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Journal articles on the topic 'Spline finite element'

Author: Grafiati

Published: 3 June 2025

Last updated: 19 July 2025

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1

Schneider, Teseo, Jérémie Dumas, Xifeng Gao, Mario Botsch, Daniele Panozzo, and Denis Zorin. "Poly-Spline Finite-Element Method." ACM Transactions on Graphics 38, no. 3 (2019): 1–16. http://dx.doi.org/10.1145/3313797.

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2

Kahn-Jetter, Zella L., Eugene Hundertmark, and Suzanne Wright. "Comparison of Torque Transmitting Shaft Connectivity Using a Trilobe Polygon Connection and an Involute Spline." Journal of Mechanical Design 122, no. 1 (2000): 130–35. http://dx.doi.org/10.1115/1.533556.

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The results of a finite element analysis of a trilobe polygon shaft connection used as an alternative for a spline for torque transmission is presented. These results are compared to the results of a finite element analysis previously performed on an involute spline. It is shown that the tensile stress in the polygon shaft is significantly smaller than in the involute spline and is smaller than all the other stresses in both the shaft and the hub in the polygon connection. Furthermore, the magnitudes and distributions of the maximum principal compressive stress, the shear stress, and the Von Mises stress are nearly the same on the shaft and the hub. It appears that polygonal connections can be more advantageous than splined connections because of lower stresses and the lack of stress concentrations typical of splines. [S1050-0472(00)00601-2]

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3

Fang, Lishan, and Linda Stals. "Adaptive discrete thin plate spline smoother." ANZIAM Journal 62 (November 5, 2021): C45—C57. http://dx.doi.org/10.21914/anziamj.v62.15979.

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The discrete thin plate spline smoother fits smooth surfaces to large data sets efficiently. It combines the favourable properties of the finite element surface fitting and thin plate splines. The efficiency of its finite element grid is improved by adaptive refinement, which adapts the precision of the solution. It reduces computational costs by refining only in sensitive regions, which are identified using error indicators. While many error indicators have been developed for the finite element method, they may not work for the discrete smoother. In this article we show three error indicators adapted from the finite element method for the discrete smoother. A numerical experiment is provided to evaluate their performance in producing efficient finite element grids. References F. L. Bookstein. Principal warps: Thin-plate splines and the decomposition of deformations. IEEE Trans. Pat. Anal. Mach. Int. 11.6 (1989), pp. 567–585. doi: 10.1109/34.24792. C. Chen and Y. Li. A robust method of thin plate spline and its application to DEM construction. Comput. Geosci. 48 (2012), pp. 9–16. doi: 10.1016/j.cageo.2012.05.018. L. Fang. Error estimation and adaptive refinement of finite element thin plate spline. PhD thesis. The Australian National University. http://hdl.handle.net/1885/237742. L. Fang. Error indicators and adaptive refinement of the discrete thin plate spline smoother. ANZIAM J. 60 (2018), pp. 33–51. doi: 10.21914/anziamj.v60i0.14061. M. F. Hutchinson. A stochastic estimator of the trace of the influence matrix for laplacian smoothing splines. Commun. Stat. Simul. Comput. 19.2 (1990), pp. 433–450. doi: 10.1080/0361091900881286. W. F. Mitchell. A comparison of adaptive refinement techniques for elliptic problems. ACM Trans. Math. Soft. 15.4 (1989), pp. 326–347. doi: 10.1145/76909.76912. R. F. Reiniger and C. K. Ross. A method of interpolation with application to oceanographic data. Deep Sea Res. Oceanographic Abs. 15.2 (1968), pp. 185–193. doi: 10.1016/0011-7471(68)90040-5. S. Roberts, M. Hegland, and I. Altas. Approximation of a thin plate spline smoother using continuous piecewise polynomial functions. SIAM J. Numer. Anal. 41.1 (2003), pp. 208–234. doi: 10.1137/S0036142901383296. D. Ruprecht and H. Muller. Image warping with scattered data interpolation. IEEE Comput. Graphics Appl. 15.2 (1995), pp. 37–43. doi: 10.1109/38.365004. E. G. Sewell. Analysis of a finite element method. Springer, 2012. doi: 10.1007/978-1-4684-6331-6. L. Stals. Efficient solution techniques for a finite element thin plate spline formulation. J. Sci. Comput. 63.2 (2015), pp. 374–409. doi: 10.1007/s10915-014-9898-x. O. C. Zienkiewicz and J. Z. Zhu. A simple error estimator and adaptive procedure for practical engineerng analysis. Int. J. Numer. Meth. Eng. 24.2 (1987), pp. 337–357. doi: 10.1002/nme.1620240206.

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4

Peng, Hui Fen, Guang Wei Meng, Li Ming Zhou, and Zhao Long Yang. "Modal Analysis of Cracked Plate Using Interval B-Spline Wavelet Finite Element Method." Advanced Materials Research 199-200 (February 2011): 1287–91. http://dx.doi.org/10.4028/www.scientific.net/amr.199-200.1287.

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Aiming at the defects in describing stress field near the crack tip with traditional finite element method (TFEM), a new finite element method based on interval B-Spline wavelet (IBSW) is put forward, the displacement interpolation functions of plate element are constructed by using the scaling functions of IBSW, finite element model of cracked plate based on IBSW is established, and the stiffness matrixes of plate element is derived. The first four natural frequencies and mode shapes of the cracked plate are obtained by using interval B-Spline wavelet finite element (IBSWFE). Comparison of the calculated results with those by ANSYS shows that IBSWFE method can get higher calculation precision with less elements in dealing with engineering singularity problems.

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5

Sun, Jianliang, Mengqian Sun, Yunjing Jiao, and Yanan Gao. "Study on Plate Straightening Process Based on Elastic-Plastic B Spline Finite Strip Method." Journal of Mechanics 36, no. 6 (2020): 737–47. http://dx.doi.org/10.1017/jmech.2020.16.

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ABSTRACTAn elastic-plastic B spline finite strip method is proposed to investigate the continuous plate straightening process in this paper. First, the B spline displacement function that satisfies the boundary conditions of simply supported end and free end of the strip element is established, and then the stress-strain matrix is established. Second, the set method of total stiffness matrix based on B spline finite strip method for plate straightening problem is proposed, and the influence of nodal line number and strip elements number on the sparsity of total stiffness matrix is analyzed. Third, the loads on the strip elements are taken as linear uniform distribution, and the transformation matrix between the equivalent linear load and the actual load of the strip element is established. At last, the plate straightening simulation of 11 rolls straightening machine is made based on the elastic-plastic B spline finite strip method, the calculated results agree with the measured results, which approves that the elastic-plastic B spline finite strip method established can be applied to the plate straightening process.

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6

Fan, S. C., and M. H. Luah. "New Spline Finite Element for Plate Bending." Journal of Engineering Mechanics 118, no. 6 (1992): 1065–82. http://dx.doi.org/10.1061/(asce)0733-9399(1992)118:6(1065).

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7

Ding, Hanshan, Rongguang Shao, and Dajun Ding. "A spline finite element method on mapping." Structural Engineering and Mechanics 4, no. 4 (1996): 415–24. http://dx.doi.org/10.12989/sem.1996.4.4.415.

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8

Kahn-Jetter and, Zella L., and Suzanne Wright. "Finite Element Analysis of an Involute Spline." Journal of Mechanical Design 122, no. 2 (2000): 239–44. http://dx.doi.org/10.1115/1.533573.

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Two finite element analyses of an involute spline are performed; one is axisymmetrically loaded and the other is nonaxisymmetrically loaded. An entire cross section of both an internal and external pair is analyzed for both models. It is shown that on the axisymmetrically loaded spline the highest stress experienced is the maximum compressive contact stress although the tensile stress in the shaft is also quite high. It is also shown that stress concentrations exist at the root and top of the tooth for both models. Furthermore, on the nonaxisymmetrically loaded spline at low torque, only a few teeth make initial contact, however, as torque is increased, more teeth come in contact. All the stresses remain relatively constant under increasing torque as more teeth are engaged. Once all teeth are in contact stress increases with higher torques. However, the maximum tensile stress (arising from stress concentrations) remains fairly constant, even at high torques, because the stress concentrations that occur at tooth roots appear to be relatively independent of the number of teeth in contact. [S1050-0472(00)00102-1]

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9

Rajashekar, Naraveni, Sudhakar Chaudhary, and V. V. K. Srinivas Kumar. "Approximation of p-Biharmonic Problem using WEB-Spline based Mesh-Free Method." International Journal of Nonlinear Sciences and Numerical Simulation 20, no. 6 (2019): 703–12. http://dx.doi.org/10.1515/ijnsns-2018-0298.

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Abstract We describe and analyze the weighted extended b-spline (WEB-Spline) mesh-free finite element method for solving the p-biharmonic problem. The WEB-Spline method uses weighted extended b-splines as basis functions on regular grids and does not require any mesh generation which eliminates a difficult, time consuming preprocessing step. Accurate approximations are possible with relatively low-dimensional subspaces. We perform some numerical experiments to demonstrate the efficiency of the WEB-Spline method.

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10

Lamichhane, Bishnu P., Elizabeth Harris, and Quoc Thong Le Gia. "Approximation of noisy data using multivariate splines and finite element methods." Journal of Algorithms & Computational Technology 15 (January 2021): 174830262110084. http://dx.doi.org/10.1177/17483026211008405.

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We compare a recently proposed multivariate spline based on mixed partial derivatives with two other standard splines for the scattered data smoothing problem. The splines are defined as the minimiser of a penalised least squares functional. The penalties are based on partial differential operators, and are integrated using the finite element method. We compare three methods to two problems: to remove the mixture of Gaussian and impulsive noise from an image, and to recover a continuous function from a set of noisy observations.

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11

Bettayeb, Abdellatif. "Examples of linear multi-box splines." LMS Journal of Computation and Mathematics 15 (December 1, 2012): 444–62. http://dx.doi.org/10.1112/s1461157012001167.

