Abstract
We develop a discontinuous Petrov–Galerkin scheme with optimal test functions (DPG method)
for the Timoshenko beam bending model with various boundary conditions,
combining clamped, simply supported, and free ends. Our scheme approximates
the transverse deflection and bending moment. It converges quasi-optimally
in
Funding source: Comisión Nacional de Investigación Científica y Tecnológica
Award Identifier / Grant number: Fondecyt 1190009
Award Identifier / Grant number: Fondecyt 11170050
Award Identifier / Grant number: Fondecyt 3190359
Funding statement: The research was supported by CONICYT-Chile through Fondecyt projects 1190009, 11170050, 3190359.
References
[1] M. Baccouch, The local discontinuous Galerkin method for the fourth-order Euler–Bernoulli partial differential equation in one space dimension. Part I: Superconvergence error analysis, J. Sci. Comput. 59 (2014), no. 3, 795–840. 10.1007/s10915-013-9782-0Search in Google Scholar
[2] L. Beirão da Veiga, D. Mora and R. Rodríguez, Numerical analysis of a locking-free mixed finite element method for a bending moment formulation of Reissner–Mindlin plate model, Numer. Methods Partial Differential Equations 29 (2013), no. 1, 40–63. 10.1002/num.21698Search in Google Scholar
[3] V. M. Calo, N. O. Collier and A. H. Niemi, Analysis of the discontinuous Petrov–Galerkin method with optimal test functions for the Reissner–Mindlin plate bending model, Comput. Math. Appl. 66 (2014), no. 12, 2570–2586. 10.1016/j.camwa.2013.07.012Search in Google Scholar
[4] C. Carstensen, L. Demkowicz and J. Gopalakrishnan, Breaking spaces and forms for the DPG method and applications including Maxwell equations, Comput. Math. Appl. 72 (2016), no. 3, 494–522. 10.1016/j.camwa.2016.05.004Search in Google Scholar
[5] F. Celiker, B. Cockburn and H. K. Stolarski, Locking-free optimal discontinuous Galerkin methods for Timoshenko beams, SIAM J. Numer. Anal. 44 (2006), no. 6, 2297–2325. 10.1137/050635821Search in Google Scholar
[6] J. Chan, N. Heuer, T. Bui-Thanh and L. Demkowicz, A robust DPG method for convection-dominated diffusion problems II: Adjoint boundary conditions and mesh-dependent test norms, Comput. Math. Appl. 67 (2014), no. 4, 771–795. 10.1016/j.camwa.2013.06.010Search in Google Scholar
[7] L. Demkowicz, T. Führer, N. Heuer and X. Tian, The double adaptivity paradigm (How to circumvent the discrete inf-sup conditions of Babuška and Brezzi), ICES Report 19-07, The University of Texas at Austin, 2019. Search in Google Scholar
[8] L. Demkowicz and J. Gopalakrishnan, Analysis of the DPG method for the Poisson equation, SIAM J. Numer. Anal. 49 (2011), no. 5, 1788–1809. 10.1137/100809799Search in Google Scholar
[9] L. Demkowicz and N. Heuer, Robust DPG method for convection-dominated diffusion problems, SIAM J. Numer. Anal. 51 (2013), no. 5, 2514–2537. 10.1137/120862065Search in Google Scholar
[10] T. Führer, A. Haberl and N. Heuer, Trace operators of the bi-Laplacian and applications, IMA J. Numer. Anal. (2020), 10.1093/imanum/draa012. 10.1093/imanum/draa012Search in Google Scholar
[11] T. Führer and N. Heuer, Fully discrete DPG methods for the Kirchhoff–Love plate bending model, Comput. Methods Appl. Mech. Engrg. 343 (2019), 550–571. 10.1016/j.cma.2018.08.041Search in Google Scholar
[12] T. Führer, N. Heuer and A. H. Niemi, An ultraweak formulation of the Kirchhoff–Love plate bending model and DPG approximation, Math. Comp. 88 (2019), no. 318, 1587–1619. 10.1090/mcom/3381Search in Google Scholar
[13] T. Führer, N. Heuer and F.-J. Sayas, An ultraweak formulation of the Reissner–Mindlin plate bending model and DPG approximation, Numer. Math. 145 (2020), no. 2, 313–344. 10.1007/s00211-020-01116-0Search in Google Scholar
[14] N. Heuer and M. Karkulik, A robust DPG method for singularly perturbed reaction-diffusion problems, SIAM J. Numer. Anal. 55 (2017), no. 3, 1218–1242. 10.1137/15M1041304Search in Google Scholar
[15] F. Lepe, D. Mora and R. Rodríguez, Locking-free finite element method for a bending moment formulation of Timoshenko beams, Comput. Math. Appl. 68 (2014), no. 3, 118–131. 10.1016/j.camwa.2014.05.011Search in Google Scholar
[16] L. K. Li, Discretization of the Timoshenko beam problem by the p and the h-p versions of the finite element method, Numer. Math. 57 (1990), no. 4, 413–420. 10.1007/BF01386420Search in Google Scholar
[17] A. H. Niemi, J. A. Bramwell and L. F. Demkowicz, Discontinuous Petrov-Galerkin method with optimal test functions for thin-body problems in solid mechanics, Comput. Methods Appl. Mech. Engrg. 200 (2011), no. 9–12, 1291–1300. 10.1016/j.cma.2010.10.018Search in Google Scholar
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