Статті в журналах: "Higher order finite element"

1

Oskooei, S., and J. S. Hansen. "Higher-Order Finite Element for Sandwich Plates." AIAA Journal 38, no. 3 (March 2000): 525–33. http://dx.doi.org/10.2514/2.991.

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2

Yuan, Fuh-Gwo, and Robert E. Miller. "Higher-order finite element for short beams." AIAA Journal 26, no. 11 (November 1988): 1415–17. http://dx.doi.org/10.2514/3.10059.

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3

Oskooei, S., and J. S. Hansen. "Higher-order finite element for sandwich plates." AIAA Journal 38 (January 2000): 525–33. http://dx.doi.org/10.2514/3.14442.

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4

Zhang, Qinghui, Uday Banerjee, and Ivo Babuška. "Higher order stable generalized finite element method." Numerische Mathematik 128, no. 1 (January 18, 2014): 1–29. http://dx.doi.org/10.1007/s00211-014-0609-1.

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5

Omerović, Samir, and Thomas-Peter Fries. "Higher-order conformal decomposition finite element method." PAMM 16, no. 1 (October 2016): 855–56. http://dx.doi.org/10.1002/pamm.201610416.

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6

Olesen, K., B. Gervang, J. N. Reddy, and M. Gerritsma. "A higher-order equilibrium finite element method." International Journal for Numerical Methods in Engineering 114, no. 12 (February 28, 2018): 1262–90. http://dx.doi.org/10.1002/nme.5785.

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7

Liu, Liping, Kevin B. Davies, Michal Křížek, and Li Guan. "On Higher Order Pyramidal Finite Elements." Advances in Applied Mathematics and Mechanics 3, no. 2 (April 2011): 131–40. http://dx.doi.org/10.4208/aamm.09-m0989.

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AbstractIn this paper we first prove a theorem on the nonexistence of pyramidal polynomial basis functions. Then we present a new symmetric composite pyramidal finite element which yields a better convergence than the nonsymmetric one. It has fourteen degrees of freedom and its basis functions are incomplete piecewise triquadratic polynomials. The space of ansatz functions contains all quadratic functions on each of four subtetrahedra that form a given pyramidal element.

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8

Zhang, Yi Xia, and Chun Hui Yang. "Laminated Plate Elements Based on Higher-Order Shear Deformation Theories." Advanced Materials Research 32 (February 2008): 119–24. http://dx.doi.org/10.4028/www.scientific.net/amr.32.119.

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Efficient and accurate finite elements are crucial for finite element analysis to provide adequate prediction of the structural behavior. A large amount of laminated plate elements have been developed for finite element analysis of laminated composite plates based on the various lamination theories. A recent and complete review of the laminated finite elements based on the higher-order shear deformation theories, including the global higher-order theories, zig-zag theories and the global-local higher-order theories is presented in this paper. Finally some points on the development of the laminated plate elements are summarized.

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9

LOU, ZHENG, and JIAN-MING JIN. "Higher Order Finite Element Analysis of Finite-by-Infinite Arrays." Electromagnetics 24, no. 7 (January 2004): 497–514. http://dx.doi.org/10.1080/02726340490496338.

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10

Maniatty, Antoinette M., Yong Liu, Ottmar Klaas, and Mark S. Shephard. "Higher order stabilized finite element method for hyperelastic finite deformation." Computer Methods in Applied Mechanics and Engineering 191, no. 13-14 (January 2002): 1491–503. http://dx.doi.org/10.1016/s0045-7825(01)00335-8.

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11

SUZUKI, Kohji, Isao KIMPARA, and Kazuro KAGEYAMA. "Leyerwise Higher-Order Finite Element with Penalty Method." Journal of the Japan Society for Composite Materials 25, no. 3 (1999): 109–19. http://dx.doi.org/10.6089/jscm.25.109.

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12

Yuan, Fuh-Gwo, and Robert E. Miller. "A higher order finite element for laminated beams." Composite Structures 14, no. 2 (January 1990): 125–50. http://dx.doi.org/10.1016/0263-8223(90)90027-c.

