Proposed Smoothing Method Research Articles

A particle-size dependent smoothing scheme for polydisperse Euler-Lagrange simulations

In Euler-Lagrange models for particulate systems, the information exchange of multiphase coupling terms, i.e. drag force, heat transfer, or mass transfer, is often smoothed from particles' location to nearby computational cells when the cell and particle sizes are comparable. The diffusion-based smoothing is among the most popular approach. However, when it comes to polydisperse systems, the state-of-the-art constant diffusivity approach does not consider variation in particle sizes, resulting in an even smoothing of exchanged information across all particles with different sizes.In this paper, a particle-size dependent smoothing scheme named “non-constant diffusivity approach” is proposed. This approach distributes the exchanged information based on the local Sauter diameter d32 such that the area of influence per particle varies by size. The one-dimensional analytical solution of the non-constant diffusivity approach is first derived and two control parameters L⁎ and S are identified, where L⁎ determines the characteristic length of the diffusivity and S determines the magnitude of the diffusivity. Together they control the characteristic length scale ℓ˜c of the smoothing operation. In the one-dimensional study, it is found that the proposed smoothing method is most suitable for the dilute particulate regime. The investigation is extended to a three-dimensional dilute particle flow based on the experiment of the CRIEPI (Central Research Institute of Electric Power Industry) pulverized coal burner. It is shown that using the non-constant diffusivity smoothing, the prediction of the O2 concentration is more accurate than when using the constant diffusivity smoothing approach, which assumes a universal smoothing length for all sizes of particles. The findings suggest that when it comes to polydisperse particulate systems, the area of influence per particle on the fluid depends on the particle size. The proposed non-constant diffusivity approach can describe that with the benefits of cheap computational cost, conservation of exchanged quantities, and easy-to-implement.

The Efficiency of the Proposed Smoothing Method over the Classical Cubic Smoothing Spline Regression Model with Autocorrelated Residual

Spline smoothing is a technique used to filter out noise in time series observations when predicting nonparametric regression models. Its performance depends on the choice of the smoothing parameter. Most of the existing smoothing methods applied to time series data tend to over fit in the presence of autocorrelated errors. This study aims to determine the optimum performance value, goodness of fit and model overfitting properties of the proposed Smoothing Method (PSM), Generalized Maximum Likelihood (GML), Generalized Cross-Validation (GCV), and Unbiased Risk (UBR) smoothing parameter selection methods. A Monte Carlo experiment of 1,000 trials was carried out at three different sample sizes (20, 60, and 100) and three levels of autocorrelation (0.2, 05, and 0.8). The four smoothing methods' performances were estimated and compared using the Predictive Mean Squared Error (PMSE) criterion. The findings of the study revealed that: for a time series observation with autocorrelated errors, provides the best-fit smoothing method for the model, the PSM does not over-fit data at all the autocorrelation levels considered ( the optimum value of the PSM was at the weighted value of 0.04 when there is autocorrelation in the error term, PSM performed better than the GCV, GML, and UBR smoothing methods were considered at all-time series sizes (T = 20, 60 and 100). For the real-life data employed in the study, PSM proved to be the most efficient among the GCV, GML, PSM, and UBR smoothing methods compared. The study concluded that the PSM method provides the best fit as a smoothing method, works well at autocorrelation levels (ρ=0.2, 0.5, and 0.8), and does not over fit time-series observations. The study recommended that the proposed smoothing is appropriate for time series observations with autocorrelation in the error term and econometrics real-life data. This study can be applied to; non – parametric regression, non – parametric forecasting, spatial, survival, and econometrics observations.

On some techniques of selecting spline smoothing parameters for a correlated dataset with autocorrelation structure in the residual

Residuals are minimized in a correlated dataset by selecting a smoothing parameter with optimum performance in the smoothing spline. The selection methods utilized in this study include Generalized Maximum Likelihood (GML), Generalized Cross-Validation (GCV), Unbiased Risk (UBR), and the Proposed Smoothing Method (PSM). The aim of this study is to compare the smoothing parameter selection ability of the four parameter selection methods for a correlated dataset with autocorrelation structure in the error term. To achieve this purpose, a Monte-Carlo simulation was conducted by utilizing program written in R-4.2.2. The performance of the parameter selection methods were evaluated using predictive Mean Squared Error (PMSE). Findings from the study indicated that GCV and GML were mostly affected by the presence of auto correlation in the residual and therefore had an asymptotically similar behavioural pattern. The estimators conformed to the asymptotic properties of the smoothing parameter selection methods considered; this is noticed in all the sample sizes and at all the smoothing parameters. The result also showed that; the most consistent and efficient among the four spline smoothing parameter selection methods considered in this study based on sample size and performance in the presence of autocorrelated residual error is the proposed smoothing method (PSM) because it does not undersmooth relative to the other smoothing method especially for small sample and medium sample size of 50 and 100.