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AbstractLet S1=S1(v0,…,vr+1) be the space of compactly supported C0 piecewise linear functions on a mesh M of lines through ℤ2 in directions v0,…,vr+1, possibly satisfying some restrictions on the jumps of the first order derivative. A sequence ϕ=(ϕ1,…,ϕr) of elements of S1 is called a multi-box spline if every element of S1 is a finite linear combination of shifts of (the components of) ϕ. We give some examples for multi-box splines and show that they are stable. It is further shown that any multi-box spline is not always symmetric

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12

Ye, J. Q. "Postbuckling Analysis of Plates Under Combined Loads by a Mixed Finite Element and Boundary Element Method." Journal of Pressure Vessel Technology 115, no. 3 (1993): 262–67. http://dx.doi.org/10.1115/1.2929526.

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The postbuckling behavior of thin plates under combined loads is studied in this paper by using a mixed boundary element and finite element method. The transverse and the in-plane deformation of the plates are analyzed by the boundary element method and the finite element method, respectively. Spline functions were used as the interpolation functions and shape functions in the solution of both methods. A quadratic rectangular spline element is adopted in the finite element procedure. Numerical results are given for typical problems to show the effectiveness of the proposed approach. The possibilities to extend the method developed in this paper to more complicated postbuckling problems are discussed in the concluding section.

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13

Leen, S. B., I. R. McColl, C. H. H. Ratsimba, and E. J. Williams. "Fatigue life prediction for a barrelled spline coupling under torque overload." Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering 217, no. 3 (2003): 123–42. http://dx.doi.org/10.1243/095441003322297234.

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Aeroengine spline couplings experience a wide range of loading conditions leading to contrasting service life limiting phenomena, including fatigue, fretting fatigue and fretting wear. Highly loaded couplings may employ incomplete contact axial profiles, while the contact geometry transverse to the spline axis is nominally complete with theoretical stress singularities at the contact edges. Life assessment of such components is consequently complex. The effect of torque overload conditions on the fatigue life of a barrelled, aeroengine type spline coupling is investigated experimentally. The experimental results are interpreted using three-dimensional finite element analyses, incorporating frictional contact and elastic-plastic material behaviour and the results of simple tension-tension fatigue tests. Torque-life and finite element predicted stress-life relationships are generated for spline life prediction purposes. Good correlation is obtained between the spline coupling and simple tension-tension fatigue test results, interpreted via the finite element predicted stress ranges.

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14

Sana, Madiha, and Muhammad Mustahsan. "Finite Element Approximation of Optimal Control Problem with Weighted Extended B-Splines." Mathematics 7, no. 5 (2019): 452. http://dx.doi.org/10.3390/math7050452.

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In this research article, an optimal control problem (OCP) with boundary observations is approximated using finite element method (FEM) with weighted extended B-splines (WEB-splines) as basis functions. This type of OCP has a distinct aspect that the boundary observations are outward normal derivatives of state variables, which decrease the regularity of solution. A meshless FEM is proposed using WEB-splines, defined on the usual grid over the domain, R 2 . The weighted extended B-spline method (WEB method) absorbs the regularity problem as the degree of the B-splines is increased. Convergence analysis is also performed by some numerical examples.

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15

Ronglin, Li, Ni Guangzheng, and Yu Jihui. "B-spline finite-element method in polar coordinates." Finite Elements in Analysis and Design 28, no. 4 (1998): 337–46. http://dx.doi.org/10.1016/s0168-874x(97)00044-9.

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16

Li, Chong-Jun, and Ren-Hong Wang. "A new 8-node quadrilateral spline finite element." Journal of Computational and Applied Mathematics 195, no. 1-2 (2006): 54–65. http://dx.doi.org/10.1016/j.cam.2005.07.017.

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17

Sanches, R. A. K., P. B. Bornemann, and F. Cirak. "Immersed b-spline (i-spline) finite element method for geometrically complex domains." Computer Methods in Applied Mechanics and Engineering 200, no. 13-16 (2011): 1432–45. http://dx.doi.org/10.1016/j.cma.2010.12.008.

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18

Mozgaleva, Marina, Pavel Akimov, and Taymuraz Kaytukov. "LOCALIZATION OF SOLUTION OF THE PROBLEM OF TWO-DIMENSIONAL THEORY OF ELASTICITY WITH THE USE OF B-SPLINE DISCRETE-CONTINUAL FINITE ELEMENT METHOD." International Journal for Computational Civil and Structural Engineering 17, no. 2 (2021): 83–104. http://dx.doi.org/10.22337/2587-9618-2021-17-2-83-104.

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Localization of solution of the problem of two-dimensional theory of elasticity with the use of B-spline discrete-continual finite element method (specific version of wavelet-based discrete-continual finite element method) is under consideration in the distinctive paper. The original operational continual and discrete-continual formulations of the problem are given, some actual aspects of construction of normalized basis functions of a B-spline are considered, the corresponding local constructions for an arbitrary discrete-continual finite element are described, some information about the numerical implementation and an example of analysis are presented.

