Journal articles: 'Discontinuous spline'

1

Lytvyn, Oleg, Oleg Lytvyn, and Oleksandra Lytvyn. "Analysis of the results of a computational experiment to restore the discontinuous functions of two variables using projections." Physico-mathematical modelling and informational technologies, no. 33 (September 2, 2021): 12–17. http://dx.doi.org/10.15407/fmmit2021.33.012.

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This article presents the main statements of the method of approximation of discontinuous functions of two variables, describing an image of the surface of a 2D body or an image of the internal structure of a 3D body in a certain plane, using projections that come from a computer tomograph. The method is based on the use of discontinuous splines of two variables and finite Fourier sums, in which the Fourier coefficients are found using projection data. The method is based on the following idea: an approximated discontinuous function is replaced by the sum of two functions – a discontinuous spline and a continuous or differentiable function. A method is proposed for constructing a spline function, which has on the indicated lines the same discontinuities of the first kind as the approximated discontinuous function, and a method for finding the Fourier coefficients of the indicated continuous or differentiable function. That is, the difference between the function being approximated and the specified discontinuous spline is a function that can be approximated by finite Fourier sums without the Gibbs phenomenon. In the numerical experiment, it was assumed that the approximated function has discontinuities of the first kind on a given system of circles and ellipses nested into each other. The analysis of the calculation results showed their correspondence to the theoretical statements of the work. The proposed method makes it possible to obtain a given approximation accuracy with a smaller number of projections, that is, with less irradiation.

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Koo, Ja-Yong. "Spline Estimation of Discontinuous Regression Functions." Journal of Computational and Graphical Statistics 6, no. 3 (September 1997): 266. http://dx.doi.org/10.2307/1390733.

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Koo, Ja-Yong. "Spline Estimation of Discontinuous Regression Functions." Journal of Computational and Graphical Statistics 6, no. 3 (September 1997): 266–84. http://dx.doi.org/10.1080/10618600.1997.10474742.

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Peters, Jörg. "General spline filters for discontinuous Galerkin solutions." Computers & Mathematics with Applications 70, no. 5 (September 2015): 1046–50. http://dx.doi.org/10.1016/j.camwa.2015.06.034.

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López de Silanes, M. C., M. C. Parra, M. Pasadas, and J. J. Torrens. "Spline approximation of discontinuous multivariate functions from scattered data." Journal of Computational and Applied Mathematics 131, no. 1-2 (June 2001): 281–98. http://dx.doi.org/10.1016/s0377-0427(00)00268-5.

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Liu, Cheng-ming, Ze-kun Wang, Hai-bo Pang, and Jun-xiao Xue. "Image Interpolation via Scanning Line Algorithm and Discontinuous B-Spline." Mathematical and Computational Applications 22, no. 2 (May 31, 2017): 34. http://dx.doi.org/10.3390/mca22020034.

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Saranen, J., and W. Wendland. "On the convergence of the spline collocation with discontinuous data." Mathematical Methods in the Applied Sciences 9, no. 1 (1987): 59–75. http://dx.doi.org/10.1002/mma.1670090107.

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8

Wright, R. K. "Spline fitting discontinuous functions given just a few Fourier coefficients." Numerical Algorithms 9, no. 1 (March 1995): 157–69. http://dx.doi.org/10.1007/bf02143932.

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9

Zhang, Zhimin, and Clyde F. Martin. "Convergence and Gibbs' phenomenon in cubic spline interpolation of discontinuous functions." Journal of Computational and Applied Mathematics 87, no. 2 (December 1997): 359–71. http://dx.doi.org/10.1016/s0377-0427(97)00199-4.

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Pershyna, Iuliia. "RECONSTRUCTION OF THE TWO VARIABLES DISCONTINUOUS FUNCTION BY DIFFERENT INFORMATION OPERATORS USING TRIANGULAR ELEMENTS." Bulletin of the National Technical University "KhPI". Series: Mathematical modeling in engineering and technologies, no. 2 (November 30, 2021): 84–96. http://dx.doi.org/10.20998/2222-0631.2021.02.10.

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The paper examines methods for constructing mathematical models of two variables discontinuous functions using various information about them: one-sided values at points and one-sided traces along a given system of lines. The case is considered when the domain of the required function is triangulated by right-angled triangles. If interpolation or approximation methods are used, then for their construction the values of the function at given points must be given; if we use interlination methods, then traces of the desired function along a given system of lines. In this work, we construct a discontinuous interpolation and approximation splines for approximating a discontinuous function of two variables with given one-sided values in a given system of points (in our case, at the vertices of right-angled triangles), and prove theorems on the estimation of the approximation error by constructed discontinuous structures. In the paper a discontinuous interlination spline, which uses completely different information about the discontinuous function, namely one-sided traces along a given system of lines (in our case, along the sides of right-angled triangles) is also built. Interlination of functions can find wide application in the aircraft and automobile body design automation; when receiving and processing the results of sonar and radar, when solving problems of computed tomography, in digital signal processing and in many other areas. In the paper theorems on the integral form and an estimate of the approximation error by the constructed discontinuous interlination operator are also proved. Computational experiments that compare the results of the approximation of a discontinuous function of two variables by different information operators using triangular elements are presented. In the future, it is planned to apply the constructed operators of discontinuous approximation and interlination to solve a two-dimensional problem of computed tomography with a significant use of the inhomogeneity of the internal structure of the body, which must be reconstructed.

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Medina, S., and A. V. Olver. "Regimes of Contact in Spline Couplings." Journal of Tribology 124, no. 2 (July 19, 2001): 351–57. http://dx.doi.org/10.1115/1.1403456.

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The contact between the male and female teeth of involute spline couplings connecting misaligned transmission shafts has been studied using an elastostatic contact model with stick-slip friction based on the boundary integral element method. The effect on the distribution of pressure and on the slip path during shaft rotation, of a wide range of design parameters and of applied torque and misalignment has been explored. The predicted behavior is classified according to the regime of friction (cyclic stick-slip or gross slip) and to that of the pressure history (uniform, cyclic, discontinuous, or toppled). The magnitude of the maximum tooth load, the axial skewness of the distribution of pressure and a maximum wear depth parameter are presented in terms of dimensionless design and operating parameters. The effect of tooth crowning is briefly examined. The results show a number of previously unreported features including a cyclic tooth load—which declines to zero for certain conditions—and an effective slip amplitude of around half the rigid-body value. This may affect the interpretation of laboratory fretting tests.

