Abstract
Hybrid-Trefftz finite elements combine favourable features of the Finite and Boundary Element methods. The domain of the problem is divided into finite elements, where the unknown quantities are approximated using bases that satisfy exactly the homogeneous form of the governing differential equation. The enforcement of the governing equations leads to sparse and Hermitian solving systems (as typical to Finite Element Method), with coefficients defined by boundary integrals (as typical to Boundary Element Method). Moreover, the physical information contained in the approximation bases renders hybrid-Trefftz elements insensitive to gross mesh distortion, nearly-incompressible materials, high frequency oscillations and large solution gradients. A unified formulation of hybrid-Trefftz finite elements for dynamic problems is presented in this chapter. The formulation reduces all types of dynamic problems to formally identical series of spectral equations, regardless of their nature (parabolic or hyperbolic) and method of time discretization (Fourier series, time-stepping procedures, or weighted residual methods). For non-homogeneous problems, two novel methods for approximating the particular solution are presented. The unified formulation supports the implementation of hybrid-Trefftz finite elements for a wide range of physical applications in the same computational framework.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Moldovan, I.D., Freitas, J.A.T.: Hybrid-Trefftz displacement and stress elements for bounded poroelasticity problems. Comput. Geotech. 42, 129–144 (2012)
Ruoff, G.: Die praktische Berechnung der Kombination der Trefftzschen Methode und bei flachen Schalen, pp. 242–259. Finite Elemente in der Statik, Berlin (1973)
Jirousek, J., Teodorescu, P.: Large finite elements method for the solution of problems in the theory of elasticity. Comput. Struct. 15, 575–587 (1982)
Herrera, I.: Boundary Methods - an Algebraic Theory. Pitman Advanced Publishing Program. Boston, London, Melbourne (1984)
Piltner, R.: Special finite elements with holes and internal cracks. Int. J. Numer. Methods Eng. 21, 1471–1485 (1985)
Qin, Q.H.: Postbuckling analysis of thin plates by a hybrid Trefftz finite element method. Comput. Methods Appl. Mech. Eng. 128, 123–136 (1995)
Freitas, J.A.T., Moldovan, I.D., Cismaşiu, C.: Hybrid-Trefftz displacement element for bounded and unbounded poroelastic media. Comput. Mech. 48, 659–673 (2011)
Cheung, Y.K., Jin, W.G., Zienkiewicz, O.C.: Direct solution procedure for solution of harmonic problems using complete, non-singular, Trefftz functions. Commun. Appl. Numer. Methods 5(3), 159–169 (1989)
Piltner, R.: The application of a complex 3-dimensional elasticity solution representation for the analysis of a thick rectangular plate. Acta. Mech. 75, 77–91 (1988)
Moldovan, I.D., Cao, D.T., Freitas, J.A.T.: Elastic wave propagation in unsaturated porous media using hybrid-Trefftz stress element. Int. J. Numer. Methods Eng. 97, 32–67 (2014)
Qin, Q.H., Wang, H.: MATLAB and C Programming for Trefftz Finite Element Methods. CRC Press, Boca Raton, London, New York (2009)
Moldovan, I.D., Cismaşiu, I.: FreeHyTE: a hybrid-Trefftz finite element platform. Adv. Eng. Softw. 121, 98–119 (2018)
Moldovan, I.D., Coutinho, A., Cismaşiu, I.: Hybrid-Trefftz finite elements for non-homogeneous parabolic problems using a novel dual reciprocity variant. Eng. Anal. Bound. Elem. 106, 228–242 (2019)
Tamma, K.K., Zhou, X., Sha, D.: The time dimension: A theory towards the evolution, classification, characterisation and design of computational algorithms for transient/dynamic applications. Arch. Comput. Meth. Eng. 7(2), 67–290 (2000)
Freitas, J.A.T.: Time integration and the Trefftz method. Part I - First-order and parabolic problems. CAMES 10(4), 453–463 (2003)
Freitas, J.A.T.: Time integration and the Trefftz method. Part II - Second-order and hyperbolic problems. CAMES 10(4), 465–477 (2003)
Arruda, M.R.T., Moldovan, I.D.: On a mixed time integration procedure for non-linear structural dynamics. Eng. Comput. 32(2), 329–369 (2015)
Moldovan, I.D.: Hybrid-Trefftz Finite Elements for Elastodynamic Analysis of Saturated Porous Media. PhD thesis, Universidade Técnica de Lisboa (2008)
Moldovan, I.D.: A new particular solution strategy for hyperbolic boundary value problems using hybrid-Trefftz displacement elements. Int. J. Numer. Methods Eng. 102, 1293–1315 (2015)
Moldovan, I.D.: A new approach to non-homogeneous hyperbolic boundary value problems using hybrid-Trefftz stress finite elements. Eng. Anal. Bound. Elem. 69, 57–71 (2016)
Moldovan, I.D., Radu, L.: Trefftz-based dual reciprocity method for hyperbolic boundary value problems. Int. J. Numer. Methods Eng. 106, 1043–1070 (2016)
Cho, H.A., Golberg, M.A., Muleshkov, A.S., Li, X.: Trefftz methods for time dependent partial differential equations. Comput. Mater. Continua 1(1), 1–37 (2004)
Alves, C.J.S., Martins, N.F.M., Valtchev, S.S.: Extending the method of fundamental solutions to non-homogeneous elastic wave problems. Appl. Numer. Math. 115, 299–313 (2017)
FreeHyTE distribution page (2015). https://www.sites.google.com/site/ionutdmoldovan/freehyte. Cited 29 Sep 2019
Acknowledgements
This research was supported by Fundação para a Ciência e a Tecnologia through grants PTDC/EAM- GTC/29923/2017 and UID/ECI/04625/2019.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Moldovan, I.D., Cismaşiu, I., Freitas, J.A.T.d. (2020). Unified Hybrid-Trefftz Finite Element Formulation for Dynamic Problems. In: Alves, C., Karageorghis, A., Leitão, V., Valtchev, S. (eds) Advances in Trefftz Methods and Their Applications. SEMA SIMAI Springer Series, vol 23. Springer, Cham. https://doi.org/10.1007/978-3-030-52804-1_9
Download citation
DOI: https://doi.org/10.1007/978-3-030-52804-1_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-52803-4
Online ISBN: 978-3-030-52804-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)