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19

Akimov, Pavel, Marina Mozgaleva, and Taymuraz Kaytukov. "NUMERICAL SOLUTION OF THE PROBLEM OF ISOTROPIC PLATE ANALYSIS WITH THE USE OF B-SPLINE DISCRETE-CONTINUAL FINITE ELEMENT METHOD." International Journal for Computational Civil and Structural Engineering 16, no. 4 (2020): 14–28. https://doi.org/10.22337/2587-9618-2020-16-4-14-28.

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Numerical solution of the problem of isotropic plate analysis with the use of B-spline discrete-continual finite element method (specific version of wavelet-based discrete-continual finite element method) is under consideration in the distinctive paper. The original operational continual and discrete-continual formulations of the problem are given, some actual aspects of construction of normalized basis functions of a B-spline are considered, the corresponding local constructions for an arbitrary discrete-continual finite element are described, some information about the numerical implementation and an example of analysis are presented.

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20

Sari, Murat, and Huseyin Tunc. "Finite element based hybrid techniques for advection-diffusion-reaction processes." An International Journal of Optimization and Control: Theories & Applications (IJOCTA) 8, no. 2 (2018): 127–36. http://dx.doi.org/10.11121/ijocta.01.2018.00452.

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In this paper, numerical solutions of the advection-diffusion-reaction (ADR) equation are investigated using the Galerkin, collocation and Taylor-Galerkin cubic B-spline finite element method in strong form of spatial elements using an ?-family optimization approach for time variation. The main objective of this article is to capture effective results of the finite element techniques with B-spline basis functions under the consideration of the ADR processes. All produced results are compared with the exact solution and the literature for various versions of problems including pure advection, pure diffusion, advection-diffusion, and advection-diffusion-reaction equations. It is proved that the present methods have good agreement with the exact solution and the literature.

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21

Xue, Xiaofeng, Xinhai Wang, Zhen Wang, and Wei Xue. "Wave Motion Analysis in Plane via Hermitian Cubic Spline Wavelet Finite Element Method." Shock and Vibration 2020 (August 26, 2020): 1–21. http://dx.doi.org/10.1155/2020/8752656.

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A plane Hermitian wavelet finite element method is presented in this paper. Wave motion can be used to analyze plane structures with small defects such as cracks and obtain results. By using the tensor product of modified Hermitian wavelet shape functions, the plane Hermitian wavelet shape functions are constructed. Scale functions of Hermitian wavelet shape functions can replace the polynomial shape functions to construct new wavelet plane elements. As the scale of the shape functions increases, the precision of the new wavelet plane element will be improved. The new Hermitian wavelet finite element method which can be used to simulate wave motion analysis can reveal the law of the wave motion in plane. By using the results of transmitted and reflected wave motion, the cracks can be easily identified in plane. The results show that the new Hermitian plane wavelet finite element method can use the fewer elements to simulate the plane structure effectively and accurately and detect the cracks in plane.

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22

Xue, Xiaofeng, Xuefeng Chen, Xingwu Zhang, Baijie Qiao, and Jia Geng. "Hermitian Mindlin Plate Wavelet Finite Element Method for Load Identification." Shock and Vibration 2016 (2016): 1–24. http://dx.doi.org/10.1155/2016/8618202.

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A new Hermitian Mindlin plate wavelet element is proposed. The two-dimensional Hermitian cubic spline interpolation wavelet is substituted into finite element functions to construct frequency response function (FRF). It uses a system’s FRF and response spectrums to calculate load spectrums and then derives loads in the time domain via the inverse fast Fourier transform. By simulating different excitation cases, Hermitian cubic spline wavelets on the interval (HCSWI) finite elements are used to reverse load identification in the Mindlin plate. The singular value decomposition (SVD) method is adopted to solve the ill-posed inverse problem. Compared with ANSYS results, HCSWI Mindlin plate element can accurately identify the applied load. Numerical results show that the algorithm of HCSWI Mindlin plate element is effective. The accuracy of HCSWI can be verified by comparing the FRF of HCSWI and ANSYS elements with the experiment data. The experiment proves that the load identification of HCSWI Mindlin plate is effective and precise by using the FRF and response spectrums to calculate the loads.

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23

Xue, Xiangzhen, Yifan Li, Kuan Lin, Liqi Sui, Yiqiang Jiang, and Ning Zhang. "A Study on the Influence of Nonlinear Vibration on Fretting Damage of Involute Spline Pairs in Aero-Engines." Lubricants 11, no. 12 (2023): 515. http://dx.doi.org/10.3390/lubricants11120515.