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Lytvyn, Oleg N., and Yu I. Pershina. "Approximation of Discontinuous Function of Two Variables by Approximating Discontinuous Bilinear Spline Using the Least Squares Method (Rectangular Elements)." Journal of Automation and Information Sciences 44, no. 5 (2012): 48–56. http://dx.doi.org/10.1615/jautomatinfscien.v44.i5.50.

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13

Köhler, Peter, Christoph Nehrbass-Ahles, Jochen Schmitt, Thomas F. Stocker, and Hubertus Fischer. "A 156 kyr smoothed history of the atmospheric greenhouse gases CO2, CH4, and N2O and their radiative forcing." Earth System Science Data 9, no. 1 (June 20, 2017): 363–87. http://dx.doi.org/10.5194/essd-9-363-2017.

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Abstract. Continuous records of the atmospheric greenhouse gases (GHGs) CO2, CH4, and N2O are necessary input data for transient climate simulations, and their associated radiative forcing represents important components in analyses of climate sensitivity and feedbacks. Since the available data from ice cores are discontinuous and partly ambiguous, a well-documented decision process during data compilation followed by some interpolating post-processing is necessary to obtain those desired time series. Here, we document our best possible data compilation of published ice core records and recent measurements on firn air and atmospheric samples spanning the interval from the penultimate glacial maximum ( ∼ 156 kyr BP) to the beginning of the year 2016 CE. We use the most recent age scales for the ice core data and apply a smoothing spline method to translate the discrete and irregularly spaced data points into continuous time series. These splines are then used to compute the radiative forcing for each GHG using well-established, simple formulations. We compile only a Southern Hemisphere record of CH4 and discuss how much larger a Northern Hemisphere or global CH4 record might have been due to its interpolar difference. The uncertainties of the individual data points are considered in the spline procedure. Based on the given data resolution, time-dependent cutoff periods of the spline, defining the degree of smoothing, are prescribed, ranging from 5000 years for the less resolved older parts of the records to 4 years for the densely sampled recent years. The computed splines seamlessly describe the GHG evolution on orbital and millennial timescales for glacial and glacial–interglacial variations and on centennial and decadal timescales for anthropogenic times. Data connected with this paper, including raw data and final splines, are available at doi:10.1594/PANGAEA.871273.

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Kolman, R., M. Okrouhlík, A. Berezovski, D. Gabriel, J. Kopačka, and J. Plešek. "B-spline based finite element method in one-dimensional discontinuous elastic wave propagation." Applied Mathematical Modelling 46 (June 2017): 382–95. http://dx.doi.org/10.1016/j.apm.2017.01.077.

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15

Pu, Yasong, Yaoyao Shi, Xiaojun Lin, Yuan Hu, and Zhishan Li. "C2-Continuous Orientation Planning for Robot End-Effector with B-Spline Curve Based on Logarithmic Quaternion." Mathematical Problems in Engineering 2020 (July 22, 2020): 1–16. http://dx.doi.org/10.1155/2020/2543824.

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Smooth orientation planning is beneficial for the working performance and service life of industrial robots, keeping robots from violent impacts and shocks caused by discontinuous orientation planning. Nevertheless, the popular used quaternion interpolations can hardly guarantee C2 continuity for multiorientation interpolation. Aiming at the problem, an efficient quaternion interpolation methodology based on logarithmic quaternion was proposed. Quaternions of more than two key orientations were expressed in the exponential forms of quaternion. These four-dimensional quaternions in space S3, when logarithms were taken for them, could be converted to three-dimensional points in space R3 so that B-spline interpolation could be applied freely to interpolate. The core formulas that B-spline interpolated points were mapped to quaternion were founded since B-spline interpolated point vectors were decomposed to the product of unitized forms and exponents were taken for them. The proposed methodology made B-spline curve applicable to quaternion interpolation through dimension reduction and the high-order continuity of the B-spline curve remained when B-spline interpolated points were mapped to quaternions. The function for reversely finding control points of B-spline curve with zero curvature at endpoints was derived, which helped interpolation curve become smoother and sleeker. The validity and rationality of the principle were verified by the study case. For comparison, the study case was also analyzed by the popular quaternion interpolations, Spherical Linear Interpolation (SLERP) and Spherical and Quadrangle (SQUAD). The comparison results demonstrated the proposed methodology had higher smoothness than SLERP and SQUAD and thus would provide better protection for robot end-effector from violent impacts led by unreasonable multiorientation interpolation. It should be noted that the proposed methodology can be extended to multiorientation quaternion interpolation with higher continuity than the second order.

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Vaid, Mandeep Kaur, and Geeta Arora. "Quintic B-Spline Technique for Numerical Treatment of Third Order Singular Perturbed Delay Differential Equation." International Journal of Mathematical, Engineering and Management Sciences 4, no. 6 (December 1, 2019): 1471–82. http://dx.doi.org/10.33889/ijmems.2019.4.6-116.

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In this paper, a class of third order singularly perturbed delay differential equation with large delay is considered for numerical treatment. The considered equation has discontinuous convection-diffusion coefficient and source term. A quintic trigonometric B-spline collocation technique is used for numerical simulation of the considered singularly perturbed delay differential equation by dividing the domain into the uniform mesh. Further, uniform convergence of the solution is discussed by using the concept of Hall error estimation and the method is found to be of first-order convergent. The existence of the solution is also established. Computation work is carried out to validate the theoretical results.

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17

Wright, R. K. "Local spline approximation of discontinuous functions and location of discontinuities, given low-order Fourier coefficient information." Journal of Computational and Applied Mathematics 164-165 (March 2004): 783–95. http://dx.doi.org/10.1016/s0377-0427(03)00647-2.

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18

Einkemmer, Lukas. "A performance comparison of semi-Lagrangian discontinuous Galerkin and spline based Vlasov solvers in four dimensions." Journal of Computational Physics 376 (January 2019): 937–51. http://dx.doi.org/10.1016/j.jcp.2018.10.012.