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13

Zahari, Rizal, Faizal Mustapha, Dayang Laila Abd Majid, Azmin Shakrine M. Rafie, and Thariq Hameed Sultan. "Geometric Non-Linear Analysis of Composite Laminated Plates Using Higher Order Finite Strip Element." Applied Mechanics and Materials 225 (November 2012): 165–71. http://dx.doi.org/10.4028/www.scientific.net/amm.225.165.

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A finite strip method for geometric non-linear static analysis based on the tangential stiffness matrix has been developed using the new concept of polynomial finite strip elements, with Reissner (higher order shear deformable element) plate-bending theory for composite plates. A finite strip analysis programming package, which is capable of performing non-linear analysis for composite flat panels, has also been developed with Reissner plate bending element. Good agreement with the finite element results has been observed through various test cases, confirming the accuracy and reliability of the new developed finite strip method.

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14

Gara, Fabrizio, Sandro Carbonari, Graziano Leoni, and Luigino Dezi. "Finite Elements for Higher Order Steel–Concrete Composite Beams." Applied Sciences 11, no. 2 (January 8, 2021): 568. http://dx.doi.org/10.3390/app11020568.

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This paper presents finite elements for a higher order steel–concrete composite beam model developed for the analysis of bridge decks. The model accounts for the slab–girder partial interaction, the overall shear deformability, and the shear-lag phenomenon in steel and concrete components. The theoretical derivation of the solving balance conditions, in both weak and strong form, is firstly addressed. Then, three different finite elements are proposed, which are characterised by (i) linear interpolating functions, (ii) Hermitian polynomial interpolating functions, and (iii) interpolating functions, respectively, derived from the analytical solution expressed by means of exponential matrices. The performance of the finite elements is analysed in terms of the solution convergence rate for realistic steel–concrete composite beams with different restraints and loading conditions. Finally, the efficiency of the beam model is shown by comparing the results obtained with the proposed finite elements and those achieved with a refined 3D shell finite element model.

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15

Gara, Fabrizio, Sandro Carbonari, Graziano Leoni, and Luigino Dezi. "Finite Elements for Higher Order Steel–Concrete Composite Beams." Applied Sciences 11, no. 2 (January 8, 2021): 568. http://dx.doi.org/10.3390/app11020568.

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Анотація:

This paper presents finite elements for a higher order steel–concrete composite beam model developed for the analysis of bridge decks. The model accounts for the slab–girder partial interaction, the overall shear deformability, and the shear-lag phenomenon in steel and concrete components. The theoretical derivation of the solving balance conditions, in both weak and strong form, is firstly addressed. Then, three different finite elements are proposed, which are characterised by (i) linear interpolating functions, (ii) Hermitian polynomial interpolating functions, and (iii) interpolating functions, respectively, derived from the analytical solution expressed by means of exponential matrices. The performance of the finite elements is analysed in terms of the solution convergence rate for realistic steel–concrete composite beams with different restraints and loading conditions. Finally, the efficiency of the beam model is shown by comparing the results obtained with the proposed finite elements and those achieved with a refined 3D shell finite element model.

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16

Warren, G. S., and W. R. Scott. "Numerical dispersion of higher order nodal elements in the finite-element method." IEEE Transactions on Antennas and Propagation 44, no. 3 (March 1996): 317–20. http://dx.doi.org/10.1109/8.486299.

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17

Stazi, F. L., E. Budyn, J. Chessa, and T. Belytschko. "An extended finite element method with higher-order elements for curved cracks." Computational Mechanics 31, no. 1-2 (May 1, 2003): 38–48. http://dx.doi.org/10.1007/s00466-002-0391-2.

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18

Kaveh, A., and M. J. Tolou Kian. "Efficient finite element analysis of models comprised of higher order triangular elements." Acta Mechanica 224, no. 9 (April 10, 2013): 1957–75. http://dx.doi.org/10.1007/s00707-013-0855-9.

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19

Baran, Á., and G. Stoyan. "Gauss-Legendre elements: a stable, higher order non-conforming finite element family." Computing 79, no. 1 (February 2007): 1–21. http://dx.doi.org/10.1007/s00607-007-0219-1.

