TY - CHAP

T1 - Unified hybrid-trefftz finite element formulation for dynamic problems

AU - Moldovan, Ionuţ Dragoş

AU - Cismaşiu, Ildi

AU - Freitas, João António Teixeira de

N1 - This research was supported by Funda??o para a Ci?ncia e a Tecnologia through grants PTDC/EAM-GTC/29923/2017 and UID/ECI/04625/2019.

PY - 2020

Y1 - 2020

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85091920969&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-52804-1_9

DO - 10.1007/978-3-030-52804-1_9

M3 - Chapter

AN - SCOPUS:85091920969

SN - 978-3-030-52803-4

T3 - SEMA SIMAI Springer Series

SP - 157

EP - 188

BT - Advances in Trefftz Methods and Their Applications

A2 - Alves, C.

A2 - Karageorghis, A.

A2 - Leitão, V.

A2 - Valtchev, S.

PB - Springer

CY - Cham

ER -