A Kriging-Based Uncertainty Quantification for Fracture Assessment

This paper presents a computational framework for uncertainty quantification in prognostics in the context of condition-based monitoring of aerospace systems. The different sources of uncertainty and the various uncertainty quantification activities in condition-based prognostics are outlined in detail, and it is demonstrated that the Bayesian subjective approach is suitable for interpreting uncertainty in online monitoring. A state-space model-based framework for prognostics, that can rigorously account for the various sources of uncertainty, is presented. Prognostics consists of two important steps. First, the state of the system is estimated using Bayesian tracking, and then, the future states of the system are predicted until failure, thereby computing the remaining useful life of the system. The proposed framework is illustrated using the power system of a planetary rover test-bed, which is being developed and studied at NASA Ames Research Center.

An overview is presented of the methods available to quantify uncertainties and to combine them in a systems reliability analysis. The different sources of uncertainty are identified and a discussion is provided to illustrate their different nature. The methods of reliability analysis are then described showing how they can incorporate parameter and model uncertainty. Finally Bayesian methods are presented illustrating how to update probabilistic statements with new information and how to incorporate parameter and model uncertainty in the reliability predictions.

The prediction of creep deformations of concrete and reinforced concrete structures has been a field of extensive research. Several different creep models have been developed. Most of the models were developed for constant concrete stresses; thus, in the case of varying stresses, a specific time integration is necessary. Currently, when modeling concrete creep, the engineering focus is on the application of sophisticated time-integration methods rather than choosing more appropriate creep models or considering the scatter of input parameters. For this reason, this paper presents a method to quantify the uncertainties of creep prediction originating from the selection of creep models, from the time-integration methods, and from input parameter uncertainty. By adapting variance-based global sensitivity analysis, a methodology is developed to quantify the influence of different sources of uncertainty. By application of the developed method, it is determined that selecting more appropriate creep model...

In the context of complex applications from engineering sciences the solution of identification problems still poses a fundamental challenge. In terms of Uncertainty Quantification (UQ), the identification problem can be stated as a separation task for structural model and parameter uncertainty. This thesis provides new insights and methods to tackle this challenge and demonstrates these developments on an industrial benchmark use case combining simulation and real-world measurement data. While significant progress has been made in development of methods for model parameter inference, still most of those methods operate under the assumption of a perfect model. For a full, unbiased quantification of uncertainties in inverse problems, it is crucial to consider all uncertainty sources. The present work develops methods for inference of deterministic and aleatoric model parameters from noisy measurement data with explicit consideration of model discrepancy and additional quantification of the associated uncertainties using a Bayesian approach. A further important ingredient is surrogate modeling with Polynomial Chaos Expansion (PCE), enabling sampling from Bayesian posterior distributions with complex simulation models. Based on this, a novel identification strategy for separation of different sources of uncertainty is presented. Discrepancy is approximated by orthogonal functions with iterative determination of optimal model complexity, weakening the problem inherent identifiability problems. The model discrepancy quantification is complemented with studies to statistical approximate numerical approximation error. Additionally, strategies for approximation of aleatoric parameter distributions via hierarchical surrogate-based sampling are developed. The proposed method based on Approximate Bayesian Computation (ABC) with summary statistics estimates the posterior computationally efficient, in particular for large data. Furthermore, the combination with divergence-based subset selection provides a novel methodology for UQ in stochastic inverse problems inferring both, model discrepancy and aleatoric parameter distributions. Detailed analysis in numerical experiments and successful application to the challenging industrial benchmark problem -- an electric motor test bench -- validates the proposed methods.

