Optimal spatiotemporal reduced order modeling, Part I: proposed framework

An optimal spatiotemporal reduced order modeling framework is proposed for nonlinear dynamical systems in continuum mechanics. In this paper, Part I, the governing equations for a general system are modified for an under-resolved simulation in space and time with an arbitrary discretization scheme. Basic filtering concepts are used to demonstrate the manner in which subgrid-scale dynamics arise with a coarse computational grid. Models are then developed to account for the underlying spatiotemporal structure via inclusion of statistical information into the governing equations on a multi-point stencil. These subgrid-scale models are designed to provide closure by accounting for the interactions between spatiotemporal microscales and macroscales as the system evolves. Predictions for the modified system are based upon principles of mean-square error minimization, conditional expectations and stochastic estimation, thus rendering the optimal solution with respect to the chosen resolution. Practical methods are suggested for model construction, appraisal, error measure and implementation. The companion paper, Part II, is devoted to demonstrating the methodology through a computational study of a nonlinear beam.     

Numerical Analysis of Parameters in a Laminated Beam Model by Radial Basis Functions

In this paper we investigate a thermal driven Micro-Electrical-Mechanical system which was originally designed for inkjet printer to precisely deliver small ink droplets onto paper. In the model, a tiny free-ended beam of metal bends and projects ink onto paper. The model is solved by using the recently developed radial basis functions method. We establish the accuracy of the proposed approach by comparing the numerical results with reported experimental data. Numerical simulations indicate that a light (low composite mass) beam is more stable as it does not oscillate much. A soft (low rigidity) beam results in a higher rate of deflection, when compared to a high rigidity one. Effects caused by the values of physical parameters are also studied. Finally, we give a prediction on the optimal time for the second current pulse which results in maximum rate of second deflection of the beam.     

Vibration analysis of single-walled carbon nanotubes using different gradient elasticity theories

The present work aims at investigating the vibrational characteristics of single-walled carbon nanotubes (SWCNTs) based on the gradient elasticity theories. The small-size effect, which plays an essential role in the dynamical behavior of nanotubes, is captured by applying different gradient elasticity theories including stress, strain and combined strain/inertia ones. The theoretical formulations are established based upon both the Euler–Bernoulli and the Timoshenko beam theories. To validate the accuracy of the present analysis, molecular dynamics (MDs) simulations are also conducted for an armchair SWCNTs with different aspect ratios. Comparisons are made between the aforementioned different gradient theories as well as different beam assumptions in predicting the free vibration response. It is shown that implementation of the strain gradient elasticity by incorporating inertia gradients yields more reliable results especially for shorter length SWCNTs on account of two small scale factors corresponding to the inertia and strain gradients. Also, the difference between two beam models is more prominent for low aspect ratios and the Timoshenko beam model demonstrates a closer agreement with MD results.     

An engineering design methodology with multistage Bayesian surrogates and optimal sampling

This paper presents an adaptive, surrogate-based, engineering design methodology for the efficient use of numerical simulations of physical models. These surrogates are nonlinear regression models fitted with data obtained from deterministic numerical simulations using optimal sampling. A multistage Bayesian procedure is followed in the formulation of surrogates to support the evolutionary nature of engineering design. Information from computer simulations of different levels of accuracy and detail is integrated, updating surrogates sequentially to improve their accuracy. Data-adaptive optimal sampling is conducted by minimizing the sum of the eigenvalues of the prior covariance matrix. Metrics to quantify prediction errors are proposed and tested to evaluate surrogate accuracy given cost and time constraints. The proposed methodology is tested with a known analytical function to illustrate accuracy and cost tradeoffs. This methodology is then applied to the thermal design of embedded electronic packages with five design parameters. Temperature distributions of embedded electronic chip configurations are calculated using spectral element direct numerical simulations. Surrogates, built from 30 simulations in two stages, are used to predict responses of new design combinations and to minimize the maximum chip temperature.     

