This work investigates surrogate modeling techniques for learning to approximate a computationally expensive function evaluation of 3D models. Radial Basis Functions (RBF), Kriging, and shallow 1D analogs of popular deep 2D image classification neural networks are investigated in this work. We find the nonintuitive result that departing from neural networks to decode latent representations of 3D objects into performance predictions is far more efficient than using a neural network decoder. We find that an RBF surrogate model is able to approximate the lift and drag coefficients of 234 aircraft models with a mean absolute error of 1:97e–03, and trains in only 3 seconds. Furthermore, the RBF surrogate model is able to rank a set of designs with an average percentile error of less than 8%. In comparison, a 1D ResNet achieves an average absolute error of 1:35e–03 in 38 minutes for the same test case. We validate the comparable accuracy of the four techniques through a test case involving 214 3D watercraft models, but we also find that the distribution of the performance values of the data, in particular the presence of many outliers, has a significant negative impact on accuracy. These results contradict a common perception of neural networks as an efficient “one-size-fits-all” solution for learning black box functions, and suggests that even within systems that utilize multiple neural networks, potentially more efficient alternatives should be considered for each network in the system.

Tool errors are inevitable in an actual gear-manufacturing environment and may direct affect the accuracy of machined tooth surfaces. In this paper, tool errors including spheric radius, pressure angle, rake angle, regrind angle and cutting side relief angle errors for three-face blade are defined and considered to establish the accurate tooth surface mathematical model for face-hobbed hypoid gears based on the manufacturing process and the meshing theory. The simulation flowchart for tooth surface modeling and tooth surface topography deviation analysis are proposed and performed. Results show that the tooth surface deviation is positive with positive spheric radius and rake angle errors and contrary results can be found for other three tool errors. In addition, the impact of the pressure angle error is the strongest. In addition, the rake angle error has the weakest effect and the influence of spheric radius error on the tooth surface deviation is unsubstantial. For location of tooth surface deviation, the maximum deviation is at the top on the heel and the minimum deviation is at the middle on the toe for spheric radius error. The maximum and minimum deviations are at the top and the middle tooth on the heel for other factors, respectively.

Surrogate model is a common tool to approximate system response at untested points for design space exploration. Adaptive sampling has been studied for improving the accuracy of surrogates iteratively by introducing additional sample (simulations and experiments). New samples are often selected based on the estimated uncertainty in the design space. While some surrogates such as Kriging have readily available uncertainty models for their predictions, other surrogates do not. Consequently, there have been studies of using cross-validation tools to estimate prediction uncertainty, such as the Universal Prediction Distribution(UPD). In this paper, an adaptive sampling scheme for general surrogates is proposed based on cross-validation and interquartile range (IQR). A population of surrogate models is first developed from cross-validation. The uncertainty is then estimated from the IQR of these surrogates at a given point. New samples are added iteratively at the point with maximum IQR for design space exploration. The proposed scheme is illustrated using Kriging, Radial Basis Function and Neural Network surrogates. The proposed scheme is evaluated using four algebraic test functions. For these functions, it was found to be more accurate and robust than Kriging with its own uncertainty model. The proposed scheme is more accurate than the Universal Prediction Distribution for three out of the four test functions. For a fixed number of samples, the IQR-based adaptive sampling also proved to be more accurate than all-at-once sampling in most cases even when the estimated uncertainty was only mildly correlated with prediction errors.

Prior to the development of sophisticated computer numerical control (CNC), both face milling (FM) and face hobbing (FH), the two most popular technologies for bevel gear production, required cradle-type machines with diverse and complicated mechanisms. In the last two decades, however, the gear industry has replaced these traditional machines with six-axis CNC bevel gear cutting machines that have superior efficiency and accuracy. One such machine is a vertical six-axis machine with a vertical spindle arrangement, which offers two industrially proven advantages: compact design and maximum machine stiffness. The technical details of this machine, however, remain undisclosed, so this paper proposes a mathematical model that uses inverse kinematics to derive the vertical machine's nonlinear six-axis coordinates from those of a traditional machine. The model also reduces manufacturing errors by applying an effective flank correction method based on a sensitivity analysis of how slight variations in the individual machine setting coefficients affects tooth geometry. We prove the model's efficacy by first using the proposed equations to derive the nonlinear coordinates for pinion and gear production and then conducting several cutting experiments on the gear and its correction. Although the numerical illustration used for this verification is based only on FM bevel gears produced by an SGDH cutting system, the model is in fact applicable in the production of both FM and FH bevel gears.

