Guest Editorial

J. Comput. Nonlinear Dynam. 2019;14(2):020301-020301-2. doi:10.1115/1.4042262.

As modeling and simulation of physical phenomena are taking a central role in the design, assessment, and optimization of engineering systems, engineers must often answer questions such as: How well does a mathematical model capture the relevant physical phenomena? What confidence can be placed on simulation results? How far from the nominal design can computational results be extrapolated? What are the impacts of inherent model parameter variability and imprecise measurements on the calculated results? How can a design be made robust to imperfections and uncertainties, with performance and operational safety maximized and cost minimized? Qualitative and quantitative answers to such questions are provided by sensitivity and uncertainty analyses.

Commentary by Dr. Valentin Fuster

Research Papers

J. Comput. Nonlinear Dynam. 2019;14(2):021001-021001-11. doi:10.1115/1.4041237.

The gradient-based design optimization of mechanical systems requires robust and efficient sensitivity analysis tools. The adjoint method is regarded as the most efficient semi-analytical method to evaluate sensitivity derivatives for problems involving numerous design parameters and relatively few objective functions. This paper presents a discrete version of the adjoint method based on the generalized-alpha time integration scheme, which is applied to the dynamic simulation of flexible multibody systems. Rather than using an ad hoc backward integration solver, the proposed approach leads to a straightforward algebraic procedure that provides design sensitivities evaluated to machine accuracy. The approach is based on an intrinsic representation of motion that does not require a global parameterization of rotation. Design parameters associated with rigid bodies, kinematic joints, and beam sectional properties are considered. Rigid and flexible mechanical systems are investigated to validate the proposed approach and demonstrate its accuracy, efficiency, and robustness.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2019;14(2):021002-021002-9. doi:10.1115/1.4041827.

This work presents how the analytical sensitivity of Lyapunov characteristic exponents (LCEs) can be used in the design of nonlinear dampers, which are frequently utilized to stabilize the response of mechanical systems. The kinetic energy dissipated in the form of heat often induces nonlinearities, therefore reducing the reliability of standard stability evaluation methods. Owing to the difficulty of estimating the stability properties of equilibrium solution of the resulting nonlinear time-dependent systems, engineers usually tend to linearize and time-average the governing equations. However, the solutions of nonlinear and time-dependent dynamical systems may exhibit unique properties, which are lost when they are simplified. When a damper is designed based on a simplified model, the cost associated with neglecting nonlinearities can be significantly high in terms of safety margins that are needed as a safeguard with respect to model uncertainties. Therefore, in those cases, a generalized stability measure, with its parametric sensitivity, can replace usual model simplifications in engineering design, especially when a system is dominated by specific, non-negligible nonlinearities and time-dependencies. The estimation of the characteristic exponents and their sensitivity is illustrated. A practical application of the proposed methodology is presented, considering that the problem of helicopter ground resonance (GR) and landing gear shimmy vibration with nonlinear dampers are implemented instead of linear ones. Exploiting the analytical sensitivity of the Lyapunov exponents within a continuation approach, the geometric parameters of the damper are determined. The mass of the damper and the largest characteristic exponent of the system are used as the objective function and the inequality or equality constraint in the design of the viscous dampers.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2019;14(2):021003-021003-12. doi:10.1115/1.4041622.

The focus of this paper is on the global sensitivity analysis (GSA) of linear systems with time-invariant model parameter uncertainties and driven by stochastic inputs. The Sobol' indices of the evolving mean and variance estimates of states are used to assess the impact of the time-invariant uncertain model parameters and the statistics of the stochastic input on the uncertainty of the output. Numerical results on two benchmark problems help illustrate that it is conceivable that parameters, which are not so significant in contributing to the uncertainty of the mean, can be extremely significant in contributing to the uncertainty of the variances. The paper uses a polynomial chaos (PC) approach to synthesize a surrogate probabilistic model of the stochastic system after using Lagrange interpolation polynomials (LIPs) as PC bases. The Sobol' indices are then directly evaluated from the PC coefficients. Although this concept is not new, a novel interpretation of stochastic collocation-based PC and intrusive PC is presented where they are shown to represent identical probabilistic models when the system under consideration is linear. This result now permits treating linear models as black boxes to develop intrusive PC surrogates.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2019;14(2):021004-021004-9. doi:10.1115/1.4041960.