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Deswal, Komal, and Devendra Kumar. "Rannacher time-marching with orthogonal spline collocation method for retrieving the discontinuous behavior of hedging parameters." Applied Mathematics and Computation 427 (August 2022): 127168. http://dx.doi.org/10.1016/j.amc.2022.127168.

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20

Zhang, Xinyi, Xiaoqing Qiang, Jinfang Teng, and Wensheng Yu. "A new curvature-controlled stacking-line method for optimization design of compressor cascade considering surface smoothness." Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering 234, no. 5 (December 11, 2019): 1061–74. http://dx.doi.org/10.1177/0954410019894119.

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The paper presents an advanced parametric method of blade stacking lines in terms of sweep and lean based on controlled curvature. To the knowledge of the authors, there is no related approach reported in open literature that uses Bezier spline as the radial curvature distribution to improve the smoothness of the blade surface; most previous studies ignored the discontinuous slopes of curvature of the parametric curves. The parametric method called curvature-controlled stacking-line method (CCSLM) is performed by changing the magnitude of the sweep or lean. A fourth Bezier spline is adopted to define the curvature of spanwise stacking line directly ensuring surface smoothness. Then, the redesign cascades are created by sectional profiles stacked along the radial stacking lines which are obtained by twice integrating the Bezier spline. Then, the advanced method is conducted to optimize a high-subsonic controlled diffusion airfoil at design point, where the blade shape is generated in terms of lean. A single-objective optimization is performed using Kriging model and genetic algorithm to optimize total pressure loss, and the optimized geometry is obtained. The optimization results show that the blade design CCSLM has significant effects on the endwall flow vortex as well as radial loading distribution. The reduction of total pressure loss and secondary flow is also observed, and the aerodynamic performance is well improved compared with the original cascade.

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Gucma, Stanisław, and Paweł Zalewski. "Deterministic-Probabilistic Method of Waterway Design Parameters Determination." Annual of Navigation 19, no. 2 (December 1, 2012): 19–30. http://dx.doi.org/10.2478/v10367-012-0015-7.

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Abstract The paper presents a method of recommended waterway parameters determination, based on modified relationships derived from actual empirical research - the authors’ combination of PIANC and Canadian deterministic methods and stochastic relationships). These relationships are mostly discontinuous functions (e.g. of ship type variable, speed, method of determining the position, etc.), which can be approximated by continuous spline functions. Application built in C# allows determining the recommended minimum width of the waterway, the width increase due to the accuracy of the vessel positioning method, and the minimum radius of waterway bends. The results can be further utilized in the Decision Support System for Water Transport Safety.

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Debela, Habtamu Garoma, and Gemechis File Duressa. "Uniformly Convergent Nonpolynomial Spline Method for Singularly Perturbed Robin-Type Boundary Value Problems with Discontinuous Source Term." Abstract and Applied Analysis 2021 (October 22, 2021): 1–12. http://dx.doi.org/10.1155/2021/7569209.

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In this paper, a singularly perturbed second-order ordinary differential equation with discontinuous source term subject to mixed-type boundary conditions is considered. A fitted nonpolynomial spline method is suggested. The stability and parameter uniform convergence of the proposed method are proved. To validate the applicability of the scheme, two model problems are considered for numerical experimentation and solved for different values of the perturbation parameter, ε , and mesh size, h . The numerical results are tabulated in terms of maximum absolute errors and rate of convergence, and it is observed that the present method is more accurate and ε -uniformly convergent for h ≥ ε where the classical numerical methods fail to give good result and it also improves the results of the methods existing in the literature.

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Liu, Guo-Xiang, Meng-Meng Wang, Xiu-Li Du, Jin-Guan Lin, and Qi-Bing Gao. "Jump-detection and curve estimation methods for discontinuous regression functions based on the piecewise B-spline function." Communications in Statistics - Theory and Methods 47, no. 23 (November 27, 2017): 5729–49. http://dx.doi.org/10.1080/03610926.2017.1400061.

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Dahiya, Sumita, and Ramesh Chandra Mittal. "A modified cubic B-spline differential quadrature method for three-dimensional non-linear diffusion equations." Open Physics 15, no. 1 (July 3, 2017): 453–63. http://dx.doi.org/10.1515/phys-2017-0051.

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AbstractThis paper employs a differential quadrature scheme for solving non-linear partial differential equations. Differential quadrature method (DQM), along with modified cubic B-spline basis, has been adopted to deal with three-dimensional non-linear Brusselator system, enzyme kinetics of Michaelis-Menten type problem and Burgers’ equation. The method has been tested efficiently to three-dimensional equations. Simple algorithm and minimal computational efforts are two of the major achievements of the scheme. Moreover, this methodology produces numerical solutions not only at the knot points but also at every point in the domain under consideration. Stability analysis has been done. The scheme provides convergent approximate solutions and handles different cases and is particularly beneficial to higher dimensional non-linear PDEs with irregularities in initial data or initial-boundary conditions that are discontinuous in nature, because of its capability of damping specious oscillations induced by high frequency components of solutions.

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Xia, Qin Xiang, Zhe Lin Li, Wei Qi Zong, and He Qing Xie. "Contour Extraction for Images of High Temperature Long Shaft Heavy Forgings." Advanced Materials Research 562-564 (August 2012): 1655–59. http://dx.doi.org/10.4028/www.scientific.net/amr.562-564.1655.

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Non-contact measuring method based on CCD camera is desirable for product quality of high temperature long-shaft heavy forgings. In the light of the characteristics of RGB primary color and halation in forging image, the mean red gray value in the high temperature area is proposed as the dynamic threshold to acquire external contours. Internal edges in the image of the hot forging are blurry and discontinuous. For these characteristics, a method based on quadratic B-spline curve is employed to extract and fit the internal contours. Experiments show that this method can effectively remove pseudo features and extract accurate internal and external contours for images of high temperature squaring and chamfering forgings of 900 0C to 1050 0C.

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Katerinina, S. Yu, G. V. Voronkova, and S. S. Rekunov. "Construction of discrete analogs of differential equations for bending of plates and shells with discontinuous parameters based on spline approximation method." IOP Conference Series: Materials Science and Engineering 687 (December 10, 2019): 033037. http://dx.doi.org/10.1088/1757-899x/687/3/033037.

