Nam H. Kim

Prognostics 101: A tutorial for particle filter-based prognostics algorithm using Matlab

This paper presents a Matlab-based tutorial for model-based prognostics, which combines a physical model with observed data to identify model parameters, from which the remaining useful life (RUL) can be predicted. Among many model-based prognostics algorithms, the particle filter is used in this tutorial for parameter estimation of damage or a degradation model. The tutorial is presented using a Matlab script with 62 lines, including detailed explanations. As examples, a battery degradation model and a crack growth model are used to explain the updating process of model parameters, damage progression, and RUL prediction. In order to illustrate the results, the RUL at an arbitrary cycle are predicted in the form of distribution along with the median and 90% prediction interval. This tutorial will be helpful for the beginners in prognostics to understand and use the prognostics method, and we hope it provides a standard of particle filter based prognostics.

Practical options for selecting data-driven or physics-based prognostics algorithms with reviews

This paper is to provide practical options for prognostics so that beginners can select appropriate methods for their fields of application. To achieve this goal, several popular algorithms are first reviewed in the data-driven and physics-based prognostics methods. Each algorithm’s attributes and pros and cons are analyzed in terms of model definition, model parameter estimation and ability to handle noise and bias in data. Fatigue crack growth examples are then used to illustrate the characteristics of different algorithms. In order to suggest a suitable algorithm, several studies are made based on the number of data sets, the level of noise and bias, availability of loading and physical models, and complexity of the damage growth behavior. Based on the study, it is concluded that the Gaussian process is easy and fast to implement, but works well only when the covariance function is properly defined. The neural network has the advantage in the case of large noise and complex models but only with many training data sets. The particle filter and Bayesian method are superior to the former methods because they are less affected by noise and model complexity, but work only when physical model and loading conditions are available.

DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION OF NONLINEAR TRANSIENT DYNAMICS

A shape design sensitivity analysis (DSA) and optimization of structural transient dynamics are proposed for the finite deformation elastoplastic materials under impact with a rigid surface. A shape variation of the structure is considered using the material derivative approach in continuum mechanics. Hyperelasticitybased multiplicatively decomposed elastoplasticity is used for the constitutive model. The implicit Newmark time integration scheme is used for the structural dynamics. The design sensitivity equation is solved at each converged time step with the same tangent stiffness matrix as response analysis without iteration. The cost of sensitivity computation is more efficient than the cost of response analysis for the implicit time integration method. The efficiency and the accuracy of the proposed method are shown through the design optimization of a vehicle bumper. *Communicated by D. Tortorelli.

Design sensitivity analysis for sequential structural–acoustic problems

Abstract A design sensitivity analysis of a sequential structural–acoustic problem is presented in which structural and acoustic behaviors are de-coupled. A frequency-response analysis is used to obtain the dynamic behavior of an automotive structure, while the boundary element method is used to solve the pressure response of an interior, acoustic domain. For the purposes of design sensitivity analysis, a direct differentiation method and an adjoint variable method are presented. In the adjoint variable method, an adjoint load is obtained from the acoustic boundary element re-analysis, while the adjoint solution is calculated from the structural dynamic re-analysis. The evaluation of pressure sensitivity only involves a numerical integration process for the structural part. The proposed sensitivity results are compared to finite difference sensitivity results with excellent agreement.

Uncertainty Reduction of Damage Growth Properties Using Structural Health Monitoring

Structural health monitoring provides sensor data that can monitor fatigue-induced damage in service. This information may in turn be used to improve the characterization of material properties that govern damage growth for the structure beingmonitored. These properties are oftenwidely distributed amongnominally identicalmaterials because of differences in manufacturing processes and due to aging effects. Improved accuracy in damage growth characteristics would allowmore accurate prediction of the remaining useful life of the structural component. In this paper, a probabilistic approach using Bayesian inference is employed to progressively reduce the uncertainty in structure-specific damage growth parameters in spite of noise and bias in sensor measurements. Starting from an initial wide distribution of damage growth parameters that are obtained from coupon tests, the distribution is progressively narrowed using damage growth data between consecutivemeasurements. Detailed discussions on how to construct the likelihood function under the given noise of sensor data and how to update the distribution are presented. The approach is applied to simulated damage growth in fuselage panels due to cycles of pressurization. It is shown that the proposed method rapidly converges to the accurate damage growth parameters when the initial damage size is relatively large: e.g., 20 mm. Fairly accurate damage growth parameters are obtained even with measurement errors of 5mm. Using the identified damage growth parameters, it is shown that the 95% conservative remaining useful life converges to the true remaining useful life from the conservative side. The proposed approach may have the potential of turning aircraft into flying fatigue laboratories.

Effects of Structural Tests on Aircraft Safety

Thispaperpresents amethodology to investigatethe effects of structuraltests onaircraft safety.Inparticular, the paper focuses on the effect of the number of coupon tests and structural element tests on the final distribution of failure stress. The mean failure stress is assumed to be predicted by a failure criterion (e.g., Tsai-Wu), and the initial distribution of this mean failure stress reflects the uncertainty in the analysisprocedure that usescoupontest datato predict structural failure. In addition to the uncertainty in the mean failure stress, there is also uncertainty in its variabilityduetothe finitenumberofcoupontests.Bayesianupdatingisusedtoupdatethefailurestressdistribution basedonresultsoftheelementtests.AMonteCarlosimulationofalargenumberofuncertaintiesandthepossibletest results are used to obtain the probability of structural failure in a certification test or in actual flight. Incorporating the Bayesian updating into the Monte Carlo simulation loop is computationally prohibitive; therefore, a surrogate procedure is devised to overcome the computational challenge. A structural design following the Federal Aviation Administration regulations is considered, and the tradeoffs between the number of tests and the weight and probability of failure in the certification test and in service are explored. To make this tradeoff analysis computationally affordable, response surface approximations are used to relate the knockdown factor to the probability of failure in service and in the certification test. It is found that it is possible to do a simultaneously probabilistic design and satisfy the Federal Aviation Administration regulations for deterministic design.

