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Curve fitting from unordered noisy point samples is needed for surface reconstruction in many applications. In the literature, several approaches have been proposed to solve this problem. However, previous works lack formal characterization of the curve fitting problem and assessment on the effect of several parameters (i.e. scalars that remain constant in the optimization problem), such as control points number (m), curve degree (b), knot vector composition (U), norm degree (k), and point sample size (r) on the optimized curve reconstruction measured by a penalty function (f). The paper aims to discuss these issues.

A numerical sensitivity analysis of the effect of m, b, k and r on f and a characterization of the fitting procedure from the mathematical viewpoint are performed. Also, the spectral (frequency) analysis of the derivative of the angle of the fitted curve with respect to u as a means to detect spurious curls and peaks is explored.

It is more effective to find optimum values for m than k or b in order to obtain good results because the topological faithfulness of the resulting curve strongly depends on m. Furthermore, when an exaggerate number of control points is used the resulting curve presents spurious curls and peaks. The authors were able to detect the presence of such spurious features with spectral analysis. Also, the authors found that the method for curve fitting is robust to significant decimation of the point sample.

The authors have addressed important voids of previous works in this field. The authors determined, among the curve fitting parameters m, b and k, which of them influenced the most the results and how. Also, the authors performed a characterization of the curve fitting problem from the optimization perspective. And finally, the authors devised a method to detect spurious features in the fitting curve.

This paper provides a methodology to select the important tuning parameters in a formal manner.

Up to the best of the knowledge, no previous work has been conducted in the formal mathematical evaluation of the sensitivity of the goodness of the curve fit with respect to different possible tuning parameters (curve degree, number of control points, norm degree, etc.).

Presents a method of Max‐Fit biarc curve fitting technique to improve the accuracy of STL files and to reduce the file size for rapid prototyping. STL file has been widely accepted as a de facto standard file format for the rapid prototyping industry. However, STL format is an approximated representation of a true solid/surface model, and a huge amount of STL data is needed to provide sufficient accuracy for rapid prototyping. Presents a Max‐Fit biarc curve fitting technique to reconstruct STL slicing data for rapid prototyping. The Max‐Fit algorithm progresses through the STL slicing intersection points to find the most efficient biarc curve fitting, while improving the accuracy. Our results show that the proposed biarc curve‐fitting technique can significantly improve the accuracy of poorly generated STL files by smoothing the intersection points for rapid prototyping. Therefore, less strict requirements (i.e. loose triangle tolerances) can be used while generating the STL files.

This research aims to develop an effective and efficient algorithm for solving the curve fitting problem arising in automated manufacturing systems.

This paper takes curve fitting as an optimization problem of a set of data points. Expressing the data as a function will be very effective to the data analysis and application. This paper will develop the stochastic optimization method to apply to curve fitting. The proposed method is a combination optimization method based on pattern search (PS) and simulated annealing algorithm (SA).

The proposed method is used to solve a nonlinear optimization problem and then to implement it to solve three circular arc‐fitting problems of curve fitting. Based on the analysis performed in the experimental study, the proposed algorithm has been found to be suitable for curve fitting.

Curve fitting is one of the basic form errors encountered in circular features. The proposed algorithm is tested and implemented by using nonlinear problem and circular data to determine the circular parameters.

The developed machine vision‐based approach can be an online tool for measurement of circular components in automated manufacturing systems.

This study investigates the fitting techniques for notch fatigue curves, seeking a more reliable method to predict the lifespan of welded structures.

Building on the fatigue test results of butt and cruciform joints, this research delves into the selection of fitting methods for the notch fatigue curve of welded joints. Both empirical formula and finite element methods (FEMs) were employed to assess the notch stress concentration factor at the toe and root of the two types of welded joints. Considering the mean stress correction and weld misalignment coefficients, the notch fatigue life curves were established using both direct and indirect methods.

An engineering example was employed to discern the differences between the direct and indirect approaches. The findings highlight the enhanced reliability of the indirect method for fitting the fatigue life curve.

While the notch stress approach is extensively adopted due to its accurate prediction of component fatigue life, most scholars have overlooked the importance of its curve fitting methods. Existing literature scantily addresses the establishment of these curves. This paper offers a focused examination of fatigue curve fitting techniques, delivering valuable perspectives on method selection.

The methods used in long‐range prediction include causal models and regression on leading variables, but one method which is generally worth considering is the fitting of trend lines. It is the object of this paper to present simple techniques for fitting these lines.

When establishing a mathematical model to simulate solid mechanics, considering realistic geometries, special tools are needed to translate measured data, possibly with noise, into idealized geometrical entities. As an engineering application, wheel-rail contact interactions are fundamental in the dynamic modeling of railway vehicles. Many approaches used to solve the contact problem require a continuous parametric description of the geometries involved. However, measured wheel and rail profiles are often available as sets of discrete points. A reconstruction method is needed to transform discrete data into a continuous geometry.

The authors present an approximation method based on optimization to solve the problem of fitting a set of points with an arc spline. It consists of an initial guess based on a curvature function estimated from the data, followed by a least-squares optimization to improve the approximation. The authors also present a segmentation scheme that allows the method to increment the number of segments of the spline, trying to keep it at a minimal value, to satisfy a given error tolerance.

The paper provides a better understanding of arc splines and how they can be deformed. Examples with parametric curves and slightly noisy data from realistic wheel and rail profiles show that the approach is successful.