Training machine learning tools such as neural networks requires the availability of sizable data, which can be difficult for engineering and scientific applications where experiments or simulations are expensive. In this work, a novel multi-fidelity physics-constrained neural network is proposed to reduce the required amount of training data, where physical knowledge is applied to constrain neural networks, and multi-fidelity networks are constructed to improve training efficiency. A low-cost low-fidelity physics-constrained neural network is used as the baseline model, whereas a limited amount of data from a high-fidelity physics-constrained neural network is used to train a second neural network to predict the difference between the two models. The proposed framework is demonstrated with two-dimensional heat transfer, phase transition, and dendritic growth problems, which are fundamental in materials modeling. Physics is described by partial differential equations. With the same set of training data, the prediction error of physics-constrained neural network can be one order of magnitude lower than that of the classical artificial neural network without physical constraints. The accuracy of the prediction is comparable to those from direct numerical solutions of equations.

We address a central issue that arises within element-based topology optimization: To achieve a sufficiently well-defined material interface, one requires a highly refined finite element mesh, however, this leads to an increased computational cost due to the solution of the finite element analysis problem. By generating an optimal structure on a coarse mesh and using an artificial neural network to scale up to a larger domain, we can greatly reduce computation time. This approach resulted in time savings of up to 85% for test cases considered. This significant advantage in computation time also preserves the structural integrity when compared to a fine mesh optimization with limited error. Along with the savings in computation time, the boundary edges become more refined during the process, allowing for a sharp transition from solid to void. This improved boundary edge can also be leveraged to improve the manufacturability of the optimized designs.

The estimation of the statistical moments is widely involved in the industrial application, whose accuracy affects the reliability analysis result considerably. In this study, a novel hybrid dimension-reduction method based on the Nataf transformation is proposed to calculate the statistical moments of the performance function with correlated input variables. Nataf transformation is intrinsically the Gaussian copula, which is commonly used to transform the correlated input variables into independent ones. To calculate the numerical integration of the univariate component function in the proposed method, a normalized moment-based quadrature rule is employed. According to the statistical moments obtained by the proposed method, the probability density function of the performance function can be recovered accurately via maximum entropy method. Six examples are tested to illustrate the accuracy and efficiency of the proposed method, compared with that of Monte Carlo simulation, the conventional univariate dimension-reduction method, and the bivariate dimension-reduction method. It is confirmed that the proposed method achieves a good tradeoff between accuracy and efficiency for structural reliability analysis with correlated input variables.

The Next Generation NATO Reference Mobility Model (NG-NRMM) plays a vital role in vehicle mobility prediction and mission planning. The complicated vehicle-terrain interactions and the presence of heterogeneous uncertainty sources in the modeling and simulation (M&S) result in epistemic uncertainty/errors in the vehicle mobility prediction for given terrain and soil conditions. In this paper, the uncertainty sources that cause the uncertainty in mobility prediction are first partitioned into two levels, namely uncertainty in the M&S and uncertainty in terrain and soil maps. With a focus on the epistemic uncertainty in the M&S, this paper presents a testing design optimization framework to effectively reduce the uncertainty in the M&S and thus increase the confidence in generating off-road mobility maps. A Bayesian updating approach is developed to reduce the epistemic uncertainty/errors in the M&S using mobility testing data collected under controllable terrain and soil conditions. The updated models are then employed to generate off-road mobility maps for any given terrain and soil maps. Two types of design strategies, namely testing design for model selection and testing design for uncertainty reduction, are investigated in the testing design framework to maximize the information gain subject to limited resources. Results of a numerical example demonstrate the effectiveness of the proposed mobility testing design optimization framework.