Vibration analysis of single-walled carbon nanotubes using different gradient elasticity theories

The present work aims at investigating the vibrational characteristics of single-walled carbon nanotubes (SWCNTs) based on the gradient elasticity theories. The small-size effect, which plays an essential role in the dynamical behavior of nanotubes, is captured by applying different gradient elasticity theories including stress, strain and combined strain/inertia ones. The theoretical formulations are established based upon both the Euler–Bernoulli and the Timoshenko beam theories. To validate the accuracy of the present analysis, molecular dynamics (MDs) simulations are also conducted for an armchair SWCNTs with different aspect ratios. Comparisons are made between the aforementioned different gradient theories as well as different beam assumptions in predicting the free vibration response. It is shown that implementation of the strain gradient elasticity by incorporating inertia gradients yields more reliable results especially for shorter length SWCNTs on account of two small scale factors corresponding to the inertia and strain gradients. Also, the difference between two beam models is more prominent for low aspect ratios and the Timoshenko beam model demonstrates a closer agreement with MD results.     

Data-based stochastic subgrid-scale parametrization: an approach using cluster-weighted modelling

A new approach for data-based stochastic parametrization of unresolved scales and processes in numerical weather and climate prediction models is introduced. The subgrid-scale model is conditional on the state of the resolved scales, consisting of a collection of local models. A clustering algorithm in the space of the resolved variables is combined with statistical modelling of the impact of the unresolved variables. The clusters and the parameters of the associated subgrid models are estimated simultaneously from data. The method is implemented and explored in the framework of the Lorenz '96 model using discrete Markov processes as local statistical models. Performance of the cluster-weighted Markov chain scheme is investigated for long-term simulations as well as ensemble prediction. It clearly outperforms simple parametrization schemes and compares favourably with another recently proposed subgrid modelling scheme also based on conditional Markov chains.     

Optimal input design for online parameter estimation for aircraft with multiple control surfaces

An optimal input design method is proposed for online parameter estimation for aircraft with multiple control surfaces. The optimal input is designed considering the input and output constraints. These constraints are constructed based on a military standard, MIL-STD-8785C, viz., the flying qualities of a piloted aircraft. The accuracy of parameter estimation using the optimal input is compared with that of parameter estimation using conventional doublet/3211 inputs. Two online parameter estimation schemes are also considered to evaluate the performance of the designed optimal input: a Bayesian method based on the time domain and an equation-error method based on the frequency domain. The recursive form of the Bayesian method is also derived. Numerical simulations are performed, and the performance, convergence, and accuracy of two online parameter estimation schemes are compared.     

Evaluation of crack locations in beam using artificial neural network-based modified curvature damage index

Although the frequency response-curvature methodology is commonly used to detect irregularities in mechanical and civil structures, the artificial neural network-based frequency response-curvature damage index method may have good efficacy in the detection and localization of structural damages. By utilizing experimental data sets, a novel method is proposed to pinpoint a saw-cut damage location and the degree of damage in beam models. Using a dynamic data logger, the frequency response function of a beam model is obtained for altering damage levels at different positions. As frequency response data contains environmental and operational noise, the accuracy of obtained results may get reduced. To improve the accuracy by reducing the noise effect, the experimentally obtained frequency response data is trained through an artificial neural network. Using central difference approximation, the sets of trained modal data are utilized to determine the improved mode shape curvature. The curvature damage index is then obtained by using the improved mode shape curvature for different damaged scenarios to ultimately identify structural damages.     

On the computation of sound by large-eddy simulations

The effect of the small scales on the source term in Lighthill's acoustic analogy is investigated, with the objective of determining the accuracy of large-eddy simulations when applied to studies of flow-generated sound. The distribution of the turbulent quadrupole is predicted accurately, if models that take into account the trace of the SGS stresses are used. Its spatial distribution is also correct, indicating that the low-wave-number (or frequency) part of the sound spectrum can be predicted well by LES. Filtering, however, removes the small-scale fluctuations that contribute significantly to the higher derivatives in space and time of Lighthill's stress tensor T _ij. The rms fluctuations of the filtered derivatives are substantially lower than those of the unfiltered quantities. The small scales, however, are not strongly correlated, and are not expected to contribute significantly to the far-field sound; separate modeling of the subgrid-scale density fluctuations might, however, be required in some configurations.     