Algorithms for the sensitivity analysis of multibody systems are quickly maturing as computational and software resources grow. Indeed, the area has made substantial progress since the first academic methods and examples were developed. Today, sensitivity analysis tools aimed at gradient-based design optimization are required to be as computationally efficient and scalable as possible. This paper presents extensive verification of one of the most popular sensitivity analysis techniques, namely the direct differentiation method (DDM). Usage of such method is recommended when the number of design parameters relative to the number of outputs is small and when the time integration algorithm is sensitive to accumulation errors. Verification is hereby accomplished through two radically different computational techniques, namely manual differentiation and automatic differentiation, which are used to compute the necessary partial derivatives. Experiments are conducted on an 18-degree-of-freedom, 366-dependent-coordinate bus model with realistic geometry and tire contact forces, which constitutes an unusually large system within general-purpose sensitivity analysis of multibody systems. The results are in good agreement; the manual technique provides shorter runtimes, whereas the automatic differentiation technique is easier to implement. The presented results highlight the potential of manual and automatic differentiation approaches within general-purpose simulation packages, and the importance of formulation benchmarking.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2019;14(2):021005-021005-9. doi:10.1115/1.4042015.

Manufacturing tolerances and other uncertainties may play an important role in the performance of parallel manipulators since they can affect the distance to a singular configuration. Motion planning strategies for parallel manipulators under uncertainty require decision making approaches for classifying reliable regions within the workspace. In this paper, we address fail free and reliable motion planning for parallel manipulators. Failure is related to parallel kinematic singularities in the motion equations or to ill-conditioning of the Jacobian matrices. Monte Carlo algorithm is employed to compute failure probabilities for a dense grid of manipulator workspace configurations. The inverse condition number of the Jacobian matrix is used to compute the distance between each configuration and a singularity. For supporting motion planning strategies, not only failure maps are constructed but also reliable and failure-free workspaces are obtained. On the one hand, the reliable workspace is obtained by minimizing the failure probabilities subject to a minimal workspace area. Differently, a failure-free workspace is found by maximizing the workspace area subject to a probability of failure equal to zero. A 3RRR manipulator is used as a case study. For this case study, the usage of the reliable strategy can be useful for robustifying motion planning algorithm without a significant reduction of the reliable regions within the workspace.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2019;14(2):021006-021006-11. doi:10.1115/1.4041578.

The creep phenomenon has enormous effect on the stress and displacement distribution in the structures. Redistribution of the stress field is one of these effects which is called stress relaxation. The importance of stress relaxation in the design of structures is increasing due to engineering applications especially in high temperature. However, this phenomenon has remained absent from the structural optimization studies. In the present study, the effect of stress relaxation due to high temperature creep is considered in topology optimization (TO). Internal element connectivity parameterization (I-ECP) method is utilized for performing TO. This method is shown to be effective to overcome numerical instabilities in nonlinear problems. Time-dependent adjoint sensitivity formulation is implemented for I-ECP including creep effect. Several benchmark problems are solved, and the optimum layouts obtained by linear and nonlinear methods are compared to show the efficiency of the proposed method and to show the effect of stress relaxation on the optimum layout.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2019;14(2):021007-021007-9. doi:10.1115/1.4042140.

An innovative approach to topology optimization of dynamic system is introduced that is based on the system transfer-function H-norm. As for the structure, the proposed strategy allows to determine the optimal material distribution that ensures the minimization of a suitable goal function, such as (an original definition of) the dynamic compliance. Load uncertainty is accounted for by means of a nonprobabilistic convex-set approach (Ben-Haim and Elishakoff, 1990, Convex Models of Uncertainty in Applied Mechanics, Elsevier Science, Amsterdam). At each iteration, the worst load is determined as the one that maximizes the current dynamic compliance so that the proposed strategy fits the so-called worst case scenario (WCS) approach. The overall approach consists of the repeated solution of the two steps (minimization of the dynamic compliance with respect to structural parameters and maximization of the dynamic compliance with respect to the acting load) until convergence is achieved. Results from representative numerical studies are eventually presented along with extensions to the proposed approach that are currently under development.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2019;14(2):021008-021008-15. doi:10.1115/1.4041580.