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To meticulously examine the repercussions of nonlinear vibrations on fretting damage within aero-engine involute spline pairs, a dynamic model was constructed rooted in well-established theories and methodologies. MATLAB was engaged to resolve the model, where the vibration displacement function was treated under Fourier transformation. The emergent sub-model was then integrated into finite element analysis software to scrutinize the distribution curves of fretting damage over the external spline tooth surface. The analysis included a comprehensive comparison of the axial and radial distributions, in addition to scenarios with and without vibration interferences. Further, an empirical platform was devised to authenticate the outcomes harvested through finite element simulation. The results indicate that the principal mode of fretting damage failure in aero-engine involute spline pairs fundamentally comprises fretting wear. This wear occurs throughout the rotational period of the fretting cycle and reciprocally interacts with fretting fatigue phenomena. Significantly, it was ascertained that acute nonlinear vibrations escalate the magnitude of fretting damage and the quantity of worn teeth within aero-engine spline pairs. Beyond that, angular misalignment was recognized as an aggravating factor that compounds fretting damage in the secondary bond teeth of involute spline pairs. These newfound insights are of paramount significance for the strategic design of involute splines to combat wear.

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24

Zhang, Meng, Li Yong Zhang, Yin He Bao, Fei Gao, and Cong Wang. "Analysis on Deformation and Relieving of Spline in 1.5MW Wind Turbine Gearbox." Applied Mechanics and Materials 86 (August 2011): 908–13. http://dx.doi.org/10.4028/www.scientific.net/amm.86.908.

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For the sun gear shaft in wind turbine gearbox, the deformation will occur on the inner spline because of the thermal inserting of ring gear. So, the spline pair should be tooth relieved to avoid decrease of backlash and conflict of copulate parts. Based on the finite element analysis results, the deformation caused by the thermal inserting on the inner spline was calculated, and the theoretical tooth relieving curve was got. The practical thermal inserting deformation on the spline was measured. And the results indicate that the measuring deformations are in good consistency with the deformation calculated by finite element analysis method. The spline deformation and tooth relieving calculation method can provide a reference for the tooth relieving of the spline on long structure sun gear shaft in 1.5MW wind turbine gearbox.

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25

Iqbal, Azhar, Nur Nadiah Abd Hamid, and Ahmad Izani Md. Ismail. "Numerical Solution of Nonlinear Schrödinger Equation with Neumann Boundary Conditions Using Quintic B-Spline Galerkin Method." Symmetry 11, no. 4 (2019): 469. http://dx.doi.org/10.3390/sym11040469.

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This paper is concerned with the numerical solution of the nonlinear Schrödinger (NLS) equation with Neumann boundary conditions by quintic B-spline Galerkin finite element method as the shape and weight functions over the finite domain. The Galerkin B-spline method is more efficient and simpler than the general Galerkin finite element method. For the Galerkin B-spline method, the Crank Nicolson and finite difference schemes are applied for nodal parameters and for time integration. Two numerical problems are discussed to demonstrate the accuracy and feasibility of the proposed method. The error norms L 2 , L ∞ and conservation laws I 1 , I 2 are calculated to check the accuracy and feasibility of the method. The results of the scheme are compared with previously obtained approximate solutions and are found to be in good agreement.

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26

Tian, Zhuo, and Bai Cheng Li. "A Finite Element Analysis of the Rectangle Spline Broach." Applied Mechanics and Materials 687-691 (November 2014): 163–66. http://dx.doi.org/10.4028/www.scientific.net/amm.687-691.163.

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In the traditional design,the impact tools of broach maybe unable to achieve the requirements for the processing precision,or even occur the situation of partial fracture due to a exaggerated partial deformation.In this article,by using Pro/E we complete the 3D solid modeling and the dimensional parameterization of the impact tools for rectangle spline broach,then import the 3D model into ANSYS,to analyze and solve the whole process of load and deformation at work.It can effectively improve the machining accuracy and the reliability of broach, shorten the design cycle and reduce cost.

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27

Sundararajan, S., and S. Amin. "Finite-element analysis of ring gear/casing spline contact." Journal of Propulsion and Power 7, no. 4 (1991): 602–6. http://dx.doi.org/10.2514/3.23368.

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28

Carminelli, A., and G. Catania. "Spline finite element updating of a reinforced concrete beam." Journal of Physics: Conference Series 305 (July 19, 2011): 012080. http://dx.doi.org/10.1088/1742-6596/305/1/012080.

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29

Han, Jian-Gang, Wei-Xin Ren, and Yih Huang. "A spline wavelet finite-element method in structural mechanics." International Journal for Numerical Methods in Engineering 66, no. 1 (2006): 166–90. http://dx.doi.org/10.1002/nme.1551.

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30

Sum, W. S., S. B. Leen, E. J. Williams, R. Sabesan, and I. R. McColl. "Efficient Finite Element Modelling For Complex Shaft Couplings Under Non-Symmetric Loading." Journal of Strain Analysis for Engineering Design 40, no. 7 (2005): 655–73. http://dx.doi.org/10.1243/030932405x30858.

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A number of key aspects of the three-dimensional finite element (FE) modelling of spline couplings for fretting and fatigue assessment are discussed. The primary issue addressed is the development of an efficient and accurate modelling technique for non-symmetric shaft loading of the couplings, which is an important mode of loading for fretting fatigue assessment. An improved method is presented for implementing an axial modification of the contact geometry, commonly referred to as barrelling, which is also important for fretting fatigue assessment of splines.