Stochasticity in staged models of epidemics: quantifying the dynamics of whooping cough

Although many stochastic models can accurately capture the qualitative epidemic patterns of many childhood diseases, there is still considerable discussion concerning the basic mechanisms generating these patterns; much of this stems from the use of deterministic models to try to understand stochastic simulations. We argue that a systematic method of analysing models of the spread of childhood diseases is required in order to consistently separate out the effects of demographic stochasticity, external forcing and modelling choices. Such a technique is provided by formulating the models as master equations and using the van Kampen system-size expansion to provide analytical expressions for quantities of interest. We apply this method to the susceptible–exposed–infected–recovered (SEIR) model with distributed exposed and infectious periods and calculate the form that stochastic oscillations take on in terms of the model parameters. With the use of a suitable approximation, we apply the formalism to analyse a model of whooping cough which includes seasonal forcing. This allows us to more accurately interpret the results of simulations and to make a more quantitative assessment of the predictions of the model. We show that the observed dynamics are a result of a macroscopic limit cycle induced by the external forcing and resonant stochastic oscillations about this cycle.     

Robust optimization of an airplane component taking into account the uncertainty of the design parameters

A slat track, structural component of an aircraft wing that transfers the aerodynamical loads, excited by operational forces can result in excessive displacement levels if not properly designed. The design parameter values are not always precisely known but can contain a level of uncertainty to some extent due to, for example dimensional variation. During the different optimization approaches, the slat track geometry is optimized in order to limit the maximum vertical displacement, taking into account the variability of the design parameters. Application and comparison of different optimal, robust and generalized optimization approaches is presented and applied on the slat track finite element model, making use of mean and variance response functions to model the uncertainty on the finite element displacement values. Next to validating different objective function statements, a comparison is also made on the level of accuracy and practicability concerning the different response function models, based on regression techniques and Monte Carlo simulations, optimization and transmissibilities and regressive techniques and vibration reduction over a frequency range. Copyright © 2008 John Wiley & Sons, Ltd.     

Empirical modeling methods using partial data

Methods were developed to calculate empirical models for device error behavior from data sets with missing data. These models can be used to develop reduced test point testing procedures for the devices. Normally, models are built from only full data measurement sets, and partial data sets are discarded. For models built from noisy data, the accuracy of the models improves as more data is used. This paper explores methods to use partial data sets. Both real and simulated data results are described. Simulations show that the proposed partial data methods improve the accuracy of the models for some test points. When these methods are applied to real data where the underlying model has changed, the improvement is less than the simulations predict.     

Hybrid finite element for analysis of functionally graded beams

A hybrid finite element model is presented, where stiffness and mass distributions over a beam with functionally graded material (FGM) are accurately modeled for both elastic and inelastic material responses. Von Mises and Drucker-Prager plasticity models are implemented for metallic and ceramic parts of FGM, respectively. Three-dimensional stress-strain relations are solved by a general closest point projection algorithm, and then condensed to the dimensions of the beam element. Numerical examples and verification studies on a proposed element demonstrate accuracy and robustness under inelastic material response as well as capturing fundamental, higher, and mix modes of vibration frequencies and shapes.     

Feasibility study for the simulation of beam propagation: consideration of coherent lidar performance

To analyze the effects of atmospheric refractive turbulence on coherent lidar performance in a realistic way it is necessary to consider the use of simulations of beam propagation in three-dimensional random media. The capability of the split-step solution to simulate the propagation phenomena is shown, and the limitations and numerical requirements for a simulation of given accuracy are established. Several analytical theories that describe laser beam spreading, beam wander, coherence diameters, and variance and autocorrelation of the beam intensity are compared with results from simulations. Although the analysis stems from a study of coherent lidar performance, the conclusions of the method are applicable to other areas related to beam propagation in the atmosphere.     

负刚度吸振器对有限长弹性梁的抑振效果研究

考虑不同形式负刚度动力吸振器对有限长弹性简支梁动态响应的影响,提出并建立"弹性梁-负刚度动力吸振器"耦合系统动力学模型。基于模态叠加法,推导得到各阶模态对应幅频响应解析表达式。以弹性梁第1阶振动模态作为振动抑制目标,结合固定点理论和最大值最小化优化准则得到各类型动力吸振器的最优设计参数。以功率流作为振动控制效果的评价指标,建立"弹性梁-动力吸振器"耦合系统的导纳功率流理论模型。在此基础上,计算得到安装动力吸振器前后弹性梁的总功率流和净功率流,以及动力吸振器消耗的功率流,研究不同形式动力吸振器的振动抑制效果。最后,选择振动控制效果最显著的动力吸振器作为研究对象,针对部分主要设计参数展开研究。计算结果表明：在目标控制模态频率附近,负刚度动力吸振器对弹性梁动态响应的控制效果较好,且多个振动模态响应均被有效控制;当阻尼元件和负刚度元件同时接地对弹性梁动态响应的控制效果最佳;众多设计参数均存在最优值。     