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27

Zhang, Lixiang, Yi'an Zhu, Wei Lu, Jie Wen, and Junyun Cui. "A detection and restoration approach for vessel trajectory anomalies based on AIS." Xibei Gongye Daxue Xuebao/Journal of Northwestern Polytechnical University 39, no. 1 (February 2021): 119–25. http://dx.doi.org/10.1051/jnwpu/20213910119.

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In view of the continuous increase in the amount of AIS data at sea and the existence of more abnormal points, it is difficult to construct ship trajectories based on AIS data. Aiming at this problem, a new method for identifying and repairing abnormal points in trajectories only based the AIS data of the ship itself is proposed. Longitude and latitude, speed, acceleration, direction and other parameters are comprehensively used to identify and repair the abnormal points in the method proposed. Compared with the methods based on single location data, it can effectively reduce the missed judgement of outliers. Compared with the methods based on trajectories clustering to judge singular point, this method does not require the data of historical trajectories to expand the application scope. The cubic spline method is used to interpolate points for the discontinuous segments to further improve the continuity and integrity of the ship trajectory. The results of AIS data processing and analysis on ships in actual sea areas verify the feasibility and effectiveness of the proposed method.

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Kölbl, G., K. Kalcher, and K. J. Irgolic. "Computerized data treatment for an HPLC-GFAAS system for the identification and quantification of trace element compounds." Journal of Automatic Chemistry 15, no. 2 (1993): 37–45. http://dx.doi.org/10.1155/s1463924693000082.

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Liquid chromatographs, coupled with graphite furnace atomic absorption spectrometers, have been widely used for the identification and quantification of trace element compounds. The quantification of the discontinuous signals from the spectrometer defining a chromatographic band is very much a matter of judgement and therefore prone to error. This paper describes a system which links a high-performance liquid chromatograph via a ‘Brinckman’ flowthrough cup to a Hitachi Zeeman graphite furnace atomic absorption spectrometer equipped with an autosampler. The introduction of aliquots from the column effluent and the analysis sequence is computer-controlled through a home-built interface. The signals from the spectrometer are passed through an analoguedigital converter and processed by selectable algorithms. The software offers a variety of options for processing the chromatographic data, such as data smoothing, Gaussian or spline interpolation, and trapezium or Simpson integration.This system was used to separate and determine selenite and selenate in aqueous solution with absolute detection limits (3 σ) of 23 ng Se for selenite and 16 ng Se for selenate. This system can be adapted to other spectrometers, provided that the required connections to the electronics can be made.

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Minbashian, Hadi, Hojatollah Adibi, and Mehdi Dehghan. "An adaptive space-time shock capturing method with high order wavelet bases for the system of shallow water equations." International Journal of Numerical Methods for Heat & Fluid Flow 28, no. 12 (December 3, 2018): 2842–61. http://dx.doi.org/10.1108/hff-08-2017-0320.

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PurposeThis paper aims to propose an adaptive method for the numerical solution of the shallow water equations (SWEs). The authors provide an arbitrary high-order method using high-order spline wavelets. Furthermore, they use a non-linear shock capturing (SC) diffusion which removes the necessity of post-processing.Design/methodology/approachThe authors use a space-time weak formulation of SWEs which exploits continuous Galerkin (cG) in space and discontinuous Galerkin (dG) in time allowing time stepping, also known as cGdG. Such formulations along with SC term have recently been proved to ensure the stability of fully discrete schemes without scarifying the accuracy. However, the resulting scheme is expensive in terms of number of degrees of freedom (DoFs). By using natural adaptivity of wavelet expansions, the authors devise an adaptive algorithm to reduce the number of DoFs.FindingsThe proposed algorithm uses DoFs in a dynamic way to capture the shocks in all time steps while keeping the representation of approximate solution sparse. The performance of the proposed scheme is shown through some numerical examples.Originality/valueAn incorporation of wavelets for adaptivity in space-time weak formulations applied for SWEs is proposed.

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Dong, Linwei, Zhiyong Yang, Zhenyong Wang, Yaowen Ding, and Weiqiang Qi. "Study on Internal Force of Tunnel Segment by Considering the Influence of Joints." Advances in Materials Science and Engineering 2020 (October 30, 2020): 1–13. http://dx.doi.org/10.1155/2020/1020732.

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The mechanical performance of segments is an important aspect of the safety of tunnel structures. Study on the internal force of tunnel segment by considering the influence of joints is beneficial for obtaining a better understanding of the influence of various factors on the internal force of the segments. Based on the mechanical characteristics of shield segment joints, in which the displacements and stiffness are discontinuous, a mechanical model of the segment component under the constraints of elastic support was established. The elastic centre method and the principle of superposition were used to quantify the influence of joint displacements on the internal force of the segment component. Combined with a practical engineering application, the internal force of the segment component with joint rotation and dislocation was analysed. The displacements of the segment joints cause an unloading effect of the corresponding internal force of the joints, leading to internal force redistribution of each segment cross section. According to the spline interpolation results of the load test data of the segment joints, the internal force of the segment component under an external load is solved by the iterative method.

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Wang, X. Q., F. K. Cui, X. J. He, and H. Y. Xu. "Forming Principle and Error Analysis of Ballscrew Manufactured by Cold Rolling." Advanced Materials Research 670 (March 2013): 85–90. http://dx.doi.org/10.4028/www.scientific.net/amr.670.85.

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Cold rolling is an advanced splinterless processing method, which has been applied to process many parts, like spur gears, helical gears, spline shaft and ballscrew, etc. Ballscrew which determines the motion characteristics has been used as key transmission part of ballscrew pair. According to differential geometry and meshing theory, the mathematic model of ballscrew manufactured by cold rolling is built, and the normal section equation of ballscrew is derivated in this paper. The cold forming principle of ballscrew is investigated; the meshing model, the contact line and theoretical errors of the ballscrew are constructed. The objective moving laws between roller and workpiece are revealed by analyzing the meshing model and the contact line, the processing method of cold-rolling is considered as the generating method which can lay the foundation for roller design by analyzing forming principle. By analyzing the profile error and the error of the actual helix of the centrode of ball, the error generation causes are discovered, and the calculation methods of the profile error and the spiral error of ballscrew are proposed. By analyzing the error evolution and taking a certain type of ballscrew for example, the results indicate that the profile error is brought by roller designing, and the spiral error is caused by discontinuous contact.