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20

Rui, Xi, Jun Hu, and Qing Huo Liu. "HIGHER ORDER FINITE ELEMENT METHOD FOR INHOMOGENEOUS AXISYMMETRIC RESONATORS." Progress In Electromagnetics Research B 21 (2010): 189–201. http://dx.doi.org/10.2528/pierb10031605.

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21

Schilling, Nathanael, Gary Froyland, and Oliver Junge. "Higher-order finite element approximation of the dynamic Laplacian." ESAIM: Mathematical Modelling and Numerical Analysis 54, no. 5 (July 28, 2020): 1777–95. http://dx.doi.org/10.1051/m2an/2020027.

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The dynamic Laplace operator arises from extending problems of isoperimetry from fixed manifolds to manifolds evolved by general nonlinear dynamics. Eigenfunctions of this operator are used to identify and track finite-time coherent sets, which physically manifest in fluid flows as jets, vortices, and more complicated structures. Two robust and efficient finite-element discretisation schemes for numerically computing the dynamic Laplacian were proposed in Froyland and Junge [SIAM J. Appl. Dyn. Syst. 17 (2018) 1891–1924]. In this work we consider higher-order versions of these two numerical schemes and analyse them experimentally. We also prove the numerically computed eigenvalues and eigenvectors converge to the true objects for both schemes under certain assumptions. We provide an efficient implementation of the higher-order element schemes in an accompanying Julia package.

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22

Farthing, Matthew W., Christopher E. Kees, and Cass T. Miller. "Mixed finite element methods and higher-order temporal approximations." Advances in Water Resources 25, no. 1 (January 2002): 85–101. http://dx.doi.org/10.1016/s0309-1708(01)00022-7.

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23

Garcia-Donoro, Daniel, Adrian Amor-Martin, and Luis E. Garcia-Castillo. "Higher-Order Finite Element Electromagnetics Code for HPC environments." Procedia Computer Science 108 (2017): 818–27. http://dx.doi.org/10.1016/j.procs.2017.05.239.

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24

Gaspoz, Fernando D., and Pedro Morin. "Approximation classes for adaptive higher order finite element approximation." Mathematics of Computation 83, no. 289 (December 17, 2013): 2127–60. http://dx.doi.org/10.1090/s0025-5718-2013-02777-9.

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25

Ferradi, Mohammed Khalil, Xavier Cespedes, and Mathieu Arquier. "A higher order beam finite element with warping eigenmodes." Engineering Structures 46 (January 2013): 748–62. http://dx.doi.org/10.1016/j.engstruct.2012.07.038.

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26

Haasdonk, B., M. Ohlberger, M. Rumpf, A. Schmidt, and K. G. Siebert. "Multiresolution Visualization of Higher Order Adaptive Finite Element Simulations." Computing 70, no. 3 (June 2003): 181–204. http://dx.doi.org/10.1007/s00607-003-1476-2.

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27

Ahmed, Nesar U., and Prodyot K. Basu. "Higher-order finite element modelling of laminated composite plates." International Journal for Numerical Methods in Engineering 37, no. 1 (January 15, 1994): 123–39. http://dx.doi.org/10.1002/nme.1620370109.

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28

Eid, R. "Higher order isoparametric finite element solution of Stokes flow." Applied Mathematics and Computation 162, no. 3 (March 2005): 1083–101. http://dx.doi.org/10.1016/j.amc.2004.01.014.

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29

Ribarić, Dragan, and Gordan Jelenić. "Higher-order linked interpolation in triangular thick plate finite elements." Engineering Computations 31, no. 1 (February 25, 2014): 69–109. http://dx.doi.org/10.1108/ec-03-2012-0056.