In the study of the dynamics of mechanical systems with distributed parameters, the discretization of partial differential equations by the finite element method, finite difference method, or boundary element method is widely used. This makes it possible to simplify the analysis of dynamic processes by reducing it to solving a system of ordinary differential equations. The aim of this paper is to mathematically substantiate the method of direct discretization of long-dimensional elastic links as applied to free and forced longitudinal oscillations of a straight rod as a mechanical system with distributed parameters, and to investigate the convergence of the method in determining the natural frequencies and amplitudes of forced oscillations of mechanical systems. The method of direct discretization is considered in the application to a rod with distributed parameters, which performs longitudinal oscillations. It is shown that by mathematical discretization of the wave equation, the boundary value problem can be reduced to the analysis of oscillatory phenomena in a chain mechanical system of material points. The formulas for determining the masses of the inertial elements of the computational model and the stiffness coefficients of the elastic connections are given. The convergence of the method is investigated by determining the natural frequencies of oscillations of a rod with free ends, as well as with one end clamped and the other end free, and comparing the approximate results of calculations with the exact results. The effect of the number of degrees of freedom, rod length, and material density on the eigenfrequencies is investigated. It is found that to ensure sufficient accuracy for engineering practice in determining the three or four lowest natural frequencies, computational models with 8-10 degrees of freedom can be used. It is emphasized that the cross-sectional area of the rod does not affect the characteristics of the frequency spectrum of the mechanical system. The zero value of the lowest natural frequency of an elastic rod with free ends can be explained by the fact that this value corresponds not to the oscillatory but to the translational motion of a mechanical system that is not connected to the base. Forced longitudinal oscillations of an elastic rod, one end of which is clamped at the base and the other end is free, under power and kinematic excitation are considered. The amplitude-frequency characteristics of mechanical systems are constructed on the examples of the dependences of the movement amplitude of one of the material points and the amplitude of the longitudinal force in one of the elastic links of the computational model on the frequency of forced oscillations. Since the dissipation of the mechanical energy of the system during its oscillations is not taken into account, as the frequency of forced oscillations approaches one of the natural frequencies of the system, the movement amplitudes, as well as the amplitudes of internal forces, tend to infinity. The actual values of resonant amplitudes are not of practical interest, since resonant modes of operation are not expected for the vast majority of machine units. The scientific novelty of the results obtained is that the mathematical substantiation of the method of direct discretization of the rod in its application to its longitudinal oscillations, the creation of computational algorithms and the study of the convergence of the results of determining the natural frequencies and amplitudes of forced oscillations can be considered as a development of the computer methodology for the dynamic analysis of mechanical systems with distributed parameters. The practical significance and scope of further research of the results of this work are based on the wide possibilities of practical application of the direct discretization method in the study of dynamic processes in rather complex mechanical systems of such technical objects as drilling rigs, cranes, mine hoisting machines, conveyors, rotary kilns, etc. This is facilitated by the simplicity and computational efficiency of the method in solving problems of dynamics of both linear and nonlinear mechanical systems. The method can be applied in the study of both stationary and transient dynamic processes in machine units.

This paper deals with the modelling and the prediction of the dynamic instabilities for a rubbing system. Two hybrid approaches are introduced for dynamic instability analysis and applied to a reduced disc brake system. The methods are based on stochastic algorithms coupled with the finite element method (FEM) using the complex eigenvalue analysis technique. By considering the input parameters as random variables, the uncertainty analysis is performed through two approaches to predict the unstable frequencies of a braking system (1) Monte-Carlo (MC) using Mersenne-Twister (MT19937) algorithm and (2) periodic sampling technique. Since the mechanism of brake squeal involves many design parameters, stochastic finite element approaches will be coupled with sensitivity algorithms, e.g. Variance-Based Sensitivity Analysis and Fourier Sensitivity Analysis Test, to analyze the contribution of each random variable on the dynamic instabilities. First, a comparison between the two stochastic algorithms is performed on standard analytical models. The objective is to validate the accuracy and to assess the numerical efficiency that FAST presents to (1) propagate the uncertainties upstream of the model and (2) to compute the partial variances of the model output. Secondly, the coupling of the previous stochastic algorithms with FEM is carried out and tested through a reduced brake system consisting of a rotating disc with two flat pads. Results show that the hybrid approach FAST-FE is more robust and computationally more efficient compared to the widely used MC-FE for these types of problems. FAST-FE solver converges, within a reasonable computing time, either to approximate the probability density function of the random variables or to compute the partial variances of the dynamic instabilities. Hence, it can be considered as an efficient numerical method for squeal instability analysis in order to reduce squeal noise of such a mechanical system.