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Guo, Hewei, Wei Chen, Xinbiao Jiang, Duoyu Jiang, and Xinyi Zhang. "A discontinuous finite element method based on B-spline wavelet on the interval for solving first-order neutron transport equation with discrete ordinate (SN) angular discretisation." Annals of Nuclear Energy 162 (November 2021): 108511. http://dx.doi.org/10.1016/j.anucene.2021.108511.

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EVANS, JOHN A., and THOMAS J. R. HUGHES. "ISOGEOMETRIC DIVERGENCE-CONFORMING B-SPLINES FOR THE DARCY–STOKES–BRINKMAN EQUATIONS." Mathematical Models and Methods in Applied Sciences 23, no. 04 (January 31, 2013): 671–741. http://dx.doi.org/10.1142/s0218202512500583.

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We develop divergence-conforming B-spline discretizations for the numerical solution of the Darcy–Stokes–Brinkman equations. These discretizations are motivated by the recent theory of isogeometric discrete differential forms and may be interpreted as smooth generalizations of Raviart–Thomas elements. The new discretizations are (at least) patchwise C0 and can be directly utilized in the Galerkin solution of Darcy–Stokes–Brinkman flow for single-patch configurations. When applied to incompressible flows, these discretizations produce pointwise divergence-free velocity fields and hence exactly satisfy mass conservation. In the presence of no-slip boundary conditions and multi-patch geometries, the discontinuous Galerkin framework is invoked to enforce tangential continuity without upsetting the conservation or stability properties of the method across patch boundaries. Furthermore, as no-slip boundary conditions are enforced weakly, the method automatically defaults to a compatible discretization of Darcy flow in the limit of vanishing viscosity. The proposed discretizations are extended to general mapped geometries using divergence-preserving transformations. For sufficiently regular single-patch solutions, we prove a priori error estimates which are optimal for the discrete velocity field and suboptimal, by one order, for the discrete pressure field. Our estimates are in addition robust with respect to the parameters of the Darcy–Stokes–Brinkman problem. We present a comprehensive suite of numerical experiments which indicate optimal convergence rates for both the discrete velocity and pressure fields for general configurations, suggesting that our a priori estimates may be conservative. The focus of this paper is strictly on incompressible flows, but our theoretical results naturally extend to flows characterized by mass sources and sinks.

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EVANS, JOHN A., and THOMAS J. R. HUGHES. "ISOGEOMETRIC DIVERGENCE-CONFORMING B-SPLINES FOR THE STEADY NAVIER–STOKES EQUATIONS." Mathematical Models and Methods in Applied Sciences 23, no. 08 (April 22, 2013): 1421–78. http://dx.doi.org/10.1142/s0218202513500139.

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We develop divergence-conforming B-spline discretizations for the numerical solution of the steady Navier–Stokes equations. These discretizations are motivated by the recent theory of isogeometric discrete differential forms and may be interpreted as smooth generalizations of Raviart–Thomas elements. They are (at least) patchwise C0 and can be directly utilized in the Galerkin solution of steady Navier–Stokes flow for single-patch configurations. When applied to incompressible flows, these discretizations produce pointwise divergence-free velocity fields and hence exactly satisfy mass conservation. Consequently, discrete variational formulations employing the new discretization scheme are automatically momentum-conservative and energy-stable. In the presence of no-slip boundary conditions and multi-patch geometries, the discontinuous Galerkin framework is invoked to enforce tangential continuity without upsetting the conservation or stability properties of the method across patch boundaries. Furthermore, as no-slip boundary conditions are enforced weakly, the method automatically defaults to a compatible discretization of Euler flow in the limit of vanishing viscosity. The proposed discretizations are extended to general mapped geometries using divergence-preserving transformations. For sufficiently regular single-patch solutions subject to a smallness condition, we prove a priori error estimates which are optimal for the discrete velocity field and suboptimal, by one order, for the discrete pressure field. We present a comprehensive suite of numerical experiments which indicate optimal convergence rates for both the discrete velocity and pressure fields for general configurations, suggesting that our a priori estimates may be conservative. These numerical experiments also suggest our discretization methodology is robust with respect to Reynolds number and more accurate than classical numerical methods for the steady Navier–Stokes equations.

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Kulkarni, Rekha P., and N. Gnaneshwar. "Spectral refinement using a new projection method." ANZIAM Journal 46, no. 2 (October 2004): 203–24. http://dx.doi.org/10.1017/s1446181100013791.

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AbstractIn this paper we consider two spectral refinement schemes, elementary and double iteration, for the approximation of eigenelements of a compact operator using a new approximating operator. We show that the new method performs better than the Galerkin, projection and Sloan methods. We obtain precise orders of convergence for the approximation of eigenelements of an integral operator with a smooth kernel using either the orthogonal projection onto a spline space or the interpolatory projection at Gauss points onto a discontinuous piecewise polynomial space. We show that in the double iteration scheme the error for the eigenvalue iterates using the new method is of the order of , where h is the mesh of the partition and k = 0, 1, 2,… denotes the step of the iteration. This order of convergence is to be compared with the orders in the Galerkin and projection methods and in the Sloan method. The error in eigenvector iterates is shown to be of the order of in the new method, in the Galerkin and projection methods and in the Sloan method. Similar improvement is observed in the case of the elementary iteration. We show that these orders of convergence are preserved in the corresponding discrete methods obtained by replacing the integration by a numerical quadrature formula. We illustrate this improvement in the order of convergence by numerical examples.

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Zhang, Zhiwei, Rajendra Singh, and Ashley R. Crowther. "Limitations of Smoothening Functions for Automotive Vibro-Impact Problems." Shock and Vibration 18, no. 1-2 (2011): 397–406. http://dx.doi.org/10.1155/2011/102720.