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Purpose – In this work, the authors aim to employ the so-called linked-interpolation concept already tested on beam and quadrilateral plate finite elements in the design of displacement-based higher-order triangular plate finite elements and test their performance. Design/methodology/approach – Starting from the analogy between the Timoshenko beam theory and the Mindlin plate theory, a family of triangular linked-interpolation plate finite elements of arbitrary order are designed. The elements are tested on the standard set of examples. Findings – The derived elements pass the standard patch tests and also the higher-order patch tests of an order directly related to the order of the element. The lowest-order member of the family of developed elements still suffers from shear locking for very coarse meshes, but the higher-order elements turn out to be successful when compared to the elements from literature for the problems with the same total number of the degrees of freedom. Research limitations/implications – The elements designed perform well for a number of standard benchmark tests, but the well-known Morley's skewed plate example turns out to be rather demanding, i.e. the proposed design principle cannot compete with the mixed-type approach for this test. Work is under way to improve the proposed displacement-based elements by adding a number of internal bubble functions in the displacement and rotation fields, specifically chosen to satisfy the basic patch test and enable a softer response in the bench-mark test examples. Originality/value – A new family of displacement-based higher-order triangular Mindlin plate finite elements has been derived. The higher-order elements perform very well, whereas the lowest-order element requires improvement.

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30

Himeur, Mohammed, Hamza Guenfoud, and Mohamed Guenfoud. "A higher order triangular plate finite element using Airy functions." Advances in Mechanical Engineering 12, no. 11 (November 2020): 168781402097190. http://dx.doi.org/10.1177/1687814020971906.

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The present paper describes the formulation of a new moderately thick plate bending triangular finite element based on Mindlin–Reissner plate theory. It is called a Great Triangular Moderately Thick Plate Finite Element, or GTMTPFE. The formulation is based on the strain approach, on solution of Airy’s function and on the analytical integration in the construction of the stiffness matrix. The strengths associated with this approach consist of: • automatic verification of equilibrium conditions and kinematic compatibility conditions, • the enrichment of the degrees of the interpolation polynomials of displacements, strains and constraints (refinement p), • the consideration distortions sections related to Poisson effects, • the treatment of blocking phenomena related to transverse shear. In general, this approach results in a competitive, robust and efficient new moderately thick plate finite element. This is visible, on the one hand, through its stability against patch tests (constant twists, state of constants moments, transverse shear locking phenomenon, isotropy test). This is visible, through its good response to the patch tests to which it is subjected (constant torsions, state of constant moments, phenomenon of blocking in transverse shears, isotropy test). As has excellent convergence to the reference solution. Thus, it exhibits better performance behavior than other existing plate elements in the literature, particularly for moderately thick plates and for thin plates (L/h ratio greater than 4).

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31

Weißer, Steffen. "Higher order Trefftz-like Finite Element Method on meshes with L-shaped elements." PAMM 14, no. 1 (December 2014): 31–34. http://dx.doi.org/10.1002/pamm.201410009.

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32

Rezaiee-Pajand, Mohammad, Mohammadreza Ramezani, and Nima Gharaei-Moghaddam. "Using Higher-Order Strain Interpolation Function to Improve the Accuracy of Structural Responses." International Journal of Applied Mechanics 12, no. 03 (April 2020): 2050026. http://dx.doi.org/10.1142/s175882512050026x.

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It is widely known that the accuracy of the finite element method has a direct relation with the type of elements and meshes. Another issue which has remained less treated is the impact of loading type on the accuracy of responses. Changing the applied forces from concentrated to distributed loading has a great effect on the accuracy of certain types of elements and this action can greatly reduce their accuracy. Particularly in the coarse meshes, it creates a critical situation. Some elements do not have the ability to provide the exact answers in stated conditions. For example, the well-known plane element, LST, demonstrates promising performance under concentrated shear and bending loading as well as surface traction. In the case of distributed loads and coarse meshes, its accuracy diminishes considerably. To remedy this defect, in this paper, a new higher-order triangular element is proposed by using natural assumed strain approximation. Various numerical examples demonstrate high accuracy and efficiency of the element in comparison with common well-known finite elements in analysis of structures under distributed loading.

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33

Rezaiee-Pajand, Mohammad, Nima Gharaei-Moghaddam, and Mohammadreza Ramezani. "Higher-order assumed strain plane element immune to mesh distortion." Engineering Computations 37, no. 9 (April 13, 2020): 2957–81. http://dx.doi.org/10.1108/ec-09-2019-0422.

