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Journal articles on the topic 'Least-squares finite element'

Author: Grafiati

Published: 4 June 2021

Last updated: 3 February 2022

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1

Musivand-Arzanfudi, M., and H. Hosseini-Toudeshky. "Moving least-squares finite element method." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 221, no. 9 (September 1, 2007): 1019–36. http://dx.doi.org/10.1243/09544062jmes463.

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A new computational method here called moving least-squares finite element method (MLSFEM) is presented, in which the shape functions of the parametric elements are constructed using moving least-squares approximation. While preserving some excellent characteristics of the meshless methods such as elimination of the volumetric locking in near-incompressible materials and giving accurate strains and stresses near the boundaries of the problem, the computational time is decreased by constructing the meshless shape functions in the stage of creating parametric elements and then utilizing them for any new problem. Moreover, it is not necessary to have knowledge about the full details of the shape function generation method in future uses. The MLSFEM also eliminates another drawback of meshless methods associated with the lack of accordance between the integration cells and the problem boundaries. The method is described for two-dimensional problems, but it is extendable for three-dimensional problems too. The MLSFEM does not require the complex mesh generation. Excellent results can be obtained even using a simple mesh. A technique is also presented for isoparametric mapping which enables best possible mapping via a constrained optimization criterion. Several numerical examples are analysed to show the efficiency and convergence of the method.

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2

Keith, Brendan, Socratis Petrides, Federico Fuentes, and Leszek Demkowicz. "Discrete least-squares finite element methods." Computer Methods in Applied Mechanics and Engineering 327 (December 2017): 226–55. http://dx.doi.org/10.1016/j.cma.2017.08.043.

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3

Duan, Huo-Yuan, and Guo-Ping Liang. "Nonconforming elements in least-squares mixed finite element methods." Mathematics of Computation 73, no. 245 (March 27, 2003): 1–18. http://dx.doi.org/10.1090/s0025-5718-03-01520-5.

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4

Bochev, Pavel B., and Max D. Gunzburger. "Finite Element Methods of Least-Squares Type." SIAM Review 40, no. 4 (January 1998): 789–837. http://dx.doi.org/10.1137/s0036144597321156.

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5

Chen, T. F., and G. J. Fix. "Least squares finite element simulation of transonic flows." Applied Numerical Mathematics 2, no. 3-5 (October 1986): 399–408. http://dx.doi.org/10.1016/0168-9274(86)90042-5.

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6

Bedivan, D. M. "Error estimates for least squares finite element methods." Computers & Mathematics with Applications 43, no. 8-9 (April 2002): 1003–20. http://dx.doi.org/10.1016/s0898-1221(02)80009-8.

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7

Brannick, J., C. Ketelsen, T. Manteuffel, and S. McCormick. "Least-Squares Finite Element Methods for Quantum Electrodynamics." SIAM Journal on Scientific Computing 32, no. 1 (January 2010): 398–417. http://dx.doi.org/10.1137/080729633.

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8

Kiousis, Panos D. "Least-Squares Finite-Element Evaluation of Flow Nets." Journal of Geotechnical and Geoenvironmental Engineering 128, no. 8 (August 2002): 699–701. http://dx.doi.org/10.1061/(asce)1090-0241(2002)128:8(699).

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9

Bochev, Pavel, Leszek Demkowicz, Jay Gopalakrishnan, and Max Gunzburger. "Minimum Residual and Least Squares Finite Element Methods." Computers & Mathematics with Applications 68, no. 11 (December 2014): 1479. http://dx.doi.org/10.1016/j.camwa.2014.11.005.

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10

Chen, Fuchen, Eric Chung, and Lijian Jiang. "Least-squares mixed generalized multiscale finite element method." Computer Methods in Applied Mechanics and Engineering 311 (November 2016): 764–87. http://dx.doi.org/10.1016/j.cma.2016.09.010.

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11

Jiang, Bo-Nan, and Louis A. Povinelli. "Least-squares finite element method for fluid dynamics." Computer Methods in Applied Mechanics and Engineering 81, no. 1 (July 1990): 13–37. http://dx.doi.org/10.1016/0045-7825(90)90139-d.