The Efficiency of the Proposed Smoothing Method over the Classical Cubic Smoothing Spline Regression Model with Autocorrelated Residual

Spline smoothing is a technique used to filter out noise in time series observations when predicting nonparametric regression models. Its performance depends on the choice of the smoothing parameter. Most of the existing smoothing methods applied to time series data tend to overfit in the presence of autocorrelated errors. This study aims to determine the optimum performance value, goodness of fit and model overfitting properties of the proposed Smoothing Method (PSM), Generalized Maximum Likelihood (GML), Generalized Cross-Validation (GCV), and Unbiased Risk (UBR) smoothing parameter selection methods. A Monte Carlo experiment of 1,000 trials was carried out at three different sample sizes (20, 60, and 100) and three levels of autocorrelation (0.2, 05, and 0.8). The four smoothing methods' performances were estimated and compared using the Predictive Mean Squared Error (PMSE) criterion. The findings of the study revealed that: for a time series observation with autocorrelated errors, provides the best-fit smoothing method for the model, the PSM does not over-fit data at all the autocorrelation levels considered ( the optimum value of the PSM was at the weighted value of 0.04 when there is autocorrelation in the error term, PSM performed better than the GCV, GML, and UBR smoothing methods were considered at all-time series sizes (T = 20, 60 and 100). For the real-life data employed in the study, PSM proved to be the most efficient among the GCV, GML, PSM, and UBR smoothing methods compared. The study concluded that the PSM method provides the best fit as a smoothing method, works well at autocorrelation levels (ρ=0.2, 0.5, and 0.8), and does not over fit time-series observations. The study recommended that the proposed smoothing is appropriate for time series observations with autocorrelation in the error term and econometrics real-life data. This study can be applied to; non – parametric regression, non – parametric forecasting, spatial, survival, and econometrics observations.

CFD modelling of biomass ash deposition under multiple operation conditions using a 2D mass-conserving dynamic mesh approach

A dynamic CFD model with conjugate heat transfer that considers particle sticking and erosion is developed and used for predicting the effect of multiple operation conditions on ash deposition for a model biomass fly ash species (K2Si4O9) in a lab-scale entrained flow reactor. In order to achieve stable dynamic mesh morphing, a globally mass-conserving smooth method is proposed using a multiple-point weighted moving average algorithm and a growth scaling factor. A new method to estimate particle count for Lagrangian particle tracking is also proposed, which is based on particle impaction efficiency and face count of the mesh of deposition tube. Particle sticking is predicted based on the two-body collision method and particle erosion is evaluated using the empirical model that is dependent on temperature, particle diameter and velocity, and impaction angle. The proposed smoothing method incorporated with the particle count estimated can accomplish stable dynamic mesh morphing for all the 37 deposition cases without the need to loop the smoothing process and sub-group/averaging the growth rate. The prediction results using the proposed ash deposition model agree reasonably with the experimental observation of the effects of flue gas temperature, tube surface temperature, flue gas velocity, fly ash flux and deposition time on deposit formation rate. R2 of the predicted and measured deposit formation rates is approximately 0.68. The results also show that the predicted erosion rate essentially correlates negatively with the experimental deposit formation rate. Interestingly, coarse particles are numerically seen to reduce the deposition rate caused by smaller particles via erosion.

Non-Linear Gaussian Smoothing With Taylor Moment Expansion

This letter is concerned with solving continuous-discrete Gaussian smoothing problems by using the Taylor moment expansion (TME) scheme. In the proposed smoothing method, we apply the TME method to approximate the transition density of the stochastic differential equation in the dynamic model. Furthermore, we derive a theoretical error bound (in the mean square sense) of the TME smoothing estimates showing that the smoother is stable under weak assumptions. Numerical experiments are presented in order to illustrate practical use of the method.