Infilling strategies have been proposed for decades and are widely used in engineering problems. It is still challenging to achieve an effective trade-off between global exploration and local exploitation. In this paper, a novel decision-making infilling strategy named the Go-inspired hybrid infilling (Go-HI) strategy is proposed. The Go-HI strategy combines multiple individual infilling strategies, such as the mean square error (MSE), expected improvement (EI), and probability of improvement (PoI) strategies. The Go-HI strategy consists of two major parts. In the first part, a tree-like structure consisting of several subtrees is built. In the second part, the decision value for each subtree is calculated using a cross-validation (CV)-based criterion. Key factors that significantly influence the performance of the Go-HI strategy, such as the number of component infilling strategies and the tree depth, are explored. Go-HI strategies with different component strategies and tree depth are investigated and also compared with four baseline adaptive sampling strategies through three numerical functions and one engineering case. Results show that the number of component infilling strategies exerts a larger influence on the global and local performance than the tree depth; the Go-HI strategy with two component strategies performs better than the ones with three; the Go-HI strategy always outperforms the three component infilling strategies and the other four benchmark strategies in global performance and robustness, and saves much computational cost.

This paper presents a set of motion transmission error data for a family of helical gears having different profile and lead modifications operated under both low-speed (quasi-static) and dynamic conditions. A power circulatory test machine is used along with encoder and accelerometer-based transmission error measurement systems to quantify motion transmission behavior within wide ranges of torque and speed. Results of these experiments indicate that the tooth modifications impact the resultant static and dynamic transmission error amplitudes significantly. A design load is shown to exist for each gear pair of different modifications where static transmission error amplitude is minimum. Forced response curves and waterfall plots are presented to demonstrate that the helical gear pairs tested act linearly with no signs of nonlinear behavior such as tooth contact separations. Furthermore, static and dynamic transmission error amplitudes are observed to be nearly proportional, suggesting that static transmission error can be employed in helical gear dynamic models as the main gear mesh excitation. The data presented here is intended to fill a void in the literature by providing means for validation of load distribution and dynamic models of helical gear pairs.

A new active ease-off topography modification approach is proposed to improve the meshing performance of hypoid gears based on a fourth-order predesigned transmission error (PTE) model and a modified error sensitivity analysis method. Ease-off topography modifications that describe local deviations of pinion tooth surfaces can be conducted by converting the fourth-order PTE into equivalent deviations of pinion tooth surfaces. The modified error sensitivity analysis method is developed to investigate the effects of misalignments on the moving velocity of a contact point of a hypoid gear pair. The moving velocity of the contact point can describe transmission error (TE) curve shapes of ease-off tooth surfaces. The ease-off topography modification approach can achieve TE precontrol and modification curvature adjustment of the pinion for stable meshing performance of the hypoid gear pair. Moreover, pinion ease-off tooth surfaces are finished by a five-axis computer numerical control swarf-cutting machine tool. Swarf-cutting tests and TE measurement tests are conducted on hypoid gear pair specimens to demonstrate the feasibility and effectiveness of the proposed methodology.

A multitude of studies in economics, psychology, political and social sciences have demonstrated the wisdom of crowds (WoC) phenomenon, where the collective estimate of a group can be more accurate than estimates of individuals. While WoC is observable in such domains where the participating individuals have experience or familiarity with the question at hand, it remains unclear how effective WoC is for domains that traditionally require deep expertise or sophisticated computational models to estimate objective answers. This work explores how effective WoC is for engineering design problems that are esoteric in nature, that is, problems (1) whose solutions traditionally require expertise and specialized knowledge, (2) where access to experts can be costly or infeasible, and (3) in which previous WoC studies with the general population have been shown to be highly ineffective. The main hypothesis in this work is that in the absence of experts, WoC can be observed in groups that consist of practitioners who are defined to have a base familiarity with the problems in question but not necessarily domain experts. As a way to emulate commonly encountered engineering problem-solving scenarios, this work studies WoC with practitioners that form microcrowds consisting of 5–15 individuals, thereby giving rise to the term the wisdom of microcrowds (WoMC). Our studies on design evaluations show that WoMC produces results whose mean is in the 80th percentile or better across varying crowd sizes, even for problems that are highly nonintuitive in nature.