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12

Chaudhry, Jehanzeb H., Luke N. Olson, and Peter Sentz. "A Least-Squares Finite Element Reduced Basis Method." SIAM Journal on Scientific Computing 43, no. 2 (January 2021): A1081—A1107. http://dx.doi.org/10.1137/20m1323552.

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13

Singh, Krishna M., and Manjeet S. Kalra. "Least-squares finite element schemes in the time domain." Computer Methods in Applied Mechanics and Engineering 190, no. 1-2 (October 2000): 111–31. http://dx.doi.org/10.1016/s0045-7825(99)00417-x.

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14

Yang, Suh-Yuh, and Jinn-Liang Liu. "Least-squares finite element methods for the elasticity problem." Journal of Computational and Applied Mathematics 87, no. 1 (December 1997): 39–60. http://dx.doi.org/10.1016/s0377-0427(97)00174-x.

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15

Truty, A. "A Galerkin/least‐squares finite element formulation for consolidation." International Journal for Numerical Methods in Engineering 52, no. 8 (November 20, 2001): 763–86. http://dx.doi.org/10.1002/nme.224.

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16

Fried, Isaac. "The condition of the least-squares finite element matrices." International Journal for Numerical Methods in Engineering 106, no. 9 (November 3, 2015): 760–70. http://dx.doi.org/10.1002/nme.5146.

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17

Pratap, Rudra, and Tribikram Kundu. "A least squares finite element formulation for elastodynamic problems." International Journal for Numerical Methods in Engineering 26, no. 8 (August 1988): 1883–91. http://dx.doi.org/10.1002/nme.1620260813.

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18

Westphal, Chad. "A least-squares finite-element method for viscoelastic fluids." PAMM 7, no. 1 (December 2007): 1025101–2. http://dx.doi.org/10.1002/pamm.200700141.

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19

Jiang, Bo-Nan, and G. F. Carey. "Least-squares finite element methods for compressible Euler equations." International Journal for Numerical Methods in Fluids 10, no. 5 (April 1, 1990): 557–68. http://dx.doi.org/10.1002/fld.1650100504.

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20

Bao, Gang, and Hongtao Yang. "A Least-Squares Finite Element Analysis for Diffraction Problems." SIAM Journal on Numerical Analysis 37, no. 2 (January 1999): 665–82. http://dx.doi.org/10.1137/s0036142998342380.

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21

Carstensen, Carsten, and Johannes Storn. "Asymptotic Exactness of the Least-Squares Finite Element Residual." SIAM Journal on Numerical Analysis 56, no. 4 (January 2018): 2008–28. http://dx.doi.org/10.1137/17m1125972.

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22

Chen, Fuchen, Eric Chung, and Lijian Jiang. "Adaptive Least-Squares Mixed Generalized Multiscale Finite Element Methods." Multiscale Modeling & Simulation 16, no. 2 (January 2018): 1034–58. http://dx.doi.org/10.1137/17m1138844.

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23

Gerritsma, Marc, Carsten Carstensen, Leszek Demkowicz, and Jay Gopalakrishnan. "Minimum Residual and Least Squares Finite Element Methods II." Computers & Mathematics with Applications 74, no. 8 (October 2017): 1922. http://dx.doi.org/10.1016/j.camwa.2017.07.021.

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24

Lee, Hsueh-Chen, and Tsu-Fen Chen. "Adaptive least-squares finite element approximations to Stokes equations." Journal of Computational and Applied Mathematics 280 (May 2015): 396–412. http://dx.doi.org/10.1016/j.cam.2014.11.041.

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25

Jiang, Bo-Nan, and Louis A. Povinelli. "Optimal least-squares finite element method for elliptic problems." Computer Methods in Applied Mechanics and Engineering 102, no. 2 (January 1993): 199–212. http://dx.doi.org/10.1016/0045-7825(93)90108-a.

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26

Carstensen, Carsten, Eun-Jae Park, and Philipp Bringmann. "Convergence of natural adaptive least squares finite element methods." Numerische Mathematik 136, no. 4 (February 23, 2017): 1097–115. http://dx.doi.org/10.1007/s00211-017-0866-x.

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27

Pontaza, J. P., and J. N. Reddy. "Least-squares finite element formulation for shear-deformable shells." Computer Methods in Applied Mechanics and Engineering 194, no. 21-24 (June 2005): 2464–93. http://dx.doi.org/10.1016/j.cma.2004.07.041.