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Purpose This paper aims to propose a new robust membrane finite element for the analysis of plane problems. The suggested element has triangular geometry. Four nodes and 11 degrees of freedom (DOF) are considered for the element. Each of the three vertex nodes has three DOF, two displacements and one drilling. The fourth node that is located inside the element has only two translational DOF. Design/methodology/approach The suggested formulation is based on the assumed strain method and satisfies both compatibility and equilibrium conditions within each element. This establishment results in higher insensitivity to the mesh distortion. Enforcement of the equilibrium condition to the assumed strain field leads to considerably high accuracy of the developed formulation. Findings To show the merits of the suggested plane element, its different properties, including insensitivity to mesh distortion, particularly under transverse shear forces, immunities to the various locking phenomena and convergence of the element are studied. The obtained results demonstrate the superiority of the suggested element compared with many of the available robust membrane elements. Originality/value According to the attained results, the proposed element performs better than the well-known displacement-based elements such as linear strain triangular element, Q4 and Q8 and even is comparable with robust modified membrane elements.

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34

Klimczak, Marek, and Witold Cecot. "Higher Order Multiscale Finite Element Method for Heat Transfer Modeling." Materials 14, no. 14 (July 8, 2021): 3827. http://dx.doi.org/10.3390/ma14143827.

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In this paper, we present a new approach to model the steady-state heat transfer in heterogeneous materials. The multiscale finite element method (MsFEM) is improved and used to solve this problem. MsFEM is a fast and flexible method for upscaling. Its numerical efficiency is based on the natural parallelization of the main computations and their further simplifications due to the numerical nature of the problem. The approach does not require the distinct separation of scales, which makes its applicability to the numerical modeling of the composites very broad. Our novelty relies on modifications to the standard higher-order shape functions, which are then applied to the steady-state heat transfer problem. To the best of our knowledge, MsFEM (based on the special shape function assessment) has not been previously used for an approximation order higher than p = 2, with the hierarchical shape functions applied and non-periodic domains, in this problem. Some numerical results are presented and compared with the standard direct finite-element solutions. The first test shows the performance of higher-order MsFEM for the asphalt concrete sample which is subject to heating. The second test is the challenging problem of metal foam analysis. The thermal conductivity of air and aluminum differ by several orders of magnitude, which is typically very difficult for the upscaling methods. A very good agreement between our upscaled and reference results was observed, together with a significant reduction in the number of degrees of freedom. The error analysis and the p-convergence of the method are also presented. The latter is studied in terms of both the number of degrees of freedom and the computational time.

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35

Perego, Mauro, Max Gunzburger, and John Burkardt. "Parallel finite-element implementation for higher-order ice-sheet models." Journal of Glaciology 58, no. 207 (2012): 76–88. http://dx.doi.org/10.3189/2012jog11j063.

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AbstractHigher-order models represent a computationally less expensive alternative to the Stokes model for ice-sheet modeling. In this work, we develop linear and quadratic finite-element methods, implemented on parallel architectures, for the three-dimensional first-order model of Dukowicz and others (2010) that is based on the Blatter-Pattyn model, and for the depth-integrated model of Schoof and Hindmarsh (2010). We then apply our computational models to three of the ISMIP-HOM benchmark test cases (Pattyn and others, 2008). We compare results obtained from our models with those obtained using a reliable Stokes computational model, showing that our first-order model implementation produces reliable and accurate solutions for almost all characteristic length scales of the test geometries considered. Good agreement with the reference Stokes solution is also obtained by our depth-integrated model implementation in fast-sliding regimes and for medium to large length scales. We also provide a comprehensive comparison between results obtained from our first-order model implementation and implementations developed by ISMIP-HOM participants; this study shows that our implementation is at least as good as the previous ones. Finally, a comparison between linear and quadratic finite- element approximations is carried out, showing, as expected, the better accuracy of the quadratic finite-element method.