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31

Mozgaleva, Marina, and Pavel Akimov. "LOCALIZATION OF SOLUTION OF THE PROBLEM OF THREE-DIMENSIONAL THEORY OF ELASTICITY WITH THE USE OF B-SPLINE DISCRETE-CONTINUAL FINITE ELEMENT METHOD." International Journal for Computational Civil and Structural Engineering 19, no. 3 (2023): 155–64. http://dx.doi.org/10.22337/2587-9618-2023-19-3-155-164.

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Localization of solution of the problem of three-dimensional theory of elasticity with the use of B-spline discrete-continual finite element method (specific version of wavelet-based discrete-continual finite ele-ment method) is under consideration in the distinctive paper. The original operational continual and discrete-continual formulations of the problem are given, some actual aspects of construction of normalized basis func-tions of a B-spline are considered, the corresponding local constructions for an arbitrary discrete-continual finite element are described, some information about the numerical implementation and an example of analysis are presented.

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32

Cui, Minchao, Shengdun Zhao, Chao Chen, Dawei Zhang, and Yongyi Li. "Finite element modeling and analysis for the integration–rolling–extrusion process of spline shaft." Advances in Mechanical Engineering 9, no. 2 (2017): 168781401668858. http://dx.doi.org/10.1177/1687814016688585.

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An integration–rolling–extrusion process is raised for the manufacture of spline shaft in this study. First, the principle and procedures of integration–rolling–extrusion process are described. Next, the finite element model with a simplified sector blank is established to obtain a practical method for the simulation of integration-rolling-extrusion process. Through the simulation results, the plastic forming mechanisms are clearly revealed. During the integration–rolling–extrusion process, the equivalent stress, deformation degree, and material flow behavior mainly distribute on the surface layer of the blank and then gradually decrease along the radial inward direction. In the core region of the blank, there are almost no effective stress distribution, deformation degree, and material flow behavior. Next, the experiments are carried out on a specialized forming equipment to verify the finite element model. The results are measured and compared with finite element results. The finite element results show a good agreement with experiments; thus, the finite element analysis on the integration–rolling–extrusion process is credible. In addition, the measurement results show that the dimensions meet the requirement of heavy truck application. It indicates that the integration–rolling–extrusion process is feasible for the manufacture of spline shaft. However, the surface quality of the formed spline shaft is not satisfying, which needs to be discussed further.

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33

Wang, Aizeng, Ling Li, Wei Wang, et al. "Linear Independence of T-Spline Blending Functions of Degree One for Isogeometric Analysis." Mathematics 9, no. 12 (2021): 1346. http://dx.doi.org/10.3390/math9121346.

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Linear independence of the blending functions is a necessary requirement for T-spline in isogeometric analysis. The main work in this paper focuses on the analysis about T-splines of degree one, we demonstrate that all the blending functions of such T-spline of degree one are linearly independent. The advantage owned by one degree T-spline is that it can avoid the problem of judging whether the model is analysis-suitable or not, especially for occasions that need a quick response from the analysis results. This may provide a new way of using T-spline for a CAD and CAE integrating scenario, since one degree T-spline still guarantees the topology flexibility and is compatible with the spline-based modeling system. In addition, we compare the numerical approximations of isogeometric analysis and finite element analysis, and the experiment indicates that isogeometric analysis using T-spline of degree one can reach a comparable result with classical method.

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Tanaka, Satoyuki, and Hiroshi Okada. "An Adaptive Wavelet Finite Element Method with High-Order B-Spline Basis Functions." Key Engineering Materials 345-346 (August 2007): 877–80. http://dx.doi.org/10.4028/www.scientific.net/kem.345-346.877.

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In this paper, an adaptive strategy based on a B-spline wavelet Galerkin method is discussed. The authors have developed the wavelet Galerkin Method which utilizes quadratic and cubic B-spline scaling function/wavelet as its basis functions. The developed B-spline Galerkin Method has been proven to be very accurate in the analyses of elastostatics. Then the authors added a capability to adaptively adjust the special resolution of the basis functions by adding the wavelet basis functions where the resolution needs to be enhanced.

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Okuzono, Takeshi, Toru Otsuru, Reiji Tomiku, and Noriko Okamoto. "A finite-element method using dispersion reduced spline elements for room acoustics simulation." Applied Acoustics 79 (May 2014): 1–8. http://dx.doi.org/10.1016/j.apacoust.2013.12.010.

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Koubaiti, Ouadie, Said EL Fakkoussi, Jaouad El-Mekkaoui, Hassan Moustachir, Ahmed Elkhalfi, and Catalin I. Pruncu. "The treatment of constraints due to standard boundary conditions in the context of the mixed Web-spline finite element method." Engineering Computations 38, no. 7 (2021): 2937–68. http://dx.doi.org/10.1108/ec-02-2020-0078.