A numerical method of structural dynamic shape optimization is presented by using reproducing kernel particle method (RKPM), by which the mesh distortion that exists in shape optimal method based on finite element can be eliminated completely and the optimal model for structural dynamic optimization design is built. The discreteness-based design sensitivity analysis in both natural frequency and dynamic response is proposed by using direct differentiation method and discrete derivatives on the basis of structural dynamic analysis, in which the penalty method is employed into imposing the essential boundary conditions, and the derivatives of shape functions with respect to design variables are derived. The algorithm of dynamic sensitivity analysis is testified by numerical example, and the numerical results obtained are in good agreement with those obtained using semi-analytical method and global finite differences method. Finally, by integrating the algorithm mentioned based on RKPM with parameterized descriptive method of boundary shape, two examples for structural dynamic shape optimization are performed.

A procedure for seismic fragility assessment, suitable for application to non ductile RC structures is presented, which is based on the estimate of a response surface that gives the probability of failure of the structure as a function of the random variables that affect the response. The seismic fragility or risk is then evaluated through numerical integration. The method considers different sources of uncertainty: (i) in the seismic input, through the use of different accelerograms for the dynamic analysis; (ii) in the structural response, through the use of a refined nonlinear finite element model; and (iii) in the ultimate state capacity, taking into account the different modes of failure which may occur, for which a random mechanical capacity model is available. Aiming at reducing the number of simulation analyses, the uncertainties on the seismic input and on the mechanical parameters governing the response are treated according to the response surface methodology, while the limit-state randomness is treated explicitly during the simulations. Using the proposed procedure, the seismic safety of two reinforced concrete bridges from Italian highway network, with simply supported deck and either single stem or frame piers, is evaluated. The results are expressed in terms of fragility curves as function of spectral acceleration. The obtained results highlight the influence of material randomness on reliability and the relative importance of seismic input with respect to mechanical and epistemic uncertainty.

Direct probability integral method for stochastic response analysis of static and dynamic structural systems

A major challenge in the identification of material properties is handling different sources of uncertainty in the experiment and the modelling of the experiment for estimating the resulting uncertainty in the identified properties. Numerous improvements in identification methods have provided increasingly accurate estimates of various material properties. However, characterizing the uncertainty in the identified properties is still relatively crude. Different material properties obtained from a single test are not obtained with the same confidence. Typically the highest uncertainty is associated with respect to properties to which the experiment is the most insensitive. In addition, the uncertainty in different properties can be strongly correlated, so that obtaining only variance estimates may be misleading. A possible approach for handling the different sources of uncertainty and estimating the uncertainty in the identified properties is the Bayesian method. This method was introduced in the late 1970s in the context of identification [1] and has been applied since to different problems, notably identification of elastic constants from plate vibration experiments [2]-[4]. The applications of the method to these classical pointwise tests involved only a small number of measurements (typically ten natural frequencies in the previously cited vibration test) which facilitated the application of the Bayesian approach. For identifying elastic constants, full field strain or displacement measurements provide a high number of measured quantities (one measurement per image pixel) and hence a promise of smaller uncertainties in the properties. However, the high number of measurements represents also a major computational challenge in applying the Bayesian approach to full field measurements. To address this challenge we propose an approach based on the proper orthogonal decomposition (POD) of the full fields in order to drastically reduce their dimensionality. POD is based on projecting the full field images on a modal basis, constructed from sample simulations, and which can account for the variations of the full field as the elastic constants and other parameters of interest are varied. The fidelity of the decomposition depends on the number of basis vectors used. Typically even complex fields can be accurately represented with no more than a few dozen modes and for our problem we showed that only four or five modes are sufficient [5]. To further reduce the computational cost of the Bayesian approach we use response surface approximations of the POD coefficients of the fields. We show that 3rd degree polynomial response surface approximations provide a satisfying accuracy. The combination of POD decomposition and response surface methodology allows to bring down the computational time of the Bayesian identification to a few days. The proposed approach is applied to Moire interferometry full field displacement measurements from a traction experiment on a plate with a hole. The laminate with a layup of [45,- 45,0]s is made out of a Toray® T800/3631 graphite/epoxy prepreg. The measured displacement maps are provided in Figure 1. The mean values of the identified properties joint probability density function are in agreement with previous identifications carried out on the same material. Furthermore the probability density function also provides the coefficient of variation with which the properties are identified as well as the correlations between the various properties. We find that while the longitudinal Young's modulus is identified with good accuracy (low standard deviation), the Poisson's ration is identified with much higher uncertainty. Several of the properties are also found to be correlated. The identified uncertainty structure of the elastic constants (i.e. variance co-variance matrix) has potential benefits to reliability analyses, by allowing a more accurate description of the input uncertainty. An additional advantage of the Bayesian approach is that it provides a natural way (in the form of the prior probability density function) for accounting for prior information that may be available on the material properties thought. This is of great interest for reducing the uncertainty on properties that can only be determined with low confidence from the plate with a hole experiment, such as Poisson's ratio or transverse Young's modulus in our case.