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Nonlinear torsional models are used to analyze automotive transmission rattle problems and find solutions to reduce noise, vibration and dynamic loads. The torsional stiffness and inertial distribution of such systems show that the underlying mathematical problem is numerically stiff. In addition, the clearance nonlinearities in the gear meshes introduce discontinuous functions. Both factors affect the efficacy of time domain integration and smoothening functions are widely used to overcome computational difficulties and improve the simulation. In this paper, alternate smoothening functions are studied for their influence on the numerical solutions and their impact on global convergence and computation times. In particular, four smoothening functions (arctan, hyperbolic-cosine, hyperbolic-tan and quintic-spline) are applied to a five-degree-of-freedom generic torsional system with two backlash (clearance) elements. Each function is assessed via a global convergence metric across an excitation map (a design of experiment). Regions of the excitation map, along with multiple solutions, are studied and the implications to assessing convergence are critically examined. It is observed that smoothening functions do not lead to better convergence in many cases. The smoothening parameter needs to be carefully selected, or over-smoothened solutions may be found. The system studied is representative of a typical automotive rattle problem and it was found that benefits were limited from applying such smoothening functions.

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Lu, Lei, Lei Zhang, Shijun Ji, Dunlan Song, and Ji Zhao. "A segment feedrate profile fitting method for parametric interpolation." Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture 232, no. 2 (April 19, 2016): 267–80. http://dx.doi.org/10.1177/0954405416641325.

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There are many researches in scheduling an optimal feedrate profile under various constraints by numerical calculation. A large number of discrete feedrate data points are obtained. They are inconvenient for the parametric interpolator. Therefore, these discrete feedrate data points need to be fitted by parameter curves. Different from the regular curve fitting, the inappropriate feedrate fitting method can generate larger acceleration and jerk that seriously affect the machining accuracy and stability, although the feedrate satisfies the error requirements. In order to generate a suitable feedrate profile, a segment feedrate profile fitting method using B-spline is proposed in this article. The discrete feedrate data points are segmented in the jerk discontinuous points. In each segment, the feedrate profile is fitted by the linear least squares method. These fitted feedrate profiles are combined to generate a unified feedrate profile. The unified fitted feedrate profile and the tool path trajectory are used in the controller to command the axis. In this article, the process of parametric interpolation is separated into the arc-length calculation process and the curve parameter calculation process. Using parallel computation, the two processes are calculated simultaneously in the controller, and the computational efficiency is improved. Both simulation and experiment are carried out to verify that the fitted feedrate profile satisfies the error requirements, and the novel interpolation can be applied to the controller appropriately.

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Salazar, Jacob, Jaime Mora, and Leszek Demkowicz. "Alternative Enriched Test Spaces in the DPG Method for Singular Perturbation Problems." Computational Methods in Applied Mathematics 19, no. 3 (July 1, 2019): 603–30. http://dx.doi.org/10.1515/cmam-2018-0207.

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AbstractWe propose and investigate the application of alternative enriched test spaces in the discontinuous Petrov–Galerkin (DPG) finite element framework for singular perturbation linear problems, with an emphasis on 2D convection-dominated diffusion. Providing robust {L^{2}} error estimates for the field variables is considered a convenient feature for this class of problems, since this norm would not account for the large gradients present in boundary layers. With this requirement in mind, Demkowicz and others have previously formulated special test norms, which through DPG deliver the desired {L^{2}} convergence. However, robustness has only been verified through numerical experiments for tailored test norms which are problem-specific, whereas the quasi-optimal test norm (not problem specific) has failed such tests due to the difficulty to resolve the optimal test functions sought in the DPG technology. To address this issue (i.e. improve optimal test functions resolution for the quasi-optimal test norm), we propose to discretize the local test spaces with functions that depend on the perturbation parameter ϵ. Explicitly, we work with B-spline spaces defined on an ϵ-dependent Shishkin submesh. Two examples are run using adaptive h-refinement to compare the performance of proposed test spaces with that of standard test spaces. We also include a modified norm and a continuation strategy aiming to improve time performance and briefly experiment with these ideas.

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Li, Feng, Haihong Zhu, Zhenwei Luo, Hang Shen, and Lin Li. "An Adaptive Surface Interpolation Filter Using Cloth Simulation and Relief Amplitude for Airborne Laser Scanning Data." Remote Sensing 13, no. 15 (July 26, 2021): 2938. http://dx.doi.org/10.3390/rs13152938.

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Separating point clouds into ground and nonground points is an essential step in the processing of airborne laser scanning (ALS) data for various applications. Interpolation-based filtering algorithms have been commonly used for filtering ALS point cloud data. However, most conventional interpolation-based algorithms have exhibited a drawback in terms of retaining abrupt terrain characteristics, resulting in poor algorithmic precision in these regions. To overcome this drawback, this paper proposes an improved adaptive surface interpolation filter with a multilevel hierarchy by using a cloth simulation and relief amplitude. This method uses three hierarchy levels of provisional digital elevation model (DEM) raster surfaces with thin plate spline (TPS) interpolation to separate ground points from unclassified points based on adaptive residual thresholds. A cloth simulation algorithm is adopted to generate sufficient effective initial ground seeds for constructing topographic surfaces with high quality. Residual thresholds are adaptively constructed by the relief amplitude of the examined area to capture complex landscape characteristics during the classification process. Fifteen samples from the International Society for Photogrammetry and Remote Sensing (ISPRS) commission are used to assess the performance of the proposed algorithm. The experimental results indicate that the proposed method can produce satisfying results in both flat areas and steep areas. In a comparison with other approaches, this method demonstrates its superior performance in terms of filtering results with the lowest omission error rate; in particular, the proposed approach retains discontinuous terrain features with steep slopes and terraces.

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40

Korotkiy, Viktor. "Cubic Curves in Engineering Geometry." Geometry & Graphics 8, no. 3 (November 24, 2020): 3–24. http://dx.doi.org/10.12737/2308-4898-2020-3-24.