The developed methods have theoretical value. Furthermore, they have practical value since the approximation approach is better suited to deal with the reconstruction of wheel/rail profiles than interpolation, which most methods use to some degree.

The purpose of the article is to develop an equation to accurately represent OCV as a function of SoC with reduced computational burden. Dependency of open circuit voltage (OCV) on state of charge (SoC) is often represented by either a look-up table or an equation developed by regression analysis. The accuracy is increased by either a larger data set for the look-up table or using a higher order equation for the regression analysis. Both of them increase the memory requirement in the controller. In this paper, Gaussian exponential regression methodology is proposed to represent OCV and SoC relationships accurately, with reduced memory requirement.

Incremental OCV test under constant temperature provides a data set of OCV and SoC. This data set is subjected to polynomial, Gaussian and the proposed Gaussian exponential equations. The unknown coefficients of these equations are obtained by least residual algorithm and differential evolution–based fitting algorithms for charging, discharging and average OCV.

Root mean square error (RMSE) of the proposed equation for differential evolution–based fitting technique is 35% less than second-order Gaussian and 74% less than a fifth-order polynomial equation for average OCV with a 16.66% reduction in number of coefficients, thereby reducing memory requirement.

The knee structure in the OCV and SoC relationship is accurately represented by Gaussian first-order equation, and the exponential equation can accurately describe the linear relation. Therefore, this paper proposes a Gaussian exponential equation that accurately represents the OCV as a function of SoC. The results obtained from the proposed regression methodology are compared with the polynomial and Gaussian regression in terms of RMSE, mean average, variance and number of coefficients.

This paper aims to summarize the influence law of hybrid deposited and micro-rolling (HDMR) technology on the bead morphology and overlapping coefficient. A better bead topology positively supports the overlapping deposited in multi-beads between layers while actively assisting the subsequent layer's deposition in the wire and arc additive manufacturing (WAAM). Hybrid-deposited and micro-rolling (HDMR) additive manufacturing (AM) technology can smooth the weld bead for improved surface quality. However, the micro-rolling process will change the weld bead profile fitting curve to affect the overlapping coefficient.

Weld bead contours for WAAM and HDMR were extracted using line lasers. A comparison of bead profile curves was conducted to determine the influence law of micro-zone rolling on the welding bead contour and fitting curve. Aiming at the optimized overlapping coefficient of weld bead in HDMR AM, the optimal HDMR overlapping coefficient curve was proposed which varies with the reduction based on the best surface flatness. The mathematical model for overlapping in HDMR was checked by comparing the HDMR weld bead contours under different rolling reductions.

A fitting function of the bead forming by HDMR AM was proposed based on the law of conservation of mass. The change rule of the HDMR weld bead overlapping spacing with the degree of weld bead rolling reduction was generated using the flat-top transition calculation for this model. Considering the damming-up impact of the first bead, the overlapping coefficient was examined for its effect on layer surface flatness.

Using the predicted overlapping model, the optimal overlapping coefficients for different rolling reductions can be achieved without experiments. These conclusions can encourage the development of HDMR technology.

The purpose of this study is to establish a new temperature set for characterizing the frictional temperature rise (FTR) of disc brakes. The FTR produced by braking is an important factor which directly affects the tribological properties of disc brakes. Presently, most existing researches characterize the FTR only by several static parameters such as average temperature or maximum temperature, which cannot reflect accurately the dynamic characteristics of temperature variation in the process of braking. In this paper, a new temperature parameter set was extracted and the influences of braking conditions on these parameters were investigated by experiments.

First, several simulated braking experiments of disc brakes were conducted to reveal the dynamic variation rules and mechanisms of the FTR in braking. Second, the characteristic parameter subset of the FTR was extracted with five significant parameters, namely, initial temperature, average temperature, end temperature, maximum temperature and the ratio of maximum temperature time. Furthermore, the fitting parameter subset of the FTR was constructed based on the temperature rise curve. Finally, the influence and mechanisms of initial braking velocity and braking pressure on the new temperature parameter set were investigated through braking experiments.

This paper extracted a new temperature parameter set including a characteristic parameter subset and a fitting parameter subset and revealed the influences of braking conditions on it by experiments.

The results showed that the new temperature parameter set extracted in this paper can characterize the dynamic characteristics of disc brake’s FTR variations more objectively and comprehensively. The research results will provide a theoretical basis for extracting the fault feature of friction properties.

Standard value/cost flow models (often referred to as S‐curves) are widely used in cash flow forecasting, particularly at the tender stage. A substantial amount of research has concentrated on improving the accuracy of these curves. Categorizing construction projects into groups and subgroups has helped, but the fact remains that different construction projects possess different profiles of cost flow. This paper is an attempt at assessing the extent of influence of planning and programming the work on the cost flow curves. One real project was used as a case study and four planners were independently asked to produce programmes for executing the project. These programmes were analysed and converted to cost flow curves using one database of productivity and unit cost rates. Results confirmed that the variations in programmes produced less variations in cost flow curves than the errors to be expected from the use of average curves derived from project groups (mean SDY 2.88 compared to previous studies of 5.5, 8.5 and 10.67). The results suggest, taking due account of the limitations of the scope of the study, that further effort at the categorisation of projects into subgroups will result in a reasonable improvement in accuracy.