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Purpose This paper aims to propose a new boundary condition and a web-spline basis of finite element space approximation to remedy the problems of constraints due to homogeneous and non-homogeneous; Dirichlet boundary conditions. This paper considered the two-dimensional linear elasticity equation of Navier–Lamé with the condition CAB. The latter allows to have a total insertion of the essential boundary condition in the linear system obtained; without using a numerical method as Lagrange multiplier. This study have developed mixed finite element; method using the B-splines Web-spline space. These provide an exact implementation of the homogeneous; Dirichlet boundary conditions, which removes the constraints caused by the standard; conditions. This paper showed the existence and the uniqueness of the weak solution, as well as the convergence of the numerical solution for the quadratic case are proved. The weighted extended B-spline; approach have become a much more workmanlike solution. Design/methodology/approach In this paper, this study used the implementation of weighted finite element methods to solve the Navier–Lamé system with a new boundary condition CA, B (Koubaiti et al., 2020), that generalises the well-known basis, especially the Dirichlet and the Neumann conditions. The novel proposed boundary condition permits to use a single Matlab code, which summarises all kind of boundary conditions encountered in the system. By using this model is possible to save time and programming recourses while reap several programs in a single directory. Findings The results have shown that the Web-spline-based quadratic-linear finite elements satisfy the inf–sup condition, which is necessary for existence and uniqueness of the solution. It was demonstrated by the existence of the discrete solution. A full convergence was established using the numerical solution for the quadratic case. Due to limited regularity of the Navier–Lamé problem, it will not change by increasing the degree of the Web-spline. The computed relative errors and their rates indicate that they are of order 1/H. Thus, it was provided their theoretical validity for the numerical solution stability. The advantage of this problem that uses the CA, B boundary condition is associated to reduce Matlab programming complexity. Originality/value The mixed finite element method is a robust technique to solve difficult challenges from engineering and physical sciences using the partial differential equations. Some of the important applications include structural mechanics, fluid flow, thermodynamics and electromagnetic fields (Zienkiewicz and Taylor, 2000) that are mainly based on the approximation of Lagrange. However, this type of approximation has experienced a great restriction in the level of domain modelling, especially in the case of complicated boundaries such as that in the form of curvilinear graphs. Recently, the research community tried to develop a new way of approximation based on the so-called B-spline that seems to have superior results in solving the engineering problems.

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Yi, Tae-Hyeong, and Francis X. Giraldo. "Vertical Discretization for a Nonhydrostatic Atmospheric Model Based on High-Order Spectral Elements." Monthly Weather Review 148, no. 1 (2019): 415–36. http://dx.doi.org/10.1175/mwr-d-18-0283.1.

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Abstract This study addresses the treatment of vertical discretization for a high-order, spectral element model of a nonhydrostatic atmosphere in which the governing equations of the model are separated into horizontal and vertical components by introducing a coordinate transformation, so that one can use different orders and types of approximations in both directions. The vertical terms of the decoupled governing equations are discretized using finite elements based on either Lagrange or basis-spline polynomial functions in the sigma coordinate, while maintaining the high-order spectral elements for the discretization of the horizontal terms. This leads to the fact that the high-order model of spectral elements with a nonuniform grid, interpolated within an element, can be easily accommodated with existing physical parameterizations. Idealized tests are performed to compare the accuracy and efficiency of the vertical discretization methods, in addition to the central finite differences, with those of the standard high-order spectral element approach. Our results show, through all the test cases, that the finite element with the cubic basis-spline function is more accurate than the other vertical discretization methods at moderate computational cost. Furthermore, grid dependency studies in the tests with and without orography indicate that the convergence rate of the vertical discretization methods is lower than the expected level of discretization accuracy, especially in the Schär mountain test, which yields approximately first-order convergence.

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DEMYANOVICH, YU K. "SPLINE APPROXIMATIONS ON MANIFOLDS." International Journal of Wavelets, Multiresolution and Information Processing 04, no. 03 (2006): 383–403. http://dx.doi.org/10.1142/s0219691306001324.

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A method of construction of the local approximations in the case of functions defined on n-dimensional (n ≥ 1) smooth manifold with boundary is proposed. In particular, spline and finite-element methods on manifold are discussed. Nondegenerate simplicial subdivision of the manifold is introduced and a simple method for evaluations of approach is examined (the evaluations are optimal as to N-width of corresponding compact set).

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Provatidis, Christopher G. "Equivalence between C1 -continuous Cubic B-splines and Cubic Hermite Polynomials in Finite Element and Collocation Methods." WSEAS TRANSACTIONS ON SYSTEMS 23 (December 31, 2024): 585–97. https://doi.org/10.37394/23202.2024.23.60.

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In this paper, we will show that the C 1 -continuous B-spline functional set of polynomial degree p = 3 , can be written as a linear transformation of the well-known piecewise cubic Hermite polynomials. This change of functional basis means that the global B-spline finite element solution is equivalent to that of usual piecewise finite elements in conjunction with cubic Hermite polynomials, with two degrees per nodal point, like those used in beam-bending analysis. In this context, we validate the equivalence between the global Bspline solution and the piecewise solution in boundary-value and eigenvalue problems, for collocation and RitzGalerkin methods.

