A global Arnoldi method for the model reduction of second-order structural dynamical systems

Abstract In this paper we consider the reduction of second-order dynamical systems with multiple inputs and multiple outputs (MIMO) arising in the numerical simulation of mechanical structures. In commercial software for the kind of application considered here, modal reduction is commonly used to obtain a reduced system with good approximation abilities of the original transfer function in the lower frequency range. In recent years new methods to reduce dynamical systems based on (block) versions of Krylov subspace methods emerged. This work concentrates on the reduction of second-order MIMO systems by the global Arnoldi method, an efficient extension of the standard Arnoldi algorithm for MIMO systems. In particular, a new model reduction algorithm for second order MIMO systems is proposed which automatically generates a reduced system of given order approximating the transfer function in the lower range of frequencies. It is based on the global Arnoldi method, determines the expansion points iteratively and the number of moments matched per expansion point adaptively. Numerical examples comparing our results to modal reduction and reduction via the block version of the rational Arnoldi method are presented

MODEL UPDATING AND STRUCTURAL HEALTH MONITORING OF HORIZONTAL AXIS WIND TURBINES VIA ADVANCED SPINNING FINITE ELEMENTS AND STOCHASTIC SUBSPACE IDENTIFICATION METHODS

Wind energy has been one of the most growing sectors of the nation’s renewable energy portfolio for the past decade, and the same tendency is being projected for the upcoming years given the aggressive governmental policies for the reduction of fossil fuel dependency. Great technological expectation and outstanding commercial penetration has shown the so called Horizontal Axis Wind Turbines (HAWT) technologies. Given its great acceptance, size evolution of wind turbines over time has increased exponentially. However, safety and economical concerns have emerged as a result of the newly design tendencies for massive scale wind turbine structures presenting high slenderness ratios and complex shapes, typically located in remote areas (e.g. offshore wind farms). In this regard, safety operation requires not only having first-hand information regarding actual structural dynamic conditions under aerodynamic action, but also a deep understanding of the environmental factors in which these multibody rotating structures operate. Given the cyclo-stochastic patterns of the wind loading exerting pressure on a HAWT, a probabilistic framework is appropriate to characterize the risk of failure in terms of resistance and serviceability conditions, at any given time. Furthermore, sources of uncertainty such as material imperfections, buffeting and flutter, aeroelastic damping, gyroscopic effects, turbulence, among others, have pleaded for the use of a more sophisticated mathematical framework that could properly handle all these sources of indetermination. The attainable modeling complexity that arises as a result of these characterizations demands a data-driven experimental validation methodology to calibrate and corroborate the model. For this aim, System Identification (SI) techniques offer a spectrum of well-established numerical methods appropriated for stationary, deterministic, and data-driven numerical schemes, capable of predicting actual dynamic states (eigenrealizations) of traditional time-invariant dynamic systems. As a consequence, it is proposed a modified data-driven SI metric based on the so called Subspace Realization Theory, now adapted for stochastic non-stationary and timevarying systems, as is the case of HAWT’s complex aerodynamics. Simultaneously, this investigation explores the characterization of the turbine loading and response envelopes for critical failure modes of the structural components the wind turbine is made of. In the long run, both aerodynamic framework (theoretical model) and system identification (experimental model) will be merged in a numerical engine formulated as a search algorithm for model updating, also known as Adaptive Simulated Annealing (ASA) process. This iterative engine is based on a set of function minimizations computed by a metric called Modal Assurance Criterion (MAC). In summary, the Thesis is composed of four major parts: (1) development of an analytical aerodynamic framework that predicts interacted wind-structure stochastic loads on wind turbine components; (2) development of a novel tapered-swept-corved Spinning Finite Element (SFE) that includes dampedgyroscopic effects and axial-flexural-torsional coupling; (3) a novel data-driven structural health monitoring (SHM) algorithm via stochastic subspace identification methods; and (4) a numerical search (optimization) engine based on ASA and MAC capable of updating the SFE aerodynamic model

Automatic and Sampling-Free Parametric Model Order Reduction of Vibro-Acoustic Systems

Recently, a novel parametric model order reduction formulation has been derived for vibroacoustic systems that allows for the reduction of systems with low-rank parametric changes [1]. This scheme does not require sampling of the parameter space, in contrast to conventional parametric model reduction techniques. This means that a single reduction basis, obtained with conventional non-parametric model order reduction schemes, can be used for a wide range of parameter values. This is done by rewriting the system in a non-parametric form, in which the low-rank contributions act as inputs. A disadvantage of this scheme is that the size of the input matrix scales with the amount of chosen parameters, leading to a potentially large reduced basis when many parameters are considered. Therefore, in [2] an automatic Krylov reduction scheme has been proposed that utilizes the similarity in the reduced bases for inputs which are spaced closely together to still get a small reduced basis with a large number of inputs. This is done by using a combination of block second order Arnoldi with a singular value decomposition acting on the resulting basis. The algorithm includes an error estimator that uses a complementary approximation to calculate the error. The main advantages of this algorithm as compared to the commonly used iterative rational Krylov approach [3] are that only a small amount of system inversions are required and that the final reduced order model has the desired predefined relative error in the specified frequency band. In this paper the automatic Krylov reduction scheme and low-rank parametric model order reduction approach are combined and a suitable error estimator is derived, to arrive at compact but accurate parametric reduced order models. The effectiveness is shown with several examples

Model Order Reduction

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This three-volume handbook covers methods as well as applications. This third volume focuses on applications in engineering, biomedical engineering, computational physics and computer science

Numerical methods for parametric model reduction in the simulation of disk brake squeal

We present numerical methods for model reduction in the numerical simulation of disk brake squeal. Automotive disk brake squeal is a high frequency noise phenomenon based on self excited vibrations. Our method is based on a variation of the proper orthogonal decomposition method and involves the solution of a large scale, parametric eigenvalue problem. Several important challenges arise, some of which can be traced back to the finite element modeling stage. Compared to the current industrial standard our new approach is more accurate in vibration prediction and achieves a better reduction in model size. This comes at the price of an increased computational cost, but it still gives useful results when the traditional method fails to do so. We illustrate the results with several numerical experiments, some from real industrial models, some from simpler academic models. These results indicate where improvements of the current black box industrial codes are advisable

The Biglobal Instability of the Bidirectional Vortex

State of the art research in hydrodynamic stability analysis has moved from classic one-dimensional methods such as the local nonparallel approach and the parabolized stability equations to two-dimensional, biglobal, methods. The paradigm shift toward two dimensional techniques with the ability to accommodate fully three-dimensional base flows is a necessary step toward modeling complex, multidimensional flowfields in modern propulsive applications. Here, we employ a two-dimensional spatial waveform with sinusoidal temporal dependence to reduce the three-dimensional linearized Navier-Stokes equations to their biglobal form. Addressing hydrodynamic stability in this way circumvents the restrictive parallel-flow assumption and admits boundary conditions in the streamwise direction. Furthermore, the following work employs a full momentum formulation, rather than the reduced streamfunction form, accounting for a nonzero tangential mean flow velocity. This approach adds significant complexity in both formulation and implementation but renders a more general methodology applicable to a broader spectrum of mean flows. Specifically, we consider the stability of three models for bidirectional vortex flow. While a complete parametric study ensues, the stabilizing effect of the swirl velocity is evident as the injection parameter, kappa, is closely examined