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A Locking-Free DPG Scheme for Timoshenko Beams

Thomas Führer ORCID logo, Carlos García Vera and Norbert Heuer ORCID logo

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 L 2 and is locking free. In particular, it behaves well (converges quasi-optimally) in the limit case of the Euler–Bernoulli model. Several numerical results illustrate the performance of our method.

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. Search 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. Search 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. Search 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. Search 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. Search 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. Search 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. Search 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. Search 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. Search 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. Search 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. Search 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. Search 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. Search 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. Search 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. Search 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. Search in Google Scholar

Received: 2020-04-01
Revised: 2020-07-07
Accepted: 2020-07-09
Published Online: 2020-08-05
Published in Print: 2021-04-01

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