On-board dc grid systems have been proposed in the marine industry to reduce specific fuel consumption and to improve dynamic performance and manoeuvrability. The primary energy sources of an on-board dc system are variable frequency operating diesel engine generators with an active front-end converter system. Such systems might suffer disturbances from the engine side as well as from the electrical load side. These dynamic conditions result in torque overloads in the mechanical system, resulting in failures to sensitive driveline components. To understand the dynamics associated with disturbances and to propose a robust and reliable system, a detailed modeling approach is required. This paper presents complete modeling of a diesel generator system, including cylinder pressure, mechanical driveline system, electrical generator characteristics, and control system of an active front-end converter. The detailed model is used to develop a novel power electronic-based controller to reduce elevated load levels induced during disturbances. The controller increases the damping ratio of the first natural frequency by using the speed difference between a generator and a flywheel. The generator speed is estimated while fly wheel inertia speed is measured from an existing speed sensor used for monitoring and electronic control, and their difference is used to improve the reliability under disturbance conditions. Simulation with the detailed model has indicated excessive drivetrain loads and potential damage to critical components. The proposed controller enhances the damping of the system by transferring the torsional energy to the electrical system, reducing the shaft torque amplitude during resonance. This paper presents simulation results demonstrating the fidelity of the diesel generator model as well as the effectiveness of the proposed controller.

Synchronization phenomenon appears in several natural systems being associated with physical, chemical and biological processes. In brief, synchronization may be understood as a harmonization of two or more system behaviors following some patterns. This paper deals with synchronization analysis of a mechanical pendulum-chart system composed by a hierarchical network of three pendula coupled to each other through their own chart that receives continuous supply of energy via a base excitation. Dynamical patterns are classified and investigated in order to understand the conditions to each one of them. Asynchronous behaviors are analyzed including the chimera state defined as a symmetry break of the behavior of identical oscillator network. Numerical simulations indicate that patterns do not have a direct relation with periodicity. An energetic approach is proposed in order to define a measure of synchronized states, providing new insights about the origin of chimera state and its relationship with natural frequencies.

The coupling of finite element simulations to mathematical optimization techniques has contributed significantly to product improvements and cost reductions in the metal forming industries. The next challenge is to bridge the gap between deterministic optimization techniques and the industrial need for robustness. This paper introduces a generally applicable strategy for modeling and efficiently solving robust optimization problems based on time consuming simulations. Noise variables and their effect on the responses are taken into account explicitly. The robust optimization strategy consists of four main stages: modeling, sensitivity analysis, robust optimization and sequential robust optimization. Use is made of a metamodel-based optimization approach to couple the computationally expensive finite element simulations with the robust optimization procedure. The initial metamodel approximation will only serve to find a first estimate of the robust optimum. Sequential optimization steps are subsequently applied to efficiently increase the accuracy of the response prediction at regions of interest containing the optimal robust design. The applicability of the proposed robust optimization strategy is demonstrated by the sequential robust optimization of an analytical test function and an industrial V-bending process. For the industrial application, several production trial runs have been performed to investigate and validate the robustness of the production process. For both applications, it is shown that the robust optimization strategy accounts for the effect of different sources of uncertainty onto the process responses in a very efficient manner. Moreover, application of the methodology to the industrial V-bending process results in valuable process insights and an improved robust process design.