Application of the digital model thermal errors of machine tools in automated production

The article presents a predicting method for a machine tool thermal error based on a nonlinear autoregressive neural network with an external input, as well as methods for smoothing experimental data obtained from measuring devices by approximation using polynomial regression and the gray systems theory. The development of accurate and robust thermal models is a critical step in achieving high productivity in thermal deformation reduction techniques on machine tools. Because thermal deformations of the machine structure caused by temperature increase often lead to thermal errors and reduce the accuracy of machining parts. The use of neural networks is a promising direction in solving forecasting problems. The authors propose a block diagram of a thermal process digital twin based on a neural network, which can be used in automated production. The results of the experiment carried out for the machine model 400V are obtained in the form of an assessment of approximation quality and accuracy of the forecasting model. The results show that the use of the proposed smoothing methods and a model for predicting a machine tool thermal error based on a neural network can improve the forecast accuracy.

A circumscribed local interpolation methodology for CNC machining along linear tool paths

ABSTRACT The G01 tool path, which consisted of linear segments, is commonly used in NC machining. There is the tangential and curvature discontinuity at the junctions of adjacent linear segments, which can cause the fluctuations of feedrate and deteriorate surface quality. In this paper, a novel circumscribed local interpolation method is proposed to smooth the discrete path and consequently improve the kinematic performance of the machine tool. The polygons are firstly constructed with a constraint of the maximum approximation error. Then, the double cubic Bezier curves are used to smooth the corner between two linear segments and the control points are derived according to the G2 continuity condition and the length constraint. Further, the motion profile along the blended tool path is generated by utilizing the quantic polynomial feedrate scheduling strategy. Finally, the proposed smoothing method is validated by simulations and experiments.

An Integrated Quasi-Monte Carlo Method for Handling High Dimensional Problems with Discontinuities in Financial Engineering

Quasi-Monte Carlo (QMC) method is a useful numerical tool for pricing and hedging of complex financial derivatives. These problems are usually of high dimensionality and discontinuities. The two factors may significantly deteriorate the performance of the QMC method. This paper develops an integrated method that overcomes the challenges of the high dimensionality and discontinuities concurrently. For this purpose, a smoothing method is proposed to remove the discontinuities for some typical functions arising from financial engineering. To make the smoothing method applicable for more general functions, a new path generation method is designed for simulating the paths of the underlying assets such that the resulting function has the required form. The new path generation method has an additional power to reduce the effective dimension of the target function. Our proposed method caters for a large variety of model specifications, including the Black-Scholes, exponential normal inverse Gaussian L\'evy, and Heston models. Numerical experiments dealing with these models show that in the QMC setting the proposed smoothing method in combination with the new path generation method can lead to a dramatic variance reduction for pricing exotic options with discontinuous payoffs and for calculating options' Greeks. The investigation on the effective dimension and the related characteristics explains the significant enhancement of the combined procedure.

Accurate Smoothing Methods for State Estimation of Continuous-Discrete Nonlinear Dynamic Systems

In this note, we extend the accurate continuous-discrete extended-cubature Kalman filter and continuous-discrete cubature Kalman filter to deal with the problem of Bayesian optimal smoothing in nonlinear dynamic systems. The dynamics can be modeled with nonlinear stochastic differential equations (SDEs) and the noise corrupted measurements are obtained at discrete time instants. To be consistent with the literature, the resulting nonlinear smoothers are referred to as the accurate continuous-discrete extended-cubature Kalman smoother and the continuous-discrete cubature Kalman smoother, respectively. We first present two approximation methodologies to solve the SDE encountered in the prediction step of continuous-discrete filter. Then, two types of novel Gaussian approximation smoothing methods are derived based on the fixed-interval Rauch–Tung–Striebel smoother, which computes the smoothing solution according to the stored filtering results. The performances of the proposed smoothing methods are demonstrated in a simulated application and the numerical results show that the newly presented approaches are more flexible and robust than other smoothing algorithms with lower computational cost.