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In this paper are considered historically the first (the 60’s of the 20th century) computational methods for algebraic cubic curves constructing. The analysis of a general cubic curve equation r(t)=a3t3+a2t2+a1t+a0 has been carried out. As an example has been considered the simplest cubic curve r(t)=it3+jt2+kt. Based on the general cubic curve equation have been obtained equations of a cubic curve passing through two predetermined points and having predetermined tangents at these points. The equations have been presented both in Ferguson and Bézier forms. It has been shown that the cubic curve vector equation (for example, the standard equation of a Bezier curve) can be represented in a point form. Have been considered examples for constructing segments of cubic curves meeting the given boundary conditions. The generalized cubic curve equation, containing weight coefficients, has been obtained by the method of exit into four-dimensional space. Has been considered a vector parametric equation of a conical section, passing through two given points and touching predetermined straight lines at these points. The conical section is considered as a special case of a cubic curve. Curvature can be specified as an additional boundary condition. Has been considered the possibility for constructing a cubic curve with fixed positions of contacting planes at end points and given radii of curvature. Has been proposed an algorithm for constructing a plane cubic curve with a given curvature at the end points. Have been considered algorithms for constructing smooth compound Ferguson-Bezier curves. Smoothness conditions are imposed on a compound curve: 1) at any of its points, the curve must have a tangent (no fractures are allowed), 2) the curvature vector must be changed continuously from point to point (no discontinuous jump in the curvature vector is allowed neither in modulus no in direction). Have been proposed examples for constructing compound Ferguson-Bézier curves. Has been performed comparison of polynomial cubic spline with compound parametrically defined curves. Have been given examples for constructing cubic splines with fastened and free ends. The paper is for educational purposes, and intended for in-depth study of computer graphics basics.

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Pershyna, Iuliia. "Approximation of discontinuous functions of three variables by discontinuous interpolation splines." Physico-mathematical modelling and informational technologies, no. 33 (September 4, 2021): 99–104. http://dx.doi.org/10.15407/fmmit2021.33.099.

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In this paper, discontinuous interpolation splines of three variables are constructed and a method for reconstructing of the discontinuous internal structure of a three-dimensional body by constructed splines is proposed. It is believed that a three-dimensional object, which is described by a function of three variables with discontinuities of the first kind on a given grid of nodes, is completely covered by a system of parallelepipeds. The experimental data are the one-sided value of the discontinuous function in a given grid of nodes. In the article, theorems on interpolation properties and the error of the constructed discontinuous structures are formulated and proved. Moreover, the constructed discontinuous interpolation splines include, as a special case, classical continuous splines. The developed approximation method can be applied in three-dimensional mathematical modeling of discontinuous processes, including in computed tomography.

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Nordam, Tor, and Rodrigo Duran. "Numerical integrators for Lagrangian oceanography." Geoscientific Model Development 13, no. 12 (December 1, 2020): 5935–57. http://dx.doi.org/10.5194/gmd-13-5935-2020.

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Abstract. A common task in Lagrangian oceanography is to calculate a large number of drifter trajectories from a velocity field precalculated with an ocean model. Mathematically, this is simply numerical integration of an ordinary differential equation (ODE), for which a wide range of different methods exist. However, the discrete nature of the modelled ocean currents requires interpolation of the velocity field in both space and time, and the choice of interpolation scheme has implications for the accuracy and efficiency of the different numerical ODE methods. We investigate trajectory calculation in modelled ocean currents with 800 m, 4 km, and 20 km horizontal resolution, in combination with linear, cubic and quintic spline interpolation. We use fixed-step Runge–Kutta integrators of orders 1–4, as well as three variable-step Runge–Kutta methods (Bogacki–Shampine 3(2), Dormand–Prince 5(4) and 8(7)). Additionally, we design and test modified special-purpose variants of the three variable-step integrators, which are better able to handle discontinuous derivatives in an interpolated velocity field. Our results show that the optimal choice of ODE integrator depends on the resolution of the ocean model, the degree of interpolation, and the desired accuracy. For cubic interpolation, the commonly used Dormand–Prince 5(4) is rarely the most efficient choice. We find that in many cases, our special-purpose integrators can improve accuracy by many orders of magnitude over their standard counterparts, with no increase in computational effort. Equivalently, the special-purpose integrators can provide the same accuracy as standard methods at a reduced computational cost. The best results are seen for coarser resolutions (4 and 20 km), thus the special-purpose integrators are particularly advantageous for research using regional to global ocean models to compute large numbers of trajectories. Our results are also applicable to trajectory computations on data from atmospheric models.

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Wen, Xue, Jian-ping Tan, Su-qi Liu, and Xin-he Li. "Research on monitoring method for thinning spinning process of the ultra-thin-walled cylinder based on the drum shape." Science Progress 103, no. 1 (October 10, 2019): 003685041987772. http://dx.doi.org/10.1177/0036850419877727.

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In order to perceive the state of the process of reverse thinning spinning of the ultra-thin wall tube, a monitoring method based on drum shape is proposed. The method uses the non-uniform rational basis spline curve fitting method based on the moving least squares method to reconstruct the outer contour data of the spinning zone collected by the line laser displacement sensor and extracts the outer contour curve of the drum zone according to the spinning characteristics. Then, the dynamic time warping algorithm is used to analyze the similarity between the current and previous drum curves in order to judge whether the current spinning state is abrupt or not. At the same time, the current spinning state is judged by combining the curvature comb of the drum curve, the number of convex areas of the drum curve, the drum ratio, and the drum change trend. Finally, the drum shape detection and the spinning condition monitoring experiments are carried out. The experimental results show that this method can obtain the outer contour shape of the drum section at one time, and the accuracy is 0.05 mm with respect to the measured value of the three coordinates. The method can realize the visual monitoring of the abrupt change of the spinning process by judging the similarity of the drum curve at the adjacent time under different spinning states. In the state of instability, the curvature comb of the drum curve is discontinuous, the curve is G0 continuous, the drum ratio exceeds the critical drum ratio 2.0, and there is more than one convex region, which realizes the monitoring of spinning state more accurately. The experimental results verify the correctness of the proposed method and prove that the method can be used as a new method to judge the stability of the ultra-thin-walled cylinder spinning process.

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44

Mezhuyev, V., O. M. Lytvyn, I. Pershyna, and O. Nechuiviter. "APPROXIMATION OF DISCONTINUOUS FUNCTIONS OF TWO VARIABLES BY DISCONTINUOUS INTERLINATION SPLINES USING TRIANGULAR ELEMENTS." Journal of the Serbian Society for Computational Mechanics 14, no. 1 (June 30, 2020): 75–89. http://dx.doi.org/10.24874/jsscm.2020.14.01.07.