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36

Guzmán, Johnny, Manuel A. Sánchez, and Marcus Sarkis. "Higher-order finite element methods for elliptic problems with interfaces." ESAIM: Mathematical Modelling and Numerical Analysis 50, no. 5 (September 2016): 1561–83. http://dx.doi.org/10.1051/m2an/2015093.

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37

Youn, Sung-Kie, and Sang-Hoon Park. "A new direct higher-order Taylor-Galerkin finite element method." Computers & Structures 56, no. 4 (August 1995): 651–56. http://dx.doi.org/10.1016/0045-7949(94)00561-g.

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38

Nguyen, Steven H. "A higher-order finite element scheme for incompressible lubrication calculations." Finite Elements in Analysis and Design 10, no. 4 (February 1992): 307–17. http://dx.doi.org/10.1016/0168-874x(92)90018-8.

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39

Coyle, Joe. "Overlapping solution finite element method – Higher order approximation and implementation." Applied Numerical Mathematics 62, no. 12 (December 2012): 1910–24. http://dx.doi.org/10.1016/j.apnum.2012.07.005.

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40

Burger, Martin, Christina Stöcker, and Axel Voigt. "Finite Element-Based Level Set Methods for Higher Order Flows." Journal of Scientific Computing 35, no. 2-3 (June 2008): 77–98. http://dx.doi.org/10.1007/s10915-008-9204-x.

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41

Kocak, S., and H. Hassis. "A higher order shear deformable finite element for homogeneous plates." Engineering Structures 25, no. 2 (January 2003): 131–39. http://dx.doi.org/10.1016/s0141-0296(02)00061-5.

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42

Tessler, Alexander. "A higher-order plate theory with ideal finite element suitability." Computer Methods in Applied Mechanics and Engineering 85, no. 2 (January 1991): 183–205. http://dx.doi.org/10.1016/0045-7825(91)90132-p.

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43

Luo, Xiao-Juan, Mark S. Shephard, Lie-Quan Lee, Lixin Ge, and Cho Ng. "Moving curved mesh adaptation for higher-order finite element simulations." Engineering with Computers 27, no. 1 (February 27, 2010): 41–50. http://dx.doi.org/10.1007/s00366-010-0179-5.

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44

Murthy, M. V. V. S., D. Roy Mahapatra, K. Badarinarayana, and S. Gopalakrishnan. "A refined higher order finite element for asymmetric composite beams." Composite Structures 67, no. 1 (January 2005): 27–35. http://dx.doi.org/10.1016/j.compstruct.2004.01.005.

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45

Shu, S., D. Sun, and J. Xu. "An Algebraic Multigrid Method for Higher-order Finite Element Discretizations." Computing 77, no. 4 (May 2, 2006): 347–77. http://dx.doi.org/10.1007/s00607-006-0162-6.

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46

Karpik, Anna, Francesco Cosco, and Domenico Mundo. "Higher-Order Hexahedral Finite Elements for Structural Dynamics: A Comparative Review." Machines 11, no. 3 (February 24, 2023): 326. http://dx.doi.org/10.3390/machines11030326.

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Анотація:

The finite element method (FEM) is widely used in many engineering applications. The popularity of FEM led to the development of several variants of formulations, and hexahedral meshes surged as one of the most computationally effective. After briefly reviewing the reasons and advantages behind the formulation of increasing order elements, including the serendipity variants and the associated reduced integration schemes, a systematic comparison of the most common hexahedral formulations is presented. A numerical benchmark was used to assess convergency rates and computational efficiencies when solving the eigenvalue problem for linear dynamic analysis. The obtained results confirmed the superior performances of the higher-order brick element formulations. In terms of computational efficiency, defined as the ratio between achievable accuracy and computational execution time, quadratic or cubic formulations exhibited the best results for the stages of FE model assembly and solution computation, respectively.

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47

Liu, Liping, Michal Křížek, and Pekka Neittaanmäki. "Higher order finite element approximation of a quasilinear elliptic boundary value problem of a non-monotone type." Applications of Mathematics 41, no. 6 (1996): 467–78. http://dx.doi.org/10.21136/am.1996.134338.