A New Smoothing Method for Time Series Data in the Presence of Autocorrelated Error

Spline Smoothing is used to filter out noise or disturbance in an observation, its performance depends on the choice of smoothing parameters. There are many methods of estimating smoothing parameters; most popular among them are; Generalized Maximum Likelihood (GML), Generalized Cross-Validation (GCV), and Unbiased Risk (UBR), this methods tend to overfit smoothing parameters in the presence of autocorrelation error. A new Spline Smoothing estimation method is proposed and compare with three existing methods in order to eliminate the problem of over fitting associated with the presence of Autocorrelation in the error term. It is demonstrated through a simulation study performed by using a program written in R based on the predictive Mean Score Error criteria. The result indicated that the predictive mean square error (PMSE) of the four smoothing methods decreases as the smoothing parameters increases and decreases as the sample sizes increases. This study discovered that the proposed smoothing method is the best for time series observations with Autocorrelated error because it doesn’t over fit and works well for large sample sizes. This study will help researchers overcome the problem of over fitting associated with applying Smoothing spline method time series observation.

Accurate Smoothing for Continuous-Discrete Nonlinear Systems With Non-Gaussian Noise

In this letter, an accurate Gaussian sum-smoothing approach is derived for the continuous-discrete systems, where the dynamics can be modeled with nonlinear It $\hat{\mathrm{o}}$ -type stochastic differential equations and the measurements are obtained at discrete sampling times with non-Gaussian noise. The proposed smoothing method is derived by applying a bank of parallel accurate continuous-discrete extended-cubature Kalman filters used in the classical Gaussian state estimation to approximate the non-Gaussian estimation densities as a finite number of weighted sums of Gaussian densities. The performances of the proposed method are compared with the recently presented filters based on the maximum correntropy criterion in a simulated application and the numerical results show that the new approach is more accurate and robust than others.

A general mesh smoothing method for finite elements

This paper presents a general mesh smoothing method for finite elements. The method deals with two and three-dimensional meshes with virtually any type of element. To evaluate the quality of different element types, the paper also introduces a broad quality function. The proposed method works by solving a standard deformation analysis where nodal forces aim to deform each element into an optimally placed reference element. A detailed algorithm of the proposed smoothing method is presented which is suitable for a straightforward computer implementation. Several application examples in two and three-dimensions are presented and analyzed in order to demonstrate the capabilities of the proposed algorithm. In all cases very good quality improvements were obtained with very few iterations. Also, the proposed method provided greater improvements when compared to conventional Laplacian-based methods.

Method of smoothing photoluminescence spectra

The smoothing method for experimental photoluminescence spectra, which uses Tikhonov regularization method, is proposed. Smoothing problem statement is performed as a statement of the minimization problem based on the least squares criterion. The functional to be minimized is a compound of two terms. The first one is the basic term, which determines the quality of data approximation provided by smooth function. The second one is the regularization term, which sets a constraint on the energy by smooth function derivative of a given order. A solution of the smoothing problem is obtained in the closed form analytically. The solution can be implemented by a simple array of matrix operations; one of those is an operation of matrix inversion. The necessity of matrix inversion imposes constraints on the order of derivative used in the proposed method and on noise level. It is related to the increase of either the order of derivative at the given noise level in data or the noise level at the given order of derivative, that deteriorates the conditionality of a matrix to be inverted and limits the scope of the proposed method, accordingly. Efficiency comparison of the proposed smoothing method, the Savitzky-Golay method, and the moving average method was performed by the numerical simulation of two models of photoluminescence spectra. Their advantages, disadvantages, and also the scopes of applicability are noted. The best values of operating parameters of these methods for the different noise levels in data were obtained. The conditions, under which the potential capabilities of the proposed smoothing method excel the potential possibilities of Savitzky-Golay method, were indicated. The results of smoothing the experimental photoluminescence spectra obtained for ZnO:Mn nanocrystals and ZnS:Mn crystals are provided.

Combining multi-scale composite windows with hierarchical smoothing strategy for fingerprint orientation field computation