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28

Jiang, B. N., and G. F. Carey. "Adaptive refinement for least-squares finite elements with element-by-element conjugate gradient solution." International Journal for Numerical Methods in Engineering 24, no. 3 (March 1987): 569–80. http://dx.doi.org/10.1002/nme.1620240308.

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29

Huo-Yuan, Duan, and Liang Guo-Ping. "Normal and tangential continuous elements for least-squares mixed finite element methods." Numerical Methods for Partial Differential Equations 20, no. 4 (2004): 609–23. http://dx.doi.org/10.1002/num.20002.

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30

BENSOW, RICKARD E., and MATS G. LARSON. "DISCONTINUOUS/CONTINUOUS LEAST-SQUARES FINITE ELEMENT METHODS FOR ELLIPTIC PROBLEMS." Mathematical Models and Methods in Applied Sciences 15, no. 06 (June 2005): 825–42. http://dx.doi.org/10.1142/s0218202505000595.

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Least-squares finite element methods (LSFEM) are useful for first-order systems, where they avoid the stability consideration of mixed methods and problems with constraints, like the div-curl problem. However, LSFEM typically suffer from requirements on the solution to be very regular. This rules out, e.g., applications posed on nonconvex domains. In this paper we study a least-squares formulation where the discrete space is enriched by discontinuous elements in the vicinity of singularities. The weighting on the interelement terms are chosen to give correct regularity of the solution space and thus making computation of less regular problems possible. We apply this technique to the first-order Poisson problem, show coercivity and a priori estimates, and present numerical results in 3D.

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31

Kumar, Rajeev, and Brian H. Dennis. "Bubble-Enriched Least-Squares Finite Element Method for Transient Advective Transport." Differential Equations and Nonlinear Mechanics 2008 (2008): 1–21. http://dx.doi.org/10.1155/2008/267454.

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The least-squares finite element method (LSFEM) has received increasing attention in recent years due to advantages over the Galerkin finite element method (GFEM). The method leads to a minimization problem in theL2-norm and thus results in a symmetric and positive definite matrix, even for first-order differential equations. In addition, the method contains an implicit streamline upwinding mechanism that prevents the appearance of oscillations that are characteristic of the Galerkin method. Thus, the least-squares approach does not require explicit stabilization and the associated stabilization parameters required by the Galerkin method. A new approach, the bubble enriched least-squares finite element method (BELSFEM), is presented and compared with the classical LSFEM. The BELSFEM requires a space-time element formulation and employs bubble functions in space and time to increase the accuracy of the finite element solution without degrading computational performance. We apply the BELSFEM and classical least-squares finite element methods to benchmark problems for 1D and 2D linear transport. The accuracy and performance are compared.

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32

Pontaza, J. P., and J. N. Reddy. "Least-squares finite element formulations for one-dimensional radiative transfer." Journal of Quantitative Spectroscopy and Radiative Transfer 95, no. 3 (October 2005): 387–406. http://dx.doi.org/10.1016/j.jqsrt.2004.11.015.

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33

Jou, Jang, and Suh-Yuh Yang. "Least-squares finite element approximations to the Timoshenko beam problem." Applied Mathematics and Computation 115, no. 1 (October 2000): 63–75. http://dx.doi.org/10.1016/s0096-3003(99)00139-3.

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34

Zaki, S. I. "A least-squares finite element scheme for the EW equation." Computer Methods in Applied Mechanics and Engineering 189, no. 2 (September 2000): 587–94. http://dx.doi.org/10.1016/s0045-7825(99)00312-6.

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35

Pontaza, J. P., Xu Diao, J. N. Reddy, and K. S. Surana. "Least-squares finite element models of two-dimensional compressible flows." Finite Elements in Analysis and Design 40, no. 5-6 (March 2004): 629–44. http://dx.doi.org/10.1016/s0168-874x(03)00100-8.

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36

TANG, L. Q., and T. T. H. TSANG. "A LEAST-SQUARES FINITE ELEMENT METHOD FOR DOUBLY-DIFFUSIVE CONVECTION." International Journal of Computational Fluid Dynamics 3, no. 1 (January 1994): 1–17. http://dx.doi.org/10.1080/10618569408904497.