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48

Hu, Jun, та Shangyou Zhang. "Finite element approximations of symmetric tensors on simplicial grids in ℝn: The lower order case". Mathematical Models and Methods in Applied Sciences 26, № 09 (26 липня 2016): 1649–69. http://dx.doi.org/10.1142/s0218202516500408.

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In this paper, we construct, in a unified fashion, lower order finite element subspaces of spaces of symmetric tensors with square-integrable divergence on a domain in any dimension. These subspaces are essentially the symmetric tensor finite element spaces of order [Formula: see text] from [Finite element approximations of symmetric tensors on simplicial grids in [Formula: see text]: The higher order case, J. Comput. Math. 33 (2015) 283–296], enriched, for each [Formula: see text]-dimensional simplex, by [Formula: see text] face bubble functions in the symmetric tensor finite element space of order [Formula: see text] from [Finite element approximations of symmetric tensors on simplicial grids in [Formula: see text]: The higher order case, J. Comput. Math. 33 (2015) 283–296] when [Formula: see text], and by [Formula: see text] face bubble functions in the symmetric tensor finite element space of order [Formula: see text] from [Finite element approximations of symmetric tensors on simplicial grids in [Formula: see text]: The higher order case, J. Comput. Math. 33 (2015) 283–296] when [Formula: see text]. These spaces can be used to approximate the symmetric matrix field in a mixed formulation problem where the other variable is approximated by discontinuous piecewise [Formula: see text] polynomials. This in particular leads to first-order mixed elements on simplicial grids with total degrees of freedom per element [Formula: see text] plus [Formula: see text] in 2D, 48 plus 6 in 3D. The previous record of the degrees of freedom of first-order mixed elements is, 21 plus 3 in 2D, and 156 plus 6 in 3D, on simplicial grids. We also derive, in a unified way which is completely different from those used in [D. Arnold, G. Awanou and R. Winther, Finite elements for symmetric tensors in three dimensions, Math. Comput. 77 (2008) 1229–1251; D. N. Arnold and R. Winther, Mixed finite element for elasticity, Number Math. 92 (2002) 401–419], a family of Arnold–Winther mixed finite elements in any space dimension. One example in this family is the Raviart–Thomas elements in one dimension, the second example is the mixed finite elements for linear elasticity in two dimensions due to Arnold and Winther, the third example is the mixed finite elements for linear elasticity in three dimensions due to Arnold, Awanou and Winther.

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Noh, Hyuk-Chun, and Phill-Seung Lee. "Higher order weighted integral stochastic finite element method and simplified first-order application." International Journal of Solids and Structures 44, no. 11-12 (June 2007): 4120–44. http://dx.doi.org/10.1016/j.ijsolstr.2006.11.013.

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50

Ahmed, Sidrah, and Saqib Zia. "The Higher-Order CESE Method for Two-dimensional Shallow Water Magnetohydrodynamics Equations." European Journal of Pure and Applied Mathematics 12, no. 4 (October 31, 2019): 1464–82. http://dx.doi.org/10.29020/nybg.ejpam.v12i4.3538.

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The numerical solution of two-dimensional shallow water magnetohydrodynamics model is obtained using the $4^{th}$-order conservation element solution element method (CESE). The method is based on unified treatment of spatial and temporal dimensions contrary to the finite difference and finite volume methods. The higher-order CESE scheme is constructed using same definitions of conservation and solution elements that are used for $2^{nd}$-order CESE scheme formulation. Hence it is more convenient to increase accuracy of CESE methods as compared to the finite difference and finite volume methods. Moreover the scheme is developed using the conservative formulation and do not require change in the source term for treating the degenerate hyperbolic nature of shallow water magnetohydrodynamics system due to divergence constraint. The spatial and temporal derivatives have been obtained by incorporating $4^{th}$-order Taylor expansion and the projection method is used to handle the divergence constraint. The accuracy and robustness of the extended method is tested by performing a benchmark numerical test taken from the literature. Numerical experiment revealed the accuracy and computational efficiency of the scheme.

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