Purpose: In particle therapy, treatment planning and evaluation are frequently based on biological models to estimate the relative biological effectiveness (RBE) or the equivalent dose in 2 Gy fractions (EQD2). In the context of the linear-quadratic model, these quantities depend on biological parameters (α, β) for ions as well as for the reference radiation and on the dose per fraction. The needed biological parameters as well as their dependency on ion species and ion energy typically are subject to large (relative) uncertainties of up to 20–40% or even more. Therefore it is necessary to estimate the resulting uncertainties in e.g. RBE or EQD2 caused by the uncertainties of the relevant input parameters. Methods: We use a variance-based sensitivity analysis (SA) approach, in which uncertainties in input parameters are modeled by random number distributions. The evaluated function is executed 10{sup 4} to 10{sup 6} times, each run with a different set of input parameters, randomly varied according to their assigned distribution. The sensitivity S is a variance-based ranking (from S = 0, no impact, to S = 1, only influential part) of the impact of input uncertainties. The SA approach is implemented for carbon ion treatment plans on 3D patient data,more » providing information about variations (and their origin) in RBE and EQD2. Results: The quantification enables 3D sensitivity maps, showing dependencies of RBE and EQD2 on different input uncertainties. The high number of runs allows displaying the interplay between different input uncertainties. The SA identifies input parameter combinations which result in extreme deviations of the result and the input parameter for which an uncertainty reduction is the most rewarding. Conclusion: The presented variance-based SA provides advantageous properties in terms of visualization and quantification of (biological) uncertainties and their impact. The method is very flexible, model independent, and enables a broad assessment of uncertainties. Supported by DFG grant WI 3745/1-1 and DFG cluster of excellence: Munich-Centre for Advanced Photonics.« less

Purpose – The purpose of this paper is to address the practically important problem of the load dependence of transverse vibrations for helical springs. At the beginning, the author develops the equations for transverse vibrations of the axially loaded helical springs. The method is based on the concept of an equivalent column. Second, the author reveals the effect of axial load on the fundamental frequency of transverse vibrations and derive the explicit formulas for this frequency. The fundamental natural frequency of the transverse vibrations of the spring depends on the variable length of the spring. The reduction of frequency with the load is demonstrated. Finally, when the frequency nullifies, the side buckling spring occurs. Design/methodology/approach – Helical springs constitute an integral part of many mechanical systems. A coil spring is a special form of spatially curved column. The center of each cross-section is located on a helix. The helix is a curve that winds around with a constant slope of the surface of a cylinder. An exact stability analysis based on the theory of spatially curved bars is complicated and difficult for further applications. Hence, in most engineering applications a concept of an equivalent column is introduced. The spring is substituted for the simplification of the basic equations by an equivalent column. Such a column must account for compressibility of axis and shear effects. The transverse vibration is represented by a differential equation of fourth order in place and second order in time. The solution of the undamped model equation could be obtained by separation of variables. The fundamental natural frequency of the transverse vibrations depends on the current length of the spring. Natural frequency is the function of the deflection and slenderness ratio. Is the fundamental natural frequency of transverse oscillations nullifies, the lateral buckling of the spring with the natural form occurs. The mode shape corresponds to the buckling of the spring with moment-free, simply supported ends. The mode corresponds to the buckling of the spring with clamped ends. The author finds the critical spring compression. Findings – Buckling refers to the loss of stability up to the sudden and violent failure of seed straight bars or beams under the action of pressure forces, whose line of action is the column axis. The known results for the buckling of axially overloaded coil springs were found using the static stability criterion. The author uses an alternative approach method for studying the stability of the spring. This method is based on dynamic equations. In this paper, the author derives the equations for transverse vibrations of the pressure-loaded coil springs. The fundamental natural frequency of the transverse vibrations of the column is proved to be the certain function of the axial force, as well as the variable length of the spring. Is the fundamental natural frequency of transverse oscillations turns to be to zero, is the lateral buckling of the spring occurs. Research limitations/implications – The spring is substituted for the simplification of the basic equations by an equivalent column. Such a column must account for compressibility of axis and shear effects. The more accurate model is based on the equations of motion of loaded helical Timoshenko beams. The dimensionless for beams of circular cross-section and the number of parameters governing the problem is reduced to four (helix angle, helix index, Poisson coefficient, and axial strain) is to be derived. Unfortunately, that for the spatial beam models only numerical results could be obtained. Practical implications – The closed form analytical formulas for fundamental natural frequency of the transverse vibrations of the column as function of the axial force, as well as the variable length of the spring are derived. The practically important formulas for lateral buckling of the spring are obtained. Originality/value – In this paper, the author derives the new equations for transverse vibrations of the pressure-loaded coil springs. The author demonstrates that the fundamental natural frequency of the transverse vibrations of the column is the function of the axial force. For study of the stability of the spring the author uses an alternative approach method. This method is based on dynamic equations. The new, original expressions for lateral buckling of the spring are also obtained.