A new one-stage type cycloid drive reducer named China Bearing Reducer (CBR) is designed and its tooth contact analysis is investigated. First, the CAD model is built in SolidWorks and the structure of CBR reducer is introduced. Its advantages and disadvantages are compared with rotor vector (RV) reducer and harmonic drive reducer. Second, the mathematic model of cycloid profile and modified cycloid profile are established based on gear meshing and differential geometry, and the tooth shapes of three different modifications are compared with CBR25 reducer. Third, the conventional TCA method is described and a new TCA method by using discretized points is proposed to calculate transmission error and contact force of cycloid drive. Both the methods are used to compute the unloaded transmission error of CBR25 reducer to compare the computational efficiency. Finally, three different modified methods of tooth profile are investigated by using the new TCA method to calculate transmission error and contact force of CBR25 reducer, and the results show that the negative isometric and negative offset modified method is best for CBR25 reducer to reduce transmission error and contact force. The prototype of CBR25 reducer is manufactured and the transmission error and hysteresis curve are measured by using a testing bench. The results show that the new TCA method can be used to calculate transmission error and help better design the CBR reducer.

Least-square problems arise in multiple application areas. The numerical algorithm intended to compute offline the minimum (Euclidian)-norm approximation to an overdetermined system of linear equations, the core of least squares, is based on Householder reflections . It is self-understood, in the application of this algorithm, that the coefficient matrix is dimensionally homogeneous, i.e., all its entries bear the same physical units. Not all applications lead to such matrices, a case in point being parameter identification in mechanical systems involving rigid bodies, whereby both rotation and translation occur; the former being dimensionless and the latter bearing units of length. Because of this feature, dual numbers have found extensive applications in these fields, as they allow the analyst to include translations within the same relations applicable to rotations, on dualization 2 of the rotation equations, as occurring in the geometric, kinematic, or dynamic analyses of mechanical systems. After recalling the basic background on dual numbers and introducing reflection matrices defined over the dual ring , we obtain the dual version of Householder reflections applicable to the offline implementation of parameter identification. For the online parameter identification, recursive least squares are to be applied. We provide also the dual version of recursive least squares. Numerical examples are included to illustrate the underlying principles and algorithms.

The construction of surrogate models, such as radial basis function (RBF) and Kriging-based surrogates, requires an invertible (square and full rank matrix) or pseudoinvertible (overdetermined) linear system to be solved. This study demonstrates that the method used to solve this linear system may result in up to five orders of magnitude difference in the accuracy of the constructed surrogate model using exactly the same information. Hence, this paper makes the canonic and important point toward reproducible science: the details of solving the linear system when constructing a surrogate model must be communicated. This point is clearly illustrated on a single function, namely the Styblinski–Tang test function by constructing over 200 RBF surrogate models from 128 Latin Hypercubed sampled points. The linear system in the construction of each surrogate model was solved using LU, QR, Cholesky, Singular-Value Decomposition, and the Moore–Penrose pseudoinverse. As we show, the decomposition method influences the utility of the surrogate model, which depends on the application, i.e., whether an accurate approximation of a surrogate is required or whether the ability to optimize the surrogate and capture the optimal design is pertinent. Evidently the selection of the optimal hyperparameters based on the cross validation error also significantly impacts the utility of the constructed surrogate. For our problem, it turns out that selecting the hyperparameters at the lowest cross validation error favors function approximation but adversely affects the ability to optimize the surrogate model. This is demonstrated by optimizing each constructed surrogate model from 16 fixed initial starting points and recording the optimal designs. For our problem, selecting the optimal hyperparameter that coincides with the lowest monotonically decreasing function value significantly improves the ability to optimize the surrogate for most solution strategies.

