### Computational Methods in Stochastic Dynamics: Volume 2

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The second approach in order to reduce the computational costs is the parallelization of the analysis. The importance of parallel computing in stochastic structural mechanics has long been emphasized see e. In the field of structural dynamics, Johnson et al.

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Applications of parallel processing in context with uncertainty quantification in structural dynamics can be found in e. The third approach is concerned with the minimization of the computational costs of one simulation. This can be achieved by using local approximation methods based on the perturbation method or Neumann expansion series which are widely used in computational stochastic analysis [28]. The perturbation method is based on a Taylor expansion of the structural matrices, the load vector and the vector of the structural response. The computational efforts grow with the number of uncertain parameters and with the order of the expansion.

In case of small coefficients of variation, a linear approximation of the response turns out to predict the structural response sufficiently accurately. The Neumann expansion takes advantage — similarly to the perturbation approach — from a reference solution and is based on the approximation of the solution using a few basis vectors which span the preconditioned stochastic Krylov subspace as shown in [35]. Hence, the Neumann method constitutes a special case of a stochastic reduced basis method as proved in [27].

Another stochastic reduced basis method which approximates the solution in the random dimension — and not in space as described above — is given by the Polynomial Chaos decomposition scheme [18]. The key ingredient of the chaos expansion is to represent the random entities by an orthogonal basis defined in a Hilbert space.

Applications of Polynomial Chaos in dynamics, specifically for the random eigenvalue problem, are shown in e.

### Kundrecensioner

A comparison of different subspace projection techniques can be found in [44]. The goal of reducing the computational costs of one single simulation can also be reached by using the substructure techniques described in Sect. Hence, recent advances and new developments in the combination of uncertainty quantification methods and model reduction techniques are addressed in the following. In addition, the substructures are usually statistically independent since they are manufactured by different companies and finally assembled. Hence, this makes it possible to independently analyze the components with the option of applying different uncertainty propagation procedures to the components.

Component mode synthesis is also particularly advantageous for the application of perturbation approaches in order to estimate the variability at the global level. Due to the almost linear relationship between the structural parameters and the component modal properties, as well as between the component modal properties and global modal properties, it is possible to accurately approximate the variability of the global modal properties and frequency response functions by linear approximations. Another approach for the assessment of the variability of the eigenfrequency and eigenvectors using the advantages of substructuring is presented in [41], where a special form of CMS is proposed which is oriented towards an efficient computation of eigenvectors and eigenvalues.

This novel approach will be discussed in detail in the next section. In addition, a second approach will be proposed that uses the advantages of substructuring by randomly combining independently simulated substructures.

### Computational Methods in Stochastic Dynamics - Volume 2 | Manolis Papadrakakis | Springer

Hence, in addition to the probability density functions of the physical parameters, also the random combination, expressed by discrete uniform distributions for the set of substructure matrices, introduces variability in order to capture the uncertainty by which the structure is affected. Finally, the combination of the approximation methods and substructuring constitutes a straight forward combination of advanced techniques for uncertainty propagation and model reduction schemes.

These approaches will be described in the following. The special form of the Craig-Bampton substructure matrices with its high sparsity see Eq.

In order to calibrate the uncertain reduced substructures, samples of substructures are generated by considering the scatter of the physical parameters. Based on this sample set of each substructure, the second order statistics of the entries of the reduced matrices can be estimated. When generating the calibration set for the modal mass matrix it should be noted that in contrast to the matrices mBB and kBB the coupling matrix mIB is a function of the normal modes.

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Hence, the entries of this matrix obtained from two different simulations are only physically comparable if the sequence of the mode shapes is the same in both simulations. Hence, mode pairing has to be adopted, which can be performed using e. The coefficients 12 G. The extraction of the important component can be carried out by projecting the eigenvectors and constraint modes of the current simulation onto the subspace X, i.

## SIAM Journal on Numerical Analysis

The same procedure is employed for the constraint modes. If the eigenvector or constraint mode can not be represented with a certain accuracy by the current basis vectors, then the subspace is augmented. An eigenvalue solution of the covariance matrix yields the basis vectors for reproducing the uncertainty in the calibration set.

Hence, in each simulation, the vector x. These substructure matrices are assembled to the global reduced matrices according to the Craig-Bampton reduction scheme see Eqs. An example-based verification of this approach where the results are compared with direct Monte Carlo simulation will be presented in Sect. Model Reduction and Uncertainties in Structural Dynamics 13 4. The idea consists of the generation of a pool of random substructure matrices and to combine these substructure matrices randomly. More specifically, J.

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These samples are generated using direct Monte Carlo simulation. In order to represent the variability with a few samples it can also be advantageous to use Latin Hypercube sampling.

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The random combination of the substructures introduces some additional variability in the structural response which makes the approach particularly advantageous in case of many substructures. This method aims to imitate the randomness associated with random sampling. Once the set of substructures is selected, the global reduced structural matrices are assembled by applying the Craig-Bampton method see Eqs. It should be noted that this approach can also be used if mode switching occurs and is thus also applicable for the higher frequency range. An example-based verification of this approach will be given in Sect.

The most timeconsuming part concerns the solution of the eigenvalue problem. Therefore it is advantageous to apply approximate solution schemes. Among these approximation methods, the Stochastic Reduced Basis Approximation SRBA introduced in [36], which constitutes a reduced basis formulation for the efficient solution of largescale eigenvalue problems, will be in the focus of this manuscript. In the present application, these structural matrices do not refer to the full system, but to the substructures of the investigated structure.

Hence, instead of taking one simulated matrix for each substructure randomly from the sample pool, these matrices are approximated using two previously generated matrices. In this way, a meta-model for constructing efficiently structural matrices and for solving the corresponding eigenanalysis in a reduced space can be constructed. Model Reduction and Uncertainties in Structural Dynamics 15 4. Hence, Eq.

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The associated structural stiffness and mass matrices are now defined as M. In the following, however, it is not assumed that these values are accurate, but that Q. This assumption implies that the correct eigenvectors are a linear combination of the vectors in the matrix Q. This model is divided into the following ten substructures shown in Fig.

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The uncertainties of the structural parameters, i. A fine mesh is adopted to model the build-up structure leading to approximately 36, DOFs. This size of the structural matrices makes it possible to explore also the scalability of the proposed algorithms in addition to investigating their accuracy. The quantities of interest are the statistics of the eigenvalues. For the methods that are based on a random combination of substructures, a set of 50 substructure matrices is generated and combined randomly in order to assess the statistical description of the output quantities of interest.

In order to capture the randomness with few samples, Latin Hypercube sampling is adopted.