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You, Feng Xiang, Fei Zhang, and Buo Lei Zuo. "The Optimum Design of Composite Laminated Plate Based on Spline Finite Element Analysis." Applied Mechanics and Materials 138-139 (November 2011): 199–209. http://dx.doi.org/10.4028/www.scientific.net/amm.138-139.199.

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Geometrical parameters of composite laminated plates in engineering structure tend to have stochastic properties. It would be very significant on how to study the random parameter laminated plates, and the parameters optimization analysis of mechanical properties, influencing on the correctly-estimated reliability of structure design. Based on the classical theory advocated by Kirchhoff, which indicates that with spline finite element method, cubic b-spline function constitutes the spline to antisymmetry multi-layer angle laid against laminated plates on the three independent displacement and the interpolation which can deduce composite laminated plate stiffness matrix, quality array type, the damping array type, and the dynamic equations of laminates is derived by Lagrange equation and a characteristic equation established by Rayleigh-Ritz theory. On the basis of Kirchhoff hypothesis, the laminated plates mechanics characteristic analysis with spline collocation method can lead to the resolution of the structural displacement, and dynamic response of velocity and acceleration, further to comparing with Newmark method. As to laminated plates of nonlinear bending, its mechanical properties will be under siscussion. The optimum design of of laying layer of composite laminated plate horn based on the spline finite element analysis will be conducted. The numerical column verifies the effectiveness of the proposed algorithm.

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Li, Chong-Jun, Vittoria Demichelis, and Catterina Dagnino. "Finite-part integrals over polygons by an 8-node quadrilateral spline finite element." BIT Numerical Mathematics 50, no. 2 (2010): 377–94. http://dx.doi.org/10.1007/s10543-010-0262-8.

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Uhm, Tae-Kyoung, Ki-Seung Kim, Yu-Deok Seo, and Sung-Kie Youn. "T-spline Finite Element Method for CAD/CAE Integrated Approach." Transactions of the Korean Society of Mechanical Engineers A 33, no. 2 (2009): 127–34. http://dx.doi.org/10.3795/ksme-a.2009.33.2.127.

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You, Feng Xiang, Fei Zhang, and Buo Lei Zuo. "Spline-Based Finite Element Analysis in Composite Laminates Mechanical Properties." Applied Mechanics and Materials 138-139 (November 2011): 673–80. http://dx.doi.org/10.4028/www.scientific.net/amm.138-139.673.

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The geometric parameters of the composite laminate in the engineering structure tend to have random properties. It is of great significance on how to study sensitivity of random parameters of laminated plates and carry on the optimized analysis to the parameteranalys when accurately estimating the reliability of structural design. According to the first order shear deformation theory, by using the spline finite element method, we can infer and the establish a laminated plate vibration equation, the stiffness matrix, mass matrix, proportional damping matrix, before making solution of the antisymmetric laminated plates response sensitivity formula, and analyzing the normal displacement, the sensitivity, the natural frequency of compound materials laminated plate. The Numerical examples verify the effectiveness of this algorithm.

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Chaudhary, Sudhakar, Vimal Srivastava, V. V. K. Srinivas Kumar, and Balaji Srinivasan. "WEB-Spline-based mesh-free finite element approximation forp-Laplacian." International Journal of Computer Mathematics 93, no. 6 (2015): 1022–43. http://dx.doi.org/10.1080/00207160.2015.1016923.

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45

Ni Guangzheng, Xu Xiaoming, and Jain Baidun. "B-spline finite element method for eddy current field analysis." IEEE Transactions on Magnetics 26, no. 2 (1990): 723–26. http://dx.doi.org/10.1109/20.106420.

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Fan, S. C., and M. H. Luah. "New Spline Finite Element for Analysis of Shells of Revolution." Journal of Engineering Mechanics 116, no. 3 (1990): 709–26. http://dx.doi.org/10.1061/(asce)0733-9399(1990)116:3(709).

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Gupta, A., J. Kiusalaas, and M. Saraph. "Cubic B-spline for finite element analysis of axisymmetric shells." Computers & Structures 38, no. 4 (1991): 463–68. http://dx.doi.org/10.1016/0045-7949(91)90042-k.

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Lan, Peng, and Ahmed A. Shabana. "Integration of B-spline geometry and ANCF finite element analysis." Nonlinear Dynamics 61, no. 1-2 (2009): 193–206. http://dx.doi.org/10.1007/s11071-009-9641-6.

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Kagan, Pavel, Anath Fischer, and Pinhas Z. Bar-Yoseph. "Mechanically based models: Adaptive refinement for B-spline finite element." International Journal for Numerical Methods in Engineering 57, no. 8 (2003): 1145–75. http://dx.doi.org/10.1002/nme.717.

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Han, Jian-Gang, Wei-Xin Ren, and Yih Huang. "A spline wavelet finite element formulation of thin plate bending." Engineering with Computers 25, no. 4 (2009): 319–26. http://dx.doi.org/10.1007/s00366-009-0124-7.

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