When simulations are expensive and multiple realizations are necessary, as is the case in uncertainty propagation, statistical inference, and optimization, surrogate models can achieve accurate predictions at low computational cost. In this paper, we explore options for improving the accuracy of a surrogate if the modeled phenomenon presents symmetries. These symmetries allow us to obtain free information and, therefore, the possibility of more accurate predictions. We present an analytical example along with a physical example that has parametric symmetries. Although imposing parametric symmetries in surrogate models seems to be a trivial matter, there is not a single way to do it and, furthermore, the achieved accuracy might vary. We present four different ways of using symmetry in surrogate models. Three of them are straightforward, but the fourth is original and based on an optimization of the subset of points used. The performance of the options was compared with 100 random designs of experiments (DoEs) where symmetries were not imposed. We found that each of the options to include symmetries performed the best in one or more of the studied cases and, in all cases, the errors obtained imposing symmetries were substantially smaller than the worst cases among the 100. We explore the options for using symmetries in two surrogates that present different challenges and opportunities: Kriging and linear regression. Kriging is often used as a black box; therefore, we consider approaches to include the symmetries without changes in the main code. On the other hand, since linear regression is often built by the user; owing to its simplicity, we consider also approaches that modify the linear regression basis functions to impose the symmetries.

The array structure is widely used in precise electronic products such as large phased array antennas and large optical telescopes, the main components of which are a large surface base and a large number of high-precision discrete elements mounted on the surface base. The geometric error of discrete elements is inevitable in the manufacturing process and will seriously degrade the product performance. To deal with the tolerance design of discrete elements, a region-division-based tolerance design method is proposed in this paper. The whole array was divided into several regions by our method and the tolerance of discrete elements was correlated with the region importance on the performance. The method specifically includes the following steps: first, the sensitivity of the product performance to geometric errors was analyzed and the statistical relationship between the performance and geometric errors was established. Then, based on the sensitivity matrix, the regional division scheme was developed, and the corresponding tolerance was optimized according to the established relationship function. Finally, the optimal tolerance was selected among the multiple solutions to achieve the best performance. Taking a large phased array as an example, a simulation experiment was performed to verify the effectiveness of the proposed method.

Model-based reliability analysis may not be practically useful if reliability estimation contains uncontrollable errors. This paper addresses potential reliability estimation errors from model bias together with model parameters. Given three representative scenarios, reliability analysis strategies with representative methods are proposed. The pros and cons of these strategies are discussed and demonstrated using a tank storage problem based on the finite element model with different fidelity levels. It is found in this paper that the confidence-based reliability analysis considering epistemic uncertainty modeling for both model bias and model parameters can make reliability estimation errors controllable with less conservativeness compared to the direct reliability modeling using the Bayesian approach.

Reliability-based design optimization (RBDO) aims at optimizing the design of an engineered system to minimize the design cost while satisfying reliability requirements. However, it is challenging to perform RBDO under high-dimensional uncertainty due to the often prohibitive computational burden. In this paper, we address this challenge by leveraging a recently developed method for reliability analysis under high-dimensional uncertainty. The method is termed high-dimensional reliability analysis (HDRA). The HDRA method optimally combines the strengths of univariate dimension reduction (UDR) and kriging-based reliability analysis to achieve satisfactory accuracy with an affordable computational cost for HDRA problems. In this paper, we improve the computational efficiency of high-dimensional RBDO by pursuing two new strategies: (i) a two-stage surrogate modeling strategy is adopted to first locate a highly probable region of the optimum design and then locally refine the accuracy of the surrogates in this region; and (ii) newly selected samples are updated for all the constraints during the sequential sampling process in HDRA. The results of two mathematical examples and one real-world engineering example suggest that the proposed HDRA-based RBDO (RBDO-HDRA) method is capable of solving high-dimensional RBDO problems with higher accuracy and comparable efficiency than the UDR-based RBDO (RBDO-UDR) and ordinary kriging-based RBDO (RBDO-kriging) methods.

The performance of a product varies with respect to time and space if the associated limit-state function involves time and space. This study develops an uncertainty analysis method that quantifies the effect of random input variables on the performance (response) over time and space. The combination of the first order reliability method (FORM) and the second-order reliability method (SORM) is used to approximate the extreme value of the response with respect to space at discretized instants of time. Then the response becomes a Gaussian stochastic process that is fully defined by the mean, variance, and autocorrelation functions obtained from FORM and SORM, and a sequential single loop procedure is performed for spatial and random variables. The method is successfully applied to the reliability analysis of a crank-slider mechanism, which operates in a specified period of time and space.