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37

Winterscheidt, Daniel, and Karan S. Surana. "p-version least-squares finite element formulation of Burgers' equation." International Journal for Numerical Methods in Engineering 36, no. 21 (November 15, 1993): 3629–46. http://dx.doi.org/10.1002/nme.1620362105.

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38

Carey, G. F., and Yun Shen. "Least-squares finite element approximation of Fisher's reaction-diffusion equation." Numerical Methods for Partial Differential Equations 11, no. 2 (March 1995): 175–86. http://dx.doi.org/10.1002/num.1690110206.

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39

Gu, Haiming, and Ning Chen. "Least-squares mixed finite element methods for the RLW equations." Numerical Methods for Partial Differential Equations 24, no. 3 (2008): 749–58. http://dx.doi.org/10.1002/num.20285.

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40

Carey, G. F., and B. N. Jiang. "Least-squares finite element method and preconditioned conjugate gradient solution." International Journal for Numerical Methods in Engineering 24, no. 7 (July 1987): 1283–96. http://dx.doi.org/10.1002/nme.1620240705.

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41

Chang, Shin-Perng, and Tsu-Fen Chen. "Adaptive least-squares finite element approximations to transonic-flow problems." PAMM 7, no. 1 (December 2007): 2100035–36. http://dx.doi.org/10.1002/pamm.200700257.

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42

Schwarz, Alexander, Karl Steeger, and Jörg Schröder. "A Least-Squares Mixed Finite Element for Quasi-Incompressible Elastodynamics." PAMM 10, no. 1 (November 16, 2010): 219–20. http://dx.doi.org/10.1002/pamm.201010102.

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43

Gardner, L. R. T., G. A. Gardner, and A. Dogan. "A least-squares finite element scheme for the RLW equation." Communications in Numerical Methods in Engineering 12, no. 11 (November 1996): 795–804. http://dx.doi.org/10.1002/(sici)1099-0887(199611)12:11<795::aid-cnm22>3.0.co;2-o.

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44

Cai, Zhiqiang, Xiu Ye, and Huilong Zhang. "Least-squares finite element approximations for the Reissner-Mindlin plate." Numerical Linear Algebra with Applications 6, no. 6 (September 1999): 479–96. http://dx.doi.org/10.1002/(sici)1099-1506(199909)6:6<479::aid-nla172>3.0.co;2-k.

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45

Cao, Yanzhao, and Max D. Gunzburger. "Least-Squares Finite Element Approximations to Solutions of Interface Problems." SIAM Journal on Numerical Analysis 35, no. 1 (February 1998): 393–405. http://dx.doi.org/10.1137/s0036142996303249.

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46

Starke, Gerhard. "Multilevel Boundary Functionals for Least-Squares Mixed Finite Element Methods." SIAM Journal on Numerical Analysis 36, no. 4 (January 1999): 1065–77. http://dx.doi.org/10.1137/s0036142997329803.

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47

LEFEBVRE, D., J. PERAIRE, and K. MORGAN. "LEAST SQUARES FINITE ELEMENT SOLUTION OF COMPRESSIBLE AND INCOMPRESSIBLE FLOWS." International Journal of Numerical Methods for Heat & Fluid Flow 2, no. 2 (February 1992): 99–113. http://dx.doi.org/10.1108/eb017483.

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48

Carstensen, Carsten, and Eun-Jae Park. "Convergence and Optimality of Adaptive Least Squares Finite Element Methods." SIAM Journal on Numerical Analysis 53, no. 1 (January 2015): 43–62. http://dx.doi.org/10.1137/130949634.

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49

Kim, Keehwan, Sangdong Kim, and Sangsik Shin. "Finite element least-squares methods for a compressible stokes system." International Journal of Mathematics and Mathematical Sciences 2004, no. 72 (2004): 3965–74. http://dx.doi.org/10.1155/s0161171204204124.

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The least-squares functional related to avorticityvariable or avelocity fluxvariable is considered for two-dimensional compressible Stokes equations. We show ellipticity and continuity in an appropriate product norm for each functional.

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50

Cai, Xian-Xin, Bonan Jiang, and Guojun Liao. "Adaptive grid generation based onthe least-squares finite-element method." Computers & Mathematics with Applications 48, no. 7-8 (October 2004): 1077–85. http://dx.doi.org/10.1016/j.camwa.2004.10.006.

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