Stability of a Time-Delayed System With Parametric Excitation

This paper investigates two different temporal finite element techniques, a multiple element (h-version) and single element (p-version) method, to analyze the stability of a system with a time-periodic coefficient and a time delay. The representative problem, known as the delayed damped Mathieu equation, is chosen to illustrate the combined effect of a time delay and parametric excitation on stability. A discrete linear map is obtained by approximating the exact solution with a series expansion of orthogonal polynomials constrained at intermittent nodes. Characteristic multipliers of the map are used to determine the unstable parameter domains. Additionally, the described analysis provides a new approach to extract the Floquet transition matrix of time periodic systems without a delay.

Comparison Between Elastic Foundation and Contact Force Models in Wear Analysis of Planar Multibody System

In this paper, two procedures to analyze planar multibody systems experiencing wear at a revolute joint are compared. In both procedures, the revolute joint of interest includes a clearance whose shape and size are dictated by wear. The procedures consist of coupled iterative analyses between a dynamic system analysis with nonideal joints and a wear prediction to determine the evolution of the joint clearance. In the first procedure, joint forces and contact pressures are estimated using the elastic foundation model with hysteresis damping via the dynamic analysis. In the second procedure, a contact force model with hysteresis damping is used to estimate the joint forces. In the latter case, however, the contact pressure is estimated using a finite element method (FEM). A comparison in performance of the two models is facilitated through the use of an experimental slider-crank mechanism in which wear is permitted to occur at one of the joints. It is observed that the two procedures provide similar estimates for the dynamic response and wear volumes but substantially different predictions on the wear profiles. Additionally, experimental results show that while predictions on the wear volume from both models are reasonably accurate, the FEM-based model produced more accurate predictions on the wear profile.

Shape Design Sensitivity Analysis and Optimization of Elasto-Plasticity with Frictional Contact

A shape design sensitivity analysis and optimization are proposed for the ine nitesimal elasto ‐plasticity with a frictional contact condition. Rate-independent plasticity is considered with a return mapping algorithm and a von Mises yield criterion. The contact condition is formulated using the penalty method and the modie ed coulomb friction law. A continuum-based shape design sensitivity formulation is developed for structural and frictional contact variational equations. The direct differentiation method is used to compute the displacement sensitivity, and the sensitivities of various performance measures are computed from the displacement sensitivity. Path dependency of the sensitivity equation due to the constitutive relation and friction is discussed. It is shown that no iteration is required to solve the sensitivity equation. Response analysis and the proposed sensitivity formulation are implemented using the mesh-free method where the mesh distortion problem can be resolved. Numerical examples show accurate results of the proposed method compared to the e nite difference method. Dife culties in the sensitivity formulation for the e nite deformation problem are discussed. I. Introduction B ECAUSEoftherecentdevelopmentofcomputationalmechanics, it is now possible to analyze practical examples of complicated structural problems. Many design engineers, who are not satise ed with response analysis alone, have keen interests in the methodology of the design. For more than two decades, signie cant research efforthasbeen focused on the rate of response with respect to the changes in structural shape under shape design sensitivity analysis (DSA). Analysis of the design sensitivity information is the most important and costly procedure in the automated optimum design process. It supplies useful quantitative information to the design engineer about the direction of the desired design change. In a classical linear problem, DSA research has proved the differentiability of the solution of the response analysis using the linear operator theory and has derived specie c sensitivity expressions for variousproblems. 1 Aresultworthy of attention in linear DSAis that theoriginalresponseandthesensitivityoftheresponsebelongtothe samekinematicallyadmissiblespaceandhavethesameregularities. Owing to the development of the response analysis capability, engineers have directed their interest to the nonlinear problems that are dealt with efe ciently. Because many design application problems are accompanied by plastic deformation, the design sensitivity of nonlinear problems has been actively developed, and many research results are reported. In the procedure of nonlinear response analysis, the projection, called a return mapping, of the elastic trial stress is carried out to satisfy the variational inequality (VI)through an iteration in the stress space. 2 The DSA, on the other hand, computes the rate of change of the projected response in the tangential direction of the constraint set without iteration. Note that the sensitivity analysis is linear and can be computed without iteration even thoughtheresponseanalysisisnonlinear. 3 Unlikethenonlinearelastic problem, the sensitivity equation of the plastic problem requires sensitivity of the stress and internal evolution variables at the previous time. The sensitivity equation is solved at each time without iteration, and the sensitivity of the stress and evolution variables are

Identification of correlated damage parameters under noise and bias using Bayesian inference

This article presents statistical model parameter identification using Bayesian inference when parameters are correlated and observed data have noise and bias. The method is explained using the Paris model that describes crack growth in a plate under mode I loading. It is assumed that the observed data are obtained through structural health monitoring systems, which may have random noise and deterministic bias. It was found that a strong correlation exists (a) between two parameters of the Paris model, and (b) between initially measured crack size and bias. As the level of noise increases, the Bayesian inference was not able to identify the correlated parameters. However, the remaining useful life was predicted accurately because the identification errors in correlated parameters were compensated by each other. It was also found that the Bayesian identification process converges slowly when the level of noise is high.