main restriction in the general applicability of modal reduction techniques has been the lack of a proper dominancy analysis as well as the lack of a guaranteed bound for the approximation error. In this study, a modal dominancy approach for reduction of dynamical systems is presented. A quadratic-metric, introduced based on modal contribution to the ℋ2ℋ2-norm of the frequency response function matrix, is given in closed-form formulation. Briefly stated, a performance and error analysis of the proposed modal dominancy procedure is carried out, the problem of metric non-uniqueness as well as the treatment of structural non-minimality for a class of systems with multiple eigenvalues is described, and a method to circumvent this problem is proposed. In treating problems with high-dimensional input space, such as in moving and/or distributed loading problems, the presented method is an improvement as it incorporates information extracted from the structural and spectral properties of the input force in the modal dominancy analysis. In addition, the method׳s performance is validated for reduction of a large-scale finite element model, originated from a moving load problem in railway mechanics, and the results are compared with the balancing approach