BackgroundOrientation field (OF) plays a very significant role in automatic fingerprint recognition systems. Many algorithms have been proposed for the estimation of fingerprints’ OF but it is hard to solve the dilemma of correcting spurious ridge structure and avoiding singularity location deviation, especially for poor images. So far, the following drawbacks still need to be solved for OF construction methods for practical application: (1) How to adaptively choose block scales to resolve the contradiction between accuracy and anti-noise, since small scale is beneficial to accuracy but is sensitive to noise, while large scale is more resistant to noise, but the accuracy is deteriorated. (2) How to construct the genuine OF in the areas close-by singular points and to evade singularity location deviation? Current block based methods give spurious OF estimates in the area near singular points because these areas have large curvature thus the detected singular points deviate from the genuine localizations. When these singular points are used as the anchor for referencing minutiae, it makes the average error of matching or recognition even larger. Therefore, it is essentials to construct the genuine OF in the areas close-by singular points and to evade singularity deviation.MethodsTo overcome the above-mentioned limitations, a novel method, combining a weighted multi-scale composite window (WMCM) with a hierarchical smoothing strategy has been proposed for the computation of fingerprint OF. This method mainly contains two procedures: the approximate OF estimation and the hierarchical OF smoothing. In the first procedure, a series of OFs are established under multiple scales of composite windows by using a gradient based method then a coarse OF is estimated using the weight of each scale determined by a squared gradient consistency. In the second procedure, the OF is first quantized into a two-digitized orientation zone and a two-orientation-zone filtering strategy is adapted to the OF blocks based on a filtering mask obtained after eliminating the isolated blocks. In the end a similar three-digitized orientation zone is performed to obtain an accurate and smooth OF. To validate the performance, the proposed method has been applied to OF computation using the FVC2004 databases and three experiments are designed. Experiment 1 aims to validate whether the weighted multi-scale composite window can balance the dilemma of accuracy and robustness more effectively than the previous works do. Experiment 2 is designed to examine whether the hierarchical smoothing method can correct the spurious ridge flow and preserve the genuine localization of singular points. The purpose of experiment 3 is to test the performance of the proposed method on OF reconstruction in low quality fingerprint images. The fingerprint databases FVC 2004 DB1–DB4 are employed in this study.ResultsThe results of experiment I shows that the proposed method is capable to extract the information of OF reliably and it is more robust against singularity localization deviation in comparison with the other three gradient based methods. The results of experiment II indicates that the proposed smoothing method can balance the contradiction in correcting spurious ridge structures and preserving genuine singularity localization. The results of experiment III illustrates that our approach combing WMCW with the hierarchical smoothing method is capable to extract the information of OF ridge reliably and it is more robust against singularity deviation in comparison with the other three gradient based methods. In a word, the experiment results demonstrate that the proposed method can correct spurious ridge structure and meanwhile avoid singularity deviation compared with the previous works.ConclusionsA novel gradient based algorithm has been proposed which is more reliable for the estimation of the ridge information for fingerprint OF and is more accurate in preserving the singularity localization. Compared with the previously proposed gradient based methods, the advantages of the proposed RBSF lie in three aspects. Firstly a weighted multi-scale composite window is put forward to replace the single window used by conventional gradient based methods and to adaptively choose the scales of the blocks. Secondly, a hierarchical smoothing strategy is proposed to enhance the OF by using the two-orientation-zone filtering and the three-orientation-zone filtering, aiming to correct the spurious ridges and preserving the genuine location of singular points. Finally, three experiments are designed to test the proposed algorithm together with other popular gradient based methods on real fingerprint images, which are selected from different categories and all are suffering from obvious noise effects. All the experiment results show that the proposed method is superior with respect to reliable OF construction and avoiding singularity localization deviation.

Depth Contour Smoothing Based on the Fitting of Multi-Segment Bezier Curves

ABSTRACTTo overcome smoothness inconsistencies in depth contours interpolated from sounding data, this paper proposes a smoothing method based on the fitting of multi-segment Bezier curves considering navigational safety. The basic principle of this method is the division of a generated depth contour into a series of bends categorized as either convex bends or concave bends. Different smoothing strategies are designed for different bends while considering the safety constraint. Convexity-preserving Bezier curves are used to fit convex bends. For concave bends, the line segment of the bend is utilized as a boundary to construct a control polygon within the interior of the bend, after which a Bezier curve is constructed based on the polygon. Experiments with real data are implemented, and the results are compared with those of other methods. The results show that the smoothing procedure for depth contours with angularities is effective and respects the safety constraint; moreover, the shape characteristics are preserved, and high positional accuracy is achieved. In addition, the proposed smoothing method is free of topological inconsistencies.