The mechanisms with uncertain parameters may exhibit multiple dynamic response patterns. As a single surrogate model can hardly describe all the dynamic response patterns of mechanism dynamics, a new computation methodology is proposed to study multiple dynamic response patterns of a flexible multibody system with uncertain random parameters. The flexible multibody system of concern is modeled by using a unified mesh of the absolute nodal coordinate formulation (ANCF). The polynomial chaos (PC) expansion with collocation methods is used to generate the surrogate model for the flexible multibody system with random parameters. Several subsurrogate models are used to describe multiple dynamic response patterns of the system dynamics. By the motivation of the data mining, the Dirichlet process mixture model (DPMM) is used to determine the dynamic response patterns and project the collocation points into different patterns. The uncertain differential algebraic equations (DAEs) for the flexible multibody system are directly transformed into the uncertain nonlinear algebraic equations by using the generalized-alpha algorithm. Then, the PC expansion is further used to transform the uncertain nonlinear algebraic equations into several sets of nonlinear algebraic equations with deterministic collocation points. Finally, two numerical examples are presented to validate the proposed methodology. The first confirms the effectiveness of the proposed methodology, and the second one shows the effectiveness of the proposed computation methodology in multiple dynamic response patterns study of a complicated spatial flexible multibody system with uncertain random parameters.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2019;14(2):021009-021009-13. doi:10.1115/1.4041238.

This paper presents a probabilistic framework for discrepancy prediction in dynamical system models under untested input time histories, based on information gained from validation experiments. Two surrogate modeling-based methods, namely observation surrogate and bias surrogate, are developed to predict the bias of a dynamical system simulation model under untested input time history. In the first method, a surrogate model is built for the observed experimental output, and the model bias for the untested input is obtained by comparing the output of the observation surrogate with the output of the physics-based model. The second method constructs a surrogate model for the bias in terms of the inputs in the conducted experiments. The bias surrogate model is then used to correct the simulation model prediction at each time-step under a predictor–corrector scheme to predict the model bias under untested conditions. A neural network-based surrogate modeling technique is employed to implement the proposed methodology. The bias prediction result is reported in a probabilistic manner, in order to account for the uncertainty of the surrogate model prediction. An air cycle machine case study is used to demonstrate the effectiveness of the proposed bias prediction framework.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2019;14(2):021010-021010-9. doi:10.1115/1.4041772.

We explore the use of generalized polynomial chaos (GPC) expansion with stochastic collocation (SC) for modeling the uncertainty in the noise radiated by a plate subject to turbulent boundary layer (TBL) forcing. The SC form of polynomial chaos permits re-use of existing computational models, while drastically reducing the number of evaluations of the deterministic code compared to Monte Carlo (MC) sampling, for instance. Further efficiency is attained through the application of new, efficient, quadrature rules to compute the GPC expansion coefficients. We demonstrate that our approach accurately reconstructs the statistics of the radiated sound power by propagating the input uncertainty through the computational physics model. The use of optimized quadrature rules permits these results to be obtained using far fewer quadrature nodes than with traditional methods, such as tensor product quadrature and Smolyak sparse grid methods. As each quadrature node corresponds to an expensive deterministic model evaluation, the computational cost of the analysis is seen to be greatly reduced.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2019;14(2):021011-021011-14. doi:10.1115/1.4041473.