Dynamics analysis of distributed parameter system subjected to a moving oscillator with random mass, velocity and acceleration

The problem of calculating the response of a distributed parameter system excited by a moving oscillator with random mass, velocity and acceleration is investigated. The system response is a stochastic process although its characteristics are assumed to be deterministic. In this paper, the distributed parameter system is assumed as a beam with Bernoulli–Euler type analytical behaviour. By adopting the Galerkin's method, a set of approximate governing equations of motion possessing time-dependent uncertain coefficients and forcing function is obtained. The statistical characteristics of the deflection of the beam are computed by using an improved perturbation approach with respect to mean value. The method, useful to gathering the stochastic structural effects due to the oscillator–beam interaction, is simple and leads to results very close to Monte Carlo simulation for a wide interval of variation of the uncertainties.     

Non‐linear spatial Timoshenko beam element with curvature interpolation

The paper presents a spatial Timoshenko beam element with a total Lagrangian formulation. The element is based on curvature interpolation that is independent of the rigid‐body motion of the beam element and simplifies the formulation. The section response is derived from plane section kinematics. A two‐node beam element with constant curvature is relatively simple to formulate and exhibits excellent numerical convergence. The formulation is extended to N‐node elements with polynomial curvature interpolation. Models with moderate discretization yield results of sufficient accuracy with a small number of iterations at each load step. Generalized second‐order stress resultants are identified and the section response takes into account non‐linear material behaviour. Green–Lagrange strains are expressed in terms of section curvature and shear distortion, whose first and second variations are functions of node displacements and rotations. A symmetric tangent stiffness matrix is derived by consistent linearization and an iterative acceleration method is used to improve numerical convergence for hyperelastic materials. The comparison of analytical results with numerical simulations in the literature demonstrates the consistency, accuracy and superior numerical performance of the proposed element. Copyright © 2001 John Wiley & Sons, Ltd.     

Leveraging the nugget parameter for efficient Gaussian process modeling

Gaussian process (GP) metamodels have been widely used as surrogates for computer simulations or physical experiments. The heart of GP modeling lies in optimizing the log‐likelihood function with respect to the hyperparameters to fit the model to a set of observations. The complexity of the log‐likelihood function, computational expense, and numerical instabilities challenge this process. These issues limit the applicability of GP models more when the size of the training data set and/or problem dimensionality increase. To address these issues, we develop a novel approach for fitting GP models that significantly improves computational expense and prediction accuracy. Our approach leverages the smoothing effect of the nugget parameter on the log‐likelihood profile to track the evolution of the optimal hyperparameter estimates as the nugget parameter is adaptively varied. The new approach is implemented in the R package GPM and compared to a popular GP modeling R package ( GPfit) for a set of benchmark problems. The effectiveness of the approach is also demonstrated using an engineering problem to learn the constitutive law of a hyperelastic composite where the required level of accuracy in estimating the response gradient necessitates a large training data set.     

A small cantilever beam test for determination of creep properties of materials

The application of small specimen creep test techniques in the evaluation of creep properties of materials in‐service has been increasing. To obtain the creep data accurately and conveniently, a new creep test method with small cantilever beam specimens is proposed. Analytical equations are derived that can convert the load to equivalent uniaxial stress and the displacement rate to equivalent uniaxial strain rate. Three types of the cantilever beam specimens are designed. The optimal configuration of the cantilever beam specimens is recommended with the aid of finite element method, which is further validated by the cantilever beam and uniaxial specimen tests. The results show that parameters obtained from the cantilever beam tests correspond reasonably well with those from uniaxial tests at low stress levels. With a relatively large equivalent gauge length, the cantilever beam specimen allows the small creep strain rate data obtained with a high accuracy.     

Iterative method for the design of a nonperiodic grating-assisted directional coupler

We propose and investigate the optimal design of a nonperiodic grating-assisted directional coupler by iterative methods using the beam propagation method. Computer simulations were carried out at wavelengths of 0.8, 1.3, and 1.5 mum, which are often used in optical communications and networking. We found that the complete power coupling lengths can be reduced considerably in comparison with those in the case of the periodic grating-assisted waveguides with the same set of parameters.