Speckle removal and change preservation by distance-driven anisotropic diffusion of synthetic aperture radar temporal stacks

We propose a method that smoothes spatially and uses, but also preserves, temporal information of synthetic aperture radar image stacks. To provide smoothing of such imagery without effacing temporal changes in the scene, we put forth an anisotropic diffusion technique, using a PDE approach. This approach smoothes uniform areas and preserves and enhances edges, such as roads or other features. In order to use this anisotropic diffusion technique, a homogeneous region must first be selected in the image to calculate statistics of the speckle noise. We propose a method that uses the temporal information to automatically select a homogeneous region prior to smoothing. Our proposed smoothing method calculates the distance between either pixel time-series or pixel CDFs as a measure of similarity, or uniformity. Results demonstrate the efficacy of the approach on real and synthetic data, showing lower mean squared ratio and higher structural similarity index than leading methods, as well as temporal change preservation.

Real-time local smoothing for five-axis linear toolpath considering smoothing error constraints

Abstract Five-axis linear toolpaths (or G 01 blocks) are widely used in CNC machine tools. The tangential and curvature discontinuities at the corners of linear toolpath result in feed fluctuation and deteriorate the machining efficiency and quality. Several methods have been proposed to locally smooth the corners for three-axis toolpath, but local smoothing for five-axis toolpath is still challenging due to two main difficulties: control of the tool orientation smoothing error and parameter synchronization between tool position and tool orientation. This paper proposes a real-time G 2 continuous local smoothing method by replacing the corners of tool position and tool orientation paths with cubic B-splines. The two difficulties in five-axis local smoothing are both resolved in simple and analytical ways. With a two-step method, the tool orientation smoothing error is directly and analytically constrained. By converting the remaining linear segments into B-splines, the C 1 continuity of the smooth tool position and smooth tool orientation paths is achieved. The parameter synchronization is realized by sharing the parameter of the tool orientation with that of the tool position. Compared with the existing analytical methods, the proposed method has a higher computation efficiency and a tighter tolerance in control of the orientation smoothing error. We have developed an open-source benchmark which validates the computation efficiency and error control ability of the proposed method. After smoothing and synchronization, the inserted B-spline of tool position is traversed with constant feedrate and the feedrate of the remaining linear segment is planned with jerk-bounded trajectory profile. The proposed smoothing method can be implemented on-line and has been integrated into a self-developed open-architecture CNC system. Its effectiveness for on-line generating smooth motion has been validated via simulations and experiments.

MARITAL STATUS AS A RISK FACTOR IN LIFE INSURANCE: AN EMPIRICAL STUDY IN TAIWAN

AbstractGender and age are the top two risk factors considered in pricing life insurance products. Although it is believed that mortality rates are also related to other factors (e.g. smoking, overweight, and especially marriage), data availability and marketing often limit the possibility of including them. Many studies have shown that married people (particularly men) benefit from the marriage, and generally have lower mortality rates than unmarried people. However, most of these studies used data from a population sample; their results might not apply to the whole population. In this study, we explore if mortality rates differ by marital status using mortality data (1975–2011) from the Taiwan Ministry of the Interior. In order to deal with the problem of small sample sizes in some marital status groups, we use graduation methods to reduce fluctuations in mortality rates. We also use a relational approach to model mortality rates by marital status, and then compare the proposed model with some popular stochastic mortality models. Based on computer simulation, we find that the proposed smoothing methods can reduce fluctuations in mortality estimates between ages, and the relational mortality model has smaller errors in predicting mortality rates by marital status. Analyses of the mortality data from Taiwan show that mortality rates differ significantly by marital status. In some age groups, the differences in mortality rates are larger between marital status groups than between smokers and non-smokers. For the issue of practical consideration, we suggest modifications to include marital status in pricing of life insurance products.

A contact smoothing method for arbitrary surface meshes using Nagata patches

Abstract This paper presents a contact surface smoothing method combined with the node-to-segment discretization technique to solve large deformation frictional contact problems between deformable bodies. The Nagata patch interpolation is used to smooth the surface mesh, providing a master surface with quasi- G 1 continuity between patches. Moreover, the local support of the interpolation method allows to deal with surface meshes of arbitrary topology (regular and irregular finite element discretizations), as well as hybrid meshes. The non-physical oscillations in the contact force evolution, induced by the faceted contact surface representation, are reduced using the proposed smoothing method. Furthermore, the smooth representation of the master surface allows a more accurately evaluation of the resulting stresses and forces, while providing an important improvement in convergence behaviour. Four representative numerical examples are used to demonstrate the advantages of the proposed contact smoothing method. The results show a significant improvement in the accuracy, robustness and performance of the numerical simulations using the smoothing approach, when compared with the piecewise faceted contact surface description.