This paper develops a framework for propagation of uncertainties, governed by different probability distribution functions in a stochastic dynamical system. More specifically, it deals with nonlinear dynamical systems, wherein both the initial state and parametric uncertainty have been taken into consideration and their effects studied in the model response. A sampling-based nonintrusive approach using pseudospectral stochastic collocation is employed to obtain the coefficients required for the generalized polynomial chaos (gPC) expansion in this framework. The samples are generated based on the distribution of the uncertainties, which are basically the cubature nodes to solve expectation integrals. A mixture of one-dimensional Gaussian quadrature techniques in a sparse grid framework is used to produce the required samples to obtain the integrals. The familiar problem of degeneracy with high-order gPC expansions is illustrated and insights into mitigation of such behavior are presented. To illustrate the efficacy of the proposed approach, numerical examples of dynamic systems with state and parametric uncertainties are considered which include the simple linear harmonic oscillator system and a two-degree-of-freedom nonlinear aeroelastic system.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2019;14(2):021012-021012-10. doi:10.1115/1.4041350.

A framework for generation of reliability-based stochastic off-road mobility maps is developed to support the next generation NATO reference mobility model (NG-NRMM) using full stochastic knowledge of terrain properties and modern complex terramechanics modeling and simulation capabilities. The framework is for carrying out uncertainty quantification (UQ) and reliability assessment for Speed Made Good and GO/NOGO decisions for the ground vehicle based on the input variability models of the terrain elevation and soil property parameters. To generate the distribution of the slope at given point, realizations of the elevation raster are generated using the normal distribution. For the soil property parameters, such as cohesion, friction, and bulk density, the min and max values obtained from geotechnical databases for each of the soil types are used to generate the normal distribution with a 99% confidence value range. In the framework, the ranges of terramechanics input parameters that will cover the regions of interest are first identified. Within these ranges of input parameters, a dynamic kriging (DKG) surrogate model is obtained for the maximum speed of the nevada automotive test center (NATC) wheeled vehicle platform complex terramechanics model. Finally, inverse reliability analysis using Monte Carlo simulation is carried out to generate the reliability-based stochastic mobility maps for Speed Made Good and GO/NOGO decisions. It is found that the deterministic map of the region of interest has probability of only 25% to achieve the indicated speed.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2019;14(2):021013-021013-9. doi:10.1115/1.4041680.

In the mechanical engineering world, there is a growing interest in being able to create so-called “digital twins” to assess the impact to performance or response. Part of the challenge is to be able to include and assess manufactured geometries as opposed to nominal design intent, particularly for components that are sensitive to small shape variations. In this paper, we show how the update of digital models adopted in computer aided engineering (CAE) can be conducted according to a mesh morphing workflow based on radial basis functions (RBF). The CAE mesh of the nominal design is updated onto the actual one as acquired from surveying a manufactured individual. The concept is demonstrated on a practical application, the wing structure of the RIBES experiment, showing how the new proposed method compares with a traditional one based on the reconstruction of the geometrical model.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2019;14(2):021014-021014-8. doi:10.1115/1.4041773.

Batch fabrication processes used to produce micro-electro-mechanical systems (MEMS) are prone to uncertainties in the system geometrical and contact parameters as well as material properties. However, since the common design method for these systems is typically based on precise deterministic assumptions, it is necessary to get more insight into their variations. To this end, understanding the influences of uncertainties accompanied by these processes on the system performance and reliability is warranted. The present paper focuses on predictions of uncertainty measures for MEMS switches based on the transient dynamic response, in particular, the bouncing behavior. To understand and quantify the influence of pertinent parameters on the bouncing effects, suitable mathematical model that captures the bouncing dynamics as well as the forces that are dominant at this micron scale are employed. Measure of performance in terms of second-order statistics is performed, particularly for the beam as well as beam tip parameters since excessive tip bounce is known to degrade switch performance. Thus, the present study focusses on the influence of uncertainties in the beam tip geometry parameters such as beam tip length/width as well as contact asperity variables such as the area asperity density and the radius of asperities. In addition to beam tip parameters, this study quantifies the effects of uncertainties in Young's modulus, beam thickness as well as actuation voltage. These influences on significant switch performance parameters such as initial contact time and maximum bounce height have been quantified in the presence of interactive system nonlinearities.

Commentary by Dr. Valentin Fuster

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