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The paper develops a method for approximation of the discontinuous functions of two variables by discontinuous interlination splines using arbitrary triangular elements. Experimental data are one-sided traces of a function given along a system of lines (such data are commonly used in remote methods, in particular in tomography). The paper also proposes a method for approximating the discontinuous functions of two variables taking into account triangular elements having one curved side. The proposed methods improve approximation of the discontinuous functions, allowing an application to complex domains of definition and avoiding the Gibbs phenomenon.

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45

Laurent, P. J. "Inf-convolution spline pour l'approximation de données discontinues." ESAIM: Mathematical Modelling and Numerical Analysis 20, no. 1 (1986): 89–111. http://dx.doi.org/10.1051/m2an/1986200100891.

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Hou, Wenbin, Kai Jiang, Xuefeng Zhu, Yuanxing Shen, and Ping Hu. "Extended isogeometric analysis using B++ splines for strong discontinuous problems." Computer Methods in Applied Mechanics and Engineering 381 (August 2021): 113779. http://dx.doi.org/10.1016/j.cma.2021.113779.

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Zlotnik, A., and O. Kireeva. "Error Bounds for Finite Element Methods with Generalized Cubic Splines for a 4-th Order Ordinary Differential Equation with Nonsmooth Data." Computational Methods in Applied Mathematics 9, no. 2 (2009): 203–18. http://dx.doi.org/10.2478/cmam-2009-0012.

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Abstract A boundary value problem for a 4-th order self-adjoint ordinary differential equation is considered in the case where the coefficients of the equation and its right-hand side can be nonsmooth (discontinuous, concentrated or rapidly oscillating functions). Generalized cubic splines of deficiency 1 depending on the major coefficient of the equation are applied. An error analysis of finite element methods exploiting such splines is presented in detail including superconvergence error bounds. .

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48

Miller, S. I., and D. F. Wirth. "trans splicing in Leishmania enriettii and identification of ribonucleoprotein complexes containing the spliced leader and U2 equivalent RNAs." Molecular and Cellular Biology 8, no. 6 (June 1988): 2597–603. http://dx.doi.org/10.1128/mcb.8.6.2597.

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The 5' ends of Leishmania mRNAs contain an identical 35-nucleotide sequence termed the spliced leader (SL) or 5' mini-exon. The SL sequence is at the 5' end of an 85-nucleotide primary transcript that contains a consensus eucaryotic 5' intron-exon splice junction immediately 3' to the SL. The SL is added to protein-coding genes immediately 3' to a consensus eucaryotic 3' intron-exon splice junction. Our previous work demonstrated possible intermediates in discontinuous mRNA processing that contain the 50 nucleotides of the SL primary transcript 3' to the SL, the SL intron sequence (SLIS). These RNAs have a 5' terminus at the splice junction of the SL and the SLIS. We examined a Leishmania nuclear extract for these RNAs in ribonucleoprotein (RNP) particles. Density centrifugation analysis showed that the SL RNA is predominantly in RNP complexes at 60S, while the SLIS-containing RNAs are in complexes at 40S. We also demonstrated that the SLIS can be released from polyadenylated RNA by incubation with a HeLa cell extract containing debranching enzymatic activity. These data suggested that Leishmania enriettii mRNAs are assembled by bimolecular or trans splicing as has been recently demonstrated for Trypanosoma brucei. Furthermore, we determined the partial sequence of the Leishmania U2 equivalent RNA and demonstrated that it cosediments with the SL RNA at 60S in a nuclear extract. These RNP particles may be analogous to so-called spliceosomes that have been demonstrated in other systems.

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Miller, S. I., and D. F. Wirth. "trans splicing in Leishmania enriettii and identification of ribonucleoprotein complexes containing the spliced leader and U2 equivalent RNAs." Molecular and Cellular Biology 8, no. 6 (June 1988): 2597–603. http://dx.doi.org/10.1128/mcb.8.6.2597-2603.1988.

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The 5' ends of Leishmania mRNAs contain an identical 35-nucleotide sequence termed the spliced leader (SL) or 5' mini-exon. The SL sequence is at the 5' end of an 85-nucleotide primary transcript that contains a consensus eucaryotic 5' intron-exon splice junction immediately 3' to the SL. The SL is added to protein-coding genes immediately 3' to a consensus eucaryotic 3' intron-exon splice junction. Our previous work demonstrated possible intermediates in discontinuous mRNA processing that contain the 50 nucleotides of the SL primary transcript 3' to the SL, the SL intron sequence (SLIS). These RNAs have a 5' terminus at the splice junction of the SL and the SLIS. We examined a Leishmania nuclear extract for these RNAs in ribonucleoprotein (RNP) particles. Density centrifugation analysis showed that the SL RNA is predominantly in RNP complexes at 60S, while the SLIS-containing RNAs are in complexes at 40S. We also demonstrated that the SLIS can be released from polyadenylated RNA by incubation with a HeLa cell extract containing debranching enzymatic activity. These data suggested that Leishmania enriettii mRNAs are assembled by bimolecular or trans splicing as has been recently demonstrated for Trypanosoma brucei. Furthermore, we determined the partial sequence of the Leishmania U2 equivalent RNA and demonstrated that it cosediments with the SL RNA at 60S in a nuclear extract. These RNP particles may be analogous to so-called spliceosomes that have been demonstrated in other systems.

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Jamal, Yasir S., Enas I. Raml, Mazen O. Kurdi, Ettedal A. Aljahdali, and Abdullah Y. Jamal. "Discontinuous-type splenogonadal fusion in abdominoscrotal hydrocele." Journal of King Abdulaziz University - Medical Sciences 25, no. 1 (April 1, 2018): 45–49. http://dx.doi.org/10.4197/med.25-1.7.

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Splenogonadal fusion is a rare congenital malformation, which is an abnormal connection between the spleen and gonads or mesonephric remnants. It usually presents with left cryptorchidism, scrotal mass or left inguinal hernia. Here we present the first case report of splenogonadal fusion as large left scrotal mass leading to abdominoscrotal hydrocele in a 2-year-old boy.

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Journal articles on the topic 'Discontinuous spline'

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