Cut finite element methods for linear elasticity problems. (English) Zbl 1390.74180
Bordas, Stéphane P. A. (ed.) et al., Geometrically unfitted finite element methods and applications. Proceedings of the UCL workshop, London, UK, January, 6–8, 2016. Cham: Springer (ISBN 978-3-319-71430-1/hbk; 978-3-319-71431-8/ebook). Lecture Notes in Computational Science and Engineering 121, 25-63 (2017).
Summary: We formulate a cut finite element method for linear elasticity based on higher order elements on a fixed background mesh. Key to the method is a stabilization term which provides control of the jumps in the derivatives of the finite element functions across faces in the vicinity of the boundary. We then develop the basic theoretical results including error estimates and estimates of the condition number of the mass and stiffness matrices. We apply the method to the standard displacement problem, the frequency response problem, and the eigenvalue problem. We present several numerical examples including studies of thin bending dominated structures relevant for engineering applications. Finally, we develop a cut finite element method for fibre reinforced materials where the fibres are modeled as a superposition of a truss and a Euler-Bernoulli beam. The beam model leads to a fourth order problem which we discretize using the restriction of the bulk finite element space to the fibre together with a continuous/discontinuous finite element formulation. Here the bulk material stabilizes the problem and it is not necessary to add additional stabilization terms.
For the entire collection see [Zbl 1392.65006].
For the entire collection see [Zbl 1392.65006].
MSC:
74S05 | Finite element methods applied to problems in solid mechanics |
74C05 | Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials) |
65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |